Temperature-Controlled, Single-Molecule Kinetic
Investigations of Nucleic Acid Folding Motifs

byDavid Alexander Nicholson
B.S., Furman University, 2014

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Chemical Physics
2022

Committee Members:
David Nesbitt, Chair
Joseph Falke
Meredith Betterton
Ralph Jimenez
Carl Lineberger

ii
Nicholson, David A. (Ph.D. Chemical Physics)
Temperature-Controlled, Single-Molecule Kinetic Investigations of Nucleic Acid Folding Motifs
Thesis directed by Professor David J. Nesbitt

Nucleic acids serve many biological functions beyond information storage. These
functions depend on the ability of a nucleic acid to adopt a specific spatial configuration in aprocess called folding. The folding of each nucleic acid sequence is unique, but there exist
common folding patterns, motifs, that are shared by many nucleic acids. Folding typically occurs
through a stepwise process in which motifs in a nucleic acid fold independently of each otherbefore interacting to form higher order structures. The purpose of this dissertation is to provide a
detailed account of the kinetic and thermodynamic properties of select motifs to give greater
insight into how nucleic acids fold into functional forms.
The primary tool used in this work is single-molecule fluorescence resonance energy
transfer (smFRET), which permits the observation of nucleic acid conformational dynamics inreal-time and the concomitant determination of the first-order rate constants for folding andunfolding. Furthermore, performing kinetic measurements under controlled temperature enables
the evaluation of the thermodynamic properties (e.g., enthalpy and entropy) of the folded andunfolded states, as well as of the transition state along the folding pathway. The nucleic acidmotifs surveyed in this work are a DNA hairpin, an RNA tetraloop-tetraloop receptor, and an
intramolecular DNA G-quadruplex. Temperature-controlled smFRET experiments on thesemotifs are used to probe the mechanism of motif folding, with a focus on the nature of charged
ligand uptake during folding. Furthermore, the smFRET approach for studying conformational

iii
dynamics has been developed herein to expand the range of measurable rate constants and toobtain higher order thermodynamic information on nucleic acids.

iv
Acknowledgements

Though this dissertation’s title page credits me as its sole author, there are in truth
countless contributors who made this work possible by their immense support and guidance. I
will now do my best to express my gratitude to those people, acknowledging that this limited
space permits only abridged summaries of the full contributions of each.
Along the road to graduate school have been numerous educators who have prepared theway for me, and prepared me for the way. I found science to be a dull topic until my 8th grade
science teacher Mr. Johnsen described light to the class and sparked a fascination that continues
to this day. My high school chemistry instructor Dr. Veronica Price gave me the freedom to
explore a topic of my own choosing in an open-ended end-of-year assignment which opened myeyes to the expansiveness of scientific knowledge, and, during the presentation for thatassignment, corrected my pronunciation (of “denature”) in the most kind and tactful way I haveever seen. My general physics instructor, Mr. Murray, showed me the power of mathematics inthe natural sciences, as well as the importance of humor in education.
Numerous educators stand out from my undergraduate studies at Furman University,
including Dr. Paul Wagenknecht (inorganic chemistry and photophysics), Dr. Lon Knight(physical chemistry), and Dr. Brian Goess and Dr. Greg Springsteen (bioorganic chemistry). Drs.
Goess and Springsteen redefined my understanding of what chemistry education can be, and I
hope one day to bring their style to my own classroom. I’d also like to thank several fantastic
instructors, Dr. Ken Kolb (sociology) and Dr. Aaron Simmons (philosophy) for pushing mymind to its limits in the search for what truth is, as well as the chaplain Maria Swearingen for ourconversations on what it means to be human.

v
Undergraduate research played a pivotal role in steering me toward majoring inchemistry. For this, I have extreme gratitude for Dr. Sandy Wheeler, who took pity on a
freshman that lost the class enrollment lottery to get into General Chemistry II, and then,unprompted, recommended I apply for a summer research fellowship. That fellowship shaped therest of my undergraduate career, starting by connecting me with Dr. Karen Buchmueller, my first
mentor. Dr. Buchmueller instructed me in the fundamentals of laboratory research and gave me
career advice during and after my time at Furman University, for which I am extremely grateful.
I’d also like to thank my second research advisor, Dr. Jeffrey Petty, who showed me the powerof biophysical chemistry and the beauty of single-molecule fluorescence, which had me instantly
hooked. How any chemist can resist the allure of studying molecules one at a time is a mystery
to me.
I am indebted to many people at the University of Colorado, and I’d like to express mygratitude to those who made my research possible from behind the scenes. Navigating graduate
school can be difficult, which is why I’m thankful for the Chemistry graduate program managers
I’ve had, Cora Fagan-Edminster, Otha Barrow, and Matt Bohn, as well as their assistant Kayla
Jones. The Molecular Biophysics Training Program has had a tremendous impact on mygraduate career, and it wouldn’t exist without the tireless efforts of Dr. Joe Falke and Lin Pharris.
The members of the JILA computing, electronics, and machine shops, as well as the metrology
laboratory, were crucial in my work. I’d especially like to thank Corey Keasling for help with thecompute cluster, J.R. Raith for building my data collection computer, David Alchenberger for
training me on multiple instruments in the metrology lab, and Calvin Schwadron for teaching(and patiently re-teaching) me how to machine. The JILA administrators also made my life mucheasier, and I want to give special thanks to Kim Monteleone for making JILA a warmer, moreviwelcoming place, and Agnieszka Lynch for handling my exceedingly complicated tuition andpayroll. I’d also like to thank the JILA supply office, especially Randall Holliness for handlingmy nucleic acid orders.
Not enough can be said to thank the current and past members of the Nesbitt group who
have made enormous contributions to my graduate school experience. In particular, I’d like tothank Abhi Sungupta and Hsuan-Lei Sung for training me on the single-molecule confocal
microscope, Jake Pettine for being the resident physicist of the single-molecule laboratory andtherefore fielding all my physics questions, Tim Livingston-Large for his generosity with histime and his unmistakable laugh, Brian Riggs for his calming presence and the occasional jam
session, Mia Zutz for early sage advice on surviving graduate school, and Dan Lesko for ahandsome photo-shopped portrait. Special thanks go to those who worked on the biophysics
project before me, including Julie Fiore and Erik Holmstrom, whose invaluable theses werelifesavers in my first year when I felt I was barely staying afloat as I struggled to understand my
instrument. Last, I want to express my enormous appreciation for Andrea Marton Menendez who
shaped my graduate student career in a way that no one else could have. I have gone to Andrea
for advice on nearly every topic in my life, both in and out of the laboratory, about my pipetting
or about my baking. I am especially grateful to Andrea for inviting me to “fumble forward” into
reimagining what science and academia can be when kindness towards others is taken seriously.
My research advisor, David Nesbitt, is yet another person for whom I cannot fully
express the extent of their impact on me. David’s graduate students are his highest priority, andhe shows this through his generosity with his time. The breadth of David’s scientific knowledge
is matched by few, and, ever the teacher, he freely shares his wisdom. He communicates with
unimpeachable clarity of thought which serves as an example for all his students. Early on,

viiDavid recognized my unrefined mathematical passion and helped me to cultivate the “practical”
mathematician in me. David taught me the importance of intuition, which I now understand is
not divinely inspired guesswork but an engraving of hard-earned experience into the mind. I am
inspired by his boundless curiosity, his enthusiasm for science, and his keen ability to evaluatescientific arguments for the “ring of truth.” From David, I have learned to be an ardent critic of
my own work and to never be content using a tool that I understand only in part. As my researchadvisor, David gave me the freedom to pursue my interests at my own pace, and I’ll always bethankful for his mentorship and guidance.
The trials and tribulations of graduate school were made bearable with the help of myfriends. I had a wonderful cohort of Chemistry graduate students, including Kristina
Vrouwenvelder, Jenny Berry, Kangmin Kim, Shelby Beer, John Malecha, Emily Hopkins,
Franklin Maharaj, Austin Cano, Alex Gilligan, Ryan Dill, Izzy Lattke, and Peyton Cline. During
the especially challenging first year of difficult classwork and burdensome teaching load, myfellow cohort-mates lifted my spirits through a combination of mind-refreshing hikes, midnight
study sessions at Denny’s, and dinner parties full of dancing and joy. My gratitude goes to
Steven Sartor for the late-night ping pong and Andrew Kantor for the long-distance chess games.
I also indebted to my dance partner Jackie Fleming, who gave me an important lesson inchanging my mind. Finally, in the last years of my Ph.D., I’ve had the pleasure of working withand getting to know Bri Dobson and Max O’Connor, who renew my hope for a future that is justand equitable.
I could not be where I am today without the support of my family. My mother Lori has
consistently encouraged me to follow my passion and is the highest example I can think of forunconditional love. My father John has always encouraged my curiosity about the world; heviii
introduced me the acronym “ATP” when I was ten years old, and in that moment started me
down the path to a molecular understanding of life. My sister Becca inspired me to take up
music, which has played a critical role in maintaining my mental wellbeing during my graduate
career. My stepmother Joy is a paragon of generosity whom I can only hope to emulate.
Finally, I want to express my appreciation for my partner and the love of my life, Devon
Reynolds. Devon brings unending joy and laughter into my life, and her support has helped me
endure the difficult moments of graduate school. When my research feels stagnant or tedious, she
expands my perspective and opens my mind to new horizons. When I obsess over an issue in myexperiments, she pulls me back into balance by reminding me of the world beyond the lab. WhenI feel I can do nothing right in the lab, her love envelops me and heals my spirit. For these thingsand so much more, I am endlessly grateful.

ix
Table of Contents

Chapter 1
1.1
1.2
1.3
1.4

1.5

1.6
1.7
Chapter 2
2.1

2.2
2.3
2.4
2.5

2.6
2.7
Chapter 3

3.1

Introduction ............................................................................................................1
Nucleic acids in biology.................................................................................................1
Nucleic acid structure ....................................................................................................3
Intramolecular DNA/RNA folding: Hierarchy and motif ..............................................6
Nucleic acid folding dynamics.....................................................................................10
1.4.1 The thermodynamics of folding .....................................................................11
1.4.2 The kinetics of folding ...................................................................................13
Single-molecule microscopy of nucleic acid conformational dynamics .....................141.5.1 Fluorescence resonance energy transfer (FRET) ...........................................15
1.5.2 Total internal reflection (TIR) and single-molecule fluorescencemicroscopy .....................................................................................................18
Thesis overview ...........................................................................................................20
References ....................................................................................................................21
Methods .................................................................................................................27
Single-molecule TIRF microscope ..............................................................................27
2.1.1 TIRF vs. confocal microscopy for single-molecule experiments...................27
2.1.2 TIRF microscope overview: Laser excitation and fluorescence detection .....28
2.1.3 Strategy for optical alignment pre- and post-sample......................................35
2.1.4 Camera sensitivity, data acquisition rate, and image magnification ..............37
2.1.5 System collection efficiency and sources of noise .........................................40
Sample cells and sample preparation ...........................................................................44
Temperature control .....................................................................................................47
Stroboscopic illumination ............................................................................................49
Data analysis ................................................................................................................50
2.5.1 Extraction of single-molecule trajectories from movies ................................50
2.5.2 Determination of folding rate constants from trajectories ..............................58
2.5.3 Dependence of rate constants on temperature and cosolute concentration ....62
Complementing experiments with molecular dynamics simulations ..........................63
References ....................................................................................................................66
Amino Acid Stabilization of Nucleic Acid Secondary Structure:
Kinetic Insights from Single Molecule Studies ..................................................67
Abstract ........................................................................................................................67

x
3.2
3.3
3.4
3.5
3.6
3.7
Chapter 4
4.1
4.2
4.3

4.4

4.5
4.6
4.7
4.8
4.9
Chapter 5

5.1
5.2
5.3

Introduction ..................................................................................................................68
Methods........................................................................................................................70
Results ..........................................................................................................................71
Discussion ....................................................................................................................79
Summary and conclusion .............................................................................................84
References ....................................................................................................................87
Chirality-Dependent Amino Acid Modulation of RNA Folding .....................91
Abstract ........................................................................................................................91
Introduction ..................................................................................................................92
Methods........................................................................................................................94
4.3.1 RNA FRET construct .....................................................................................94
4.3.2 Sample preparation .........................................................................................96
4.3.3 Single-molecule instrumentation ....................................................................97
4.3.4 Data analysis...................................................................................................99
4.3.5 Temperature control .......................................................................................99
4.3.6 Molecular dynamics simulations ..................................................................101
Experimental results...................................................................................................102
4.4.1 RNA tertiary folding demonstrates a strong chiral sensitivity to arginine ...102
4.4.2 Thermodynamic studies of chirally sensitive D-,L-arginine assisted
unfolding ......................................................................................................104
4.4.3 In search of other amino acid chiral sensitivities in RNA
folding/unfolding ..........................................................................................109
Molecular dynamics simulations ...............................................................................112
Discussion ..................................................................................................................120
Conclusion .................................................................................................................124
Supporting information ..............................................................................................125
References ..................................................................................................................126
Pushing Camera-Based Single Molecule Kinetic Measurements to the
Frame Acquisition Limit with Stroboscopic smFRET ...................................131
Abstract ......................................................................................................................131
Introduction ................................................................................................................131
Methodology ..............................................................................................................134
5.3.1 Theory of single-molecule kinetics ..............................................................134
5.3.2 Photon-by-photon trajectory simulation .......................................................135
5.3.3 Single molecule microscopy ........................................................................137
5.3.4 Single molecule trajectory analysis ..............................................................137

xi

5.4

5.5
5.6
5.7
5.8
5.9
Chapter 6

6.1
6.2
6.3
6.4

6.5
6.6
6.7
Chapter 7

7.1
7.2
7.3

5.3.4.1 Dwell time analysis .........................................................................137
5.3.4.2 Hidden Markov modeling ...............................................................138
5.3.4.3 Time correlation function analysis..................................................139
5.3.4.4 Probability distribution analysis .....................................................140
Simulation results.......................................................................................................141
5.4.1 Fast rate constants are systematically underestimated in smFRET
experiments ..................................................................................................141
5.4.2 Stroboscopic illumination eliminates rate constant underestimation ...........145
5.4.3 Camera averaging artifacts cannot be resolved by modified experimental
conditions .....................................................................................................148
5.4.4 Optimal excitation duty cycle and maximum measurable rate constants ....151
Experimental tests ......................................................................................................154
Discussion ..................................................................................................................156
Conclusion .................................................................................................................163
Acknowledgements ....................................................................................................163
References ..................................................................................................................164
Measuring Excess Heat Capacities of DNA Folding
at the Single Molecule Level..............................................................................168
Abstract ......................................................................................................................168
Introduction ................................................................................................................168
Methods......................................................................................................................171
Results ........................................................................................................................173
6.4.1 Heat capacity in van’t Hoff analysis ............................................................173
6.4.2 smFRET measurements reveal nonzero ΔCP in DNA folding .....................174
6.4.3 Arrhenius analysis of rate constants: Transition state excess heat capacity .177
Discussion ..................................................................................................................181
Summary and conclusion ...........................................................................................185
References ..................................................................................................................187
Kinetic and thermodynamic control of G-quadruplex polymorphism by Na+
and K+ cations ....................................................................................................192
Abstract ......................................................................................................................192
Introduction ................................................................................................................192
Methods......................................................................................................................1957.3.1 G-quadruplex FRET construct .....................................................................1957.3.2 Sample preparation and single-molecule microscopy ..................................1957.3.3 Hidden Markov modeling ............................................................................196

xii

7.4

7.5
7.6
7.7

7.3.4 Three-state Hill fit ........................................................................................197
7.3.5 van’t Hoff and Arrhenius analyses ...............................................................198
7.3.6 Prediction of FRET efficiencies ...................................................................199
Results ........................................................................................................................200
7.4.1 G-quadruplex smFRET construct exhibits multistate folding ......................200
7.4.2 Sodium ions drive the folding equilibrium to the parallel/hybrid topology .203
7.4.3 K+ has a much higher affinity than Na+ for the G-quadruplex and
exclusively promotes the antiparallel topology ............................................208
7.4.4 Temperature-dependence of G-quadruplex folding .....................................210
7.4.5 Thermodynamics of G-quadruplex folding at elevated monovalent cation
concentrations ...............................................................................................212
Discussion ..................................................................................................................217
Summary and conclusion ...........................................................................................221
References ..................................................................................................................222

Chapter 8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8

Postmortem .........................................................................................................227
Introduction ................................................................................................................227
Determination of folding volume change with molecule dynamics (MD) ................227
Cold shock protein and mRNA ..................................................................................231
Modulation of RNA folding dynamics by small peptides .........................................232
Surface tethered kinetic studies of Brome mosaic virus ............................................233
RNA conformational dynamics in liquid RNA droplets............................................233
Summary of work and open research questions ........................................................235
References ..................................................................................................................238

Bibliography ...............................................................................................................................239
Appendices
Appendix 1
Appendix 2
Appendix 3
Appendix 4

The rate constant for fluorescence resonance energy transfer (FRET) ..........266
Signal-to-noise ratio (S/N) degradation due to noisy amplifiers ...................269
Analytic correction to rate constants determined by dwell time analysis ......272
Model-free, graphical analysis of ligand binding stoichiometry using
preferential interaction coefficients ...............................................................274

xiii
List of Tables

Table 4.1 ......................................................................................................................................107
Thermodynamic values from van’t Hoff and Arrhenius analyses for tetraloop-tetraloop
receptor (TL-TLR) folding in 100 mmol/L L- vs. D-arginine.
Table 6.1 ......................................................................................................................................180
Fit results of temperature-dependent measurements of hairpin folding to heat capacitymodified van’t Hoff and Arrhenius models.
Table 7.1 ......................................................................................................................................203
Results of fitting G-quadruplex population data vs. [Na+] and [K+] to 3-state Hill model.
Table 7.2 ......................................................................................................................................215
Thermodynamic parameters for G-quadruplex folding determined by van’t Hoff andArrhenius analysis of temperature dependent kinetic measurements at multiple cation
concentrations.
Table 8.1 ......................................................................................................................................230
Folding volume changes for a DNA hairpin for multiple water models in MD.

xiv
List of Figures

Figure 1.1 ........................................................................................................................................4
The chemical structure of DNA.
Figure 1.2 ........................................................................................................................................6
Depiction of nucleic acid base pairing.
Figure 1.3 ........................................................................................................................................7
The three levels of nucleic acid structure: primary, secondary, and tertiary.
Figure 1.4 ......................................................................................................................................10
Common nucleic acid secondary structure motifs.
Figure 1.5 ......................................................................................................................................12
Schematic of free energy landscape for a nucleic acid hairpin.
Figure 1.6 ......................................................................................................................................17
Design and use of nucleic acid constructs for conformational detection.
Figure 1.7 ......................................................................................................................................19
Sample single-molecule fluorescence trajectory from a DNA hairpin.
Figure 2.1 ......................................................................................................................................29
TIRF microscope excitation path before entering the microscope.
Figure 2.2 ......................................................................................................................................32
Detection tree used to separate donor and acceptor emission and refocus image onto CCD
detector.
Figure 2.3 ......................................................................................................................................34
Truncation of fluorescence image into strips to maximize used CCD area.
Figure 2.4 ......................................................................................................................................40
Magnification calibration and on-chip CCD pixel binning.
Figure 2.5 ......................................................................................................................................43
Angular dependence of evanescent field penetration depth and intensity.
Figure 2.6 ......................................................................................................................................44“Sandwich” style flow cell for microscopy.
Figure 2.7 ......................................................................................................................................44
Schematic of surface functionalization used for nucleic acid construct tethering.

xv
Figure 2.8 ......................................................................................................................................48
Composition of dual TEC temperature controller.
Figure 2.9 ......................................................................................................................................51
The OpenTIRF program written to extract smFRET trajectories from CCD movies.
Figure 2.10 ....................................................................................................................................58
Dwell time analysis.
Figure 3.1 ......................................................................................................................................70
DNA hairpin construct used in folding studies.
Figure 3.2 ......................................................................................................................................72
Single molecule time traces for hairpin folding with and without 100 mM ArginineHCl.
Figure 3.3 ......................................................................................................................................74
Effects of amino acids on DNA hairpin folding kinetics and equilibrium.
Figure 3.4 ......................................................................................................................................76
van’t Hoff plot of the hairpin folding equilibrium’s temperature dependence
Figure 3.5 ......................................................................................................................................78
Arrhenius analysis for hairpin folding rate coefficient as a function of temperature.
Figure 3.6 ......................................................................................................................................84
Comparison of positively-charged amino acids with different side chain moieties.
Figure 3.7 ......................................................................................................................................85
Concentration-dependence of hairpin folding and unfolding rate constants forenantiomers of ArginineHCl.
Figure 3.8 ......................................................................................................................................86
Impact of negative amino acids on hairpin folding.
Figure 4.1 ......................................................................................................................................95
Structure of the 3-stranded TL-TLR smFRET construct used in this chapter.
Figure 4.2 ......................................................................................................................................98
TIRF instrument schematic.
Figure 4.3 ....................................................................................................................................100
Temperature control apparatus.
Figure 4.4 ....................................................................................................................................103
Sample single-molecule TIRF trajectories for TL-TLR folding with 300 mM L-arginineand 300 mM D-arginine.

xvi
Figure 4.5 ....................................................................................................................................104
Rate constants for TL-TLR folding and unfolding in the presence of L- and D-arginine.
Figure 4.6 ....................................................................................................................................106
van’t Hoff plot and Arrhenius plots for TL-TLR folding with 100 mM L- and D-arginine.
Figure 4.7 ....................................................................................................................................108
Temperature-dependent results for TL-TLR folding with 200 mM L- and D-arginine.
Figure 4.8 ....................................................................................................................................110
Rate constants for TL-TLR folding and unfolding in the presence of various L-aminoacids and their D-enantiomers.
Figure 4.9 ....................................................................................................................................113
Visualization of all-atom, explicit solvent simulation of arginine condensation onto the
TL-TLR system.
Figure 4.10 ..................................................................................................................................115
MMPBSA energies of arginine binding to RNA computed frame-by-frame for MD
trajectories.
Figure 4.11 ..................................................................................................................................117
Histogram of MMPBSA-computed binding free energies for L- and D- enantiomers ofarginine and lysine with the undocked TL, the undocked TLR, and the docked TL-TLR.
Figure 4.12 ..................................................................................................................................118
Representative radial distribution function g(r) of the distance between nucleic acid andarginine heavy atoms for the TL system.
Figure 4.13 ..................................................................................................................................119
Free energy isosurfaces (G ≤ -5 kBT) for arginine binding to the docked TL-TLR
structure.
Figure 5.1 ....................................................................................................................................143
Rate constants determined from simulated data of a two-state folding system.
Figure 5.2 ....................................................................................................................................144
Cartoon demonstration of how blurred FRET states arise from mapping of continuoustime dynamics into discrete-time measurements.
Figure 5.3 ....................................................................................................................................146
Experimental rate constants determined by dwell time distributions after discarding theshortest dwell times.

xvii
Figure 5.4 ....................................................................................................................................150
Rate constants determined from simulated data show that camera averaging artifacts arerobust to experimental parameters.
Figure 5.5 ....................................................................................................................................152
Determining the optimal strobe duty cycle by plotting rate constant bias as a function of
duty cycle.
Figure 5.6 ....................................................................................................................................153
Bias and uncertainty (standard deviation) of rate constants determined by dwell time
analysis for a variety of rate constants and stroboscopic duty cycles.
Figure 5.7 ....................................................................................................................................154
Experimental implementation of stroboscopic smFRET.
Figure 5.8 ....................................................................................................................................156
Experimental validation of stroboscopic method by measuring DNA hairpin folding
dynamics over a range of camera frame rates.
Figure 5.9 ....................................................................................................................................158
Systematic underestimation of rate constants is a shared feature of multiple smFRET
analysis methods including dwell time analysis, hidden Markov modeling (HMM), andtime correlation function (TCF) fitting.
Figure 5.10 ..................................................................................................................................159
Rate constants determined by hidden Markov modeling of simulated 3-state system withand without stroboscopic excitation.
Figure 5.11 ..................................................................................................................................160
Experimental measurement of DNA hairpin folding at Δtframe = 10 ms.
Figure 5.12 ..................................................................................................................................162
Comparison of analysis of simulated data by trajectory-based approach (dwell time
analysis) and FRET histogram fitting (PDA).
Figure 6.1 ....................................................................................................................................171
Schematic of DNA hairpin construct used in this chapter.
Figure 6.2 ....................................................................................................................................175
Sample single-molecule FRET time trajectories of hairpin folding at 13 °C and 33 °C.
Figure 6.3 ....................................................................................................................................176
van’t Hoff analysis of smFRET-derived equilibrium constants with and without
modifications for nonzero excess heat capacity (ΔCP).

xviiiFigure 6.4 ....................................................................................................................................177
Arrhenius analysis of unfolding rate constant kU.
Figure 6.5 ....................................................................................................................................180
Arrhenius analysis of folding rate constant kF.
Figure 6.6 ....................................................................................................................................184
Excess heat capacity folding landscape.
Figure 7.1 ....................................................................................................................................201
G-quadruplex smFRET construct design and sample smFRET trajectory with FRET
histogram and proposed structural assignment.
Figure 7.2 ....................................................................................................................................204
Influence of Na+ on G4 folding equilibrium for [Na+] = 50−300 mM.
Figure 7.3 ....................................................................................................................................206
Influence of Na+ on G4 folding kinetics with 6-state binding model.
Figure 7.4 ....................................................................................................................................209
Influence of K+ on G4 folding for 0−20 mM K+.
Figure 7.5 ....................................................................................................................................211
Temperature-dependence of G4 folding rate constants and equilibrium constants in 100
mM Na+.
Figure 7.6 ....................................................................................................................................213
Folding landscape for G4 in 100 mM Na+ as determined by van’t Hoff and Arrhenius
analyses.
Figure 7.7 ....................................................................................................................................214
Temperature-response of G4 folding under multiple cationic conditions: 100 mM Na+,
200 mM Na+, and 100 mM Na+ with 4 mM K+.
Figure 7.8 ....................................................................................................................................216
Determination of cation binding thermodynamics.
Figure 7.9 ....................................................................................................................................220
Proposed scheme of Na+ and K+ binding to TAA-GQ.
Figure 8.1 ....................................................................................................................................229
Schematic of strategy for determining nucleic acid folding volume in MD
Figure 8.2 ....................................................................................................................................231
DNA hairpin volume change at different simulation pressures.

xix
Figure 8.3 ....................................................................................................................................234
FRET histograms of brome mosaic virus (BMV) single-molecule construct from surface
tethered measurements.
Figure A4.1 .................................................................................................................................272
General binding model of ligand L binding to U and F receptor configurations.

1
Chapter 1
Introduction

1.1

Nucleic acids in biology
Nucleic acids are one of the main categories of polymers in biology, alongside proteins.

Nucleic acids come in two flavors: ribonucleic acids (RNA) and deoxyribonucleic acids (DNA).
In high school biology classes, the functions of these three polymers (RNA, DNA, and proteins)are laid out in a tidy system of relationships in which DNA transfers information to RNA(transcription) which is then used by the ribosome to synthesize proteins (translation). In this socalled “central dogma” of molecular biology, DNA and RNA are purely information carriers:
DNA serves as a long-term memory storage system that keeps a permanent blueprint (a gene) of
each protein in the organism, and RNA is a transient messenger which delivers an accurate
genetic transcript to the ribosome before being recycled for further messenger duties. In thissimplified picture, DNA and RNA exist solely to create proteins, while proteins carry out the real
work of the cell, such as catalyzing chemical reactions, orchestrating cellular transport, andproviding cellular structure.
Today, DNA and RNA are known to serve cellular functions that go far beyond simply
carrying genetic information. A key historical moment in the understanding of nucleic acids was
the discovery by Thomas Cech and coworkers in the 1980s of the tetrahymena hammerhead
intron which catalyzes its own cleavage without any assistance from proteins.1-2 RNA molecules
called ribozymes which perform catalysis are now widely documented across manyorganisms/domains of life.3-4 Indeed, every living cell on earth relies on RNA in the ribosome tocatalyze peptide bond formation required for protein synthesis.5 This ability of RNA to act as

2both genetic storage and catalyst inspired the development of the “RNA world” hypothesis
which postulates that early life relied exclusively on RNA to perform all cellular functions, andthat DNA and protein arrived later in evolutionary history.6 In contemporary biology, RNA has
been found to play a profound role in genetic regulation: the determination of whether or notgenes are translated into protein.7 RNA-based genetic regulation occurs through a variety ofmeans. For instance, some organisms use riboswitches8-9 which are short RNA sequences
positioned at the start of an RNA transcript and that can bind to specific small molecules which
may or may not be present in the cell. Then, depending on whether the small molecule is bound,
the riboswitch will block or allow synthesis of the gene. In this way, the riboswitch acts as asensor that detects a specific metabolite and determines how the cell will respond (e.g., bypermitting the synthesis of a protein that metabolizes the riboswitch’s target molecule). Other
examples of genetic control by RNA include RNA interference,10 which plays an important rolein the eukaryote immune response, and transcript splicing11 in which non-coding portions of
nascent RNA transcripts called introns are removed before translation.
DNA, though not as diverse in biological functionality as RNA, also goes beyond mere
information storage. For instance, DNA plays an essential role providing the structure oftelomeres and centromeres, which are two large-scale structures in the chromosome.12-13 Much of
the interest in DNA functionality, however, stems from its potential use in biotechnology due tothe improved biochemical stability of DNA over RNA. Over the last several decades, researchers
have shown that DNA can reproduce most RNA capabilities: deoxyribozymes can catalyze
reactions and DNA aptamers can bind to small molecules with high affinity and specificity.14-15
Furthermore, the structural capacity of DNA has been exploited to create arbitrarily shaped DNA
objects termed “DNA origami.”16-17 Clearly, the functional capability of DNA is firmly

3established in the literature, and it is likely that even more nucleic acid functionalities will be
discovered as the scientific community continues to explore the 98% of the human genome that
does not encode for protein sequences.18-19

1.2

Nucleic acid structure
Understanding the many functions of nucleic acids requires first understanding thestructure of nucleic acids. As previously stated, nucleic acids are polymers, meaning that they
consist of monomers linked together to form a linear chain. In the case of nucleic acids, themonomers are nucleotides, and nucleotides are themselves composed of a ribose or deoxyribose
sugar, for DNA or RNA respectively, a phosphate group, and a variable nucleobase (Figure1.1).20 In most nucleic acids, there are four possible nucleobases; in DNA, these are adenine,
thymine, guanine, and cytosine, denoted as A, T, G, and C, respectively. In RNA, the same
nucleobases are present except for thymine (T) which is replaced by the structurally similar
uracil (U) that has a hydrogen at the C5 position instead of a methyl group. A, T, G, C, and U are
the five so-called canonical nucleobases; however, there are several modified versions of these
nucleobases that are regularly incorporated into nucleic acids in nature.21-22 For instance,
methylated nucleobases in DNA are part of an extremely common mode of epigenetic
regulation.23-24 The genetic versatility of DNA comes from the ability to freely swap among the 4
nucleobases, which in turn gives each position in a DNA sequence 2 bits of information. Thismeans that the human genome, which is 3 billion nucleotides long, contains slightly less than 1
gigabyte of information (however, not all nucleotides in the human genome are highly
conserved, so the true information content is smaller).
There are several important physical properties of nucleic acids that are relevant to this4

Figure 1.1 Chemical structure of DNA with highlighted nucleobases guanine, cytosine, adenosine,and thymine. In RNA, thymine is replaced uracil, and the 2’ position in the sugar ring contains ahydroxyl (-OH) group.

dissertation. First, it is important to note that nucleic acid sequences are polar: the two ends of a
nucleic acid are chemically distinct, and the nucleotide sequence is therefore directional.
Specifically, the phosphodiester linkage between two adjacent nucleotides always, in nature,connects to the 3’ carbon of one nucleotide to the 5’ carbon of the other (Figure 1.1). Therefore,
no matter how long a nucleic acid is, it will always have a 3’ terminus and a 5’ terminus, and any
nucleic acid sequence (e.g., CAG) is ambiguous unless the polarity is specified (e.g., 5’-CAG-3’
vs 3’-CAG-5’; it is conventional to write sequences in the 5’ to 3’ direction, which aligns withthe direction of nucleic acid synthesis in nature). This directionality is chemically subtle but has

5a significant impact on the ability of a nucleic acid to interact with itself or other nucleic acids.
The second noteworthy physical property is that nucleic acids are highly electrically charged.
Each nucleotide contributes a phosphate group and therefore a charge of −1 e, where e is theelementary charge (1.6×10−19 C), and the total nucleic acid will therefore have a large, negative
charge distributed evenly along its sugar-phosphate backbone. This net negative charge meansthat nucleic acids inherently repel one another, and that therefore any inter- or intra-strand
interactions, which are the focus of this dissertation, would be impossible were it not for thepresence of positive counterions such as K+, Na+, and Mg2+ which reduce the repulsive forces
between phosphate groups through electrostatic screening.25-26 The importance of nucleic acidcation interactions cannot be understated and indeed forms a recurring theme in this dissertation.
At the heart of nucleic acid functionality is the ability of two individual strands to bind
one another to form a duplex, mediated by a noncovalent interaction called base pairing. Basepairing occurs when two nucleobases form hydrogen bonds between them (Figure 1.2A). Due tothe structural and geometric requirements of hydrogen bonding, not all possible base pairs havethe same strength, with the strongest being the canonical Watson-Crick-Franklin base pairs: C
pairing with G, and A pairing with T or U. Other non-canonical base pairs, such as Hoogsteenbase pairs and wobble base pairs, are possible and do form in nature, but their stability is far lessthan that of the canonical CG and A(T/U) base pairs.27-29 As a result of the specificity of basepairing, two nucleic acid strands are best able to bind to one another and form a duplex, or
hybridize, if their sequences are complementary, meaning that they maximize the number ofcanonical base pairs formed between them. In addition, the geometry of the canonical base pairs
requires the strands to be antiparallel, denoting that the direction of the 5’ to 3’ progression of
one strand is the opposite of the other. In this optimal case, every adjacent nucleotide will be6

Figure 1.2 Nucleic acid base pairing. (A) Structures of the guanine-cytosine (GC), adeninethymine (AT), and adenine-uracil (AU) base pairs. (B) Complementary, antiparallel nucleic acids
can stack adjacent base pairs to form a double-helix. (PDB record 1ZEW)

part of a base pairing interaction, and the resulting structure is a ladder of base pairs. To
minimize the surface area of the hydrophobic nucleobases exposed to water, this ladder thentwists, forming the famous double helix (Figure 1.2B). Nucleic acid duplexes can take on several
helical geometries with varying degrees of helical twist, including A-form (11 base pairs perturn, 2.6 A rise per base pair, right-handed helix), B-form (10 base pairs per turn, 3.4 A rise perbase pair, right-handed helix), and Z-form (12 base pairs per turn, 3.7 A rise per base pair, lefthanded helix) geometries.30 DNA duplexes primarily populate the B-form conformation, whereas
RNA duplexes and mixed RNA-DNA duplexes tend toward A-form conformations.

1.3

Intramolecular DNA/RNA folding: Hierarchy and motif
Beyond forming intermolecular duplexes, nucleic acids can also self-interact thoughintramolecular base-pairing to generate complex, three-dimensional structures capable of various
biological functions, examples of which have been described above. The process by which DNA

7
or RNA forms these structures is known as folding,31 a name which invokes the image of a
nucleic acid bending back upon itself to create base pairs, like a rope bending around itself toform a knot. The terminology of folding is borrowed from the protein literature, where the“protein folding problem” has a long history,32-34 and indeed the framework of nucleic acidfolding that I am about to present draws many concepts from the field of protein folding.
First, nucleic acid structure can be divided into three levels: primary, secondary, andtertiary,20 which are shown schematically in Figure 1.3. The primary structure of a nucleic acid issimply its nucleotide sequence, including any modified nucleobases. The secondary structure is
the network of base pairs between nucleotides, which is often represented as a two-dimensionalbase pair map (e.g., Figure 1.3, middle). Finally, tertiary structure is the full, three-dimensional
shape of a nucleic acid as determined by its tertiary interactions, which in turn are variously
defined as (i) any interactions besides base pairing contributing to the overall structure or (ii) the

Figure 1.3 The three levels of nucleic acid structure (left to right): primary, secondary, and tertiary.
Primary structure is the nucleic acid sequence, secondary structure is the base pairing, and tertiary
structure is the 3-dimensional shape. Sequence shown is the yeast phenylalanine tRNA (PDB
record 1EHZ).

8interactions between secondary structure elements.* It is this latter definition that encapsulates
the next development of nucleic acid folding theory: the modular nature of folding.
The three nucleic acid structural levels underlie a hierarchical model of nucleic acidfolding in which information flow between the levels is one-directional: the primary structure(sequence) determines the secondary structure (hybridized regions) which in turn determine thetertiary structure.35-37 This hierarchical approach is supported by experimental evidence that, ingeneral, secondary structures form quickly, on a timescale of microseconds to milliseconds, andare extremely stable while tertiary interactions are slow to form (milliseconds to seconds) andmuch less stable. Therefore, nucleic acid folding is thought of sequentially as a rapid formationof secondary structures which then rearrange as tertiary interactions slowly emerge. Any
conformational flexibility stems from the flighty tertiary interactions, while the stable core of thenucleic acid’s structure is provided by immutable secondary structural elements. Of course, thisrigid presentation of the hierarchical model is an overly simplistic picture. In truth, theinteraction between the secondary and tertiary levels has some bidirectional character. For
instance, the additional stabilization provided by tertiary interactions may be the energetictiebreaker between two similarly stable secondary structures. Second, secondary structures are
not the permanent, eternal, “rocks” that the hierarchical model supposes them to be. An exampleof this is shown in Chapter 3, where an intramolecular double helix with eight contiguous basepairs is observed to become completely unfolded multiple times per second. Similarly short oreven shorter double helical regions are extremely common in complex nucleic acids, which*

Note that these definitions are heuristic in nature and do not form a logically complete set. For instance, howshould we classify the kissing loop interaction? It brings together two secondary structure elements (the two stemloops), so it seems to be a tertiary interaction, and it is usually described in the literature as such. However, the loops
themselves interact by base pairing. So, kissing loops also meet the definition of a secondary structure. Anotherexample is the pseudoknot, for which the literature is inconsistent as to whether pseudoknots are secondary or
tertiary structures. This inconsistency can be confusing for those new to the field.

9
implies that the DNA/RNA can experience significant fluctuations in their secondary structurethat may interfere or interact with tertiary interactions. Clearly, a static image of secondary
structure fails to capture the true dynamical richness of base paired regions in nucleic acids.†
With those caveats stated, the hierarchical model nevertheless provides a useful framework forunderstanding nucleic acid folding as a network of secondary structure elements called folding
motifs that can be reliably predicted and individually characterized.38-39
Nucleic acid secondary structure motifs are numerous, and a few that are particularly
important to this dissertation are sketched in Figure 1.4. The most basic and ubiquitous structureis the hairpin (Figure 1.4A), also known as a stem-loop, in which a nucleic acid bends backwards
to position two complementary or near-complementary sequences in proximity for basepairing.40-42 The base-paired region is the stem, and the leftover single-stranded region is theloop. The stability of a hairpin depends primarily on the number base pairs in the stem, withmore base pairs providing more stability. The length of the loop matters as well, with a loop of
fewer than three nucleotides being sterically impossible. Some secondary structure featuresderive from helical regions containing defects: an internal loop occurs in a stem with contiguousbase pair mismatches, while a bulge or kink describes a stem where one strand has one or more
unpaired, excess nucleotides (Figure 1.4B−D). If a hairpin has a long enough loop, the loop
nucleotides can base pair with another portion of the nucleic acid, creating two overlappingstem-loops called a pseudoknot.43 Finally, there are higher order structures that involve morethan two strands, such as minor/major groove triplexes, four-way junctions, i-motifs, and Gquadruplexes.44-49 G-quadruplexes are featured in the final work of this dissertation (Ch. 7) and†In contrast, protein secondary structure is reliably quite stable. Protein tertiary structure, which are the interactions
between amino acid side chains, may be a better comparison to nucleic acid secondary structure, as base pairing is
an interaction between nucleobases, which are like the “side chains” of nucleotides Clearly, the parallelism between
nucleic acid and protein folding is somewhat stretched.

10

Figure 1.4 Common nucleic acid secondary structure motifs. (A) The hairpin consists of stem andloop, and variations on the stem-loop include (B) the internal loop, (C) the bulge/kink, and (D) the
pseudoknot. 4-stranded nucleic acid helices include (E) the G-quadruplex and (F) the i-motif. In
the G-quadruplex, 4 guanine residues form a planar “tetrad,” similar to planar arrangement ofhands in the “wrist-grab star” figure (E, right) in contra dancing, an American folk dance. In
contrast, the i-motif consists of cytosine pairs which alternate along the helical axis, which is aconfiguration similar to the “hands-across star” contra dance figure (F, right).

are characterized by a unique guanine tetrad structure with hydrogen bonds occupying both the
Watson-Crick-Franklin and Hoogsteen edges of each guanine (Figure 1.4E).

1.4

Nucleic acid folding dynamics
A nucleic acid may have access to many possible structures, or conformations, depending

on its length and sequence. How then does a newly synthesized RNA molecule “choose” fromamong its possible conformations? And how quickly does the RNA molecule fold? Does it
remain in that configuration indefinitely? These questions of stability and dynamics are
addressed in this dissertation using two important theoretical frameworks, statistical
thermodynamics and chemical kinetics, which I will now introduce.

11
1.4.1 The thermodynamics of folding
The field of thermodynamics, with its origins in the flow of heat in engines, may at firstglance appear not to have any relationship with nucleic acid folding; however, the principles ofclassical thermodynamics laid down by Carnot and others can be extended into the molecular
realm using the tools of statistical mechanics.50 At the microscopic level, nucleic acids do in factexchange minute quantities of heat with their environment during conformational rearrangements
due to the making and breaking of bonds—not covalent bonds, but other noncovalentinteractions like hydrogen bonding and ion-dipole interactions. These energy changes in turnplay a key role in determining which configurations a nucleic acid will adopt.
A central result of statistical thermodynamics is that the states of any system, such as theconformations of a nucleic acid, exist in a dynamic equilibrium. Some states are populated moreof the time than others, and the relative population of two states is given by the Boltzmann
distribution,
𝑃2

−

∆G°

= 𝑒 𝑘𝐵𝑇 .
𝑃
1(Eq. 1.1)

In this equation, P1 and P2 are the fraction of molecules in the states labeled 1 and 2,
respectively, e is Euler’s number, kB is the Boltzmann constant, T is the absolute temperature,and ΔG° is the Gibbs free energy difference between the two states. This is a remarkably simple
result that reduces any arbitrarily complex process down to a single intrinsic quantity: the freeenergy change ΔG° for the process. Importantly for this work, if an experiment can measure
relative populations of states, then ΔG° can be determined. It is possible, therefore, to assign freeenergies the network of possible nucleic acid configurations and thereby construct the so-called
free energy “landscape” for folding (Figure 1.5).51 Furthermore, the Gibbs free energy change isitself composed of two competing thermodynamic contributions:12

Figure 1.5 Schematic of free energy landscape for a nucleic acid hairpin.
∆G° = ∆H° − 𝑇∆S°,(Eq. 1.2)

where ΔH° enthalpy, or the heat absorbed, and ΔS° is the change in entropy, which is a measureof system disorder. Nucleic acid folding generally releases heat (ΔH° < 0) and is therefore
enthalpically favorable, but the folded nucleic acid is usually conformationally constricted,
which is entropically unfavorable (−TΔS° > 0).52 This competition between enthalpy and entropy
is a hallmark of folding53-54 and is discussed considerably in the results (Chapter 3−7). Notable in
Eq. 1.2 is the temperature T, which is a parameter that can be controlled in an experiment. In
fact, by intentionally changing T and recording how ΔG° changes, one can determine theenthalpy ΔH° and the entropy ΔS° for nucleic acid folding, and this strategy is at the heart ofmuch of this dissertation.55

13
1.4.2 The kinetics of folding
Chemical kinetics is the study of chemical reaction rates and, similarly tothermodynamics, may not obviously have connections with nucleic acid conformational
dynamics. After all, there are no covalent chemical bonds being broken or formed during a
conformational transition. However, just as two chemical reactants must randomly collide withthe correct energy and orientation for a reaction, so too must a nucleic acid find the correct
conditions for a conformational transition to occur via random collisions with its environment.
Both chemical reactions and nucleic acid conformational transitions can be described as Poissonprocesses, meaning that the likelihood of the process happening is independent of time.56
Therefore, these processes can be described by a single quantity, the rate constant, which is theprobability of the process occurring per molecule per unit time. For instance, the folding
transition from conformation A to B can be written down schematically as
𝑘𝐴𝐵

𝐴 → 𝐵,(Eq. 1.3)

where kAB is the rate constant for the A to B transition. The result of this kinetics approach is thatthe enormous complexity of nucleic acid structural dynamics is reduced to a single numericalvalue for each conformational transition.57 Of course, the folding rate constant may depend on
environmental factors, such as temperature or solute concentrations, and this can be used to makeinferences about the mechanism of nucleic acid folding. For instance, if a folding rate constantchanges upon an increase in the concentration of an ion in solution such as potassium, then it canbe inferred that folding involves the binding or releasing of that ion from the nucleic acid. Thesetypes of environmental manipulations are used frequently in this work in as a way of proposing
possible mechanisms for folding and testing the validity of putative folding mechanisms.

14
1.5

Single-molecule microscopy of nucleic acid conformational dynamics
Nucleic acids have largely been studied using ensemble methods: experiments in which

many, many molecules are examined simultaneously. These “bulk” experiments have provided
and continue to provide invaluable information on nucleic acid structure and conformational
dynamics. For instance, it is now routinely possible to measure the precise position of individual
atoms in nucleic acid structures using x-ray crystallography or nuclear magnetic resonance(NRM) spectroscopy58-59. The thermodynamics of nucleic acid folding (i.e., enthalpy andentropy) can be determined with high precision either directly by microcalorimetry in isothermal
titration calorimetry (ITC) and differential scanning calorimetry (DSC) or indirectly bytemperature-controlled ultraviolet-visible (UV-vis) spectroscopy.60-62 These thermodynamic
measurements have been integrated into “nearest neighbor” models for prediction of secondary
structure thermodynamics,52, 63 which in turn forms the basis of numerous software packages thatcan rapidly predict secondary structures for arbitrary nucleic acid sequences.64
In recent decades, it has become possible to observe nucleic acid folding one molecule ata time, and these single-molecule methods allow experimentalists to access information
previously unavailable to ensemble measurements.65-73 First, an ensemble experiment by its
nature requires averaging over many molecules, and as a result, information on molecule-tomolecule differences, or heterogeneity, is lost.74-75 The existence of subpopulations must be
inferred out by indirect means. In contrast, observing molecules one at a time permits immediatecharacterization of sample heterogeneity and the presence of subpopulations. Second, and more
importantly for this dissertation, most ensemble methods are incapable of reporting on thetimescale for folding. This is because, at equilibrium, the numerous molecules in a bulk
measurement are experiencing conformational transitions in an uncoordinated way, with an equal

15
number of molecules folding and unfolding at any given moment. As a result, the data from anensemble experiment is typically static and unchanging. The exceptions to this are ensemble
experiments that are not carried out at equilibrium, such as the temperature jump and thestopped-flow methods in which the system is rapidly displaced from equilibrium and thetimescale of the relaxation to a new equilibrium is recorded.76-78 However, converting relaxation
times into rate constants is only possible under certain assumptions, such as that the folding
equilibrium consisting of only two states, or that the dynamics far from equilibrium are
representative of those at equilibrium. Single molecule experiments, on the other hand, are able
to detect structural fluctuations in a system at equilibrium and are therefore well-suited to
explore nucleic acid conformational dynamics, even in many-state systems.
1.5.1 Fluorescence resonance energy transfer (FRET)At the heart of the single molecule method used in this dissertation is the photophysicalprocess of fluorescence‡ resonance energy transfer (FRET).79 FRET begins with a “donor”
fluorophore, which is a molecule that can absorb light of one color and then emit light of adifferent color. In photophysical terms, the donor absorbs a photon of light to enter anenergetically “excited” state before releasing its excess energy (“relaxing”) either (i) as light(fluorescence), (ii) as heat (non-radiative relaxation), or (iii) by transferring its energy to a
nearby “acceptor” fluorophore. This last possibility is termed FRET, and the probability that
FRET occurs (the “FRET efficiency” or EFRET) depends on how large the rate of energy transfer(kFRET) is compared to the rates for fluorescence (kf) and non-radiative relaxation (knr):‡

In the literature, the FRET acronym is sometimes spelled out as “Forster resonance energy transfer” after Theodore
Forster, the theorist who gave the first derivation of the physics behind FRET. In this dissertation, I prefer“fluorescence” over “Forster” for historical reasons: the FRET acronym was originally defined using“fluorescence,” and it was only later that the acronym was redefined to include Forster’s name. Additionally, while
naming a phenomenon after someone is respectful, it is generally unhelpful to the neophyte, who I think would
prefer the more descriptive term “fluorescence” to help them remember that FRET experiments are light-based
measurements.

16
𝑘

𝐸𝐹𝑅𝐸𝑇 = 𝑘 +𝑘 𝐹𝑅𝐸𝑇
+𝑘
𝑓

𝑛𝑟

𝐹𝑅𝐸𝑇

.

(Eq. 1.4)

Certain conditions must be met for FRET to occur. First, to conserve energy, the energy gap fordonor fluorescence and acceptors absorbance must match. Second, as FRET originates fromelectric dipole-dipole interactions, the magnitude of kFRET depends on the relative orientation of
the donor and acceptor and the distance between them (kFRET ∝ R-6).§ Therefore, the EFRET
expression can be written as a function of distance:
𝐸𝐹𝑅𝐸𝑇 (𝑅) =

1
1+(

𝑅 6
)
𝑅0

,

(Eq. 1.5)

where R is the donor-acceptor distance, and R0 is a characteristic length named the Forster
distance. R0 depends on the properties of the donor and acceptor but is usually 1−10 nm, which isserendipitously the same length scale of nucleic acid movement in many conformationalchanges.80 From Eq. 1.5, knowledge of EFRET implies knowledge of the donor-acceptor distance,
which is why FRET is sometimes referred to as a “nanoscopic ruler.” The experimentalist canmeasure EFRET by examining the fluorescence spectrum: when EFRET is small, fluorescencecomes from the donor only; however, when EFRET is large, the donor’s fluorescence is quenched,
and fluorescence comes from the acceptor instead. Intermediate FRET efficiencies can be
determined by the relative amount of donor fluorescence (FD) and acceptor fluorescence (FA):**𝐹𝐴𝐸𝐹𝑅𝐸𝑇 = 𝐹 +𝐹.
𝐴

𝐷

(Eq. 1.6)

Therefore, by monitoring the fluorescence of the donor and acceptor, on can in principle
determine the distance between the two fluorophores, a property which is utilized herein for
nucleic acid structural studies.
§

The full expression for kFRET is derived in Appendix 1.
Eq. 1.6 neglects several important factors that the experimentalist must contend with to accurately determine the
FRET efficiency, including donor and acceptor differences in fluorescence quantum yields and detector efficiency.
For a more thorough discussion, see Chapter 2.
**

17
Using FRET to measure inter-nucleotide distances in DNA and RNA, and by proxy the
nucleic acid’s conformation, requires specially designed nucleic acid “constructs” (Figure 1.6).
First, as nucleic acids are not fluorescent, the nucleic acid of interest must be chemicallymodified to include a pair of fluorophores to act as FRET donor and acceptor. The selection of
the fluorophores is critical, as a good FRET donor-acceptor pair must exhibit several properties,
including strong light absorbance, high fluorescent quantum yields (probability of fluorescing
versus non-radiatively relaxing), and a large spectral overlap between donor emission andacceptor absorbance.81 The cyanine-based dyes Cy3 and Cy5 are a popular FRET pair and areused extensively in this work. Equally important are the labeling positions of the donor andacceptor on the nucleic acid, as the distance between the two fluorophores needs to change
during the anticipated conformational transition(s). Structural information from the literature(e.g., x-ray crystallography) can assist in determining appropriate labeling positions, but
occasionally trial and error is required as FRET efficiency prediction is still a developing field.82

Figure 1.6 Design and use of nucleic acid constructs for conformational detection. The nucleic
acid is labeled with donor (D) and acceptor (A) fluorophores. When the nucleic acid is unfolded(left), the donor and acceptor are far apart and excitation of the donor (hνex) leads only to donoremission (hνem,D). However, when the nucleic acid is folded (right), the excited donor can transfer
energy to the acceptor via FRET, which results in acceptor emission (hνem,A).

18
Fortunately, the challenge of synthesizing nucleic acid constructs is made simpler through rapid
and relatively inexpensive commercial sources that can generate labeled nucleic acids ofarbitrary sequence up to a few hundred nucleotides in length (“oligonucleotides”) with highpurity.
1.5.2 Total internal reflection (TIR) and single-molecule fluorescence microscopy
Essential for performing single-molecule FRET (smFRET) experiments is the ability tomeasure fluorescence from individual molecules. Traditional, ensemble fluorescencemeasurements use light to excite many molecules simultaneously and observe the bulk emission.
Translating ensemble fluorescence techniques into the single-molecule realm requires severely
limiting the excitation of fluorophores so that only one is observed at a time. In this work, this isaccomplished using the phenomenon of total internal reflection (TIR).83
TIR occurs when light intersects an interface at an angle greater than or equal to thecritical angle, which is the incidence angle that produces a transmitted angle of 90° in Snell’s lawof refraction,𝑛𝑖 sin(𝜃𝑖 ) = 𝑛𝑡 sin(𝜃𝑡 ),(Eq. 1.7)

where ni (nt) and θi (θt) are the incident (transmitted) index of refraction and angle,
respectively. Substituting θt = π/2, the critical angle θc is𝑛sin(𝜃𝑐 ) = 𝑛𝑡 ,𝑖(Eq. 1.8)

which has a solution only if ni > nt. Beyond this critical angle, no light is transmitted, and thelight is said to totally internally reflect. However, the electromagnetic field at the interface
cannot discontinuously go to zero, and a more thorough electromagnetic treatment reveals thatthe forbidden medium contains a so-called “evanescent wave” with an electric field magnitude
that decays exponentially with distance from the interface (for more detail, see Chapter 2). The

19length scale (1/e) of the evanescent decay is the same order of magnitude as the wavelength of
the incident light, which in the case of visible light is less than 10-6 m. Therefore, while it is true
under TIR conditions that no light is transmitted in the “far field,” there is a “near field” effectwhich permits excitation of chromophores near the interface. This fact is exploited in a TIRF
microscope by illuminating a glass-water interface under TIR conditions to obtain a fluorescence
image with greatly reduced background fluorescence when compared to a typical widefield
fluorescence microscope. This low-background, single-molecule fluorescence is extremely dim
but can be detected using highly sensitive scientific cameras that record a “movie” documenting
the fluorescence of each molecule in its field of view over time (Figure 1.7).

Figure 1.7 (A) Single-molecule fluorescence trajectory from a DNA hairpin labeled with Cy3 andCy5 fluorophores, obtained at room temperature in buffer with 80 mM Na+. The fluorescenceshows alternative periods of high Cy3 fluorescence (green line) and high Cy5 fluorescence (redline). (B) FRET efficiency (EFRET) trajectory calculated from fluorescence intensities using Eq.
1.6. When Cy3 emission is high, EFRET is low. This implies that the distance between donor andacceptor is large, and the hairpin is in the unfolded state. Conversely, periods of high Cy5 emission
correspond to high EFRET, a small donor-acceptor distance, and occupancy of the folded hairpin
state.

20

1.6

Thesis overview
In this work, I use temperature-controlled single-molecule FRET microscopy toinvestigate the kinetics and thermodynamics of a several fundamental nucleic acid folding
motifs. In Chapter 2, I describe the single-molecule FRET microscope in detail as well as sample
preparation, data acquisition, and data analysis. In Chapters 3 and 4, I probe the ability of aminoacids to act as nucleic acid folding “chaperones”, which has implications for the evolution of life.
34, 84This work is split by structural level: Chapter 3 deals with amino acid effects on secondary

structure using a DNA hairpin, while Chapter 4 looks at tertiary interactions via a tetralooptetraloop receptor.85 Amino acids are found to have quite distinct effects on the secondary andtertiary levels, with tertiary structure being far more impacted by amino acids. Furthermore,tertiary structure is sensitive to amino acid chirality, and most of Chapter 4 is dedicated to
exploring the origin of this chiral sensitivity. Chapters 5 and 6 are devoted to extending themethodology of smFRET measurements. Chapter 5 introduces the use of stroboscopic
illumination as a means to extend the upper limit of rate constants determined by smFRET.
Chapter 6 pushes the limits of temperature-controlled smFRET experiments to obtain not only
the enthalpy and entropy of nucleic acid folding but also the change in heat capacity between
two nucleic acid conformations.86 Chapter 7 examines the influence of monovalent ions on themultistate folding dynamics of G-quadruplexes, where K+ and Na+ are found to exert control
over G-quadruplex polymorphism through distinct mechanisms. Finally, Chapter 8 presents
possible directions for future work based on preliminary results from several unfinished projects.

21

1.7

References

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Cech, T. R., Self-splicing and enzymatic activity of an intervening sequence RNA from
Tetrahymena (Nobel Lecture). Angew. Chem., Int. Ed. Engl. 1990, 29 (7), 759-768.

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Cech, T. R.; Zaug, A. J.; Grabowski, P. J., In vitro splicing of the ribosomal RNA
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Müller, S.; Appel, B.; Balke, D.; Hieronymus, R.; Nübel, C., Thirty-five years of research
into ribozymes and nucleic acid catalysis: Where do we stand today? F1000Research
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Weinberg, C. E.; Weinberg, Z.; Hammann, C., Novel ribozymes: Discovery, catalytic
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Cech, T. R., The ribosome Is a ribozyme. Science 2000, 289 (5481), 878-879.

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Joyce, G. F., The antiquity of RNA-based evolution. Nature 2002, 418 (6894), 214-21.

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Morris, K. V.; Mattick, J. S., The rise of regulatory RNA. Nat. Rev. Genet. 2014, 15 (6),423-37.

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Sherwood, A. V.; Henkin, T. M., Riboswitch-mediated gene regulation: Novel RNA
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Chapter 2
Methods

2.1

Single-molecule TIRF microscopeSingle-molecule measurements requires a specialized microscope capable of recording

the minute fluorescence from individual molecules. The single-molecule measurements
presented in this dissertation were primarily obtained using a total-internal reflection
fluorescence (TIRF) microscope. The exception to this is the first results chapter (Chapter 3) inwhich a confocal microscope was used while I was building the TIRF microscope. Both the
TIRF and confocal microscopes are given brief descriptions in our papers,1-4 but the details arecompact and sparse. Here, I will elaborate on the design and operation of the TIRF microscope.
For a more thorough description of the confocal apparatus, see the excellent theses of Dr. Julie
Fiore and Dr. Eric Holmstrom.
2.1.1 TIRF vs. confocal microscopy for single-molecule experiments
Before going over the technical details of the TIRF microscope, it is worth noting thedifferences between TIRF and confocal microscopy for single-molecule measurements, as theseare both excellent methods which have complementary uses. These two flavors of singlemolecule microscopy are named for the different type of illumination they use: in confocal
microscopy, light is focused to a diffraction limited spot in confocal microscopy, while in TIRF,
an evanescent field is created through total internal reflection. However, the capabilities andlimitations of these methods differ due to the different detector types in each. A single-molecule
confocal instrument contains a single-photon counting detector, usually an avalanche photodiode(APD), which responds to single-photon impingement on the detector by emitting an electrical28
current pulse for each photon detected. With the appropriate auxiliary equipment, these current
pulses can be recorded on a computer as a sequence of individual photon detection events withhigh absolute time precision (typically ≈ 50 ns due to APD dead time). This excellent time
resolution has allowed confocal microscopes to measure dynamics on the microsecond timescale.
However, this impressive speed comes at the cost of low throughput: APDs can only be used toobserve one molecule at a time. This is a small price to pay when the system of interest has fast
dynamics, but for a system with slow dynamics, the experimentalist can be required to spend
many hours, or days, to acquire the necessary amount of data for analysis. A TIRF microscope
instead uses a charge-coupled device (CCD) which consists of a grid of independent lightdetectors called pixels. Because each pixel measures light independently, the CCD detector canbe used to observe many fluorescent molecules simultaneously, typically 100−1000. This greatly
increases the rate of data acquisition over the confocal arrangement. However, digitizing the
CCD’s many pixels for computer storage is a time-consuming process which can only be done
about once per millisecond with state-of-the-art CCD cameras.* Any dynamics faster than this“frame rate limit” are lost to the experimentalist. Therefore, the confocal method has better time
resolution, while the TIRF method has higher throughput. This tradeoff of time resolution vs.
throughput means that neither confocal nor TIRF microscopes are the “silver bullet” which isbest for all circumstances, and that they are instead complementary approaches.
2.1.2 TIRF microscope overview: Laser excitation and fluorescence detection
This section will provide an overview of the flow of information in the TIRF microscope,from excitation to detection. Specific details for alignment of optics and system characterization*

The ever-evolving nature of CCD technology means that this digitization time will likely be made even shorter inthe future. In addition, there are ways to increase a CCD’s acquisition rate by decreasing the number of pixels to bedigitized (see section 2.1.4).

29
are provided in Sections 2.1.3−5. The light source in the TIRF microscope is a 532 nm
frequency-doubled, continuous wave (CW) Nd:YAG laser with 200 mW power. Light at thiswavelength allows for efficient excitation of Cy3 (donor) molecules with little Cy5 (acceptor)
excitation (σCy5/σCy3 = 3% at 532 nm). The raw laser output must be manipulated before going
into the microscope, as schematically shown in Figure 2.1A. First, residual 1064 nm light isFigure 2.1 (A) Schematic of the TIRF microscope excitation path before entering the microscope.
(B) Schematic depicting translation of periscope mirror to deflect laser beam. (C) Photograph of
optical arrangement.

30
removed with a 532 nm bandbass filter with an optical density (OD) of 7, pure linear polarizationof the laser output is obtained using a half-wave plate with a polarizing beamsplitter cube, andthe beam power is attenuated using neutral density (ND) filters. The beam is then expanded to 1
cm in diameter in a telescope consisting of plano-convex lenses with focal lengths f = 30 mm andf = 300 mm (magnification = 10). The lenses are oriented so that the collimated beam enters orexits the curved face of each lens to minimize spherical aberration. A pair of mirrors in aperiscope configuration vertically displace the beam to direct it into the rear port of an inverted
microscope (Zeiss Axiovert 135). Inside the microscope, the beam reflects off a long-passdichroic mirror (λ = 550 nm) before entering a 60X, high numerical aperture (NA = 1.4) oilimmersion microscope objective. To achieve widefield illumination, a f = 400 mm“epifluorescence” lens placed in the beam path before the microscope focuses the beam onto theback focal plane of the microscope objective (Figure 2.1B). The resulting collimated beamtransmitted by the objective illuminates a 40 μm diameter spot on the sample. This spot size isdetermined by the input beam size, the focal length of the epifluorescence lens, and themagnification of microscope objective.
To bring about total internal reflection (TIR), the laser must strike the sample at an
oblique angle beyond the critical angle. For experiments performed in aqueous solution withmolecules attached to a glass coverslip, the incident medium is glass (n = 1.52) and thetransmitted medium is water (n = 1.33); therefore, the critical angle is sin-1(1.33/1.52) ≈ 61°.
There are two common methods to obtain an angle of incidence which exceeds this critical angle.
The first, used in this work, is “through-objective TIRF,” in which the excitation laser passes
through the microscope objective intentionally off-axis to cause the laser to hit the sample at anangle (Figure 2.1B), just as translating a lens in front of a laser to deflect the beam. This method

31
is relatively straightforward to implement, as the excitation spot and detection region areautomatically aligned. However, there are drawbacks to this method. First, to generate TIR, themicroscope objective must have a sufficiently large numerical aperture so that the maximum
collection half-angle is greater than the critical angle. This criterion is met if the objective’s NA
is greater than the index of refraction of the transmitted medium, in this case water (n = 1.33).
Such a high numerical aperture can only be obtained using expensive, oil-immersion objectives.
Second, the laser inevitably interacts with the objective’s internal optics to generate
autofluorescence which results in additional background signal. The second method is “prismbased TIRF,” in which the laser strikes the sample on the side opposite the microscope objective,
where the angle of incidence is controlled by a prism. This method has an improved signal-tobackground ratio because the laser never enters the objective, so there is virtually no
autofluorescence. However, prism-based TIRF has its own drawbacks, such as needing to alignthe excitation spot with the objective’s field of view, and requiring an objective with a highworking distance to focus through the sample to the far interface. We weighed these issues andultimately decided against the prism-based approach for a practical reason: through-objective
TIRF allowed us to leave one side of the sample free to attach a temperature control module (seeSection 2.3), which would have been much more awkward to accomplish in a prism-based setup.
We obtain through-objective TIR by mounting one of the periscope mirrors onto a
micrometer translation stage (Figure 2.1B). As the stage is moved, the beam is translated off the
optical axis of the objective which continuously increases the angle of incidence. Sample slides
are mounted on a homebuilt translation stage to allow scanning of the sample surface. The stage
consists of a pair of translation micrometers, one each for the x and y directions, that are attached
to a cantilevered aluminum plate which suspends the sample above the microscope objective.

32
This design should be improved in future implementations, as the cantilever requires meticulous
tuning to align the sample plane with the objective’s imaging plane. Fluorescence from thesample is collected by the microscope objective, transmitted through the long-pass dichroic
mirror in the microscope, focused by the microscope’s internal tube lens, and directed out theside port into the final section of the experiment, the detection tree (Figure 2.2).
The fluorescent image output by the microscope comes to a focus approximately 10 cm
beyond the microscope side port. This image is then relayed to a charge-coupled device (CCD)
by a f = 75 mm lens that collimates the light and a second f = 400 mm lens which focuses theimage onto the camera (to avoid chromatic aberration, all lenses used in the detection tree are
achromatic cemented doublets). This lens pair also acts as a telescope to further magnify the

Figure 2.2 Detection tree used to separate donor and acceptor emission and refocus image onto
CCD detector. Donor emission is in yellow, acceptor emission is in red, and combined emission
before the dichroic is in orange. Two filters remove excitation laser light: a 547 nm long-pass (OD7) at the excitation wavelength before the dichroic mirror and a 1064 nm short-pass (OD 6) toremove residual un-doubled light from the Nd:YAG after the dichroic mirror. (LP = long-pass).

33
image. Within this image relay, there are several additional optical elements. First, a periscope
lowers the beam to the optical table. Second, a 547 nm long-pass filter (OD 7) removes scatteredlaser light from the image. Third, and most importantly, a long-pass dichroic mirror (λ = 645 nm)
separates the Cy3 and Cy5 emission into independent channels. The two channels are thenrecombined by a second, identical dichroic mirror before focusing onto the CCD. The splitting
and then recombining of the Cy3 and Cy5 emission is used to introduce a translational offset in
the two channels. Therefore, two images (donor and acceptor) are projected side-by-side onto the
CCD. An alternative method is to use two CCD cameras, one for each channel, but the onecamera approach avoids needing to synchronize the two cameras for data analysis.
One should note two things about this detection setup. First, in this arrangement, there are
two f = 400 mm lenses, one for each channel. Using two focusing lenses provides even greater
mitigation of chromatic aberration by permitting slightly different path lengths for each channel,
which ensures that both Cy3 and Cy5 images come to the same focus. Second, it is essential thatthe donor and acceptor images do not overlap on the CCD. When the images are circular, as isnatural when working with a circular laser beam, projecting the images on the CCD in a nonoverlapping way results in a significant unused area of the CCD. However, truncating thecircular image into a vertical strip permits nearly full use of the CCD’s pixels, which in turnallows imaging of more molecules simultaneously (Figure 2.3). We perform thistruncation by inserting a slit at the image focus output of the microscope (Figure 2.2).
The CCD used in single-molecule fluorescence measurements must be capable ofoperating in low-light conditions with rapid readout and low noise. In this experiment, the CCD
used for detection is a so-called intensified CCD, or ICCD. In an ICCD, light strikes amultichannel plate (MCP) to generate photoelectrons which are then multiplied ~105-6 times

34

Figure 2.3 Fluorescence image is truncated into strips to maximize used CCD area. Slit is placedat the microscope output tube lens focus (see Figure 2.2).

before being converted back into light using a phosphor screen, and this amplified light isdirected by a fiber optic bundle to strike a ¼ megapixel CCD (512 x 512 pixels). The CCD
operates in the usual way (more details are given below in Section 2.1.4): photoelectrons areaccumulated (“integrated”) during the exposure time, after which the pixel grid is shifted column
by column into a shift register which delivers each pixel to a fast readout register (5 MHz) for
analog-to-digital conversion (ADC) before computer storage. The key principle for CCD
operation in low light is the use of signal amplification to bring the signal level (in CCD
electrons) above the readout noise level of the ADC. For ICCDs, the MCP plays this role. A
popular alternative to ICCDs for smFRET measurements is the electron multiplying CCD(EMCCD), which uses an electron multiplying shift register on the CCD chip to achieve signal
amplification. By swapping the MCP for on-chip electron multiplication, the EMCCD
architecture has a higher photon detection quantum yield and therefore a higher signal level.
However, unlike the ICCD, the EMCCD also amplifies unwanted electrons in the CCD, such as
thermally generated “dark electrons,” which results in greater noise. As a result, the signal-to-35noise ratio (SNR) of the two camera types is similar. A third camera type gaining interest in the
smFRET field is the scientific complementary metal-oxide-semiconductor (sCMOS) camera.
Vendors developing sCMOS cameras promise decreased readout noise, higher frame rates, andmore pixels than ICCD or EMCCD cameras at comparable prices. This relatively newtechnology will be one to watch.
2.1.3 Strategy for optical alignment pre- and post-sample
I will now outline the step-by-step process used to align the various optical elements in
the TIRF microscope. Alignment is split into pre-sample laser alignment and post-sample
emission collection alignment. Laser alignment occurs in three steps: (i) align for confocal
illumination, (ii) switch to epifluorescence (widefield) mode, and (iii) generate TIR. For confocalmode, the laser light must be collimated and on-axis when entering the microscope objective.
Therefore, I first ensure that the beam is collimated after leaving the expansion telescope (Figure2.1), which is done simply by temporarily inserting a mirror in the beam path and projecting thebeam off the optical table to a far wall (after, of course, taking all safety precautions needed forworking with a laser off-table). The beam is well collimated if the beam diameter is unchanged
from the table to the wall. Then, to align the beam to the center of the microscope objective, I
replace the objective with a long (~0.5 m) lens tube with an iris attached to each end, and I use
the two periscope mirrors to align to the two irises through the spot projected onto the ceiling. It
is helpful here to use a tight iris early in the beam path to “clamp down” on the beam, which
creates a diffraction pattern that is easier to use when determining if the beam is centered on eachiris. At this point, the system is aligned to create a diffraction-limited spot (“confocal mode”).
For widefield illumination (epifluorescence), the laser must be focused onto the backfocal plane of the objective, which is accomplished by the epi lens (Figure 2.1). First, the spot

36
location on the ceiling is marked and the lens tube is replaced by the microscope objective. Then,an empty sample chamber (see below) is placed on the microscope stage above the objective, anda flip-mounted epi lens is introduced into the beam path. The epi lens position perpendicular tothe beam is adjusted until the ceiling spot is centered on the marked confocal spot location,
which indicates that the lens is centered and the beam has not been deflected. The epi lensposition along the beam propagation direction is adjusted until the spot size is minimized on theceiling, as a beam properly focused onto the objective’s back focal plane will have the minimum
divergence angle. At this point, in epifluorescence mode, the sample is illuminated by awidefield spot that transmits through the sample.
To enter TIR mode, the translation-state mounted periscope mirror is moved to deflect
the beam until the critical angle is reached. However, as determining the optimal angle is best
done during fluorescence imaging, one should align the fluorescence detection system (described
below) in epifluorescence mode first before attempting to enter TIR mode. Optimizing the TIR
angle is done by observing how a fluorescence image changes as the angle of incidence isincreased: when TIR is obtained, the background fluorescence will sharply diminish and thefluorescence from surface attached molecules will “flash” brighter before becoming
progressively dimmer as the critical angle is surpassed. The origin of this flash is described in
Section 2.1.5; it suffices here to say that the angle of incidence at which the flash is observed isthe optimal angle that maximizes the signal to background ratio.
Alignment of the fluorescence imaging system is done through several levels of
increasing refinement. For coarse alignment, it is useful to have a fluorescence source that isvisible to the naked eye, such as concentrated (millimolar) rhodamine 6G in ethanol. Better yet isa fluorophore whose emission is split by the dichroic that separates Cy3 and Cy5 emission, such

37
as 4-dicyanomethylene-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran (DCM). Under strong
enough laser illumination (~100 mW), the fluorescence image obtained by the microscope
objective from these laser dyes can be seen on white paper in a dark room, which facilitates
positioning optical elements to obtain rough alignment. The next step is to turn on the CCD
camera and image bright, fluorescent particles larger than the diffraction limit, such as 2 μm
fluorophore impregnated beads. These beads are useful because they are extremely bright objects
that photobleach very slowly and can thus survive extended imaging. Care should be taken at thisstep to adjust the laser power so as not to overload the CCD. For 2 μm fluorescent beads, thepower level at which CCD saturation occurs is approximately 0.1 mW. First, the acceptor
channel is blocked, and a focus is obtained on the donor channel. Then, the donor channel isblocked, and the acceptor channel is brought into focus by translating the acceptor focusing lens(Figure 2.2). One may then switch to imaging photostable objects smaller than the diffraction
limit, such as 100 nm diameter fluorescent beads or quantum dots, and further refine the coalignment of the two channels. At this point, the slit can be introduced at the focal plane of the
tube lens output from the microscope. A simple slit construction is two razor blades mounted
parallel on optical posts. When positioned correctly, the edges of the slit will appear sharply infocus on the CCD, and the transmitted image will center on the brightest part of the laser
illumination spot. Finally, the donor and acceptor channels are simultaneously uncovered, andthe offset of the two images is adjusted until the camera field of view is fully occupied. (e.g.,
Figure 2.3).
2.1.4 Camera sensitivity, data acquisition rate, and image magnification
The ICCD camera used in these experiments reports the light level per pixel in ADUs
(analog-to-digital units), which is an arbitrary unit. To convert ADUs per frame into a physically38
meaningful quantity, the photon detection rate, we need to know the ADU-to-photon conversion
rate κ. For an average number of photons detected by the camera per frame N, the average CCD
signal S is
𝑆 = 𝜅𝑁 + 𝑅 + 𝐷(Eq. 2.1)

where R is the ADC readout offset, and D is the CCD dark charge. The ADC readout offset R is
specific to the ADC and is a constant value for all pixels, measured as R = 140 ADU/pixel. The
dark charge D is signal due to thermally generated electron-hole pairs in the CCD pixels. D
scales linearly with the time between pixel readouts, and by using an ICCD cooled to −30 °C, thedark current is reduced to D = 4 ADU/s/pixel, which is negligible at all frame rates used in thisdissertation. While the average values of R and D can be simply subtracted from each pixel, thenoise that R and D contribute cannot and must be accounted for. As N, R, and D are independent,
the corresponding uncertainty (standard deviation) of the signal σS is the quadrature sum of theuncertainty of each contribution:
𝜎𝑆 = √𝜅 2 𝐺 2 𝜎𝑁2 + 𝜎𝑅2 + 𝜎𝐷2 = √𝜅 2 𝐺 2 𝑁 + 𝜎𝑅2 + 𝐷 ≈ 𝜅𝐺√𝑁

(Eq. 2.2)

Here σN, σR, and σD, are the uncertainties in the photon rate, the readout noise, and the darkcharge, respectively, and G is a multiplicative increase in the photon detection uncertainty due toimperfect gain (see Appendix 2), which for the exponential gain in the MCP is approximately√2. Photon detection and dark charge accumulation are both Poisson processes whose variances
are equal to their averages, which is the basis of the substitutions σN = N1/2 and σD = D1/2. The
read noise uncertainty is measured as σR = 4 ADU. At high N, Eq. 2.2 reduces to a simple squareroot relationship between σS and N, and the ratio of the CCD variance to the signal level is
𝜎𝑠2

=
𝑆

𝜅2𝐺2 𝑁𝜅𝑁= 𝐺 2𝜅

(Eq. 2.3)

39
Therefore, measuring the ADU/photon conversion rate can be done by taking a CCD movie at a
relatively high but non-saturating signal level and computing the uncertainty per pixel squared
divide by twice the average per pixel. For the current ICCD camera, κ = 1.9 ADU/photon.
The rate at which CCD frames can be acquired places important limits on the measurable
rate constants for this experiment. The frame rate ϕframe is determined by the per-pixel
digitization time τADC, the shift register time τSR which occurs once per digitized row, and theframe shift time τFS, as well as the number of rows Nx and columns Ny to be digitized:−1
𝜙𝑓𝑟𝑎𝑚𝑒= 𝑁𝑥 𝑁𝑦 𝜏𝐴𝐷𝐶 + 𝑁𝑥 𝜏𝑆𝑅 + 𝜏𝐹𝑆(Eq. 2.4)

The measured parameters of the ICCD used in this work are (τADC)-1 = 5 MHz, τSR = 7 μs perrow, and τFS = 5.6 ms. Therefore, the full frame readout time (Nx = Ny = 512 pixels) is 61 ms (ϕframe = 16 Hz). Higher frame rates can be achieved by “cropping” the field of view to have fewer

columns or especially rows, but cropping comes at the cost of decreased throughput, as a smaller
imaging area means fewer molecules observed simultaneously. Even if the entire field of view iscropped (Nx = Ny = 0), the frame shift must occur once per frame and sets an upper limit of ϕframe= 180 Hz. Dynamics occurring faster than the frame rate are lost due to averaging, and therefore
the ICCD in these studies is most well-suited to study rate constants slower than 10 s-1 but can be
adjusted as needed and can in principle be used to examine dynamics on the single-digit
millisecond timescale.
The net magnification of the CCD image is an important quantity to select for in theexperimental design. Due to diffraction, each fluorescent molecule cannot be imaged as a point
light source but will instead appear spatially broadened across multiple CCD pixels as a finitewidth point spread function (PSF). The PSF depends on the details of the optical system, but acommonly used PSF is the Airy pattern, which is the Fourier transform of a circular aperture.

40
One may define the minimum resolvable distance as the first zero of the Airy pattern, which is
the Rayleigh criterion (rRC = 1.22 λ / 2 NA, where λ is the light wavelength and NA is thenumerical aperture of the microscope objective), or alternatively as the Airy pattern’s full widthat half max (FWHMAiry = 1.03 λ / 2 NA), which is easier to compare to a Gaussian fit(FWHMGaussian = 2.35 σ, where σ is the radial standard deviation). For Cy3 emission centerednear 600 nm, rRC = 260 nm and FWHMAiry = 220 nm. This resolution determines the appropriate
pixel resolution dpixel, which is the size of pixels in object space. If dpixel is larger than the PSF
width, molecules will appear as single-pixel points, and all width information is lost (forinstance, one cannot fit the PSF width to determine if the “molecule” is in fact more than one
molecule superimposed on the same pixel). As dpixel is decreased, individual fluorophores will
occupy more pixels on the CCD. However, when summing pixels to determine total intensity,each pixel contributes additional read noise, and therefore the signal-to-noise ratio degrades as
dpixel decreases. Furthermore, the multiplexing capability of the CCD diminishes as eachmolecule take up more space. Therefore, an ideal system magnification produces a PSF with adiameter of 2 or 3 pixels. For the current system, the magnification, as measured with acalibration slide of a grid of 5 μm squares (see Figure 2.4A), is 78 nm/pixel. Upon binning pixels
into 2×2 or 3×3 “superpixels,” the desired 3-pixel wide PSF is obtained (see Figure 2.4B). As of
the writing of this dissertation, the system magnification is being altered so as to utilize the full,unbinned 512×512 pixel grid to increase throughput.
2.1.5 System collection efficiency and sources of noise
The fluorescence emitted by molecules in the single-molecule microscope undergo
unavoidable losses before being detected by the camera. These losses are quantified by thecollection efficiency η, which is the ratio of the detected photon rate ϕE to the true fluorophore

41

Figure 2.4 (A) Image of calibration grid of squares with side length 5 μm for magnification
determination. (B) Use of on-chip pixel binning to decrease system PSF width (image is of a 100
nm diameter quantum dot).
photon emission rate ϕEm (η = ϕEm / ϕE). In the absence of acceptor, the donor emission rate canbe calculated as the product of the photon flux density in the TIR evanescent field with theabsorbance cross section and fluorescence quantum yield of the donor (Cy3):𝐼

𝜙𝐸𝑚 = ℎ𝜈 𝜎𝐶𝑦3 𝑄𝐶𝑦3

(Eq. 2.5)

where I is the electromagnetic field intensity (in W/cm2), hυ is the energy per photon (3.74 x
10-19 J at 532 nm), σCy3 is the absorbance cross section of Cy3 (3.49 x 10-16 cm2) and QCy3 is thequantum yield of Cy3 fluorescence (≈ 0.25†). The intensity I is not the same for all imaged
molecules due to the laser’s Gaussian beam profile,2

2

𝐼(𝑟) = 𝐼0 𝑒 −2𝑟 /𝑤 ,†(Eq. 2.6)

This quantum yield is for Cy3 at room temperature conjugated to a nucleic acid. The quantum yields of Cy3 andCy5 are greatly enhanced by stacking on nucleobases in DNA and RNA. This has the undesirable effect ofstabilizing nucleic acid secondary structure. A purportedly less perturbative FRET pair is Alexa Fluor 555 and 648,
which could be used without changing any of the optical components in this work (e.g., dichroic beamsplitters).

42
where I0 is the intensity at r = 0 and w is the 1/e electric field radius of the beam, equal to 20 μm
for this optical setup.‡ For this signal calculation, consider a molecule at the center of the beam.
The laser power P, which is easy to measure, is related to the peak intensity I0 by integrating thebeam profile:
2𝜋

∞

1

𝑃 = ∫0 𝑑𝜙 ∫0 𝑟 𝑑𝑟 𝐼(𝑟) = 2 𝐼0 𝜋𝑤 2 .

(Eq. 2.7)

At a typical operating laser power P = 10 mW, this yields a peak intensity I0 = 1.6 kW/cm2.
However, this is the intensity of the freely propagating laser, not of the evanescent field that thefluorophores experience, which will now be discussed.
The behavior of light at interfaces is described in general by the Fresnel equations, whichshow in the case of total internal reflection (TIR) that there is an evanescent wave generated in
the low-index medium which oscillates at the same frequency as the incident wave and has afield intensity that decays exponentially with the distance from the interface:5
𝐼𝑡 (𝑧) = 𝐼0 𝑒 −𝑧/𝑑 .

(Eq. 2.8)

This evanescent wave is characterized by its intensity at the interface I0 and its penetration depth
d. The subscript t stands for the “transmitted” medium, though no light is transmitted in the far
field sense. For incident light polarized perpendicular to the plane of incidence (s-polarization),
the evanescent field is linearly polarized, while incident light polarized parallel to the plane ofincidence (p-polarization) produces an elliptically polarized evanescent field. The penetration
depth depends on the laser frequency and angle of incidence:
𝜆𝑑 = 4𝜋0‡1√𝑛𝑖2 sin2 𝜃𝑖 −𝑛𝑡2

(Eq. 2.9)This uneven illumination results in a fluorescence image with a “vignette”—molecules near the edge fluoresce lessthan those near the center. This has the effect of sampling many different signal-to-noise levels in one movie, which
unfortunately means that not all parts of the field of view are equally useable. Adopting “flat field” illumination
methods would therefore be a useful improvement in future developments.

43
where λ0 is the light wavelength in vacuum, θi is the angle of incidence, and ni and nt are theindices of refraction for the incident and transmitted media, respectively. For this work, theincident medium is glass (ni = 1.52) and the transmitted medium is water (nt = 1.33). The
intensity at z = 0 (I0) depends on the angle of incidence and the laser polarization:
𝐼0⊥ = 𝐼𝑖⊥

𝐼0∥ = 𝐼𝑖∥

4 cos2 𝜃𝑖2𝑛
1−( 𝑡⁄𝑛𝑖 )2𝑛
4 cos2 𝜃𝑖 [2 sin2 𝜃𝑖 −( 𝑡⁄𝑛𝑖 ) ]42𝑛𝑛( 𝑡⁄𝑛𝑖 ) +sin2 𝜃𝑖 −( 𝑡⁄𝑛𝑖 )(Eq. 2.10)(Eq. 2.11)

where the superscript symbols ⊥ and ∥ refer to s-polarized and p-polarized light, respectively,
and Ii⊥,∥ is the light intensity in the incident medium. The angular dependence of d, I0⊥, and I0∥
for 532 nm light at the glass-water interface are shown in Figure 2.5. Note that for incidence

Figure 2.5 Angular dependence of evanescent field penetration depth d and intensity I0⊥,∥ for 532
nm light at the water-glass interface (nt/ni = 1.33/1.52).
angles θi near the critical angle θc, the evanescent intensity at the surface is greater than theintensity of the incident wave (I0⊥,∥ > Ii⊥,∥); this explains why fluorophores become brighter when
TIR is established. The intensity increase is slightly greater for p-polarized light than s-polarized
light; however, the microscope in this work is configured to use s-polarization rather than ppolarization to avoid the transmission of imperfectly collimated laser light striking the sample atthe Brewster’s angle. The angle of incidence in the experiment is difficult to determine precisely

44
but is narrowly bounded from 61° to 67°. The lower bound is the glass-water critical angle θc =61°, and the upper bound is the maximum acceptance angle of the objective θmax = 67°, which is
calculated from the objective’s numerical aperture (NA = 1.4),
𝑁𝐴 = 𝑛 sin 𝜃𝑚𝑎𝑥 ,(Eq. 2.12)

where n is the index of refraction of glass (n = 1.52). A reasonable estimate for θi is 64°, forwhich the penetration depth d is 140 nm and the ratio of transmitted to incident surface intensity(It / Ii) is 3.3. The fluorophores are tethered to the surface at a distance z ≈ 10 nm, yielding thefluorophore’s true local electric field intensity I = 4.9 W/cm2. Multiplication of I by σCy3 gives an
expected excitation rate of 4.6 MHz, for an average time between excitations of 220 ns. Since
this time is much larger than the fluorescence lifetime of Cy3 (τf ≈ 5 ns), there is no concern forCy3 saturation, even up to the present 200 mW laser power limit, and the linear absorbanceassumption in Eq. 2.5 is valid. Multiplying by the fluorescence quantum yield gives an emission
rate ϕEm = 1.2 MHz. The measured photon fluorescence rate under these conditions is ϕF = 80
kHz, and therefore the system collection efficiency η is 7%.

2.2

Sample cells and sample preparation
Disposable glass flow cells of a cheap and simple “sandwich-style” design act as sample

cells which allow for easy optical access and solution exchange. The flow cell, shown in Figure
2.6, consists of #1.5 glass coverslip (CGI Life Sciences) attached to a 75 mm x 25 mm glass
slide (VWR) via double-sided tape (3M). It is essential for the coverslip surface to be clean andfree of any fluorescent material, so the coverslip is soaked in acetone overnight (>12 hours)
before being cleaned in a UV-ozone oven (Jelight Mo. 42). The UV-ozone treatment oxidizes
any residual material on the coverslip, and the oxidized products are removed by later wash

45

Figure 2.6 “Sandwich” style flow cell for microscopy.

steps. Two parallel strips of double-sided tape are used to create a channel approximately 2 mm
wide, 22 mm long, and 0.1 mm deep. This channel can then be used for solution exchange by
pipetting solution droplets onto one channel entrance and applying a tissue to the other to draw
up solution via capillary action (i.e., wicking). One must be careful when wicking not to
introduce bubbles into the channel. Bubbles can become stuck in the channel, which impedes
solution flow and limits usable surface area for microscopy. Additionally, even if the bubble
clears the channel, it can disturb the surface protein coatings (described next) that permit thetethering of constructs to the surface.
Once a clean glass coverslip incorporated into a flow cell, the coverslip surface is thenfunctionalized to enable tethering of nucleic acid constructs (see Figure 2.7). This is carried out
by first coating the surface in bovine serum albumin (BSA, 10 mg/mL) which serves to “block”
the glass surface so that the nucleic acids do not adhere to the glass. In addition, 10% of the BSA
are labeled with biotin (vitamin B7), a small molecule that has an extremely high-affinity
noncovalent interaction with the protein streptavidin (Kd ≈ 10 fM, unbinding rate kdiss = 10-4 s-1).
Therefore, treating the BSA-coated surface with streptavidin (0.2 mg/mL) will create BSAbiotin-streptavidin sites. Because streptavidin is a tetrameric protein complex with four bindingsites for biotin, each tethered streptavidin is available to bind more biotin molecules. So, a final

46

Figure 2.7 Surface functionalization for nucleic acid construct tethering. A clean glass surface iscoated in a mixture of bovine serum albumin (BSA) and biotinylated BSA (BSA in gray, biotin inyellow). Streptavidin (green) binds to biotin-BSA sites on the surface, and finally a biotin-labeledconstruct (purple) is added which attaches to available biotin binding sites in the tetrameric
streptavidin complex. The result is a surface decorated in nucleic acid construct with a surface
density controlled by the concentration of the proteins and nucleic acid.

wash with a biotin-labeled nucleic acid construct (100 pM) will attach the construct to thestreptavidin to complete the surface tethering process. These three solutions (BSA + biotin-BSA,
streptavidin, and nucleic acid construct) are each incubated with the surface for 10 minutes inbuffer (50 mM hemisodium HEPES, pH 7.6). This incubation time is significantly longer thanthe ≈ 1 minute required for molecules to diffuse to the surface, as predicted from the Einstein
relation for diffusion in 1 dimension, Δx2 = 2Dt, where D is the diffusion coefficient (≈ 10-6
cm2/s for a 60 kDa protein in water), Δx is the distance traveled (= 0.1 mm), and t is the diffusion
time. The measured construct surface density σ = 1 molecules/μm2 is smaller than the theoreticalmaximum for 100 pM construct of σ = 3 molecules/μm2, and the surface density increases if the47construct is incubated for longer, saturating after ≈20 minutes. This clearly indicates the presenceof additional kinetic barriers to streptavidin-biotin binding besides simple diffusion. Note that theresulting surface density is highly variable (± 50%) between samples and even between locations
on a given sample. Having a reproducible surface density is important for efficient datacollection, so it may be worthwhile to explore more stringent coverslip cleaning methods oralternative suppliers for coverslips, double-sided tape, etc.
One issue with the sandwich-style sample holders is sample evaporation. The channelends are open to atmosphere, and evaporation of the sample becomes visually noticeable after
approximately 5 minutes. This problematic for many reasons, including that the loss of solvent
will increase the concentration of solutes in the imaging solution during data collection. If data
can be collected in 1−2 minutes, such as for relatively rapidly folding constructs, sample
evaporation is not a concern. However, for slower nucleic acid folding, or for operation at hightemperatures (see below), evaporation poses a serious problem. To halt solution evaporation, thechannel ends are sealed with RTV silicone which sets sufficiently 10 minutes after application.
The silicone sealant is highly effective and allows a sample to be used for several hours with nonoticeable evaporation. It is possible that some of the RTV components, which include acetic
acid, may mix with the imaging solution, but a test showed no difference between rate constants
measured from samples with or without RTV. Once sealed, the imaging solution in the flow cell
can no longer be exchanged, which means that comparing folding dynamics at multiple cosoluteconcentrations requires creating a sample holder for each concentration. Ideal would be thefuture development of an alternative flow cell design which has limited evaporative losses
without the use of sealants.

48
2.3

Temperature control
Essential to the work in this dissertation is the ability to perform single-molecule

microscopy at a controllable sample temperature. Previously in the group, temperature control on
the confocal instrument has been accomplished in two ways: (i) resistively heating a heatingstage and objective collar, and (ii) locally heating a column of water around the microscopefocus with an infrared laser.6 Both are capable only of increasing the sample temperature above
ambient temperature. For the TIRF microscope, I have instead opted to use thermoelectriccoolers (TECs) to allow for both cooling and heating of the sample. TECs work via the Peltier
effect, whereby a voltage across a thermoelectric material creates a thermal gradient and thereby
heat flow across the TEC. By changing the sign of the applied voltage, the direction of heat flow
can be switched, which allows the TEC to both heat and cool.
In detail, temperature control is established using an arrangement of two TECs, one
resting on top of the sample slide, and one attached to the microscope objective (Figure 2.8).
Heating the objective is important, as the objective is in thermal contact with the sample via theimmersion oil. Furthermore, the use of two TECs helps eliminate thermal gradients across the

Figure 2.8 Composition of dual TEC temperature controller.

49
sample. The TECs are constructed from TEC modules (TE Technology HP-127-1.0-0.8)
compression-mounted between two home-machined aluminum plates using nylon screws to
reduce thermal backflow, and the TECs are coupled to the sample/objective via thermal pads
(EN-Laboratories). Temperature stabilization is accomplished via a PI feedback loop. In thisloop, the sample temperature is measured by thermistors (Vishay) attached to the aluminum
plates, which feed into the error signal of temperature controllers (TE Technology TC-720) thatdrive the TEC modules.
The rate of cooling that can be achieved by TECs is limited. This is due to the resistive
heating of the TEC modules as electrical current is applied as well as the imperfect thermal
isolation of the two sides of the TEC devices. The TEC attached to the objective has anespecially difficult task as the objective itself is connected to the microscope body, and therefore
the entire microscope acts as a thermal reservoir which replaces heat removed by the TEC. To
combat these issues, the TECs are cooled by heat sinks containing frozen reservoirs. Using
frozen water (ice), the system is capable of cooling to ≈ 10 °C, whereas solid carbon dioxide (dryice) can cool the sample well below 0 °C.§
Tuning of the PI feedback parameters was initially carried out using the Ziegler-Nichols(ZN) method, in which the proportional term P is increased until temperature oscillation occurs(at ω = 2.1 min-1 for this system), then the integral term I is selected based on the oscillation
period and the P term is reduced by a prescribed amount. However, the ZN parameters were
found to be too aggressive. In particular, the large temperature overshoot when changing the set
point would occasionally cause the system to become unstable. Therefore, the PI parameters§

The ability to operate at subzero temperatures opens up interesting research avenues. For instance, one could
examine folding under conditions of freezing point depression, such as in salty water.

50
were manually decreased until reaching a satisfactory temperature stability (± 0.05 °C) and amodest settling time (< 1 min) without the risk of runaway heating.

2.4

Stroboscopic illumination
Folding rate constants that are faster than 10% of the camera frame rate are subject

systematic underestimation in smFRET measurements with constant illumination. The origin of
this effect, and the explanation of how a flashing light source (i.e., a strobe) can resolve thisissue, are described in detail in Chapter 5. Here, I provide a technical description of how
stroboscopic illumination is produced and synchronized with CCD data acquisition. An ideal
strobe is a pulse train with one pulse per data acquisition frame and a duty cycle of 10-20%. The
first attempt at light pulsing was to send a TTL signal to the Nd:YAG laser driver. However, theresulting laser pulse has a rise time of ~10 ms while lasing was being reestablished, which
prevented going to the fastest camera frame rates. The second arrangement was to use an opticalchopper wheel driven by a frequency-stabilized feedback controller (ThorLabs). This ensureseven intensity through the strobe pulse, and 1 ms pulse times are easily achieved. The strobe
pulse must be synchronized to the camera frame acquisition, which is done by sending thereference out signal of the chopper controller to the CCD trigger input. In the CCD control
software, WinView, the camera’s acquisition settings are selected to make the exposure time lessthan the time between pulses, so that no triggers are missed, and the timing mode is changed
from the internally-timed “free-run” mode to “external trigger.” Rate constants determined bystroboscopic experiments require a simple correction to account for the unobserved time (theportion of each frame when the laser is off); this correction is described in Chapter 5.

51
2.5

Data analysis

2.5.1 Extraction of single-molecule trajectories from movies
Once recorded, movies of the construct-decorated surface are processed in order to obtainsingle-molecule fluorescence trajectories. This analysis is carried out using OpenTIRF.exe, aprogram written by myself in the LabWindows/CVI C-language programming environment (seeFigure 2.9). OpenTIRF processes each movies through a sequence of steps to extract, for eachmolecule, the donor and acceptor fluorescence intensity trajectories (D(t) and A(t), in photon
counts per unit time) as well as the estimated FRET trajectory (E(t)). I will now describe each ofthese processing steps.
First, the movies are loaded into memory. Movies are decoded from the SPE file format
used by WinView, the Princeton Instruments software that controls the CCD camera. The SPE
file consists of a header, which contains information about the data collection conditions(number of frames, exposure time, pixel dimensions, etc.), and the body, which contains the 12bit digitized output per pixel per frame. Because of the information in the header, no informationis required from the user to load the movie, unless stroboscopic illumination is used, in whichcase the frame rate is not saved in the header and must be input manually.
Second, the program identifies the locations of molecules in the donor and acceptor
channels. The first step is to average together a range of frames as a representative image to use
in particle detection. This averaging is necessary because constructs in folding states withextremal FRET efficiencies (EFRET ≈ 0 or 1) will only appear in one of the two channels (donoror acceptor). Therefore, using a single frame for particle detection can result in the identificationof donors without acceptors, or vice versa. Instead, averaging over the folding equilibration timeof the smFRET construct ensures that molecules appear on both donor and acceptor channels.

52

Figure 2.9 Screenshots of OpenTIRF program written by the author to extract smFRET trajectories
from CCD movies. (A) OpenTIRF loads movies with metadata, averages frames for molecule
detection, quickly cycles through molecules in a movie, extracts donor and acceptor fluorescenceintensities, calculates FRET trajectories, and can threshold FRET trajectories for dwell time
analysis. (B) The image thresholding utility automatically finds molecules with several algorithms
available. (C) The 2D fitting utility models particle PSFs as 2D Gaussians for position refinement
and filtering.

53
Additionally, this averaging increases the signal to noise ratio, which reduces the likelihood ofsingle bright pixels (due to cosmic rays) being identified as molecules. Next, this averaged image
is subjected to a local thresholding algorithm from the LabWindows/CVI machine vision
package, and pixel regions above threshold are labeled as molecules (Figure 2.9B; a manualthreshold option is available in case the automated thresholding algorithm fails). The center ofmass of each above-threshold region is used as a rough estimate of the particle’s position. Next,
the rough position estimate is refined by fitting each putative molecule to a 2D Gaussian (Figure
2.9C),𝐴

1

2

𝐺2𝐷 (𝑥, 𝑦) = 2𝜋𝜎2 exp (− 2𝜎2 [(𝑥 − 𝜇𝑥 )2 + (𝑦 − 𝜇𝑦 ) ]) + 𝐵,

(Eq. 2.13)

where A is the amplitude, B is the background per pixel, σ is the standard deviation (the full
width half max is calculated as FWHM ≈ 2.35 σ), and μx and μy are the Gaussian’s x and ycenters, respectively. This fit is restricted to the local pixel neighborhood surrounding themolecule to avoid having the fit converge to a nearby brighter molecule. As a result, B is a local
background value, which is useful for computing background-corrected FRET values, described
below. This 2D fit is also used to filter out particles whose fit parameters fail to converge orconverge to unusual values, such as negative amplitudes, or standard deviations larger than thefit window. Note that this 2D fit function is circularly symmetric, with a single, isotropic widthparameter σ. Fitting the PSFs to a full bivariate normal distribution with independent x and ywidths and an xy correlation term shows negligible ellipticity (major to minor axis ratio a/b =1.02 ± 0.06). Therefore, to reduce model complexity and speed up fitting, the simpler, isotropic
Gaussian is used instead. However, it may be worthwhile to explore using a bivariate normal fit
as a way of obtaining additional parameters to use in particle filtering (e.g., particles that appear
elliptical may in fact be two molecules in close proximity). Lastly, particles are assigned as

54
donors or acceptors by their location on the CCD, i.e., whether they lie on the right or left half,
respectively.
With molecule locations determined, the single-molecule fluorescence trajectories can becalculated. There are several options in OpenTIRF of how to calculate intensity trajectories. In
each case, the fitted local background value is first subtracted from each pixel. The first method
is simple pixel summation: pixels within a user-specified radius of the molecule’s center position
are summed together for each frame (I(t) = ∑pi(t)). This method is robust and fast but requires
tuning by the user to find the optimal radius that maximizes the signal to noise ratio. The second
option is a weighted pixel sum: each pixel in the molecule’s neighborhood is multiplied by thefitted 2D Gaussian value for that pixel and then summed together (I(t) = ∑pi(t)·G(xi,yi)). Thismethod gives less weight to pixels far from the molecule center and therefore includes lessbackground noise and can handle overlapping molecules better than simple pixel summation.
However, this is a heuristic which should be developed further, preferably under an informationtheoretic lens, before being used in literature publications. The final option is to fit the pixel datafrom each frame to a 2D Gaussian in which only the amplitude is allowed to vary, and theamplitude is recorded as the fluorescence intensity (I(t) = ∫∫G(xi,yi) = A). This option suffers frombeing slow to compute, which is why the analyses in this work have instead utilized simple pixel
summation. A thorough comparison of these three methods, and in particular the signal-to-noise
ratios of the resulting trajectories, would provide a useful guide as to which method, if any, issuperior.
Once donor and acceptor intensity trajectories have been extracted from the movies, thenext step is to match donor and acceptor molecules with each other so that the FRET trajectory
E(t) can be calculated. This pairing is accomplished with a spatial mapping function M(r) which

55
maps locations in the donor channel rD to the corresponding location in the acceptor channel rA.
For a given donor, the mapping is used to identify the expected location of its acceptor, and theacceptor molecule which is closest to this expected location is chosen as its most likely pair. The
discrepancy between the acceptor’s predicted and true location, |M(rD) – rA|, is typically lessthan 2 pixels, and larger values alert the user to a poor match. Initially, M(r) was determined bycalibration: 2 known donor-acceptor pairs (“fiduciaries”) identified manually are used todetermine the parameters of an affine transformation which consists of a translation, rotation,and magnification. However, this requires recalibration each time the optics of the detection tree
are aligned, and the rotation and magnification terms were routinely negligible. This inspired thelater implementation of a translation-only automatic mapping based on image cross correlation:
the donor and acceptor portions of the full CCD image are cross correlated, and the vector thatmaximizes the value of the cross-correlation is the vector which best translates the donor image
onto the acceptor image. The mapping is calculated this way for each movie and requires nocalibration but cannot be used if there are rotations or magnifications. In the case of small but
nonnegligible rotations or magnifications, a local image registration method could be employed
instead.7
Occasionally, the translation stage will drift during data collection, which invalidates themolecule locations identified by thresholding. This drift can be corrected for by translating eachframe to align with the first frame of the movie. The translation vector is computed using image
cross correlation similarly to calculation of the donor-acceptor translation vector. In this case, theframes of the movie are grouped into sections of equal length (usually 50-100 frames per
section), and each section is averaged (averaging into sections is done to decrease the length ofthe calculation). Then the average of each section is cross-correlated with the first section, and56
the vector that maximizes the cross-correlation is the drift vector for that section. Reversing thisvector yields the drift correction vector. This correction is applied immediately upon loading themovie into memory, before any other processing has occurred.
Finally, with the donor and acceptor matched and their fluorescence intensities extracted
from the movie, the FRET efficiency trajectory can be computed. The background-subtracted
fluorescence intensities for the donor and acceptor, ID and IA, respectively, are
𝐼𝐷 = 𝜙𝑒𝑥,𝐷 (1 − 𝐸)𝑄𝐷 𝜂(Eq. 2.14)

𝐼𝐴 = 𝜙𝑒𝑥,𝐷 𝐸𝑄𝐴 𝜂,(Eq. 2.15)

andwhere ϕex,D is the donor excitation rate, E is the FRET efficiency (the probability of energy
transfer from the donor to the acceptor per excitation), QD and QA are the donor and acceptor
fluorescence quantum yields, respectively, and η is the collection efficiency. Solving these
expressions for the FRET efficiency yields
𝐸=

𝐼𝐴
𝑄
𝑄𝐷

𝐼𝐴 + 𝐴 𝐼𝐷.

(Eq. 2.16)

If the donor and acceptor quantum yields are equal, then this equation simplifies to the acceptor
fraction of emission (E = IA / (IA + ID)). Furthermore, the total emission IA + ID is a constant. The
quantum yield ratio for the Cy3-Cy5 pair is estimated as QA/QD = 0.3/0.25 = 1.2, which is closeto achieving constant total signal. The quantum yield ratio can depend on a number of factors,
such as fluorophore interactions with nucleic acid, the presence of quenching cosolutes, andvariation in temperature, and the constancy of IA + ID should be checked by measuring whether
the total signal depends on FRET efficiency. It is worth noting that Eqs. 2.14 and 2.15 assume
equal donor and acceptor collection efficiencies and no detector crosstalk, which is notguaranteed. Clearly, measuring absolute FRET efficiencies is quite challenging; however, for the

57
purpose of determining transition rates between FRET states, the naïve FRET efficiency
calculation (Eq. 2.16) is sufficient.
Analyzing the molecules in a smFRET movie requires examining each molecule by handto evaluate if the molecule should be used for data analysis. Valid reasons to reject molecules
include (i) rapid photobleaching before dynamics can be observed, (ii) low FRET trajectorysignal-to-noise ratio due to molecule positioning on dim outer ring of Gaussian illumination area,(iii) the overlap of two or more molecules which convolutes the FRET trajectory, and (iv) thelack of either donor and acceptor fluorophore on the construct due to incomplete labeling.
Additionally, one may observe molecules which do not exhibit the same dynamical behavior as
most other molecules in the movie, such as a molecule showing no dynamics at all. This may
indicate an incorrectly synthesized construct, or a molecule whose local environment is unusual(e.g., a gap in the BSA layer with an exposed glass surface), though these “exceptional”
molecules should not be discarded lightly and may reflect true heterogeneity in the population.
To speed up the tedious process of looking at every molecule in the smFRET movie,
OpenTIRF has an optional feature to sort molecules by likelihood to be used for analysis using a
support vector machine (SVM).8 SVMs use supervised machine learning to perform binaryclassification of multidimensional input. This binary classification is done by dividing themultidimensional space with an oriented hyperplane (the “decision boundary”), such that the
input is classified as “positive” or “negative” (in this case, “likely to be used for data analysis” or“likely not used,” respectively) depending on which side of the hyperplane it lies, with input
farther from the hyperplane being more confidently classified. This calculation is carried out by asimple dot product between the input vector vi and the vector orthogonal to the hyperplane woalong with an offset b:

58
𝑆𝑐𝑜𝑟𝑒 = ⃗⃗⃗𝑣𝑖 ∙ 𝑤⃗⃗ 0 + 𝑏.

(Eq. 2.17)

Determining the optimal hyperplane is done using a training set, which OpenTIRF accumulates
automatically as it is used. The training set includes, for each analyzed molecule, the donor andacceptor brightness and signal-to-noise ratio, the donor-acceptor intensity correlation constant,the donor and acceptor PSF widths, the donor and acceptor decorrelation time, the donoracceptor spatial mapping error, and of course whether or not the molecule was used for analysis.
This training set is fed into MATLAB, where an SVM library is used to efficiently determine wo
and b. This process should be repeated periodically as the training set grows over time. The SVM
is not perfectly accurate (the problem is not linearly separable), so each molecule must still be
examined by hand, but the user can move more quickly through the molecules at the bottom of
the SVM ranking, which makes data analysis faster.
2.5.2 Determination of folding rate constants from trajectories
Nucleic acid conformational dynamics are fundamentally stochastic, and the timerequired to transition between states is random. If the folding dynamics are memoryless, in otherwords, the nucleic acid’s probability of transitioning between states depends only on its current
state, then the conversion between conformations is a Poisson process and can be describedusing first order kinetics. The transition rate from conformation A to conformation B is governedby the rate constant kA→B, which is the probability per unit time per molecule that A transitions toB, and the time rate of change of the concentration of A is the sum of all processes creating anddepleting A:𝑑[𝐴]𝑑𝑡= ∑𝑖 𝑘𝑖→𝐴 [𝑖] − ∑𝑖 𝑘𝐴→𝑖 [𝐴].

(Eq. 2.18)

Traditional chemical kinetics involves solving these equations, analytically or numerically, forthe concentration of a species as a function of time and fitting to experimental results to obtain

59
the rate constants. Single-molecule kinetics must take a different route, as concentration is an
ensemble phenomenon. Two approaches to determining rate constants from single-molecule data
are used in this work, dwell time analysis and hidden Markov modeling, which will now be
described.
Dwell time analysis involves examining the distribution of time spent in eachconformation to determine folding rate constants (Figure 2.10). First, FRET trajectories are split
into periods of time spent in each state, or “dwells,” by applying a threshold to the trajectory. A
high signal to noise ratio is required to avoid random fluctuations in E(t) crossing the thresholdand being measured as short dwells. Alternative methods for dwell time extraction exist which
are supposedly more robust to FRET noise, such as minimum description length (MDL)
analysis,9-10 but these are beyond the scope of this work. The dwell times from many
molecules are then used to construct the complementary cumulative distribution function, also
called the survival function, S(t), for dwell times in each state, which is the probability that an
observed dwell time in that state is longer than a time t. Given a kinetic model, the dwell time
distribution for a given state or manifold of states can be determined by solving the kinetics for amodified kinetic system in which all rate constants generating the state(s) are set to zero. For a

Figure 2.10 Dwell time analysis is carried out by (A) applying a threshold (red line) to the FRET
trajectory to determine dwell times {τ} in each state. (B) The survival function S(τ) = P(t>τ) is
computed by combining the dwell times from many molecules. For a two-state system, the rate
constant k can be determined by fitting S(τ) to an exponential decay.

60
two-state system (A↔B), this is particularly simple and results in an exponential decay:𝑑[𝐴]𝑑𝑡= −𝑘𝐴→𝐵 [𝐴],[𝐴](𝑡)

𝑆(𝑡) = [𝐴](𝑡=0) = 𝑒 −𝑘𝐴→𝐵 𝑡 .

(Eq. 2.19)(Eq. 2.20)

Fitting the dwell time distribution for a state to an exponential decay therefore yields the rate
constant kA→B. Uncertainty estimation can be done by splitting the dwell times into N subsets(three or more), fitting each subset, and calculating the standard error of the mean𝜎𝑆𝐸𝑀 =

1√𝑁

𝑠𝑡𝑑(𝑘𝑖 ),(Eq. 2.21)

where std(ki) is the standard deviation of the rate constants for the N subsets.
A second approach to rate constant determination, hidden Markov modeling (HMM),
describes the data probabilistically and can therefore be used on FRET trajectories with low
signal to noise ratio for which dwell time thresholding fails. HMM uses the principal of
maximum likelihood: the best model for explaining a set of data is the model for which theprobability of observing the data is greatest. In other words, one selects model parameters θwhich maximize the probability of observing the data {xi} given those model parameters,P({xi}| θ), which is also called the likelihood function L. The most important task in maximum
likelihood estimation is the computation of the likelihood function, which requires a precise
statement of the model being used. In HMM, the model consists of a set of “hidden” states {j}(conformations) that cannot be directly observed but are instead inferred from an observable (theFRET efficiency E), with each state having its own “emission” probability distribution P(E | j).
The system can transition between states in discrete time as governed by a set of transition
probabilities per frame Tj→k, which can be conveniently arranged as a matrix T. Therefore, thelikelihood function L can be computed using the so-called forward algorithm:
𝐿 = 𝟏𝑇 [ ∏𝑖≥2 𝐎(𝐸𝑖 )𝐓 ]𝐎(𝐸1 )𝒑𝑒𝑞 .

(Eq. 2.22)

61This matrix product is read right to left. First, one determines the equilibrium probability vector
peq, which is the unique eigenvector of T with eigenvalue 1. One then multiplies this vector by
the observation matrix O, which is a diagonal matrix whose diagonal elements are theprobabilities of observing the ith FRET value Ei for each state (O = diag(P(Ei | j)). A simple
model for the observation probabilities P(Ei | j) is each state having a Gaussian FRET
distribution. Next, multiplication by T advances time to the next frame, and the processcontinues until all frames have been used. Finally, one multiplies by 1T, a row vector of all ones,
which serves to sum the elements of the remaining column vector.
As a sample computation, consider a FRET trajectory which begins E = {0.5, 0.65, …}
⅓
0.5 0.25
for a two-state system with 𝐓 = [] (for which 𝒑𝑒𝑞 = [ ]) and Gaussian FRET
⅔
0.5 0.75
distributions with centers μ1 = 0.3 and μ2 = 0.6 and standard deviations σ1 = σ2 = 0.15. The
probability density of observing the first FRET value (E1 = 0.5) in state 1 is P(E1 |1) =1
𝜎1 √2𝜋1 𝐸 −𝜇1 2
)
𝜎1

− ( 1

𝑒 2

= 1.09, and likewise is 2.13 for state 2. Therefore, the observation matrix 𝐎(𝐸1 )

1.09
0
0.36
is [
]. Therefore, the likelihood product after the first frame is 𝐎(𝐸1 ) ∙ 𝒑𝑒𝑞 ≈ [
].
0
2.13
1.42
Next, time is advanced to the second frame by multiplication by T to give 𝐓 ∙ 𝐎(𝐸1 ) ∙ 𝒑𝑒𝑞 ≈[0.54]. The observation matrix for the second FRET value (E2 = 0.65) is computed like the first
1.24

to give 𝐎(𝐸2 ) = [

0.17
0

0
0.09
] which in turn yields 𝐎(𝐸2 ) ∙ 𝐓 ∙ 𝐎(𝐸1 ) ∙ 𝒑𝑒𝑞 ≈ []. This
2.52
3.14

process of alternating multiplication by T and O(Ei) is continued until the end of the trajectory,
at which point L is calculated by summation of the vector elements. For example, the probability
0.09
density of observing this trajectory after two frames is 𝐿 = [1 1] [] = 0.09 + 3.14 = 3.23.
3.14

62
Note that the forward algorithm calculation almost always results in extremely large orsmall values for L which risk numerical overflow or underflow, respectively. Therefore, thevector should be periodically renormalized, and the logarithm of L should be computed instead.
By describing the first order kinetic system using a rate matrix K, where the off-diagonal
element Kij is the rate constants for the transition from state j to state i (kj→i) and the diagonal
elements Kii are the negative sum of rate constants leaving that state (-∑i(ki→j)), we can calculate
T as
𝐓 = exp(𝑲𝑡𝑓𝑟𝑎𝑚𝑒 ),(Eq. 2.23)

where tframe is the length of each frame in the trajectory, and the exp() operator is the matrix
exponential. It is possible to optimize L using the elements of T, as is done in many HMM
packages for smFRET analysis, but determining K from T requires either computing the matrix
logarithm of T, which can be ill-defined, or using the assumption T ≈ I + Ktframe, which is only
true for small tframe. Instead, I perform the optimization directly on the rate constants in K, which
obviates these issues. The code for this analysis is written in MATLAB; faster implementation ina lower-level language like C would have significant speed improvements.
2.5.3 Dependence of rate constants on temperature and cosolute concentration
A powerful tool for illuminating the mechanism and thermodynamics of nucleic acidfolding is the measurement of folding rate constants under a variety of environmental conditions.
Two such environmental variables employed heavily in this dissertation are temperature and theconcentration of cosolutes such as monovalent cations. By changing the temperature, one canlearn about the relative enthalpy H and entropy S (and occasionally heat capacity Cp) of thereactants, products, and the transition state. The calculation of these quantities is done by van’t
Hoff and Arrhenius analysis (for a thorough description, see Chapter 6). Care must be taken

63
when interpreting measured thermodynamic changes, as H and S depend on the nucleic acid aswell as its environment (e.g., bound waters and diffuse ions in the nucleic acid’s ion
atmosphere).
The concentration-dependence of rate constants can elucidate mechanistic details forprocesses that are otherwise unobservable. For instance, the binding of a ligand to a nucleic acidduring folding can occur via two pathways: (i) the ligand can bind to the nucleic acid which
induces it to fold (“induced fit”), or (ii) the ligand can stabilize a conformation which the nucleic
acid can sample without ligand (“conformational selection”). These mechanisms have distinct
kinetic behaviors which can be observed by changing the ligand concentration (for an example,
see Chapter 7). Such a kinetic analysis requires solving a particular kinetic model; more generalis the model-free measurement of the net stoichiometry of ligand binding using preferential
interaction coefficients, as described in Appendix 4.

2.6Complementing experiments with molecular dynamics simulations
The results in this dissertation are primarily derived from single-molecule experimental

measurements. However, thanks to the development of easy-to-use computational tools by
theorists, it has become increasingly painless** for experimentalists to turn to computation toaddress questions that their experiments cannot answer. Nucleic acids are far too large forquantum mechanical calculations, but they are suitable for molecular dynamics (MD)simulations, in which a classical force field is applied to determine the time-evolution of theatomic positions of the nucleic acid. The principles of MD are described in many excellent**

Even with user-friendly software, the learning curve for MD can be steep. Fortunately, a quick internet search
reveals a number of online tutorials that can expediate the learning process.

64
review articles;11 here, I will describe the process for preparing, running, and analyzing MD
calculations as pertains to this thesis.
One begins with an initial structure of the desired nucleic acid. This is usually a crystal
structure from the online protein database (PDB), but some simple structures such as hairpins orduplexes can be generated by tertiary structure prediction software. Then, the structure must be
prepared for simulation. This includes removing any unnecessary crystalizing agents that may bein the PDB, possibly modifying the identify of cations††, and perhaps mutating residues if
desired. Additionally, the nucleic acid is placed in a box of water molecules,‡‡ and ions are added
to achieve a desired cation concentration.§§ The next key step is parameterize the simulation, i.e.,
to determine the forces between each atom. In this work, I use the Amber force field (OL3 for
RNA and OL15 for DNA) with the TIP3P rigid water model and Cheatham-Young ion
parameters. For other cosolutes, there occasionally exist optimized force fields in the literature orelse the general Amber force field (GAFF) can be applied. Manual editing of the starting
structure is done in VMD, and parameterization and water box creation are done using LEaP in
the AmberTools suite. Finally, the simulation is run in the freely available NAMD program on††

Most force fields treat atoms as point charges, and this approach is well-known to poorly describe polarizable
molecules, notably multivalent ions. Therefore, it is recommended that any multivalent ions that are not essential to
the structure be removed.
‡‡

The choice of water model is an important decision in the design of an MD calculation. To reduce thecomputational cost of MD calculations, water is often simplified to a rigid 3-point model with the hydrogens having
no van der Waals interactions (e.g., TIP3P). There are numerous water models available (e.g., TIP4P, OPC, OPC3,
SPC/Fw), each with their advantages and disadvantages. For instance, currently, the OPC and OPC3 water models
best reproduce the behavior of bulk water, at the cost of highly non-physical water geometries.
§§

There is a common practice of adding only enough ions to neutralize the simulation (e.g., if the nucleic acid has acharge of −30 e, then 30 K+ ions are added). This results in a bulk cation concentration of zero, which far removed
from cytoplasmic concentrations. An ionic strength of zero may be fine for proteins with low net charge, but highly
charged nucleic acids are known to behave quite unusually at low bulk ion concentrations. Therefore, I strongly
recommend simulating at no less than 50 mM ionic strength (i.e., if using KCl, add enough ion pairs so that [Cl -] >50 mM).

65
the JILA compute cluster. Depending on the size of the simulation (typically 30,000-100,000
atoms) and number of compute cores used (8-32), simulations run at 5-25 ns per day.
The final step is to analyze the simulation trajectory. Though the effective computational
speed can be increased by running multiple simulations in parallel, it is clearly infeasible to reachtimescales long enough to observe nucleic acid folding (typically longer than 10 ms), and evenstate-of-the-art calculations rarely exceed one millisecond. Therefore, MD calculations must be
used to observe other processes, such as the interaction of a nucleic acid with a ligand. There are
numerous options available for MD analysis, each tailored to answer different researchquestions. One especially useful calculation within the reach of microsecond-limited simulations
is the measurement of ligand binding free energies. This can be done either by observing ligand
binding in an unbiased simulation (i.e., with no added forces), as in Chapter 4, or by enhanced
sampling methods (e.g., umbrella sampling, collective variable biasing, and replica exchange), orby alchemical transformations (e.g., free energy perturbation). The latter two options require
additional modifications to the system preparation or simulation settings in NAMD. Whatever
the purpose of the MD calculation, establishing an analysis plan before performing runningsimulations is an important step to avoid wasted compute time.

66
2.7

References1.

Nicholson, D. A.; Sengupta, A.; Nesbitt, D. J., Chirality-dependent amino acidmodulation of RNA folding. J. Phys. Chem. B. 2020, 124 (51), 11561-11572.

2.

Nicholson, D. A.; Nesbitt, D. J., Pushing camera-based single-molecule kinetic
measurements to the frame acquisition limit with stroboscopic smFRET. J. Phys. Chem.
B 2021, 125 (23), 6080-6089.

3.

Hodak, J. H.; Fiore, J. L.; Nesbitt, D. J.; Downey, C. D.; Pardi, A., Docking kinetics andequilibrium of a GAAA tetraloop-receptor motif probed by single-molecule FRET. Proc.
Natl. Acad. Sci. U.S.A. 2005, 102 (30), 10505.

4.

Fiore, J. L.; Hodak, J. H.; Piestert, O.; Downey, C. D.; Nesbitt, D. J., Monovalent anddivalent promoted GAAA tetraloop-receptor tertiary interactions from freely diffusing
single-molecule studies. Biophys. J. 2008, 95 (8), 3892-3905.

5.

Axelrod, D.; Burghardt, T. P.; Thompson, N. L., Total internal reflection fluorescence.
Ann. Rev. Biophys. Bioeng. 1984, 13, 247-268.

6.

Holmstrom, E. D.; Dupuis, N. F.; Nesbitt, D. J., Pulsed IR heating studies of singlemolecule DNA duplex dissociation kinetics and thermodynamics. Biophys. J. 2014, 106,220-231.

7.

Brown, L. G., A survey of image registration techniques. ACM Comput. Surv. 1992, 24(4), 325–376.

8.

Cervantes, J.; Garcia-Lamont, F.; Rodríguez-Mazahua, L.; Lopez, A., A comprehensive
survey on support vector machine classification: Applications, challenges and trends.
Neurocomputing 2020, 408, 189-215.

9.

Shuang, B.; Cooper, D.; Taylor, J. N.; Kisley, L.; Chen, J.; Wang, W.; Li, C. B.;
Komatsuzaki, T.; Landes, C. F., Fast step transition and state identification (STaSI) for
discrete single-molecule data analysis. J. Phys. Chem. Lett. 2014, 5 (18), 3157-3161.

10.

Hu, B.; Rakthanmanon, T.; Hao, Y.; Evans, S.; Lonardi, S.; Keogh, E., Using theminimum description length to discover the intrinsic cardinality and dimensionality oftime series. Data Min. Knowl. Discov. 2015, 29 (2), 358-399.

11.

Sponer, J.; Bussi, G.; Krepl, M.; Banas, P.; Bottaro, S.; Cunha, R. A.; Gil-Ley, A.;
Pinamonti, G.; Poblete, S.; Jurecka, P., et al., RNA structural dynamics as captured bymolecular simulations: A comprehensive overview. Chem. Rev. 2018, 118 (8), 41774338.

67
Chapter 3
Amino Acid Stabilization of Nucleic Acid Secondary Structure:
Kinetic Insights from Single Molecule Studies*

3.1

Abstract
Amino-acid and nucleic acid interactions are central in biology and may have played arole in the evolutionary development of protein-based life from an early “RNA Universe.” To
explore the possible role of single amino acids in promoting nucleic acid folding, singlemolecule Förster resonance energy transfer (smFRET) experiments have been implemented witha DNA hairpin construct (7 nucleotide double strand with 40A loop) as a simple model forsecondary structure formation. Exposure to positively charged amino acids (arginine and lysine)
is found to clearly stabilize secondary structure. Kinetically, each amino acid promotes folding
by generating a large increase in the folding rate with little change in the unfolding rate. From
analysis as a function of temperature, arginine and lysine are found to significantly increase theoverall exothermicity of folding while imposing only a small entropic penalty on the folding
process. Detailed investigations into the kinetics and thermodynamics of this amino acid-induced
folding stability reveal arginine and lysine to interact with nucleic acids in a manner reminiscentof monovalent cations. Specifically, these observations are interpreted in the context of an ion
atmosphere surrounding the nucleic acid, in which amino acid salts stabilize folding qualitatively
like small monovalent cations, but also exhibit differences due to the composition of their side
chains.

*This chapter is adapted from: Nicholson, D. A.; Sengupta, A.; Sung, H.-L.; Nesbitt, D. J. J. Phys. Chem. B. 2018,
122 (43), 9869-9876.

68
3.2

Introduction
Interactions between amino acids and nucleic acids are biologically ubiquitous, lying at

the core of numerous protein-nucleic acid interactions, such as histone-driven chromatin
dynamics1 and nucleic acid packaging into viral particles.2-3 Individual amino acids have beenfound to influence RNA functionality; for instance, many of the riboswitches discovered to datehave specific binding sites for amino acids,4 including glycine,5-6 lysine7-8 and arginine.9 RNA
catalysis can also be affected by amino acids, as demonstrated in pioneering work by Yarus andcoworkers, in which the self-splicing capability of the Tetrahymena ribozyme is inhibited by thepresence of L-arginine.10 Indeed, amino acid influence over RNA catalytic properties has
prompted an interesting suggestion for the mechanism by which early RNA-based life11-12transitioned into the modern “protein world” via an intermediate step of amino acids acting as
RNA-folding chaperones.13 In this hypothesis, early RNA-based structures which struggle toreach or remain in their biologically competent conformations may have benefited from thepresence of amino acids which assist in the folding of RNA. This chaperoning could have begun
with abiotically-generated amino acids, with an evolutionary path eventually leading to the
biological synthesis of amino acids and small peptides, and ultimately to the protein world of
today. This mechanism is speculative yet intriguing, and clearly deserves quantitative evaluation.
Stimulated by the biological implications of amino acid-nucleic acid interactions,crystallographic14-16 and computational16-19 studies have been performed, yielding valuable
structural information, but the influence of amino acids on nucleic acid folding kinetics is lesswell-studied. Recently, single-molecule fluorescence microscopy tools have been applied to thestudy the amino acid-based promotion or inhibition of the ubiquitous tetraloop-tetraloop receptor
tertiary interaction.20 Indeed, amino acids (arginine in particular) were found to destabilize the69
tertiary folding equilibrium by diminishing folding rates via competitive binding to the GAAA
tetraloop. Even the simplest amino acid glycine exhibited a significant decrease in unimolecularunfolding rates for the tetraloop receptor. The present work both builds on and extends thesesingle molecule efforts, focusing on secondary structure rather than tertiary structure. Research
into nucleic acid secondary structure is quite mature, complete with apparatus for prediction of
hybridization thermodynamics (e.g. nearest-neighbor models21), representing an area of nucleic
acid behavior that is well-characterized.22 Exploring the influence of amino acids on secondary
structure formation allows us to isolate dynamical effects in a folding process which is better
understood than tertiary folding. To perform these experiments, we employ a DNA hairpin as asimple model for nucleic acid secondary structure, which can be interrogated for its
conformational status by single-molecule fluorescent resonant energy transfer (FRET)
microscopy.23-25 In contrast to the previous studies on tertiary folding, only the charged aminoacids (lysine and arginine, rather than simple zwitterionic species such as glycine) alter thehairpin’s folding dynamics. Furthermore, these charged amino acids act to stabilize the folded
conformation, which is exactly the opposite effect noted for tertiary structures. Implementationof temperature-dependent studies permits folding and transition state free energies to be
deconstructed into enthalpic and entropic components.25 The combination of kinetic andthermodynamic data suggests an ionic screening mechanism in which charged amino acids
alleviate the electrostatic cost of nucleic acid folding. This electrostatic interaction is the
predominant source of amino acid-based secondary structure stabilization, though the sensitivityof the nucleic acid to the molecular detail of the side chain confers additional, amino acidspecific contributions.

70
3.3

Methods
The DNA hairpin construct used in these studies has three components (Figure 3.1): (1) a7 base-pair stem (5’-CTTCAGT-3’), (2) a loop consisting of 40 adenine residues, and (3) an 11
adenine tail at the 3’ terminus, end-labeled with biotin for surface attachment. This specifichairpin has no biological significance; rather, the sequence is designed to avoid misfolding or theformation of higher-order structures, as well as to achieve kinetic rates within the current
dynamic range of the instrument (< 100 s-1). For spectroscopic readout of conformation, acyanine-based FRET fluorophore pair is used, consisting of Cy5 at the 5’ terminus and internal
Cy3 joining the 3’ end of the loop with the stem. The complete DNA oligonucleotide (5’-Cy5CTTCAGT-A40-Cy3-ACTGAAG-A11-Biotin-3’) is purchased from Integrated DNA
Technologies.
This protocol yields a surface density of ~1/m2, which is sufficiently well-resolved for
optical microscopy with a high numerical aperture (NA) objective. Measurements are taken inbuffer (50 mM HEPES, pH 7.6) with an enzymatic oxygen scavenging system (2 mM TROLOX,
100 nM PCD, 5 mM PCA) to improve fluorophore photostability.27 All experiments include a

Figure 3.1 DNA hairpin construct used in folding studies.

71
background of 100 mM monovalent ions (70 mM NaCl and 30 mM KCl for concentrationdependent studies; 100 mM NaCl for temperature-dependent studies). Other solutes are added as
described below, using compounds purchased from Sigma.
Single DNA molecules are illuminated in a scanning confocal microscope,28 aspreviously described.20 Briefly, 532 nm laser light is focused through a 1.2 NA water immersion
microscope objective to a 270 nm diffraction-limited spot. The spot is scanned over the coverslip
surface by a piezoelectric stage to locate DNA molecules. Emission from fluorophores iscollected by the microscope objective, split by dichroic filter into donor and acceptor colors, anddirected onto single-photon-counting avalanche photodiodes. Concentration-dependent studies
are performed at 24 °C, and temperature studies are carried out using a stage heater (HSC60,
Instec) and a microscope objective heating collar (Bioptechs) with 0.1 °C temperature accuracy
and stability.29

3.4

Results
DNA hairpin folding in the absence of amino acids demonstrates that the construct

behaves according to expectations in several ways. First, as demonstrated in Figure 3.2a,fluorescence trajectories depict switching between two states, indicating that the hairpin isconverting between an unfolded low-FRET state and a folded high-FRET state without
significantly sampling additional conformations. Second, the high-FRET state has an averageenergy transfer efficiency of 0.88, corresponding to an interfluorophore distance of ~38 Å, which
agrees well with an estimated distance of 34 Å for a 7 base-pair DNA duplex with B-form
dimensions.30 Third, the fractional residence time in each state can be used to compute a folding
equilibrium constant (Keq = 0.74), from which the free energy of folding (G° = -RT ln(K)) is72

Figure 3.2 (a) Single molecule time traces for hairpin folding with and without 100 mM
ArginineHCl, exhibiting two-state behavior. Green and red lines correspond to donor (Cy3) andacceptor (Cy5) fluorescence, respectively. Residence durations are used to construct survival time
distributions for (b) folding and (c) unfolding. Exponential fits are used to extract rate constants.
Data are obtained in a background of 70 mM Na+ and 30 mM K+.

determined to be 0.75 kJ mol-1. This is comparable to 1.9 kJ mol-1 nearest-neighbor model
prediction21 for this hairpin at 24 °C in 100 mM Na+. Indeed, this ~1 kJ mol-1 discrepancy isquite small and attributable to differing conditions in which nearest-neighbor parameters were
measured.31 Fourth, the residence time distributions in each state are well-fit to a first-orderkinetic model, as seen in Figure 3.2b, which indicates that folding and unfolding are eachgoverned by a single rate-limiting step, in agreement with literature observations of the folding
kinetics of short hairpins.32-33

73
Next, the influence of arginine on hairpin folding is examined. Shown in Figure 3.2a, theaddition of 100 mM arginine hydrochloride (ArginineHCl) results in the DNA hairpin spending
dramatically less time in the unfolded state while remaining in the folded state for an unchanged
amount of time. These qualitative observations can be quantitatively confirmed by analysis of theresidence time distributions (Figure 3.2b), in which the folding rate constant is increased by morethan an order of magnitude from 2.6 ± 0.2 s-1 in plain buffer to 26.7 ± 1.5 s-1 with 100 mM
ArginineHCl. In contrast, the unfolding rate constant is statistically unchanged (2.3 ± 0.1 s-1 to
2.3 ± 0.3 s-1). By selectively increasing the rate of folding, arginine achieves stabilization of the
DNA hairpin.
To explore the mechanism underlying hairpin stabilization, arginine is compared side-byside with glycine and lysine. Glycine isolates the contribution of the zwitterionic backbone,while lysine represents swapping arginine’s guanidinium-based side chain for an ammonium
moiety. As seen in Figure 3.3, glycine has little influence on either the folding or unfolding of
the hairpin, establishing that the zwitterionic backbone is not sufficient for stabilization of thesecondary structure. Lysine hydrochloride (LysineHCl), on the other hand, is able to stabilize thehairpin, again primarily by increasing the folding rate with only a minor slowing of unfolding, ina similar fashion to arginine. However, the magnitude of lysine’s influence is not as large: 100
mM ArginineHCl speeds up folding over nine-fold, compared to the four-fold increase in 100
mM LysineHCl. Thus, the positively-charged side chain in lysine is sufficient to stabilize thehairpin, but the guanidinium group in arginine clearly provides an additional stabilizing
contribution.
The net positive charge of arginine and lysine at the experimental pH suggests that themode of interaction with DNA may be primarily electrostatic. To evaluate this possibility,

74

Figure 3.3 Effects of amino acids on DNA hairpin folding. (a) The folding rate constant is strongly
increased, while (b) the unfolding rate constant is slightly decreased. (c) The equilibrium constantshows little sensitivity to glycine, while positively-charged amino acids stabilize secondary
structure. Sodium chloride (dashed line) is shown for reference.

hairpin folding measurements have been carried out in the presence sodium chloride, a simple
monovalent cation (Figure 3.3, dashed line; note that the concentration represents additional
sodium chloride beyond the 100 mM NaCl present in all samples). Sodium chloride exhibits
hairpin stabilization via the same kinetic signature as the amino acid salts, viz. increasing thefolding rate with a small effect on the unfolding rate. This is in agreement with previous studies
of monovalent cation influence on secondary structure formation.34 However, the net
stabilization () illustrated by the equilibrium constants in Figure 3.3b was different for eachspecies, resulting in an order of stabilization: ArgHCl > NaCl > LysHCl. Though their effects aredifferent in magnitude, all three cations display a similar kind of influence on folding kinetics for75
the DNA hairpin, in general corroborating the heuristic connection between amino acid charge
and hairpin stabilizing power.
Further insight comes from temperature-dependent measurements, which permit thedeconstruction of free energies into entropic and enthalpic contributions. The van’t Hoff plot in
Figure 3.4a displays the influence of temperature on the DNA hairpin’s folding equilibrium
constant with and without 100 mM ArginineHCl, LysineHCl, and NaCl over the range of 24–29°C. Linear fits to these data yield the entropy (S°) and enthalpy (H°) of folding, according to
the van’t Hoff equation,
∆G°

∆H°

1

∆S°

ln(K) = - RT = - R ( T ) + R ,(Eq. 3.1)

where T is the absolute temperature and R is the gas constant. The enthalpies and entropies offolding, derived from these slopes and intercepts respectively, are displayed in Figure 3.4b.
Folding of the hairpin without any additional solute (leftmost column) is driven by significant
enthalpic release (H° = -138 ± 21 kJ mol-1 ) offset by a large entropic penalty (S° = -470 ± 70
J mol-1 K-1), in agreement with previous observations of hairpin formation.35-36 Each of the added
amino acids enhances the enthalpic favorability (average ° = -46 ± 23 kJ mol-1; dashed line
in upper panel of Figure 3.4b), while also increasing the entropic penalty associated with folding(average S° = -130 ± 80 J mol-1 K-1). The net result between these opposing effects isincreased stabilization (G° < 0); i.e. the enthalpic benefit provided by each of the solutes isstronger than the additional entropic cost incurred. However, any differential effect between
these three species is statistically indistinguishable from zero within experimental uncertainty(see Figure 3.4b).

76

Figure 3.4 (a) van’t Hoff plot of the hairpin folding equilibrium’s temperature dependence.
Colored symbols have 100 mM solute. Black squares have sodium chloride added to make thesame ionic strength as colored squares, i.e., 200 mM total [NaCl]. Linear fits are used to extract
H and S contributions, shown in (b). Dashed lines indicate average H and S for added
solutes.

77
Similarly, the temperature dependence of the rate constants for folding may be used to
extract thermodynamic parameters for approaching the folding transition state. To achieve this,the transition state theory expression for the rate constant may be recast into the moreconvenient, activated complex form:ln(𝑘) = ln(𝜐) +

ΔS‡
R

−

ΔH‡
R

1

( T ),(Eq. 3.2)

where 𝜐 is the attempt frequency along the reaction coordinate, and ΔS‡ and ΔH‡ are changes inentropy and enthalpy, respectively, required to reach the transition state. In a plot of ln(k) against
1/T, the slope of a linear fit reveals ΔH‡ unambiguously, whereas the intercept reflects
information on both ΔS‡ as well as 𝜐. Since the attempt frequency 𝜐 is not known a priori, theArrhenius plot does not provide the absolute activation entropy. However, if 𝜐 is assumed to beconstant over the range of conditions in this study, then the solute-induced change in theactivation entropy (ΔΔS‡) can be determined unambiguously. Figure 3.5a reveals activatedcomplex theory Arrhenius plots for the folding rate constant of the DNA hairpin, with the leastsquares fitting results summarized in Figure 3.5b. Similar to what was noted in the above van’t
Hoff analysis, it is difficult to distinguish the more detailed differences between ArginineHCl,
LysineHCl, or NaCl. However, all three salts do act in the same general fashion: the enthalpic
requirement to reach the transition state is significantly diminished (average ΔΔH‡ = -54 ± 15 kJ
mol-1), while the transition state becomes entropically less accessible (average ΔΔS‡ = -160 ± 50
J mol-1 K-1). Notably, the thermodynamic parameters of the folding reaction are similarly
impacted by arginine and lysine (Figure 3.5b), which in turn accounts for their comparable
influences on the folding kinetics.

78

Figure 3.5 (a) Arrhenius plot for hairpin folding rate coefficient as a function of temperature.
Linear fits yield activation enthalpies and entropies in (b). Dotted line indicate averaged enthalpies
and entropies, not including “no amino acid.”

79
3.5

Discussion
Previous work has shown the ability of amino acids, especially L-arginine, to alterribozyme activity.10 The suggestion that the origin of such inhibitory behavior lies in amino acids
modification of nucleic acid folding properties has prompted this series of single-molecule
investigations into the kinetic behavior of nucleic acids in amino acid environments. Prior efforts
examined an isolated tertiary interaction, the tetraloop-tetraloop receptor, revealing the ability ofglycine, lysine, and especially arginine to destabilize the tertiary structure, each in an aminoacid-specific manner.20 The current study aims to bring the same analysis to bear on secondary
structure formation kinetics, modeled by a DNA hairpin.
First, we have explored the ability of arginine, lysine, and glycine to influence secondary
structure. Of these three amino acids, both arginine and lysine are found to stabilize the hairpin
formation, whereas glycine has little or no effect. The insensitivity of hairpin folding to thepresence of glycine suggest that the amino acid zwitterionic backbone is relatively unimportantin the mechanism by which arginine and lysine promote the folded state. Rather, it is the side
chain which determines the effectiveness of the promoting interaction. ArginineHCl and
LysineHCl have qualitatively similar influences on the hairpin’s folding kinetics (Figure 3.3a)and thermodynamics (Figures 3.4 and 3.5), but the magnitude of these effects is different, witharginine offering more pronounced stabilization. That arginine and lysine have a shared kinetic
and thermodynamic signature suggests that they also have a shared mechanism for stabilization,with the overall effect modulated by subtle differences in residue composition. In particular,
arginine and lysine share the structural feature of a charged side chain, suggesting that charge
state is the primary source of stabilization. If charge determines stabilizing power, then the
addition of other monovalent cations should affect hairpin folding in a similar fashion. Indeed,80
sodium chloride titrations produce kinetic and thermodynamic modifications comparable to thoseof the amino acid salts.
The ability of monovalent cations to promote nucleic acid folding is well known34, 37 andcan be explained by considering the role of the ionic atmosphere in folding.38 Nucleic acids arehighly charged polyanions which recruit electrolytes from solution to form an ionic cloudsurrounding the polynucleotide, known as the “ionic atmosphere.” A nucleic acid undergoing a
conformational change will concomitantly experience a reorganization in its ionic atmosphere.3940

Importantly, folding events typically make the nucleic acid more compact, resulting in the net

recruitment of ions from the bulk.34, 41 Consequently, increasing salt concentration serves, bymass action, to shift the equilibrium to favor more compact, folded states. Our kinetic
observations can be explained by examining the ionic atmosphere of the transition state.
According to a “kinetic zipper” picture,32, 42 the transition state for the DNA hairpin ought to be
structurally similar to the folded state, with the loop already formed (and entropic costs alreadypaid) and held together by a few nascent base pairs. Therefore, the transition state and the folded
state will have similar ionic atmospheres, which would explain why the unfolding rate is foundto be relatively insensitive to salt levels. On the other hand, the initial folding process requires
net uptake of ions to reach the transition state, so the folding rate constant is anticipated to bestrongly dependent on salt and/or charged amino acid concentration.
This salt-mediated promotion of secondary structure is partially opposed by an additional
entropic cost arising from the salt-dependence of the polynucleotide flexibility. Repulsive forces
between phosphates along the nucleic acid backbone tend to straighten the polynucleotide,limiting the conformational phase space sampled by the polymer. Increasing salt levels improveionic screening, which diminishes phosphate repulsion down the backbone and decreases the

81polymer’s persistence length.43-45 In turn, the nucleic acid can sample a greater region ofconformational phase space, thereby increasing its entropy. For the DNA hairpin, both the folded
and unfolded states enjoy this entropic increase, however the effect is greater in the unfolded
state which has a longer single-stranded region with unconstrained ends. Therefore, the folding
reaction suffers a net entropic cost due to the increase in bulk salt concentrations. The present
temperature-dependent experiments have confirmed the prediction of a salt-induced entropic costin the folding of the hairpin (S° = -130 ± 80 J mol-1 K-1, 100 mM added salt). Furthermore,
the structural similarity of the folded state and the transition state implies that there should be asimilar cost in the activation entropy for folding, and indeed the Arrhenius analysis reveals anactivation entropy increase of equal magnitude (ΔΔS‡ = -160 ± 50 J mol-1 K-1). It is worth notingthat the salt-dependence of DNA duplex formation has been historically observed to be primarily
entropic in origin46 (i.e. S > 0; H ≈ 0), though recent studies have identified non-negligible
enthalpic contributions.34, 47 That the DNA hairpin in these studies responds to monovalent salt
levels in a thermodynamically distinct fashion (S < 0; H < 0) is a reminder of the ongoingneed for research into the thermodynamic properties of salt-mediated nucleic acid folding (e.g.
bimolecular vs. intramolecular processes, long vs. short hairpin loops, long vs. short stems). In
particular, we anticipate the entropic effect to be different in hairpins with shorter loops, which
are known to be less sensitive to salt overall.41, 48
Though arginine and lysine influence hairpin folding in a way that is qualitatively similar
to a monovalent cation, the additional stabilizing ability of arginine over lysine suggests that themolecular detail of the side chain further modulates these amino acid-nucleic acid interactions.
To isolate the effects of the amino acid side chain without the backbone, we have also explored
DNA hairpin folding in the presence of guanidine hydrochloride (GuHCl), which imitates

82
arginine’s guanidinium group, and ammonium chloride, which imitate lysine’s ammonium group(Figure 3.6). Both salts are observed to stabilize the hairpin, with GuHCl having a stronger effectthan NH4Cl. This is unsurprising given guanidine’s known strong interactions with nucleotides
due to hydrogen bonding.49-50 In addition, GuHCl and NH4Cl were both found to have greater
stabilizing power than their respective amino acid counterparts. Interestingly, the kinetic sourceof GuHCl’s greater stabilization of the hairpin is depression of the unfolding rate constant (3-foldat 100 mM GuHCl) beyond what any of the other salts in this study achieve, so the guanidinium
ion stabilizes the folded state relative to the transition state more effectively than any other cationin these experiments.
These observations lead to a simple picture explaining the relative stabilizing power oflysine, arginine, ammonium, and guanidium based on monovalent cation size41, 51 and hydrogenbonding ability. Lysine interacts with the nucleic acid via a mechanism similar to that of anammonium cation, but with additional molecular bulk which prevents lysine from approaching
the hairpin as closely as ammonium. Lysine therefore cannot contribute to the ionic atmosphereas efficiently as ammonium, resulting in a restricted ability to stabilize secondary structure(ηLysHCl < ηNH4 Cl). Arginine, on the other hand, interacts with the hairpin like guanidinium,
which is dominated by the molecule’s positive charge but made even stronger due to thehydrogen-bonding abilities of the guanidinium moiety (ηGuHCl > ηNH4Cl ; ηArgHCl > ηLysHCl ).
However, arginine suffers from the same steric hindrance as lysine, so arginine cannot stabilize
the secondary structure of the nucleic acid as effectively as guanidinium (ηGuHCl > ηArgHCl ).
Several control tests of this model have been carried out. First, we examined the role ofamino acid chirality in imparting stability to the hairpin. If the dominant mode of interaction

83

Figure 3.6 Comparison of positively-charged amino acids with their side chain moieties, for (a)folding rate constants (b) unfolding rate constants, and (c) equilibrium constants. ArginineHCl andGuanidineHCl have similar effects on folding kinetics, but GuanidineHCl has a stronger influence
on unfolding rates. NH4Cl has a slightly larger influence on both folding and unfolding over
LysineHCl.

84
between amino acids and nucleic acids is electrostatic, then the folding of the hairpin should be
insensitive to the chirality of the amino acid. In addition, the guanidinium group in arginine isachiral, so the hydrogen-bonding ability of arginine should not depend of the chirality of the
distant -carbon. As shown in Figure 3.7, smFRET experiments comparing L- and D-reagents
reveal no difference between the enantiomers of ArginineHCl or of LysineHCl, in agreementwith the model. Second, we tested to see if negatively-charged amino acids can participate insecondary structure stabilization. Since nucleic acids are highly negatively-charged, a negative
amino acid like aspartate would not be expected to interact strongly with the hairpin. However,the negative amino acid must be accompanied by a positive counter-ion, which should participate
in same stabilizing interactions as other cations. To test this, we compared the impact of sodium
aspartate and sodium chloride on the folding kinetics of the DNA hairpin, depicted in Figure 3.8.
The data confirm that the effect of sodium aspartate is indistinguishable from that of sodium
chloride, supporting our model for nucleic acid-induced secondary structure stabilization.

3.6

Summary and conclusion
In this work, single-molecule analysis has been used to examine the ability of aminoacids to influence the formation of secondary structure in nucleic acids. Positively charged aminoacids (arginine and lysine) are found to stabilize secondary structure, while the zwitterionicamino acid glycine has no significant effect. The kinetic origin of the stabilization is primarily alarge increase in the folding rate constant, with only a minor change in the unfolding rate.
Thermodynamic analysis reveals that both arginine and lysine increase the reaction
exothermicity (H° < 0) while also imposing a slight entropic penalty (S° < 0), yielding a
net stabilizing effect (G° < 0). Furthermore, the barrier to folding is modified, characterized

85

Figure 3.7 Hairpin folding is insensitive to amino acid chirality. (a) Concentrationdependence of folding (filled) and unfolding (unfilled) rate constants for enantiomers of
ArginineHCl. (b) van’t Hoff plots for ArginineHCl (red) and LysineHCl (green) enantiomers.
by a transition state which is enthalpically more accessible (ΔΔH‡ < 0) but entropically lessapproachable (ΔΔS‡ < 0). The kinetic and thermodynamic signature of arginine and lysine isshared by other monovalent cations, suggesting that the mechanism of secondary structurestabilization is predominantly electrostatic. A model is proposed to explain these observations,86

Figure 3.8 Negative amino acids behave according to their positively-charged counterion. Sodium
chloride and sodium aspartate have indistinguishable effects on hairpin folding kinetics.
by examining how amino acids contribute to a nucleic acid’s ionic atmosphere. In an
evolutionary context, significant stabilization effects were observed only at relatively high aminoacid concentrations (tens of millimolar), which disfavors secondary structure stabilization as amajor driving force for amino acid incorporation into biology. However, secondary structureformation could prove to be far more selectively sensitive to small di- and tripeptides, which
could serve as an interesting direction for further experiments.

87
3.7

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interactions. Annu. Rev. Biochem. 2014, 83, 813-841.

39.

Trachman, R. J., 3rd; Draper, D. E., Divalent ion competition reveals reorganization of an
RNA ion atmosphere upon folding. Nucleic Acids Res. 2017, 45, 4733-4742.

40.

Gebala, M.; Giambasu, G. M.; Lipfert, J.; Bisaria, N.; Bonilla, S.; Li, G.; York, D. M.;
Herschlag, D., Cation-anion interactions within the nucleic acid ion atmosphere revealed
by ion counting. J. Am. Chem. Soc. 2015, 137, 14705-14715.

90
41.

Vieregg, J.; Cheng, W.; Bustamante, C.; Tinoco, I., Jr., Measurement of the effect ofmonovalent cations on RNA hairpin stability. J. Am. Chem. Soc. 2007, 129, 1496614973.

42.

Kuznetsov, S. V.; Ansari, A., A kinetic zipper model with intrachain interactions applied
to nucleic acid hairpin folding kinetics. Biophys. J. 2012, 102, 101-111.

43.

Murphy, M. C.; Rasnik, I.; Cheng, W.; Lohman, T. M.; Ha, T., Probing single-stranded
DNA conformational flexibility using fluorescence spectroscopy. Biophys. J. 2004, 86,
2530-2537.

44.

Chen, H.; Meisburger, S. P.; Pabit, S. A.; Sutton, J. L.; Webb, W. W.; Pollack, L., Ionic
strength-dependent persistence lengths of single-stranded RNA and DNA. Proc. Natl.
Acad. Sci. U.S.A. 2012, 109, 799-804.

45.

Bosco, A.; Camunas-Soler, J.; Ritort, F., Elastic properties and secondary structureformation of single-stranded DNA at monovalent and divalent salt conditions. Nucleic
Acids Res. 2014, 42, 2064-2074.

46.

Privalov, P. L.; Ptitsyn, O. B.; Birshtein, T. M., Determination of stability of DNA double
helix in an aqueous medium. Biopolymers 1969, 8, 559-571.

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Dupuis, N. F.; Holmstrom, E. D.; Nesbitt, D. J., Single-molecule kinetics reveal cationpromoted DNA duplex formation through ordering of single-stranded helices. Biophys. J.
2013, 105, 756-766.

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Tan, Z. J.; Chen, S. J., Salt dependence of nucleic acid hairpin stability. Biophys. J. 2008,
95, 738-752.

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Biophys. Res. Commun. 1968, 33, 384-390.

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Blanco, F.; Kelly, B.; Sanchez-Sanz, G.; Trujillo, C.; Alkorta, I.; Elguero, J.; Rozas, I.,
Non-covalent interactions: Complexes of guanidinium with DNA and RNA nucleobases.
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Lambert, D.; Leipply, D.; Shiman, R.; Draper, D. E., The influence of monovalent cation
size on the stability of RNA tertiary structures. J. Mol. Biol. 2009, 390, 791-804.

91
Chapter 4
Chirality-Dependent Amino Acid Modulation of RNA Folding*

4.1

Abstract
The preponderance of a specific D-or L- chirality in fats, sugars, amino acids, nucleic

acids, etc. is ubiquitous in nature, yet the biological origin of such chiral dominance (i.e., with
one enantiomer overwhelmingly present) remains an open question. One plausible proposal forthe predominance of L-chirality isn amino acids could be through evolutionary templating ofchiral RNA-folding via chaperone activity. In order to help evaluate this possibility, single
molecule fluorescence experiments have been performed that measure the chiral dependence ofchaperone folding dynamics for the simple tetraloop-tetraloop receptor (TL-TLR) tertiarybinding motif in the presence of a series of chiral amino acids. Specifically, D- vs. L- arginine arefound to accelerate the unfolding of this RNA motif in a chirally selective fashion, withtemperature-dependent studies of the kinetics performed to extract free energy, enthalpy, andentropy landscapes for the underlying thermodynamics. Furthermore, all-atom molecular
dynamics (MD) simulations are pursued to provide additional physical insight into this chiralsensitivity, which reveal enantiomer-specific sampling of nucleic acid surfaces by D- vs. Larginine and support a putative mechanism for chirally-specific denaturation of RNA tertiary
structure by arginine but not other amino acids.

* This chapter is adapted from: Nicholson, D. A.; Sengupta, A.; Nesbitt, D. J. J. Phys. Chem. B. 2020, 124(51), 11561-11572.

92
4.2

Introduction
From the overwhelming abundance of homochiral species in nature, it appears that

biology abhors racemic mixtures. The tendency toward homochirality (or chiral purity) is a longappreciated hallmark of biological systems,1 characterizing the biosynthesis of proteins, nucleic
acids, saccharides, and small molecules.2-3 Consider the case of homochirality in proteins, forwhich all amino acids more complex than glycine have chiral centers, yet biology almost
exclusively favors L-amino acids.4-5 This remarkable evolution toward homochirality in proteins,for example, demands to be accounted for in any complete model of the origin of life, for which
a number of mechanisms have been proposed. One family of theories suggests that homochirality
initially developed in nucleic acids during the RNA world6 and was subsequently transmitted toamino acids. For instance, protein synthesis in a D-ribose RNA world could have an inherent bias
toward L-amino acids, as demonstrated in the chiral discrimination of amino acids during
uncatalyzed aminoacylation of tRNA, a key step in protein synthesis.7-8
An alternative possibility explored herein is that evolutionary selection of amino acidchirality occurred before the development of protein synthesis, based instead on the role ofamino acids in the RNA world, possibly as nucleic acid-folding chaperones.9-10 RNA molecules
are notoriously inefficient at folding into biologically competent form11, with slow folding
kinetics and therefore a tendency to adopt long-lived misfolded states (with lifetimes of order 1–1000 s)10. Modern biology in the current proteomic era addresses such issues through the use ofproteins12 as well as osmolytes13 as chaperones to guide the pathway toward correct RNA
folding. It is thus entirely reasonable to expect that early life would explore similar strategies,
utilizing amino acids as small molecule chaperones. Indeed, experiments have recently
demonstrated the ability of amino acids to influence RNA folding at secondary14 and tertiary15

93
structure levels (and at even higher ribozyme16 levels of complexity) at 10–100 mM
concentrations (1mM = 1mmol/L). Although the corresponding affinities are relatively low, suchconcentrations are well within the limits of osmolyte cellular levels, which can be in excess of 1
M.17-18 If early life relied on amino acids to modulate the thermodynamic stability and/or kinetics
of RNA folding, and if the capacity of amino acids to chaperone RNA folding depends on aminoacid chirality, then the resulting evolutionary pressure could have promoted an amino acidsynthesis mechanism that favors the now ubiquitous L-amino acid.
This is a bold statement, requiring experimental evidence to evaluate as even a putative
evolutionary pathway. Many studies on the chiral dependence of amino acid-nucleic acidinteractions have focused on aminoacylation of tRNA structure es, with a sizable body of work
established.7-8, 19-20 Amino acid chirality effects have also been examined in other contexts, suchas codon binding,21-23 histidine selection,24 chromatographic applications,25-26 and ribozyme
activity.27 Indeed, the study of Yarus and Majerfeld represents an especially intriguing andrelevant case in point, in which they identified an arginine-specific binding pocket in the
Tetrahymena ribozyme that exhibited a tenfold higher selectivity for L-arginine over D-arginine,with clear catalytic implications. These ensemble studies provided substantial first insights into
how nucleic acid structures might sense amino acid chirality. However, few experiments havebeen performed at the single molecule level, which can in principle provide even deeper levels ofphysical understanding into a mechanism for chirally sensitive RNA folding.28
In a previous work, we reported14 on single-molecule experiments investigating theeffects of amino acids on nucleic acid secondary structure folding kinetics, for which nodependence on amino acid chirality was observed. We have also examined15 how the presence ofamino acids can influence the formation of RNA tertiary structure, for which significant

94
chaperone effects were indeed observed but without further exploration into any dependence onamino acid chirality. To test for such chiral chaperone effects, an additional stratum of work isnecessary, specifically a single molecule study into chiral amino acid influence on chiral RNA
tertiary folding, which represents the focus of the present paper.
Here, we utilize single molecule methods in a search for amino acid chiral dependency on
RNA tertiary folding dynamics. Specifically, we explore the ubiquitous 11-nt GAAA tetralooptetraloop receptor (TL-TLR) motif29 as a model tertiary folding system, exploiting single
molecule fluorescence resonance energy transfer (smFRET) to probe conformational changes in
the RNA construct. Rate constants for TL-TLR folding (kfold) and unfolding (kunfold) aresystematically investigated for a series of enantiomeric pairs over a wide range of amino acidconcentrations. Interestingly, we observe a strong chiral dependence for D-, L-arginine inhibition
of RNA TL-TLR tertiary folding and yet no evidence of a similar chirally sensitive signature forany other chiral species selected from 5 representative classes of soluble amino acids (lysine,
histidine, alanine, serine, and proline). In addition, single molecule folding studies of theseamino acid-RNA constructs are explored under temperature-controlled conditions, in order toextract both chirally sensitive and chirally insensitive enthalpic and entropic contributions to thefolding free energy landscape. Finally, these single molecule experiments are complemented by
all-atom molecular dynamics (MD) simulations, in efforts to provide additional insight andmicroscopic perspective into the chiral dependence of the amino acid assisted folding andunfolding kinetics.

4.3

Methods

4.3.1 RNA FRET construct

95
To examine the influence of amino acid chirality in modulating RNA tertiary folding, we
employ an RNA construct bearing an isolated tetraloop-tetraloop receptor (TL-TLR) folding
motif (figure 1), which has been successfully used in previous studies to examine various kinetic
and thermodynamic aspects of TL-TLR folding at the single molecule level.15, 29-30 The constructconsists of three strands: (1) a biotinylated DNA oligonucleotide for surface attachment (5’biotin-CGACTCGTCTCGAG-3’), (2) an RNA sequence labeled with Cyanine 5 (Cy5) at the 5’
position (5’-Cy5-GCCGAUAUGGACGACACGCCCUCGAGACGAGUCG-3’), and (3) an
RNA strand containing a GAAA tetraloop, an internal hexameric polyethylene glycol (PEG)
linkage, and a Cyanine 3 (Cy3) fluorophore at the 5’ terminus (5’-Cy3-GGCGAAGCC-PEG6CGUGUCGUCCUAAGUCGGC-3’). All sequences are commercially synthesized and HPLC-

Figure 4.1 Structure of the 3-stranded TL-TLR smFRET construct.

96
purified (Integrated DNA Technologies), where companies/sources identified herein do notindicate product support. Strands (1) and (2) hybridize to form a stable fourteen base-pair RNADNA duplex, while strands (2) and (3) bind to generate the tetraloop receptor (TLR) domain
flanked by double helices to help maintain structural stability (see figure 1). The flexible PEG
linker in strand (3) largely avoids potential persistence length issues due to base-stacking in an
ssRNA linker, which could in turn hinder diffusion of the GAAA tetraloop.31 The strands areannealed by heating a 10 μmol/L mixture to 85 °C, slowly cooling to room temperature at 1 °C
per minute, prior to long term storage at -20 °C. The annealed stock is used without purification,
since constructs lacking the biotinylated strand do not attach to the surface, and constructs
lacking the Cy3-labeled strand are not excited by the laser (σCy5/σCy3 = 3% at 532 nm).
4.3.2 Sample preparation
Samples are prepared in a home configured glass flow cell, for which a #1.5 glass
coverslip (CGI Life Sciences) is soaked overnight in acetone and cleaned in a UV-ozone oven
(Jelight Mo. 42) for 30 minutes. The clean coverslip is attached to a glass slide (VWR) by twoparallel strips of double-sided tape, forming a channel approximately 2 mm wide, 22 mm long,and 0.1 mm deep to act as a flow cell for rapid solution exchange. The coverslip surface islabeled with the RNA FRET construct using biotin-streptavidin interactions, as previously
described.15, 32-33 Three solutions, all buffered with 50 mM HEPES at pH 7.6, are sequentially
incubated in the sample chamber for 10 minutes in the following order: i) 10 mg/mL bovine
serum albumin (BSA) with 1 mg/mL biotinylated-BSA to block and biotinylate the glass surface,ii) 0.2 mg/mL streptavidin to bind the biotin with streptavidin tetramers, and iii) 1 nM
biotinylated TL-TLR construct to bind to the streptavidin, resulting in a surface RNA coverageof ~0.1/μm2 . For each experiment, the sample chamber is washed with buffer prior to filling97
with an imaging solution consisting of a buffer (50 mM HEPES, pH 7.6), an oxygen scavenging
cocktail34 (100 nM protocatechuate 3,4-dioxygenase, 5 mM protocatechuic acid in equimolar
sodium hydroxide, and 2 mM Trolox), a metal chelating agent (0.1 mM EDTA), monovalent salt
(50 mM KCl; total [M+] = 80 mM), and the desired D/L-amino acid (see results: note that
arginine and lysine are added as chloride salts to maintain near neutral pH). HEPES has beenpurchased from MP Biomedicals, EDTA and L-histidine from Fluka, D-histidine and L-serinefrom Alfa Aesar, D-alanine and D-proline from Fischer, and with all other compounds obtained
from Sigma-Aldrich.
4.3.3 Single-molecule instrumentation
Single-molecule measurements are carried out by total internal reflection fluorescence(TIRF) microscopy (schematically outlined in figure 2), with a similar design to other throughobjective TIRF microscopes.35 Light from a 532 nm diode-pumped solid-state laser (MeshTel
GSF32-300PS) is directed into an inverted microscope (Zeiss Axiovert 135) where it is focused
onto the back focal plane of a 1.4 NA, oil-immersion microscope objective (Olympus
PLAPON60XOSC2; Olympus Type-F immersion oil). To achieve total internal reflection,
moving a mirror translates the beam laterally away from the optical axis, thereby increasing
angle of incidence at the sample until the critical angle is reached. Fluorescence from the sample
is collected by the same objective before passing through a 550 nm LP dichroic mirror (Chroma)
to filter out reflected and scattered excitation light, with the transmitted photons collected on acharge-coupled device (CCD) video camera. First, Cy3 and Cy5 fluorescence is separated into
two channels by 645 nm LP dichroic mirror (Chroma DRLP), which are recombined by a second
dichroic mirror to laterally offset the two images and project them side-by-side onto the CCD.
Second, the images are magnified by a 4x telescope, with Cy3 and Cy5 emission focused

98

Figure 4.2 (a) TIRF instrument schematic. 532 nm light is focused by lens L1 onto the back focal
plane of the microscope objective. Emission collected by the objective passes through dichroic
mirror DM1 to remove excitation light and is spatially filtered by a 1.5 mm slit placed at the focusof the tube lens L2. Cy3 and Cy5 emission are separated and recombined as offset images bydichroic mirrors DM2 and DM3. In addition, the images are magnified by 4x telescope (L3:L4/L4’) before detection by CCD camera.
independently using separate lenses (L4 and L4’) to reduce chromatic aberration. Finally, theemission is transmitted through a ~1.5 mm slit placed at the focus of the microscope’s tube lens,truncating the circular image to maximize use of the CCD’s square imaging area. Light isdetected and recorded as greyscale movies by an intensified CCD camera (Princeton Instruments
I-PentaMAX 512-EFT) with on-chip 3x3 pixel binning to increase the acquisition framerate to
40 Hz.

99
4.3.4 Data analysis
Single-molecule movies are analyzed using custom analysis software written in
LabWindows/CVI (National Instruments). The first ~100 frames of each movie are averaged tosmooth over conformational fluctuations, and local thresholding is applied to the averaged image
to locate particles, whose (x,y) centroid positions are then refined by 2D Gaussian fitting.
Summation over a 4 pixel diameter circle yields single particle fluorescence traces as a functionof time, which are converted to actual photon count rates with calibration factors obtained fromthe variance-mean ratio method.36 Single molecule trajectories are sorted into donors andacceptors by lateral position on the CCD camera, with donor-acceptor pairs identified bycalibrated affine mapping.37 Subsequent FRET data analysis follows protocols we have used inprevious studies:38 donor and acceptor fluorescence traces are combined into single-molecule
FRET (smFRET) trajectories, from which state-to-state transition times are extracted bythresholding. Cumulative dwell time distributions are then subjected to single exponential leastsquares fit analysis, thereby yielding rate constants for both folding and unfolding of the single
molecule constructs.
4.3.5 Temperature control
Temperature control is established using dual thermoelectric coolers (TEC), depicted in
Figure 4.3, in which two TECs (upper and lower) are used as a distributed source of cooling tominimize thermal gradients across the sample. TEC modules (TE Technology HP-127-1.0-0.8)
are compression-mounted between aluminum plates and coupled to the sample via thermal pads
(EN-Labs). The upper TEC assembly lies directly on top of the sample slide, while the lower one
embraces the microscope objective, which is in thermal contact with the sample through theimmersion oil. Precise temperature stabilization (±0.05 °C,  <1 min response time, Figure 4.3c)

100

Figure 4.3 Temperature control apparatus. (a) To minimize thermal gradients, the sample iscontacted by two TEC assemblies, one directly on the glass slide, and the other mediated by the
microscope objective. (b) Schematic of TEC control system. (c) Sample response time andtemperature stability (inset) of TEC system.

is achieved via digital servo loop control. Specifically, sample and objective temperatures aremeasured by thermistors (Vishay) which feed into bidirectional temperature controllers (TETechnology TC-720) that drive the upper and lower TEC modules, respectively. The upper
operational temperature of the system (~40 °C) is constrained by softening of optical cement in

101
the microscope objective at ~45 °C, while the lowest temperature is limited by air coolingefficiency to the heat sink. The coldest temperature data reported herein (~10 °C) required
supplemental cooling of the TEC heat sink by contact with an enclosed frozen liquid reservoir(e.g., ice).
4.3.6 Molecular dynamics simulations
To complement these smFRET experimental results, we have in parallel pursued all-atom
molecular dynamics (MD) simulations with explicit solvent. Initial structures for the nucleic acidconstructs are used directly from the protein database (PDB), with the exception of the docked
TL-TLR (PDB 1GID), which is truncated to residues belonging to the tetraloop (TL) andtetraloop receptor (TLR) residue indices (148–155, 220–229, and 245–253) to reduce systemsize. In the AmberTools19 software suite,39 the three structures of interest (TL, TLR, TL-TLR)
are each solvated in a 15 Å padded TIP3P water box,40 for which the total charge is neutralized
by K+ ions and with an additional 150 mM KCl to mimic the experimental ionic strength. Aminoacids are added as chloride salts using the Visual Molecular Dynamics (VMD)41 software
package, with initial positions selected at random. Dynamics are based on Amber force fields:
χOL3 for RNA,42 Cheatham and Young’s parameters for the ions,43 the TIP3P model for water,and Horn’s zwitterionic amino acid parameterization for D- and L- amino acids.44 Simulations
have been performed on the Nanoscale Molecular Dynamics (NAMD)45 platform using periodic
boundary conditions and a 2 fs timestep, made possible by the constraining NH, CH, and OH
bonds as rigid. Constant temperature and pressure conditions are maintained by Langevindynamics at 300 K (1 ps-1 damping rate), with a Berendsen barostat at 1 atm (100 fs relaxation
time, 4.57 x 10-5 bar-1 compressibility) and simulation snapshots stored every 10 ps. All other102
parameters are initialized at NAMD default values, with VMD software used for simulation
analysis and visualization.46

4.4

Experimental results

4.4.1 RNA tertiary folding demonstrates a strong chiral sensitivity to arginine
Single molecule experiments have been performed to evaluate the ability of amino acids
to promote or inhibit RNA tertiary folding in a chirally sensitive fashion. We select an RNA
construct bearing a tetraloop-tetraloop receptor (TL-TLR) motif and a FRET fluorophore pair
(Figure 4.1) as a representative tertiary folding system.29-30 Surface-tethered TL-TLR molecules
are observed in a TIRF microscope to monitor and extract the single molecule folding dynamics,
with sample smFRET trajectories shown in Figure 4.4a. The trajectories demonstrate clear twostate behavior, with two distinct FRET efficiencies: Elow ≈ 0.4 and Ehigh ≈ 0.7 representing the
TL-TLR unfolded and folded states, respectively.30 Under control conditions (50 mM HEPES
buffer, 80 mM M+, pH 7.6), the TL-TLR construct favors the folded state (Keq = 2.9 ± 0.2, where
the uncertainty represents the standard deviation of the mean (m)). Addition of 300 mM Larginine (as L-arginine chloride) strongly destabilizes the folding equilibrium by nearly twofold(Keq = 1.51 ± 0.08), in agreement with previous observations.15 More relevantly to the current
study, 300 mM D-arginine exerts a 50% greater destabilization effect than L-arginine (Keq = 0.99
± 0.04). Clearly, folding in the TL-TLR tertiary binding motif is capable of kinetically sensing
chirality in amino acids, as well as vice versa.
The robustness of such chiral “anti-chaperone” behavior is next probed by examining theunimolecular folding and unfolding rate constants (kfold, kunfold) over a wide range of arginine
concentrations. TL-TLR folding and unfolding rate constants are determined by dwell time

103

Figure 4.4 (a) Sample single-molecule TIRF trajectories for TL-TLR folding with 300 mM Larginine (red, top row) and 300 mM D-arginine (blue, bottom row). FRET histograms with double
Gaussian fits are shown to the right. (b) Survival functions for TL-TLR dwell times. Lines are
single exponential fits, from which rate constants can be extracted. Data are obtained in a
background of 50 mM K+ and 30 mM Na+.
analysis (Figure 4.4b)38 and plotted as a function of [arginine] (Figure 4.5). In general, thefolding rate constants follow qualitatively similar trends for both arginine chiral enantiomers.
Specifically, both the D- and L-folding rate constants (kfold) decrease modestly (-30%) and thenincrease (+30%) with respect to [arginine]. By way of contrast, the D- and L- unfolding rate
constants (kunfold) monotonically increase by nearly 200-300%, which thereby corresponds to a
net destabilization of the TL-TLR binding motif with increasing [arginine]. Of interest here is adirect comparison between the two chiral enantiomers, for which the folding rate constant reveals
remarkably little sensitivity to the substrate chirality. Instead, the chiral differences between

104

Figure 4.5 Rate constants for TL-TLR folding (a) and unfolding (b) in the presence of L- and Darginine. Experiments are carried out in buffer (pH 7.6) with 80 mM background [M+].

enantiomers manifest primarily in the unfolding rate constant, for which D-arginine issignificantly more effective than L-arginine at enhancing the rate.
4.4.2 Thermodynamic studies of chirally sensitive D-,L-arginine assisted unfolding
To probe the thermodynamics of RNA unfolding by the enantiomers of arginine, we turn now totemperature-dependent folding studies. Changes in temperature modify the free energy of folding
by the Gibbs expression∆G° = ∆H° − 𝑇∆S° = −𝑅𝑇 ln(𝐾𝑒𝑞 ),(Eq. 4.1)

where ΔG° is the folding free energy change, ΔH° is the enthalpy of folding, T is the absolutetemperature, ΔS° is the entropy of folding, R is the gas constant, and Keq is the folding
equilibrium constant. Rearranged into the standard van’t Hoff form,

105
∆H° 1

∆S°

ln(𝐾 𝑒𝑞 ) = − R (T) + R ,(Eq. 4.2)

a plot of ln(Keq) vs. 1/T can be used to recover the enthalpy and entropy of folding via linear
least squares analysis. Similarly, an Arrhenius-type expression can be derived from Kramers’
theory for the temperature dependence of rate constants,47-48
∆H‡ 1

∆S‡

ln(𝑘fold/unfold ) = − R (T) + R + ln(𝜅 ∙ 𝜈),(Eq. 4.3)

where k is the rate constant, ΔH‡ and ΔS‡ are the activation enthalpy and entropy, respectively, ν
is the attempt frequency along the reaction coordinate, and κ is the transmission coefficientaccounting for deviation from transition state (TS) theory49 due to diffusive barrier recrossing.
The prefactor product κ·ν depends on the exact details of the folding energy landscape, but an
estimate of κ·ν ~106 s-1 is plausible from previous studies of RNA and protein folding.48, 50
Furthermore, the ΔS‡ thermodynamic entropy change parameters obtained from a TS analysis
depend only logarithmically on the choice of κ·ν, while the changes in these TS entropies ΔΔS‡
are rigorously independent of the choice of κ·ν. Nevertheless, caution is warranted wheninterpreting the absolute intercept values in these Arrhenius plots, as they represent a
combination of activation entropy and prefactor contributions.
The results from temperature-dependent experiments (11–27 °C) at a fixed concentrationof 100 mM L- and D-arginine are displayed in Figure 4.6. Consistent with concentration studies,
D-arginine is found to decrease TL-TLR stability more than L-arginine (Figure 4.6a), and the

kinetic basis of this chiral-specific destabilization is primarily due to differences in unfolding
rather than folding (Figure 4.6b). Linear regression is applied, and extracted enthalpies andentropies are reported in Table 4.1. Visual inspection of folding rate constant data (Figure 4.6b,left panel; see zoomed insert) reveals small but nevertheless clear differences in slopes between
enantiomers, indicating that the enthalpic cost to reach the transition state during folding is

106Figure 4.6 Temperature dependent studies of folding and unfolding thermodynamics. (a) van’t
Hoff plot and (b) Arrhenius plots for TL-TLR folding with 100 mM L- and D-arginine. (c) Fitted
values for enthalpies (left) and entropies (right) of the folded state (F) and the transition state (TS)relative to the unfolded state (U).
greater for L-arginine (ΔH‡fold 18.9 ± 2.2 kJ/mol for L-arginine and 11.1 ± 2.8 kJ/mol for Darginine). This is intriguing, as no chiral behavior in the folding rate constant was observed in

107
‡

‡

∆H° (kJ/mol)∆Hfold (kJ/mol)∆Hunfold (kJ/mol)

L-arginine-66.2(2.2)

18.9(2.2)85(1)D-arginine-69.7(2.6)

11.1(2.8)83(3)∆∆H° (kJ/mol)∆∆Hfold (kJ/mol)∆∆Hunfold (kJ/mol)

3(3)

8(4)

2(3)

∆S° (J/mol K)∆Sfold (J/mol K)∆Sunfold (J/mol K)

L-arginine-219(8)-45(8)

174(3)

D-arginine-233(9)-71(9)

169(10)∆∆S° (J/mol K)∆∆Sfold (J/mol K)∆∆Sunfold (J/mol K)

14(12)

26(12)

5(10)Difference

Difference

‡

‡

‡

‡

‡

‡

Table 4.1. Thermodynamic values from van’t Hoff and Arrhenius analyses for TL-TLR folding in
100 mmol/L L- vs. D-arginine.

room temperature experiments (Figure 4.5). The explanation for this is that the difference in
activation enthalpy is counteracted by a difference in activation entropy (ΔΔS‡fold = -26 ± 12
J/molK), which results in a negligibly small net free energy difference at room temperature(ΔΔG‡fold = ΔΔH‡fold - TΔΔS‡fold = 0.1 ± 5 kJ/mol). Similarly, the unfolding rate constant (Figure
4.6b, right panel) appears to exhibit indistinguishable slopes but a clear vertical offset,
corresponding to more purely entropic rather than enthalpic contributions to free energies
differences between the two enantiomers. However, the data are taken only over a relatively
small (< 10%) absolute temperature range, with extrapolation resulting in significant correlation
between enthalpic (slope) and entropic (intercept) parameters. After careful inclusion ofcorrelated error propagation, the fitted unfolding activation entropies prove to be statisticallyindistinguishable (174 ± 3 J/mol K for L-arginine and 169 ± 10 J/mol K for D-arginine).
Similarly, the van’t Hoff thermodynamic parameters (ΔH0, ΔS0) are also found to be identical

108
within experimental error for both 100 mM L- and D-arginine. In order to eliminate thepossibility that such agreement is accidental for the chosen arginine concentration, temperaturedependent experiments were also performed at 200 mM arginine, but again exhibit no chiralsensitivity (outside of 1σ uncertainties) in the least squares fits (Figure 4.7). In summary,temperature-dependent measurements reveal a clear difference in folding activation enthalpies in
the presence of L- vs. D-arginine, but the thermodynamic basis of chiral differences in unfolding
activation remains smaller than our experimental resolution.

Figure 4.7 Temperature-dependent experiments for 200 mM L- and D-arginine. (a) Arrhenius
plots for unfolding (left) and folding (right) rate constants. (b) Enthalpies (ΔH) and entropies(ΔS) parameters from linear fits. Both enantiomers of arginine yield the same fit results, withinerror.

109
4.4.3 In search of other amino acid chiral sensitivities in RNA folding/unfolding
To provide further insight into the structural origin of such chiral sensitivity, we haveextended our kinetic analysis to additional amino acids (Figure 4.8). By determining which otheramino acids, if any, interact with the TL-TLR construct with chiral specificity, we might glean
key molecular characteristics that induce enantiomer-dependent amino acid chaperoning of RNA
tertiary folding, namely whether amino acid side chains are hydrophilic or hydrophobic, charged
or uncharged, saturated or unsaturated, etc. Previous work determined that the TL-TLR foldnecessitated amino acid concentrations of order 100 mM to observe significant changes in thefolding equilibrium,15 which restricts the present slate of amino acids based on solubility inwater.51 Nevertheless, the set of five soluble amino acids explored herein (lysine, histidine,
alanine, serine, and proline) bear side chains with a diverse range of physicochemical properties
which covers much of “amino acid space.” By exposing the RNA tertiary fold to thisrepresentative suite of five amino acids and their enantiomers, we hope to uncover general
structural qualities which give rise to the chiral behavior observed above for arginine.
As a first candidate we consider lysine, which, like arginine, also has a long side chain
with a net positive charge, but with an ammonium moiety substituting for the guanidinium
group. Despite these physical similarities, little to no chiral dependence between the folding andunfolding rate constants is observed for lysine (see Figure 4.8, top row). We next examine
histidine, whose side chain contains a conjugated hydrocarbon network similar to arginine, butwhich is predominantly unprotonated (~2% cationic) under our experimental conditions (pH =7.6). Again, despite these qualitative similarities, the results reveal no distinction between L- andD-histidine behavior. We additionally investigated amino acids with other side chain types:

alanine, with a small and hydrophobic side chain; serine, with a polar hydroxymethyl side chain,

110

Figure 4.8 Rate constants for TL-TLR folding (left) and unfolding (right) in the presence of Lamino acids (light symbols) and their D-enantiomers (dark symbols). Concentration ranges foreach amino acid are constrained by solubility.

111
and proline, whose amine group is incorporated into an aliphatic heterocycle. The TL-TLR
response to each of these amino acids proves to be independent of chirality (Figure 4.8, rows 35). Indeed, among all of the amino acids tested, arginine and arginine alone induces chiralspecific modulation of tertiary folding rates in the TL-TLR construct.
Nevertheless, the different classes of amino acids tested do provide some insight into thesalient aspects of arginine’s effect on the TL-TLR. Firstly, all amino acids, regardless of side
chain, can down regulate the folding rate. One simple explanation for this is that all amino acids
engage in nonspecific binding to the surface of the nucleic acid, such that the decrease in surface
area upon TL-TLR folding would be thermodynamically disfavored. However, note that not allamino acids have the same effectiveness in this regard; 100 mM histidine, for example, achieves
the same degree of slowdown as 500 mM alanine, a five-fold greater sensitivity. Second, only
the positively charged amino acids can, at higher concentrations, increase the folding rate. Thisbehavior likely derives from simple ionic shielding of the nucleic acid phosphate backbone byelectrolytes, similar to addition of monovalent atomic cations, whereby decreased repulsion
between phosphate groups lowers the barrier to folding and increases the folding rate constant.30,
52-53

Finally, and most importantly, arginine is unique in its ability to speed up the rate ofunfolding. Furthermore, the chiral specific response to arginine is exclusively contained in theunfolding rate constant behavior. Structurally, these data would suggest that arginine’s
guanidinium-based side chain is responsible its chiral specificity. This is quite curious, however,as the guanidinium group itself is achiral and is 6–7 covalent C–C bonds removed fromarginine’s chiral center. This immediately raises many intriguing questions, such as how does
arginine’s chiral carbon affect the binding mode of the arginine molecule, and why does this notoccur in the other tested amino acids, which undoubtedly also bind to RNA?

112
4.5

Molecular dynamics simulations
To provide insight into the mechanistic origin of such a chiral sensitivity to amino acidassisted RNA folding/unfolding, we have additionally performed computational modeling of theinteractions between the TL-TLR single molecule construct and the two chiral enantiomers ofarginine. At the outset, we first considered a “molecular docking” strategy,54 for which a varietyof candidate ligand-macromolecule binding “poses” are heuristically generated and scored to
predict binding affinities to specific binding pockets. However, the use of molecular docking topredict relative binding free energies of enantiomeric pairs has been shown to be insufficiently
reliable.55 Therefore, the molecular docking approach was not selected for these studies. Instead,
we choose to employ all-atom, explicit solvent molecular dynamics (MD)56-57 as our
computational strategy for assessing amino acid binding to the TL-TLR fold, exploiting welltested computational tools such as Nanoscale Molecular Dynamics (NAMD) and Visual
Molecular Dynamics (VMD) for MD simulation as parallel in silico experiments. The key
limitation of such calculations is the simulation timescale, which for the present construct sizes istypically < 10 s, though longer run times can in principle be accessed with enhanced sampling
methods56, 58 or coarse-grained models.59-60 However, as the TL-TLR construct used in thesestudies folds on the 10–100 millisecond timescale, efforts to simulate the full equilibrium
dynamics of TL-TLR tertiary folding in real time would be computationally prohibitive. We
instead take a conceptually simpler approach, performing three separate shorter timescale
simulations with which to evaluate arginine distributions in equilibrium with each of the TLTLR components. Specifically, we run three NPT simulations in parallel for 100 ns, probing Land D-arginine attachment onto (i) the GAAA tetraloop (initial structure: PDB 1ZIF), (ii) the 11nt tetraloop receptor (initial structure: PDB 1TLR), and (iii) the docked tetraloop-tetraloop

113
receptor contained in the P4-P6 domain of the Tetrahymena thermophilia intron (initial structure:
PDB 1GID, truncated to TL-TLR residues; see methods). Each of the three initial TL, TLR, and
TL-TLR structures is placed in a water box with periodic boundary conditions and appropriate
potassium and chloride ion densities to mimic ionic strength values used experimentally, plus
100 mM of either L- or D-arginine (Figure 4.9; see methods for details). In effect, this piecewise

Figure 4.9 All-atom, explicit solvent simulation of arginine condensation onto the TL-TLR
system. Simulations consist of nucleic acid (docked TL-TLR shown; TL in red, TLR in purple)
solvated in TIP3P water with 100 mM L/D-arginine (yellow).

strategy serves to model arginine interaction with the TL-TLR in the vicinity of its folding andunfolding free energy minima, which circumvents the need for ultralong simulation timescales.
The corresponding penalty is being unable to observe the folding transition state, which limits

114
dynamical interpretation. Nevertheless, these MD simulations offer first insights into thethermodynamics of the overall folding process.
As the first step in our analysis, we evaluate the interaction energy between the three
nucleic acid constructs and their surrounding arginine ions. Free energies corresponding toindividual snapshots 200 ps apart for each of the MD trajectories are calculated by the Molecular
Mechanics Poisson-Boltzmann Surface Area (MMPBSA) method.61 In MMPBSA, the binding
free energy ΔGbind is computed by combining the change in three energy contributions upon
ligand binding:∆Gbind = ∆EMM + ∆Gsolv,PB + ∆Gsolv,SA .

(Eq. 4.4)

In this expression, EMM is molecular mechanics force field energy, Gsolv,PB is the solvation freeenergy under a Poisson-Boltzmann treatment, and Gsolv,SA is a semiempirical term accounting forthe free energy of the solute-solvent interface and assumed proportional to surface area. One
important caveat to the use of MMPBSA is that the entropic contribution to the binding freeenergy is incomplete. Specifically, the ligand-receptor conformational entropy change (ΔSconf) is
unaccounted for and cannot be rigorously incorporated. While there are a number of approximate
methods for calculating ΔSconf, in practice such computations fail to improve the accuracy andthus are not included in our estimates of ΔGbind.61 Despite such potential concerns, MMPBSA
has previously been shown to accurately reproduce experimental relative binding free energies ofenantiomer pairs,62-63 motivating its use here.
These MMPBSA computations reveal two important features for arginine binding onto
the single molecule nucleic acid construct. First, as depicted in Figure 4.10, the MMPBSA
energies exhibit an initial, fast relaxation of trelax ~10 ns, with an additional longer timescale
relaxation which goes beyond our simulation time limits. The fast relaxation agrees reasonably

115

0

TL

D-Arg
L-Arg

-20
-40

MMPBSA Energy (kJ/mol)

-60
0

50

100

Time (ns)
0

TLR

-20
-40
-60
0

50

100

Time (ns)0

TL-TLR

-20
-40
-60
0

50 100 150 200

Time (ns)Figure 4.10 MMPBSA energies of arginine binding to RNA computed frame-by-frame for MD
trajectories. Binding is with GAAA tetraloop (TL, top), 11-nt tetraloop receptor (TLR, middle),
and docked tetraloop-tetraloop receptor (TL-TLR, bottom).

well with the predicted diffusion times for arginine to reach to the surface of the nucleic acid(τdiffusion ~ R2/D ~ 13 ns), based on the typical distance between the arginine and nucleic acid (R~ 30 Å) and the diffusion coefficient of arginine in water (D = 80 µm2/s).64 Energetic relaxation

116
appears to converge most quickly for the smallest simulated system, the lone tetraloop (TL). On
the other hand, the TLR and docked TL-TLR constructs do not reach full equilibration after 100
ns, with arginine molecules continuing to explore a variety of binding poses, some morefavorable than others. Even if we provide an additional +100 ns to the simulation, the TL-TLR
construct still does not reach the equilibrium ergodic limit. It is not surprising that the binding
free energies are not yet fully converged in ~ 200 ns, as small molecule docking to specificbinding sites can take multiple microseconds to occur.56-57
Secondly, and of primary importance, the MD simulations reveal that MMPBSA energies
of arginine interactions with nucleic acids to be both D-, L-enantiomer- and folding state-specific.
In particular, the MMPBSA energies computed for L- and D-arginine interactions with i) TL, ii)
TLR, and iii) the docked TL-TLR are plotted as a histogram in Figure 4.11. The MD simulations
indicate only marginal differences in the binding affinities of L- and D-arginine to the TL(ΔΔGbind = ΔGbind,D-Arg − ΔGbind,L-Arg = −1.3 ± 1.9 kJ/mol) and TLR (ΔΔGbind = +1.6 ± 4.1
kJ/mol). The sum in the two chiral free energy differences is therefore ΔΔGbind,unfolded = 0.3 ± 4.5,
which is zero within the simulation uncertainties. However, for the docked TL-TLR simulation,
there is a much larger difference in binding energies for L- and D-arginine, favoring D-arginine
by ΔΔGbind,folded = −5.3 ± 3.4 kJ/mol.
To better appreciate both the statistical significance and uncertainties of these chirally
sensitive results, we have explored identical MD simulations with D/L-lysine in place of D/Larginine (Figure 4.11, right panels). In particular, recall that lysine experimentally exhibits nochirally specific behavior in excess of the uncertainty limit (Figure 4.8), from which we would
expect to see statistically vanishing chiral sensitivity in the molecular dynamics simulations. In
terms of absolute magnitudes, we find that lysine displays ~2-fold smaller binding free energies

117

Figure 4.11 Histogram of MMPBSA-computed binding free energies for L- and D- enantiomersof arginine (left) and lysine (right) with (a) the undocked TL, (b) the undocked TLR, and (c) the
docked TL-TLR. Solid curves are Gaussian fits, and vertical dashed lines are histogram centers.
than arginine, with chirally dependent differences only on the order of 1-2 kJ/mol (ΔΔGbind(TL)= +1.3 kJ/mol, ΔΔGbind(TLR) = +2.1 kJ/mol, ΔΔGbind(TL-TLR) = −1.4 kJ/mol). Indeed, fromthese values we can estimate roughly that the computational measurement uncertainties are on
the same 1-2 kJ/mol order. Most importantly, we can conclude that arginine binding to theunfolded construct components (TL, TLR) is chirally insensitive within our MMPBSA

118
uncertainties, whereas arginine binding to the fully folded TL-TLR shows a distinct preference
for D- vs L-arginine.
We can take these MD simulations one step further in an effort to identify the spatial
region of preferred attachment for chiral D- vs L-arginine. To do this, 3D probability distributionsof the arginine atoms p(x,y,z) are computed by aligning MD trajectories and binning arginine
atom locations using a 0.5 Å grid spacing. In the equilibrium thermodynamic limit, theseprobabilities can be converted into a 3D free energy distribution ΔG(x,y,z) via:65
𝑝(x, y, z)∆G(x, y, z) = −k B T ln ()
𝑝0where p0 is the bulk concentration at the periphery of the simulation cell, as determined by theasymptotic value of the arginine-RNA radial distribution function (Figure 4.12). By way of
example, the resulting two free energy isosurfaces for L- and D-arginine binding to the docked
TL-TLR construct are plotted in Figure 4.13, corresponding to a relatively strong binding slice at

300

10

g(r)

200

5

100

0
-5

0

5

10

15

20

25

30

0
35

Number density (mmol/L)

15r(NA-Arg) / Angstroms

Figure 4.12 Representative radial distribution function g(r) of the distance between nucleic acidand arginine heavy atoms for the TL system. Large enrichment at medium distances is observed,
corresponding to arginine cation recruitment into the ion atmosphere of the nucleic acid. For othersystems, similar local concentrations of 10–25x the bulk concentration are observed. Asymptotic
values of the number density (right axis) are used to compute the effective bulk concentration.

119G = −12 kJ/mol and therefore revealing the most significantly occupied binding pockets (Wenote that many weaker and stronger binding locations exist across the full surface of the nucleic
acid. For a full 3D view, including isosurfaces for other binding pocket depths and nucleic acids,
see Movies S1−S3 at https://pubs.acs.org/doi/10.1021/acs.jpcb.0c07420?goto=supporting-info).
Interestingly, most of these binding pockets are shared relatively equally by both L- and Darginine. However, there is at least one notably large region at the cleft interface between the TL
and TLR constructs to which D-arginine binds with significantly greater propensity than Larginine, as identified with an arrow in Figure 4.13. Most importantly, MD simulations for the
undocked TL and TLR, by way of contrast, show no binding pockets with chiral preference for
L- or D-arginine (as evident in Movies S1 and S2). Furthermore, for parallel MD simulations

with lysine binding, no chirally-specific binding pockets are observed for any of the TL, TLR,

Figure 4.13 Free energy isosurfaces (G ≤ -5 kBT) for arginine binding to the docked TL-TLR
structure. Arrow indicates the presence of an arginine-binding pocket preferential for the D- vs Lenantiomer at the interface of TL and TLR. For animated perspectives and additional free energy
isosurfaces, go to https://pubs.acs.org/doi/10.1021/acs.jpcb.0c07420?goto=supporting-info.

120
and TL-TLR constructs simulated systems (Movies S4−S6), consistent with the complete lackof chiral sensitivity observed experimentally in amino acid induced folding/unfolding. Thus,
although these simulations are run only for 100–200 ns and therefore do not fully sample the true
equilibrium state, these MD trajectories do already reveal chiral enantiomer-specific binding
interactions on the sub-microsecond timescale, qualitatively consistent with our single molecule
experimental results. It is of course possible that even longer MD simulations would provide
additional examples of stereospecific binding of arginine to this TL-TLR tertiary folding motif.

4.6

Discussion
This paper speculates that homochirality may have been transmitted to amino acids fromRNA through chiral differences in amino acid-based modification of RNA folding behavior. To
help assess such a conjecture, we have used single-molecule FRET measurements coupled with
all-atom MD simulations, focusing on the 11-nt GAAA tetraloop-tetraloop receptor as a modelRNA tertiary fold. The TL-TLR folding/unfolding of this tertiary motif is found to clearly
depend on amino acid chirality for arginine (Figure 4.5), with only negligible sensitivities
exhibited for each of the other classes (hydrophobic/hydrophilic, charged/uncharged,
saturated/unsaturated) of the 6 amino acids investigated (Figure 4.8). Outside of simple empiricalobservation, the question remains: how does the chiral nature of D- vs L-arginine influence its
interaction with the TL-TLR tertiary binding motif, and why might structurally similar aminoacids not also exhibit a similarly strong chiral effect? We examine the evidence for relevant
clues.
Firstly, the concentration-dependent data (Figures 4.5 and 4.7) reveal that arginine isunique among the 6 amino acids tested, specifically impacting the rate constant for TL-TLR

121
unfolding and indeed, buried therein lies all chiral dependence. Each of the other amino acids
tested modify the folding rates but have only minimal effect on the unfolding rates. Lysine evenshares arginine’s nonmonotonic influence on the folding rate constant, likely a positive charge
effect, but only in the presence of arginine does the TL-TLR unfolding rate constant change
significantly, specifically increasing from kunfold = 5.4 ± 0.4 s-1 to 8.0 ± 0.4 s-1 with 100 mM Larginine. Furthermore, such clear TL-TLR sensitivity to arginine chirality is manifested
exclusively in the unfolding process, with D-arginine accelerating the unfolding rate constant by
50% more than L-arginine at 100 mM. This suggests a connection between arginine’s special
ability to enhance TL-TLR unfolding and its chiral-specific influence. Molecular dynamics (MD)simulations strongly support this connection, as any differential L- vs. D-arginine binding to the
TL and TLR constructs is only observed in the folded (TL-TLR) state, whereas the unfolded
state would appear to bind both enantiomers of arginine equivalently.
Secondly, results from temperature-dependent experiments reveal small but significant
differences between the folding activation enthalpy of the TL-TLR motif with L- vs. D-arginine,with ΔΔH‡fold (D- vs. L-) = 8 ± 4 kJ/mol in favor of the D-enantiomer. The differential binding ofarginine to the folding, unfolding, and transitions states result in changes in the relative freeenergies of these states, with greater binding leading to greater free energy stabilization.
Therefore, the chiral difference in activation enthalpy could either be due to preferential bindingof D-arginine to the transition state of the TL-TLR, or superior L-arginine binding to the unfolded
state, or a combination of the two. As the MMPBSA calculations suggest that each enantiomer ofarginine binds equivalently to the separated TL and TLR constructs, the more consistent
explanation would seem to be that D-arginine binds to the TL-TLR folding transition state morefavorably than L-arginine, thereby reducing the activation enthalpy for folding . It is worth noting122
that this reduced activation enthalpy is almost exactly compensated by an additional entropy cost
(ΔΔS‡fold = −26 ± 12 J/molK, raising the free energy barrier for D-arginine), presumably due tothe loss of translational and conformational entropy with one or more arginine molecules
binding. This results in vanishingly small differences in activation free energies (ΔΔG‡fold =ΔΔH‡fold − TΔΔS‡fold) at room temperature, and therefore the folding rate constants for L- and Darginine become largely indistinguishable. Less can be inferred about the unfolding process fromtemperature-dependent experiments, as the measured thermodynamic parameters for L- and Darginine are within experimental uncertainty (ΔΔH‡unfold < 3 kJ/mol, ΔΔS‡ unfold < 10 J/molK).
That the thermodynamic differences are modest is not surprising; D-arginine’s 30% faster
unfolding rate constant kunfold at 100 mM would from transition state theory require a differencein activation free energy, ΔΔG‡ = 0.6 kJ/mol, which is relatively small compared to the thermalenergy product kBT = 2.4 kJ/mol.
Our MD simulations have revealed that TL and TLR docking creates an interstitial cleft
to which D-arginine binds more effectively than L-arginine. However, it must be the case thatthese simulations cannot fully explore all binding interactions, as a prediction of greater Darginine stabilization of the folded state from MD would be in direct contradiction with theexperimental results, for which we find that destabilization of the TL-TLR is found to be greaterfor D-arginine than L-arginine. One possible reason for such a discrepancy is the limited
simulation time (100 ns) explored, which prevents the system from surmounting large kinetic
barriers, for instance arginine displacement of a tightly bound potassium ion or large structural
rearrangements in the nucleic acid. Presumably, at longer timescales an additional binding
pocket preferential to L-arginine would emerge, and therefore long simulations and/or enhanced
sampling techniques are necessary to resolve the ultimate cause of greater destabilization of the

123
TL-TLR fold by D-arginine. Nevertheless, it is interesting that even relatively weak transientbinding can display strong chiral preferences, an observation which we might expect to begeneralizable to other RNA-ligand systems. Furthermore, the fact that the chirality-sensitive
binding pocket is localized at the interface of the TL and TLR suggests that RNA tertiary
interfaces in general might hold the conformational key to generating such enantiospecificbehavior.
MD simulations suggest that arginine’s positive charge in combination with its
sidechain’s especially strong interactions with nucleic acids is responsible for its special ability
to modulate TL-TLR folding in a chiral specific fashion. Due to the cationic nature of thearginine residue, it readily participates in the anionic atmosphere of the nucleic acid.66 Indeed,
the arginine-nucleic acid radial distribution function (Figure 4.12) show that arginine
concentration near nucleic acids is enhanced by 10-25 fold over the bulk concentration. One
might therefore expect such large local concentrations to enable arginine to sample weak,chirality-specific binding sites inaccessible to neutral or negatively-charged amino acids. With its
positively charged amide side chain, lysine would also be expected to experience a similar
increase in local ion atmosphere near the polyanionic backbone. However, the present MMPBSA
calculations suggest and are at least consistent with the lysine side chain having a greatly
reduced ability for binding to RNA than the guanidinium cation group in arginine.
The primary motivation of this study is to investigate a potential mechanism by which
nucleic acid chirality was imprinted upon amino acids, and a few remarks on the evolutionary
implications of these results are appropriate. In the hypothesis considered here, the role of aminoacids in early biology was to promote correct RNA folding, perhaps by destabilizing misfolded
states. This is similar to the role of modern chaperone proteins, which unfold misfolded proteins

124
to allow them a second chance to fold correctly. L-amino acids may have acted as superior
chaperones to D-amino acids, leading to increased fitness for cells containing an enantiomeric
excess of L-amino acids, which would drive evolution toward homochiral synthesis of L-aminoacids. In our experiments, only one amino acid, arginine, is found to exhibit chiral-specificchaperone properties, while all other tested amino acids modify RNA folding independent ofchirality. This casts doubt on the chaperone-homochirality hypothesis, as it seems unlikely that asingle amino acid’s interaction with RNA would drive synthesis of all amino acids to L-chirality.
Furthermore, arginine is usually considered to have been a late arrival to the amino acidalphabet, based on its low prebiotic availability.67 However, prebiotic availability may not be an
appropriate criterion for the development of the genetic code, and recently Blanco et al. havehighlighted arginine as a likely candidate for an early amino acid based on RNA-protein
binding.68 If arginine or another biophysically-related amino acid was highly enriched in ancientbiology, then the chiral sensitivity observed in these studies may have been significant enough tosteer evolution toward L-amino acids.

4.7

Conclusion

Single molecule experiments have been used to explore the potential sensitivity of RNA tertiary
folding rate constants and equilibria to amino acid chirality. Of the 4 classes of amino acids
studied, only arginine exhibits any chiral-specific influence on the RNA folding equilibrium for
the TL-TLR. Both enantiomers of arginine increase the unfolding rates and thereby destabilize
the RNA, but the non-natural enantiomer (D-arginine) is found to be more strongly-destabilizing,by up to a factor of 50% difference in equilibrium constant at 300 mM. Kinetically the source of
this effect is exclusively rooted in changes the unfolding rate constant, with essentially no

125measurable difference in the corresponding folding rate constant with D- vs L-arginine chirality.
From temperature dependent studies of the rate constants, these chiral sensitivities can be traced
back to simple thermodynamical variables, specifically with a reduced activation enthalpy forfolding (H‡ = 8 ± 4 kJ/mol) in the presence of D-arginine. In order to obtain additional
preliminary insights into the mechanism for such chiral sensitivity, we have explored molecular
dynamics (MD) simulations for the folding and unfolding events with NAMD and VMD
computer platforms, specifically using 100 ns long trajectories to discover chirally sensitive freeenergy differences in D- vs L- arginine binding to the TL, TLR and TL-TLR constructs.
Interestingly, the results indicate a complete lack of chiral sensitivity to the unfolded TL and
TLR species, but with a clear differential chiral effect on D- vs. L-arginine binding to the fully
folded TL-TLR construct. Furthermore, the probability distributions from the MD simulations
have been used to generate a 3D free energy landscape for binding of D- and L-arginine to thefolded TL-TLR tertiary motif, revealing that RNA-RNA tertiary interfaces may provide a moregeneral source of chirality-sensing binding pockets for enhanced D- vs L-arginine attachment.
These preliminary computational results corroborate many but not all of the chiral experimental
findings, offering first insights in support of a putative mechanism for RNA chirality influencing
and being influenced by associated amino acid chirality.

4.8

Supporting information

Animations of full 3D amino acid binding distributions S1−S6 are available online athttps://pubs.acs.org/doi/10.1021/acs.jpcb.0c07420?goto=supporting-info.

126

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Chapter 5
Pushing Camera-Based Single Molecule Kinetic Measurements to
the Frame Acquisition Limit with Stroboscopic smFRET*

5.1

Abstract
Single-molecule fluorescence resonance energy transfer (smFRET) experiments permit

detailed examination of microscopic dynamics. However, kinetic rate constants determined by
smFRET are susceptible to systematic underestimation when the rate constants are comparable tothe data acquisition rate. We demonstrate how such systematic errors in camera-based TIRF
experiments can be greatly reduced by using stroboscopic illumination/detection, allowing
accurate rate constant determination up to the data sampling rate and yielding an order ofmagnitude increase in dynamic range. Implementation of these stroboscopic smFRET ideas isstraightforward and the stroboscopically obtained data are compatible with multiple trajectory
analysis methods, including dwell time analysis and hidden Markov modeling. Such
stroboscopic methods therefore offer a remarkably simple yet valuable addition to the smFRET
toolkit, requiring only relatively trivial modification to the normal data collection and analysis
procedures.

5.2

Introduction
Single-molecule microscopy is a powerful tool for examining kinetic systems atotherwise inaccessible levels of detail. From the folding of biopolymers,1-2 to single enzyme
catalysis,3-4 to single DNA replication5 and transcription,6 both in vitro and in vivo,7-9 as well as*

This chapter is adapted from: Nicholson, D. A.; Nesbitt, D. J. J. Phys. Chem. B 2021, 125 (23), 6080-6089.

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nonbiological applications,10-11 the ability to probe at the sub-ensemble level clarifies underlying
mechanisms in ways that are impossible in bulk studies. Especially powerful in the field of
biophysics has been the use of single-molecule fluorescence resonance energy transfer(smFRET),12-14 in which spatial motion on biologically relevant length scales (1–10 nm) can beconverted into a colorimetric ratio, the FRET efficiency (EFRET). By monitoring the timedependence of EFRET for surface-tethered molecules, structural rearrangements can be observedin real time. Stochastic state-to-state hopping in EFRET trajectories can then be analyzed to extract
dynamical information, e.g., the number of thermally accessible states and the rate constants forinterconversion between those states.
smFRET experiments perform well when rate constants are slow compared to the dataacquisition rate, but rate constants that are comparable to or exceed the sampling rate pose
additional challenges. In particular, these faster rate constants are susceptible to systematic
underestimation, with bias becoming significant even for as little as k ≥ 10% of the samplingrate.15 This systematic error/bias arises from well-known distortion of the FRET distributions by“camera averaging” or “camera blurring.”16-17 Similar to “motional blur” in conventional still
photography, the multiple FRET states become temporally averaged over (i.e., “blur”) withincreasingly larger exposure times, introducing artifactual density in the FRET histograms andintercalation between the true FRET values. These blurred FRET values present problems for
accurate kinetic analysis of the data and can even incorrectly suggest the existence of additional,
nonphysical states.15
The simplest way to avoid such bias is obviously to increase the acquisition rate, which
for camera-based detection, options include cropping the field of view or performing on-chip
pixel binning. To push hardware limitations even further, Tang et al. have achieved increased

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effective frame rates through clever use of sparse imaging and a galvo-mirror.18 However, eachof these options requires sacrificing some multiplexing capacity of the experiment, which isindeed a primary benefit of widefield methods over non-multiplexed detection such as scanningconfocal microscopy.19-20 Instead, it would be more desirable to mitigate camera averaging
effects while still maintaining the highest possible parallel throughput of data.
More sophisticated smFRET data analysis methods incorporate such camera averaging
effects directly. For instance, much work has been put into modeling the resulting distortions inthese FRET distributions,21 as in probability distribution analysis (PDA).22-24 A more agnostic
approach is taken by Chen et al.25 in which the experimentally-determined (underestimated) rate
constants are least squares compared with those derived from simulated data subject to the same
camera averaging artifact. The parameters of the kinetic model are then iteratively varied until
the simulated results match the experiment. While effective, this simulate-compare-iterate
approach requires accurate modeling of the experimental system and is therefore prone tointroducing additional systematic errors.
In the present work, we propose a relatively simple experimental solution based on
stroboscopic illumination to eliminate camera averaging effects. Stroboscopes have been used
before in single-molecule microscopy, especially in single-particle tracking, where a flashinglight source can reduce diffusional blurring to increase accuracy in position determination.26-28
Our interest in the strobe is not to reduce motional blur, but rather to reduce temporal blur. Ourdevelopment builds on the work of Farooq et al.,29 who used stroboscopic imaging to mimic thefluorescence burst data from confocal diffusing studies and then analyzed the FRET histogramusing PDA. Here, we show that stroboscopic data can be analyzed as trajectories, which has thebenefit of utilizing the full information content of the smFRET data and permits rate constant

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determination up to the data acquisition rate. Under typical smFRET experimental conditions,this translates into roughly an order of magnitude enhancement in data collection bandwidth for
trajectory-based analysis, free from systematic underestimation of rate constants.

5.3

Methodology

5.3.1 Theory of single-molecule kinetics
The kinetic systems studied in single-molecule experiments can be abstractly represented
as a finite set of discrete states {Si} which undergo state-to-state transitions governed by firstorder kinetics. Specifically, the state transitions are considered Poisson processes, with kij as theunimolecular rate constants for state Sj converting to state Si. The rate of change of the
population in state Si is the sum of the total loss rate and the total production rate:𝑑

𝑆 = − ∑𝑗≠𝑖 𝑘𝑗𝑖 𝑆𝑖 + ∑𝑗≠𝑖 𝑘𝑖𝑗 𝑆𝑗 .
𝑑𝑡 𝑖(Eq. 5.1)

One may compactly express the total rate of change of all states in matrix-vector form byforming the state vector S = {S1, S2, …} and the rate matrix K (with off-diagonal elements Kij =kij and on-diagonal elements Kii = -∑j≠i kji) which satisfy the master equation𝑑𝑑𝑡

𝐒 = 𝐊𝐒.

(Eq. 5.2)

From Eq. 5.1 it is clear that the sum of any column of K vanishes, which is equivalent toconservation of molecular number. The formal solution to this system of differential equations is
𝐒(𝑡) = 𝑒 𝐊𝑡 𝐒0 = 𝐓(𝑡)𝐒0(Eq. 5.3)

where S0 is the initial state distribution at t = 0 and T(t) is the time evolution operator. In Eq. 5.3,
the matrix exponential operator is evaluated by Taylor series expansion as eKt = (I + Kt + (Kt)2/2+ (Kt)3/(3!) + …), where Kt simply multiplies each element by t and I is the identity matrix.

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To connect this formalism, which is based on ensemble chemical kinetics, to the
dynamics of single-molecule systems, T(t) is interpreted as the matrix of transition probabilities
at lag time t. Specifically, the matrix element Tji(t-t’) is the probability that the system will be instate j at time t after being in state i a time t’ earlier. Note that this does not specify which paththe system took to reach the final state j. Indeed, Tji(t) includes all possible trajectories thesystem could take to go from i to j (e.g., i→j, i→k→j, i→j→i→j). For stroboscopic experiments
this has important implications, discussed below, in which the experimentalist is “blind” forsome fraction of the observation time window.
5.3.2 Photon-by-photon trajectory simulation
Single-molecule trajectories are simulated using a photon-by-photon approach, similar tothat of Szabo and Gopich.30 This is carried out in two steps: i) simulation of the state of themolecular system as a function of time S(t), followed by ii) simulation of the fluorescently
emitted and detected photons. First, the initial state S0 (e.g., 0 or 1 for i=1, n) for the molecule israndomly selected from an equilibrium probability vector peq, corresponding to the uniqueeigenvector of the rate matrix K with an eigenvalue of zero. Time is then iteratively forwardpropagated by randomly choosing a dwell time for the current state i, which is exponentiallydistributed with a time constant associated with the total loss rate from that state, i.e., -1/Kii. The
next state index is then randomly selected, with a probability weighted by the branching ratio
into the selected target state j, P(i→j) = Kji/∑j≠i Kji, with this exponential time propagation
continued until the desired total simulation time is achieved.
After the state trajectory S(t) is generated, a sequence of individual photon detection
events is produced. Similar to the case for state transitions, photon detection is assumed to be a
Poisson process, which is accurate for fluorophore excitation rate small compared to the136
fluorescence rate Ifluor = 1/τfluor, where τfluor is the fluorescence lifetime. In this limit, single
photons will arrive exponentially distributed in time as determined by the photon detection rate
kphoton. In general, kphoton can be a function of the conformational state of the molecule, forexample, due to a FRET pair with different quantum yields. However, many FRET dye pairs, inparticular the Cy3-Cy5 pair used in the present experiments, have very similar quantum yields,
which motivates treating kphoton as constant, though deviations could be easily incorporated into
the kinetic model. The color of the emitted photon is probabilistically distributed based on theconformation (i.e., FRET state) of the molecule at the time of the excitation event. Specifically,
the photon is labelled as an “acceptor” with probability Ei or a “donor” with probability (1-Ei),
where Ei is the FRET efficiency of the molecule in state i. As with the state trajectories, theprocess of exponential time-jumping and assignment of photon color is continued until the fulldesired simulation window is achieved. Lastly, photons are binned at the frame time Δtframe, togenerate donor and acceptor average intensities as a function of time, D(t) and A(t) respectively.
Note that all noise considered in the modeled data arises exclusively from quantum fluctuations(“shot noise”) in the photon counting process. While other sources such as dark count noise andread noise31 are also present in camera-based smFRET, the experimental conditions we areinterested place the system well within the limit where shot noise dominates over all othersources of noise. This simulation method can be extended to include background donor andacceptor average countrates (BD and BA); however, the effect of adding background photons canbe equivalently achieved by (1) shifting the FRET efficiencies and (2) increasing the relative
noise by decreasing the photon detection rate, and therefore for simplicity we have set thebackground intensities to zero in these studies. MATLAB code for trajectory simulation andanalysis (see below) are available freely upon request.

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5.3.3 Single molecule microscopy
smFRET experiments are performed on a DNA hairpin by total internal reflection
fluorescence (TIRF) microscopy, as described previously.32 Briefly, a 7 bp DNA hairpin33 with a
40-adenine loop is biotinylated at the 3’ end and attached to a glass surface via streptavidinbiotin binding.34 The hairpin is labeled with Cy3 and Cy5 for FRET-based conformational
detection. The DNA construct is imaged in 50 mM HEPES buffer (pH = 7.6) with 70 mM total
monovalent cations (K+ and Na+) and a PCA-PCD-TROLOX oxygen scavenging and triplet
quenching cocktail for enhanced fluorophore photostability.35-36 A diode-driven Nd:YAG laser
illuminates the surface-attached DNA construct in a through-objective TIRF configuration.37
Fluorescence is separated by a dichroic mirror into donor (Cy3) and acceptor (Cy5) channels,
which are each focused onto one half of an intensified charge-coupled device (ICCD) array. To
achieve stroboscopic (gated) illumination, the diode light output is modulated by a variable duty
cycle square wave current profile generated in LabVIEW (NI, Austin, TX), which also triggers
the ICCD to initiate frame acquisition. Movies are analyzed using homebuilt software
programmed in LabWindows/CVI to extract single molecule trajectories. Particles are located bybrightness thresholding and then sorted into donors and acceptors and paired based on relative
location. Finally, integration inside a 2-pixel radius around particle centers then generates donor
and acceptor intensity trajectories, D(t) and A(t).
5.3.4 Single molecule trajectory analysis
5.3.4.1 Dwell time analysis
In the simplest mode of analysis, time dependent FRET trajectories are computed fromthe binned donor (D(t)) and acceptor (A(t)) intensities by FRET(t) = A(t) / (A(t) + D(t)), where
we assume any corrections for differential donor vs. acceptor quantum yields to be negligible.

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Simple two-state thresholding at the arithmetic mean of low and high FRET values is applied to
the FRET trajectories to determine the state function, S(t), with the temporal duration between
threshold crossings yielding a histogram of dwell times N(Δtdwell). This dwell time histogram isthen re-expressed as the “survival probability” P(Δtdwell) for a given conformational state lasting
longer than Δtdwell, with single exponential fits for folded state dwell times (Δtfold) yielding theunfolding rate constant kunfold (and vice versa).
5.3.4.2 Hidden Markov modeling
As a second analysis scheme, Hidden Markov modeling (HMM) is performed bycomputing the likelihood function (L) of observing an experimental FRET trajectory given astring of model parameters, and then finding the string of parameters that maximizes L. For a
FRET trajectory En of temporal length NΔtframe, the scalar probability L is computed as
𝐊∗∆𝑡frame ]𝐿 = 𝟏T ∗ [∏𝑁∗ 𝐎(E1 ) ∗ 𝐩eq .
𝑛=2 𝐎(E𝑛 ) ∗ 𝑒(Eq. 5.4)

This equation is read right to left, where peq is the equilibrium probability vector, O(En) is the
diagonal matrix of observation probabilities for the nth observed FRET value, K is the ratematrix, Δtframe is the time between frames, and 1T is the row vector (1, 1, …) of length equal to
the number of states in the system. Diagonal elements of the observation matrix (Oii) are theprobabilities of observing a FRET value in the state i, which we model as Gaussian variables, i.e.
Oii(E) ∝ exp(-(E-Ei)2/2σi2) with center Ei and width σI. Gradient descent is used to determine theset of parameters (kij, Ei, σi) which maximize L, where L is periodically renormalized to avoid
instabilities due to numerical underflow.38. Note that this probability is maximized by
optimization of the rate matrix elements Kij rather than the transition probabilities Tij, which
avoids inaccuracies and instabilities due to computation of the matrix logarithm K =ln(T)/Δtframe.

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5.3.4.3 Time correlation function analysis
In the third kinetic analysis approach, time correlation functions (TCFs) are computed forthe four possible combinations of donor and acceptor intensities: <D(t)D(t+τ)>, <D(t)A(t+τ))>,<A(t)D(t+τ)>, <A(t)A(t+τ)>. For example, <D(t)A(t+τ))> reflects the donor-acceptor crosscorrelation function, given by〈𝐷(𝑡)𝐴(𝑡 + 𝜏)〉 = ∑𝑖 𝑃𝑖 𝐷𝑖 (∑𝑓 𝑃𝑖→𝑓 (𝜏) 𝐴𝑓 ).

(Eq. 5.5)

In Eq. 5.5, Pi is the equilibrium probability of initial state i, Di the average donor intensity
in state i, 𝑃𝑖→𝑓 (τ) is the transition probability from i to f in lag time τ, Af is the average acceptor
intensity in the final state f, and the summation is over all initial and final states. Additional
contributions to the TCF from fast dynamics (e.g. , fluorophore blinking) are not included, as thetime resolution of these CCD measurements is typically in the 10s of milliseconds domain, farlonger than the timescales of such photophysical behavior in Cy3 and Cy536 (in effect, thesecontributions are statically incorporated into Ai and Di). We may therefore generalize to any
combination of donor and acceptor TCFs by writing Eq. 5.5 in matrix form as〈𝐶 1 (𝑡)𝐶2 (𝑡+𝜏)〉𝐼2= 𝟏T ∗ 𝐄(𝐶2 ) ∗ 𝐓(𝜏) ∗ 𝐄(𝐶1 ) ∗ 𝐩eq(Eq. 5.6)

where the channel Ck is now labelled by either donor D or acceptor A, peq is the equilibrium
probability vector, E(Ck) is a diagonal matrix whose diagonal elements are FRET efficiencies ineach conformational state (i.e., Ei if Ck = A or (1-Ei) for Ck = D), T(τ) is the transition probability
matrix at lag time τ equal to exp(Kτ), and 1T is the row vector (1, 1, …) of length equal to thenumber of states in the system. Here we have normalized the TCFs by the square of the total
intensity (I2 = (Di + Ai)2), which for comparable quantum yields is only weakly dependent on thesystem state, and have exploited the simple definition of the FRET efficiency as Ei = Ai / (Ai +Di). The four TCF’s are evaluated at nonnegative integer multiples of the frame time Δtframe and140
simultaneously fit to equation 6 by minimizing the sum of the squares of the residuals in
MATLAB.
5.3.4.4 Probability distribution analysis
In probability distribution analysis (PDA), the observed FRET histogram is compared toa simulated FRET histogram, which depends on model parameters such as rate constants andstate FRET efficiencies, as well as the exposure time of each frame. The model parameters arethen optimized to achieve the best match between observed and simulated FRET histograms.
To simulate FRET histograms, we use code from Farooq and Hohlbein, who havedescribed the method in detail. In brief, the FRET histogram of a 2-state system is modeled using
6 parameters: forward and reverse rate constants (k12 and k21), the FRET efficiencies of the states(E1 and E2), and an excess FRET width for each state (σ1 and σ2) which accounts for FRET
broadening due to contributions other than shot noise. Then, for each frame in the observed
smFRET data, a simulated FRET value is generated as follows: first, the amount of time spent ineach state is randomly drawn from a Monte Carlo simulation using the rate constants; second, thenumber of acceptor and donor photons from each state are drawn from binomial distributions,where the total number of simulated photons is equal to the number of measured photons in theobserved frame, and the FRET value of each state is drawn from a normal distribution (e.g. the
FRET probability density in state 1 is N(E1, σ12)); finally the FRET efficiency, EFRET = A/(A+D)
is computed. Once all frames have been simulated, a FRET histogram is generated with bin size
ΔEFRET = 0.01. The process is then repeated K times (here, K = 8) and the K simulated FRET
histograms are averaged. A goodness-of-fit statistic is then calculated to compare the simulated
and observed FRET histograms:1𝜒𝑟2 = 𝑛−𝑚 ∑𝑛𝑖=1[Freqobs (𝐸𝑖 )−Freqsim (𝐸𝑖 )]2
Freqobs (𝐸 𝑖 )(Eq. 5.7)

141
where n is the number of nonzero bins, m is the number of model parameters, Freqobs(Ei) and
Freqsim(Ei) are the frequency of FRET values in bin Ei for the observed and simulated data sets,
respectively.
The best choice of model parameters (k12, k21, E1, E2, σ1, σ2) that will be that which
minimizes χr2. A common method of minimization is gradient descent; however, since thesimulated FRET histogram is random, χr2 is itself a random variable. Because of this,determining the gradient using finite-differences is prone to errors, and we found gradientdescent to be challenging. Instead, we opted for a derivative-free grid-based approach, in which
each parameter θi is sequentially optimized one at a time by sampling a set of points centered on
the current best parameter value θi,best (20 points evenly distributed from θi,best − 50% to θi,best +50%). If a parameter value is found that decreases the χr2 statistic below the current lowest χr2,
then that parameter value becomes the new estimate θi,best. The process is then repeated until noparameter changes by more than 0.1% in an iteration, with a minimum of 12 iterations. We
performed minimization on Keq = k12 / k21 and τ = (k12 + k21)-1 rather than k12 and k21, as we foundworking in this transformed parameter space to increase the rate of convergence. This
optimization method is admittedly inefficient but circumvents the issues of deterministic
gradient-descent approach; a promising alternative to improve efficiency is stochastic
approximation gradient descent.

5.4

Simulation results

5.4.1 Fast rate constants are systematically underestimated in smFRET experiments
To establish the effects of bin time on smFRET-derived rate constants, we begin byanalyzing our simulation (“synthetic”) single-molecule fluorescence trajectories. The primary

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benefit of such an approach is that the actual simulation parameters are known precisely,
allowing the computation of absolute and asymmetrical errors. Additionally, use of thesimulations permit rapid testing of these kinetic analysis methods over the full parameter space,including modifying parameters not typically under experimental control, such as rate constants.
There are numerous protocols for modeling single-molecule trajectories;30-31 we opted for a
photon-by-photon approach, in which both state-to-state transitions and photon detection events
are treated as continuous-time processes, with the simulated photon arrivals binned by time/color
into frames (Δtframe) to generate discrete-time fluorescence trajectories of a typical CCD-based
smFRET experiment. In doing so, we make no assumptions as to how photons are distributed pertime bin, nor are requirements imposed that a molecule remain in a given state for the entire bin.
Since camera averaging is a result of mapping from continuous-time to discrete-time processes,
this photon-by-photon approach is ideal for capturing and modeling such artifacts.
We introduce our model for camera averaging by considering the two-state system in
Figure 5.1a, which interconverts between two well-resolved FRET states (Elow = 0.2, Ehigh = 0.8)
with forward rate kF = 20.0 s-1 and reverse rate kR = 10.0 s-1, and achieves a total fluorescencesignal-to-noise ratio (SNR = I/σI, where σI is the standard deviation in the brightness I) of 10:1
(100 photons per frame). Simulated trajectories of 200 s in length are binned with Δtframe from 5
ms to 150 ms and are subjected to conventional dwell time analysis39 to determine the apparent
rate constants plotted in Figure 5.1b. Both kF and kR are accurately determined at short bin times
but are systematically underestimated as these rate constants become comparable to the frame
rate (kframe = 1/Δtframe). As Figure 5.1b clearly highlights, the magnitude of systematic rate
constant errors smoothly increase in transition to the fast rate constant regime (kF,kR ≈ kframe).

143

Figure 5.1 Systematic underestimation of rate constants in a model system. (a) Markov schematicof two-state system governed by rates kF = 20.0 s-1 and kR = 10.0 s-1. Simulated trajectories areanalyzed by dwell time analysis to extract rate constants as a function of camera frame length for(b) continuous illumination and (c) stroboscopic illumination (20% duty cycle). Shaded regions
represent ±1σ uncertainty bands. Simulation conditions: 2000 s, Elow = 0.2, Ehigh = 0.8, 100 photons
per frame (SNR = 10), repeated for each Δtframe until standard error of the mean σSEM = σ/√Nreached 0.5%.

The obvious source of this systematic rate constant underestimation is due to errors in thetransformation between continuous-time dynamics and discrete-time measurements. The true

144

Figure 5.2 Mapping of continuous-time dynamics onto discrete-time results in blurred FRET
states. (a) Sample true (hidden) state trajectory over six camera frames. (b) and (c) Ideal, noiseless
FRET values averaged over time window during which laser is on (green blocks) for (b)continuous illumination and (c) stroboscopic illumination.

state of a system is a continuous function of time (Figure 5.2a), which is converted into asequence of discrete frames by temporal binning (Figure 5.2b). The ideal, noiseless FRET value
equals the time-weighted average of all FRET values visited in Δtframe, with frames consisting ofa single state i exhibiting a constant Ei FRET value. Conversely, frames for which the system has
traversed multiple states in Δtframe present an average FRET value contaminated by each of theindividual FRET values. As a result of this “FRET averaging,” 16-17 accurately assigning whichstate or combination of states the system occupies in each frame becomes difficult. However,
FRET averaging is not a major concern as long as frames with more than one state occur
infrequently. This will be true if the sampling frame rate is much faster than the rates of145interconversion between states (−Kii << kframe). Indeed, the measured rate constants for thesynthetic data agree well with the extracted rate constants for sufficiently fast (kframe > 10 kF, kR)
frame rates (Figure 5.1b). However, when the frame rate is within an order of magnitude of kF or
kR, the probability of a frame containing multiple states results in many frames with incorrectstate assignments. Since the theoretical basis of most analyses of single-molecule trajectories
assumes single-state occupancy per frame, faster rate constants lead to a greater fraction offrames that are unmodeled. In addition to making accurate state assignment difficult, fast rate
constants and/or slow frame rates result in fundamentally “missed dynamics.” Short excursions
to a state lasting less than one frame will only result in a slightly altered FRET efficiency, which
will be misinterpreted as if no transition had occurred at all. The number of threshold-crossing
transitions in a given smFRET trajectory will be lower than expected, resulting again in
systematic underestimation of the rate constants. We note that simply discarding the shortest
dwell times (< 1−2 frames), as is sometimes used to account for spurious short dwell times,39 isinsufficient to correct the underestimation of rate constants (Figure 5.3).

5.4.2 Stroboscopic illumination eliminates rate constant underestimation
To help resolve this issue, we propose an extremely simple solution which uses
stroboscopic illumination to effectively address both FRET- averaging and missed dynamics asfundamental causes of rate constant underestimation. In a stroboscopic smFRET experiment, theexcitation time window is compacted into a small fraction of the full frame. This reduction in
duty cycle can be accomplished in a variety of ways, including modulating the excitation laser
current, gating the CCD detection, or simply using an analog optical chopper wheel. Reducing
the time per frame that the molecule is interrogated increases the probability that the molecule

146
occupies only one state for the duration of the observation pulse (Figure 5.2c). For example, in a
two-state system with kframe ≈ kF, kR, the probability of remaining in a single state for an entire
frame is on average only 1/e = 37%. The same system with a 10% duty cycle strobe will have a

Figure 5.3 Experimental rate constants determined by dwell time distributions after discarding theshortest dwell times. Fitting the full dwell time distribution (red) produces similar dependence on
choice of bin time as when fitting only dwell times lasting longer than 1 (yellow), 2 (green), or 4
(purple) frames. The analyzed data is the same non-stroboscopic data set as in Figure 5.8 in themain text.

147
survival probability exp(−kΔtstrobe) = 90%, which makes state assignment more accurate andlowers the probability of both FRET averaging and missed dynamics.
The experimental “cost” of implementing stroboscopic methods is that one is blind to themolecular state behavior during the increasing window duration when the illumination is off. We
address this with the second component of the proposed technique, which involves analyticalcorrection to the measured rate constants accounting for missed transitions. This correction
incurs negligible computational cost and in fact becomes exact in the limit of vanishing strobe
pulse width. In the interest of space, we briefly motivate and describe how to calculate thecorrection, with a complete derivation to be found in the Appendix. The first correction is
specific to dwell time analysis, in which each state’s survival function is fit to an exponential
distribution to obtain an apparent rate constant kiapp. For fast frame rates, kiapp is approximately
the total rate constant for leaving state i, but more generally kiapp is related to the single-frame
self-transition probability Tii byapp

𝑇𝑖𝑖 = 𝑒 −𝑘𝑖∗∆𝑡frame(Eq. 5.8)

One first computes the diagonal elements Tii of the transition probability matrix T, from which
the full matrix T can be constructed. This is particularly trivial for a two-state system, as each of
the columns of T must sum to 1. For more complex systems, the empirical branching ratios are
required, as addressed in the Appendix. The second step in the correction applies to any methodthat measures the transition probability matrix T (e.g., hidden Markov Modeling), in which therate constant matrix K is determined from T by computing the matrix logarithmln(𝐓)𝐊 = ∆𝑡frame.

(Eq. 5.9)

which yields the desired corrected rate constants as matrix elements Kij. Note that the common
approximation K = (1/Δtframe)(T – I) is only accurate in the slow rate constant regime (– KiiΔtframe

148<<1), with the full matrix logarithm required when measuring rate constants comparable to thedata acquisition rate. Fortunately, there are a number of computationally efficient algorithms
available for performing this matrix logarithm.40-41
Combining stroboscopic illumination with the matrix logarithm correction abrogates thesystematic underestimation of rate constants in our simulated smFRET data, as seen in Figure
5.1c. The forward and reverse rate constants determined by dwell time analysis under a 20%
duty cycle strobe recapitulate the simulated values quite well and are independent of the choiceof bin time. Neither the strobe nor analytical correction alone is sufficient to completely remove
systematic error, with only the combination correctly addressing the underlying complications ofstate-averaging and missed dynamics. Additional random (nonsystematic) errors are also
observed at large frame times (shaded uncertainty bands in Figure 5.1c; note that these
uncertainties are standard deviations σ, whereas the standard error of the mean σSEM = σ/√N canbe decreased to arbitrary precision by repeating the measurement N times and averaging). In thisregime, the average dwell time is shorter than a single frame, and the dwell time distributions
cover only a few frames in time, which increases uncertainty in fitting. Said differently, theinformation content of the trajectory diminishes as frame-to-frame correlations become
sufficiently weak. This loss of correlation generates an upper limit on the rate constants
measurable with this combination of techniques for a given frame time, as will be empiricallyverified in Sec III.D.
5.4.3 Camera averaging artifacts cannot be resolved by modified experimental conditions
One is tempted to think it is possible to reduce or eliminate such underestimation of rate
constants by control of experimental parameters (e.g., laser intensity). However, this is not thecase. To explicitly address this possibility, we have performed data simulations over a wide

149
variety of “experimental” conditions, as summarized in Figure 5.4a-e and which reveal no
impact on the rate constant deviations. By way of example, the results shown in Figure 5.4a
represent simulations under identical conditions as in Figure 5.1, but with a 1:1 rather than 2:1
ratio of forward:reverse rate constants. Notice that the rate constants bias becomes significant(i.e., > 5%, as indicated by the yellow banded region) at kΔtframe ≥ 0.2, in agreement withprevious observations.15 As a second example, we can increase the signal to noise ratio (SNR) by
4x (Figure 5.4b), which improves the accuracy of state identification, but has little effect on
avoiding rate constant underestimation error. This is because binning-induced broadening of
FRET values is fundamentally not related to shot noise on the photon stream but rather on thestate-transition dynamics. Thirdly, we can increase the total duration of the FRET trajectories(e.g., by obtaining data from more molecules), which reduces statistical noise in the measured
rate constants, but once again has little effect on the systematic bias (Figure 5.4c). As a fourth
example, we might hope to modify the 2-state FRET efficiencies to be better resolved (Figure5.4d), which could be experimentally achieved by redesigning the single-molecule construct tooptimize placement of the fluorophores. However, an increase in ΔEFRET from 0.6 to 1.0 leaves
the rate constant underestimation errors remarkably similar. Finally, increasing the equilibrium
constant from Keq = 1 to 4 (in Figure 5.4e by reducing the reverse rate kR by 4x) leads to only
small differences in the errors for kF and kR, with the average magnitude of the bias unchanged.
In summary, systematic underestimation of fast rate constants by camera averaging and missing
dynamics proves to be remarkably insensitive to the choice of parameters potentially underexperimental control.

150

Figure 5.4 Camera averaging artifacts are robust to experimental parameters. (a) Bias as percent
error in rate constants for simulated two-state system with equal forward and backward rate
constants. Simulation conditions: 2000 s, Elow = 0.2, Ehigh = 0.8, kF = kR, 100 photons per frame(SNR = 10), repeated 24 times to determine uncertainties (standard error of the mean). For
reference, the yellow region demarks a ±5% bias. (b–e) Same as (a) but simulation conditions are
modified as follows: (b) 4x greater signal to noise ratio, (c) 10x more simulated time, (d) increased
FRET separation from ΔEFRET = Ehigh – Elow = 0.6 to 1.0, and (e) increased equilibrium constantfrom Keq = 1 to 4 by decreasing kR.

151
5.4.4 Optimal excitation duty cycle and maximum measurable rate constants
We have shown that introducing a low duty cycle (i.e., stroboscopic) light source canhelp eliminate binning-related artifacts in smFRET studies, but this obviously can result in lossof signal, signal to noise ratio, and reduction in performance. To compensate for loss of signal,
the experimentalist may wish to increase laser power, but this also has obvious limits due tophotophysics (e.g., photobleaching, triplet state formation) and nonlinearity in fluorophorebrightness when the time interval between photoexcitation events approaches the fluorescence
lifetime. Therefore, it is a useful exercise to identify the largest duty cycle that still reduces
systematic errors to acceptable levels. To determine this optimal strobe duty cycle, we havesimulated smFRET data over a range of duty cycles, rate constants, and frame acquisition rates(Figures 5.5 and 5.6). For a fixed kΔtframe (Figure 5.5a), decrease in strobe duty cycle
monotonically reduces systematic error. However, under real world experimental conditions,
single-molecule rate constants might have typical statistical uncertainties of several percent orlarger; hence, reduction of these systematic errors to < 3% is usually unwarranted. For rate
constants equal to the frame rate (kΔtframe = 1), we observe in Figure 5.5a that a strobe duty cycleof 10–20% already has reduced systematic error in the extracted rates to < 3%. Obviously for
rate constants lower than the frame rate (kΔtframe < 1), one achieves this 3% error limit target
even more quickly with reduction in duty cycle. Alternatively summarized, below a 10% duty
cycle, one is discarding signal for little gain in extracted rate constant accuracy.
The corresponding upper limit in the measured rate constants is determined by the dataacquisition rate. Trajectory-based analyses of rate constants rely on frame-to-frame correlations,and these methods fail when kinetic relaxation occurs on the timescale of the frame rate. Indeed,this mode of failure is readily observable in the shaded uncertainty regions in the right-hand side

152

Figure 5.5 Determining the optimal strobe duty cycle. (a) Rate constant bias as a function of duty
cycle, for fixed kΔtframe (1.0, red; 0.5, green; 0.1, blue). Duty cycle of 100% corresponds to no
strobe. Shaded bands indicate uncertainties (±1σ). Bias band of ±1.5% highlighted in yellow, andrecommended duty cycle (< 20%) indicated by vertical dashed line. (b) Rate constant bias at fixed
duty cycles. The quality of results decreases for kΔtframe > 1. Simulation conditions: 8000 s, kF =kR, Elow = 0.2, Ehigh = 0.8, 100 photons per frame (SNR = 10). For 2D heatmaps of bias anduncertainty for combinations of duty cycle and framerate, see Figure 5.6.

of Figure 5.1c, for which even the stroboscopic data produce results with high levels of statisticalnoise when the rate constants exceed ≈ 1.5 kframe. Simply stated, this is due to low information
content per frame when the system decorrelation is fast. In this limit, individual data points in the
FRET trajectory therefore become uncorrelated and can provide only non-dynamical information(e.g., equilibrium constants). This same behavior can also be noted in Figure 5.5b, where even153
for low duty cycle data acquisition the quality of results decays quickly at kF,kR > kframe. Insummary, stroboscopic smFRET methods can help rescue rate constants up to the frame rate, butfaster rate constants require alternative approaches, such as stroboscopic probability distribution
analysis discussed below.29

Figure 5.6 (a) Bias and (b) uncertainty (standard deviation) of rate constants determined by
dwell time analysis for a variety of rate constants and stroboscopic duty cycles, where a duty
cycle of 100% is equivalent to no strobe. Of interest are the regions of bias near zero (blue in(a)) and uncertainty near zero (blue in (b)). Uncertainties for all simulations increase rapidly for
kΔt > 1. Simulation conditions: 2000 s, 100 photons per bin (SNR = 10), kF = kR, Elow = 0.2, Ehigh= 0.8, repeated 48 times to determine uncertainty.

154
5.5

Experimental tests
To demonstrate that the above results and predictions based on simulated data areexperimentally valid, we have performed a series of smFRET experiments measuring kinetic rate
constants with and without stroboscopic illumination (Figures 5.7 and 5.8). Specifically, wechoose to examine a very simple single-molecule construct consisting of a DNA hairpin with a 7

Figure 5.7 Experimental evaluation of stroboscopic smFRET. (a) Cartoon depiction of surfacetethered, two-state DNA hairpin investigated in these studies. (b) Sample smFRET trajectory (left)taken at a fast frame rate (30 ms frame-1) compared to underlying dynamics. FRET histogram(right) exhibits two well-resolved peaks. (c) Same as (b), but for a slow frame rate (300 ms frame1). Many frames contain intermediate FRET values due to camera averaging. (d) Same as (c) but
using stroboscopic illumination at 20% duty cycle. Data are obtained at room temperature in 40
mM K+ and 30 mM Na+.

155
base-pair stem and 40-adenine loop which we have studied previously (Figure 5.7a).33 This
construct exhibits well-behaved two-state single-exponential kinetics (with kfold = 1.2 s-1, kunfold =2.2 s-1) and therefore represents a useful model system with which to test these stroboscopic
analysis methods. The data collection is performed on a total internal reflection fluorescence(TIRF)37 microscope apparatus with charge-coupled device (CCD) camera detection, and theincident laser power is increased or decreased to maintain a constant signal-to-noise ratio (SNR)≈ 7 at all acquisition rates. As shown in Figure 5.7b at fast frame rate, smFRET trajectories undercontinuous illumination exhibit switching between two states, with the integrated FRET
histogram (right panel) clearly depicting two distinct FRET peaks. On the other hand, continuous
illumination at this slower frame rate (Figure 5.7c) yields a notably smeared smFRET histogram,with spurious density appearing between the two FRET states. By way of contrast, however, theuse of a 10% duty cycle strobe at this same slow frame rate (Figure 5.7d) removes camera
averaging artifacts from the FRET histogram (far right), therefore restoring distinct two-statebehavior.
Furthermore, the rate constants from the smFRET trajectories were measured by dwell
time analysis over a systematic range of data acquisition bandwidths (40 Hz to 1.3 Hz, Figure5.8). Without stroboscopic illumination, the extracted rate constants exhibit the behavior
characteristic of our simulated results, i.e., with the measured rate constants (filled squares)
systematicallydependent on frame rate and underestimated with increasing Δtframe. Inclusion of a 10% duty
cycle strobe (open diamonds) completely removes this dependence on frame rate, in agreementwith our simulations. Indeed, the kinetic measurements remain faithful even up to kΔtframe = 1.7,
which agrees with our upper limit of kΔtframe ≈ 1.5 predicted above.

156

Figure 5.8 Experimental validation of stroboscopic method by measuring DNA hairpin folding
dynamics over a range of camera frame rates. (a) Folding rate constants. Without stroboscopic
illumination (red), the measured folding rate constant is a function of frame rate. Application of a
10% duty cycle strobe (purple) removes the dependence. (b) Same as (a), but for the unfolding rate
constants. Each data point represents an analysis of 2000-3000 observed dwell times from ~50
molecules. Error bars are standard errors of the mean as determined by bootstrapping.

5.6

Discussion
smFRET has proven itself to be an invaluable technique for measuring the detailed

dynamics of biological processes.13-14 However, this work demonstrates that rate constants

157
determined in such smFRET experiments can be prone to systematic underestimation,
particularly when the conformational state-to-state transition rates are comparable to the dataacquisition rate (kΔtframe ⪆ 0.1). Herein, we have presented an extremely simple “stroboscopic”
method for eliminating such “camera averaging” artifacts by restricting fluorescence collection
to a reduced but contiguous portion of each time bin, for instance, by gating the excitation lightsource. We have demonstrated that the use of stroboscopic data collection, together with a simple
mathematical correction, can accurately recover rate constant information up to the dataacquisition rate, validating these methods on both synthetic (Figure 5.1) and experimental(Figure 5.8) data. As a result, stroboscopic smFRET methods in principle extend the upper limitof measurable rate constants by up to an order of magnitude over that of conventional smFRET
without resorting to reducing the instrument throughput by cropping the field of view to increase
frame rates.
We can extend the use of stroboscopic smFRET one step further. The issue of systematic
rate constant underestimation due to time binning is ubiquitous and generates similar constraints
for more sophisticated smFRET trajectory analysis methods such as hidden Markov modeling(HMM)39, 42-44 and time correlation function (TCF) fitting.45-46 Consequently, each of these andother analysis methods might also benefit from stroboscopic data collection. To explore this in
more detail, we have analyzed simulated smFRET trajectories using each of three methods i)
dwell time analysis, ii) HMM, and iii) TCF fitting, under both continuous and stroboscopic
illumination conditions (Figure 5.9). As clearly evident in Figure 5.9b–c (filled symbols), neither
HMM nor TCF fitting escapes this fundamental issue of underestimating fast rate constants. Thisis quite simply because both methods make the assumption, as in dwell time analysis, that thesystem occupies a single state throughout each observation point. This assumption becomes

158

Figure 5.9 Systematic underestimation of rate constants is a shared feature of multiple smFRET
analysis methods. Simulated conditions are the same as in Figure 5.4a. (a) Dwell time analysis. (b)Hidden Markov modeling (HMM). (c) Time correlation function (TCF) fitting. Each analysis isperformed on data simulated under full-frame illumination (solid symbols) as well as 25% duty
cycle stroboscopic illumination (open symbols). Error bars are ±1σ.
more nearly correct when fractional duty cycle “stroboscopic” illumination is applied, with all
methods accurately estimate the rate constant for a 10% duty cycle excitation (Figure 5.9, open
symbols). Consequently, the data in Figure 5.9 demonstrate that stroboscopic smFRET improves
the accuracy of rate constant determination by other analysis methods besides dwell time

159

Figure 5.10 Rate constants determined by hidden Markov modeling of simulated 3-state system(1 ↔ 2 ↔ 3), with (red) and without (blue) stroboscopic excitation. Shaded regions are 1-σuncertainties. Under continuous illumination, rate constants are systematically underestimated atslow frame rates (large Δtframe), whereas stroboscopic data yields accurate rate constants acrossall frame rates. Simulation conditions: 2000 s, 300 photons per bin, k12 = 10 s-1, k21 = 7 s-1, k23 =3 s-1 , k32 = 5 s-1, E1 = 0.2, E2 = 0.5, E3 = 0.8, repeated 192 times to determine uncertainties.

analysis. In addition, the use of HMM enables an especially facile extension of the stroboscopic
method to systems of more than 2 states, as demonstrated in Figure 5.10.
Stroboscopic smFRET requires greater laser power than conventional smFRET to obtainthe same signal level, which can potentially exacerbate issues of fluorophore saturation andphotobleaching. To examine this effect, we collected stroboscopic data with a 20% duty cycle at
100 frames per second (Figure 5.11), which is approximately the full-frame capture rate ofcurrent CCD technology. Working at our maximum available laser fluence (≈50 mW over a 20
μm diameter TIR spot), we observed a 50% reduction in signal-to-noise ratio compared to nonstroboscopic data at the same conditions (SNR = 3 vs. 6; Figure 5.11a). Despite the loss of SNR,
the quality of the data is sufficient for dwell time analysis, and the extracted rate constants areequivalent to those obtained without stroboscopic illumination (Figure 5.11b−c). These results

160

Figure 5.11 Experimental measurement of DNA hairpin folding at Δtframe = 10 ms (100 Hz
frame rate). (a) Normalized FRET histogram of data acquired at the same laser power (50 mW)
with 20% duty cycle strobe (blue) and without strobe (red). Though the FRET widths are larger
for the strobe data due to the short exposure time (Δtexposure = 2 ms), the FRET peaks are resolved
sufficiently to permit dwell time analysis, shown in (b) and (c) for folding and unfolding times,
respectively. The extracted rate constant for folding (kF) and unfolding (kU) are statisticallyequivalent for the case of data acquired with strobe (blue) and without strobe (red). Experimental
conditions: 75 mM total M+ (K+ and Na+), 20.0 °C.

show that, with the help of oxygen removal35 and triplet quenching,36 fluorophore photophysics
do not impede stroboscopic smFRET operation at acquisition rates up to 100 Hz.

161
This article builds off the work of Farooq and Hohlbein29 who first demonstrated the useof stroboscopic smFRET to measure fast rate constants. However, instead of analyzing
stroboscopic data by trajectory-based analyses, as in these studies, Farooq and Hohlbein usedprobability distribution analysis (PDA).22-24 In PDA, data points are binned into a FRET
histogram, which is then fit to a model that includes kinetic parameters. Therefore, PDA treats
each observed FRET value as an independent measurement. In contrast, trajectory-based analysis(e.g., dwell time analysis, HMM, TCF fitting) make use of the correlation between data points,
which has the potential advantage of utilizing a greater portion of the information content of thedata. Therefore, intuitively, the trajectory-based method presented in this paper should yield rate
constants with smaller uncertainties than those determined by PDA. Indeed, when simulated data
are analyzed by both PDA and dwell time analysis, the results from dwell time analysis have
≈50% smaller uncertainties (Figure 5.12). This reduction in uncertainty may make the present
method more attractive to some smFRET researchers, especially those who already rely on
trajectory-based analysis and do not wish to switch to PDA, and entirely different form ofanalysis. However, as shown by Farooq and Hohlbein, the combination of stroboscopic smFRET
and PDA has in principle access to faster dynamics than stroboscopic trajectory analysis, as
stroboscopic PDA is limited by the exposure time rather than the frame duration. Due to thiscombination of factors, we consider the trajectory-based approach and the PDA-based approach
to be complementary methods for analyzing stroboscopic smFRET data.
We emphasize that stroboscopic smFRET is quite straightforward to incorporate into any
existing smFRET experiment, with stroboscopic illumination implementable at relatively low
cost. Furthermore, stroboscopic data does not require adopting new analysis methods, as it isfully compatible with conventional Dwell Time, Hidden Markov Model (HMM), and Time

162

Figure 5.12 Analysis of simulated data by trajectory-based approach (dwell time analysis) and
FRET histogram fitting (PDA). Under the simulated conditions (2000 s, 100 photons per bin, kF= kR, Elow = 0.2, Ehigh = 0.8), dwell time analysis (a) yields accurate rate constants up to the frame
rate with an uncertainty (shaded band; standard deviation) of ≈4% which grows rapidly beyond
kΔtframe ≈ 1. (b) PDA accurately determines rate constants, with uncertainties of ≈8% that arelargely independent of kΔtframe.

Correlation Function (TCF) analysis. Therefore, any smFRET laboratory can readily incorporate
stroboscopic methods while largely maintaining the current experimental arrangement andanalysis. In the interest of completeness and kinetic rigor, we recommend reporting frame
acquisition rates along with published rate constants, as well as closely monitoring the ratio of
smFRET-measured rate constants to this acquisition rate. If any rate constants exceed 10–20% of
the frame rate, then FRET averaging is a significant concern, for which application ofstroboscopic methods offers a reduction in rate constant systematic errors down to < 3% level.
Finally, while this paper has focused on using smFRET to determine conformational dynamics,

163such a stroboscopic approach should also be extendable to any experimental method based on a
fluorescence measurement of discrete state sampling, such as transitions between diffusional
states in single molecule diffusion studies7 or binding processes studied by protein induced
fluorescence enhancement (PIFE).47

5.7

Conclusion
We have developed, tested, and presented a stroboscopic solution to the underestimationof rate constants in time-binned smFRET experiments. Our work demonstrates that kinetic rate
constants are significantly underestimated when comparable to the sampling rate upper limit,with > 5% systematic errors in the measured rate constant if within 10% of the ultimate frame
acquisition limit. These deviations cannot be avoided by modifying experimental parameterssuch as light intensity or duration of trajectory. The core cause of the artifact is the mapping ofcontinuous time dynamics onto a discrete time domain of binned data, resulting in blurring of
FRET states and missed dynamics. These fundamental issues can be addressed through the
combined use of i) stroboscopic illumination and ii) analytic mathematical correction to the rate
constants. Stroboscopic smFRET is thereby capable of measuring rate constants up to at least thedata acquisition rate, which for a 5% threshold of measurement accuracy amounts to a tenfold
increase in dynamic range. This allows one to measure single molecule rate constants up to theframe acquisition rate limit without compromising any other multiplexing capacity of themeasurement.

5.8

Acknowledgements
We would like to thank Johannes Hohlbein for providing code for PDA calculations.

164
5.9

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Chapter 6
Measuring Excess Heat Capacities of DNA Folding at the Single Molecule Level*

6.1

Abstract
Measurements of the thermodynamic properties of biomolecular folding (ΔG°, ΔH°, ΔS°,etc.) provide a wealth of information on the folding process and have long played a central rolein biophysical investigations. In particular, the excess heat capacity of folding (ΔCP) is
particularly crucial, as typically measured in bulk ensemble studies by differential scanningcalorimetry (DSC) and isothermal titration calorimetry (ITC). Here, we report the firstmeasurements of ΔCP at the single molecule level using single-molecule fluorescence resonanceenergy transfer (smFRET), as well as the very first measurements of the heat capacity change
associated with achieving the transition state (ΔC‡P) for nucleic acid folding. The DNA hairpin
used in these studies exhibits an excess heat capacity for hybridization (ΔCP = −340 ± 60
J/mol/K per base pair) consistent with the range of literature expectations (ΔCP = −100 to −420
J/mol/K per base pair). Furthermore, the measured activation heat capacities (ΔC‡P) for such
hairpin unfolding are consistent with a folding transition state containing few fully formed basepairs, in agreement with prevailing models of DNA hybridization.

6.2

Introduction
The thermodynamics of nucleic acid folding/hybridization are of primary interest to thebiophysics community.1-4 It is widely understood that the temperature stability profile of a DNA
duplex requires knowledge of the folding enthalpy ΔH° and entropy ΔS°, as, for instance, often*This chapter is adapted from: Nicholson, D. A.; Jia, B.; Nesbitt, D. J. J. Phys. Chem. B 2021, 125, 34, 9719–9726.

169
predicted from nearest-neighbor models.5-6 Less well investigated are heat capacities and inparticular heat capacity differences between hybridized and unhybridized states, the “excess”
heat capacity (ΔCP) for a given conformational transition,2, 7-13 which directly impacts any
predictions for temperature-dependence in ΔH° and ΔS°. As a result, the neglect of such excess
heat capacity effects results in erroneous extrapolation of nucleic acid folding to conditions otherthan the reference temperature at which thermodynamic parameters have been determined. Sucherrors can arise, for example, in the prediction of polymerase chain reaction (PCR) primer
stabilities, due to requisite operation at much higher than body temperatures.7 As a second
example, attempts to unify disparate sets of nearest neighbor model parameters for nucleic acidstabilities have proven challenging due to the need for comparison/extrapolation between
different temperature conditions.13 Even beyond such technological concerns, a knowledge ofexcess heat capacities offers first insights into the microscopic dynamics of solvent restructuring
during the folding/hybridization event that are inaccessible from measurements of ΔH° and ΔS°
alone.2, 7, 9-10, 14
The existence of such finite excess heat capacities (ΔCP  0) in nucleic acid folding iswell-established, with the majority of ΔCP measurements obtained from precision calorimetricmethods under bulk ensemble conditions. Specifically, isothermal titration calorimetry (ITC)
represents a precision tool for measuring ΔCP in systems of medium to high affinity bimolecularassociation,2, 12, 15-17 with differential scanning calorimetry (DSC) able to sensitively report on
ΔCP even for weak association processes as well as unimolecular folding dynamics.17-20 UV
absorption optical detection methods offer yet another alternative, whereby hypochromic shifts
in DNA absorbance can be used to determine folding equilibrium constants (Keq), which aresubsequently analyzed via temperature dependent van’t Hoff theory in order to extract ΔCP.2

170
Each of these approaches offers a powerful and sensitive window into heat capacity differences
associated with biomolecular conformational change, though up until now constrained toequilibrium bulk ensemble conditions.
The last two decades have witnessed considerable interest in extending thermodynamic
study of variables such as ΔG°, ΔH°, and ΔS° based on single-molecule methods,21-26 which
offer a unique opportunity to probe thermodynamic “landscapes” down at the ultimate singlemolecule sensitivity limit. Application of such measurement techniques to ΔCP, however, have
proven far more challenging and been notably absent in the single-molecule literature. In
particular, temperature-dependent single-molecule folding studies are now relatively
straightforward1, 27 but to date have focused exclusively on measurements of folding freeenergies, enthalpies, and entropies rather than excess heat capacities. Indeed, despite an initialreport by Williams et al. in 2001 establishing the feasibility of determining ΔCP via opticaltweezers,28 twenty years have passed without the publication of any articles utilizing single
molecule methods to extract ΔCP.
In this work, we revisit the prospect for determination of ΔCP at the single molecule
level, based on high precision temperature dependent studies with single-molecule fluorescenceresonance energy transfer (smFRET).23, 26 We acknowledge at the outset that ITC and DSC
measurements under bulk conditions offer greater sensitivity and precision; our goal is simply toestablish smFRET measurement of ΔCP as a viable, alternative tool in the single-molecule
toolbox. Although this proves more experimentally challenging than conventional van’t Hoff
temperature measurements of ΔH° and ΔS°, we note that such an approach nevertheless does
offer unique new insights into oligomer hybridization dynamics. In particular, the single
molecule FRET methods described herein provide additional access to folding kinetics

171
unavailable to ITC and DSC approaches, from which we can report first measurements of excess
heat capacities involved in accessing the transition state (i.e., ΔC‡P) for hybridization in nucleic
acid oligomers.

6.3

Methods
The smFRET construct used in these studies is a DNA hairpin consisting of a 7-base pairstem and a 40-(dA) adenosine loop, which we have previously characterized.29-30 As depicted
schematically in Figure 6.1, the construct is fluorescently labeled with Cy3 and Cy5 for FRET
measurements, and a biotin moiety is attached to the 3’ end for surface tethering. The full DNA
sequence (5’-Cy5-CTTCAGT-A40-Cy3-ACTGAAG-A11-biotin-3’) is purchased in highperformance liquid chromatography (HPLC) purified form from Integrated DNA Technologies.
(Any company names listed herein are in the interest of completeness, not as endorsement ofvendor).

Figure 6.1 Schematic of DNA hairpin construct used in these studies.

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Glass surfaces are cleaned and decorated with the smFRET construct in accordance withpreviously described protocols.29, 31 Simply summarized, glass coverslips are soaked in acetone
and treated in a UV-ozone cleaner for 30 minutes. The cleaned coverslips are then exposed, insequence, to i) 10%-biotinylated bovine serum albumin (10 mg/mL), ii) streptavidin (0.2mg/mL), and finally iii) the DNA construct (100 pM; 1 pM = 1 pmol/L). The resulting sample
has a surface density of ~1 construct per 10 μm2, yielding approximately 100 molecules in the
microscope field of view. Molecules are observed in buffer (50 mM HEPES, pH 7.6) with 120
mM monovalent (K+) ions. To decrease the rate of fluorophore photobleaching, an enzymatic
oxygen scavenging cocktail32 (100 nM protocatechuate 3,4-dioxygenase, 5 mM protocatechuicacid, and 2 mM Trolox) is also present.
Single-molecule fluorescence measurements are taken on a through-objective totalinternal reflection fluorescence (TIRF) microscope.33,31 Briefly, laser light at 532 nm is focused
onto the back focal plane of a high numerical aperture oil-immersion microscope objective, witha pair of mirrors translating the beam off axis to increase the angle of incidence at the sample
until total internal reflection is achieved. Widefield fluorescence is collected by the objective andseparated by dichroic mirror into Cy3 and Cy5 emission channels, which are each focused onto acharge-coupled device (CCD) video camera. To increase accuracy when measuring folding rate
constants comparable to the CCD frame rate, we exploit stroboscopic excitation methods bymodulating the laser to excite the sample for the first 20% of each frame.34 Single molecule
FRET trajectories are extracted from CCD movies by31 i) applying a brightness threshold tolocate particles, ii) summing pixels in a circular neighborhood around each particle to obtainfluorescence rates as a function of time, iii) subtracting local background, iv) spatially pairing
Cy3 and Cy5 particles by a calibrated affine map,35 and v) computing the FRET efficiency,

173
FRET(t) = ICy5(t)/(ICy3(t)+ICy5(t)). FRET trajectories are analyzed by dwell time analysis withcorrection for stroboscopic excitation36 to yield folding and unfolding rate constants (kF and kU)
as well as equilibrium constants (Keq = kF/kU). Agreement between Keq values measured by ratiosof rate constants to those determined from integrated fractional time spent in the unfolded (TU)vs folded (TF) state (Keq = TF/TU) is routinely at the < 2% level.
Sample temperature is controlled using thermoelectric cooling/heating modules under
servo loop control, as previously established.31 To prevent thermal gradients, the sample isheated from above by a module in direct contact with the sample and from below by a module
attached to the microscope objective, which maintains thermal contact with the sample by theimmersion oil. The system temperature is measured by thermistors, which we have calibrated byin situ measurements of the sample temperature using a thin wire thermocouple. The thermistor
readings are used as input to the thermoelectric modules for computer-based feedback
stabilization with a 0.015 Hz bandwidth, resulting in 0.1 °C temperature accuracy and stabilityon the 1-minute time scale.

6.4

Results

6.4.1 Heat capacity in van’t Hoff analysis
Measurement of the equilibrium constant at multiple temperatures provides insight into
the thermodynamic parameters (enthalpy/entropy) of the process. Specifically, the van’t Hoff
equation2 describes the dependence of the equilibrium constant Keq as a function of overall ΔH°
and entropy ΔS°:ln(𝐾eq ) = −∆𝐻°(𝑇) 1𝑅(𝑇) +∆𝑆°(𝑇)
𝑅(Eq. 6.1)

174
where T is the absolute temperature and R is the gas constant. Notably, if ΔH° and ΔS° are
independent of temperature, then the resulting van’t Hoff plot of ln(Keq) vs. 1/T is predicted to belinear, with slope −ΔH°/R and intercept ΔS°/R.
If products and reactants have different heat capacities CP, then the overall reaction will
have a nonzero excess heat capacity ΔCP = CP (products) – CP (reactants). The excess heat
capacity is related to ΔH° and ΔS° by the fundamental thermodynamic expressions2
∆𝐶𝑃 =

𝜕
𝜕𝑇

(∆𝐻°)𝑃(Eq. 6.2)

𝜕∆𝐶𝑃 = 𝑇 𝜕𝑇 (∆𝑆°)𝑃(Eq. 6.3)

If ΔCP is approximated as constant over the measured temperature range, we may integrate theseequations and substitute the resulting temperature-dependent ΔH°(T) and ΔS°(T) into Eq. 6.1 toproduce the ΔCP-modified van’t Hoff equation,ln(𝐾eq ) = −

o
∆𝐻𝑟𝑒𝑓

𝑅

1

(𝑇) +

o
∆𝑆𝑟𝑒𝑓

𝑅

∆𝐶

− 𝑅𝑃 [(𝑇−𝑇𝑟𝑒𝑓𝑇𝑇) + ln ( 𝑟𝑒𝑓)]𝑇(Eq. 6.4)

where Tref is an arbitrary reference temperature and ΔH°ref and ΔS°ref are ΔH°(T) and ΔS°(T)evaluated at Tref, respectively. Most importantly, the additional logarithmic term introduces nonlinearity into the single molecule van’t Hoff plot, which provides a simple experimental
diagnostic for nonzero ΔCP.

6.4.2 smFRET measurements reveal nonzero ΔCP in DNA folding
In order to test these predictions, we have measured the folding kinetics of a FRETlabeled DNA hairpin (see Figure 6.1) as a function of temperature. Across all temperatures (13–35.5 °C), smFRET trajectories exhibit clear switching between two distinct FRET states (seeFigure 6.2), representing unhybridized (EFRET  0) and hybridized (EFRET  0.8) conformations.

175

Figure 6.2 Sample single-molecule FRET time trajectories (A) and dwell time distributions (B)for DNA hairpin folding at 13 °C and 33 °C. All data are obtained in 90 mM K+ and 30 mM Na+.
As expected for an exothermic folding process (H0 < 0), the population of the folded state
decreases with increasing temperature relative to the unfolded state, corresponding to heat
induced denaturation or “melting” of the DNA hairpin.3, 6 In order to extract thefolding/unfolding dynamics in more quantitative detail, we perform dwell time analysis on the
smFRET trajectories.37 At each temperature, the folding and unfolding cumulative dwell time
distributions are well-fit by single exponential decays, indicative of simple two-state kinetics
governed by the unimolecular rate constants kF and kU. From these rate constants, we can directlycompute the folding equilibrium constant Keq = kF / kU as a function of temperature.

176

Figure 6.3 van’t Hoff analysis of smFRET-derived equilibrium constants with and without
modifications for nonzero excess heat capacity (ΔCP). (A) Linear van’t Hoff fit. (B) Fit with ΔCP
term to account for curvature. Fit residuals for the linear fit (C) show systematic deviations awayfrom linearity, while residuals for ΔCP -modified fit (D) exhibit no temperature dependence.
The temperature-dependence of Keq for the DNA hairpin is presented as a van’t Hoff plot
in Figure 6.3. By way of a first order treatment, we have analyzed these data with a simple linear
fit (Figure 6.3A, red line), which is equivalent to setting ΔCP = 0 in Eq. 6.4. From the fit
residuals (Figure 6.3C), however, the data clearly deviate systematically from such a linear
model, with the fit overestimating Keq at both the lowest and highest temperatures. To accountfor such curvature in the van’t Hoff plot, we therefore fit the data to the modified van’t Hoff
equation (Eq. 6.4), permitting ΔCP  0. The residuals now exhibit little to no systematic
temperature dependence, suggesting a more satisfactory model fit. From a non-linear weighted

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least squares fit, we infer the excess heat capacity to be ΔCP = −2.5 ± 0.4 kJ/mol/K, where theuncertainty represents 1 standard deviation. The small fractional uncertainty (< 20%) in ΔCP
provides additional support for data consistency with nonzero excess heat capacity (ΔCP  )
upon DNA oligo hybridization.

6.4.3 Arrhenius analysis of rate constants: Transition state excess heat capacity
Temperature dependent equilibrium constants provide clear evidence for the influence offinite excess heat capacity (ΔCP  0) on the overall DNA hybridization event, as observed hereinat the single molecule level. However, there is also additional thermodynamic information
encoded in the kinetics of such single molecule data, which is not readily accessible in bulk ITC
or DSC studies. Specifically, we can exploit Arrhenius plots of ln(k) vs. 1/T to further obtainthermodynamic information about the transition state for DNA hybridization. For example, thekinetic data is often fit to the standard Eyring transition state theory (TST) result38-39
ln(𝑘) = −∆𝐻 ‡ (𝑇) 1
𝑅(𝑇 ) +∆𝑆 ‡ (𝑇)𝑅+ ln(𝜈 ‡ )(Eq. 6.5)

where k is the folding or unfolding rate constant, ΔH‡ is the activation enthalpy, ΔS‡ is theactivation entropy, and ν‡ is the attempt frequency for the system to reach the transition state.
Similar to the van’t Hoff analysis (Eq. 6.2−6.4), we can extend our TST expression in Eq. 6.5 toinclude non-zero excess heat capacity between transition state and the reactants, yielding:ln(𝑘) = −

‡
∆𝐻𝑟𝑒𝑓

𝑅

1

(𝑇) +

‡
∆𝑆𝑟𝑒𝑓

𝑅

∆𝐶 ‡

− 𝑅𝑃 [(𝑇−𝑇𝑟𝑒𝑓𝑇

𝑇𝑟𝑒𝑓) + ln ( 𝑇 )] + ln(𝜈 ‡ )(Eq. 6.6)

where ΔH‡ref and ΔS‡ref are ΔH‡(T) and ΔS‡(T) evaluated at the reference temperature Tref, and
ΔC‡P is the excess activation heat capacity. Just as in our van’t Hoff analysis, a nonzero ΔC‡P
results in curvature in the Arrhenius plot, which for ΔC‡P = 0 would be perfectly linear. We note

178
that ΔS‡ref and ln(ν‡) cannot be determined independently from such an analysis, since bothparameters are perfectly correlated and simply account for a vertical offset. However, suchpotential ambiguity in ΔS‡ref and ln(ν‡) has no effect on ΔC‡P, which is a function only of
curvature in the temperature dependent Arrhenius plot. We also note that the applicability of
TST to nucleic acid folding is complicated by the presence of multiple folding pathways, as is
posited for instance in the kinetic zipper model of DNA hybridization40-42 (see Discussion). In
such a scenario, the measured ΔH‡ref, ΔS‡ref, and ΔC‡P are instead interpreted as an average overthe multiple folding pathways.
The temperature-dependent unfolding kinetic measurements for the DNA hairpin arereported in Figure 6.4A, where the black squares represent experimental data and the solid red

Figure 6.4 Arrhenius analysis of unfolding rate constant kU. (A) Data are fit to linear (red) andnonlinear (blue) models. Linear fit residuals (B) exhibit systematic errors compared to nonlinear
fit residuals (C), in support of non-negligible CP.

179

and black lines reflect the linear (ΔC‡P = 0) and higher order, non-linear approximations (ΔC‡P 0), respectively. The rate constants increase by more than two orders of magnitude over thetemperature range tested, from 0.28 ± 0.01 s-1 at 13 °C to 48 ± 4 s-1 at 35.5 °C. We have fit thesedata with both a ΔC‡P = 0 two-parameter fit (Eq. 6.5) and variable ΔC‡P  0 three-parameter fit(Eq. 6.6), with differences between the two fits highlighted by examining the residuals (Figure
6.4B−C). The residuals for the ΔC‡P = 0 fit clearly show systematic temperature-dependent
deviations, with the linear fit underestimating measured rate constants at both low and hightemperatures. Allowance for a nonzero ΔC‡P results in significant improvement, though not as
dramatically as for the van’t Hoff analysis (Figure 6.3C−D). To help evaluate the quality of fit(and suitability of the underlying model), we have examined the reduced χ2 statistic, which is theaverage of the square of the residuals divided by their variance (χ2 = <(yi-yi,fit)2/σi2>), whichshould be  1 for a physically correct model.43 For the linear fit (ΔC‡P = 0), we find χ2 = 2.7,
while for the non-linear fit (ΔC‡P  0), χ2 = 0.98, which signals clear justification for the higherlevel analysis. In summary, the non-linear least squares fits yield ΔC‡P = 2.3 ± 0.7 kJ/mol/K forthe excess activation heat capacity between the i) fully folded and ii) transition state forunfolding of the smFRET construct.
A similar analysis can of course be performed on kF, though the curvature in such an
Arrhenius plot (see Figure 6.5) is much reduced, with a non-zero value of ΔC‡P on the thresholdof our experimental resolution. Inspection of the fit residuals shows negligible visible
improvement in the residuals upon introducing a nonzero ΔC‡P term. Indeed, this is confirmed bythe χ2 values for the folding rate constant statistics, which are close to unity for both linear (χ2 =1801.7) and nonlinear (χ2 = 1.3) fits. Nevertheless, an error-propagated estimate of ΔC‡P, −0.7 ± 0.4
kJ/mol/K, for unfolding kinetic data appears finite, though zero within 95% (2) uncertainty.
The results from all fits of the equilibrium and rate constants to van’t Hoff and Arrhenius models
with finite excess heat capacity are summarized in Table 6.1.

Figure 6.5 Arrhenius analysis of folding rate constant kF. (A) Linear (red) and nonlinear (blue) fits
for kF are visually indistinguishable. (B) Residuals of the linear fit are unstructured, and (C)introduction of a nonzero ΔCP in fitting has little effect on residual errors.

Folding
Unfolding

ΔH°ref (kJ/mol)

ΔS°ref (J/mol K)

ΔCP (kJ/mol K)-88(2)-294(8)-2.5(4)ΔH‡ref (kJ/mol)

ΔS‡ref (J/mol K)

ΔC‡P (kJ/mol K)

85(2)176(4)

57(7)357(13)-0.7(4)2.3(7)

Table 6.1. Fit Results of Temperature-Dependent Measurements. The reference temperature isTref = 300 K and the attempt frequency is ν‡ = 6 x 1012 s-1. See text for details.

181
6.5

Discussion
In this work, we have explored the capacity of single molecule FRET microscopy formeasuring excess heat capacities in biomolecular folding. As a proof-of-concept demonstration,
we have measured the folding dynamics of a DNA hairpin test system29-30 (Figure 6.1) from 13°C to 35.5 °C on a single-molecule TIRF microscope. Deviations from linear behavior in van’t
Hoff plots of Keq and Arrhenius plots of kU (Figs. 3 and 4), both of which indicate clear effects
due to non-zero excess heat capacities. From the curvature in these plots, we have extracted
differential heat capacities for the overall folding reaction (ΔCP) and for the approach to thetransition state (ΔC‡P) from the reactants and products (Table 6.1).
The ability to perform such measurements for excess heat capacity ΔCP requires the
combination of a large temperature range and precise rate constant measurements. For example,
maximal deviations from linearity in the Arrhenius plot for the unfolding rate constant kU (Figure6.4) are only of order ≈ 0.1 logarithmic units, which in turn demands <10% fractional
uncertainties to observe. We achieve this level of precision by acquiring 20,000 dwell times from
300 molecules for each of 10 different temperatures. Obviously, a high-throughput method suchas widefield TIRF microscopy greatly facilitates this level of data acquisition in comparison to
confocal microscopy.25 Such demands on precision can be relaxed if the temperature range is
expanded, as the deviations from linearity grow quadratically with T. However, theexperimentally available range of temperatures is restricted by the stability of the folding, since
smFRET kinetic measurements are in practice typically limited37 to systems with 0.1  Keq  10.
It is worth noting that this restriction only applies when measuring equilibrium constants via Keq= kF/kU, with the dynamic range of measurable Keq values 1−2 orders of magnitude greater if one
instead integrates FRET histograms to determine population ratios. The measurement of rate

182
constants is clearly worth the additional effort, however, since such kinetic data provide theadditional capacity to measure excess activation heat capacities for accessing the transition state,
which represents one unique advantage of using single molecule methods over ITC or DSC.
Excess heat capacities in protein and nucleic acid folding are phenomenologically linked
to changes in surface area during folding.2, 7, 9-10, 12, 14 The surface area effect has been ascribed tosolvent interactions with the nucleic acid, including the hydrophobic effect, perturbations of
intramolecular vibrations, and hydrogen bonding.13 Proteins undergo significant surface area loss
during folding, whereas the higher charge densities of polyanionic nucleic acids prevents suchcompactification. Thus, nucleic acids tend to have smaller specific excess heat capacities thanproteins.13 As a consequence, excess heat capacities in nucleic acids have been more challenging
to measure than for proteins and were indeed once thought to be identically zero.13 Nonzero ΔCP
values for nucleic acid folding are now well-established,7, 44-45 with burial of hydrophobic
nucleobases during base pairing considered the primary contributor.2
The current single molecule data for excess heat capacities of nucleic acid folding can be
usefully compared with results from bulk calorimetric studies. The specific DNA hairpin
construct explored in these studies experiences a decrease in heat capacity upon folding (ΔCP =−2.4 ± 0.4 kJ/mol/K), which is consistent with the general literature consensus of ΔCP < 0 due toa decrease in accessible surface area. Since the surface area loss is proportional to the number ofbase pairs formed, and since ΔCP values are at least thought to have only a minor sensitivity to
DNA sequence,12, 20 it is convenient to report ΔCP values per base pair, which for our 7 base pair
hairpin is −340 ± 60 J/mol/K/bp (bp = base pair). Previous literature values for ΔCP from bulk
studies exhibit a rather broad range from −100 to −420 J/mol/K/bp with a recent metastudy7
recommending −130 J/mol/K/bp, against which the current single molecule measurement is high183
but nevertheless qualitatively consistent. Indeed, the 3-fold dynamic range of these literature
results is an indication of the significant experimental challenges involved even in bulk studies ofDNA folding as well as additional sensitivity to the nature of the DNA construct.
As a particularly important conformational distinction, most bulk studies have explored
duplex formation for bimolecular rather than unimolecular folding, as is the case for the 7 bp
DNA hairpin construct connected by a 40-adenine loop utilized in the current single molecule
efforts. Indeed, if we further limit the previous literature comparison only to measurements on
unimolecular hairpins, the reported ΔCP values grow significantly to −300 J/mol/K/bp, which is
now in even better agreement with our single molecule results.46-48 Other aspects of single
molecule experimental design can influence duplex stability, including surface tethering49 andfluorophore incorporation,50 which may limit comparison with bulk studies of freely-diffusing,unlabeled oligonucleotides. Finally, the polyadenosine (poly-dA) loop in the hairpin is known toform single-stranded base-stacked structures51 which are likely disrupted during folding, leading
to an increase in surface area and a concomitant increase in CP. Therefore, loops sequences thatare less prone to base-stacking, such as polythymidine (poly-dT), may have a more negative
ΔCP for folding. These comparisons serve to further highlight the level of complexity in suchmeasurements, with expectations for a possible dependence on loop sequence and GC/AT
composition in our smFRET construct.
In addition to the overall heat capacity change during folding (ΔCP), we have measured
the heat capacity change required to reach the folding transition state (ΔC‡P). While activation
heat capacities have been reported for proteins,52 this work provides, to the best of ourknowledge, the first such measurement of ΔC‡P for nucleic acid folding. As one simple check,we confirm that the overall folding heat capacity (ΔCP = −2.5 ± 0.4 kJ/mol/K) is within

184
uncertainty of the sum of the activation heat capacities (ΔC‡P,Fold + -ΔC‡P,Unfold = −3.0 ± 0.8
kJ/mol/K), as required since CP is a state function. We can combine this information in ΔCP and
ΔC‡P to construct a profile of the system heat capacity “landscape” along the reaction coordinate(Figure 6.6). If negative excess heat capacities in nucleic acid folding arise primarily from theloss of accessible surface area during base pair formation, as is the literature consensus,2, 13 then
one consistent expectation is that the heat capacity of the transition state should lie between thefolded and unfolded values, since the transition state cannot form more base pairs than the folded
state nor less base pairs than the unfolded state. Indeed, the experimental CP landscape profile
reveals an intermediate transition state heat capacity between that of the unfolded and folded
states, in good agreement with this expectation. Interestingly, the transition state excess heat
capacity C‡P appears to be closer to that of the completely unhybridized unfolded state. If one

Figure 6.6 Excess heat capacity “landscape” changes along the folding reaction coordinate,
relative to the folded state. U = unfolded state, TS = transition state, F = folded state. Note themonotonic decrease in this excess heat capacity upon hybridization of the 7 bp DNA stem and therelatively small activation heat capacity in approaching the transition state from the unfolded
direction. The right axis indicates the number of base pairs formed at each state, based on a simple
linear model for heat capacity proportional to solvent accessible surface area. Such behavior
suggests minimal base pairing at the transition state, which is entirely consistent with “zippering”
models for DNA hybridization.41, 59 Error bars reflect one standard error of the mean.

185
assumes linearity in these excess heat capacities with hybridization sequence length, this would
suggest that relatively few base pairs (n‡bp = 1  1) are fully formed in the transition state,certainly including zero within error.
The conclusion that few base pairs are formed at the DNA hairpin’s folding transition
state is in good agreement with the current kinetic zipper model of nucleic acid hybridization.4042, 53-54

In the zipper model, the free energy barrier for folding is primarily entropic, as the two

DNA strands must “pre-align” in a correct orientation before forming one or more key base paircontact(s).55 This nucleation site then catalyzes the formation of base pairs at adjacent sites in an
enthalpically-downhill cascade to the folded state. The number of base pairs in the transition
state is a subject of ongoing investigation, with evidence ranging for a transition state containing
a single base pair,56 a minimum of 2 base pairs,41 or the absence of any nascent base pairs at
all.57-58 Clearly, more work is warranted, and while the uncertainty in our results is too large toresolve these possibilities (n‡bp = 1  1), our heat capacity-based measurement provides further
confirmation of the kinetic zipper prediction of an “early” transition state for hybridization; i.e., atransition state having few if any fully formed base pairs.

6.6

Summary and conclusion
In this report, we demonstrate the use of smFRET to measure excess and activation heat

capacities in biomolecular folding. By combining a large temperature range with high level datacollection statistics, our experiment achieves the necessary precision to resolve nonzero heat
capacities in the folding of a DNA hairpin. The measured value of ΔCP (340 ± 60 J/mol/K perbase pair) is in good agreement with literature calorimetric results on similar constructs, which
provides confirming evidence for the accuracy of our method. Furthermore, the present single

186
molecule studies provide kinetic evidence for finite excess heat capacities upon approach to thetransition state (ΔC‡P), which provides novel confirmation that the transition state contains only
few fully formed base pairs (n‡bp = 1  1) and support of the “kinetic zipper” model for DNA
hybridization kinetics. In summary, heat capacity measurements at the single molecule level
offer a useful complement to calorimetric bulk methods and are uniquely capable of inspecting
heat capacities of transition states.

187
6.7

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Chapter 7
Kinetic and thermodynamic control of G-quadruplex polymorphism by Na+ and K+
cations*

7.1

Abstract
G-quadruplexes (G4s) are ubiquitous nucleic acid folding motifs that exhibit structural

diversity dependent on cationic conditions. In this work, we exploit temperature-controlled
single-molecule fluorescence resonance energy transfer (smFRET) to elucidate the kinetic andthermodynamic mechanisms by which monovalent cations (K+, Na+) impact folding topologies
for a simple G-quadruplex sequence (5’-GGG-(TAAGGG)3-3’). Kinetic measurements indicatethat Na+ and K+ influence G4 formation in two distinctly different ways: the presence of Na+
modestly enhances a parallel/hybrid G4 topology through an induced-fit (IF) mechanism withlow affinity (Kd = 228 ± 26 mM), while K+ drives G4 into an antiparallel topology via a
conformational selection (CS) mechanism with a much higher affinity (Kd = 1.9 ± 0.2 mM).
Furthermore, temperature dependent studies of the equilibrium folding ratio reveal distinctly
opposing thermodynamic signatures for binding of K+ (ΔH°bind > 0, ΔS°bind > 0) vs. Na+ (ΔH°bind< 0, ΔS°bind < 0) to G4, which further illuminates the diversity of possible pathways formonovalent facilitation of G-quadruplex folding.

7.2

Introduction
G-quadruplexes (G4s) represent ubiquitous nucleic acid secondary folding motifs that canform in G-rich DNA and RNA.1 Stable under both in vitro and in vivo conditions, G4s play

* Submitted for publication.

193
important roles in numerous biological contexts, including replication, transcription, andchromosomal stabilization.2 Recently, G4s have drawn particular attention as potential drug
targets for antiviral and anticancer therapies,3-6 while G4 folds have been incorporated as
modular components in DNA nanostructures.7-8 Predictive understanding and control of G4s inthese diverse biological and biotechnological areas has proven elusive and requires acomprehensive knowledge of the underlying biophysical issues regarding structure, kinetics, andthermodynamics for nucleic acid folding.
In contrast with canonical nucleic acid duplexes, G4s represent a highly structurally
diverse secondary folding motif.9-10 The signature element of G4 structure is the presence of Gtetrads, planar arrangements of four noncanonically base-paired (and non-contiguous) guanine
residues, which then stack (n ≈ 3, 4) to stabilize multiple G4 planes through adjacent π-πinteractions. The guanine residues in a G4 can come from multiple nucleic acids as part of ahigher order molecular complex, or from a single contiguous nucleic acid, in which case the G4
is referred to as a unimolecular fold. In such a unimolecular fold G4, sets of 4 guanine residues,
called G-tracts, are spaced by loop sequences (usually 2-3 residues) to sterically allow the Gtracts to interact in a common plane. These loops permit adjacent G-tracts to have the same or
opposite polarity (5’ to 3’ orientation), which allows G4s to fold into one of many topologies,
each defined by a specific combination of the G-tract polarities.11 G4 topologies are therefore
classified as parallel (all strands share the same polarity), antiparallel (strands alternate
polarities) or hybrid. Whether a G4 folds into a single topology or multiple topologies(polymorphism) depends on multiple factors, including loop sequence.12-13 Beyond the multiple
possible folded state topologies, G4 structural diversity is further increased by the presence of194partially folded intermediates such as G-triplexes (G3) or G-duplexes (G2) which can also be
significantly populated and interconvert.14-15
G4 folding is accompanied by obligate binding of cations to the central channel formedby the stacked G-tetrads.16 This binding site is capable of binding polyvalent cations but greatly
favors monovalent cations, especially potassium (K+). Importantly, the structure adopted by a G4
is influenced by the identity of its bound cation(s). For example, the human telomere G4
sequence folds into a parallel or hybrid topology in K+ solution and an antiparallel topology in
Na+ solution.17-18 The polymorphism in G4 sequences is clearly regulated by cationicinteractions, which has been a subject of much research activity. The influence of ions on G4
folding has been studied extensively by bulk methods; 10, 12-13, 19-25 however, the multi-statepolymorphism of G4s can make such ensemble results difficult to interpret. Single-molecule
experiments, on the other hand, are more ideally suited to resolve state-specific properties in
multistate folding systems.26-27 In particular, single-molecule methods have been used to
examine many aspects of G4 folding28 including G4 folding kinetics29-33 and temperaturedependence of the folding equilibrium constants.34 Interestingly, temperature dependent studiesof the folding kinetics, which in principle can provide first access to thermodynamic information
on transition states for G4 formation, have yet to be reported in single-molecule experiments.
As a focus of the present work, we use temperature-dependent single-molecule
microscopy to investigate the influence of monovalent cations (K+ and Na+) on the kinetics andthermodynamics of G4 folding. Specifically, we study the folding of the 5’-GGG-(TAAGGG)33’ sequence, which is known to form a polymorphic G4.32 We examine this system using
fluorescence resonance energy transfer (FRET) to monitor the dynamics of G4 folding with asingle-molecule FRET (smFRET) microscope. Transitions between the polymorphic network of

195
G4 states are observed in real time, from which we determine the equilibrium constants and rate
constants for these transitions. By analyzing the dependence of such folding dynamics on cation
concentration and temperature, we obtain first information on the kinetic mechanism forpolymorphism control by K+ and Na+ as well as the thermodynamics of cation binding to G4.

7.3

Methods

7.3.1 G-quadruplex FRET construct
The single-molecule construct used in these studies consists of two annealed DNA
oligomers. The first oligomer is a Cy3-labeled DNA G-quadruplex fold (5’-GGG-(TAAGGG)33’) connected by 8 thymine residues to an 18-nucleobase duplex-forming domain (full sequence:
5’-TGGCGACGGCAGCGAGGC-T8- GGG-(TAAGGG)3-Cy3-3’). The second oligomer iscomplementary to the duplex-forming domain and is labeled with Cy5 and biotin (5’-Cy5GCCTCGCTGCCGTCGCCA-Biotin-3’) for surface tethering by biotin-streptavidin interactions.
Both oligomers are purchased in HPLC-purified form from Integrated DNA Technologies andused as is. (Company names listed herein are in the interest of completeness and do not reflect
endorsement of a particular vendor.) The oligomers are annealed by mixing 10 μM aliquots (1 M
= 1 mol/L) and heating to 85 °C to disrupt intramolecular base-pairing before slowly cooling toroom temperature at 1 °C/min. The annealed construct is used without purification, as constructs
lacking the biotinylated strand do not adhere to the surface and constructs lacking the Cy3labeled strand are inefficiently excited by the laser (σCy3/σCy5 = 3% at 532 nm).
7.3.2 Sample preparation and single-molecule microscopy
Sample preparation and single-molecule microscopy are carried out according topreviously described protocols.35-36 In brief, a glass coverslip is cleaned by 24 hr soaking in

196
acetone followed by a 30 min treatment with UV-generated ozone. The coverslip surface isincorporated into a flow cell, where it is functionalized by sequential 10-minute exposure tobuffered solutions of (i) 10 mg/mL bovine serum albumin (BSA) with 1 mg/mL biotinylatedBSA, (ii) 0.2 mg/mL streptavidin, and (iii) 150 pM smFRET construct. This process produces asurface decorated with approximately one smFRET construct per 10 μm2. The surface-tetheredconstructs are then imaged in 50 mM HEPES buffer (pH 7.6) with an oxygen scavenging
cocktail to extend observation time (100 nM protocatechuate 3,4-dioxygenase, 5 mM
protocatechuic acid) and 2 mM Trolox to increase fluorophore brightness by quenching offluorophore triplets to the ground state.37-38 Sodium and potassium levels are controlled bytitration with NaCl and KCl solutions, respectively.
The prepared sample is observed on a though-objective total internal reflection
fluorescence (TIRF) microscope,39 which permits 532 nm excitation of Cy3 molecules within≈100 nm of the surface. Fluorescence from the smFRET constructs is collected and separated bydichroic mirror into Cy3 and Cy5 channels before being directed onto an intensified chargecoupled device (CCD) camera operating at 10 frames per second. Movies are analyzed using
software written in LabWindows which identifies particle locations by thresholding and extracts
raw donor and acceptor fluorescence trajectories by integration over the local pixel neighborhoodfor each frame. Background fluorescence levels are determined by fits to a delocalized 2D
Gaussian, which permits calculation of background-corrected donor and acceptor trajectories,
D(t) and A(t), with FRET trajectories determined from EFRET(t) = A(t) / (D(t) + A(t)).
Temperature control is achieved through a thermoelectric element in thermal contact with thesample and capable of heating and cooling with 0.1 °C stability, as previously described.36
7.3.3 Hidden Markov modeling197
Rate constants for conformational transitions are determined by hidden Markov modelingof FRET trajectories. The model consists of n states, each with a Gaussian-distributed FRET
observable, P(Ei, σi) ∝ exp[-(E-Ei)/2σi2], where Ei and σi are the FRET center and width of statei, respectively. Transitions between states occur according to first-order kinetics as contained in
the rate matrix K, where Kij is the rate constant for the transition from state j to state i, and Kii = Σj≠i Kij. The transition probability per frame is calculated via the matrix exponential T =exp(Ktframe), where tframe is the experimental time between adjacent frames (0.1 s). The likelihood
function for a single trajectory is therefore described by the matrix product
𝐿 = 𝟏𝑇 [∏𝑖≥2 𝐎(𝐸𝑖 )𝐓]𝐎(𝐸1 )𝒑𝑒𝑞 ,(Eq. 7.1)

where peq is the equilibrium probability vector (K·peq = 0), Ei is the ith observed FRET value,
O(E) is an n × n diagonal matrix with diagonal entries Oii(E) equal to the probability density ofobserving the FRET value E in the ith state, 1T is the row vector (1, 1, …) of length equal to thenumber of states n, and the product is taken over all frames in the trajectory except the first. For
multiple trajectories, the total likelihood function is the product of the single-trajectorylikelihood functions (Ltotal = ∏i Li). The maximum likelihood estimate of Kij, Ei, and σi is
obtained by performing gradient ascent on Ltotal using MATLAB, and bootstrapping analysis isperformed to estimate uncertainties.40
7.3.4 Three-state Hill fit
Population versus concentration data for the G4 are fit to a Hill model. The standard Hillequation describes ligand binding in systems with two distinguishable receptor configurations: aligand-bound state and a ligand-free state.41 However, the G4 construct in this work has threeobservable states, so we have instead used a modified Hill analysis which includes three states.
The binding model consists of two competing ligand association processes which convert a

198
receptor R into one of two distinguishable ligand-bound species B1 or B2, each with a uniquestoichiometry (n1 and n2) and dissociation constant (Kd,1 and Kd,2):
+𝑛1 𝐿

1:

𝑅→

2:

𝑅→

+𝑛2 𝐿

𝐵1

(K d,1 )𝐵2(K d,2 )

𝑛1

𝑛2

[𝑅]

= [𝐵 ] [𝐿]𝑛1(Eq. 7.2)

1[𝑅]= [𝐵 ] [𝐿]𝑛2 .

(Eq. 7.3)

2

The above dissociation constant expressions can be readily manipulated to yield the fractional
populations in each state as a function of ligand concentration:
[𝑅]

1

1

𝑛1
𝑛2
[𝐿]
[𝐿]
) +(
)
1+(
Kd,1
Kd,2

𝜃𝑅 ([𝐿]) = [𝑅]+[𝐵 ]+[𝐵 ] =2[𝐵 ]𝜃𝐵1 ([𝐿]) = [𝑅]+[𝐵 1]+[𝐵 ] =
1

2

𝑛1
[𝐿]
)
Kd,1
𝑛1
𝑛2
[𝐿]
[𝐿]
) +(
)
1+(
Kd,1
Kd,2

(Eq. 7.5)

𝑛2
[𝐿]
)
Kd,2
𝑛1
𝑛2
[𝐿]
[𝐿]
) +(
)
1+(
Kd,1
Kd,2

(Eq. 7.6)

(

(

[𝐵2 ]

𝜃𝐵2 ([𝐿]) = [𝑅]+[𝐵 ]+[𝐵 ] =1(Eq. 7.4)

2.

This model is then applied to data for sodium and potassium titrations of the G-quadruplex,
where L = Na+ or K+, R is the low FRET state, B1 is the middle FRET state, and B2 is the high
FRET state. Dissociation constants (Kd,i) and apparent stoichiometries (ni) are determined bysimultaneously fitting all three populations to this model using the weighted, nonlinear leastsquares regression tools available in OriginPro.
7.3.5 van’t Hoff and Arrhenius analyses
Temperature-dependent equilibrium constants determined from hidden Markov modelingare then subjected to van’t Hoff analysis,42-43 whereby the Gibbs free energy∆𝐺° = ∆𝐻° − 𝑇∆𝑆° = −𝑅𝑇 ln(K 𝑒𝑞 )is rewritten in the van’t Hoff form

(Eq. 7.7)

199
ln(𝐾𝑒𝑞 ) =−∆𝐻° 1∆𝑆°(𝑇) + 𝑅 ,𝑅

(Eq. 7.8)

with Keq as the equilibrium constant, R the gas constant (8.314 J/molK), T the absolutetemperature, ΔH° the reaction enthalpy, and ΔS° the reaction entropy. In accord with such avan’t Hoff model, we assume ΔH° and ΔS° to be approximately constant over the temperature
range explored, with a linear fit of ln(Keq) vs. 1/T to extract ΔH° and ΔS° via the slope andintercept, respectively. In analogous fashion, the temperature-dependent rate constants can beanalyzed in the context of Kramer’s theory44-46 to yield an Arrhenius-type expression
‡

‡

−∆𝐻 1
∆𝑆
ln(𝑘⁄𝜐) = 𝑅 (𝑇) + 𝑅 ,(Eq. 7.9)

where k is the rate constant, ν is the attempt frequency along the reaction coordinate, ΔH‡ and
ΔS‡ are the activation enthalpy and entropy, respectively. The attempt frequency can beapproximately estimated as 1 x 106 s-1 from previous RNA folding studies,44-45 but its actual
value is dependent on the structure of the free energy landscape. As a consequence, theactivation entropy ΔS‡ is not determined absolutely, though any reported differential changes inthis entropy (ΔΔS‡), for instance, due to a change in ionic conditions, are rigorously independentof the choice of ν.
7.3.6 Prediction of FRET efficiencies
FRET efficiencies of the proposed G-quadruplex structures are predicted by combining
structural data on G-quadruplex folds with the worm-like chain (WLC) model47 applied tosingle-stranded regions of the DNA construct. The WLC result for the average square end-to-end
distance 〈R2〉 of a polymer of contour length L and persistence length L0 is
𝐿〈𝑅 2 〉 = 2𝐿0 𝐿 [1 − 0 (1 − 𝑒 −𝐿/𝐿0 )].
𝐿(Eq. 7.10)

Literature-reported values for single-stranded polyT at 100 mM ionic strength48-49 vary from L =5−7 Å per nucleotide (nt) and L0 = 15−25 Å. If we assume average values of <L> = 6 ± 1 Å/nt

200
and L0 = 20 ± 5 Å, the dT8 spacer in the single-molecule construct would be expected to exhibit aroot-mean-square (RMS) end-to-end distance of Rrms = 35 ± 5 Å.
We then consider states with a fully folded G-quadruplex, for which there are multiple
folding topologies. For a G-quadruplex topology in which the first and last guanine residue arepart of the same G-tetrad, as in an antiparallel G-quadruplex fold, the Cy3 and Cy5 fluorophores
will be positioned especially close to each other (here labeled the proximal configuration). For
such a proximal topology, the average Cy3-Cy5 distance is approximately the length of the dT8
ssDNA spacer in the construct (Rrms = 35 ± 5 Å), yielding an expected FRET efficiency of
Eproximal = 0.89. On the other hand, a topology in which the first and last guanine residues are on
opposing sides of the G-quadruplex, such as a parallel topology or a hybrid topology, the Cy3
and Cy5 are further separated by the G-quadruplex fold (labeled distal topology). Based on the
crystal structure of a similar 3-tetrad G-quadruplex sequence as reference (PDB 1K8P),17 the Gquadruplex fold in the smFRET construct may be approximated as a rigid cylinder of diameter18 Å and length 7 Å, and adding the cylinder length to the Rrms of dT8 yields Edistal = 0.75. We
approximate the fully unfolded GQ as having the same persistence length as polythymine to give
EFRET = 0.07, with the accuracy of this assumption limited by sequence dependence of the
ssDNA persistence lengths.

7.4

Results

7.4.1 G-quadruplex smFRET construct exhibits multistate folding
To probe the thermodynamics of G-quadruplex folding at the single-molecule level, we
have designed a fluorophore-labeled DNA construct for FRET microscopy based on the
smFRET construct used by Tippana et al. (Figure 7.1A).32 The folding domain of the construct

201

Figure 7.1 (A) G-quadruplex smFRET construct design, labeled with Cy3 and Cy5 for FRET
detection of conformational dynamics. (B) Sample smFRET trajectory taken in buffer with 100
mM Na+. Fitted state trajectory from hidden Markov modeling with three states overlaid in orange.
(C) FRET histogram (blue) aggregated from N = 83 trajectories. Gaussian fits (black) are used to
the FRET efficiencies and relative abundance of the three states. (D) Cartoons indicate putative
structures of the three states: a G-triplex (EFRET = 0.44) and two G-quadruplex topologies, one with
Cy3 and Cy5 in a distal arrangement (EFRET = 0.73), and one in a proximal arrangement (EFRET =0.97). See text for details.
consists of a repeated six-nucleotide sequence (5’-TAAGGG-3’) which forms a G-quadruplex
with three G-tetrads. The construct is labeled with the Cy3-Cy5 FRET pair38 such that folding of
the G-quadruplex brings Cy3 and Cy5 in closer proximity, increasing the energy transfer
efficiency (EFRET) between the fluorophores. The full construct is surface-tethered by biotinstreptavidin interactions50 to enable observation of the G-quadruplex in a single-molecule
fluorescence microscope. Temporal trajectories in 100 mM Na+ buffer reveal multiple, distinct

202
EFRET states which interconvert/equilibrate on the 100 ms to second timescale (Figure 7.1B). A
histogram of EFRET values gathered from N = 83 molecules is well-fit as a sum of three Gaussian
distributions (Elow = 0.44, Emiddle = 0.73, Ehigh = 0.97), which corresponds to three constructconformations with resolved interfluorophore distances (Figure 7.1C). This observation ofconformational polymorphism is consistent the work of Tippana et. al on the same G4-forming
sequence which also exhibited the formation of three FRET states.32
Structural assignment of the three observed FRET states is performed by comparison topredicted EFRET values for the smFRET construct under several different G-quadruplex folding
topologies. Our approach is to combine structural data on G-quadruplex folding with a wormlike chain (WLC) treatment of single-stranded regions of the construct (see Methods). Thismodel is simple but provides a reasonable starting point for consideration of possible construct
structures, which could be refined by methods such as molecular dynamics simulations.51 The Gquadruplex in the construct can fold to place the Cy3 and Cy5 fluorophores on the same side of
the G-quadruplex (a proximal configuration, as in an antiparallel topology) or on opposite sidesof the G-quadruplex (a distal configuration, as in a parallel or hybrid topology), leading to two
distinguishably different FRET efficiencies (Eproximal = 0.89, Edistal = 0.75). We next consider
contributions from the fully unfolded G-quadruplex construct, in which the Cy3 and Cy5 areseparated by a 29-nucleotide ssDNA region, but the calculated FRET efficiency (Eunfolded = 0.10)
is much smaller than the lowest FRET states experimentally observed (Elow = 0.44). Therefore,
we instead consider a partially (un)folded construct in which one of the G-tracts does notparticipate in the G-quadruplex, also called a G-triplex (G3). Previous work by Hou et al.
demonstrates that G3 structures are populated, metastable folding intermediates in smFRET
constructs containing G-quadruplex sequences.15 We calculate a FRET efficiency EFRET = 0.40

203
or 0.46 for the G3 in a proximal or distal topology, respectively. Therefore, we tentatively assign
the Elow state as G3 (T), the Emiddle state as parallel/hybrid G4 (P), and the Ehigh state as
antiparallel G4 (A) and include these three putative structures in Fig. 1D.
7.4.2 Sodium ions drive the folding equilibrium to the parallel/hybrid topology
We first explore G-quadruplex folding as a function of Na+ concentration at ambient
temperature (23 °C), which from the histograms in Figure 7.2A reveal an overall shift from theG3 state to the parallel/hybrid and antiparallel G4 topologies with increasing [Na+]. Integration
over these histogram peaks yields the subpopulations over 50−300 mM Na+ concentration (Fig.
2B), which can then be fit globally to a 3-state Hill model (Eqs. 7.4−7.6) that explicitly includes
all 3 interconversion routes. The least squares fits to such a multistate model are shown as solidlines in Fig. 2B, yielding the stoichiometry for Na+ binding (Δn) and associated dissociation
constants (Kd) for each interconversion pathway, as summarized in Table 7.1. Interestingly, theG3 state gains only one Na+ ion when folding into either G4 state (ΔnT→P ≈ ΔnT→A ≈ 0.87 ±0.09), though Na+ has a 2-fold stronger affinity for parallel/hybrid G4 (Kd = 132 ± 9 mM) thanfor antiparallel G4 (Kd = 228 ± 26 mM). Conversely, there is no difference within experimental
uncertainty in the number of Na+ bound to the two G4 conformations. With the G4 construct

Elow → Emiddle
Elow → Ehigh
Emiddle → EhighΔnKd (mM)ΔnKd (mM)ΔnKd (mM)Na+
0.87 ± 0.09
132 ± 90.87 ± 0.09
228 ± 26
0.00 ± 0.09—

K+
-0.05 ± 0.05—
0.62 ± 0.05
1.9 ± 0.2
0.67 ± 0.07
2.9 ± 0.3

Table 7.1 Results of fitting G-quadruplex population data vs. [Na+] and [K+] to 3-state Hill model.
Uncertainties are reported as standard errors of the mean. For processes with Hill coefficients (Δn)
equal to zero within uncertainty, dissociation constants (Kd) are not reported.

204

Figure 7.2 Influence of Na+ on G4 folding equilibrium. (A) FRET histograms at 50, 100, and
200 mM Na+. (B) Population of the three G4 conformational states in [Na+] = 50−300 mM. Fits
to a 3-state Hill model shown as solid curves.

205
populating both the parallel/hybrid and antiparallel topologies even at [Na+] = 300 mM, it is clearthat Na+ has only a modest impact on G4 polymorphism.
Additional mechanistic insight can be obtained from kinetic analysis of the Na+dependence of G-quadruplex folding events in the time domain. Rate constants for folding aredetermined by hidden Markov modeling (HMM) of smFRET trajectories52-54 using a 3-statemodel which includes all 6 rate constants describing forward and reverse interconversion
between each FRET pair (Figure 7.3A). The rate constants for G4 folding as a function of [Na+]= 50−300 mM are shown in Figure 7.3B. As a rigorous check on the validity of these extracted
rate constants, we can calculate the free energy sum around the three-state cycle,𝑘𝑘𝑘∆𝐺𝑐𝑦𝑐𝑙𝑒 = −𝑘𝑇 ln (𝑘𝑇→𝑃 𝑘𝑃→𝐴𝑘𝐴→𝑇 ),𝑃→𝑇 𝐴→𝑃 𝑇→𝐴(Eq. 7.11)

at each Na+ concentration and plot the values in Figure 7.3C. By detailed balance, the free energy
sum around any cycle of states must equal zero within uncertainty, and indeed, the calculated
ΔGcycle/kT values vanish at all Na+ concentrations. In Figure 7.3B, the rate constants for theconversion of G3 to and from G4 conformations have the largest magnitude, while transitions
directly between the parallel/hybrid and antiparallel G4 topologies are a minor, though notforbidden, kinetic process. Dependence on [Na+] is most evident in the “forward” folding rate
constants, kT→P and kT→A, while the “reverse” unfolding rate constants, kP→T and kA→T, appear
less impacted by Na+ concentration.
To additionally explore the effects of Na+ cation on G-quadruplex folding, we havefurther developed an explicit 6-state kinetic model for Na+ binding to the G-quadruplex (Figure
7.3D). The model consists of three DNA conformations, each with ligand-free and ligand-boundforms. In accordance with 1:1 Na+ binding stoichiometry from the Hill fit of the population data
(Table 7.1), the kinetic model includes a single Na+ ion in each ligand-bound state. Ligand

206

Figure 7.3 Influence of Na+ on G4 folding kinetics. (A) Left schematic: structure of hidden
Markov model (HMM) used to fit smFRET trajectories to obtain G4 folding rate constants. Right
schematic: the orientation of the 3-state cycle used to calculate the free energy change around theloop, ΔGcycle = ΔGT→P + ΔGP→A + ΔGA→T. (B) The rate constants for G4 folding in 50−300 mM
Na+. Fits to the 6-state Na+-binding model depicted in (D) are shown as solid curves. (C) Plot ofΔGcycle at each [Na+]. To satisfy detailed balance, ΔGcycle should equal zero. (D) 6-state binding
model for Na+ influence on G4 folding kinetics. The dominant pathways through this network areshown as red arrows, which coincide with a “bind-then-fold” mechanism.

207
binding is assumed to happen rapidly compared to G4 conformational dynamics, which allows
all ligand-binding steps to be treated as a quasi-equilibrium governed by a single binding
constant[𝐶 ]𝑖
𝐾𝑑,𝑖 = [𝐶 ∙𝐿][𝐿],(Eq. 7.12)

𝑖where i labels the conformational state (triplex T, parallel/hybrid G4 P, or antiparallel G4 A), Ci
is the ligand-free conformation, Ci·L is the ligand-bound conformation, L is the ligand (in thiscase, Na+), and square brackets indicate concentration of the enclosed species. The fraction ofligand-bound molecules is therefore[𝐶 ∙𝐿][𝐿]𝑖𝜃𝑏𝑜𝑢𝑛𝑑,𝑖 = [𝐶 ]+[𝐶
= [𝐿]+𝐾 .
∙𝐿]𝑖𝑖(Eq. 7.13)

𝑑,𝑖Transitions between conformations are governed by first order rate constants depending on
ligand occupancy:
𝑘𝑖𝑗,𝑎𝑝𝑜𝐶𝑖 →𝑘𝑖𝑗,ℎ𝑜𝑙𝑜𝐶𝑖 ∙ 𝐿 →𝐶𝑗(Eq. 7.14)

𝐶𝑗 ∙ 𝐿,(Eq. 7.15)

where kij,apo and kij,holo are rate constants for transition from state i to state j in the ligand-free(“apo”) and ligand-bound (“holo”) forms, respectively. The observed rate constants kij,obs fortransitions between conformational states are population-weighted averages of the apo and holo
rate constants:[𝐿]𝑘𝑖𝑗,𝑜𝑏𝑠 = 𝑘𝑖𝑗,𝑎𝑝𝑜 (1 − [ ]𝐿 +𝐾𝑑,𝑖[𝐿]) + 𝑘𝑖𝑗,ℎ𝑜𝑙𝑜 ([ ]𝐿 +𝐾𝑑,𝑖).

(Eq. 7.16)

For the 6-state kinetic model in Figure 7.3D, Eq. 7.16 forms a system of 6 equations, where thenumber of free parameters in the model can be reduced from 15 to 11 by imposing detailedbalance constraints (ΔGcycle = 0) on all cycles. The resulting constrained equations are thenglobally fit to the kinetic data in Figure 7.3C, as summarized in solid line fits.

208
The kinetic behavior G-quadruplex folding reveals that Na+ drives folding by binding to
the G3 state which subsequently folds into either G4 conformation in a “bind-then-fold”
mechanism (Figure 7.3D, pathway highlighted in red).55-57 There are several key indicators insupport of this mechanism: first, the forward folding rate constants kT→P and kT→A vanish at low
Na+ (Figure 7.3B, filled red and blue symbols). Folding, evidently, is impossible without Na+(kTP,apo = kTA,apo = 0 s-1). Additionally, kT→P and kT→A saturate at high [Na+] (kTP,holo = 0.7 ± 0.3 s1, kTA,holo = 1.0 ± 0.4 s-1), which indicates that binding of Na+ to G3 is driven to completion.

However, there is a clear albeit weaker dependence of the unfolding rate constants kP→T and kA→T
on [Na+], so other pathways besides the bind-then-fold pathway are permitted to a lesser degree.
7.4.3 K+ has a much higher affinity than Na+ for the G-quadruplex and exclusively
promotes the antiparallel topology
Formation of G-quadruplexes is known to be particularly sensitive to K+, which we next
explore to measure K+ vs Na+ specific effects on G4 folding (Figure 7.4). Interestingly, K+ drives
the conformational equilibrium of the G4 smFRET construct exclusively toward the antiparallelG4 topology (Figure 7.4A), in essence by depleting both the G3 and parallel/hybrid G4 states.
Furthermore, this dramatic increase in folding occurs at only a few mM K+ concentrations, incontrast to the 100-fold higher concentrations of Na+ required (Figure 7.2B). Clearly, the G4
construct has a much higher affinity for K+ than Na+ but only in achieving the antiparallel G4
conformation. Fitting the full suite of K+ dependent population data to a 3-state Hill model
quantitatively confirms these qualitative observations, as summarized in Table 7.1. In this case,no K+ uptake occurs during the transition from the triplex to the parallel/hybrid G4 (ΔnT→P = 0.05 ± 0.05), which indicates that K+ does not participate in this process. Meanwhile,
approximately one K+ binds to during folding into the antiparallel G4 (ΔnT→A = 0.62 ± 0.05)

209

Figure 7.4 Influence of K+ on G4 folding. State populations (A) and folding rate constants (B)determined in 100 mM Na+ with 0−20 mM K+. Data are fit to a 3-state Hill model in (A) and a 6state ligand binding model in (B). (C) Calculation of ΔGcycle at each K+ concentration to check
whether the rate constants satisfy detailed balance. (D) 6-state model of K+ binding to G4. Red
arrws indicate the highest flux pathways (“fold-then-bind” mechanism).

210
with a binding affinity (Kd = 1.9 ± 0.2 mM) that is 2 orders of magnitude greater than Na+ inachieving the same conformational state (Kd = 228 ± 26 mM). That the Hill coefficient is lessthan unity indicates that the antiparallel G4 can form in the absence of K+ cation, consistent withthe observation of antiparallel G4 in Na+-only buffer (Figure 7.2). By driving the G4 constructexclusively to the antiparallel topology, K+ exerts much stronger control of G4 polymorphism
than Na+.
Similar to analysis of the Na+ kinetics, smFRET trajectories are subjected to hidden
Markov modeling to extract rate constants as a function of [K+] (Figure 7.4B). As before, we canrigorously confirm these kinetic results by verifying that the rate constants explicitly obey
detailed balance (ΔGcycle = 0) at all K+ concentrations (Figure 7.4C). Five of the six folding rate
constants show little to no dependence on K+ concentration, while the rate constant for unfolding
from the antiparallel G4 to G3 (kA→T) is strongly suppressed by the presence of K+. To explore
the mechanistic implications of these data, the folding rate constants are fit to the 6-state kinetic
model (Eq. 7.16) and overlaid on Figure 7.4B as solid lines. The fit correctly predicts that kA→T
goes to zero in the high [K+] limit (kAT,holo = 0.01 ± 0.03 s-1), consistent with a conformational
selection (CS, “fold-then-bind”) model in which the apo DNA pre-folds to a transient antiparallelG4 state before binding K+ with high affinity (Kd,A = 2.6 ± 1.2 mM) to stabilize the antiparallel
configuration. This K+-induced folding pathway (highlighted in Figure 7.4D) contrasts strongly
from the induced fit (IF, “bind-then-fold”) mechanism seen for Na+-stabilized formation of theG4 quadruplex and originates from a much stronger and topology-specific binding interaction.
7.4.4 Temperature-dependence of G-quadruplex folding
Temperature-controlled single-molecule experiments can be used to deconstruct theoverall free energy of G-quadruplex folding into enthalpic and entropic contributions.43 The

211
temperature-dependence of the folding rate constants and equilibrium constants in 100 mM Na+
over a temperature range of 13−29 °C is captured in Figure 7.5. We note that this is, to ourknowledge, the first use of smFRET to examine temperature-dependent kinetics of a system withmore than two conformations. We observe first that the majority of folding events start or end in
the G3 state (Fig. 5A, B, left), with folding/unfolding between the parallel/hybrid G4 andantiparallel G4 being a minor kinetic process at all temperatures (Fig. 5C, left). Second, theequilibrium constants for folding from the G3 state to both G4 states (Keq,TP and Keq,TA) decreaseat high temperature, which shows that the overall folding process is enthalpically favorable. The
kinetic origin of this behavior can be understood by examining the corresponding rate constants:

Figure 7.5 Temperature-dependence of G4 folding rate constants (left panels) and equilibrium
constants (right panels) in 100 mM Na+ for (A) G3 → parallel/hybrid G4, (B) G3 → antiparallel
G4, and (C) parallel/hybrid G4 → antiparallel G4. Lines indicate Arrhenius and van’t Hoff fits forrate constants and equilibrium constants, respectively, which are used to extract the enthalpy andentropy change for each process.

212
the forward rate constants (kT→P and kT→A) are largely temperature insensitive, indicating
negligible activation enthalpy, while the reverse rate constants (kP→T and kA→T) are strongly
increased at high temperature, indicating an enthalpic barrier for unfolding.
A more quantitative analysis is carried out by fitting rate and equilibrium constants toArrhenius and van’t Hoff models, respectively, to determine overall and transition stateenthalpies (ΔH0, ΔH‡, from slopes) and entropies (ΔS0, ΔS‡, from intercepts) from the linear fits.
The thermodynamic values obtained from such 3-state fits can in turn be used to construct 3D
folding energy landscapes (see Fig. 6) which depicts the enthalpy, entropy, and free energy foreach of the G-quadruplex conformations and the intervening transition states between them. The
enthalpic landscape (Figure 7.6A) shows that both folding processes (G3→ parallel/hybrid G4
and G3 → antiparallel G4) are enthalpically favorable (ΔH°TP = −66 ± 7 kJ/mol, ΔH°TA = −84 ±9 kJ/mol) with a negligible barrier (ΔH‡TP = −4 ± 6 kJ/mol, ΔH‡TA = −13 ± 6 kJ/mol).
Meanwhile, the enthalpy landscape (Figure 7.6B) indicates that folding is entropically costly(ΔS°TP = −240 ± 20 J/mol K, ΔS°TA = −300 ± 30 J/mol K), where the entropic penalty is partiallypaid upon reaching the transition state (ΔS‡ TP = −140 ± 20 J/mol K, ΔS‡ TA = −170 ± 20 J/mol
K). The overall free energy changes for folding (Figure 7.6C) are small in comparison to theindividual contributions from enthalpy (ΔH) and entropy (−TΔS) which nearly cancel, an
example of enthalpy-entropy compensation.58-59
7.4.5 Thermodynamics of G-quadruplex folding at elevated monovalent cation
concentrations
We have repeated these temperature-dependent studies of the G-quadruplex undermultiple cation conditions to probe the effects of Na+ and K+ cations on the folding
thermodynamics. The temperature dependence for G-quadruplex folding under ambient (100

213

Figure 7.6 Folding landscape for G4 in 100 mM Na+ as determined by van’t Hoff and Arrhenius
analyses. (A) Enthalpy of each stable conformation (black points) and transition state (grey points)
defined relative to the G3 state. (B) same as (A), but for the enthalpic contribution to the freeenergy (−TΔS) evaluated at 300 K. (C) same as (A), but for the free energy (ΔG = ΔH −TΔS)
evaluated at 300 K.

214mM) Na+, high (200 mM) Na+, and 100 mM Na+ with 4 mM K+ are shown in Figure 7.7, where
the latter two data sets can be compared to the first to isolate impact due to Na+ and K+,
respectively. Arrhenius and van’t Hoff analyses are performed on the datasets, and the fitted
entropic and enthalpic parameters for the three cationic conditions (100 mM Na+, 200 mM Na+,and 100 mM Na+ with 4 mM K+) are shown side-by-side in Table 7.2.
The dependence of folding enthalpies and entropies on cation concentration can be used
to determine the thermodynamic signature of Na+ and K+ binding. For example, consider thechange in the folding enthalpy of the G3 to antiparallel G4 transition upon an increase in [Na+]from 100 mM to 200 mM, defined as∆∆H°𝑇𝐴 (Na+ ) ≡ ∆H° 𝑇𝐴 (200 mM Na+ ) − ∆H° 𝑇𝐴 (100 mM Na+ ).

(Eq. 7.17)

Figure 7.7 Temperature-response of G4 folding under multiple cationic conditions (100 mM Na+
in black, 200 mM Na+ in red, and 100 mM Na+ with 4 mM K+ in blue) for the three G4 dynamical
processes: (A) G3 → parallel/hybrid G4, (B) G3 → antiparallel G4, and (C) parallel/hybrid G4 →antiparallel G4. Arrhenius plots for the forward and reverse rate constants are shown in the left
and middle panels, respectively, while the van’t Hoff plot for the equilibrium constants are in theright panel.

215
Our results show that this quantity is positive in value (ΔΔH°TA(Na+) = 40 ± 11 kJ/mol). Na+interacts with the G4 construct through a bind-then-fold mechanism, for which ΔΔH°TA(Na+) is
opposite in sign to the enthalpy of Na+ binding, ΔH°bind(Na+), as depicted in Figure 7.8A.
Therefore, since ΔΔH° TA(Na+) > 0, we infer that Na+ binding is enthalpically favorable(ΔH°bind(Na+) < 0). We also observe that the Na+-induced change in folding entropy for thisprocess is positive (ΔΔS°TA(Na+) = 140 ± 40 J/mol K), so by a similar logic Na+ binding must
ΔH (kJ/mol)
G3 →G4 (parallel/hybrid)
G3 →G4 (antiparallel)
G4 (parallel/hybrid)→
G4 (antiparallel)

100 mM Na+

200 mM Na+

100 mM Na+, 4 mM K+

ΔH°TP-66 ± 7-16 ± 9*-12 ± 14*

‡

ΔH TP

-4 ± 629 ± 6*

8±6

‡

ΔH PT

62 ± 245 ± 7*20 ± 13*ΔH°TA-85 ± 9-44 ± 7*-56 ± 6*‡

ΔH TA

-13 ± 621 ± 4*18 ± 4*ΔH‡AT

72 ± 766 ± 674 ± 4

ΔH°PA-34 ± 40-40 ± 25

-29 ± 24‡

ΔH PA

9 ± 2328 ± 19-5 ± 18ΔH‡AP

43 ± 33

68 ± 1525 ± 16

100 mM Na+

200 mM Na+

100 mM Na+, 4 mM K+

ΔS°TP-240 ± 20-50 ± 30*-50 ± 50*

ΔS‡TP-140 ± 20-30 ± 20*-100 ± 20

‡

ΔS PT

90 ± 1030 ± 20*-60 ± 40*

ΔS°TA-300 ± 30-150 ± 20*-190 ± 20*

‡

ΔS TA

-170 ± 20-50 ± 10*-70 ± 10*ΔS‡AT

130 ± 20100 ± 20120 ± 10

ΔS°PA-110 ± 140-140 ± 80-90 ± 80‡

ΔS PA

-100 ± 80-40 ± 70-160 ± 60

ΔS‡AP

10 ± 110100 ± 50-60 ± 50ΔS (J/mol K)
G3 →G4 (parallel/hybrid)
G3 →G4 (antiparallel)
G4 (parallel/hybrid)→
G4 (antiparallel)

Table 7.2 Thermodynamic parameters for G-quadruplex folding determined by van’t Hoff andArrhenius analysis of temperature dependent kinetic measurements at multiple cation
concentrations (Figure 7.7). Uncertainties are reported as standard errors of the mean. Asterisks
indicate statistical significance (evaluated at the p = 0.01 level) of parameters when compared to
the 100 mM Na+ case.

216

Figure 7.8 Determination of cation binding thermodynamics. (A) For Na+, which follows a bindthen-fold model, the measured enthalpy/entropy ΔH/S at low [Na+] contains contributions fromboth Na+ binding (ΔH/Sbind) and the conformational change (ΔH/Sconf). However, at high [Na+],the G3 conformation saturates with Na+, and ΔH/S reflects only ΔH/Sconf. Therefore, ΔΔH/S =ΔH/S(high [Na+]) – ΔH/S(low [Na+]) has the opposite sign of ΔH/Sbind. (B) For K+, which operates
by a fold-then-bind mechanism, the differential enthalpy/entropy ΔΔH/S = ΔH/S(high [K+]) –ΔH/S(low [K+]) has the same sign of ΔH/Sbind.
result in a loss of entropy (ΔS°bind(Na+) < 0). Together, ΔH°bind(Na+) and ΔS°bind(Na+) indicatethat Na+ binding is characterized by the formation of energetically stable contacts with G3 at theentropic cost of restricting the translational freedom of Na+ and/or the conformational flexibilityof the triplex.
Similarly, the thermodynamics of K+ binding to the antiparallel G4 can be determined byexamining the differential folding enthalpy (ΔΔH°TA(K+)) and entropy (ΔΔS°TA(K+)) upon
addition of K+, in this case defined as∆∆H°𝑇𝐴 (𝐾 + ) ≡ ∆H° 𝑇𝐴 (100 mM Na+ , 4 mM K + ) − ∆H° 𝑇𝐴 (100 mM Na+ ) (Eq. 7.18)

217
and
∆∆S°𝑇𝐴 (𝐾 + ) ≡ ∆S° 𝑇𝐴 (100 mM Na+ , 4 mM K + ) − ∆S° 𝑇𝐴 (100 mM Na+ ). (Eq. 7.19)
In contrast to Na+, K+ interacts with the G-quadruplex through a fold-then-bind mechanism, andas a result ΔΔH°TA(K+) and ΔΔS°TA(K+) have the same sign as ΔH°bind(K+) and ΔS°bind(K+),
respectively (Figure 7.8B). The observed ΔΔH°TA(K+) and ΔΔS°TA(K+) are both positive, whichindicates that K+ binding is enthalpically unfavorable (ΔH°bind(K+) > 0) and entropicallyfavorable (ΔS°bind(K+) > 0). Interestingly, this thermodynamic signature is exactly the opposite ofthat for Na+-binding, a fact which reinforces the wholly distinct nature of Na+ and K+ binding to
the G-quadruplex construct.

7.5

Discussion
In this work, we have used the 5’-GGG-(TAAGGG)3-3’ DNA G-quadruplex (TAA-G4)

as a model system for understanding the thermodynamic influence of monovalent cations Na+
and K+ on G-quadruplex folding. The TAA-G4 is incorporated into a DNA construct designed
for single-molecule FRET (smFRET) microscopy. The observed smFRET data reveal that the
TAA-G4 populates three interconverting FRET states (Figure 7.1) in agreement with previous
single-molecule measurements by Tippana et al. on a similar TAA-G4 construct.32 From thesmFRET trajectories, equilibrium constants and rate constants for transitions between the threestates are measured, and the dependence of these quantities on cation concentration andtemperature have been obtained.
We used simple structure prediction to assign structures to the FRET states of the TAAG4 construct. The middle and high FRET state efficiencies (Emiddle = 0.73, Ehigh = 0.97) areconsistent with folded G-quadruplexes with the fluorophores positioned on the opposite side

218
(distal) or the same side (proximal) as the G-quadruplex fold, respectively. The proximal
configuration would be observed in a G-quadruplex with antiparallel topology (chair or basket),while the distal configuration corresponds to the parallel and hybrid topologies which are
indistinguishable in our experiment. For an assignment of the low FRET state (Elow = 0.44), we
considered but ruled out a fully unfolded G-quadruplex for two reasons. First, the predicted
FRET efficiency is too low (Eunfolded ≈ 0.10). Second, from the crystal structures of similar Gquadruplexes containing 3 G-tetrads,10, 16-17 we expect the folded G-quadruplex to bind twomonovalent cations, while the fully unfolded state should bind none. Therefore, the binding
stoichiometry should be two ions; however, the measured Na+ and K+ binding stoichiometries(Figure 7.2B and Figure 7.4A) are more consistent with single ion binding. Therefore, we instead
propose a G-triplex (G3) structure as a partially folded species whose predicted FRET efficiency(Etriplex = 0.40–0.46) is much closer to Elow. Furthermore, G3 structures have been reported tobind to a single monovalent cation,14 which would produce the correct binding stoichiometry.
Nevertheless, alternative structural assignments are certainly plausible due to the uncertainty in
our simple EFRET calculations, especially for less-structured conformations such as the G3 state
for which inaccuracies in the worm-like chain model will be magnified.
Monovalent cations drive the TAA-G4 folding equilibrium from G3 into G4 states in acation-specific fashion. K+ exclusively promotes the antiparallel configuration (Figure 7.4A),
while Na+ prefers the parallel/hybrid G4 with some sampling of the antiparallel G4 (Figure 7.2),
i.e., the TAA-G4 is slightly polymorphic in Na+ solution. This result is surprising in light ofstudies showing that G-quadruplexes adopt parallel or hybrid topologies in K+ solution andantiparallel topologies in Na+ solution.10, 16 As stated above, it is possible that the FRET
efficiencies have been incorrectly predicted by our simple structural model; however, it is219
unlikely that the order of the FRET efficiencies is incorrect (i.e. Eantiparallel > Eparallel/hybrid).
Instead, this disagreement is more likely due to the exquisite sensitivity of G-quadruplex folding
equilibria to their nucleic acid context, i.e., loop sequence, adjacency to single-stranded or
double-stranded nucleic acids, composition (RNA vs. DNA), and strand polarity (see ref. 15 foran especially dramatic demonstration of this sensitivity). Here, we are using the relatively
unstudied TAA loop sequence which is especially polymorphic,32 and we have introduced an
ssDNA spacer between the TAA-G4 and the hybridizing sequence for surface tethering, unlikethat used in the Tippana et al. construct upon which ours is based. Therefore, the structural
impact of K+ and Na+ on the TAA-G4 construct may differ from that of other G-quadruplex
constructs in the literature.
Measurement of concentration-dependent kinetics (Figures 7.3C and 7.4C) reveal that
Na+ and K+ promote G-quadruplex folding through distinct kinetic mechanisms. Na+ binds to theG-triplex before the ligand-bound complex folds into one of the G-quadruplex configurations(Figure 7.3D, a “bind-then-fold” mechanism). In contrast, the TAA-G4 sequence must first foldinto the antiparallel G4 topology before K+ will bind to form a stabilized complex (Figure 7.4D,
a “fold-then-bind” mechanism). That these two monovalent cations act on the TAA-G4 in suchdifferent mechanisms reinforces how sensitive G-quadruplexes are to their environment, or inthis case, cation size. We present a simplified, combined kinetic model of Na+- and K+-induced
folding of G-quadruplexes in Figure 7.9. In the model, Na+ binding is a prerequisite for triplex toquadruplex transitions. Then, the antiparallel configuration of the G4 can subsequently bind a K+
ion, perhaps by replacement of Na+ (as shown in the figure) or by attachment to an alternative
binding site. It is possible and even likely that the identified Na+-binding site has a similaraffinity for other monovalent cations, including K+, but the much higher affinity of the K+-

220

Figure 7.9 Proposed scheme of Na+ and K+ binding to TAA-GQ.
specific binding site means that our K+ titrations saturate before the low affinity binding to thetriplex can be observed.
To learn about the influence of Na+ and K+ on the thermodynamics of TAA-G4 folding,we used temperature-dependent experiments to obtain the folding enthalpies and entropies atmultiple cation concentration conditions. By considering the change in thermodynamic
parameters after an increase in [Na+] or [K+], we can infer the thermodynamics of cation binding.
Interestingly, the folding thermodynamics of the TAA-G4 construct respond similarly to theaddition of either 100 mM Na+ or 4 mM K+, as summarized in Table 7.2. However, because Na+
and K+ interact with the TAA-G4 through different mechanisms (bind-then-fold vs. fold-thenbind), we conclude that thermodynamic signature of Na+ binding is the opposite of that for K+
binding (Figure 7.8). Specifically, Na+ binds to the G3 conformation driven by enthalpicallyfavorable interactions (ΔH°bind(Na+) < 0) that outweigh a loss of entropy (ΔS°bind(Na+) < 0)which is perhaps due to G-triplex conformational restriction upon ion binding. In contrast, K+
binds to the antiparallel G4 topology in an entropically driven process (ΔS°bind(K+) > 0) that
incurs an enthalpic cost (ΔH°bind(K+) > 0). One explanation of this thermodynamic signature isthat K+ replaces the poorly fitting Na+ in the binding pocket to form a more compact Gquadruplex which results in the expulsion of surface bound water molecules. Such a release ofwater molecules would be entropically favorable but energetically costly due to breaking

221hydrogen bonds between the water molecules and the nucleic acid. This hypothesis of Gquadruplex compaction and water release upon K+ binding is generally supported by literature onion dependence of G-quadruplex folding volumes,60-61 though one study reported that K+
produces larger G4s than Na+.21 Clearly, further results are needed to clarify this issue, such as
could be obtained by pressure-dependent single-molecule folding experiments.62-63

7.6

Summary and Conclusion
In this work, we have explored the influence of monovalent cations on G-quadruplex

folding through single-molecule, temperature-controlled kinetic measurements. To ourknowledge, this represents the first use of smFRET to measure the temperature-dependent
folding kinetics on a system with more than two FRET conformations. We observe 3-state Gquadruplex folding, which we ascribe to conformational transitions between a G-triplex and two
G-quadruplex topologies. Concentration-dependent measurements show that the Na+ binds to theG-triplex and remains bound while G-quadruplex topologies are sampled. In contrast, K+ binds
exclusively to the antiparallel G-quadruplex conformation and has a binding affinity
approximately 100 times larger than that of Na+. The thermodynamics of Na+ binding areenthalpically-driven (ΔH°bind(Na+) < 0, ΔS°bind(Na+) < 0), while entropy is the driving force of
K+ binding (ΔH°bind(K+) > 0, ΔS°bind(K+) > 0), likely due to G4 compaction and concomitantwater release upon K+ binding.

222
7.7

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227
Chapter 8
Postmortem

8.1

Introduction
In professional chess, at the end of a game between two masters, there is a tradition called

the “postmortem” analysis. The two opponents, after hours of exhausting mental effort spent inadversarial silence, come together for a friendly discussion of what happened during the game.
They talk about the mistakes they made, the places they could have improved, the critical
moments that shaped the outcome. In fact, most of what they discuss were moves not played in
the game. They look at the hypothetical directions, the “what-if’s” of where the game could havegone had they made different choices.
So far, this dissertation has only included research which ‘worked’, the projects which
produced results that could pass the inspection of peer review and merited presentation to thescientific community. In this concluding chapter, I will perform a ‘postmortem’ analysis of mygraduate career, exploring the “what-if’s” of my research. These are the efforts which nevermade it to the literature, the partially completed projects with inconclusive or confounding
results, and the possible research paths that I did not have the time to travel along. I write this in
the hope of inspiring some future researcher to take up these lost projects, someone with theexpertise which I lacked, or the insight which eluded me, or simply the time which I ran out of.
8.2

Determination of folding volume change with molecule dynamics (MD)During my time in the Nesbitt group, Hsuan-Lei Sung, a graduate student, implemented

pressure-controlled single-molecule FRET measurements. Nucleic acids can experience a change
in volume during folding, which affects the Gibbs free energy via pressure volume work:

228𝜕(∆𝐺°)∆𝑉°=−
𝜕𝑃
𝑅𝑇
Where ΔG° is the folding free energy, P is the pressure, ΔV° is the folding volume, R is the gas
constant, and T is the absolute temperature. These folding volumes are typically small (on theorder of the volume of one water molecule), so the contribution to ΔG° at atmospheric pressureis usually negligible. However, at elevated pressures, such as those in the deep ocean, thepressure-volume work is significant. By changing the pressure in smFRET experiments, HsuanLei Sung has been able to measure the volume change for various nucleic acids.1-2 One of theseis the “40-adenine hairpin” used in Chapter 3, for which ΔV° was measured as +23 ± 2 mL/mol.
The positive sign indicates that the folded state of the hairpin is larger than the unfolded state, asomewhat counterintuitive result which is nevertheless generally true for nucleic acids as well asproteins.
After using molecular dynamics (MD) to study the chirality-dependent binding of aminoacids to nucleic acids (Chapter 4), and inspired by Hsuan-Lei’s results, I set out to measure ΔV°
for the 40-adenine hairpin using MD. Rather than measure pressure dependence of the folding
free energy, which is inaccessible to short time MD simulations, my strategy was to use the factthat volume is a readily available quantity in MD to instead measure the absolute volume of thefolded and unfolded configurations and take the difference. Specifically, I first created the folded
40-adenine hairpin structure using the RNAComposer webserver, converted the structure into
DNA, and solvated the system in TIP3P water and 100 mM KCl. I then created an unfolded
structure by generating the linear sequence in the Ambertools program LEaP, collapsing thestructure in 1 M salt for 30 ns, then stripping away the solvent and resolvating so that the folded
and unfolded simulations had exactly the same number of water molecules, ions, etc. (Figure8.1). These simulations were then run in NAMD at constant pressure P = 1 atm, which means

229that the volume of the unit cell for the periodic simulation is changed to maintain a constantpressure. The unit cell volume versus time trajectory output by NAMD were then used to
calculate an average system volume. To estimate uncertainty in the volume, the autocorrelation
function of the volume trajectory was calculated to determine the volume decorrelation time τ,and the standard error of the mean σSEM was calculated from the raw standard deviation in thevolume σV as σSEM = 𝜎𝑉 /√𝑡/𝜏 , where t is the total simulation time. The calculations were also
run 5x in parallel, and the uncertainties derived using the 5 independent simulations were inagreement with σSEM. Usually, ~10-25 ns of total simulation time were sufficient to reach ±5
mL/mol uncertainty, which is approximately ⅓ the volume of a single water molecule.

Figure 8.1 Schematic of strategy for determining nucleic acid folding volume in MD.
The results of the MD calculations yielded a volume change ΔV° = −73 ± 5 mL/mol,which is similar in magnitude to the experimental result but opposite in sign (i.e., in MD the230folded state is larger than the unfolded state). This is a qualitatively incorrect result whichsuggests something quite wrong with the calculation. My advisor and I were particularly
suspicious of the water model, TIP3P, which is a 3-point rigid model. Interaction with solvent isthought to be key in folding volume changes, so an accurate water model may be important.
Unfortunately, despite trying many different water models, including other 3-point models, 4point rigid models, and a flexible water model, every simulation showed a negative value for
ΔV° (Table 8.l). Concerned that the long 40-adening loop in the hairpin might be poorly
converged in MD, I switched to a short 4-adenine loop, but the negative sign remained (ΔV° =−35 ± 4 mL/mol). Finally, I tried the calculation at multiple pressures, for which we expected
ΔV° to be constant, but it was not (Figure 8.2). No closer to understanding these confounding
results, I abandoned this effort. Future attempts may find it fruitful to look at alternative folding
systems, more refined water models including polarizable models, or extending simulation time
to more thoroughly investigate the convergence properties of these calculations.

water Model folded(A3) SEM
SPC/E
43128421
3-point
OPC34598428TIP3P4369855unfolded(A3)431385459955437081

SEM (A3)
17
-101
22
-113
5
-96

SEM
27
23
8

4-point

OPC
TIP4P

4343644247011214434386424753

17
16

-22
-52

20
22

Flexible

SPC/Fw

42732717427427

16

-100

23

+30

10

experiment

Table 8.1 Folding volume changes for 40-adenine hairpin for multiple water models in MD.

231

60

V (A3)

40
20
0
-20
-40
Water Model: OPC

1 atm

3 kbar

6 kbar

9 kbarPressure
Figure 8.2 40-adenine hairpin volume change at different simulation pressures.

8.3

Cold shock protein and mRNAOne under-utilized element of the TIRF system is its ability to cool samples below room

temperature. Cooling was used in this thesis as simply a tool for extending the temperature range
for van’t Hoff and Arrhenius analysis, but there is some interesting biology to explore near thefreezing point of water. A particularly intriguing and relatively simple area to explore is that ofcold shock proteins (CSPs).3-4 When exposed to low temperatures, a cell’s mRNA transcripts
may form unwanted secondary structure elements that are too stable to permit ribosome
processing. CSPs are nucleic acid folding chaperones which are released to destabilize thoseunwanted secondary structures.
The proposed experiment is quite simple: observe secondary structure formation kinetics,
perhaps for a DNA hairpin, using smFRET at low temperatures with and without the presence of
CSPs. Despite its simplicity, this experiment would provide a deeper mechanistic understandingof how exactly CSPs modify RNA folding dynamics than has previously been available. I was232
prevented from attempting this experiment due to the difficulty in obtaining purified CSPs,
which to my knowledge are not commercially available. Collaboration with researchers who areexperienced in protein synthesis may be necessary. Fortunately, most CSPs are relatively shortproteins with fewer than 100 amino acids.
An alternative but related experiment is measuring the folding dynamics of the mRNA
that encodes for the CSP protein. It has been proposed5 that this mRNA has evolved to avoid
forming secondary structure even at low temperatures, which solves the problem of the CSP
protein itself not being able to be synthesized at low temperatures due to secondary structureformation in its mRNA transcript. Again, this is an experiment I have not performed, which
would require in vitro RNA ligation to use the full mRNA transcript, or else selection of asubsequence of interest which is short enough to be commercially synthesized without the needfor ligation. Additionally, relevant FRET pair labeling locations would need to be determined.
8.4

Modulation of RNA folding dynamics by small peptides
As seen in Chapters 3 and 4, even single amino acids can affect RNA folding behavior. In

those chapters, this result was used to propose that, in the history of evolution, the first role ofamino acids in biology was to serve as RNA folding chaperones. The next logical step in thisresearch is to explore whether the evolution of peptide synthesis could have been driven by theability of peptides to act as even better RNA folding chaperones. My preliminary work here islimited to glycine-based peptides of increasing length (Gly2 and Gly3), and my results showedthat diglycine and triglycine have approximately the same influence on folding as glycine. Goingfurther in this area is challenging, as the chemical space of peptides increases in size
exponentially with the peptide length, and exhaustively exploring even the dipeptides would be
extremely challenging. However, there are certain amino acids which are believed to be older

233
member of the codon code than others, and these amino acids are good candidates for
exploration. In particular, the four amino acids glycine, alanine, valine, and aspartic acid would
be a good choice for a subspace of 16 possible dipeptides to explore. Furthermore, beyond aninterest in the evolution of life, there are simple peptides which are of interest for their RNAbinding abilities that may have clinical uses, such as stop codon binders and the HIV TAR
binder.6-7
8.5

Surface tethered kinetic studies of Brome mosaic virus
A less ambitious project could be the re-initiation of the Nesbitt group effort to study the

kinetics of the Brome mosaic virus (BMV) in single-molecule microscopy. A previous
postdoctoral researcher, Dr. Mario Vieweger, created a smFRET construct for diffusing studies
on the confocal microscope.8 From this construct, three FRET conformations were observed, andtheir population dependence on salt-levels was measured. However, no kinetic data was
obtained, as surface tethering was not performed. I restarted this project by tethering an old
BMV construct to a surface to try to observe the three FRET states at E = 0.1, 0.3, and 0.8.
However, my trajectories showed little population of the high FRET state at all salt conditions
used (Figure 8.3). Furthermore, the high FRET states were inconsistent from molecule tomolecule. This suggests that the RNA sample has degraded and is no longer kineticallyhomogenous. Obtaining a new construct would be recommended; however, this construct islonger than is commercially available to synthesize and therefore requires a ligation design.
8.6

RNA conformational dynamics in liquid RNA droplets
When stressed, some cells halt mRNA translation through the creation of liquid RNA

droplets.9 These RNA droplets, or membraneless organelles, or stress granules, are composed ofnegatively charged RNA and positively charged proteins and they behave similarly to liquid

234

Figure 8.3 FRET histograms of the BMV construct from surface tethered measurements in 75 mM
monovalent cation. Two dominant FRET states were observed at E = 0.2 and E = 0.4 (verticaldashed lines), with high magnesium driving the equilibrium to the high FRET state. However,there is no highly populated high FRET state as observed in Vieweger’s diffusion studies (expected
population at 0.5 mM Mg2+ = 50%). Instead, many low population FRET states at high E areobserved.
crystals. To my knowledge, there has not been a single-molecule FRET study of RNA
conformational dynamics inside RNA droplets, and such a work could probe the way that RNA
molecules organize themselves in membraneless organelles. My graduate career ended before I
could explore this possibility, but in principle, the experiment is fairly simple: follow developed
procedures for generating RNA droplets in vitro and dope the droplets with a small concentration

235
of FRET-labeled RNA molecules for single-molecule microscopy. There are certainly challenges
to this work, such as the possibility of a continuum of FRET states or extremely rapid structuralfluctuations. However, CU has a number of experts on RNA droplets who could be excellent
collaborators, such as Dr. Roy Parker.
8.7

Summary of work and open research questions
We may also find possible future research endeavors by examining the results found in

Chapters 3−7. Each chapter has more detail in its discussion and summary sections, but I will
now compile the major findings and what future research questions those findings raise.
In Chapters 3 and 4, I examine the effect of amino acids on nucleic acid folding. These
series of experiments are inspired by a proposed evolutionary scenario in which the earliest roleof amino acids was not incorporation into peptides but rather acting as RNA folding chaperones.
In my experiments, I found that secondary structure responds only minimally to amino acids(Chapter 3), while in contrast tertiary structure is far more sensitive to amino acids, even capableof discerning amino acid chirality via specific binding of select amino acids (Chapter 4). As
stated in section 8.4, the clear next step is to explore dipeptides and tripeptides, as an evenstronger RNA chaperoning by small peptides, if observed, could be the evolutionary pressure
responsible for the development of peptide synthesis. Additionally, the work in Chapter 4 was
performed on a single tertiary fold, a tetraloop-tetraloop receptor, but other tertiary structures
should be explored to evaluate the generality of strong amino acid sensitivity.
Chapters 5 and 6 contain efforts to extend the methodological capabilities of singlemolecule FRET. Chapter 5 explores the origin of an artifact called “camera blurring” which
occurs when FRET dynamics happen on the same timescale as the FRET data acquisition rate.
Use of stroboscopic illumination was found to eliminate the artifact and thereby increase the236
upper limit on measurable rate constants in smFRET by a factor of 5−10. In Chapter 6, I use
temperature to probe thermodynamics of nucleic acid folding and obtain not only the usualenthalpy and entropy of folding but also the heat capacity ΔCP of folding. This heat capacity
measurement was used to obtain novel structural information about the folding transition state ofa DNA hairpin. The ideal next step is to find additional nucleic acid folding systems which are
well suited for heat capacity measurements. This is a challenging task, as heat capacities infolding are typically small due to the small surface area changes in nucleic acid folding (ascompared to, say, surface area changes in globular protein folding). Perhaps a good choice would
be a folding system which encapsulates a ligand during folding, such a riboswitch.
In Chapter 7, I turn to the polymorphic folding of G-quadruplexes. Rather than a systemwith 2 folding states, as is found in most smFRET kinetic studies, this work examines a Gquadruplex sequence which has 3 observable FRET states. Assigning a physical structure tothese three states proved quite challenging, and utilization of higher quality FRET prediction
calculations is recommended for future efforts. In this chapter, I performed the first temperaturedependent smFRET kinetic measurements of G-quadruplex folding. From my results, I deduced
that the tested G-quadruplex sequence bound to both Na+ and K+, but that the two cations
exhibited entirely different kinetic mechanisms, affinities, and thermodynamics of binding. The
examined G-quadruplex sequence was selected because of its known propensity for polymorphic
folding; the clear next step is to examine if the observed behaviors depend on G-quadruplex
sequence, such as the number of G-tetrads or the loop length/sequence. Of particular interest is
the human G-quadruplex sequence, which differs from the sequence used here by a T→Amutation in the loop sequence. Another sequence of interest is the putative G-quadruplex

237
structure found in the SARS-CoV-2 viral genome which is a possible drug target of highrelevance to the current COVID-19 pandemic.10

238
8.8

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Chapter 6

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Chapter 7

Kinetic and thermodynamic control of G-quadruplex polymorphism by Na+
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Chapter 8

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Zhao, C.; Qin, G.; Niu, J.; Wang, Z.; Wang, C.; Ren, J.; Qu, X., Targeting RNA Gquadruplex in SARS-CoV-2: A promising therapeutic target for COVID-19? Angew.
Chem. Int. Ed. 2021, 60 (1), 432-438.

Appendix 3
1.

Analytic Correction to Rate Constants Determined by Dwell Time Analysis

Shuang, B.; Cooper, D.; Taylor, J. N.; Kisley, L.; Chen, J.; Wang, W.; Li, C. B.;
Komatsuzaki, T.; Landes, C. F., Fast step transition and state identification (STaSI) for
discrete single-molecule data analysis. J. Phys. Chem. Lett. 2014, 5 (18), 3157-3161.

266
Appendix 1
The rate constant for fluorescence resonance energy transfer (FRET)In this appendix, I will provide a derivation of the rate constant for FRET, which is𝑘𝐹𝑅𝐸𝑇 =

9×103 ln(10)𝑘𝑟𝑎𝑑,𝐷 𝜅 2
6128𝜋 5 𝑛4 𝑟𝐷𝐴
𝑁𝐴

𝐽,(Eq. A1.1)

where krad,D is the radiative rate constant of the donor excited state, 𝜅 2 is the donor-acceptor
orientation factor, J is the donor-acceptor spectral overlap integral, n is the index of refraction,rDA is the distance between the donor and acceptor, and NA is Avogadro’s number. The
expression for 𝜅 2 is
𝜅 2 = (cos 𝜃𝐴𝐷 − 3 cos 𝜃𝐷 cos 𝜃𝐴 )2,(Eq. A1.2)

where 𝜃𝐴𝐷 is the angle between the donor and acceptor transition dipole moments, 𝜃𝐷 (𝜃𝐴 ) is theangle between the donor (acceptor) transition dipole moment and the displacement vector fromthe donor to acceptor. The overlap integral J is∞

𝐽 = ∫0 𝑑𝜆 𝐹𝐷 (𝜆)𝜀𝐴 (𝜆)𝜆4,(Eq. A1.3)

Where 𝐹𝐷 (𝜆) is the donor emission spectrum normalized to an area of 1, and 𝜀𝐴 (𝜆) is theacceptor absorbance molar extinction coefficient. In this appendix, I will derive the FRET rate
constant expression using a semiclassical approach based on Fermi’s golden rule.
Fermi’s golden rule states that the transition rate 𝛤𝑖→𝑓 for transitions due to a perturbing
̂ between states i and f isquantum Hamiltonian 𝐻
2𝜋̂ |𝑖〉|2,
𝛤𝑖→𝑓 = ℏ2 |〈𝑓|𝐻(Eq. A1.4)

where ℏ is the reduced Planck’s constant (ℏ = ℎ/2𝜋), |i⟩ and |f⟩ are the initial and final quantum
̂ is the coupling Hamiltonian. In FRET, the initial state |i⟩ is the directstates, respectively, and 𝐻

267
product of the donor excited state |D*⟩ and acceptor ground state |A⟩ and the final state |f⟩ is the
direct project of the donor ground state |D⟩ and acceptor excited state |*A⟩:|𝑖⟩ = |𝐷∗ ⟩ × |𝐴⟩(Eq. A1.5)|𝑓⟩ = |𝐷⟩ × |𝐴∗ ⟩(Eq. A1.6)

The coupling Hamiltonian is the donor-acceptor dipole-dipole interaction, i.e., the energy of thedipole moment of the acceptor in the electric field generated by the donor dipole:11
̂ = 𝑉̂𝑑𝑖𝑝𝑜𝑙𝑒−𝑑𝑖𝑝𝑜𝑙𝑒 = −𝝁𝐻
̂ 𝐴 ∙ 𝑬𝐷 = −𝝁
̂ 𝐴 ∙ (4𝜋𝜀 𝑛2 ) (𝑟 3 ) [3(𝝁̂ 𝐷 ∙ 𝑟̂𝐷𝐴 )𝑟̂𝐷𝐴 − 𝝁
̂𝐷 ]
0

1

1

0

𝐷𝐴

𝐷𝐴

= 𝜇̂ 𝐴 𝜇̂ 𝐷 (4𝜋𝜀 𝑛2 ) (𝑟 3 ) [cos⏟ 𝜃𝐴𝐷 − 3 cos 𝜃𝐷 cos 𝜃𝐴 ],(Eq. A1.7)

𝜅

where 𝜇̂ 𝐷 and 𝜇̂ 𝐴 are the dipole moment operators for the donor and acceptor, respectively, 𝜀0 isthe vacuum permittivity, and n is the index of refraction of the medium.* Evaluating the matrix
̂𝑖,𝑓 and taking the square modulus yields
element 𝐻
2

̂ |𝑖〉|2 = 𝜇𝐴2 𝜇𝐷2 ( 1 2) ( 16 ) 𝜅 2 .
|〈𝑓|𝐻
4𝜋𝜀 𝑛
𝑟
0

(Eq. A1.8)

𝐷𝐴

Next, we treat the rate constant for FRET as the contribution of wavelength-specific∞transition rates: 𝑘𝐹𝑅𝐸𝑇 = ∫0 𝑑𝜆 𝛤(λ), where 𝛤(λ) is the hypothetical transition rate if donor andacceptor electronic transitions occur at a single wavelength λ. So far, by substituting A1.8 into
A1.4, we have
2𝜋

2

1

𝜅2

𝛤(λ) = ( ℏ2 ) (4𝜋𝜀 ) (𝑛4𝑟 6 ) 𝜇𝐴2 𝜇𝐷2 .
0(Eq. A1.9)

𝐷𝐴* Normally, one would write 4𝜋𝜀 𝑛2 as 4𝜋𝜀(𝜔), where 𝜀(𝜔) is the (frequency-dependent) dielectric constant of the0

medium. However, the symbol 𝜀(𝜔) is already used to describe the extinction coefficient, so I have replaced 𝜀(𝜔)by 𝜀0 𝑛2 (which one would normally do at the end of the derivation), using the fact that the relative dielectric
constant, 𝜀𝑅 (𝜔) ≡ 𝜀(𝜔)/𝜀0 , is approximately the square of the index of refraction n of the medium at the opticalfrequency of the transition.

268
We now replace the transition dipole moments with experimentally accessible quantities using
several well-known expressions:†
3𝜀 ℎ𝑐 30
𝜇𝐷2 = ( 2𝜔
) 𝑘𝑟𝑎𝑑,𝐷 𝐹𝐷 (λ), and
3

3𝜀 ℏ𝑐

0
𝜇𝐴2 = ( 𝜋𝜔) 𝜎𝐴 (λ),(Eq. A1.10)(Eq. A1.11)

where c is the speed of light in vacuum, 𝜔 is the angular frequency, 𝑘 𝑟𝑎𝑑,𝐷 is the donor radiative
rate constant, 𝐹𝐷 (λ) is the fluorescence spectrum of the donor, normalized so that∞∫0 𝑑𝜆 𝐹𝐷 (λ) = 1, and 𝜎𝐴 (λ) is the acceptor absorbance cross section. To convert the absorbancecross section (in cm2) into a molar extinction coefficient 𝜀𝐴 (𝜆) (in M-1cm-1), we use the relation
𝜎𝐴 (λ) =103 ln(10)𝑁𝐴𝜀𝐴 (𝜆),(Eq. A1.12)

where NA is Avogadro’s number. Substituting Eqs. A1.10−12 into Eq. A1.9, we obtain
9×103 ln(10)𝛤(λ) = (

8𝜋𝑁𝐴

𝜅2𝑘

𝜆4

) ((2𝜋)4) 𝐹𝐷 (λ)𝜀𝐴 (𝜆),) ( 𝑛4𝑟𝑎𝑑,𝐷
𝑟6(Eq. A1.13)

𝐷𝐴

𝑐 4

𝜆

4

where I have used the fact that (𝜔) = (2𝜋) . Finally, integrate over all wavelengths to produce
9×103 ln(10)𝑘𝐹𝑅𝐸𝑇 = ( 128𝜋5𝑁

𝐴

𝜅2𝑘

∞

) ∫0 𝑑𝜆 𝐹𝐷 (λ)𝜀𝐴 (𝜆)𝜆4,) ( 𝑛4𝑟𝑎𝑑,𝐷
𝑟6(Eq. A1.14)

𝐷𝐴which is the desired result.
Note that the key terms in kFRET come from the physics of the dipole-dipole interaction:−6
the 𝑟𝐷𝐴
factor is the square of the 𝑟 −3 decay of the dipole field, the 𝜅 2 factor is the square of thedipole-dipole coupling projection, n4 is the square of the relative dielectric constant at opticalfrequency, and finally krad,D and 𝜀𝐴 (𝜆) are related to the donor and acceptor transition dipole
moments, respectively.

†

Hilborn, R. C., Einstein coefficients, cross sections, f values, dipole moments, and all that. Am. J. Phys. 1982, 50(11), 982-986.

269
Appendix 2
Signal-to-noise ratio (S/N) degradation due to noisy amplifiers

An ideal amplifier multiplies an input signal by a fixed factor G, which is the gain. This
increases this signal but also equally the noise, and the net result is that the signal-to-noise ratio(S/N) of the input signal is unchanged by the amplifier. However, a nonideal amplifier has a gain
factor G which is variable and as a result introduces a loss of S/N. In this appendix, I will
evaluate the S/N for nonideal amplification of an input signal with Poisson noise (“shot noise”).
This model is relevant to this thesis as it describes how light is amplified by the intensifier in the
charge-coupled device (CCD).
Consider a Poisson random variable N, i.e., N has an expectation value 〈𝑁〉 and standard
deviation √〈𝑁〉. N determines the number of times that a random variable G (the single-countgain) is added to itself to produce the output signal S:
𝑆 = 𝐺1 + 𝐺2 + ⋯ + 𝐺𝑁 = ∑𝑁
𝑖=1 𝐺𝑖 .

(Eq. A2.1)

In other words, for N detected counts, the amplified output is the sum of N independent andidentically distributed random variables Gi. This is known as a compound Poisson process. The
gain distribution G can be any distribution with finite expectation value 〈𝐺〉 and variance 𝜎𝐺2 .
First, compute the average signal 〈𝑆〉. The conditional expectation of S given n counts is〈𝑆|𝑛〉 = 〈∑𝑛𝑖=1 𝐺𝑖 〉 = ∑𝑛𝑖=1〈𝐺𝑖 〉 = 𝑛〈𝐺〉,(Eq. A2.2)

where the expectation value can be exchanged with the summation because n is not a randomvariable. Next, use the conditional expectation 〈𝑆|𝑛〉 to calculate the expectation value of S:∞〈𝑆〉 = ∑∞
𝑛=0〈𝑆|𝑛〉𝑃𝑁 (𝑛) = ∑𝑛=0 𝑛〈𝐺〉𝑃𝑁 (𝑛)= 〈𝐺〉 ∑∞
𝑛=0 𝑛𝑃𝑁 (𝑛) = 〈𝐺〉〈𝑁〉.

(Eq. A2.3)

270This is an intuitive result: the average signal is the average number of counts multiplied by theaverage gain per count.
Now, calculate the noise in the signal. This is done using the law of total variance,22〉 + 𝜎〈𝑆|𝑛〉𝜎𝑆2 = 〈𝜎𝑆|𝑛,(Eq. A2.4)

which states that the total variance 𝜎𝑆2 is equal to the expectation of the conditional variance
2
2
〈𝜎𝑆|𝑛
〉 plus the variance of the conditional expectation 𝜎〈𝑆|𝑛〉. The conditional variance is2𝜎𝑆|𝑛= 𝑛 𝜎𝐺2 ,(Eq. A2.5)

Because 𝑆|𝑛 is the sum of n independent copies of G each with variance 𝜎𝐺2 . The expectation of
the conditional variance is therefore (by a calculation similar to that for 〈𝑆〉 above)2〈𝜎𝑆|𝑛〉 = 𝜎𝐺2 〈𝑁〉.

(Eq. A2.6)

2
The variance of the conditional expectation 𝜎〈𝑆|𝑛〉is calculated from the definition of varianceand the fact that 〈𝑆|𝑛〉 = 𝑛〈𝐺〉 from Eq. A2.2:
22
𝜎〈𝑆|𝑛〉= 𝜎𝑛〈𝐺〉= 〈(𝑛〈𝐺〉 − 〈𝑛〈𝐺〉〉)2 〉 = 〈(𝑛〈𝐺〉 − 〈𝐺〉〈𝑁〉)2 〉= 〈𝐺〉2 〈(𝑛⏟ − 〈𝑁〉)2 〉 = 〈𝐺〉2 〈𝑁〉.

(Eq. A2.7)

2≡𝜎𝑁

Therefore, the variance in S is𝜎𝑆2 = 𝜎𝐺2 〈𝑁〉 + 〈𝐺〉2 〈𝑁〉 = 〈𝑁〉(〈𝐺〉2 + 𝜎𝐺2 ),(Eq. A2.8)

Finally, compute the signal-to-noise ratio S/N of the amplified signal,
〈𝑆〉
𝜎𝑆

=

〈𝐺〉〈𝑁〉
2)√〈𝑁〉(〈𝐺〉2 +𝜎𝐺

= √𝑁

1
𝜎

2

,

(Eq. A2.9)√1+( 𝐺 )〈𝐺〉

and compare this to the S/N of the Poisson input signal,
〈𝑁〉
𝜎𝑁

= √𝑁.

(Eq. A2.10)

271
−1
𝜎𝐺 2
The output S/N is equal to the input S/N multiplied by a factor (√1 + (〈𝐺〉) ) , which is thegain noise factor. This value is always less than or equal to 1, meaning that the amplifier can
only decrease the S/N. For an ideal amplifier, 𝜎𝐺 = 0 and the gain noise factor is equal to 1.
However, many gain sources such as photomultiplier tubes and multichannel plates haveexponential gain distributions for which 𝜎𝐺 = 〈𝐺〉. In this case, the gain noise factor is 1/√2, orapproximately a 30% loss in signal-to-noise ratio.

272
Appendix 3
Analytic correction to rate constants determined by dwell time analysis

Consider a kinetic system governed by first order rate processes with rate matrix K. For
state i, the survival function Si(n) is the probability that the system is observed in state i for morethan n consecutive frames, P(ni > n). If the underlying dynamics are Markovian, then thisprobability can be expanded in a product of single frame survival probabilities
𝑆𝑖 (𝑛) = 𝑃(𝑛𝑖 > 𝑛) = [𝑃𝑖→𝑖 (∆𝑡frame )]n,(Eq. A3.1)

where 𝑃𝑖→𝑖 (∆𝑡frame ) is the probability that the system is observed to be in state i in one frame
and again in state i one frame later. If we rearrange Eq. A3.1 and change variables from numberof frames to time (n = t/Δtframe), the survival function becomes(1

𝑆𝑖 (𝑡) = 𝑒 ∆𝑡frame) ln(𝑃𝑖→𝑖 (∆𝑡frame ))𝑡app= 𝑒 −𝑘𝑖∗𝑡,(Eq. A3.2)

which decays with apparent rate constant kiapp.
The single frame self-transition probability 𝑃𝑖→𝑖 (∆𝑡frame ) is the diagonal element Tii ofthe transition matrix T evaluated at lag time Δtframe:
𝑃𝑖→𝑖 (∆𝑡frame ) = 𝐓ii (∆𝑡frame ) = (𝑒 𝐊∆𝑡frame )ii .

(Eq. A3.3)

Combining Eqs. A3.2 and A3.3, we arrive at the following relationship between kiapp and Tii:
app

𝑇ii = 𝑒 −𝑘𝑖∗∆𝑡frame.

(Eq. A3.4)

Armed with the diagonal matrix elements Tii, we now construct the full transition matrix T.
Since the columns of T must each sum to one, this is simple for a two-state system: Tji = 1-Tii.
For a system of more than two states, one must use the empirical branching ratios ηji fortransitions from state i to each other state j, in which case the off-diagonal elements arecomputed as Tji = ηji (1-Tii). Determination of the branching ratios requires a more sophisticated273change point analysis than the simple thresholding method used here (e.g. STaSI).1 Aftercomputing T, the final step is to recover K using the matrix logarithm,ln(𝐓)𝐊 = ∆𝑡frame,(Eq. A3.5)

which completes our derivation for the analytic correction. Note that for small frame times, thetransition matrix can be well approximated by T ≈ I + KΔtframe. In this case, the single frametransition probabilities simplify to Tii =1-∑j≠i kij, and the apparent rate constant kiapp will be thesum of the rate constants leaving state i. Therefore, in the slow rate constant regime (−Kii <<kframe), the apparent rate constant will accurately reproduce the true rate of depletion of eachstate, and no correction is necessary.
Finally, it may be of interest to some to instead consider the histogram of dwell times
Hi(n), which consists of the number of dwell times in state i of length n. Dividing Hi(n) by the
total number of dwell times yields the probability mass function P(ni = n), which represents theprobability of a dwell time in state i lasting exactly n bins. Similarly to our above treatment ofSi(n), P(ni = n) can be expanded into a product of single frame self-transition probabilities asTii(n-1)(1-Tii), where the factor (1-Tii) accounts for the termination of the dwell period. Therefore,
the histogram Hi(n) will also decay exponentially according to the same apparent rate constant
kiapp, with the remainder of the derivation identical as for analysis of the survival function Si(n).

References
1.

Shuang, B.; Cooper, D.; Taylor, J. N.; Kisley, L.; Chen, J.; Wang, W.; Li, C. B.;
Komatsuzaki, T.; Landes, C. F., Fast step transition and state identification (STaSI) for
discrete single-molecule data analysis. J. Phys. Chem. Lett. 2014, 5 (18), 3157-3161.

274
Appendix 4
Model-free, graphical analysis of ligand binding stoichiometry using preferential
interaction coefficients

Nucleic acid folding is often accompanied by the binding or release of ligands, which
may be ions, small molecules, or proteins. By measuring the dependence of the folding
equilibrium constant Keq on the concentration of a binding partner, an experimentalist can learn
about the strength (e.g., dissociation constant Kd) and stoichiometry (i.e., number of boundligands) of the underlying binding process. Often, this analysis is carried out by fitting a curve toa plot of Keq versus the concentration of ligand [L]. The functional form for the fitting curve
depends on which binding model is selected to represent the binding interaction. For instance,
one may use a “bind-then-fold” model,
𝑈 + 𝑛𝐿 → 𝑈 ∙ 𝐿𝑛 → 𝐹 ∙ 𝐿𝑛 ,(Eq. A4.1)

𝑈 + 𝑛𝐿 → 𝐹 + 𝑛𝐿 → 𝐹 ∙ 𝐿𝑛 .

(Eq. A4.2)

or a “fold-then-bind” model,

Here, U is the unfolded state, F is the folded state, L is the ligand, n is the ligand binding
stoichiometry, and U·Ln and F·Ln are the ligand-bound unfolded and folded states, respectively.
These two models have different Keq behavior at [L] = 0 and [L] = ∞ and can therefore be
experimentally distinguished. However, these are only two of infinitely many possible binding
models, and other models may have similar dependences on concentration. Therefore, whilesome mechanistic information can be inferred from model-dependent fitting, resolving the true
binding model with certainty is extremely difficult.

275
The method of preferential interaction coefficients is a way to determine ligand binding
stoichiometry using only the slope on a log-log plot of Keq vs. [L]. In this method, one sacrifices
mechanistic detail (i.e., the exact set of states in the binding model) to instead obtain an averaged
ligand binding stoichiometry, i.e., the difference in the average number of ligands bound to theunfolded and folded states, respectively. This method relies on the key result that𝑑 ln 𝐾𝑒𝑞∆𝑛 = 𝑑 ln [𝐿] ,(Eq. A4.3)

which states that the average ligand binding stoichiometry (also called the preferential interaction
coefficient), Δn, is the logarithmic derivative of the folding equilibrium constant, Keq, withrespect to the logarithm of ligand concentration, [L]. This powerful expression is often statedwithout proof, and the proofs given in the literature usually invoke thermodynamic arguments
which are highly abstract and, as a result, unfortunately difficult to understand for those not wellversed in thermodynamics. In this appendix, I will provide an alternative derivation of this result
which relies only on the law of mass action, and which therefore will be accessible to anyone
who has taken a university-level course in general chemistry.
Consider the binding model shown in Figure A4.1, which includes every possible binding
stoichiometry between a ligand L and two receptor configurations U and F. This is the mostgeneral possible binding model, and therefore any result derived from this model will be true ofany binding model. Each process in this network must satisfy a corresponding equilibrium

Figure A4.1 General binding model of ligand L binding to U and F receptor configurations. Notethat free ligands are omitted from this scheme for clarity.

276
constant expression. Let Kn,U be the equilibrium constant for the binding of n ligands to theunfolded state (U + nL → U·Ln). From the law of mass action, write[𝑈∙𝐿 ]

𝐾𝑛,𝑈 = [𝑈][𝐿]𝑛𝑛.

(Eq. A4.4)

This expression assumes ideal (dilute) conditions. To be rigorously correct, one should replaceconcentrations with thermodynamic activities in this expression and all that follow, but themathematics remains the same. These ligand-binding equilibrium constants Kn,U fully
characterize the distribution of ligand-bound states in the U manifold of states. Similarly, the F
manifold of states is characterized by another infinite set of equilibrium constants Kn,F for ligand
binding to the folded configuration. Finally, for each ligand-bound unfolded state U·Ln, there is afolding process U·Ln → F·Ln which has its own equilibrium constant. However, there is only one
degree of freedom for these equilibrium constants, as it suffices to specify one of theseequilibrium constants before the remainder are determined by the fact that the product ofequilibrium constants around any cycle of states must equal 1, due to free energy being a statefunction. For the purpose of this appendix, the one specified equilibrium constant is the noligand (“apo”) folding equilibrium constantKapo = [F] / [U].

(Eq. A4.5)

Note that this is not equal to the empirically observed folding equilibrium constant,
Keq = [Ftotal] / [Utotal],(Eq. A4.6)

which is the ratio of total folded population (i.e., [Ftotal] = [F] + [F·L] + [F·L2] + …) to the total
unfolded population (i.e., [Utotal] = [U] + [U·L] + [U·L2] + …).
Begin by computing the average number of bound ligands in the manifold of unfolded
states:〈𝑛〉𝑈 = 𝑃(𝑈) ∗ 0 + 𝑃(𝑈 ∙ 𝐿) ∗ 1 + 𝑃(𝑈 ∙ 𝐿2 ) ∗ 2 + ⋯

277
= ∑∞
𝑛=1 𝑃(𝑈 ∙ 𝐿𝑛 ) ∗ 𝑛 .

(Eq. A4.7)

This is the average over each state U·Ln of the number of ligands bound to that state n weighted
by the probability of being in that state P(U·Ln). Now using the fact that[𝑈∙𝐿𝑛 ]

𝑃(𝑈 ∙ 𝐿𝑛 ) = [𝑈,(Eq. A4.8)

𝑡𝑜𝑡𝑎𝑙 ]

along with the equilibrium constant expressions (Eq. A4.4), write〈𝑛〉𝑈 =
0+𝐾

0[𝑈]+1[𝑈∙𝐿]+2[𝑈∙𝐿2 ]+⋯[𝑈]+[𝑈∙𝐿]+ [𝑈∙𝐿2 ]+⋯[𝐿]+2𝐾∑∞[𝐿]2 +⋯

𝑛𝐾𝑛,𝑈 [𝐿]𝑛𝑛.
𝑛=1 𝐾𝑛,𝑈 [𝐿]

𝑛=0= 1+𝐾1,𝑈 [𝐿]+ 𝐾 2,𝑈[𝐿]2+⋯ = 1+∑
∞
1,𝑈

2,𝑈

(Eq. A4.9)

Now utilize the fact that the numerator is the derivative of the denominator with respect to thelogarithm of [L], and that the denominator is equal to [Utotal]/[U]:〈𝑛〉𝑈 =𝑛𝑑⁄(1+∑∞
𝑛=1 𝐾𝑛,𝑈 𝐿 )𝑑𝑙𝑛[𝐿]
𝑛1+∑∞
𝑛=1 𝐾𝑛,𝑈 𝐿𝑑𝑛= 𝑑𝑙𝑛[𝐿] ln(1 + ∑∞𝑛=1 𝐾𝑛,𝑈 𝐿 ) =[𝑈])𝑑 ln( 𝑡𝑜𝑡𝑎𝑙[𝑈].

𝑑𝑙𝑛[𝐿](Eq. A4.10)

A similar expression can be derived for the folded state manifold.
The derivation is finished by calculating the difference of these two binding averages andsimplify to get the desired result:
𝑑

[𝐹

]

[𝑈𝑡𝑜𝑡𝑎𝑙 ]∆𝑛 ≡ 〈𝑛〉𝐹 − 〈𝑛〉𝑈 = 𝑑 ln[𝐿] [𝑙𝑛 ( 𝑡𝑜𝑡𝑎𝑙) − 𝑙𝑛 (
[𝐹]
𝑑

[𝐹

]

[𝑈]

)]

[𝐹]

= 𝑑 𝑙𝑛[𝐿] [𝑙𝑛 ([𝑈𝑡𝑜𝑡𝑎𝑙 ]) − 𝑙𝑛 ([𝑈])].
𝑡𝑜𝑡𝑎𝑙(Eq. A4.11)

Here, recognize [Ftotal] / [Utotal] as the observed equilibrium constant Keq, and [F] / [U] as theequilibrium constant for folding in the apo state Kapo which is independent of ligand
concentration. Therefore,𝑑𝑑 ln 𝐾𝑒𝑞∆𝑛 = 𝑑 𝑙𝑛[𝐿] [𝑙𝑛 𝐾𝑒𝑞 − 𝑙𝑛 𝐾𝑎𝑝𝑜 ] = 𝑑 ln [𝐿] ,(Eq. A4.12)

278
which completes the derivation. Note that Δn is itself a function of [L] and is not guaranteed to
be an integer.