Human TATA-Binding Protein Interactions with DNA
Characterized by Stabilized, Axial Optical Trappingby
Amanda E. Carpenter
B.S., Louisiana Tech University, 2002
M.S., University of Colorado, 2007

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
Department of Physics
2011This thesis entitled:
Human TATA-Binding Protein Interactions with DNA Characterized by Stabilized, Axial
Optical Trapping
written by Amanda E. Carpenterhas been approved for the Department of Physics

Thomas Perkins

John Cumalat

Date

The final copy of this thesis has been examined by the signatories, and we find that boththe content and the form meet acceptable presentation standards of scholarly work in theabove mentioned discipline.

Carpenter, Amanda E. (Ph.D., Physics)
Human TATA-Binding Protein Interactions with DNA Characterized by Stabilized, Axial
Optical Trapping
Thesis directed by Professor Thomas Perkins

TATA-Binding Protein (TBP) binding to DNA is a critical step in gene expression.
In the first step of gene expression, transcription of RNA from DNA, TBP binds DNA andrecruits the transcription machinery. TBP binds DNA at a preferred sequence called the
TATA box and bends the DNA at ∼100◦ . In this work we characterized TBP dynamicallybending DNA at the single-molecule level using an actively stabilized, axial optical trap.
To perform this characterization, several challenges led us to develop a customized
biophysical assay. Nonspecific interactions of TBP with sample surfaces required coating
surfaces with polyethylene glycol. A small predicted signal size (∼5 nm) and slow kinetics(∼10−2 s−1 ) demanded high stability in our optical trap assay, leading us to implement
an actively stabilized instrument. An apparent affinity by TBP for non-TATA box DNA
sequences led us to use very short DNA molecules (92 nm) with carefully controlled sequences.
This short length of DNA demanded the development of a novel axial trapping and detection
technique, which also has the advantage of improved spatiotemporal resolution (integrated
noise <1 nm over 100 s at 0.5 pN, 5 fN/nm trap stiffness, ∆f = 0.03-3.2 Hz). Our final
assay involved custom surface-chemistry, an actively stabilized optical trap, short DNA withcarefully engineered sequences, and a novel axial detection method.
Using this novel assay, we measured TBP and TATA-box dependent extension changes
of DNA at the single-molecule level. Under optimized conditions (short, carefully chosen
DNA; optimized [TBP], [MgCl2 ] and [KCl]), we obtained step-wise, consistently-sized TBPdependent extension changes. By hidden Markov modeling analysis, we quantified the extension changes and rates for bending and unbending, and we performed simulations to verify

iv
our analysis methods. We applied three different forces (0.3, 0.5 and 0.8 pN) to test theaffect of force on the extension change and rates. We found a constant extension changeof 3 nm, with dynamics on the scale of tens of seconds. The developed assay directly measures dynamic bending of DNA by TBP, and the techniques developed here have potential
to be broadly useful for high-spatiotemporal resolution studies of many other DNA-protein
interactions.

Dedication

To my Marty, my Mom, and my Dad.

Acknowledgements

To my teachers, colleagues, friends, and family members, thank you for the inspirationand support that made completion of this work possible.
A big thank you to my advisor, Tom Perkins, the source of endless knowledge andideas. Thank you for your support, for always ensuring that we learn something, and forcaring that we get things right.
Thank you to the amazing Perkins Lab postdocs: Lora Nugent-Glandorf, Yeonee Seol,
Hern Paik, and Gavin King. Lora got me started in everything from optics to hands-on
biochemistry to LabVIEW programming. Yeonee was a major player in building, troubleshooting, and maintaining our instrument, development of the axial detection technique,and biochemical-assay development for the Perkins lab. Gavin was always an inspiration
and a cheerleader for me and for science. Hern lent me his expertise for everything fromsurface chemistry to instrumental trouble-shooting and was an excellent role model for being
a logical, analytical experimentalist.
Thank you to the past and present Perkins Lab graduate students: Ashley Carter,
Martha Hosotani, Justin Kuczynski, and Allison Churnside. Ashley learned the ropes ofbiophysics and Ph.D. research with me, sharing countless hours on everything from homework
sets to Tuesday night learning sessions with Tom, from training me on optics to letting
me train her on biochemistry, and from hearty arguments about science to celebrationsof successes. Martha Hosotani, generous, with a creative, analytical mind, was an amazing
listener who always asked the hard questions that made me think and helped me in countless

vii
tangible and intangible ways. Justin provided many ideas and much needed energy during
a frustrating phase of the project, and Allison was a resource of support and positivity.
Thank you to Wayne Halsey, biochemistry guru. I am eternally grateful for yourexpertise, for all the DNA that was essential for my project, and for everything that kept
our lab running smoothly.
Thank you to other students I’ve had the pleasure of working with in the Perkins Lab:
Betsy Siewert, Tara Turner, Theresa Ulrich, Nicole Kennerly, Jenni Adams, Andy Leifer,
Joseph D’Silva, and Jessica Rouge. Special thanks to George Emanuel, who converted thedrift-correcting hidden Markov modeling into a user-friendly Igor format.
Thank you to my collaborators: Jim Goodrich, Meredith Betterton, Aaron Hieb, Jinyu
Li and Tatiana Kuriabova. Jim and Aaron provided proteins, plasmids, and transcription
expertise; and I’m also grateful for help from Jennifer Kugel, Linda Drullinger and BenjaminGilman in the Goodrich lab. Meredith, Jinyu, and Tatiana provided modeling and theoreticalanalysis and expertise. Jim and Meredith were not only collaborators, but served as scientificmentors as well.
Thank you to the amazing technical support staff at JILA. Thank you to Hans Green,
Blaine Horner, Kim Hagen, Tracey Keep, and Tom Foote, all who had a hand in custom
machining of parts for the stabilized optical-trapping instrument; and thank you to David
Alchenberger who provided essential training on Keck lab instruments. Thank you to Jan
Hall, Terry Brown, Carl Sauer, Mike Whitmore, and Dave Tegart, who each played a rolein custom electronics design and development. Thank you to the JILA Computing staff,
especially Jim McKown, Mike Page, J.R. Raith and Pete Ruprecht, who kept my computers
running smoothly and saved me countless hours of frustration with their tips, tricks, andmiscellaneous computer help.
Thank you to the administrative staff at JILA. Special thanks to Agnieszka Lynch,
Maryly Dole, Julie Phillps, Brian Lynch, Jeff Sauter, Randall Holliness and past member
Ed Holliness for many positive interactions and making JILA a great place to work.

viii
Thank you to my defense committee: Tom Perkins, Jim Goodrich, Matt Glaser, John
Cumalat, and Margaret Murnane, for your time and especially for excellent questions andfeedback that are guaranteed to improve the quality of our publication on this project.
Special thanks to the final readers, Tom Perkins and John Cumalat, for your extra time andfor your comments on the text.
I’m grateful for the funding for my work, provided by the NIH Biophysics Training
Grant and by multiple other grants through my advisor, Tom Perkins.
Thank you to many teachers along the way. To my undergraduate advisor at Louisiana
Tech, Lee Sawyer, thank you for all the time, effort, and heart you put into preparing me andall of your students. Also at Louisiana Tech, thank you to the professors who taught me theessentials: Lee Sawyer, Ramu Ramachandran, Zeno Greenwood, Steve Wells, Natalia Zotov,
Neven Simicevic, and John K. Price. At CERN, thank you to my group leader Thomas
Meyer and to my advisor/mentor Joachim Baechler for encouragement and for providing anamazing research experience. And thank you to excellent teachers from earlier years: Ms.
Gore, Ms. Stocker, Ms. Stanescu, Mr. Gamble, Coach Way and Ms. Mosteller.
Thank you to my friends, not only for providing the support that was so needed tocomplete my research and thesis, but also for making sure I had fun during the process.
Thank you to Colleen Crego, Sara Bellinger Rylander, Allison Fox Swain, Holly Wilkins
Johnson, Kathryn Goode McCarty, and Jenny Walling Neville: friends from long before
grad school. Thank you to all my Boulder friends, especially Juan Pino, Kirk Ullmann, Zach
Walters, Keith and Allison Ulmer, Chris and Carly Keller, John Obrecht, Ashley Carter,
Christine Davidson Frank, Erin White, Juli Baumblatt, Sebastian Blatt and Irina, Martha
Hosotani, Stephanie Meyer, Lora Nugent-Glandorf, Julie Fiore, and Haemi Lee. Thank you
to the Ye Lab, especially the strontium group, for lunch breaks, good conversation, andsharing Marty with me. An extra special thank you to Katarina Cicak and Kevin Moll forgiving me not just housing, but an actual home with all the emotional support a true homeentails. And another extra special thank you to the friends who took such good care of me

ix
for that last year in Boulder: Martha, Julie, Stephanie, Lora and Haemi. Thank you forbeing such good friends. Thank you also to my newest California-made friends, especially
Stacey Booth, Mike Matthews, and Kristin Zumwalt. You kept me sane, kept me from being
completely swallowed by my thesis, and helped keep some non-thesis related fun in my life
during this last year.
To my family, thank you for providing the love, perspective, balance, stability, advice,countless hours listening, and understanding that allowed me to successfully undertake thisendeavor. You gave me inspiration and motivation when I didn’t think I could possibly findany more. Thank you to my mom for talking with me almost every day, always listening
sympathetically, and providing encouragement that kept me motivated and grounded. Thankyou also to my mom for passing on an analytical bent and penchant for being precise thathave served me well in science. Thank you to my dad for compassion and guidance, andfor instilling in me a need to question everything and a desire to understand the way things
work, two more qualities that have served me well. Thank you to my brothers and my sister,
Scott, Matt, and Sarah, and to my siblings’ significant others, Robin and Craig. You keep
my world wide, help me talk things through, and set examples that always make me strive to
be a better person. Thank you to the Boyds and the Boyles, especially Marilyn, Joe, Nancy,
Kate, and Scott, and especially for taking excellent care of me and being understanding
and encouraging while I worked through this entire past Thanksgiving. Thank you to myGrams, my Grandma Bo, and my extended family for providing encouragement and an
extended network of support.
And finally, thank you to Martin M. Boyd. Brilliant, big-hearted and always able to
make me laugh. Thank you for being my solid ground, for always believing in me, and formaking all burdens bearable. You’re the world’s best teammate, and I’m so glad your mine.

Contents

Chapter
1 INTRODUCTION
1.1

1

AN INTRODUCTION TO GENE EXPRESSION . . . . . . . . . . . . . . .

1

1.1.1

Transcription and regulation . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

TBP and the TATA box . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.3

Architecture of the transcription preinitiation complex . . . . . . . .

5

1.2

OPTICAL TRAPPING AND SINGLE MOLECULE TECHNIQUES . . . .

6

1.3

THESIS OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2 EXPERIMENTAL TECHNIQUE
2.1

9COMPONENTS OF THE INSTRUMENT . . . . . . . . . . . . . . . . . . .

10

2.1.1

Lasers for trapping and detection . . . . . . . . . . . . . . . . . . . .

13

2.1.2

Acousto-optic modulators (AOMs) for intensity control . . . . . . . .

16

2.1.3

Optical fibers for beam refinement and pointing stability . . . . . . .

17

2.1.4

Samplers and photodiodes for intensity stabilization . . . . . . . . . .

17

2.1.5

PZT mirrors for beam steering . . . . . . . . . . . . . . . . . . . . . .

18

2.1.6

Dichroics for light integration and separation . . . . . . . . . . . . . .

18

2.1.7

Microscope for optical trapping . . . . . . . . . . . . . . . . . . . . .

20

2.1.8

Piezo stage for sample control . . . . . . . . . . . . . . . . . . . . . .

21

2.1.9

Detectors and electronics for visualization and detection . . . . . . .

21

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2.1.10 Components for passive stabilization . . . . . . . . . . . . . . . . . .

21

INTENSITY STABILIZATION . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2.1

Nonlinearity of servo response . . . . . . . . . . . . . . . . . . . . . .

23

2.2.2

Optimization of fiber coupling at the ideal AOM setting . . . . . . .

24

2.2.3

Optimization of servo electronics . . . . . . . . . . . . . . . . . . . .

26

2.2.4

Optimization for stability and dynamic range of the trapping laser . .

26

2.3

SAMPLE STABILIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.4

ALIGNMENT, CALIBRATIONS AND CHARACTERIZATION OF THE

2.2

INSTRUMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.4.1

Alignment into the microscope . . . . . . . . . . . . . . . . . . . . . .

32

2.4.2

1st-order focal-plane alignment: Aligning laser foci with imaging plane

33

2.4.3

Alignment of QPD axes with stage axes . . . . . . . . . . . . . . . .

33

2.4.4

Centering beads and beams . . . . . . . . . . . . . . . . . . . . . . .

34

2.4.5

Axial alignment by minimizing crosstalk . . . . . . . . . . . . . . . .

34

2.4.6

2nd-order focal-plane alignment: Aligning axial detection signals . . .

36

2.4.7

Minimizing trap degradation with depth by eliminating slope of axial
voltage oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.4.8

PZT-mirror calibrations . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.4.9

Detection sensitivity calibrations . . . . . . . . . . . . . . . . . . . .

43

2.4.10 Drag measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

2.4.11 Trap symmetry and ellipticity . . . . . . . . . . . . . . . . . . . . . .

45

2.4.12 Axial alignment by sensitivity offset vs. height . . . . . . . . . . . . .

48

2.4.13 3rd-order focal-plane alignment: Optimizing xy-sensitivity vs. height48

2.4.14 Laser power measurements . . . . . . . . . . . . . . . . . . . . . . . .

50

2.4.15 Trap stiffness calibrations . . . . . . . . . . . . . . . . . . . . . . . .

52

2.4.16 Stability characterizations . . . . . . . . . . . . . . . . . . . . . . . .

53

2.4.17 Force vs. extension for DNA . . . . . . . . . . . . . . . . . . . . . . .

56

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2.5

2.6

2.7

SAMPLE PREPARATION . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

2.5.1

Cleaning coverslips and slides . . . . . . . . . . . . . . . . . . . . . .

60

2.5.2

Flow cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

2.5.3

Antidigoxegenin modification of beads . . . . . . . . . . . . . . . . .

61

2.5.4

Stuck beads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

2.5.5

PEG-modification of surfaces . . . . . . . . . . . . . . . . . . . . . .

63

2.5.6

DNA-tethering beads to surfaces . . . . . . . . . . . . . . . . . . . .

65

2.5.7

TBP handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

DATA ACQUISITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

2.6.1

Computer, controllers, drivers, and electronics . . . . . . . . . . . . .

68

2.6.2

Centering lasers and beads . . . . . . . . . . . . . . . . . . . . . . . .

68

2.6.3

Setting height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

2.6.4

Detection sensitivity calibrations . . . . . . . . . . . . . . . . . . . .

69

2.6.5

Measuring trap stiffness and force . . . . . . . . . . . . . . . . . . . .

69

2.6.6

Centering over DNA tetherpoint . . . . . . . . . . . . . . . . . . . . .

70

2.6.7

TBP and control tether data acquisition: DNA extension vs. time . .

70

DATA ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

2.7.1

PZT calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

2.7.2

Force vs. extension . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

2.7.3

Stability/noise calculations . . . . . . . . . . . . . . . . . . . . . . . .

73

2.7.4

Simulation: Bead in an optical trap . . . . . . . . . . . . . . . . . . .

74

2.7.5

Calculation of axial bead position probability distribution. . . . . .

75

2.7.6

Hidden-Markov modeling analysis of TBP data . . . . . . . . . . . .

82

2.7.7

Simulation: TBP data . . . . . . . . . . . . . . . . . . . . . . . . . .

82

3 AXIAL CALIBRATION AND DETECTION
3.1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83
83

xiii
3.1.1

Geometry constraints in traditional optical trapping . . . . . . . . . .

84

3.1.2

Advantages of axial trapping and detection . . . . . . . . . . . . . . .

86

3.1.3

Axial detection in the literature . . . . . . . . . . . . . . . . . . . . .

87

3.1.4

Overview of our technique . . . . . . . . . . . . . . . . . . . . . . . .

89

3.1.5

A conceptual description of axial trapping and detection . . . . . . .

90

3.2

MATERIALS AND METHODS . . . . . . . . . . . . . . . . . . . . . . . . .

92

3.3

AXIAL CALIBRATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

3.3.1

Calibration of relative axial displacement . . . . . . . . . . . . . . . .

94

3.3.2

Determination of axial trap equilibrium . . . . . . . . . . . . . . . . .

97

3.3.3

Voltage oscillation corrections . . . . . . . . . . . . . . . . . . . . . . 100

3.3.4

Calibration of height above the surface . . . . . . . . . . . . . . . . . 102

3.3.5

Calibration of axial stiffness . . . . . . . . . . . . . . . . . . . . . . . 106

3.4

3.5

UNCERTAINTY ESTIMATES OF CALIBRATION . . . . . . . . . . . . . . 114
3.4.1

Uncertainty by measurements on beads fixed to the surface . . . . . . 114

3.4.2

Uncertainty by trapped bead deviations from equilibrium . . . . . . . 119

3.4.3

Uncertainty by trap stiffness measurements . . . . . . . . . . . . . . . 120

3.4.4

Uncertainty by DNA force and extension . . . . . . . . . . . . . . . . 121

IMPROVED AXIAL FORCE AND EXTENSION ESTIMATIONS . . . . . 124
3.5.1

Estimating force from data traces . . . . . . . . . . . . . . . . . . . . 124

3.5.2

Force estimation by equipartition . . . . . . . . . . . . . . . . . . . . 126

3.5.3

Force estimation by power spectral density . . . . . . . . . . . . . . . 129

3.5.4

Force estimation by autocorrelation . . . . . . . . . . . . . . . . . . . 132

3.5.5

Summary of final force estimates . . . . . . . . . . . . . . . . . . . . 132

3.6

DECREASED NOISE ON SHORT DNA TETHERS . . . . . . . . . . . . . 134

3.7

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

xiv
4 DIRECT MEASUREMENT OF HUMAN TBP BENDING DNA

137

4.1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.2

MATERIALS AND METHODS . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.3

TBP AND TATA-DEPENDENT EXTENSION CHANGES IN DNA . . . . 142
4.3.1

Data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.4

EXTENSION CHANGES VS FORCE . . . . . . . . . . . . . . . . . . . . . 148

4.5

KINETICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.6

SIMULATIONS TO VALIDATE ANALYSIS . . . . . . . . . . . . . . . . . . 154

4.7

DRIFT CORRECTING HIDDEN MARKOV ANALYSIS . . . . . . . . . . . 154

4.8

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5 CONCLUSIONS AND FUTURE DIRECTIONS

159

5.1

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.2

FUTURE DIRECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.2.1

Technique and instrumentation improvements . . . . . . . . . . . . . 160

5.2.2

Exploring TBP-induced spikes in DNA extension . . . . . . . . . . . 161

5.2.3

Angle vs. stability changes for bending of non-consensus TATA boxes 162

5.2.4

Geometry and stability changes with additional PIC proteins . . . . . 163

Bibliography

165

xv

TablesTable
2.1

Detection electronics settings . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.2

Intensity servo settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.3

Laser settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.4

Typical detector readings without servos . . . . . . . . . . . . . . . . . . . .

50

2.5

Typical detector readings and laser power with servos . . . . . . . . . . . . .

51

2.6

Trapped bead simulation values . . . . . . . . . . . . . . . . . . . . . . . . .

74

3.1

Uncertainty estimates of axial calibration . . . . . . . . . . . . . . . . . . . . 115

3.2

Force vs. extension errors from calibration uncertainties . . . . . . . . . . . . 122

3.3

Force estimation from data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3.4

Final force and extension estimates . . . . . . . . . . . . . . . . . . . . . . . 134

4.1

Transition rates from HMM transition probabilities . . . . . . . . . . . . . . 153

4.2

Transition rates from dwell time fits . . . . . . . . . . . . . . . . . . . . . . . 154

4.3

TBP trace simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Figures

Figure1.1

Transcription preinitiation complex . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Crystal structure of TBP and TATA DNA . . . . . . . . . . . . . . . . . . .

5

2.1

Pre-fiber instrument setup diagram . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Post-fiber instrument setup diagram . . . . . . . . . . . . . . . . . . . . . . .

12

2.3

Beam profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.4

PZT mirror beam steering diagram . . . . . . . . . . . . . . . . . . . . . . .

19

2.5

Photodiode voltage vs. AOM voltage . . . . . . . . . . . . . . . . . . . . . .

25

2.6

Intensity noise of 810 and 850 . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.7

Intensity noise of 1064 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.8

Trap intensity stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.9

x, y scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.10 z scans with x, y crosstalk . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.11 Axial scans to show relative z . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.12 Axial scan of a trapped bead . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.13 PZT mirror calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.14 Verification of PZT mirror calibration . . . . . . . . . . . . . . . . . . . . . .

42

2.15 Detection sensitivity of a trapped bead . . . . . . . . . . . . . . . . . . . . .

44

2.16 Drag-force measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

xvii
2.17 Trap ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.18 Sensitivity vs. z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.19 Stiffness calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

2.20 Verification of stiffness calibration . . . . . . . . . . . . . . . . . . . . . . . .

55

2.21 Stability of stuck beads - 1 bead . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.22 Stability of stuck beads - 2 beads . . . . . . . . . . . . . . . . . . . . . . . .

58

2.23 DNA force vs. extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

2.24 Final assay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

2.25 Process for simulating trapped bead position traces . . . . . . . . . . . . . .

76

2.26 Simulation of a trapped bead axial position trace . . . . . . . . . . . . . . .

77

2.27 Theoretical calculation of axial bead position probability distribution . . . .

81

3.1

Diagonal vs. axial geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.2

Axial intensity signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

3.3

Axial detection displacement calibration . . . . . . . . . . . . . . . . . . . .

95

3.4

Referencing from peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

3.5

Correcting for oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.6

Determining trap height relative to the coverslip surface . . . . . . . . . . . . 103

3.7

Peak adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.8

z position vs. x position of a trapped bead . . . . . . . . . . . . . . . . . . . 107

3.9

Histograms of trapped bead positions . . . . . . . . . . . . . . . . . . . . . . 108

3.10 Histograms of trapped bead positions vs. height . . . . . . . . . . . . . . . . 109
3.11 EQP asymmetry vs. height. . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.12 Axial power spectral densities . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.13 Axial trap stiffness calibrations . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.14 Accuracy of calibrated axial measurements . . . . . . . . . . . . . . . . . . . 116
3.15 Accuracy of calibrated axial measurements for fiducial . . . . . . . . . . . . . 118

xviii3.16 Force vs. extension, measured in the axial direction . . . . . . . . . . . . . . 123
3.17 Three methods for estimating force . . . . . . . . . . . . . . . . . . . . . . . 125
3.18 Simulation of three methods for estimating force . . . . . . . . . . . . . . . . 127
3.19 Histograms of axial bead position . . . . . . . . . . . . . . . . . . . . . . . . 128
3.20 DNA stiffness vs. extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.21 Power spectral density of highest force data and simulation . . . . . . . . . . 131
3.22 Autocorrelation of axial bead position traces . . . . . . . . . . . . . . . . . . 133
3.23 Stability with 92 nm tethers . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.1

Final TBP assay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.2

TBP and TATA dependent extension changes in DNA

4.3

Common events in DNA extension vs. time traces . . . . . . . . . . . . . . . 146

4.4

Extension changes show long and short dwell times . . . . . . . . . . . . . . 150

4.5

Exploring force dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.6

Histograms of extension changes for three forces . . . . . . . . . . . . . . . . 151

4.7

Force vs. extension change, theory and experiment . . . . . . . . . . . . . . 152

4.8

Transition rates calculated from transition probabilities . . . . . . . . . . . . 153

4.9

Dwell time histograms for three forces . . . . . . . . . . . . . . . . . . . . . . 155. . . . . . . . . . . . 145

4.10 Simulations to verify analysis methods . . . . . . . . . . . . . . . . . . . . . 156
4.11 Drift correcting hidden Markov modeling analysis . . . . . . . . . . . . . . . 158

Chapter 1
INTRODUCTION

1.1

AN INTRODUCTION TO GENE EXPRESSION
The flow of genetic information - from DNA to RNA to proteins - governs biology at

the most fundamental, molecular level. DNA contains plans, code, blueprints: the genetic
information that directs growth, development, response to environmental stimuli, and disease. This genetic information is encoded in DNA in the form of genes - sequences in DNA
that encode for proteins, with RNA as an intermediary. Proteins, the end result of gene
expression, are the work horses of molecular biology.
Genetic information is encoded in both DNA and RNA in the sequences of the fourbases that compose their structure. The four bases that make up DNA are adenine, cytosine,
guanine, and thymine, abbreviated A, C, G, and T respectively. The same four bases makeup RNA, except instead of thymine, RNA contains uracil, abbreviated U. In DNA and RNA
structure, each of these bases is attached to a sugar-phosphate (deoxyribose in DNA, ribose
in RNA). These base-sugar-phosphate subunits assemble linearly to form the structure of a
DNA or RNA strand. The sugar-phosphates form the backbone of the molecule; the sequenceof the bases encodes the genetic information.
Genetic information is passed from DNA to RNA by pairing of the bases that compose
the DNA and RNA structure. The bases pair specifically: A pairs with T (or U); C pairs
with G. It is through this base pairing that DNA forms its typical double-stranded structure;
the sequence in one strand is complementary (an A for every T, a C for every G, and vice2versa) to the sequence in the other strand. It is also through this base pairing that DNA
transfers genetic information to RNA. RNA is synthesized with a complementary sequence tothe DNA template sequence. The process of RNA synthesis from a DNA template is called
transcription, and is carried out by proteins called RNA polymerases. RNA polymerase is
assisted in transcription by other transcription machinery proteins, varying by organism andfunction, discussed in detail below.
The RNA synthesized in transcription encodes for a specific protein by encoding fora specific series of amino acids. Amino acids string together linearly to form the primarystructure of proteins. Different proteins are made up of different combinations of 20 different
amino acids. Each of these 20 amino acids is encoded by a different 3 base sequence in RNA.
This protein synthesis process, the transfer of genetic information from RNA to proteins, iscalled translation.
The translated proteins go on to perform functions as diverse as life itself. As aquick sampling of protein functions, proteins catalyze reactions, metabolize energy sources,
form signalling pathways. Proteins carry cellular cargo, form and move muscles, carry out
cellular reproduction and repair. Bringing the process of gene expression full circle, proteinsregulate transcription and form the transcription machinery responsible for expressing thegenetic information encoded in DNA.

1.1.1

Transcription and regulation

Human gene expression is intricate, complex and highly regulated. The human genome
contains ∼3 billion base pairs of DNA, with ∼40,000 genes [1]. Genes must be differentiallyexpressed in different cell types, during different phases of development, and in response
to different stimuli. Appropriate gene expression from this exquisitely complex genome isaccomplished in large part by regulation of transcription.
In humans and other eukaryotes, site specific transcription of genes in DNA to themessenger RNA that codes for proteins is carried out by RNA polymerase II (pol II) plus

3several general transcription factors (GTFs), including TATA binding protein, TBP, theprotein subject of this thesis. To initiate gene expression, transcription must be initiatedat the start-site for a gene, i.e. a promotor, a sequence in DNA that signals a start site fortranscription of a gene. Selection of these start sites is where TBP plays a crucial role.
RNA pol II, the polymerase responsible for the physical synthesis of RNA from DNA,lacks the sequence specificity necessary to recognize promoters [2]. Therefore to transcribe
genes in a site-specific manner, accessory proteins are critical for recognizing the appropriate
transcription start sites. The GTFs fulfill this critical role by recognizing promotor sequencesin DNA and recruiting pol II to specific start sites.
The GTFs involved in transcription by pol II are made up of several transcription
factors (TFs): TFIIA, TFIIB, TFIID, TFIIE. TFIIF, and TFIIH. Thomas and Chiang
provide an excellent review of each of these GTFs [2]. These GTFs plus pol II form thetranscription preinitiation complex (PIC), shown in Fig. 1.1, which initiates transcription
at promotor DNA sequences.

TFIIH

TFIID
TFIIA

TBP

TFIIE

TATA TFIIB
TFIIF

pol IIDNA
template

Figure 1.1: The transcription preinitiation complex (PIC), made up of RNA pol II and thegeneral transcription factors TFIIA, TFIIB, TFIID, TFIIE, TFIIF, and TFIIH is shown
bound to promotor DNA. TBP, a subunit of TFIID, is shown bound to the TATA box
sequence on the DNA.

4
1.1.2

TBP and the TATA box

Promoter sequences that are required for functional PIC formation are known as core
promoter elements; at least 7 core promoter elements have been identified in eukaryotes[2]. The TATA box is 1 of these 7 core promoter elements. It has a consensus sequence
TATA(A/T)A(A/T)(A/G) and is recognized by TATA-binding protein, TBP, a subunit of
the GTF TFIID. Of the remaining 6 core promoter elements, 2 are recognized and boundby TFIIB, the TFIIB-recognition elements BREu and BREd. The other 4 elements arerecognized and bound by various TAF (TBP-associated factor) subunits of TFIID. These 7
core promoter elements are present in various combinations to specify transcription initiationsites in the genome of humans and other eukaryotes.
Though TBP binds to the TATA box promoter element with some specificity, TBP
must also bind to non-TATA DNA sequences. Bioinformatics analyses of the human genome
has suggested that <1/3 of human gene promoters contain a TATA box (Suzuki et al., 2001
and Gershenzon and Ioshikhes, 2005). Thus, if TBP is a critical component for transcription,
it must also bind these non-TATA box promoters, likely as a component of TFIID andstabilized by other proteins.
Binding of TBP to TATA DNA has been studied extensively. Crystal structures of
TBP bound to TATA DNA showed DNA bent at a sharp angle ( ∼100◦ for human TBP
structures, Fig. 1.2 [3]). In solution measurements, TBP-TATA complexes have been studied
by fluorescence resonance energy transfer (FRET) [4, 5, 6] establishing confirmation of thebend angle as well as kinetic parameters for TBP binding and bending DNA. While exact
experimental conditions and measurements varied, these studies typically found very fastassociation half-times on the order of seconds and very slow dissociation half-times on theorder of minutes. In a single-molecule measurement on yeast TBP, video microscopy wasused to observe TBP dependent DNA bending [7]. This single-molecule study showed slower
association rates than the FRET studies, but similarly stable TBP-DNA complexes with

5very slow dissociation rates. This set of solution measurements suggest a highly stable andstatic complex of TBP bound to TATA DNA.

Figure 1.2: Crystal structure of TBP and TATA DNA. Crystal structure of human TBP(blue) bending TATA DNA (yellow ) [3] with DNA extended from original crystal structure
by Aaron Hieb [5].

A more recent FRET study by Hieb et al. established a strong KCl concentration
dependence of the dissociation rate (kof f ) of TBP-TATA complexes, finding kof f = 1.3
x 10−2 s−1 at a KCl concentration of 140 mM, compared with 2.1 x 10−3 s−1 at 50 mM
KCl [5]. The 140 mM KCl concentration is much closer to the physiological monovalentsalt concentration in cells than the 50-100 mM KCl used in most other solution studies.
This large increase in the dissociation rate at more physiological [KCl] begins to establish amore dynamic picture of TBP binding to TATA DNA in vivo, with potentially significant
consequences for models describing the function of TBP in transcription regulation.
1.1.3

Architecture of the transcription preinitiation complex

TBP bending DNA is not the only DNA architectural change thought to be important
for transcription regulation. Other members of the PIC are thought to induce and/or stabilize bending and looping of the promoter DNA. TFIIA changes the conformation of the

6
TBP-DNA complex upon binding [5]. Crosslinking studies have suggested other members of
the PIC in combination (TFIIB + TBP + TFIIF + pol II) may induce looping of the DNA
around pol II and other PIC proteins [8]. Indeed the DNA-protein architecture likely plays
a key role in the mechanism of transcription and transcription regulation.
1.2

OPTICAL TRAPPING AND SINGLE MOLECULE TECHNIQUESThe nm-scale changes in DNA extension (distance between the two ends of the DNA)

associated with bending and wrapping of DNA around PIC proteins makes the architectureof the PIC an ideal subject for study via optical trapping. In an optical trap, a focused laser
exerts forces on small particles, creating an essentially harmonic potential that traps particles
at the laser focus. Optical traps are ideal for measuring and manipulating biomolecules atthe pN-force scale and the nm-distance scale.
Optical trapping falls into the general category of single molecule techniques. Single
molecule techniques involve measurements of the behavior of individual molecules - often
molecules of DNA, RNA, proteins. Single molecule studies are in contrast to bulk methods,
which observe average behavior of a large population of molecules. A strength of single
molecule measurements is the ability to detect time-dependent changes that cannot be synchronized in a bulk population. In addition, the study of individual molecules often allows
subsets of populations with heterogeneous behavior to be identified, whereas this heterogeneity would be averaged over the entire population in bulk methods.
Single molecule optical trapping has the potential to elucidate the structure and function of TBP and the PIC. Addressing particular issues, optical trapping could potentiallyresolve a bending stability vs. bending angle conundrum for TBP bending variant sequencesof DNA. Bulk FRET has demonstrated an average angle change for TBP binding to these
non-consensus TATA sequences, but the FRET data also suggest a change in stability that
could account for or contribute to the apparent angle change (Wu...Parkhurst 2001). In addition, optical trapping could detect individual steps in DNA bending, wrapping and looping

7
around PIC proteins. This ability to detect time-dependent changes could eventually elucidate pathways for GTF assembly at the promoter, addressing the contrasting models of
PIC assembly - sequential assembly of individual transcription factors at the promotor vs.
assembly of the proteins in solution before binding DNA (Thomas Crit Rev 2006). Addressing these types of questions via direct measurement of DNA conformational changes would
yield exciting contributions to the study of gene expression.
Before these more complicated questions can be answered, a working single molecule
assay to study DNA conformational changes by human PIC proteins must be established andproven. Much of this thesis is devoted to this topic, dealing with the challenges presented
by the sensitivity and complexity of the human transcription machinery. Ultimately, human
TBP bending DNA was chosen for the studies presented, as TBP binding DNA is the firststep in PIC assembly, making the TBP-DNA complex the simplest complex for study, as
well as a complex of primary importance.
1.3

THESIS OVERVIEW
In this thesis, single-molecule TBP-dependent extension changes of DNA are reported,along with the actively stabilized, axial optical trapping assay developed to measure them.
Chapter 2 details the experimental techniques, first discussing the instrument components,alignment, calibrations, and characterization, followed by sample preparation methods, thendata acquisition and analysis details. Chapter 3 details the axial trapping and detection
technique, discussing improvements on previously developed techniques and a true novel
contribution in the form of a method for force and extension measurement that combines
the theoretical dependencies of drag coefficients near surfaces and DNA polymer theory.
Finally Chapter 4 reports direct measurements of TBP bending DNA.
The major results of this work are two-fold: 1) a stabilized, axial optical trapping
method capable of high-resolution measurements on very short DNA and 2) a direct measurement of human TBP bending DNA. The novel method of axial force and extension

8
estimation allow precision measurements on very short DNA that expands the range of possible biological phenomenon that can be studied by optical trapping. The measurement of
human TBP bending DNA is the first single molecule study to my knowledge of a componentof the human transcription machinery. The measurement establishes and quantifies TATAbox dependent extension changes of DNA, measuring characteristic bending and unbending
rates of tens of seconds. The dynamic nature of the TBP-DNA complex measured in thiswork, while in agreement with the more recent FRET measurement of a faster dissociation
at physiological salt concentrations, contrasts with some traditional models of TBP’s role intranscription as a stable, essentially static scaffold for recruiting transcription factors. This
work suggests a more dynamic role for TBP in transcription regulation.

Chapter 2
EXPERIMENTAL TECHNIQUE

One of the challenges in studying TBP bending DNA at the single-molecule level was
the development of a working biophysical assay. The working assay needed to be capable ofmeasuring single DNA bending events on the slow time scale (∼1 event per 100 s) [5] andsmall length scale (∆ ∼5 nm) [9] predicted for TBP. The assay also needed to preserve theactivity of the protein, prevent non-specific sticking, and yield an interpretable, repeatablesignal. To achieve this working biophysical assay, we built an actively stabilized opticaltrapping instrument, implemented customized non-stick surface chemistry, and developed anovel axial trapping and detection technique. Additionally, for data analysis, we adapted ahidden Markov modeling algorithm and developed a drift-correction capability to correct forbaseline drift of our data traces. Using this biophysical assay and data analysis technique,we obtained the TBP data and interpretations in this thesis.
This chapter and the next describe the instrumentation and experimental techniques
used to obtain and interpret the data in this thesis. The actively stabilized optical-trapping
instrument is described and characterized. The customized non-stick surface chemistry andother biological assays are then described, followed by the data acquisition and data analysis.
The novel axial detection method is discussed in the next chapter.

10
2.1

COMPONENTS OF THE INSTRUMENT
An actively stabilized optical-trapping microscope was the key to studying the infre-

quent, small DNA extension changes caused by TBP. In designing this instrument, based on
previous work [10, 11, 12], the goals to be achieved were three-fold: the instrument needed tohave 1) a strong, non-biologically-damaging optical trap; 2) precision detection and positioning capabilities; and 3) long-term stability for measurements. The instrument we developed
to achieve these goals is shown in the optics diagrams, Figs. 2.1 and 2.2. The figures show
three lasers: a high-powered trapping laser (1064 nm) and two lower-powered detection lasers(850 nm and 810 nm). In our experiments, the trapping laser trapped a micron-sized beadattached to DNA; one detection laser detected the position of the trapped object, while theother detection laser detected drift between the laser foci and the sample surface. By bothactive and passive stabilization techniques, we suppressed pointing, mode, polarization andintensity noise of our three lasers, and actively stabilized the sample position relative to thelasers. Through this stabilization scheme, we achieved a stable optical-trapping instrument
capable of precision measurements. The components of this instrument will be described indetail in this section.
Before detailing the components of our instrument, a quick overview of the key elements provides a framework for how each contributes to our goals. Beginning with the lasers,
each of the three lasers was guided through an AOM to control the intensity, then coupledthrough an optical fiber to refine the mode. Next, a beam sampler picked off ∼10% of eachlaser to sample the intensity, detected by a photodiode. The intensity on the photodiodewas stabilized by servo-electronics, which controlled the intensity via the AOM. Next, thetrapping laser and one detection laser were reflected by PZT mirrors, which steered thebeams in the optical-trapping microscope. The three lasers were coupled into the microscope by dichroics. The lasers were focused into the sample plane of the microscope by ahigh NA objective. A condenser collected the laser light for back-focal-plane detection with

11

Bragg Grating
Fiber
PBS

810 nmDiode Laser

Fiber
Launch810 FiberTo
Instrument

AOM

AOM

850 FiberTo
Instrument

λ/2
plate

Bragg Grating
Fiber

850 nmDiode Laser

Lens

MirrorNd:YVO4 1064 nm
Trapping laser

Beam Block
L1
75 mm

Fiber
Launch

L2
25 mm

AOM

1st Order
L3
200 mm

L4
300 mmFiber

To
Instrument

Shutter

Figure 2.1: Pre-fiber instrument setup diagram. Diagram of the pre-optical-fiber setup of thethree lasers in the optical-trapping instrument used in the work in this thesis. Abbreviations
in diagram: AOM (acousto-optic modulator), PBS (polarizing beam splitter), L (lens).

12

Quadrant
PhotodiodePD IrisImaging Light

ND
Filter810 nmLaser
90/10
Beam Sampler

Line
Filter
Notch
FilterElectronics

Electronics

1064 nm
Laser

Condenser

Specimen Plane

MirrorIris

Objective

L4

L3

Gimbalmounted
MirrorDichroic

To CCD
Light blue indicates
optically conjugate planes

PD

PZT Mirror

850 nmLaser
Electronics

PD

Figure 2.2: Post-fiber instrument setup diagram. Post-optical-fiber diagram of the opticaltrapping instrument used in the work in this thesis. Abbreviations in diagram: PD (photodiode), ND filter (neutral density filter), PZT mirror (piezo-electric mirror), L (lens), CCD(CCD camera).

13
quadrant photodiodes (QPDs), placed in a custom-built detector arm attached to the condenser column of the microscope. Precision movement of the sample was accomplished by apiezo stage. The stabilization of the microscope was achieved by several passive stabilization
techniques (described in Section 2.1.10), combined with active laser intensity stabilization
via the AOMs and intensity servo electronics and active sample stabilization by the piezo
stage and a software-based position servo. Given this framework, the details of these key
components follows.

2.1.1

Lasers for trapping and detection

The trapping laser in our optical-trapping instrument was a high-powered, 1064-nm
Nd:YVO4 laser (Spectra-Physics, Millennia IR). This laser was chosen due to commercial
availability, high-power, and a desirable wavelength for minimizing biological damage [13].
The laser has a maximum available power of 10 W; we operated at 7.8 W (reading of 7.77
W on the laser controller) to achieve high power and stability. This high power necessitatedmirrors, lenses, and other optics in the instrument be chosen with high-damage thresholds,
particularly for optics before the optical fiber where the beam diameter is small (∼2 mm).
To improve stability, the cooling system for the laser was modified. Explicitly, the internal
fan on the laser head was replaced with an external fan coupled to the laser via air ducts,
and the cooling cycle was changed from a modulated cooling cycle to continuous cooling.
The trapping laser and optics before the optical fiber are shown at the bottom of Fig. 2.1.
This trapping laser provided the high-powered laser needed for creating a strong optical trap
in our instrument.
For reference, the trapping-laser beam profile is shown at two different saturations in
Fig. 2.3. The beam profiles characterize the shape and quality of the trapping laser, including
rings which are visible in the saturating profile. Note that the beam passed through manyother optics before passing into the optical-trapping microscope, including an optical fiberthat refined the mode, creating a nicely Gaussian beam profile before forming the optical

14

Figure 2.3: Beam profile. Beam profiles taken of the 1064 laser using the Beam Profiler
from the Keck lab. The profiles were taken at medium power to be representative of typicalrunning power (typical running power is 7.77 W out of a possible 10 W). The unsaturated
profile at the top shows a nicely Gaussian beam. The saturated profile at the bottom shows
the rings that become evident further from the Gaussian center of the laser beam. These
profiles were in the beam path very close to the laser, before the AOM and fiber.

15
trap.
The detection lasers in our instrument were diode lasers: an 808-nm diode laser (Lumics, 100 mW) for detection of the trapped object, and an 845-nm diode laser (Lumics,
200 mW) for detection of drift of the sample surface. These lasers are often referred to as
the 810 (referring to the 808-nm) and the 850 (referring to the 845-nm) lasers. The laser
wavelengths were again chosen to minimize biological damage, and the separation between
the wavelengths of the three lasers in the instrument allowed for separating and combining
the lasers using dichroics. The detection lasers and optics before the optical fiber are shownin the top two boxes of Fig. 2.1. Both detection lasers were fiber-Bragg-grating stabilized.
The Bragg-grating fibers are not polarization-maintaining fibers, and the polarization of thebeams at the fiber outputs appeared sensitive to the fiber-coiling geometry and subject todrift over long periods of time (weeks to months). The polarization after the Bragg-gratingfiber was purified with polarizing beam splitters (PBS), and the polarization and power
could be tuned with half-wave plates (λ/2). These diode lasers provided the detection forstabilizing our instrument and making precision measurements.
The three-laser scheme in our instrument was one of the keys to achieving an instrument capable of precision measurements with long term stability. In contrast, many opticaltrapping instruments implement a single laser to both trap and detect. These single-laser
systems typically obtain average detection sensitivity calibrations for a given batch of beads,
then apply those average calibrations during their experiments. However, detection sensitivity varies with the volume of beads, and in a batch of beads, bead radius actually varies
with typical coefficients of variation of ∼3% according to manufacturers specifications. Thisvariation in size causes a variation in detection sensitivity of 9%, which is unaccounted forby using average detection sensitivity calibrations as in single-laser systems. On the other
hand, implementing a separate laser for detection of the trapped bead allows detection sensitivity calibrations of the individual bead to be measured in each experiment, improving the
precision of the detection calibration. As another advantage, a separate detection laser also

16
allows multiple means of calibrating the stiffness of the optical trap, which not only provides
a more trustworthy stiffness calibration, but also provides a diagnostic tool for testing thequality of the trap (see Section 2.4.15 for more details). The third laser in our system was
the key innovation for stability of our instrument. This third laser provided the means for
detecting the position of the sample relative to the lasers, which was then stabilized via thepiezo stage. Using a second laser in our system to detect the position of the trapped bead
improved the precision and reliability of our measurements, and using a third laser provided
the long-term stability necessary for our measurements.
Because separate lasers were used for trapping, detection, and stabilization, any relative
motion of the lasers relative to one another would detract from the stability and precisionof our measurements. Thus it was essential that the relative pointing of the lasers be very
stable relative to one another. This pointing stability was achieved through several passive
stabilization techniques, as discussed in Section 2.1.10.

2.1.2

Acousto-optic modulators (AOMs) for intensity control

Each laser in our instrument passed through an acousto-optic modulator (AOM, Isomet
1205C-2, lead molybdate crystal) to provide intensity control. The AOM alignment was
optimized in the procedure outlined in A. Carter’s Thesis [14]. Briefly, the laser was collimated at a 2 mm diameter to maximize the diffraction efficiency through the AOM. As
shown in Fig. 2.1, for the trapping laser two lenses collimated the beam. For the detection
lasers, diffraction efficiency was less essential, and one lens or no lens was used to adjust thebeam collimation before the AOM. The AOM driver (Isomet 232A-2) settings were set tomaximum diffraction efficiency (“bias adjust” turned completely counter-clockwise, “power
adjust” turned completely clockwise minus 1/4 turn). The AOM was then manipulated via
a tip-tilt stage until the power in the 1st-order diffracted beam was maximized, achieving∼70% efficiency.
For the trapping laser, the AOM provided dynamic control of the trap power. Dynam-

17
ically changing the power via the AOM caused heating of the AOM crystal, which resulted
in changes in beam pointing [15, 12]. To minimize the effect of these beam pointing changes,
we imaged the center of the AOM crystal onto the fiber-coupling lens (see Fig. 2.1) and
optimized the alignment after the AOM at ∼50% of the maximum diffraction power (0.5 V“video” input on the AOM-driver). Further optimization, including re-optimization of thefiber alignment at a more carefully chosen AOM control voltage, is included in Section 2.2.
The intensity control provided by the AOM allowed stabilization of the laser power for all
three lasers, as well as modulation of trap power for the trapping laser.

2.1.3

Optical fibers for beam refinement and pointing stability

Each laser in our instrument also passed through a single-mode, polarization-maintaining
optical fiber (Oz Optics). (For a How-To guide on optimizing fiber alignment in a similar
setup, see A. Carter’s thesis [14].) The fibers refined the mode of the lasers, ensuring thebeams were nicely Gaussian before entering the optical-trapping microscope. The fibers
themselves were mechanically stabilized at the fiber output to prevent motion of the fibersthat could result in pointing noise after the fiber. Because the fiber-coupling efficiency wasvery sensitive to beam pointing changes and mode noise before the fiber, the fiber also automatically converted any beam-pointing or mode noise before the fiber into intensity noise
after the fiber. After the fiber, the intensity stabilization suppressed this intensity noise.
Passing the lasers through optical fibers refined the laser mode and transferred pre-fibermode and pointing noise into intensity noise to be stabilized after the fibers.

2.1.4

Samplers and photodiodes for intensity stabilization

After the optical fiber, each laser was sampled for intensity stabilization. Each laser
first passed through a PBS to purify the polarization, causing polarization noise to be converted into intensity noise. Next a 90/10 beam sampler (Newport) diverted ∼10% of thelaser onto a photodiode (EG&G Optoelectronics Canada YAG-444A) for intensity detection.

18
Stabilization of the intensity was performed by servo-electronics, which controlled the intensity via the pre-fiber AOMs. By stabilizing the sampled intensity of each laser, we stabilized
the intensity of the our optical trap and detection lasers.

2.1.5

PZT mirrors for beam steering

The 1064 nm and 850 nm lasers were steered in the microscope by PZT mirrors (PIS330.2SL PZT mounts, Thorlabs mirrors). Applying a voltage to the PZT mirrors rotatedthe mirrors, which resulted in translations of the lasers at the imaging plane. To ensure thatrotations of the PZT mirrors resulted in pure translations at the imaging plane, the PZT
mirrors were placed at a plane optically conjugate to the back-focal-plane of the objective(see Fig. 2.4). This placement was achieved by using two lenses to form an imaging systemthat images the PZT mirrors onto the back-focal-plane of the objective.
Determining physical placement of the PZT mirrors involved measuring backwards
along the beam path from the effective back-focal plane of the microscope. We estimated
7.8 cm above the center of the dichroic at the base of the microscope as our effective focal
plane, half-way between the back aperture and the Nikon-estimated position of the backfocal plane. The lens closest to the microscope was placed its focal length (25 cm) from theeffective back-focal plane, which was 17.2 cm from the horizontal midplane of the microscope.
The lens closest to the PZT mirrors (focal length=14 cm) was placed a distance of the sumof the focal length of both lenses (39 cm) from the other lens. Each PZT mirror was placedat the focal length behind the lens closest to the PZT mirrors. With this imaging systemand placement, the PZT mirrors steered the lasers in the imaging plane of the microscope.

2.1.6

Dichroics for light integration and separation

To integrate all three lasers and the imaging light into the optical-trapping microscope,
the lasers and imaging light were combined and split via dichroic mirrors (Chroma). Notethat the imaging light coupled into the microscope in the reverse direction from the lasers.

19Light blue indicates
optically conjugate planes

Specimen Plane
Objective

Laser
Lens

Dichroic
PZT Mirror

Figure 2.4: PZT mirror beam steering diagram. The PZT mirror is in a plane opticallyconjugate to the back focal plane of the objective so that pure rotations of the PZT mirror
cause pure translations at the sample plane.

20
Two dichroics before the microscope combined the three lasers into a single beam path. A
dichroic in the base of the microscope reflected the three lasers up into the microscope andtransmitted the imaging light through to the camera and microscope eyepiece. A dichroic
above the condenser also transmitted the imaging light and reflected the laser light into thedetector arm of the microscope. Within the detector arm, two dichroics separated the threelasers so that each laser was impingent on a separate detector. The dichroics allowed thelasers and imaging light to be integrated into the microscope and separated for individual
detection.

2.1.7

Microscope for optical trapping

The optical-trapping microscope was an inverted Nikon TE2000-S modified to enablestabilized trapping and detection. The microscope had a 1.4 numerical aperture (NA) objective (Nikon CFI Plan Apochromat 100x oil IR, NA 1.4, working distance 0.13 mm) that
focused the trapping and detection lasers and collected the imaging light. A Bertrand lensallowed imaging of air bubbles in the immersion oil between the lenses and sample, whichwere avoided as they degraded the trapping and detection properties of the microscope. The
condenser (1.4 Nikon HNA Oil) collected the laser light for detection and passed the imaging
light into the imaging plane. The microscope was modified for stability by adding an aluminum plate to stabilize the condenser column and using custom, monolithic holders for thestage and for the objective. The imaging light was coupled via a liquid light guide, with thelamp placed several feet away from the microscope to minimize drift induced by heat fromthe lamp. The microscope was modified for detection of the laser light by adding a custombuilt detector arm to the condenser arm of the microscope. With these modifications, themicroscope became a stabilized optical-trapping microscope.

21
2.1.8

Piezo stage for sample control

For sample control in the microscope, a piezo stage (PI P-517.3CD) was mounted onto
the monolithic stage holder. The stage defined the axes for the experiment. The piezo
stage allowed for precise (sub-nm) computer-controlled movement of the sample, as well as
automated calibrations. Importantly for our instrument, computer control of the stage was
the key to active stabilization of the sample position.

2.1.9

Detectors and electronics for visualization and detection

Detection in our instrument was accomplished by quadrant photodiodes (QPDs) anda CCD camera. The CCD camera (DAGE-MTI CCD100S) allowed for visualization of thesample and lasers, while the QPDs allowed for precision laser detection of objects in the
microscope. The QPD’s were mounted onto the detection arm for back-focal-plane detection[16, 17]. The voltage outpur from each QPD was digitized, anti-aliased, and converted into
x, y, and z voltage signals with custom JILA electronics (see Table 2.1 for typical detection
electronics settings). The x and y voltage signals were based on the normalized difference
between halves of the QPD, while the z voltage signal was the sum of the four quadrants
modified by a separate differential gain amplifier. These detector signals, with the settings
listed in Table 2.1 represent the x, y, and z detection voltage signals shown in this thesis.

2.1.10Components for passive stabilization

To achieve the high level of stability in our optical-trapping instrument, we implemented several other modifications. We mounted the microscope and our optics on floating
optical tables (TMC). The experiments were performed in a temperature-stabilized (±0.2
C), acoustically quiet (NC30) room with a specialized air-venting system to minimize air
currents. The noisy, heat-generating pump lasers were in a separate room. All optics before
the microscope were enclosed in plexiglass boxes. Most optics mounts after the fiber were22

Table 2.1: Typical detection electronics settings.

x-y gainx-y balance
post amp gainb
diff amp gain
antialiasingz diff amp gainda1064a
2 (0.5 in window)
5 in windowmax
×1
DAQ rate/2c
×50e

850
0.5
5 in window
×1.5 (looks like 2 on dial)
×1
DAQ rate/2
×100

We only detected with 1064 for troubleshooting.
Optimized for bead size.
c1064 antialiasing filter did not filter.
d
Was optimized for bead size.
e1064 DC offset amplifier gain was too high even at the lowest setting.
b810
0
5 in window
×2
×1
DAQ rate/2
×100

23
customized, wide-based mounts, and cables and fibers were stabilized with clamps and supports. The post-fiber optics were on an elevated breadboard to minimize the height of themounts. These extra modifications added stability to our instrument, particularly stabilizing
the pointing of the lasers relative to one another, which was critical to our active-stabilization
scheme.
2.2

INTENSITY STABILIZATION
Many of the components mentioned in the preceding section combined to provide in-tensity stabilization of the lasers in our instrument. The intensity stabilization is describedin detail in A. Carter’s thesis [14]. In brief, AOMs before the optical fibers provided active
control of the intensity of each laser. Beam samplers after the optical fiber reflected ∼10%of each laser onto a photodiode to measure the intensity. The signal from the photodiodewas optimized with ND filters, an iris, and a resistor in the photodiode. The ND filter waschosen to ensure the power onto the photodiode would not overheat the photodiode. The iris
ensured unwanted beam reflections were not projected onto the photodiode, and the resistor
was chosen so that the desired range of intensities resulted in a photodiode voltage signal
between 0-10 V (i.e., not saturating the photodiode range and matching the range available
by computer-control). JILA-made servo electronics received the photodiode signal and senta signal to the AOM to stabilize the intensity on the photodiode. The optics before the intensity measurement were designed to convert various forms of noise into intensity noise: theoptical fiber converted pointing and mode noise and the PBS converted polarization noise.
By stabilizing the intensity noise after the other noise sources were converted, we stabilized
pointing, mode, and polarization noise, as well.

2.2.1

Nonlinearity of servo response

The intensity stabilization required several steps for optimization. In addition to theinherent nonlinearity of a typical AOM response, laser intensity changes via the AOM re-24sulted in thermal changes in the AOM crystal which led to pointing changes in the beam.
Because the beam was passed through an optical fiber after the AOM, these pointing changes
resulted in changes in the fiber-coupling efficiency (but not pointing changes after the fiber).
The changes in fiber-coupling efficiency caused a nonlinearity and even an inversion of therelationship between the applied AOM voltage and the measured photodiode voltage after
the fiber, as shown in Fig. 2.5. To optimize the servo performance, operating the instrument
near the center of the linear region provided the best intensity stabilization.

2.2.2

Optimization of fiber coupling at the ideal AOM setting

For the detection lasers, which operated at a static AOM voltage, we optimized the fiber
coupling at an intermediate AOM voltage (VAOM =0.3-0.5 V) and measured the photodiodevoltage as a function of AOM voltage. Note that the location of the linear region may
depend on the fiber optimization voltage, but a quick check of this dependence by doing themeasurement at different fiber optimization voltages showed little to no dependence (datanot shown). For the detection lasers, we chose a voltage near the center of the linear region
as the static voltage for stabilization.
Once the optimum AOM voltage for stabilization of the detection lasers was chosen,
to achieve the desired laser power at the optimized AOM voltage, we had two options.
One option was to adjust the diode-laser current-temperature controllers. This option was
judged not best because the diode laser stability and other properties may also change withthe current and temperature. The second option for controlling the laser power was to adjust
the half-wave plates before the PBS cubes in the pre-fiber optics (see Fig. 2.1). We chose thisoption to dial in the desired laser power. The resulting laser powers (quantified in Section
2.4.14) were low enough power to not trap objects in the microscope, but high enough toprovide an ample signal for detection.

25

Figure 2.5: Photodiode voltage vs. AOM voltage. The plots show an approximately linear
region, as well as highly nonlinear regions. The linear region shows photodiode voltage(power) increasing with applied AOM voltage, as expected. The nonlinear regions are largely
due to pointing variations in the beam, which are translated into power variations at thephotodiode. The pointing variations are caused by thermal changes in the AOM crystal as
the applied voltage changes. The pointing variations are transferred into power variations
because the beam is fiber-coupled between the AOM and the photodiode. This fiber-coupling
efficiency is highly sensitive to pointing variations. The result is a nonlinear power response
recorded at the photodiode, but highly stable beam pointing after the fiber. For maximum
power stability, the 850 laser was coupled to the fiber at an AOM voltage in the center ofthe linear region (0.35 V applied to the 850 AOM). For stability and dynamic range, the1064 was coupled in the lower half of the linear region, at 0.4 V applied to the 1064 AOM.

26
2.2.3

Optimization of servo electronics

In the final step of servo optimization for both the diode lasers and detection lasers, theservo electronics themselves were optimized. To optimize the servo electronics, we measured
the noise using a Network Signal Analyzer (Stanford Research Systems Model SR780). Plotsof the noise vs. frequency are shown in Fig. 2.6. We adjusted the coarse and fine gain on
the servo, increasing until the servo began to oscillate, then backing off. We then adjusted
the PI corner on the servo. The goal of these adjustments was to push the noise level as
low as possible out to as high a frequency as possible. The peak in the noise level on theright hand side of Fig. 2.6 tends to move with the gain and PI corner, and by iterating
through adjustments of the gain and PI corners, this peak was pushed to as high a frequency
as possible, while also keeping the noise low at lower frequencies. The final settings for the
servo electronics are listed in Table 2.2. Fig. 2.6 shows the noise level with the servo on
was decades lower than with the servo off, up to around 60 kHz. By this series of steps,
1) optimizing fiber coupling at a mid-range AOM voltage, 2) operating at the center of thelinear region of the photodiode vs. AOM voltage response, 3) optimizing the laser power via
the pre-fiber half-wave plates, and 4) optimizing the servo gain and PI corner, we optimized
the intensity stabilization of the detection lasers.
Table 2.2: Typical intensity servo settings.
Laser
1064
850
810

2.2.4

coarse gain
max - 6 clicks
max - 1 clickmax

fine gain
max - 1 turnno info.
no info.

PI corner
max - 3 turns
no infono info.

Optimization for stability and dynamic range of the trapping laser

For the trapping laser, stability combined with a dynamic range of trapping powers
was required, slightly altering optimization from that of the detection lasers. The important

27

Figure 2.6: Intensity noise of 810 and 850. In-loop photodiode noise vs. frequency for
810 (lower plot) and 850 (upper plot) lasers, as recorded on a Network Signal Analyzer(Stanford Research Systems Model SR780). Lower frequency noise was highly suppressedwith the servo on (black ) compared to with the servo off (gray).

28
difference here between the trap-laser optimization and detection-laser optimization wasthat for the trap, the AOM was used to dynamically control the trap power, so we could not
simply operate the servo at one optimum AOM voltage. Most of the steps in optimization
were similar to those for the diode lasers, with the modifications described below.
We optimized the fiber coupling at a mid-range AOM voltage (0.6 V) and measured
the photodiode voltage vs. AOM voltage. The center of the linear region was at ∼0.5 V, but
optimizing the fiber coupling at this voltage was difficult because once the system thermallystabilized and the fiber was optimized at that voltage, the photodiode voltage was nearsaturating. So instead, we optimized the fiber coupling at an AOM voltage of 0.4 V, andthen completed the servo-electronics optimization procedure.
We measured the noise using the Network Signal Analyzer at three different servo
voltages, low (0.4 V), intermediate (1 V), and high (9 V) as shown in Fig. 2.7. Again, withthe servo on, the noise was clearly decreased as compared to with the servo off. As a final
check of the servo stability over the range of powers, as well as a check of the dynamic range
response, we measured the photodiode voltage vs. time. Fig. 2.8 shows the stability of thephotodiode voltage at a low voltage (0.4 V, top) and a high voltage (9.5 V, middle). This
figure also shows the dynamic response of the photodiode voltage as the requested locking
voltage was oscillated in a square wave between 0.4 V and 9.5 V (bottom). This figure
shows the trapping laser intensity stabilization was both stable and capable of fast changes
in power.
Over time (several months), the intensity stabilization and/or dynamic range sometimes degraded. The servos completely stopped locking, or unlocked more easily than before.
Typically this degradation could be remedied by re-optimizing the fiber coupling, choosing
the same AOM voltage for fiber coupling as before. Maintaining the intensity stabilization
described was one essential component of our actively stabilized optical-trapping microscope.

29

Figure 2.7: Intensity noise of 1064. In-loop photodiode noise vs. frequency for the 1064 laser
at 3 different photodiode voltage levels, as recorded on a Network Signal Analyzer (Stanford
Research Systems Model SR780). The magnitude of the noise changes with the voltage, but
in all cases lower frequency noise is highly suppressed with the servo on (black ) comparedto with the servo off (gray).

30

Figure 2.8: Trap intensity stabilization. In-loop photodiode voltage with intensity stabilization and intensity ramping. In the top plot, the servo voltage was set to 0.4 V; in the middle
plot, 9.5 V. For the top two plots, we quantified the peak-to-peak noise (max-min/mean)
and the average noise (std. dev./mean). The noise for the two plots shown are as follows:
at 0.4 V, peak-to-peak noise = 0.8%, average noise = 0.1%; at 9.5 V, peak-to-peak noise= 0.2% and average noise = 0.02%. The plots show the servo is able to remain locked andstable at the low and high end of the available voltage range. The bottom plot shows thein-loop photodiode response to jumps in the servo voltage from 0.4 V to 9.5 V. This plotshows the 1064 nm servo is capable of maintaining stability even while making large, fast
jumps in voltage.

31
2.3

SAMPLE STABILIZATIONThe other essential component of actively stabilizing our microscope lay in actively

stabilizing the sample position relative to the lasers. To perform this stabilization, we used
an old microscopy trick in which a fiducial mark tracks the drift of the sample. In our case,the fiducial mark was typically a micron-sized bead, fixed to the sample surface. We detected
this fiducial with the second detection laser in our system (845 nm), and we actively stabilized
the fiducial’s position using the piezo stage and a software-based servo. The software-based
servo was written in LabVIEW. The position signal from the QPD was read into LabVIEW,an error signal was generated as the difference between the current position and the desired
position, and proportional gain was applied to generate a correction signal to move the stage
and maintain the desired position. By actively stabilizing the sample position relative to thethe second detection laser, combined with all of the active and passive stabilization describedabove, we achieved a stable optical-trapping microscope.
The preceding sections describe the individual components of our actively stabilized
optical-trapping instrument, as well as details of our intensity and sample stabilization.
As a reference, the typical detector settings (Table 2.1), servo settings (Table 2.2), andlaser settings (Table 2.3) are listed in table form. With this instrument, we achieved our
instrumentation goals: a strong, non-biologically damaging optical trap capable of precision
positioning and detection, with excellent long term stability.
2.4

ALIGNMENT, CALIBRATIONS AND CHARACTERIZATION OF
THE INSTRUMENT
Careful instrument characterization was essential to making precision measurementswith our instrument, as well as understanding the limitations of these measurements. Thissection outlines the steps taken to align our optics, calibrate our measurements, and quantify
our instrumental stability.

32
Table 2.3: Typical laser settings.
Laser

Controller Setpoint1064

7.77 W setpoint(7.78 W on indicator)-275 mA, Tcoarse 26
Tfine 0 on dial, 10 in window-235 mA, Tcoarse 23
Tfine 0 on dial, 10 in window

850
810

a

AOM voltage
for fiber alignment
0.4

Servo Voltage

0.35

Manual setting
71 on dial
Manual setting
9 on dial

0.3

-4a

subject to change for desired trap stiffness

2.4.1

Alignment into the microscope

Aligning the three lasers into the microscope was a somewhat iterative process. The
first step of the process began with careful alignment of each laser as it exited the opticalfiber. The beam quality needed to be excellent (look like a perfect Gaussian by eye) at
the fiber exit. If it was not, polishing the fiber often improved the beam quality, but if
fiber-polishing failed, we replaced the fiber with a new one. After exiting the fiber, eachlaser was partially collimated by a short focal length (6-8 mm) aspheric lens (OFR). These
lenses often introduced rings or other beam imperfections; we typically tried multiple lenses
until we found one that did not mar the beam quality. Next, the laser passed through a
2nd lens to complete the collimation at the desired beam size. Collimating the beam andensuring it projected straight and level at the desired height (12.09 cm from the breadboard)
after the fiber exit provided the appropriate starting point for aligning into the microscope.
The alignment was checked to ensure the beam remained level after encountering the othermirrors before entering into the microscope.
Once the lasers were coupled into the microscope (via the system of mirrors, dichroics,
and lenses in Fig. 2.2), the lasers could be visualized on the CCD camera. A typicalsample (microscope slide with a coverslip affixed via double-sided tape) was inserted into themicroscope with the coverslip on the bottom. The microscope was aligned (objective moved

33
to focus on coverslip surface, condenser aligned for Kohler illumination). The lasers reflectedslightly off the coverslip surface, and with appropriate filters and camera adjustments, enoughlight leaked through to the CCD camera to visualize the laser foci. The laser foci weremade to appear symmetric, and alignment was adjusted to ensure the laser spots remained
symmetric even when moved via the PZT mirrors. If the spots were not symmetric, thebeams were checked for misalignment, clipping, or degradation somewhere in the beampath. Visualization of the lasers via the CCD camera was the first step in fine-tuning thealignment into the microscope.

2.4.2

1st-order focal-plane alignment: Aligning laser foci with imaging plane

The laser spots were also made to be visible at approximately the same imaging plane
as objects in the microscope sample. If the spots were not visible at the imaging plane,
the laser collimation was adjusted by adjusting the position of the fiber at the fiber launch.
In practice, we found aligning the laser spots at the imaging plane of the microscope as
visualized on the camera resulted in a less-than-optimum trap quality (possibly due to differences in wavelength of the imaging light and laser light). We therefore offset the camera
so that the camera visualized an imaging plane that coincided with the laser foci at a pointthat optimized trap-quality (Note: this alignment was only possible after iterating throughlater steps in the alignment process to determine the alignment that optimized trap quality).
Having the imaging plane and laser foci aligned is not essential for performance, but is very
convenient for the user. Characterization of the laser spots in our instrument is available onvideo [AEC VHS video]. This video shows all three lasers on a sample of beads fixed to thecoverslip surface.

2.4.3

Alignment of QPD axes with stage axes

The next step in aligning the instrument was aligning the axes of each QPD with thepiezo-stage axes. First, the QPDs were “zeroed,” i.e. centered with the laser paths. This34
centering was accomplished with a sample on the microscope with nothing in the beam path.
Then x and y micrometers on the QPDs were translated until the x and y output voltages
read 0 V. (This QPD centering was performed at least daily to guarantee precise centering of
the QPDs and lasers.) After QPD centering, the alignment of the QPD axes with the stage
axes was characterized by scanning a bead fixed to the sample surface through the lasers inthe x and y axes, using the piezo stage. Typical scans are shown in Fig. 2.9. Alignmentof the QPD with the stage axes was performed by rotating the QPD until the crosstalk
amplitudes of the non-moving axes were minimized.

2.4.4

Centering beads and beams

Once the QPD axes were aligned with the stage axes, scans like those in Fig. 2.9 could
be used for centering a laser over a bead or centering a bead under a laser. In combination,these centering routines could also center two lasers over the same bead, thus aligning thelasers with each other. This alignment was accomplished through an automated centering
routine written in LabVIEW. First, a bead was scanned in x and y through one laser via thestage to obtain signals like those in Fig. 2.9. The on-axis signals were fit to the derivativeof a Gaussian, the symmetry point was determined, and the stage was moved so that thesymmetry point was centered under the laser. Once the bead was centered via the stage
under one laser, the 2nd laser was centered on top of the bead via the PZT mirror moving
the 2nd laser. The alignment was similar to that via the stage, except now the 2nd beam was
moved over the bead while the stage remained motionless, and the beam was moved so thatthe symmetry point was under the center of the beam. Through these centering routines,
one or more lasers were centered over a bead and/or aligned with another laser.

2.4.5

Axial alignment by minimizing crosstalk

The vertical alignment of the lasers within the microscope and detection system was
the next step in alignment. To characterize this alignment, we performed axial (z) scans of35

Figure 2.9: x, y scans. The x (blue) and y (green) QPD voltage response is plotted for x
and y scans of a 330 nm dia. bead through each laser. The shape of the QPD response on
the axis of motion approximates the derivative of a Gaussian. The off-axis response shows
little change, demonstrating the alignment of the QPD axes with the stage axes. The plotsof the 1064 laser show the noise problems associated with the 1064 laser in our system. Notethat this noise is not evident in the x and y power spectra of trapped beads detected via thedetection lasers (data not shown), and thus appears not to hinder our measurements via thedetections lasers.

36
a stuck bead through the lasers via the stage (see Fig. 2.10). Optimum vertical alignment
occurred when the x and y crosstalk-signal amplitudes were minimized. This alignment was
typically adjusted by moving the 3D translation stage holding the fiber mount. However,other optics in the system may also affect this alignment, including optics in the detector
arm itself.

2.4.6

2nd-order focal-plane alignment: Aligning axial detection signals

Next in the alignment process came the relative alignment of the foci of the threelasers. As mentioned previously, the 1st-order alignment of the foci was performed using
visualization of the laser spots on the CCD camera (all three laser spots were made visible
at approximately the imaging plane of the microscope). In this step, a more precise 2nd-orderalignment was performed by scanning a stuck bead through multiple lasers simultaneously
using the stage. First the lasers were precisely aligned with each other in x and y andcentered over a bead. Once the lasers were aligned in x and y over the bead, the bead wasscanned in z via the piezo stage and the axial signals were recorded. Characterization ofthese signals is shown in Fig. 2.11. To align the foci, the inflection points of the axial signals
were aligned. Again, this alignment was typically adjusted by moving the fiber tip in thefiber-exit launch.

2.4.7

Minimizing trap degradation with depth by eliminating slope of axial
voltage oscillations

The positions of the laser foci were governed by the laser collimation. Another important impact of the laser collimation for the trapping laser was the trap quality. The trap
quality is optimum at the coverslip surface. When the trap is moved further into the sample,
the trap is degraded due to spherical abberations. This degradation is inevitable, but seems
to be minimized by adjusting the laser collimation. A true test of this effect is to test thetrap stiffness as a function of height, which we show later, but at this point in the alignment

37

Figure 2.10: z scans with x, y crosstalk. The x (blue), y (green), and z (red ) QPD voltage
response is plotted for z scans of a 330 nm dia. bead through each laser. The shape of theaxial signal is ascribed to the interference pattern that occurs as the bead passes throughthe focus of the laser, where a Gouy phase shift occurs [18]. The minimal changes in thevoltage response of the x and y axes demonstrate the alignment between the laser path andthe z axis of the stage. In other words, low crosstalk indicates the laser is vertically alignedin the microscope.

38

Figure 2.11: Axial scans to show relative z. The axial detection voltage is plotted for axial
scans of the same bead (acquired simultaneously) for two different lasers in each plot. The
relative axial alignment of the curves is an indication of the relative axial alignment of the
3 lasers. Note that these plots are also sensitive to the collimation of the laser, and thus
these plots alone should not be used for optimizing the relative axial alignment of the lasers.
Note: 1064 sum voltage is offset to more easily show relationship between two signals.

39
process, we found a quick characterization was helpful. This quick characterization involved
trapping and detecting a free bead while moving the stage axially. The signal obtained isshown in Fig. 2.12. We found adjusting the collimation to make a line drawn through theoscillating region as flat as possible provided the desired laser collimation. It is also possiblethat collimation of the detection laser affects the flatness of this signal, though I have notverified this personally.
The iterative nature of the alignment steps becomes clear with the previous step. Here,the trap collimation was adjusted to optimize the trap quality with depth, but in order tomeasure the trap quality, we had to perform all of the previous alignment steps. However,in this step, if the trap collimation was changed, the trap focus position changed, and theother lasers needed to be adjusted to follow. This step is also the point where the camera
was moved so that the imaging plane was coplanar with the laser foci. Once these changes
were made, the axial signal of a trapped bead was checked again, to ensure the oscillations
remained flat, reflecting the optimum laser collimation.

Figure 2.12: Axial scan of a trapped bead. The axial detector voltage response is plotted for atrapped bead as the stage is scanned axially. The bead was trapped by the 1064 laser. Whenthe stage was high (+1000 nm), the bead was pushed upwards in the trap and detection
lasers, above the foci. As the stage was lowered, the bead followed the stage until the beadreached the trap equilibrium point (at approximately +400 nm in this plot.) As the stage
continued to be lowered, the bead remained at trap equilibrium, and the detector voltage
response oscillated. This oscillation is credited to the bead-coverglass distance changing, andacting as a Fabry-Perot cavity [19]. Minimizing the slope of these oscillations is one method
we use to minimize trap degradation with depth.

40
2.4.8

PZT-mirror calibrations

Moving from alignment characterizations to a calibration, calibration of the PZT mirrors was next in readying the instrument for precision measurements. To calibrate the PZT
mirrors, voltages were applied to move the PZT mirrors such that the laser moved in agrid-like pattern. To follow the laser, a bead fixed to the sample surface was moved via thestage (which is factory calibrated with sub-nm precision). The stage precisely centered thebead under the laser at each position via the automated process previously described, and acorrespondence was recorded of PZT voltages (V ) to stage positions (nm). Lines were fit toeach nm vs. V cross-section, and the average of the slopes of these lines yielded the nm(V )calibration in x and y, as well as the crosstalk terms. A plot of this calibration is shown in
Fig. 2.13. The grids represent the stage positions recorded at each laser position, thus eachpoint on the grid also has a corresponding PZT mirror voltage. The calibrated quantities
obtained include x(nm/V ), y(nm/V ), x(nm)/y(V ), and y(nm)/x(V ).
In using the PZT calibration, we accounted for both the on-axis calibration and thecrosstalk using the equation
4Vx =4x(nm)/x(nm/V )))∗ y(nm)/x(V
1 − x(nm)/y(Vy(nm/V )
x(nm)

,

(2.1)

for motion requested on the x axis, and
4Vy = −4Vx ∗y(nm)/x(V )x(nm/V )(2.2)

to correct for the cross-talk on the y axis. Here, 4Vx represents the change in voltage
to apply to the PZT mirror in the x axis to move the requested distance, 4x(nm). 4Vy
represents the voltage to apply in the y axis to correct for the crosstalk that will occur. By
switching x and y in both these equations, motion along the y axis can be requested. With
this calibration and these equations, we could precisely position the laser in our sample using
the PZT mirrors.
We also verified the PZT mirror calibration to ensure the calibration was repeatable

41

Figure 2.13: PZT mirror calibration. The laser position to PZT mirror voltage correspondence for each PZT mirror was calibrated by moving the mirrors in a grid pattern of voltages.
An automated program applied the voltages (hence moving the laser) then centered a stuck
bead via the stage under the laser. These x, y stage positions are plotted. Using the corresponding applied voltages, the distance moved per applied voltage (nm/V ) was calculated
in each axis. Crosstalk terms were also calculated.

42

Figure 2.14: Verification of PZT mirror calibration. To check the accuracy of the PZT
mirror calibrations, the calibrations were used to move the PZT mirrors in a grid-pattern ofpositions. A stuck bead was then centered under the laser via the stage to provide a measured
position. The measured positions (plus sign symbols) have an average percent error of 1.9%
(std. dev. 1.5%) in each axis from the requested mirror positions (open circles).

43
and that our automated software correctly read and applied the calibration. The PZT
calibration verification program worked similarly to the original calibration program, butinstead of moving the PZT mirror in a grid of requested voltages, the mirror was moved
in a grid of requested positions, converted to voltages by the calibration described above.
The stage, with a bead fixed to the sample surface, again followed the laser and precisely
centered under the beam at each position. The obtained data is represented in Fig. 2.14,
where the requested positions are represented by open circles, and the measured positions are
represented by the plus sign symbols. The average percent error in requested vs. measured
motion was 1.9%, verifying our PZT mirror calibration.

2.4.9

Detection sensitivity calibrations

Using the PZT mirror calibrations, we calibrated the detection sensitivity for a trapped
bead in the x and y axes. Detection sensitivity calibrations could also be done with beads
fixed to the sample surface which was useful when making measurements on stuck beads.
However, obtaining the calibration of a trapped bead allowed calibration of the actual bead
to be measured in the experiment at a height in the beam similar to the measurement height.
For these reasons, detection sensitivity calibrations with a trapped bead provided for more
precise measurements. To obtain sensitivity calibrations with a trapped bead, the bead
was trapped and scanned through the detection laser in each axis using the calibrated PZT
mirror. The QPD voltages in x and y were recorded and plotted vs. the requested beadposition, as shown in Fig. 2.15. To obtain the desired calibration, each plot was fit to a
7th-order polynomial with the axes reversed such that a calibration function, nm(V ) was
obtained.
For precise sensitivity calibrations, several factors were optimized. The amplitude of
the voltage signal were maximized within the detection range of ±10V using the adjustable
gain on the detection electronics (custom JILA electronics). The curve was acquired suchthat it appeared smooth with enough data points to provide for a good fit, as in the figure.

44

Figure 2.15: Detection sensitivity of a trapped bead. For x, y detection sensitivity calibrations of a trapped bead, the bead was trapped by the 1064 laser, then swept through the810 detector laser using the 1064 PZT mirror. The resulting detector voltage for a 330 nm
bead is plotted here (open circles) vs. the bead position. The x axis is shown in blue, they axis in green. Plots like these, with the axes switched to plot position vs. voltage, were
fit to a 7th order polynomial to provide the nm(V ) detector sensitivity calibration for each
bead.

45
Also, the range of motion was chosen so that the curve did turn over at the maximum andminimum (as that would create a non-function when the axes are reversed). However, therange was large enough that the curve did extend to near the maximum and minimum sothat the 7th-order polynomial would have real curvature to fit rather than trying to fit noisein the curve. Care was also be taken to perform these calibrations quickly to minimize
drift of the system during the calibration. These sensitivity calibrations were automated in
LabVIEW, with easily adjustable data acquisition parameters so that the calibration could
be optimized.
Importantly, these sensitivity calibrations could also be acquired on a tethered bead.
With a short tether the validity of the calibration depended on ensuring the bead wascentered in the trap throughout the calibration. This was accomplished by moving thesample and the laser together during the calibration, such that the short tether did notpull the bead out of the center of the laser. Using these calibration methods, the detection
sensitivity in x and y were obtained for each bead to be measured.

2.4.10Drag measurement

Next in the steps of characterization was an essential calibration and alignment checkfor optical traps: the drag measurement. In the drag measurement, a bead was trapped
and the sample was oscillated, creating a drag force to pull the bead out of the trap center.
The drag force was calculated based on the velocity of the sample oscillations, and the beadposition was measured. A drag force (F ) vs. bead displacement (x) measurement is plottedin Fig. 2.16. This plot demonstrates the linearity of the trap. The slope of this plot also
provided a measurement of the trap stiffness (k), since for an optical trap F = −kx.

2.4.11

Trap symmetry and ellipticity

Next, another important trap symmetry check was made by plotting the x and ypositions of a trapped bead. A plot of x vs. time, y vs. time, and x vs. y is shown in46

Figure 2.16: Drag-force measurement. A 330 nm bead was trapped at 1 V on the intensity
servo, and the stage was oscillated to create a drag force on the bead. The force on thebead was calculated as the drag multiplied by the stage velocity. This force is plotted vs.
the measured bead displacement, for x in the top plot, and y in the bottom plot. The trap
stiffness at this voltage is the slope of a linear fit to these plots.

47

Figure 2.17: Trap ellipticity. Plots of bead position for a trapped bead. The top 2 plots
shows the x and y positions of a trapped bead vs. time. At the bottom, the (x, y) positionis plotted for each data point to show the shape of the optical trap. A line was fit (black,solid ) to find the slope of an ellipticity of the trap shape. The slope for this fit was 0.02.
The average slope for 5 plots with voltages 1-5 V was 0.0053.

48
Fig. 2.17. Typically the x vs. y plot for a trap is somewhat elliptical, and for simplificationof calibrations and measurements, the axes of this ellipse should align with the x and ydetection axes. To determine the ellipse axes, we fit the x vs. y plot to a line. The slope ofthe line represents one axis of the ellipse. The mean slope from fitting 5 plots was 0.0053.
This close alignment with the detection axes was accomplished by tuning the post-fiber halfwave plate in the pre-microscope optics. This measurement ensured the trap ellipse wasaligned with the detection axes and that the general shape of the trap was symmetric.

2.4.12

Axial alignment by sensitivity offset vs. height

Next was a third and final characterization and optimization of the axial positions andalignment of our lasers. For this characterization, we scanned beads on short DNA-tethers
through the trapping and detection lasers. We scanned the tethered beads in x and y ata series of axial positions, obtaining sensitivity calibrations at each axial position. As acharacterization of the axial alignment, we plotted the offset term (a0 ) vs. stage height(zstage ). A flat a0 vs. height curve represented an optimally vertical trap. Fig. 2.18 shows
the a0 terms in x and y for our instrument.
2.4.133rd-order focal-plane alignment: Optimizing xy-sensitivity vs. height

For a final optimization of the axial position of the detection laser foci relative tothe trap position, we plotted the first-order term (a1 ) of the 7th-order polynomial x and ydetection sensitivity calibration curves. The a1 sensitivity term was roughly parabolic asa function of height, with a flat peak near the laser focus, and with decreased sensitivity
further from the focus. For precision in our experiment, we wanted to ensure that when we
performed a sensitivity calibration of a trapped bead, the calibrated sensitivity would be thesame as when we actually performed our measurements. Typically, sensitivity calibrations
occurred with the bead near trap equilibrium. Measurements were performed with the beadpulled down in the trap (to exert force), below trap equilibrium. To optimize the precision

49

Figure 2.18: Sensitivity vs. z. The x and y detector sensitivity, nm(V ), was obtained fora bead on a short tether at several different heights. The sensitivity at each height was fit
to a 7th order polynomial. Here, the a0 and a1 terms are plotted for each height for the x
and y axes. The flatness of the a0 terms as a function of height (top) indicates the verticalalignment of the lasers, detectors and piezo stage axes. The a1 terms (bottom) represent thesteepness of the slope of the sensitivity curves. For accurate sensitivity calibrations, the final
measurement should occur at a height with similar sensitivity to the calibration height. We
satisfy this requirement by setting the focus of the detection laser such that a bead traps
at the high end of the flat region during calibration, and at the low end of the flat region
during the measurement. z stage = 0 is the approximate stage position where the tethered
bead is sitting at trap equilibrium. At z stage > 0, the stage is pushing the bead up in thetrap, and at z stage < 0, the stage is pulling the tethered bead down from trap equilibrium.

50
of our measurements, we moved the detection laser focus such that the trap equilibrium
point occurred near the top of the flat-peak region of the a1 vs. height curve. Thus when wecalibrated at trap equilibrium and when we measured slightly below trap equilibrium, thecalibration would be as valid as possible. Fig. 2.18 shows a1 as a function of height in our
instrument.

2.4.14Laser power measurements

Measurements of laser power were intrinsically important, as well as providing diagnostic tools for checking instrument functionality. We measured laser power at several points
in the beam path, using both photodiodes and a power meter (see Tables 2.4 and 2.5). The
in-loop photodiodes (which sampled each beam just after the optical fiber) provided a goodreference voltage for a properly functioning instrument. As another reference, we also measured the power via the QPD sum voltage. Both these sets of photodiodes provided quickreference voltages to ensure the instrument was behaving consistently from day-to-day. We
also obtained true measurements of the power via a power meter placed just before the microscope (between the last two pre-microscope lenses). From this power measurement, weestimated the power at the sample plane within the microscope.
Table 2.4: Detector voltages without servos.
Lasera
1064
850
810

VAOM b
(V)
0.4
0.35
0.3

QPD sumc
(V)
-1.92
0.4
0.648

In-loop PD
(V)
7.72
0.76
0.485

a

with typical laser settings from Table 2.3.
Voltage directly from computer to AOMs, bypassing servos.
cwith no bead in beampathb

The power at the sample plane (Psample ) was calculated from the power measured

51
Table 2.5: Laser Power with servos on.
Lasera
1064
850
810

QPD sumb(V)-1.03e0.367
0.721

In-loop PD
(V)
3.963
0.675
0.518

pre-microscopec
(mW)
421.3
0.313
0.516

At sample planed
(mW)
146.6
0.173
0.290

a

with typical laser settings from Table 2.3, except as noted.
with no bead in beampathcMeasured between last two lenses before microscope.
d
Calculated as described in the text.
e
at 1064 Vservo = -3.87 Vbbetween the two lenses before the microscope (Ppremicroscope ) as follows
Psample = Ppremicroscope ∗ Tlens ∗ Tdichroic ∗ Tobjective (∗Toverf ill ),(2.3)

where T is the transmission coefficient for each component. The subscripts for each transmission coefficient correspond as follows: lens - last lens before the microscope (Linos NIR
Doublet), dichroic - the dichroic in the base of the microscope (Chroma), objective - themicroscope objective (CFI Plan Apochromat 100x oil IR). For the trapping laser (1064), thelaser overfills the objective back aperture, leading to an additional transmission coefficient
multiplier with subscript overf ill - the overfill factor. We estimated the transmission coefficients for each component and used the above equation to calculate the final power at the
sample plane listed in Table 2.5.
Estimates for the transmission coefficients came from various sources. We used estimates from the manufacturers for the lens and dichroic: Tlens = 0.96, and Tdichroic = 0.96.
We estimated the wavelength dependent Tobjective as 0.59 for the 1064, 0.60 for the 850, and0.61 for the 810 [13]. For Toverf ill , we calculated the transmission coefficient by measuring
the diameters of our Gaussian beams and calculating the transmission through the objective
back aperture diameter (5 mm). The beam diameters at the point where the intensity had
decreased to 1/e of the maximum intensity were 4.5 mm for the 810 and 850 lasers and 7
mm for the 1064 laser. We estimated Toverf ill as follows: 1 for the 810 and 850, and 0.64 for52
the 1064. Details of the calculation are below.
To calculate Toverf ill , we assumed a Gaussian intensity profile (I) in cylindrical coordinates, I = A∗exp(−(p/w)2 ), where A is the amplitude, p is the radial cylindrical coordinate,
and w is the A/e beam radius. The power P within a radius R is given by
Z 2π Z R
PR =0

I(p, w) pdpdw = π ∗ A ∗ w2 (1 − exp(−(R/w)2 ))

(2.4)

0

The total power, evaluated at R =infinity, is then Ptotal = π ∗ A ∗ w2 . Substituting into theequation above:
PR = Ptotal ∗ (1 − exp(−(R/w)2 ))(2.5)

Using this equation we calculated the transmission of the 1064 laser through the back aperture of our objective (R = 2.5 mm), using w = 3.5/sqrt(2) for our 1064 laser with an A/e2
radius of 3.5 mm. We found the power within a radius of 2.5 mm, P2.5 = Ptotal ∗ 0.64, i.e.
Toverf ill = 0.64 for the 1064 laser.
Following the above methodology, we measured Psample = 146.6mW at a servo voltage(Vservo ) of 4 V. Generalizing this measurement to provide the relationship between Vservo andPsample , we obtain Psample = 36.65(mW/V)∗Vservo .
2.4.15Trap stiffness calibrations

After the previous careful alignment and calibration procedures, we were ready to
perform our trap stiffness calibrations. We calibrated the trap stiffness by three different
methods, equipartition theorem (EQP), power-spectral density (PSD), and drag. In the
EQP method, 21 kB T = 12 ktrap < x2 >, where kB T represents the thermal energy (4.1 pNnm
at room temperature), ktrap is the trap stiffness in a single axis, and < x2 > is the variance inthe position of a trapped bead. In the PSD method, the power spectrum of a trapped beadis recorded and fit to a Lorentzian. The trap stiffness is calculated by ktrap = 2πγf0 whereγ is the drag coefficient and f0 is the corner frequency of the Lorentzian fit. γ for a bead in
fluid is given by γ = 6πηrbd where η is the fluid viscosity (0.001kP a ∗ s for water at room

53
temperature) and rbd is the bead radius. Because most trapping is done near the sample
surface, the drag coefficient must be corrected using Faxen’s Law [13]. The drag measurement
provided the third method of trap stiffness calibration, and has been previously discussed.
These three methods provided independent means of calibrating the trap stiffness.
With automated stiffness calibration programs in LabVIEW, we acquired simultaneous
acquisition of the PSD and EQP stiffness calibrations for several different trap powers. We
typically acquired these calibrations on 5-10 beads, and an average calibration from thesebeads was calculated. For drag, typically one set of measurements over a similar power
range was acquired as a confirmation of the EQP and PSD measurements. Plots of thestiffness calibrations for our instrument are shown in Fig. 2.19. Agreement between thethree methods was better than 5%. The ktrap calibrations were valid for months at a time,
typically only needing to be redone if a change was made in the instrument.
Besides providing stiffness calibrations, these measurements also provide a diagnostic
check of overall instrumental quality [20]. If all three trap stiffness calibrations agree, the trap
is performing according to physical theory, which increases the likelihood that the alignmentis excellent; assumptions about height, temperature, and viscosity are correct; and PZT
and detector calibrations are working as they should. If the three methods do not agree,something about the instrumentation or the assumptions is not correct.
We also verified the stiffness calibrations and the software that would use the stiffness
calibrations. We used the above stiffness calibrations to request a stiffness, then measured
the stiffness, again by EQP and PSD. This verification is shown in Fig. 2.20. We obtained
a mean percent error of 5.8%.

2.4.16

Stability characterizations

Because instrumental stability was of utmost importance in our measurements, we alsocharacterized the final stability. We actively stabilized the instrument on a bead stuck to thesample surface and measured the residual noise with the other detection laser. We measured

54

Figure 2.19: Stiffness calibration. The trap stiffness was measured by three different methods, equipartition theorem (EQP), power spectral density (PSD) and drag. The measured
stiffness is plotted vs. the applied servo voltage. For EQP and PSD, the data points represent the mean measurements on 5 different beads, with error bars representing the standard
error. For drag, the data points represent 1 measurement at each voltage. The line shown
represents the calibration used during subsequent data acquisition. This calibration was
obtained by fitting the servo voltage as a function of the trap stiffness for EQP and PSD
separately, then averaging the obtained slopes and intercepts.

55

Figure 2.20: Verification of stiffness calibration. To check the stiffness calibration, kx and ky
were requested and set via the stiffness calibration. The experimental stiffness was measured
with the equipartition theorem (EQP) and the power spectral density (PSD) as describedin the text. The line in the plots is drawn through the values that would indicate a perfectmatch between requested and measured stiffness; the markers show the actual measured
values. The mean percent error for the measured values was 5.8%.

56
on the same bead (Fig. 2.21), and on a separate bead (Fig. 2.22). We obtained excellent
stability for both the single bead (means of 17 traces, integrated noise in x, y, z of 0.13, 0.063,
0.092, ∆f = 0.03-2.5 Hz) and the separate bead measurements (means of 8 traces, integrated
noise in x, y, z of 0.14, 0.067, 0.10, ∆f = 0.03-2.5 Hz) as is demonstrated by the position vs.
time plots, the PSD plots, and the integrated noise plots (the PSD noise integrated up toeach frequency).
When characterizing and optimizing the stability, a few recurring factors tended to
impact the stability. Very noisy traces with large (>2 nm) jumps tend to be caused byloose or bad cables and connectors. Replacing or stabilizing these cables and connectors was
essential for instrumental stability. Instability can also be caused by the servo electronics
themselves, either by failure to optimize the servo electronics, or electronics that may failover time. Another common culprit of instability is failed or mal-adjusted cable and fiber
stabilizers. These should be firmly stabilized, but without pulling or torquing. Finally, thefloating table is very important to stability; if the table runs out of air, the noise increases
dramatically. Though many other factors can cause noise, these factors tended to be themost common culprits of increased instrumental noise.

2.4.17Force vs. extension for DNA

As a final check of the instrumental alignment and calibrations, we obtained force
vs. extension curves of DNA. We stretched 2013 nm DNA in the x and y axes, obtaining
curves like those shown in Fig. 2.23. We obtained 6 traces (on 3 molecules). We calculated
the geometry-corrected force and extension [21] and fit the curves to a modified worm-likechain [22]. The average persistence length was 42 nm, and the average contour lengthwas 2060 nm. The accepted value for the persistence length for this DNA was 45-50 nm[[23, 24, 21, 22, 25]], and the accepted contour length was 2013 nm. These numbers provided
a reference estimation for the measurement uncertainties that remain in our instrument.

57

Figure 2.21: Stability of stuck beads - 1 bead. The x (blue), y (green), and z (red ) residual
noise during active stabilization are plotted. Both detection lasers were aligned on the same
490 nm dia. bead. One laser actively stabilized the position signals while the second laser
monitored the residual noise. The top plot displays the x, y, and z position trace vs. time,smoothed to 5 pts/sec. The middle plot shows the calculated power spectral density (PSD)
vs. frequency. The bottom plot shows the integrated noise vs. frequency. These traces mostclosely represented the mean noise from 17 individual traces (13 different beads).

58

Figure 2.22: Stability of stuck beads - 2 beads. The x (blue), y (green), and z (red ) residual
noise during active stabilization are plotted. Each detection laser was aligned on a different
490 nm (dia.) bead. One laser actively stabilized the position signals while the second
laser monitored the residual noise. The top plot displays the x, y, and z position vs. time,smoothed to 5 pts/sec. The middle plot shows the calculated power spectral density (PSD)
vs. frequency. The bottom plot shows the integrated noise vs. frequency. These traces mostclosely represented the mean noise from 8 individual traces (8 different pairs of beads).

59

Figure 2.23: DNA force vs. extension. The force vs. extension curves for 2013 nm DNA. A
330 nm dia. bead was tethered to the coverslip surface via 2013 nm DNA. The DNA was
stretched in the positive and negative directions in the x and y axes. The geometry corrected
force and extension were calculated from the bead and stage positions and the calibrated
stiffness [21]. The curves were fit to a modified worm-like-chain model [22], which returned apersistence length (P ) and contour length (L) for each trace. For 6 traces (on 3 molecules),the means were P = 42 nm and L = 2060 nm. The traces shown most closely matched themeans.

60
2.5

SAMPLE PREPARATION
For studying TBP bending DNA, several sample preparations were necessary. For

many instrumental calibrations and characterizations, we prepared samples with beads fixed
to the coverslip surface. For TBP experiments, to prevent protein sticking to the surface, we
prepared samples with customized polyethylene glycol (PEG) surfaces. We fixed beads tothese PEG surfaces to enable stabilization, and we attached DNA tethers to these surfaces
to study TBP bending the DNA. For the TBP experiments, the protein was very sensitive
to conditions (time, temperature, pipetting, storage buffer). This section details the sample
preparations and protein handling used in acquiring the data in this thesis.

2.5.1

Cleaning coverslips and slides

Cleaning the microscope coverslips (and sometimes slides) was essential for most of
our sample preparations. We cleaned by two different processes: KOH cleaning and plasma
cleaning. Sometimes both processes were used, other times only one process was used.
For KOH cleaning, we soaked slides and coverslips for 3 minutes in 5.7 M KOH(Mallinckrodt) in ethyl alcohol (Mallinckrodt, Completely Denatured) solution while sonicating in a bath sonicator (Bransonic Unltrasonic Cleaner 5200). (Note: KOH solution isextremely caustic and must be handled with care.) We rinsed with two 3-minute soaks inbeakers of double filtered, double deionized water (diH20), also placed in the bath sonicator. After each soak, we also rinsed with diH20 from a squirt bottle. We performed a final
rinse with ethyl alcohol. We dried the slides and coverslips by placing them in a microwave
oven for 1.5 minutes. We placed the slides and coverslips in closed containers to preserve
cleanliness.
For plasma cleaning, we cleaned slides and coverslips with oxygen plasma in a Reactive
Ion Etch (RIE, PlasmaSTAR AXIC). The slides and coverslips were placed in a metal holder
in the RIE. An automated program pumped the system down to vacuum (<10mT), then

61
pumped in O2 . A voltage was applied to create the oxygen plasma for 180 s. Then thechamber was pumped down to vacuum again to remove the impurities. In detail, the mass
flow controller was set to 100 sccm, the Power to 550 V, the runtime to 180 sec, and theventing to 1x. The automated program is titled ”Amandaash” on the Keck Lab RIE at
JILA, and the settings should be listed as MFC2 Setpoint 100 sccm, Power 550 W, Runtime
180 s, Vent 1x. After plasma cleaning for 180 s, the metal holder was rotated and the processrepeated to clean for another 180 s. Again, the slides and coverslips were stored in closed
containers to preserve cleanliness.

2.5.2

Flow cells

All samples were assembled on clean flow cells. A flow cell consisted of a microscope
coverslip (Corning No. 1 1/2, 22 x 40 mm) fixed to a slide (Corning Micro Slides plain
2947, Pre-Cleaned, 75 x 25 mm) with double-sided tape (3M Scotch Permanent DoubleSided Tape, 136, 1/2 in. wide) and 5-minute epoxy (Devcon). The tape was sliced in halfalong the length, and each half of the tape was placed near the center of the slide, leaving
an ∼5 mm channel between the two tape halves running across the slide along the narrowdimension. The coverslip was then fixed to the slide such that the long dimension of thecoverslip was perpendicular to the long dimension of the slide. Pressing the coverslip into
the tape provides added stability. Five minute epoxy was then mixed and quickly appliedalong the edges of the coverslip. The fluid epoxy flowed into the space between the slideand coverslip, providing added stability. Once this seepage occurred, additional epoxy wasapplied to the four corners where the slide and coverslip are joined. The edges of the coverslip
overhang the slide, conveniently allowing for fluid to be pipetted through the flow cell.

2.5.3

Antidigoxegenin modification of beads

For many DNA-tethering experiments, we modified latex beads (Invitrogen/Interfacial
Dynamics/Molecular Probes Surfactant-Free White Aldehyde/Sulfate Latex) by attaching62
antidigoxigenin. To modify these beads, we started by passivating two 1.5 mL plastic tubes
for 2 hours with 1.2 mL 2 mg/ml acetylated BSA (acBSA) in PBS buffer. The tubes wererinsed twice with diH20 and drained, and one tube was used during the protocol, while one
was set aside at 4◦ C to be used for storage of the final modified beads.
For many steps in the bead modification process, the bead size and concentration
impacts choices of centrifuge speeds and times and amounts of other reagents added. I mostcommonly used 330 nm dia. beads, so all times, speeds, and amounts in parentheses are
specific to this bead size. In the protocol tube, we washed latex beads (200 µL of 4.2%
w/v for 330 nm beads) 4x in 1 mL PBS buffer (centrifuge for 5 min. at 12krpm to forma pellet, remove supernatant, resuspend in 1 mL PBS and pipette and vortex to suspend).
After washing, we resuspended the beads in 500 uL PBS buffer and added acBSA (15 µL of
80 mg/ml acBSA). We sonicated for 40 min. or more until beads were separated. Next, weadded anti-digoxigenin fab fragments to the beads (1 mg anti-Dig in 500 uL PBS), mixedby pipetting, and incubated on a rotator (>24 hours). After this incubation, we centrifuged
the beads to form a pellet (6 min. at 12 krpm), removed the supernatant, and re-suspended
them in 1.2 mL 1.0 M glycine in PBS, incubating for 40 min. on a rotator to quench thereaction. Next we centrifuged again to form a pellet (6 mon. at 12krpm) and removed thesupernatant. We performed the final resuspension in PBS (1350 uL PBS) plus 40 µL of 80
mg/mL acBSA and 2% (w/v) sodium-azide, to form the final solution ( 0.5 nM for 330 nm
beads). The final, modified-bead solution was transferred to the other acBSA-passivatedtube. These antidigoxigenin beads were used in most of the sample preparations for myexperiments.

2.5.4

Stuck beads

For calibrating and characterizing the instrument, samples with beads fixed to thecoverslip surface were used. For preparation of these samples, we started with epoxied flow
cells with KOH and/or plasma cleaned coverslips. We flowed 20 µL of 25 pM beads diluted in

63
diH20 into the flow cell. We used several types of beads, including anti-digoxegenin-modified
latex beads (Surfactant-Free White Aldehyde/Sulfate Latex, with custom anti-digoxegenin
modifications by our lab) and Streptavidin-modified polystyrene beads (Spherotech, SpheroStreptavidin- Polystyrene Particles). The bead sizes varied from 330 nm dia. to 860 nm
dia. We then washed the flow cell with 20 µL SB buffer (10 mM Tris pH 7.5, 250 mM KCl,
20 mM MgCl2). Next we washed with 200 µL SWB buffer (10 mM Tris pH 7.5, 100 mM
MgCl2), then with 200 µL ethyl alcohol (Completely Denatured, Mallinckrodt). Finally, we
washed with 2 mL of diH20. These salt-stuck beads were well-fixed to the sample surface(see stability data in Fig. 2.22), even without a baking step, as is often used to melt thebeads onto the glass. We preferred not to use the baking/melting step so that we could
preserve the bead shape, which was important for some calibrations and characterizations.
Stuck bead samples used in these characterizations were often filled with diH20, buffer orsometimes free beads (∼0.5 pM) in buffer.
Stuck bead samples could often be reused. If the sample became sticky to free beads, we
often flushed the sample with doubly-distilled methanol followed by 2 mL diH20. Sometimes
we added BSA to these samples to prevent stickiness.

2.5.5

PEG-modification of surfaces

For TBP experiments, non-stick surfaces were essential to preventing the protein fromsticking to the coverslip surface. A common solution to protein-sticking problems is toimplement polyethylene glycol (PEG) surfaces [26]. Protocols for PEG-coating surfaces
available in the literature proved unsuitable for creating a consistently non-stick surface
for our experiments, so we developed a customized protocol for covalently attaching PEG
molecules to our slide and coverslip surfaces.
The first step in covalently attaching PEG to our surfaces was aminosilanization. As astarting point for aminosilanization, clean glass is essential. We cleaned the slides and coverslips first by plasma, then by KOH, either the day-of or day before the PEGylation protocol

64(see Section 2.5.1 for details). We also cleaned the reaction containers for aminosilanization
by performing the KOH cleaning procedure on the slides and coverslips in the same containers as would be used for the aminosilanization, hence KOH-cleaning the containers aswell as the slides and coverslips. The containers used for cleaning and aminosilanization of
the coverslips were 1 L glass beakers for each set of 10 coverslips, with the coverslips held
in a custom-made Teflon basket with a Teflon handle. The containers for the slides werecommercially available glass containers typically used for microscope slide staining, with theslides held in an accompanying glass basket with a metal handle (to prevent reactions withthe metal, we removed the handles during the cleaning and aminosilanization, only attaching them when the baskets were being moved). A final rinse with methanol on the day of
PEGylation completed the preparation of all the glassware. Cleaning the reaction containers
and other glassware after aminosilanization was also important, because glassware will itself
be aminosilanated. We cleaned after aminosilanization with methanol, followed by diH20,
followed by 1M KOH, followed by plenty of diH20.
The aminopropylsilane (N-(2-Aminoethyl)-3-Aminopropyltrimethoxysilane (UCT A0700))
is highly sensitive to air, moisture, and light, all which cause degradation of the chemical.
To prevent degradation, we stored the aminopropylsilane in a vacuum sealed bag (vacuum
sealed using a Food-Saver device) at -20◦ C, and allowed the aminopropylsilane to warm toroom temperature in the dark for 2 hrs before opening to air. Immediately after use, we
re-sealed the aminopropylsilane in a vacuum bag and placed at -20◦ C.
To aminosilanate our slides and coverslips, we incubated them in a well-mixed 2%
aminopropyl silane-methanol solution (prepared with a glass pipette, in a glass container,
mixed with a glass stir-rod). Total incubation time was 21 minutes: 10 minutes followed by 1
minute invigoration in the bath sonicator, followed by 10 more minutes. We then rinsed with
2 methanol baths, also rinsing with methanol from a squirt bottle after each bath. Finally, tocrosslink the aminosilanated surfaces, we baked at 110◦ C for 5-10 min. The aminosilanatedslides and coverslips were assembled into flow cells with no epoxy. (Epoxy may outgas during

65
storage, which is undesirable, and the epoxy may weaken or become brittle over time, so
epoxying was done just before using the final PEGylated samples.)After assembling the aminosilanated samples, we covalently attached the PEG molecules.
For samples to be used for DNA-tethering, we attached an ∼60:1 ratio mix of methoxy-PEG(mPEG) and biotinylated-PEG (bio-PEG) molecules, so that we could later specificallyattach streptavidin and biotynilated-DNA molecules to the surface while maintaining a nonstick surface. Like the aminopropylsilane, the PEG is sensitive to light, air, and moisture,
and was thus treated similarly to the aminopropyl silane, i.e. stored under vacuum at -20◦ C,
allowed to equilibrate to room temperature in the dark for 2 hours before opening to air,and quickly returned to vacuum and -20◦ C after use.
We dissolved the PEG in 0.1 M sodium bicarbonate buffer. The PEGylation process is
pH sensitive, and the pH of the sodium bicarbonate solution is time sensitive, so the sodium
bicarbonate was made within a few hours of the PEGylation process. To dissolve the PEG,
we mixed by vortexing at half-max, then centrifuged at 10,000 rpm for 1 min. We thenpipetted carefully, ∼100x, being careful to form no bubbles and evenly mix the solution
from top to bottom. We then flowed the PEG solution into the assembled, aminosilanatedsamples, and placed the samples in a humidity chamber (pipette box with a lid, with waterin the bottom). We incubated the PEG solution in the flow cells for 3-4 hours in the dark,then rinsed the flow cells with 500 µL Sodium Bicarbonate, followed by 2 mL of diH20.
We then dried the cells completely with nitrogen. We stored the PEGylated flow cells inblack-tape wrapped bottles, vacuum sealed, for up to 1 month.

2.5.6

DNA-tethering beads to surfaces

For TBP experiments, we prepared samples with DNA-tethered beads on our prepared
PEG surfaces. For stabilization during the experiments, we also fixed streptavidin-coatedbeads to the biotin-PEGs on the surface. We began this sample preparation by epoxying a
PEGylated sample. We prepared beads to be used in the sample by washing them (centrifug-

66
ing for 5 min. at 14K rpm to form a pellet, removing the supernatant, and resuspending thebeads by pipetting) 3x in 200 µL phos-tween (10 mM Na Phos, 0.4% tween-20). After washing, we suspended the beads in MTWB-no BSA (20 mM Tris, 50 mM KCl, 0.4% tween-20)
at the desired concentration (1 pM for streptavidin beads to be fixed to the sample surface,
375 pM for antidigoxegnin beads to be tethered via DNA). To separate beads that stuck
together during washing, we sonicated the beads for 5 min.
To fix beads to the surface, we first washed the sample 2x with 200 µL MTWB-no
BSA, then added 20 µL of the washed 1 pM Streptavidin-coated bead solution to the flow
cell to incubate for 10 min. After incubation, we again washed with 2 x 200 µL MTWB-no
BSA.
To form the DNA-bead tethers, we first incubated the washed 375 pM antidigoxegninbeads with 75 pM dig-bio-DNA (DNA modified at one end with digoxegenin, at the otherend with biotin) in equal volumes for 20 min, quickly pipetting and vortexing on medium
power to prevent local concentration mismatches and push the reaction towards forming
complexes with 0 or 1 DNA’s attached per bead. After the final sample wash following
the step to fix beads to the PEG surface, we added 10 µg/ml streptavidin (SA20) solutionin MTWB-noBSA to the sample and incubated for 10 min., followed by washing with 2 x
200 µL MTWB-noBSA. Finally we attached the tethers to the PEG surface by adding thebead-DNA complexes to the sample and incubating for 20 min. After the incubation, we
washed away unattached complexes with another 2 x 200 µL MTWB - no BSA wash.
For a few experiments, we prepared samples with DNA-tethered beads on non-PEG
surfaces. The preparation was very similar to that on the PEGylated surfaces, but beganwith a clean (preferably KOH), epoxied flow cell. We attached biotynilated-BSA to the clean
flow cell by flowing 50 µL of 125 µg/mL bioBSA solution in Sodium phosphate buffer andincubating for 20 min. We then washed the flow cell with a BSA solution, MTWB (20 mM
Tris, 50 mM KCl, 0.4% tween-20, 3 mg/mL BSA), and incubated for 10 min. to allow BSA
to passivate the surface. The remaining steps (attaching straptavidin beads to the surfaces,

67
attaching streptavidin to the surfaces, preparing the bead-DNA complexes and adding themto the sample) were as described for the PEGylated flow cells, but with the BSA buffer
MTWB and an added 10-min wait time between each step to allow the BSA to passivate.

2.5.7

TBP handling

For performing experiments with TBP, details of handling the protein were very important. Recombinant human TBP was provided by the Goodrich Lab, prepared as described inRef. [27]. The protein was stored at -80◦ C, and transferred from the Goodrich Lab freezer tothe Perkins Lab freezer on dry ice. When needed for an experiment, the TBP was removed
from -80◦ C and warmed in finger tips for 1-2 min. before being placed on ice (in a water-filled
cold-block). The buffers for dilution of the TBP were also on ice. The protein was added
to the buffer, pipetted gently 5-10x, then tapped in the tube to mix. The diluted TBP was
flowed into the room temperature sample on the microscope, and data was acquired for 30
min. If undiluted protein remained in the original tube, it was diluted again (after sitting
on ice for 30 min.), added to the sample on the microscope (after washing the sample with
2 x 200 µL buffer), and data was acquired for another 30 min. Any remaining protein was
discarded at this point (though care was taken to efficiently use the protein).
2.6

DATA ACQUISITION
For this thesis, we acquired several types of data, with the instrumental calibration andcharacterization, and TBP and control data on DNA tethers. This section provides detailsof the data acquisition.
For all types of data, warming up the instrument for several hours was important forstability and repeatable measurements. A minimum warm-up time was not determined,
but typically the instrument was turned on at least 3 hrs before acquiring meaningful data.
Turning on the lasers, electronics, and microscope lamp, and projecting the lasers throughthe microscope seemed to best stabilize the system.

68
2.6.1

Computer, controllers, drivers, and electronics

For precision and repeatability, data acquisition, sample positioning, laser positioning
and laser intensity control were performed via a computer using custom-written LabVIEW
software. Data was acquired on a computer via a 333 kS/s, 16-Bit, 16-Analog-Input Multifunction DAQ Board (National Instruments NI-PCI-6052E). Before being read by the DAQ
board, position data from the QPDs was anti-aliased at 1/2 the data acquisition frequency
using custom JILA electronics. The sum signal from the QPD was offset and differentially
amplified with custom JILA electronics. Voltage data was communicated from the computer
to the intensity-servo electronics, AOM drivers (when not using the intensity servo), shutters(Uniblitz) and PZT-mirror controllers (PI E-509.x3) via a 16-Bit Static Analog Voltage Output Board (National Instruments NI-PCI-6703). The stage position was both controlled andreported via a stage controller (PI E-710.P3D). Using computer-based LabVIEW software
for measurements and instrumental control allowed precision and repeatability of measurements.

2.6.2

Centering lasers and beads

For data acquisition, precisely aligning one or more lasers with a bead was often necessary. This process is described for aligning over a stuck bead in Section 2.4. As an added
step in this process, we sometimes had to manually move the fiducial detection laser via themicrometers on the PZT. This manual movement was necessary when a fiducial was beyond
the range of the available PZT motion (∼ ±5 µm).
For a trapped bead, we often wished to align the trapped bead precisely with thedetection laser. We accomplished this similarly to the process with a stuck bead, but using
the PZT mirror to translate the trap laser while detecting the position signal via the detection
laser. We acquired scans that look similar to those in Fig. 2.9, again fitting to the derivativeof a Gaussian, finding the symmetry point, but now moving the trap via the PZT mirror to69
move the symmetry point to the center of the detection laser. For trapping a bead attachedto the surface by very short tethers, we modified this process by moving the stage and PZT
mirror simultaneously to prevent pulling the bead out of the trap center. These centering
processes were all automated in LabVIEW.

2.6.3

Setting height

We also often wished to precisely control the height of the lasers in our sample for ourdata acquisition. To determine the height, we aligned a bead in our laser (either by centeringover a stuck bead or trapping a bead) and moved the stage axially. For a stuck bead, theacquired signals looked like those in Fig. 2.10. For a trapped bead, the signals looked likethose in Fig. 2.12. To precisely position the height of sample relative to the lasers, we fit aparabola to the maximum peak of these signals, and chose a height offset from that peak.

2.6.4

Detection sensitivity calibrations

Once we accomplished sample positioning in the lateral and axial directions, we typically needed to calibrate the detection sensitivity of our bead in x, y, and z. The detection
sensitivity calibration in x and y is described in Section 2.4. For the axial direction, we
simply used the axial signals, as in Fig. 2.10. We switched the axes and fit the min to max
portion of the curve to a 7th-order polynomial, obtaining a nm(V ) calibration. For trapped
beads, the axial calibration is more complicated and is described in the next chapter. Thesesensitivity calibrations were automated in LabVIEW, and typically performed on each bead
we measured.

2.6.5

Measuring trap stiffness and force

For measuring force (F ), as is typically done in optical trapping, we calibrated thetrap stiffness (ktrap ), measured the bead displacement (xbd ), then calculated force assuming
a linear force vs. bead displacement relationship, F = −ktrap ∗ xbd . The ktrap calibrations

70
are described in detail in Section 2.4. Measurement of the axial force is discussed in the next
chapter.

2.6.6

Centering over DNA tetherpoint

Many of our experiments were completed on DNA-tethered beads. For these experiments, we carefully aligned the surface-tethering point relative to the lasers. For this alignment, we performed DNA force vs. extension measurements, like those shown in Fig. 2.23.
We fit these curves with a worm-like-chain model [22] in both axes, both directions. The
symmetry point of these fits represented the surface-tethering point, which we could thenposition precisely relative to the laser center.
For many experiments, we wished to iterate through the above processes at least twice
to ensure precision. We often centered beads in our lasers, determined the height, calibrated
the sensitivity in x, y, and z, performed a tether centering routine, and repeated. For measurements on short tethers, roughly pre-centering over the tether point using average x andy sensitivity-calibration parameters was necessary.

2.6.7

TBP and control tether data acquisition: DNA extension vs. time

We acquired TBP and control data on DNA-tethered beads on PEG surfaces. Our final
assay is shown in Fig. 2.24. We tethered 330-nm dia. antidigoxigenin-modified latex beads
to the PEG surfaces with 92 nm DNA, following the protocols described in Section 2.5. After
sample preparation, we washed the sample with the buffer chosen for the experiment. Wealigned the sample on the microscope, then selected individual tethers for data acquisition.
With a custom-written LabVIEW program, we pre-stretched the DNA as described above,checking for a persistence (P ) and contour (L) length consistent with a 92-nm tether. This
screening was very preliminary because the program used average values for the sensitivity
calibration, the fit was performed without geometry corrections, and the height and centering
had only been roughly determined at this point.

71

xfid
zbd
zfid

Figure 2.24: Final assay. Shown on the left, a DNA-tethered bead is trapped directly
above the tethering point. The bead is tethered by biotynilated DNA attached to thesurface by streptavidin bound to covalently attached biotinylated-PEG molecule. The beaddisplacement is detected by a second laser (dashed curves); displacement in the axial direction
from the trap center in denoted zbd . Shown on the right, a streptavidin coated bead is
attached to the surface by biotinylated-PEG. The position (e.g. xf id , zf id ) is measured by aseparate laser and actively stabilized.

72
If the tether passed the screening criteria, we continued with data acquisition. First
we manually aligned the fiducial detection laser over a stuck bead (using the micrometer on
the PZT mirror). Next we set the height and centered over the tether-point as describedabove, then centered the fiducial-detection laser over the stuck bead with the PZT mirror.
Then we calibrated the x, y, and z detection sensitivity for both the trapped-bead and thefiducial. (See the next chapter for z detection details). Next we performed a final x and y
stretch of the DNA, before beginning data acquisition of position vs. time.
We stabilized the sample during the position vs. time data. We set the axial position
and force by changing the axial stabilization set-point until the axial bead position (zbd )moved to the desired position. We acquired data on 6 channels (x, y, z of the fiducial; x, y, zof the trapped bead) at 4 kHz, anti-aliasing at 2 kHz. We typically acquired data for 100300 s per tether at a given force, unless something happened to stop the data acquisition(e.g. tether detached or another bead fell into the trap). If data at another force or beadposition was desired, we then moved the sample-stabilization set-point and/or changed thetrap stiffness and acquired more data. On a given sample, we usually acquired at least 100s of control data on the sample before adding TBP to the sample.
2.7

DATA ANALYSIS
We analyzed our data both in real-time during data acquisition as well as afterwards

for more in-depth analysis. For real-time analysis, we used LabVIEW. We performed more
extensive analysis after acquisition using Igor.
Real-time analysis of data during acquisition was performed for centering, calibrations,and averaging of signal vs. time data to allow visualization and active sample stabilization.
Analysis for centering and calibrations was already described in Sections 2.4 and 2.6. For
sample stabilization, the LabVIEW-based software servo is also described in Section 2.3. For
stabilization, position vs. time data was box-averaged and stabilized at 100 Hz in LabVIEW.
Post-acquisition analysis in Igor was used for plotting and viewing, averaging and73
decimating, fitting, performing mathematical transformations, and simulating. We wrote
custom Igor procedures to plot and quantify essentially every type of instrumental dataacquired. To highlight the programs where Igor analysis was essential, I will discuss theprocedures used to process PZT calibration data, tether stretching data, position vs. timedata, and the step-like DNA-extension changes in TBP data. We also wrote simulations
in Igor to simulate a bead in an optical trap and to simulate TBP-induced DNA-extension
changes.

2.7.1

PZT calibrations

For calibrating the PZT mirrors, the data was loaded into Igor as a matrix of x andy stage positions with corresponding Vx and Vy PZT voltages. These were plotted and fit
in cross-section to obtain slopes, or nm/V values, in the x and y axes, as well as cross-talkterms. These slopes were averaged and the averages used as the calibration values for the
PZT mirrors.

2.7.2

Force vs. extension

For DNA force vs. extension curves, the lateral bead displacement and the lateral
tether-point displacement were recorded in LabVIEW. These values were loaded into Igor,
and a custom-written procedure calculated the geometry-corrected force and extension [21].
The curve was then fit in Igor with a modified Worm-like-chain [22] to obtain the persistence
length and contour length.

2.7.3

Stability/noise calculations

Position vs. time data was processed in Igor to show the instrumental stability andnoise. The data was Box-smoothed and the standard deviation at the smoothing rate provided one measurement of the noise. We also calculated the power spectral density using
the Wavemetrics PSD algorithm. We then calculated the integrated noise vs. frequency by

74
integrating the PSD from a minimum frequency up to the given frequency. The standard
deviation, PSD, and integrated noise were all used to characterize the noise of a given trace.

2.7.4

Simulation: Bead in an optical trap

We simulated position vs. time traces for a bead in an optical trap. Simulation werebased on the theoretical characteristics of a trapped bead trace, namely the theoretical
relationships between the trap stiffness (ktrap ) and various trace properties: the standard
deviation (σ = sqrt(kB T /ktrap ) where kB T is the thermal energy), the corner frequency ofa fit to the power spectral density (PSD) of the trace (f0 = ktrap /2πB(h) where B(h) is
the Faxen’s Law corrected drag at height h), and the autocorrelation time constant (τ =B(h)/ktrap ). Note that to fully specify these properties, we chose an input ktrap and beadheight, h. The simulation input and output values are shown in Table 2.6.
Table 2.6: Trapped bead simulation input and output values.
Value
ktrap input (pN/nm)
ktrap via EQP
ktrap via PSD
ktrap via tau
σ (nm)
f0 (Hz)τ (ms)

Simulation Input
0.0300

11.7
384
0.415

Output, Full Bandwidth

Output, Filtered

0.03000.03090.0308
11.7
395
0.404

0.03410.03100.0276
11.0
397
0.451

Fig. 2.25 outlines the process we followed for the simulation. First, we made a Gaussian
noise wave with the desired length and time scaling (10 s at 120,000 pts/sec). Next we madean exponential wave with the desired time constant, τ . We convolved the Gaussian noise
wave with the exponential wave, deleted the 2nd half of the wave that contains irrelevant
points, and scaled the remaining wave to have the correct standard deviation. Finally, we
filtered the resulting trace to 2kHz by taking the Fast-Fourier Transform (FFT), setting thecomponents with f > 2 kHz to zero, and taking the inverse FFT to yield the final filtered

75
trace. We analyzed both the full bandwidth and the filtered simulation. The resulting
traces, histograms, PSDs, and autocorrelations are shown in Fig. 2.26. The histograms
were Gaussian, the power spectral densities were Lorentzian, and the autocorrelations wereexponential. The std. dev. of the trace, corner frequency of the Lorentzian fit to the PSD(f0 ), and characteristic time (τ ) of an exponential fit to the autocorrelation are shown in
Table 2.6. We also calculated the simulation output ktrap from these values as described by
the equations above. At full bandwidth, all the simulation output values match well withthe simulation input values. However, after filtering to 2 kHz, the accuracy of the values
derived from equipartition and autocorrelation analysis decreased. In contrast, the values
derived from the power spectral density analysis remain accurate even after filtering.

2.7.5

Calculation of axial bead position probability distribution

To determine the expected distribution of positions of a trapped bead in the axial
direction, a calculation must be made of the expected probability, P (z), of axial positions ofa bead in a laser trap. This probability is related to the axial potential energy profile of thetrap (U ) and the thermal energy (kB T ) by Boltzmann statistics [13]P (z) ∝ e−U/kB T .

(2.6)

U is simply the negative integral of the total force on the trapped bead (Ftotal ),Z
U =−Ftotal dz.

(2.7)

This statement is only true for conservative forces. Ftotal is the sum of two forces, thescattering force (Fscat ) and the gradient force (Fgrad ) [28]Ftotal = Fscat + Fgrad .

(2.8)

The scattering force is not a conservative force (the integral is path-dependent), but for the
purposes of this calculation, the integral simply can be calculated for the path along x = 0,

76

Figure 2.25: Process for simulating trapped bead position traces. The goal was to simulate
a position vs. time trace with the spatio-temporal properties of a bead trapped with a
stiffness, ktrap , at a height above the surface, h. To create the simulated trace, a Gaussian
noise trace (A) was convolved with an exponential trace (B). The exponential was given by
y = exp(−t/τ ) where τ = B(h)/ktrap and B(h) is the Faxens corrected drag. The resulting
convolved trace had two distinct halves: one half was Gaussian noise with a Lorentzian power
spectral density (PSD), the other half was a constant. We truncated the convolved trace tothe relevant Gaussian noise portion, then scaled it to have the correct standard deviation (σ)
for ktrap , σ = sqrt(kB T /ktrap ), where kB T is the thermal energy. This process yielded a final
trace with Gaussian noise, a Lorentzian-shaped PSD, and an exponential autocorrelation.
(C) We then took the fast-fourier transform (FFT) of the final trace (dark gray) and filteredit by setting FFT=0 for all frequencies, f > 2 kHz (light gray).

77

A

B

C

D

Figure 2.26: Simulation of a trapped bead axial position trace. We plotted results from boththe full-bandwidth simulation (left) and the simulation filtered to 2kHz (right). (A) Axial
bead position vs. time. (B) Histogram of bead positions. (C) PSD and Lorentzian fit ofbead positions. (D) Autocorrelation and exponential fit of bead positions.

78
y = 0 [29]. Thus if we can calculate Fscat and Fgrad as functions of z, we can calculate theprobability distribution of bead positions in the axial direction.
To calculate Fscat and Fgrad for our optical trap, we must relate these forces to measurable properties of our trap. The scattering force is related to the laser intensity (I),scattering cross-section (σ), refraction of the solution (nm ), and the speed of light (c) by [13]Fscat = Iσnm /c.

(2.9)

σ is related to the bead radius (rbd ), the laser wavelength (λ), and the relative index ofrefraction (m = np /nm where np is the index of refraction of the bead) by [13]σ=(6m2 − 1 2128π 5 rbd
)(
).
3λ4
m2 + 2(2.10)The gradient force is related to the polarizability (α), the gradient of the intensity (∇I), thespeed of light, and the index of refraction of the medium (nm ) by [13]Fgrad = (2πα/cn2m ) ∗ ∇I.

(2.11)

α is related to the index of the medium, the bead radius, and the relative indices of refractionby [13]3
α = n2m rbd

m2 − 1.
m2 + 2(2.12)

Thus Fscat and Fgrad depend on several measurable properties, including the intensity profile.
To find the z dependence of these forces, we must use the axial intensity profile, Iz of ourlaser.
For calculating the axial intensity profile of our focused laser, we assume Gaussian
beam optics, i.e. the lateral intensity profile Ir is given by [30]22Ir = I0 e−2r /w .

(2.13)For the axial dependence, the beam width as a function of z, w(z) is then given by [30]w(z) = w0 [1 + (z/z0 )2 ]1/2

(2.14)

79
where w0 is the beam waist, z is the axial position from the waist, and z0 is the Rayleighrange, [30]z0 = πw02 /λ.

(2.15)

Here λ is the laser wavelength.
To relate these equations to the axial intensity profile and to the measurable power in
our trap, we use the relationship between intensity, laser power (P ) and area (A), I = P/A.
Since P remains the same at every lateral cross section, and the area changes with beamwidth as A = πw(z)2 , we obtain
Iz = Pw /(π ∗ w(z)2 )(2.16)

where Pw is the power within the beam width, w. To calculate the power within the beamwidth from the total power, we use the general equation for calculating the power within aradius, PR using Gaussian beam optics, (Newport Gaussian Beam Optics tutorial)
2

PR = Ptotal (1 − e−2(R/w) ).

(2.17)

If we substitute the beam radius w for R, we obtain Pw = Ptotal (1 − e−2 ). Substituting thisPw and the equation for w(z) (Eq. 2.14) into Eq. 2.16, we obtainIz =Ptotal (1 − e−2 ).
πw02 (1 + (z/z0 )2 )(2.18)

This intensity is the average intensity within the beam width, w. Thus with a known beam
waist (w0 ) and laser power, P , we can calculate the axial intensity profile (with intensity
being the average intensity within the beam radius w).
For the purposes of our final calculation of the axial bead position probability distribution, we only needed the relative value of the intensity profile, since any multiplying factorof the intensity would only result in an irrelevant multiplying factor of the final probability.
A normalized intensity, Inorm can be calculated based on Eq. 2.18
Inorm =

1.
1 + (z/z0 )2(2.19)

80
Using this intensity profile in the calculations of Fscat and Fgrad , then integrating gives us
the potential energy profile. We then exponentiate to obtain the probability distribution of
axial bead positions.
The values we used in the final calculation are given for reference. For the final calculation of Inorm , z0 was obtained from fit of the 1064 axial detection signal to the Pralle axial
intensity equation [18] (without the π) to give z0 = 675.144 nm. For calculating Fscat andFgrad , we ignored c since it was simply a scaling factor in both forces. We used nm = 1.33
(index of H2 0), rbd = 165 nm, λ = 1064 nm, and np = 1.591 (for our beads at 20C and 500
nm wavelength from manufacturer bead information sheet). We performed the calculations
in Igor, calculating the intensity profile Inorm given by Eq. 2.18 with 4000 points from z= -2000 to +2000 nm. We multiplied and numerically differentiated Inorm as dictated by
the equations above to obtain Fscat and Fgrad , introducing a scaling factor (10−5 ) to prevent
large-number errors when exponentiating in the final calculation of P (z). We numericallyintegrated Ftotal to obtain U . We exponentiated U to obtain P (z), using kB T = 4.1 pN nm
for the thermal energy at room temperature. The resulting normalized intensity, scatteringand gradient forces, potential energy profile, and probability distribution are shown in Fig.
2.27.
Even without calculating the actual values of the intensity curve, we see that the total
force is the addition of a hyperbolic curve shape (representing the direct relationship between
Fscat and the intensity) and the derivative of a hyperbolic curve shape (representing thederivative relationship between Fgrad and the intensity). Intuitively, the shape of this force
curve, as well as the potential energy profile derived from it and the probability distributionof trapped bead positions, will be asymmetric. This asymmetry has been noted in theliterature [31, 32, 29, 33].

81

Figure 2.27: Theoretical calculation of axial bead position probability distribution. Basedon properties from our experiment, we calculated the axial intensity profile of our trap(top). From the intensity, we calculated the scattering (Fscattering ∝ I) and gradient forces(F gradient ∝ ∇I) (2nd from top). Adding these forces together yields the total force,
which, when integrated, gives the potential energy curve, U , of our trap (3rd from top).
Finally, from U , the probability distribution of axial bead positions, P (z) ∝ exp(kB T /U )was calculated (bottom). Note the asymmetry due to adding together the scattering andgradient forces.

82
2.7.6

Hidden-Markov modeling analysis of TBP data

The analysis of the TBP-induced DNA-extension changes was based on a hiddenMarkov modeling (HMM) algorithm. We originally used a program written for processing
FRET signals, which required all values be between 0 and 1 [34]. We modified our data
for this program by normalizing all values to be between 0 and 1, processed the data with
the HMM-algorithm, then converted all values back to the original values. This analysis
yielded average extension changes, and rates of bending and unbending. The bending andunbending rates were found by either the probabilities in the transition probability matrix,
or by fitting the dwell time histograms of the most-likely path. Data was rejected if noextension changes were found. We also implemented a more sophisticated version of thisHMM-algorithm, added a drift-correction to correct for baseline drift of our data. This drift
correction was accomplished by adding Fourier modes to account for drift in the traces andusing the HMM-algorithm to find the most-likely path, including both drift and step-likechanges.

2.7.7

Simulation: TBP data

We simulated TBP data by simulating a trace with Gaussian noise and adding exponentially distributed steps to that trace. We chose the Gaussian noise level based on our data(∼1 nm noise for data smoothed to 1.67 Hz). We chose the step-size as 3 nm, based on theanalysis from our measurements. We chose the exponential rates based on the exponential
fit to the dwell times in our data. See the TBP chapter for more details.

Chapter 3
AXIAL CALIBRATION AND DETECTION

3.1

INTRODUCTION
As discussed in the previous chapters, our experimental techniques were driven by

the challenges presented by our biological system of interest: TBP bending DNA. The predictions of a small and infrequent signal for TBP bending DNA led us to implement theactively stabilized optical-trapping instrument, as well as the precise alignment and calibration techniques and the automated data acquisition described in the previous chapter.
Preserving TBP activity and preventing nonspecific binding drove the development of thesample preparation techniques. The acquired data led to the data analysis techniques described. These experimental techniques paved the way for studying TBP bending DNA at
the single-molecule level.
With the implementation of the experimental techniques described, one experimental
challenge remained. For interpretability, we wished to study TBP bending DNA in single
bending events at specific TATA box DNA sequences. While TBP has higher binding affinity
for TATA box DNA sequences, TBP also has lower binding affinity for non-TATA box DNA
sequences [2]. To ensure single bending events at specific TATA boxes (as opposed to multiple
bending events at random DNA sequences), the DNA tethers in our experiments needed to
be short with carefully designed sequences. These short DNA tethers prove challenging forstudy via optical trapping techniques due to constraints in traditional trapping geometries.
To overcome this last experimental challenge, we developed an axial calibration and detection

84
technique capable of measuring extension changes on the short DNA tethers required forstudying TBP bending DNA.

3.1.1

Geometry constraints in traditional optical trapping

In traditional optical trapping, measurements are performed primarily in the lateral(x, y) axes. For a surface-based optical-trapping assay, measuring in the lateral direction requires a diagonal experimental geometry, like that shown in Fig. 3.1. In this diagonal geometry, calibrations generally are made for the lateral bead displacement from trap equilibrium(xbd ) and the lateral trap stiffness (ktrap,x ), while the lateral location of the tetherpoint generally is obtained by a centering routine. From this tetherpoint, the pre-calibrated stage thenoffsets the sample laterally by a known amount (xstage ) to extend the tethering molecule andapply a force. With these lateral calibrations, measurements are made of lateral projectionsof the bead displacement (xbd ), force (Fx = ktrap,x ∗ xbd ), and extension (xDN A = xstage − xbd )of the tethering molecule.

Diagonal

Axial

xbd

xbd

zbd
zbd
Ext

Ext

Figure 3.1: Diagonal vs. axial geometry. The diagonal geometry is typical for surface-based
optical trapping, and the diagram shows the geometry problem that occurs with short DNA
molecules. The axial geometry offers a solution to this problem, as well as offering several
other advantages over the typical diagonal geometry.

In diagonal assays, these lateral measurements are projections of the more relevant
quantities of total force, F , and extension of the tethering molecule, Ext. Calculations of

85
F and Ext require assumptions about the axial trap stiffness, ktrap,z , and the axial extension, zDN A . Usually ktrap,z is calculated using theoretical assumptions about the axial:lateraltrap stiffness ratios [21]. For zDN A , determination generally relies on some means of determining the trap height, ztrap , and the axial bead displacement from trap equilibrium, zbd .
Determination methods for ztrap vary; examples of methods include measuring the height
dependence of the lateral drag coefficient [21], using features of the detected intensity curve
as a bead is scanned through the laser axially [35], and a combination of the two [13]. To
determine zbd , geometry is used in combination with the other known and assumed quantities. Specifically, the axial:lateral force ratio, Fz /Fx , is equated to the axial:lateral extension
ratio, zDN A /xDN A ,ktrap,z ∗ zbd
ztrap − zbd
zDN A
Fz
=
=
=
,
Fx
ktrap,x ∗ xbd
xstage − xbd
xDN A

(3.1)

and the resulting equation is solved for zbd . With the geometry fully determined, the axial
projections of the total force and the extension can be obtained, i.e. Fz = ktrap,z ∗ zbd , andzDN A = ztrap − zbd . From these and the laterally measured quantities, the total force andextension of the tethering molecule are geometrically calculated, Ext = (zDN A / sin(θ)) − rbd
and F = Fx / cos(θ) where θ is the angle between the sample surface and the tethering
molecule, θ = tan−1 (zDN A /xDN A ) [21]. In summary, the experimentally relevant quantities
of total force and total extension are calculated based on calibrated measurements in thelateral axis, combined with an estimate of the axial extension and an assumed axial trap
stiffness.
To minimize measurement uncertainties in diagonal optical-trapping assays, the angle
between the surface and the tethering molecule should be as small as possible. When thisangle is small, xDN A is large compared to zDN A and Fx is large compared to Fz . We assumezDN A and Fz have a larger uncertainty since they are derived from multiple other measurements and assumptions. Thus the larger xDN A and Fx are compared to zDN A and Fz , theless the uncertainty in the axial quantities affects the uncertainty of the final measurements.

86
When making measurements on long tethering molecules, satisfying this small angle ideal iseasily accomplished.
In contrast, when making measurements on short tethering molecules, this small angle ideal becomes much more difficult to achieve. The difficulty arises when the tethering
molecule extension becomes of similar size to the tethering bead radius, rbd , i.e. Ext ∼ rbd .
To see this mathematically, let Ext = rbd . Then θ > 45◦ for the bead to remain above thesurface. Explicitly, sin θ = (ztrap −zbd )/(Ext+rbd ). With Ext = rbd , sin θ = (ztrap −zbd )/2rbd .
For the bead to remain above the surface, ztrap − zbd > rbd , thus sin θ > rbd /2rbd = 1/2, i.e.
θ > 45◦ . These equations set a lower limit on the angle, but experimentally, this angle should
be larger than 45◦ to prevent the bead from resting on the surface and skewing the geometry.
Thus as the extension of the tethering molecule approaches the scale of the bead radius, thetrapping angle must become larger, and the uncertainty in the measurements increases as
described.

3.1.2

Advantages of axial trapping and detection

Axial trapping and detection offers a solution to the geometrical problem in traditional
optical trapping, in the form of geometrical simplification. In contrast to the diagonal
geometry of traditional surface-based trapping, the axial trapping geometry is a single axis,
vertical geometry, as shown in the axial geometry in Fig. 3.1. No angles must be calculated
or assumed; the signal measured axially is the full signal, not a geometrical projection of
the signal, and this advantage improves the signal to noise. The total bead displacement is
the axially measured bead displacement, zbd ; the total force is the axially measured force,
F = Fz ; the extension is the axially measured extension, Ext = zDN A . Additionally, axialtrapping minimizes crosstalk between axes because bead motion is directly on one axis rather
than at an angle. These geometrical advantages make a compelling case for axial detection.
The improved trapping geometry in axial trapping allows shorter biological tethers to
be studied. These shorter DNA tethers allow the detection of single binding events by DNA-

87
binding proteins with lower sequence specificity, solving the final experimental challenge in
our TBP experiments. In addition to the expansion of biological systems that can be studied,the shorter biological tethers that can be studied without a geometric restriction increase
the stiffness of the system and therefore improve the signal to noise. The signal-to-noise
advantages and addition of the biological phenomenon that can be studied on short DNA
have driven our motivation to trap and detect along the axial direction.
Axial trapping also has advantages over other methods that allow a single-axis trapping
geometry, namely dual-beam trapping and trap-pipette geometries. In a dual-beam trapping
geometry, the short biomolecules that can be studied with axial trapping would be difficult to
study because when the two traps of the dual-beams are too close, interference and/or beadhopping between traps occurs. The trap-pipette geometry would overcome this obstacle, but
the pipette in this setup is subject to mechanical noise. The mechanical noise problem could
theoretically be remedied by stabilizing the pipette tip relative to the laser, as we stabilize
our surface relative to the laser [10, 11, 12]. Given the demonstrated stability of our surfacebased assay, for the excellent signal-to-noise offered by a single-axis trapping geometry andshort biological tethers, axial trapping seems to be the best choice.

3.1.3

Axial detection in the literature

Given the potential advantages of axial trapping and detection, unsurprisingly many
examples of axial calibration exist in the literature. Earlier introduced axial calibration techniques included axial detection via an overfilled photodiode [36, 37], fluorescence detection[38, 39], evanescent wave excitation [40, 41] and video [42]. These methods tended to sufferfrom limited spatio-temporal resolution and/or require additional optics and detectors, andthus have not been widely adopted by the optical-trapping community.
The more recent axial calibration techniques, including the one introduced in thisthesis, have been based on the intensity changes that occur at a detector placed at the backfocal plane of an optical-trapping microscope when a bead is moved axially through a focused

88laser. This intensity vs. axial bead position curve was described theoretically by Pralle etal. [18] and then in even more detail by Rohrbach et al. [29]. The shape of the curve isattributed to interference of the scattered and unscattered light, which undergoes a gouyphase shift at the laser focus [18]. The dependence of this signal vs. capture angle has beentheoretically described [29, 43] and experimentally tested [44]. Neuman et al. [13, 19] used
this curve, obtained on beads fixed to the sample surface, to serve as an average calibration
curve for trapped beads. With this calibration process, axial bead position was measured[45, 13, 19], an axial force clamp was applied [45], and the effective focal shift caused by theindex mismatch at the coverslip-buffer interface was estimated [13, 19]. This axial detection
method, exploiting intensity dependence on axial bead position, has been the basis of themore recent efforts towards precision measurement in the axial direction.
Other recently introduced techniques were also based on a more complex applicationof this axial intensity detection curve. In one of the most recent and thorough treatmentsof axial calibrations, Defuel and Wang state that three parameters must be determined foraxial detection: 1)axial displacement from the trap center, 2)axial stiffness, and 3)axial
height above the surface [46]. These three quantities were determined with precision in
Deufel and Wang’s work, as well as by two other recently introduced methods [47, 48]. In
the work of Deufel and Wang, a DNA construct was unzipped to determine the average
displacement, stiffness, and height calibrations. Sample preparation and unzipping DNA
constructs for a calibration may be somewhat unwieldy, but the authors also show that using
the power spectral density to obtain the axial trap stiffness and then using this stiffness
in the equipartition equation to obtain the displacement calibration yields a reasonablecalibration of stiffness and displacement. Schaffer et al. introduce a method to calibrateaxial displacement, stiffness, and height using the power spectral density of a trapped bead
while oscillating the sample [47]. Most recently, Chen et al. introduced an axial detection
method that uses multiple trapping lasers coupled with calibrated video detection and laser
detection, and offers the advantage of the passive-force-clamp technique [48]. These methods

89
extend the detection capabilities of optical trapping to the axial direction.
We wished to build a precision axial detection method, capable of measuring force andextension changes on very short DNA, from these most recent examples in the literature.
We adopted the calibrations above that were most straightforward to perform in our actively stabilized optical- trapping instrument, in essence using Neuman’s axial displacement,
axial trap stiffness, and trap height calibrations [45, 13, 19] as a starting point. Into thedisplacement calibration, we built floating parameters that allowed the calibration to be customized for each trapped bead. We attempted to improve the displacement calibration andestimation of the trap equilibrium point by correcting for the voltage oscillations that occur
with depth. For the trap height estimation in our system, we found a stiffness-dependent
correction, and for trap stiffness, we determined a height dependent correction. Ultimately
we found these techniques provided adequate measurements of the trap stiffness and relative axial displacement, but a final connection between DNA polymer theory and the height
dependent drag was required for estimating axial force and tether extension.

3.1.4

Overview of our technique

We present here an axial calibration and detection method for determining axial displacement, stiffness, and height capable of measuring force and extension on very short (92nm) DNA. Like the other recently introduced axial calibration methods, this method is basedon the intensity distribution that occurs when a bead is moved axially through a focusedlaser. This intensity distribution is shown for a bead fixed to the sample surface (dashed line)
and for a trapped bead (gray line) in Fig. 3.2. The fit to the Pralle-described theoreticalcurve [18] is represented by the black line in Fig. 3.2. The technique involves determining
an average axial detection voltage vs. axial bead position curve with beads coupled to thecoverglass surface, then using that average curve to reconstruct full detector response curves
for trapped beads, with floating parameters that allow the calibration to be specific to theindividual bead. Unlike previously introduced methods, our method achieves high precision

90
without requiring a specialized biological construct, multiple traps, or multiple types of detection. We quantify the uncertainty in our axial calibrations, finding low uncertainty in
our relative axial displacement and trap stiffness measurements. Higher uncertainty in theabsolute height measurement and trap equilibrium position led to large errors for force andextension measurements on DNA, so we introduce a more accurate method for estimating
axially measured forces and DNA extensions. Finally, we demonstrate the improved signal tonoise afforded by axial trapping and detection of a bead attached to a short DNA molecule.

Figure 3.2: Axial intensity signal. The axial detection voltage vs. stage height is plotted foraxial scans of the sample. The stage was scanned axially, and the axial detection voltage wasrecorded for a stuck bead (black ) and a trapped bead (gray). The Pralle equation (dashedline) provides the theoretical description for these curves.

3.1.5

A conceptual description of axial trapping and detection

To provide background for understanding the calibrations and measurements in thefollowing sections, this section provides a conceptual overview of axial trapping and detection.
In this conceptual description, a focused laser propagates in the plus z direction, forming an
optical trap. The optical trap holds a bead at equilibrium, in the center of the laser laterally,slightly past the focus axially due to the balance of scattering and gradient forces. The beadis trapped within a sample, and the sample surface moves axially. As the sample surface
moves axially, because of an index mismatch between the glass surface of the sample and thebuffer inside the sample, the focal point of the laser also shifts axially. This shift is known
as the focal shift, and generates a correction factor estimated in the literature and in this

91
chapter. As the sample surface moves upward axially, it pushes the trapped bead upwards
in the laser, past equilibrium, but trapped against the sample surface. As the sample surface
moves downward axially, the bead follows the surface until it reaches equilibrium. Then thesample surface continues to drop away as the bead sits at trap equilibrium. If the beadis linked to the surface, the linker pulls the bead downward to follow the dropping sample
surface. If the linker is compliant, the extension of the linker grows as the sample surface
pulls downward. The downward force from the linker balances with the upward force form
the trap, like two springs pulling on the bead in opposite directions. This physical picture
describes axial trapping.
To describe axial detection, the laser light must be described. The laser light, after
encountering the sample and bead, continues propagating to the detector. The intensity distribution on the detector changes as the bead moves vertically in the laser. The relationshipbetween bead motion in the laser and laser intensity at the detector is given by the Pralle
equation [18].
However, for a bead near a sample surface, a secondary scattering phenomenon also
affects the intensity of the light at the detector. This secondary scattering phenomenon
is due to light interfering as it bounces between the bead and the sample surface. As thebead to sample surface distance (hbd ) changes, this phenomenon causes an oscillation in theamount of light reaching the detector, which appears as an oscillation in the intensity of thelight at the detector [19].
In the physical picture of axial detection, we therefore have two relationships governing
the intensity of light at the detector; we have the Pralle relationship describing the relationship between the intensity and the position of the bead in the laser, I(zbd ), and we havethe oscillating affect describing the relationship between the intensity and the bead-surface
distance, I(hbd ).
To complete the conceptual description of axial detection, the detection voltage must
be described. In a detection voltage picture, we examine the same physical picture as above,92
now examining the effects on the detection voltage. For a detector placed at the back-focal
plane, the axial detection voltage, Vz , is directly proportional to the laser intensity, I, on thedetector, i.e. Vz ∝ I. As described above, the intensity on the detector changes as the beadmoves vertically in the laser and as the bead-surface distance changes, I = I(zbd ) + I(hbd ).
Thus, the detector voltage changes in the same way, yielding proportionate relationships
between bead motion in the physical system and detector voltage, i.e. Vz = V (zbd ) + V (hbd ).
This physical picture and the relationship with the detection voltage provide the background information for the calibrations and measurements that follow.
3.2

MATERIALS AND METHODS
The actively stabilized optical trapping instrument and detection system were as de-scribed in the Experimental Techniques chapter. Calibrations for beam steering (PZT mirror
calibrations) and lateral trap stiffness were as described. Sample and laser positioning (centering laser foci with each other, with beads fixed to the surface, with trapped beads andrelative to tether points) and lateral displacement calibrations were also performed as described in that chapter.
For detection of bead position, we used a quadrant photodiode (QPD) for each detection laser, placed in the back-focal plane of the microscope condenser. With this arrangement, the sum of the voltage from the four quadrants is proportional to the intensity on the
QPD, which is related to the axial bead position as described in the text. In our experiment, this QPD sum voltage was further modified electronically with a tunable voltage offset
and differential amplification. For conciseness, this offset, differentially amplified QPD sum
voltage is referred to as the axial detection voltage, Vz , in the rest of this thesis.
For clarity, an explanation is useful describing the variability of the absolute valuesof Vz and axial stage position, zstage , in our experiment. The absolute value of Vz varied
due to an adjustable electronic voltage offset. In a given experiment, the absolute valueof Vz provided the means for measuring axial position and thus was not adjusted between

93
calibration and measurement or during the measurement. However, between experiments,
the offset varied. The absolute value of zstage varied due to drift. Because of the variability
in both these quantities, we built voltage and axial offsets (Vof f and zof f ) into the fits of Vz
vs. zstage curves (see text for details). To eliminate variability in plots of the Vz vs. zstage ,we subtracted the voltage and axial offsets obtained in the fits before plotting, giving Vz = 0
and z = 0 meaning only in the context of the fit. In later sections, we identify two othermeanings for z = 0. In one context, z = 0 indicates that a bead trapped at equilibrium
would rest on the sample surface; in another context, z = 0 represents the point when atrapped bead would be positioned at equilibrium (zbd = 0). The relative nature of Vz andzstage and the context-dependent meaning of z = 0 are important details for understanding
our plots and calibrations.
Sample preparation was performed as described in Experimental Techniques, as well.
Samples included flow cells with beads fixed to surfaces (by salt, not melting), beads free insolution, and beads tethered to the surface via 92 nm DNA. The DNA tethered bead samples
were either on biotynilated BSA (force vs. extension curves) or on PEGylated surfaces (allother tether data). Beads were either 330 nm dia. antidigoxigenin-modified beads or 860
nm Streptavidin coated beads, as indicated. Data was acquired in filtered, deionized H2 0,
except force vs. extension data (10 mM phos, 3 mg/ml BSA, 0.4% tween-20, [25]) and other
DNA tether data (10 mM Tris, 10 mM Hepes, 150 mM KCl, 0.5 mM MgCl2, 1 mM DTT,
0.2% tween-20, 0.4 mg/mL BSA).
Data acquisition was also as described in the Experimental Techniques chapter. In
each type of data acquired, care was taken to laterally align detection lasers with detected
beads, since the axial detection signal depends on this lateral alignment. This alignment
was accomplished via automated centering programs as described in that chapter. Lateraldetection calibrations were performed at the measurement height.
Data analysis was performed in Igor; methods are described in the text.

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3.3

AXIAL CALIBRATIONSTo quantitatively manipulate and measure in z, we calibrated the axial displacement(zbd ), the trap height above the surface (ztrap ), and the axial trap stiffness (ktrap,z ). Because
the trap equilibrium point on the calibration curve was not a priori known, the axial displacement measurement was divided into two parts: the relative axial displacement (∆zbd )and the estimated trap equilibrium position (zbd = 0). These calibrations provided measurements of 4 experimentally relevant quantities: 1) extension changes of DNA, 2) estimates of
the axial trap stiffness, 3) an initial estimate of the force on a trapped bead (F = ktrap,z ∗ zbd )and 4) an initial estimate of the height above the surface of a trapped bead (hbd = ztrap −zbd ).
3.3.1

Calibration of relative axial displacement

To calibrate the relationship between the relative axial displacement and the axial
detection voltage, ∆zbd (Vz ), we followed a three step process. First, an average axial detection
voltage vs. bead position curve was acquired by scanning beads fixed to the sample surface
through the detection laser (Fig. 3.3 A). Second, this average voltage curve was used toreconstruct the detection voltage curve for individual trapped beads (Fig. 3.3 B ). Third,
this reconstruction was fit with a 7th order polynomial (Fig. 3.3 C ) to give the relative
displacement calibration ∆zbd (Vz ) for each individual bead.
In the first step of this process, we established an average axial detection voltage vs.
bead position curve Vz (z) based on beads fixed to the sample surface. Ten beads, fixed tothe sample surface, were scanned axially through the detection laser using the calibrated
piezo stage while Vz at the back-focal plane was recorded. The obtained Vz vs. axial stage
position (zstage ) curves were each fit to the following equation, adapted from Pralle et al.
1999 [18],Vz (z) = Vof f + Ic (1 + (z − zof f 2 −1/2z − zof f))sin(π arctan(
)).
z0
z0

(3.2)

The axial position, z, is the relative bead position in the laser, which here is equivalent to95

Figure 3.3: Axial detection displacement calibration. (Top) Step 1: A master curve (blackline) is generated by fitting axial scans of 10 different stuck beads (gray dots). Each axial
scan is fit to the Pralle Equation. The Pralle fit coefficients are averaged to generate thePralle master curve. (Middle) Step 2: The Pralle master curve is used to reconstruct a
Pralle fit (black line) for free bead axial scans (gray dots). 10 different free bead scans areplotted. Each scan is fit to the Pralle master curve with floating offsets to generate thePralle reconstruction. The average Pralle reconstruction is plotted here. (Bottom) Step 3:
A 7th order polynomial is fit to the Pralle reconstruction, with axes switched, to provide thenm(V ) displacement calibration.

96
zstage since the bead is fixed to the sample surface. Ic , z0 , Vof f , and zof f are fit parameters;
π is simply a scaling factor and does not affect the fit quality. The acquired curves for 10
different 330 nm dia. beads are shown in Fig. 3.3 A (gray dots) with the voltage and axial
offsets (Vof f and zof f in Eq. 3.2) subtracted. The shape of the curves was very consistentfrom bead to bead; a calculation of variability in the curve shape fit parameters yielded
2.7% for Ic and 0.42% for z0 (100*standard deviation/mean). This consistency in the shapeparameters establishes support for the using the average fit to establish a ”master curve”
for calibrations of trapped beads. The average fit to Eq. 3.2 is plotted as a black line.
This average fit provided the ”master curve” for the next step: reconstructing the full axial
detection voltage curve for individual trapped beads.
In the second step of the relative axial displacement calibration, the ”master curve”
generated in the first step was used to reconstruct the Vz (z) curves for individual trapped
beads. For trapped beads not fixed to the sample surface, a Vz vs. zstage scan results in a
curve with two distinct regions: a peaked upper region where the bead is trapped against
the sample surface and therefore tracks the position of the surface, and a lower region where
the bead has reached the axial trap equilibrium point and the sample surface drops awayfrom the trapped bead. In the peaked upper region, since the bead is following the motionof the stage, the curve provides information about the axial bead position relationship with
Vz . In contrast, in the lower region below trap equilibrium, the bead is not closely following
the motion of the stage and does not provide adequate information about the relationshipbetween Vz and the bead position below trap equilibrium. Since most measurements involve
exerting force by lowering the sample surface and pulling the bead below trap equilibrium,
this lower region of the curve is the most relevant portion. To provide the relationshipbetween Vz and the bead position below trap equilibrium, we reconstruct the full Vz (z) curve
based on the ”master curve” determined in the first step of the calibration and customized
to fit the upper region of the Vz vs. zstage curve for the individual trapped bead.
To provide a basis for this reconstruction of the full Vz (z) curve, a Vz vs. zstage scan

97
was performed on each individual bead. The individual bead was trapped and the stage
was scanned axially while the Vz (z) curve was recorded. The upper region of this curve(from 100 nm above the peak to 325 nm below the peak) was fit to Eq. 3.2. The fit was
performed with fixed values of Ic and z0 from the ”master curve” determined in the firststep of the calibration and floating values of the voltage and axial offsets (Vof f and zof f ) toallow customization of the fit for the individual bead. The acquired curves for ten trapped
330 nm dia. beads are shown in Fig. 3.3 B (gray dots) with the voltage and z offsets
subtracted. The figure shows excellent overlap of the individual free bead curves, indicating
the reproducibility of the shape of these curves. The average reconstruction of the full Vz (z)curve (also with axial and voltage offsets subtracted) is plotted as a black line. Though theplot shows the average reconstruction with offsets subtracted, these offsets are the floating
parameters that allowed each reconstruction to be customized for the individual trappedbead.
In the third and final step of the axial displacement calibration, ∆zbd (Vz ) was given byfitting a 7th order polynomial to the reconstructed curve (Fig. 3.3). The fits were performed
with the axes of this curve exchanged, i.e. zstage vs. Vz , fitting from voltage peak-to-peak sothat the curve is single valued. This 7th-order polynomial, fit to the Pralle reconstruction
for trapped beads, was the calibration curve that provided the means for measuring ∆zbd .
3.3.2

Determination of axial trap equilibrium

Once the relationship between the relative axial displacement and the axial detection
voltage ∆zbd (Vz ) was established, to fully measure the axial bead displacement from trap
equilibrium, we still needed to determine the point on the Vz (z) curve that corresponded totrap equilibrium, i.e. where zbd = 0. For consistency, we first established a reference pointon the curve: the peak of the axial detection voltage curve. The position of this peak was
precisely determined by a parabolic fit to a region ± 200 nm from the maximum voltage
point (Fig. 3.4 A). From this peak reference point, we could determine the offset that would

98
represent trap equilibrium, the point on the curve where zbd = 0.
The point on the Vz (z) curve corresponding to zbd = 0 varied with height above thesurface. We see this height variation in Fig. 3.4 B as the voltage oscillations in the graycurve that occur as the sample surface drops away from the trapped bead. Though the beadin this figure remained at trap equilibrium as zstage dropped from 0 to -1000 nm, the voltage
representing this equilibrium position oscillated. Thus for an accurate measurement of thetrap equilibrium voltage, we customized the estimation of the zbd = 0 point for the planned
measurement height.
The height-dependent zbd = 0 estimation is illustrated graphically in Fig. 3.4 B. The
axial detection voltage for a trapped bead (gray) shows the oscillating trap equilibrium
voltage as the surface drops away from the bead. The measurement height, here 110 nm
from the surface, is illustrated by a vertical green line. The intersection of the green line andthe gray curve gives the value of the trap equilibrium voltage at the measurement height.
This voltage is illustrated by the horizontal blue line. The point where the blue line intersects
the Pralle reconstruction curve (black ) represents zbd = 0 on the curve. The zbd = 0 position
relative to the peak reference point (red dot) established an offset from the peak that could
be applied to any trapped bead Vz (z) curve.
In practice, we completed this estimation of zbd = 0 on several beads and establishedan average offset from the reference peak. To test the accuracy of this determination of
zbd = 0 we trapped a bead at equilibrium at the established height and measured the mean
axial displacement, zbd . The result is shown in Fig. 3.4 C. The mean axial displacement
for this trace was 0.25 nm, but for measurements of several beads the mean displacement
typically varied by ±5 nm. This offset from the peak established the zbd = 0 point on the
curve at the planned measurement height, completing the calibrations needed to measure
axial bead displacement, zbd .

99

Figure 3.4: Referencing axial position from the peak of the axial detection signal. (Top) The
axial detection signal from a free bead (gray) has a peak in voltage. This peak is used as areference point for axial position measurements. The position of this peak is found by fitting
to a parabola (red). The position of the peak is indicated (red dot). (Middle) Oscillations
in voltage occur in the free bead signal (gray) as the sample surface is moved away fromthe trapped bead. At a measurement depth of 110 nm (green line), a bead trapped at
equilibrium has a corresponding trap equilibrium voltage (blue line). The trap equilibrium
voltage at a given measurement height represents a zero bead displacement (zbd = 0) at thatheight. (Bottom) A bead, trapped at equilibrium, at a measurement height of 110 nm. The
calibrated average bead displacement is 0.

100
3.3.3

Voltage oscillation corrections

As a second order correction to the displacement calibrations and a more sophisticatedmeans for determining the trap equilibrium voltage, we explicitly incorporated the voltage
oscillations into our calibrations. In detail, we fit the voltage oscillations for several freebead curves, added the average of these fits onto the Pralle reconstruction, and obtained
the final calibration by fitting the 7th order polynomial fit to this sum of two curves. For
the oscillation fits, the oscillating region below trap equilibrium was fit to a combinationof 3rd order polynomial and sinusoidal functions, Pl (z − z0 ) + Pm (z − z0 )sin(f (z − z0 )),similarly to Neuman et al. [19]. The curves of 10 free beads (gray) and the average fit(dashed black line) is shown in Fig. 3.5 A. This average fit to the oscillations was added
onto the Pralle Reconstruction at each stage height (with an offset term to yield continuity
at the trap equilibrium point). The sum of these voltage curves was fit to a 7th orderpolynomial, performed at each stage height, yielding a calibration curve at each stage heightthat incorporated the voltage oscillations.
To demonstrate the improvement in the axial displacement calibration offered by fitting
the oscillations, we measured the axial displacement of a trapped bead as the sample surface
dropped away from trap equilibrium. The measured displacement, zmeas , is plotted in Fig.
3.5 B both with and without the oscillation corrections. Without the oscillation corrections(dark gray), the displacement from trap equilibrium deviated from trap equilibrium by asmuch as 74 nm, though the bead was trapped at equilibrium for the entire measurement. In
contrast, with the oscillation corrections (light gray), the maximum measured deviation fromequilibrium was < 10 nm, with a mean of -0.3 nm and std. dev. of 2.9 nm. The apparent
oscillations with height were clearly diminished in the oscillation corrected calibration. Theseoscillation incorporations provided an axial displacement calibration that incorporated thedetection voltage dependence of both the Pralle relationship, describing V (zbd ), and theoscillating affect describing V (hbd ).

101

Figure 3.5: Correcting for oscillations (A) Freely diffusing beads were trapped, and thestage was scanned axially through the detection laser. The detector response is shown for
10 trapped beads (gray dots). Eq. 3.2, with Ic and z0 values from fitting the stuck beadsignals, is used to reconstruct the full detector response curve for trapped beads (black line).
The oscillating region below zstage = 0 is fit to a combination of polynomial and sinusoidal
functions (dashed line). (B) Oscillations in the voltage signal are converted to displacements
using calibrations without oscillaiton corrections (dark gray) and with oscillation corrections(light gray). The dashed line is at zmeas = 0.

102
3.3.4

Calibration of height above the surface

We determined the trap equilibrium height relative to the coverslip surface, ztrap , using
a method that exploits the surface-distance dependence of the drag on a sphere, similar to
the method used by Neuman and Block [13]. The surface-distance (h) dependence of thelateral hydrodynamic drag (β) on a sphere is described by Faxen’s Law as
β=6πηrbd9 rbd
1 rbd 345 rbd 4
1 rbd 5 ,1 − 16 ( h ) + 8 ( h ) − 256( h ) − 16
(h)

(3.3)

where η is the viscosity and rbd is the bead radius. To determine ztrap , we measured thelateral drag on trapped beads at a series of axial stage positions as described below and fit
to Faxen’s Law; this fit determined the height of the axial trap equilibrium relative to thesample surface.
In determining ztrap , we first needed a reference point from which to measure. Again
as in Neuman and Block [13], we chose the reference point as the peak in the axial detection
voltage curve from an axial stage scan performed on a trapped bead (Fig. 3.6). Stage motion
was measured from this reference point, corresponding to a stage height of 0 in Fig. 3.6.
Once the reference point was determined, we measured the lateral drag on trapped
beads at a series of axial positions from the reference point. Explicitly, to measure thelateral drag, we estimated the lateral trap stiffness (k) by two different methods: the power
spectral density method (kP SD ), which depends on the drag, and the equipartition method(kEQP ), which does not. In these two estimations of k, kP SD = 2πβf0 , where f0 is the cornerfrequency from the Lorentzian fit to the power spectral density of the Brownian motion ofa trapped bead, and kEQP = kB T / < x2 >, where kB T is the thermal energy (Boltzmann’sconstant, kB , times the temperature, T ), and < x2 > is the mean-squared Brownian motionof the trapped bead. The equations for the two methods of trap stiffness determination were
equated and solved for β. In this way, we measured β at a series of axial positions.
To determine the relationship between our chosen reference point and the bead height
relative to the surface, we plot and fit the measured drag coefficients (Fig. 3.6). The

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Figure 3.6: Determining trap height relative to the coverslip surface. (A) The detector response as the stage is scanned axially while simultaneously detecting a trapped bead (markers) and a stuck bead (solid line, voltage divided by 2 for display). The peak of the trapped
bead detector response curve is used as a reference point for trap height determination, corresponding to 0 on the stage height axis. (B) Trap height determined by Faxens Law fit to
the normalized drag coefficient as a function of height. The normalized drag coefficient was
obtained from lateral stiffness calibrations via the power spectral density and equipartition
theorem (gray markers). The normalized drag as a function of axial stage position was fit
with Faxens Law (black line) to obtain the correspondence between stage position and trap
height. The point where the bead rests on the surface is found to be 333 nm from the peak,at the intersection of the gray and white regions of the graph. In the gray region to the rightof this point, the stage is pushing the bead upward in the trap. In the white region to theleft of this point, the stage is dropping away from the trapped bead.

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measured drag coefficients are plotted as a function of stage height from the reference peak(gray markers), normalized as β0 /β where β0 = 6πηrbd . We fit to the normalized form of
Eq. 3.3, with h = δ(z − ε). Here, z is the stage motion from the chosen peak reference pointand δ and ε are the free parameters in the fit: δ describes the focal shift (shift of the laser
foci caused by the index mismatch at the sample surface), and ε is an offset term. We find
δ = 0.834 ± 0.022 and ε = 135.7 ± 9.6 nm. The estimate and error in the focal shift are
similar to those found in the literature [13]. As a second estimate of the focal shift, we also
fit the oscillations in the free bead signal (see section on voltage oscillations, and [19]). We
found δ = 0.82 ± 0.004, within the estimated uncertainty of the focal shift found by the dragmethod. The point where the trap equilibrium is one bead radius from the surface is foundby setting h = δ(z − ε) = rbd , which gives the surface offset from the peak as 333 ± 13 nm.
By using this one time measurement of the height dependence of the hydrodynamic drag,the height of axial trap equilibrium ztrap from the surface is thus determined, referenced fromthe peak in the axial detection voltage curve of a bead.
On further investigation, we found the peaks in the axial detection curves acquired
small offsets that varied with trap stiffness (Fig. 3.7). We measured these offsets by aligning
the curves by the peak in the detection curves from stuck beads acquired simultaneously tothe trapped bead scans. Fig. 3.7 shows the detector response curves from three different
trap stiffnesses. The curves are aligned by the peak in the stuck bead scans such that thepeak in the data from Fig. 3.6 is represented by the dashed line at 333 nm and the point
where trap equilibrium would be one bead radius form the surface occurs at zstage = 0. The
offsets we measured were small (between ±30 nm, Fig. 3.7, inset), but would be significant
when trying to measure the length of a short molecule tethered to the surface.
Potentially, conditions other than trap stiffness could affect the relative position of
the peaks in the axial detection curves. We considered the possibility that buffer conditions,surface coatings, DNA-tether presence and length, and day-to-day variation could potentiallyaffect peak positions. We measured that the day-to-day variability was smaller than the the105

Figure 3.7: Changes in the position of the peak in the detector response curves of trapped
beads. The voltage of the detector response is plotted from axial stage scans performed while
trapping a bead at various trap stiffnesses. The peak positions are found by fitting a parabola
to ±200 nm from the maximum voltage point of the curves. The solid black line shows theparabola fit to the circle data points. At zstage = 0, a trapped bead at trap equilibrium
would rest on the surface, as determined in Faxen’s Law fit data. The dashed line at 333 nm
represents the peak position of the detector response in Faxen’s Law fit data. Aligning the
curves by the peaks of stuck bead scans acquired simultaneously to the trapped bead scans,the detector response curves for trapped beads are plotted for various trap stiffnesses (circles- 18.3 fN/nm , squares - 25.7 fN/nm, triangles - 40 fN/nm). The inset shows the distancefrom the peak in Faxen’s Law fit data to the peak for each stiffness. Error bars representing
the standard error on the data points are smaller than the markers.

106
statistical uncertainty on a given day (data not shown). For the other possible variables, we
did not systematically determine their affects on the peak positions, but instead obtained
the peak offset for data under each given set of conditions, eliminating the need to determine
the exact effect of each condition.

3.3.5

Calibration of axial stiffness

We calibrated the axial trap stiffness, ktrap,z , by measuring the axial Brownian motion oftrapped beads. As described in the trap height calibration section, the measured Brownian
motion provides the means to calculate k by both the equipartition method, kEQP , andthe power spectral density method, kP SD . When calibrating the stiffness by each of thesemethods, we encountered areas that required special consideration in the axial direction.
When measuring the Brownian motion of a trapped bead, we noted departures fromthe theoretical ideal of a perfectly harmonic trap. A plot of bead position, z vs. x (Fig.
3.8) shows an obvious apparent asymmetry in the axial direction. Plotted in histogramsof bead position, in the lateral axis the x position histograms showed a nicely symmetric
Gaussian distribution. In contrast in the axial direction, histograms of the z position showed
an asymmetry (Fig. 3.9) that varied with height (Fig. 3.10). A plot of the theoreticallycalculated axial probability distribution (see Experimental Techniques chapter for details)
is shown in Fig. 3.9 D. The asymmetry in the theoretical distribution is similar to the data
asymmetry, although less pronounced. We also noted that the difference between ktrap,zmeasured via the EQP and PSD methods was correlated with the asymmetries in the axial
position histograms, as measured by the chi squared value for fits to a Gaussian (Fig. 3.11).
These departures from the ideal harmonic trap are not surprising given that the additionof the scattering and gradient forces in the axial direction yields an asymmetric potential[31, 32, 29, 33].
When measuring the power spectral density of the axial Brownian motion of a trapped
bead, we found the shape of the power spectral density to be highly dependent on the op-

107

Figure 3.8: z position vs. x position of a trapped bead. Note asymmetry in the z axis.

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Figure 3.9: Top - Histogram of x positions (green markers) with a Gaussian fit (black line).
Note symmetry and excellent fit. 2nd from top - Histogram of z positions (red markers) witha Gaussian fit (black line). Note asymmetry in histogram and lack of fit. 3rd from top Theoretical calculation of the probability distribution of the axial positions of a trapped beadwith parameters chosen to simulate the experimental conditions in our trap (gray). Detailsof the theoretical calculation are in the Experimental Techniques Chapter. A gaussian fit isshown in black. Note asymmetry is similar, but more subtle than in data.

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Figure 3.10: Histograms at various heights of axial positions of a trapped bead (markers),
fit to a Gaussian (line). Note that the symmetry of the data and goodness of the Gaussian
fit is much better at 300 nm and 700 nm, and poor at 100 nm.

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Figure 3.11: EQP asymmetry vs. height. (Top) Variation with height of stiffness agreement
and chi squared from Gaussian fits to histograms. In black, the absolute value of the difference between kz calculated via PSD and EQP is plotted vs. height. Note agreement between
methods varies with height, with closer agreement at 300, 400, 700, and 800 nm. In gray, thechi squared values are plotted for each height. Note correlations between improved stiffness
agreement (small kdif f ) and goodness of Gaussian fit (small chi squared). (Bottom) Stiffness
agreement vs. chi squared values. A linear fit shows a small positive slope, suggesting somecorrelation between the values.

111
timization of the intensity stabilization. This relationship was not surprising, given thatthe axial detection voltage is a measure of intensity. Thus intensity noise would directly
manifest as axial detection voltage noise. We demonstrated this intensity stabilization dependence of the power spectral densities in Fig. 3.12, which shows power spectral densities
for four conditions. In A, an unoptimized servo caused a significant spurious peak. This
peak corresponded to a servo setting where the intensity noise peak had not been tuned toa high enough frequency, and thus the intensity noise peak showed up significantly in the
power spectrum. In B, the power spectrum was affected by residual intensity noise evenwith an optimized servo. The residual noise results in a non-Lorentzian shape, clear fromthe attempt at a Lorentzian fit (black ). In C, the power spectral density of a bead fixed
to the surface reflected residual noise despite our active stabilization. In D, the differential
power spectrum, subtracting the power spectrum of the bead fixed to the surface from thatof the trapped bead, resulted in a far more Lorentzian shape. Ultimately, the servo optimization proved important for the resulting f0 from the Lorentzian fits to the power spectral
densities. Surprisingly, however, the improvement in the Lorentzian shape in the differential
power spectrum changed the estimated kP SD by an insignificant amount (< 2%).
Keeping in mind these special considerations for the axial measurements, we calibrated
ktrap in x and z by both equipartition and power spectral density methods as a function oflaser power. We performed the calibration at an axial stage position of 1200 nm while actively
stabilizing the sample position and laser intensity. The linear relationship obtained between
ktrap and laser power (measured by the intensity servo control voltage, see Experimental
Techniques for conversion of Vservo to power) is shown in Fig. 3.13. In x, the slope of V (k)derived from the PSD and EQP measurements differed by < 3%. In z, the slope values
differed by < 2%. However, kP SD,z values were far more linear, with a very small offset from
0 when fitted with a line, whereas kEQP,z values had a significant offset from 0 when fittedwith a line without 0 fixed. Due to this offset and the significant axial asymmetries we based
our axial stiffness calibrations on the kP SD measurements.

PSD (nmRMS2)

PSD (nmRMS2)

PSD (nmRMS2)

PSD (nmRMS2)

112

Figure 3.12: Axial power spectral densities (PSDs). top - Power spectral density of a trapped
bead with an unoptimized servo. The servo was intentionally unoptimized by changing thecoarse gain. Note the peak in the PSD that results. 2nd from top - PSD of a trapped beadwith the servo optimized (gray) and the Lorentzian fit from Labview (black). 3rd from topPSD of a salt-stuck bead. 4th from top - Differential PSD (gray), calculated by subtracting
the stuck bead PSD from the trapped bead PSD. Note the improvement in the Lorentzianfit (black).

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Figure 3.13: Axial trap stiffness calibrations. (A) Trap stiffness in x (solid gray markers) andz (open markers) as a function of servo control voltage. Power spectral density measurements(triangles) and equipartition theorem measurements (squares) show good agreement between
the two different methods. (B) Trap stiffness as a function of axial stage position.

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Because the trap stiffness has been shown to change as a function of trap height [21, 13,
19, 46, 47], we also calibrated the trap stiffness as a function of height (Fig. 3.13). To apply
this height dependent calibration, we consider that the ktrap vs. power calibrations wereacquired at zstage = 1200 nm, and we typically performed measurements with zstage between
90 and 200 nm. According to this ktrap vs. height calibration, kz (1200) = 14.9pN/nm,kz (100) = 12.3pN/nm, giving a 17.4% correction to kz . We took this height-dependent
stiffness correction factor into account in our estimations of ktrap,z .
3.4

UNCERTAINTY ESTIMATES OF CALIBRATION
The previous section describes calibrations of the relative axial bead displacement(∆zbd ), the position of axial trap equilibrium (zbd = 0), the trap height (ztrap ) and the axial
trap stiffness (ktrap ). Based on these calibrations, we wished to make measurements of DNA
extension, DNA extension changes and force on a trapped bead. To know the uncertainty in
our final measurements, we quantified the uncertainty in each of these calibrated quantities.
We quantified the uncertainty in the axial calibrations by several methods. We compared the measured vs. requested motion of beads fixed to surfaces, both for the 330 nm
beads used in our trapped bead measurements and for 860 nm beads used as fiducials forstabilization in our measurements. We trapped beads at equilibrium and measured theirapparent deviations from equilibrium, both with and without incorporating the oscillation
corrections (w/ and w/out osc. cor.). We compared axial trap stiffness obtained by multiple methods. We quantified the uncertainty remaining from the fits in our ztrap calibration
data, and finally we attempted DNA force vs. extension curves. Table 3.1 summarizes theuncertainty measurements described in this section.

3.4.1

Uncertainty by measurements on beads fixed to the surface

As a test of the accuracy of ∆zbd , we compared the measured vs. requested positionsof beads fixed to the sample surface while stepping the sample back-and-forth with the115
Table 3.1: Uncertainty estimates of axial calibration.
Experiment and Analysis
fixed bead zmeasured vs. zrequested
trapped bead deviations w/out osc. cor.
trapped bead deviations w/ osc. cor.
trapped bead deviations w/out osc. cor.
kz PSD vs. EQP at 1200 nm
Fits in ztrap calibration
DNA force vs. extension

Quantity∆zbd (fixed)
zbd = 0zbd = 0∆zbd (trapped)
kztrapztrap
zbd , ztrap , kz

Estimated uncertainty
4%
65 nm
30 nm
20%
2%
±13.3 nm
uninterpretablepre-calibrated piezo stage. First we performed the three-step axial displacement calibration
method for trapped beads on 330 nm dia. salt-stuck beads. Next, the piezo stage was
oscillated in 1 nm and 5 nm steps in z. The axial detection response was recorded, and thedisplacement calibration used to give the measured axial displacement (z). The measured
axial displacement vs. time is shown in Fig. 3.14 A, smoothed and decimated to 100 Hz(light gray - 1 nm steps, dark gray - 5 nm steps) and 10 Hz (black).
We quantified the accuracy of the measured step size by two methods. In the first
method, the 10 Hz data was histogrammed in 0.1 nm bins (Fig. 3.14 A, right, light gray- 1 nm steps, dark gray - 5 nm steps). The histogram of each set of steps was fit to two
Gaussians. The average quantities obtained from 9 traces yielded peak separations (mean ±std. dev.) of 0.99±0.1 nm (1 nm steps) and 4.89±0.22 nm (5 nm steps). The second methodof determining the measured step size was to calculate the pairwise distance distribution(PDD). The PDD for the 10 Hz stepping data was calculated by subtracting each data pointfrom every other data point [49]. The histograms of the PDDs are shown in Fig. 3.14 B, with
0.1 nm bin widths (light gray - 1 nm steps, dark gray - 5 nm steps). Each PDD histogramshows three peaks, which were fit with three Gaussians (black lines) to obtain peak values(mean ± std. dev.) of ±4.9 ± 0.25 nm and ±0.96 ± 0.18nm. The average peak differencein the histograms for 9 different traces is taken as the limit of the accuracy of a relative
displacement measurement, and the std. dev. of the peak difference is the precision.

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Figure 3.14: Accuracy of calibrated axial measurements. Stuck bead signals were calibratedusing the method described for free beads. The stage was oscillated in 1 and 5 nm steps
to test the resolution and accuracy of the calibrated z measurements. (A) Examples ofmeasured axial steps smoothed to 100 Hz (1nm - light gray, 5 nm - dark gray) and 10 Hz(black). The dashed lines are spaced at 1 and 5 nm intervals. Histograms of 10 Hz stepping
data with two Gaussian fits to each set show the peak separation to be 0.99 ± 0.1 nm and4.89 ± 0.22 nm (mean ± std. dev. for 9 traces). (B) The Gaussian fits (black) of pairwise
distance distributions from 9 different traces have peaks for the 1nm steps (light gray) at±0.96 ± 0.18 nm and for the 5 nm steps (dark gray) at ±4.9 ± 0.25 nm.

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If we translate these measurements to percent differences between the measured andrequested displacements, the uncertainty in ∆zbd measured by this method is quite small(1 − 4%). However, this measurement provides no information about the uncertainty of thetrap equilibrium (zbd = 0) position. Also, this measurement is really a lower limit on theuncertainty of ∆zbd of a trapped bead, because of differences between the detection signalof a stuck bead and a trapped bead.
For completeness, we also determined axial displacement calibrations for large (860nm) stuck beads used as fiducials in stabilizing the coverslip surface relative to the laser
focus. A representative detector response vs. axial stage position curve is shown in Fig. 3.15
A (gray markers). The curve was fit to 7th order polynomials in two separate regions to
obtain a shallow calibration, (solid line) and a deep calibration (dashed line).
As a metric to test the accuracy of the fiducial axial displacement calibration, we used
these shallow and deep calibrations to convert the detector response voltage to measured
axial position (zmeas ). The measured axial position vs. requested axial stage position (zreq )is plotted for the shallow (Fig. 3.15 B ) and deep (Fig. 3.15 C ) regions (gray markers).
The measured vs. requested positions were fit to a line, obtaining a slope of 1.0057 ± 0.0022
and offset of 2.89 ± 1.1 in the shallow region, and a slope of 1.0027 ± 0.00158 and offset of
4.53 ± 2.5 in the deep region. The residuals are shown for each linear fit. The extremely
small deviation of the fitted slopes from an ideal slope of 1 represent the very small percent
uncertainty for the displacement calibrations on the fiducials used for stabilization. The
small offsets represent small systematic offsets for the measured positions for these fiducials.
Because these uncertainties were so small, and because they represent uncertainties for thefiducial rather than the trapped bead, they are not included in the table summarizing theuncertainties in the axial calibration.

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Figure 3.15: Accuracy of calibrated axial measurements for fiducial. (A) Seventh-orderpolynomials are fit to the shallow (solid line, white background) and deep (dashed line, graybackground) regions of the stuck bead curve. (B) The calibration is used to convert thevoltage signal (markers) to the measured z (zmeas ) vs. the requested z (zreq ) in the shallow
region. A linear fit yields a slope 1.0057 ± 0.0022 and offset of 2.89 ± 1.1. Residuals areshown for the linear fit. (C) Same as B, for the deeper region. The slope is 1.0027 ± 0.00158
and offset is 4.53 ± 2.5 in the deep region. The residuals are shown for each linear fit.

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3.4.2

Uncertainty by trapped bead deviations from equilibrium

As another quantification of the uncertainty in our calibrated quantities, we measured
the apparent deviations from equilibrium for a bead trapped at equilibrium as we moved thesample surface. With no external forces on the trapped bead, the average position of thebead should remain at equilibrium, i.e. zbd = 0. However, as described, the axial detection
voltage oscillates as the sample surface drops away from trap equilibrium. We quantified thisuncertainty both with and without oscillation corrections incorporated into the displacement
calibration.
Without oscillation corrections, we saw previously in Fig. 3.5 that apparent deviationfrom trap equilibrium was as large as 74 nm. Thus depending on the measurement height
and the height where zbd = 0 was determined, zbd = 0 could be off by up to the full amplitudeof that largest oscillation. I often determined zbd = 0 between 100-150 nm from the surface,which is near the bottom (-65 nm) of the first oscillation. With trap height changes, and/or
unaccounted changes in bead height due to the DNA tether, zbd = 0 could be off my as muchas 65 nm.
With oscillation corrections incorporated, we measured apparent deviations from trap
equilibrium of as much as ± 10 nm. Also, depending on the bead, the average measured zbd
position was sometimes closer to +10 or even +20 nm. This measurement means that evenwith oscillation corrections, zbd = 0 could be off by up to 30 nm (20 nm average + 10 nm
remaining oscillations).
These oscillations also illustrate the increased uncertainty in ∆zbd when measuring therelative position of a stuck bead vs. a trapped bead. For trapped beads, the largest of theseoscillations, zmeas = 65 - 25 nm = 40 nm uncorrected, occurs between 150 and 350 nm
depth. This slope (0.2) can be interpreted as a 20% potential error in a ∆zbd measurement
taken along this slope. With a stationary surface, if a trapped bead moved from 350 to
150 nm (∆ = 200 nm) in depth, it would only have appeared to move 160 nm, a 20%

120
underestimation. On another part of the curve, the motion could actually be overestimatedby 20% due to these oscillations. However, the error in ∆zbd should not be larger than 20%,
because if the bead moved from one oscillation to the next, the opposite directions of theslopes would actually cancel out some or all of the error.

3.4.3

Uncertainty by trap stiffness measurements

The next method for estimating uncertainty was a comparison of axial trap stiffness
measurements by equipartition and power spectral density, kEQP and kP SD . The difference
between ktrap,z estimated by the two methods provides an estimate of the uncertainty in
ktrap,z . For the measurements of ktrap,z as a function of intensity, the measured slopes were
within 2% of each other, yielding an estimated average uncertainty in ktrap,z of 2%. Thisaverage uncertainty is actually substantially lower than the uncertainty in ktrap,z introducedby variations in bead size. According to the manufacturer, the bead size coefficient ofvariation is 4%, which leads to a 12% variation in ktrap,z , since ktrap,z is proportional to thevolume of the beads. Thus the bead-to-bead variation in ktrap,z of 12% is larger than theuncertainty estimated by measuring ktrap,z by the two different methods.
Another uncertainty estimation based on trap stiffness measurements is based on theztrap calibration in Section 3.3.4. The estimated uncertainty in ztrap from this method is
simply the fitting error in the Faxen’s Law fit obtained to the lateral drag vs. height data.
This uncertainty estimate for ztrap was ± 13.3 nm.
The above measurements, combined with previous measurements during the course ofcalibration, provided estimates of the uncertainty of the various quantities of our calibration.
From the summary of uncertainty quantifications in Table 3.1, we can see that the uncertainty
estimate is smallest (2%) for the trap stiffness (ktrap ), and larger but still reasonable (< 20%)for the relative axial displacement (∆zbd ). The estimated uncertainty in ztrap is not large(±13.3 nm), but is difficult to manage if an experiment aims to measure quantities near
the sample surface. In contrast, the uncertainty for the trap equilibrium location (zbd = 0)121
is substantial, especially without the oscillation corrections (65 nm). From our uncertainty
estimations, we expect we can estimate ktrap and measure relative axial displacements well.
However, mostly due to the large uncertainty in zbd = 0, we would have large uncertainties
when measuring extension of molecules (Ext = ztrap − zbd ) and force (F = ktrap ∗ zbd ).
The next section described in detail how these uncertainties would be expected to impact
measurements of DNA force and extension.

3.4.4

Uncertainty by DNA force and extension

Another method for estimating the uncertainty in the axial calibration was the DNA
force vs. extension measurements. In this method, the axial force (F ) and extension (Ext)were measured on DNA molecules. Because F = −ktrap ∗ zbd and Ext = ztrap + zbd , thismethod relies on all of the calibrated quantities, zbd , ztrap , and ktrap . This method has
the potential to provide a very nice confirmation of the axial calibration if all calibrated
quantities have very small errors. However, if more than one of the calibrated quantities has
more than a very small error, the DNA force vs. extension measurement becomes virtually
uninterpretable.
Based on the relationships listed above between the calibrated quantities and the measured force and extension, we can predict the effect on the DNA force vs. extension curve.
These predicted effects are summarized in Table 3.2. An error exclusively in ktrap could be
easily identified, as the F vs. Ext curve shape would simply appear distorted in the force
axis. A systematic error in ztrap would also be easily identified, as the curve shape would
simply be offset on the extension axis. An error exclusively in trap equilibirum (zbd = 0)
could also potentially be identified, as the curve would be simply shifted towards the upper
left quadrant (bigger F , smaller Ext) or the lower right quadrant (smaller F , bigger Ext).
These three examples assume the error is constant over the relevant portion of the curve.
An error in the relative displacement measurement ∆zbd would be very difficult to identify,
as it would skew both the force axis and the extension axis, resulting in an oddly shaped

122
curve. But in the least interpretable example, if non-negligible errors occur in more than
one calibrated quantity, the result of a shifted and distorted curve would be uninterpretable.
Unfortunately, this last example seems to be the case for our data.
Table 3.2: Force vs. extension errors from calibration uncertainties.
Parameter with error
ktrap
ztrap
zbd = 0∆zbd
multiple parameters

Resulting change in F vs. Ext curvesame general curve shape, but with scaled F axis
same curve shape, but shifted along Ext axis
same curve shape, but shifted to upper left or lower right quadrantskewed F and Ext
skewed and possibly shifted F and Ext

Despite the uncertainty estimates predicted, we tested our axial measurement technique
by obtaining axial force vs. extension curves on short (92 nm) DNA molecules. The DNA
tethers were stretched by dropping the stage axially. The trap height ztrap was corrected
for the focal shift by multiplying by 0.83. The extension Ext = ztrap + zbd and the force
F = −ktrap ∗ zbd were calculated from the calibrated axial quantities. The obtained force vs.
extension curves are plotted in Fig. 3.16, along with the theoretical F (Ext) curve from the
Bouchiat-modified Marko-Siggia WLC model [22], with corrections for short DNA [25]. The
persistence length used in the calculation was 19 nm, calculated using the equation from
Seol et al. with pinf inity = 44.5 nm, as obtained in lateral stretching curves of 2013 nm DNA
under the same conditions. The curves demonstrate the expected scenario: the curves areboth skewed and shifted in such a way that no specific uncertainties can be interpreted.
Attempts at obtaining axial DNA stretching curves on long DNA proved uninterpretable as well. With long DNA (2013 nm), the force vs. extension curve curve is more wellestablished. However, the 2013 nm DNA was so long that the trap had to be at a depth of2000 nm to pull the DNA to forces where the stretching curve became steeper. At this depth,
the reliability of the axial trap calibrations became questionable, and in a way with little
relevance to the very shallow measurements we actually cared about. Thus, the axial 2013
nm DNA measurement became somewhat irrelevant. However, the lateral measurement of

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Figure 3.16: Force vs. extension, measured in the axial direction. Markers - Force vs. extension data, stretching in the axial direction and using axial detection and axial calibrations forthe bead displacement from equilibrium (zbd ), trap height (htrap ), and trap stiffness (ktrap ).
The force axis was calculated via F = ktrap ∗ zbd . The extension axis was calculated as
Ext = htrap − zbd . ktrap = 0.0332 pN/nm, using the axial stiffness calibration from the axial
stiffness figure, with the height correction at 100 nm (i.e. k100 = 0.87 ∗ k1200 ). Trap heightwas calculated using the trapped bead peak as a reference, with the surface offset from
Faxen’s Law fit and the peak corrections for stiffness, then correcting for the focal shift bymultiplying by a factor of 0.83. Black markers - The displacement calibration was performed
via the Pralle reconstruction, using the simple peak offset method for determining the trap
equilibrium voltage (as in TBP data). Gray markers - The displacement calibration was
performed by adding the oscillation corrections on top of the Pralle reconstruction. Line- Theoretical force vs. extension curve. The curve was calculated using a modified WLC
model [22] with a contour length of 92 nm and persistence length of 19 nm. The 19 nm
persistence length was calculated using the empirical formula for calculating p of short DNA[25] with p∞ = 44.5 nm (calculated from 2013 nm DNA lateral stretching in experimental
methods section).

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the 2013 nm DNA provided a good measure of the persistence length, which was needed toprovide the theoretical force vs. extension curve for the short DNA.
3.5

IMPROVED AXIAL FORCE AND EXTENSION ESTIMATIONS

3.5.1

Estimating force from data traces

Because of the larger uncertainties in the estimation of the trap equilibrium position
and trap height, we needed a different method to measure the force and extension of amolecule axially. While a general estimation of the force and extension would be useful,what is most meaningful for our data is a specific estimation of the force and extension for
the actual measurements we obtained. For our TBP data, we obtained measurements atthree different forces, nominally F = 0.08, 0.3 and 1 pN. Therefore, we needed to obtain our
separate estimation of the force and extension at these forces.
We obtained this improved estimates of the force and extension by measuring theBrownian motion of trapped, tethered beads under identical conditions to our TBP data.
We analyzed the Brownian motion of trapped, tethered beads to yield estimates of thetotal stiffness of the tethered-bead system, ktotal . From ktotal and the calibrated ktrap , wecalculated kDN A = ktotal − ktrap . From this estimated kDN A and the theoretical calculation
kDN A = dF/dExt (based on the the modified-WLC F (Ext) [22], we found the force andextension that corresponded to the experimentally derived kDN A . This force and extension
were the final estimates for the force and extension in our TBP data.
Initially, we estimated the force and extension by estimating ktotal via three different
methods: an equipartition theorem method, a power spectral density method, and an autocorrelation method. Fig. 3.17 illustrates the Brownian motion analysis of a typical trace
at the intermediate force for each of the three methods. Each method had advantages andpotential drawbacks, as detailed below.
After performing these calculations on our data, we used the obtained mean ktotal

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Figure 3.17: Three methods for estimating force. In each method, we analyzed tethered beadmotion to calculate the total stiffness of the system (ktotal ). From ktotal and the trap stiffness(ktrap ), we then calculated the DNA stiffness, kDN A = ktotal −ktrap . From kDN A , we found thecorresponding force on the theoretical DNA stiffness curve, kDN A = dF /dExt, where Force(F ) as a function of Extension (Ext) was based on the Bouchiat-modified, Marko-Siggia
WLC model for DNA. (A) Axial bead position (zbd ) vs. time (from intermediate force data).
Here, ktotal is calculated via the equipartition theorem (EQP). In the EQP method, ktotal= kB T /σ 2 , where kB T is the thermal energy of the system and σ 2 is the variance of zbd .
(B) Power spectral density (PSD) of the zbd vs. time trace (gray). Here, ktotal is calculated
via the PSD method, ktotal = 2πβf0 , where β is the drag coefficient of the bead and f0 is
the corner frequency of the Lorentzian fit (black) to the PSD. (C) Autocorrelation of thezbd vs. time trace (gray). Here, ktotal is calculated from the autocorrelation time (τ ) of an
exponential fit (black) to the autocorrelation of the data.

126
and extension (Ext = hbd ) at each force to simulate trapped bead traces. (See details
in Experimental Techniques chapter.) Fig. 3.18 shows simulated data and analysis. By
simulating these traces at full bandwidth, then filtering, we estimated the uncertainty/error
caused by the finite bandwidth in our data for each analysis method. In the simulations,analyzing the data via the power spectral density provided the closest match between input
values and output values. In fact, errors between the input forces and output forces were
< 2%. Due to this low error, we ultimately relied on the force and extension values estimatedby the power spectral density.

3.5.2

Force estimation by equipartition

In detail, for the equipartition measurement, we calculated ktotal based on the variance,σ 2 of our data traces, ktotal,EQP = kB T /σ 2 . We calculated σ 2 in 1 s windows to prevent
misestimation from residual drift. From this estimate of ktotal , we estimated the DNA force
and extension as already described.
Comparison of the equipartition analysis of the data and simulations showed that
equipartition misestimated the force and extension for at least two reasons. The simulations
showed that equipartition significantly underestimated the variance on the traces filtered to
2 kHz, which translated to an overestimate in the simulations of ktotal , Ext, and F output
values compared to simulation input values. Another source of error was the anharmonicityof the trap in the axial direction. This anharmonicity is shown in Fig. 3.19 in the histogramson the left. For these simulations, the simulated traces assumed the ideal harmonic trap, asshown by the histograms on the right. The lack of symmetry and non-Gaussian distributionsin the data degraded the accuracy of the stiffness measurement by equipartition in an unpredictable way. Ultimately equipartition analysis of the data and simulations did not follow
the same trends, which we interpreted as being due to the competing and unpredictableaffects described. Ultimately the equipartition analysis was not the best way to estimate theforce.

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Figure 3.18: Simulation of three methods for estimating force.

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Figure 3.19: Histograms of axial bead position. Histograms of data traces (left) show askewed, non-Gaussian position distribution. Histograms of simulated traces (right) are Gaussian and symmetric, as simulations were based on a harmonic trap. The asymmetry seen in
the data histograms affects the variance calculated from the traces, and therefore affects theaccuracy of the force calculated via the EQP method.

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3.5.3

Force estimation by power spectral density

For the power spectral density analysis, we obtained ktotal based on the corner frequency, f0 of the Lorentzian fit to the power spectrum of our data traces, ktotal,P SD (h) =2πβ(h)f0 , where β(h) is the height dependent drag coefficient. We fit the region 1 < f <1500 Hz. The low frequency limit was to prevent skewing by residual drift; the high frequency limit was to prevent skewing by the high f tail in our data caused by an imperfectfilter.
The key concept for estimating the force and extension by the power spectral density
method was carrying the height dependence of the drag in the stiffness measurement β(h)through to the estimation of the DNA stiffness, kDN A (h) = ktotal,P SD (h) − ktrap . By retaining
the height dependence, we estimate kDN A (h) without using the error-prone trap height ortrap equilibrium calibrations. Then, using the fact that h = Ext for a DNA molecule
stretched between the surface and a bead, we plot kDN A (h) on the same axes as kDN A =dF/dExt. Fig. 3.20 shows both an example of these curves. The intersection of the two
curves gives the resulting kDN A − Ext pair, from which we found the corresponding force.
The power spectral analysis relied on temporal properties of trace, and data was acquired at a bandwidth (4 kHz with 2 kHz antialiasing) not much faster than the cornerfrequency (f0 ) of the power spectrum of the data. Because of this low bandwidth/f0 ratio,
we feared fitting the power spectrum to obtain f0 would result in inaccuracies. Fig. 3.21
shows the power spectral density and fit of the highest force (and hence highest f0 ) data,
and the PSD and fit of the simulation of the highest force data. The percent difference
between the simulation input f0 and the simulation output f0 from the fit to the simulationwas < 5% for all forces. This small difference between simulation input and the fit to thefiltered simulation gives confidence in estimating the force and extension using the power
spectral method.

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Figure 3.20: DNA stiffness vs. extension. The PSD and autocorrelation methods for calculating ktotal from tethered bead data traces depend on the drag coefficient (β) of the bead.
β depends on the height above the surface. Thus, ktotal is a function of height, which means
kDN A calculated from a tethered bead data trace is also a function of height, kDN A (h) =ktotal (h) - kDN A . For a tethered bead, the height to the bottom of the bead (h) is the same as
the DNA extension (Ext). Therefore, kDN A (h) can be plotted as kDN A (Ext). kDN A (Ext) can
also be calculated theoretically as dF /dExt, as described in the text. Plotting kDN A (Ext)calculated from a tethered bead data trace and from theory, the intersection of the two
curves provides the correct kDN A and Ext, which can then be used to find the corresponding
force.

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Figure 3.21: Power spectral density of highest force data and simulation. Data was acquired
at a bandwidth (4 kHz with 2 kHz antialiasing) not much faster than the corner frequency(f0 ) of the power spectrum of the data (1030 Hz). Because of this low bandwidth/f0 ratio,
we feared fitting the PSD to obtain f0 may result in inaccuracies. This plot shows the PSD
and fit of the highest force (highest f0 ) data (top), and the PSD and fit of the simulation(bottom) of the highest force data. The percent difference between the input f0 for thesimulation and the fit f0 was less than 5% for every force. The agreement between thesimulation input and output demonstrates the accuracy of f0 from the PSD fit despite thelow bandwidth/f0 ratio.

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3.5.4

Force estimation by autocorrelation

For the autocorrelation method, we obtained ktotal based on the autocorrelation time,τ , obtained from an exponential fit to the autocorrelation of the data. Here, ktotal,tau (h) =β(h)/τ . Similarly to the power spectral analysis, we retained the height dependence inthe equations through the calculation of kDN A (h), plotted kDN A (h) and the theoreticalkDN A (Ext), and found the kDN A , extension, and corresponding force at the intersectionof the curves.
Simulations indicated that the autocorrelation analysis tended to underestimate ktotal(and therefore F and Ext), likely because of limited bandwidth. Data similarly showedktotal,tau values smaller than the ktotal,P SD values, which simulations showed were most accurate. However, the effect between data and simulation was not quantitatively identical,
possibly due to the small residual drift in data. This effect was most dramatic for the highest
force data, which had the most drift. On lower force data, fitting the autocorrelation function to 1 s from the autocorrelation peak provided reasonable estimates of ktotal . However,on high force data, when fitting to 1 s from the peak, the slope of the function skewed the
autocorrelation time from the fit to the point where the ktotal calculation became nonsensical. Narrowing the fit range to 0.1 s from the peak provided a reasonable estimate. I
interpret the slope in the autocorrelation function calculated from data, which is not present
in the simulation, as the cause of the non-quantitative agreement between the values of ktotal
calculated from the data and simulation.

3.5.5

Summary of final force estimates

To summarize, we estimated the force and extension from the Brownian motion of ourdata traces at three different forces. We estimated via the three methods described: equipartition, power spectral density, and autocorrelation analysis. For the power spectral and autocorrelation analyses, using the intersection of kDN A (h) by the analysis and kDN A (Ext) from

133

Figure 3.22: Autocorrelation of axial bead position traces. (Top) Autocorrelation of thehighest force data. (Bottom) Autocorrelation of the highest force simulation. Simulations
indicated that the autocorrelation analysis tended to underestimate ktotal (and therefore F
and Ext), likely because of limited bandwidth. Data also showed ktotal values smaller thankP SD values, which the simulation showed are most accurate. However, the affect between
data and simulation was not quantitatively identical, possibly due to the small residual drift
in data. This effect was most dramatic for the highest force data, which had the mostdrift. On lower force data, fitting the autocorrelation function to 1 s from the peak provided
reasonable estimates of ktotal . However, on high force data, when fitting to 1 sec from thepeak, the slope of the function skewed the autocorrelation time from the fit to the point
where the ktotal calculation became nonsensical. Narrowing the fit range to 0.1 s from thepeak provided a reasonable estimate. However, I interpret the slope in the autocorrelation
function calculated from data, which is not present in the simulation, as the cause of the nonquantitative agreement between the values of ktotal calculated from the data and simulation.

134
the theory curve provided force and extension measurements independent of the zbd = 0
and ztrap calibrations with large uncertainties. We performed simulations to quantify theaccuracy of these estimations and the affect of the low bandwidth/f0 ratio. Based on thesesimulations, the power spectral analysis yielded the most accurate estimations of force andextension, with simulation input to output errors of < 2%. This uncertainty neglects theanharmonicity of the trap that exists in the experiment but not in the simulation. The
summary of the final force and extension estimates by the above calculations is given in two
tables. Table 3.3 gives the force estimated by each of the three methods. Table 3.4 gives
the final estimates for the force and extension by power spectral analysis, judged the mostreliable method.
Table 3.3: Force estimation from data by equipartition (EQP), power spectral density (PSD)
and autocorrelation (tau) methods.
MethodEQP
PSD
tau

Low force (pN)
0.33
0.34
0.33

Intermediate force (pN)
0.40
0.47
0.37

High force (pN)
0.67
0.79
0.62

Table 3.4: Final force and extension estimates via the power spectral density method, whichyielded the most accurate results in simulations.
Relative force
Low
Intermediate
High

3.6

Force (pN)
0.34
0.47
0.79

Extension (nm)
54.5
60.4
67.8

DECREASED NOISE ON SHORT DNA TETHERS
As a demonstration of the improved signal to noise made possible by axial detection, we

obtained axial position vs. time traces for 92 nm DNA tethers (Fig. 3.23 A). The position of
the tethered bead was measured in all 3 axes while the sample was stabilized on a bead fixed135
to the sample surface. The obtained position vs. time traces, smoothed to 5 Hz, are shownin Fig. 3.23 A. The integrated noise vs. time (Fig. 3.23 B ) was obtained by integrating
the power spectral density of the axial position vs. time traces shown in Fig. 3.23 A. The
signal to noise obtained in these measurements demonstrates the power of axial trappingand detection. Because such short DNA can be used for tethering, the stiffness of the DNA
is higher and the spatiotemporal resolution is improved, leading to an integrated noise of<1nm over 0.03 - 3.2 Hz for a bead trapped at 0.47 pN.

Figure 3.23: Stability of DNA extension measured on 92 nm tethers with stabilized, axial
optical trap. (A) Axial position vs. time, smoothed to 5 Hz, for three different applied
forces (black - 0.34 pN, light gray - 0.47 pN, dark gray - 0.79 pN). (B) Integrated noise vs.
frequency for three different applied forces (same color representation as in A). (Inset) The
integrated noise (0.03 - 2.5 Hz) vs. force. With our stabilization and axial-detection scheme,
the integrated noise level is below 1 nm up to 3.2 Hz for a bead trapped at 0.47 pN (ktrap 0.00528 pN/nm).

3.7

CONCLUSIONS
We have demonstrated an axial calibration and detection method that allows stable,

precision measurements on very short DNA. Using a single laser and detection method and136stuck beads, we calibrated the axial displacement, stiffness, and height above the surface
and developed a new method of estimating axial force and extension on very short DNA
molecules. By axial trapping and detection, the trapping geometry is simplified, resulting
in measurements made with no assumptions about the off-axis terms, less cross-talk, andimproved signal-to-noise. The signal-to-noise is further improved by the ability affordedby axial trapping to make measurements on ultra-short DNA molecules, resulting in fasteraveraging of Brownian motion.
For the data in the following chapter, we used several relevant parts of the axial calibrations described in this chapter. We used the relative axial displacement calibration (without
the oscillation corrections) to measure extension changes of DNA. We also used the axial
trap stiffness calibration, combined with the described improved axial force and extension
estimations to measure the absolute force and extension.
The other portions of the axial calibrations described in this chapter remain as useful
1st-order approximations. For our experiments, we used the described estimations of trap
equilibrium and trap height to set the force and height during data acquisition. Later,
during analysis, we corrected the force and height using the improved force and extension
estimations.

Chapter 4
DIRECT MEASUREMENT OF HUMAN TBP BENDING DNA

4.1

INTRODUCTION
As described in Chapter 1, TBP binding DNA is one of the first steps in gene expres-sion: recruitment of the transcription machinery necessary for making RNA. Upon binding,human TBP induces an ∼100◦ bend in DNA. Using the techniques described in the previous
chapters, we measured TBP dynamically bending DNA at the single molecule level in order
to elucidate TBPs role in regulating transcription.
We report here direct observations of TBP-induced extension changes of DNA. The
measurements were made using a novel biophysical assay including custom PEG surfaces,
an actively stabilized optical trap, and an axial trapping geometry with a new method foraxial force and extension determination (Fig. 4.1). Calibrated forces were applied to axially
stretch the DNA and TBP-induced step-like DNA extension changes were measured. Hidden
Markov modeling was applied for quantitative analysis of the extension change as well as
rates of bending and unbending. Our results show that, under physiological KCl and MgCl2
concentrations, human TBP dynamically bends and unbends TATA box DNA in step-likechanges on time-scales of tens of seconds, which may imply a more dynamic role for TBP intranscription regulation than previously thought.

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TBP

xfid

∆zbd
zfid

Figure 4.1: Final TBP assay. A cartoon depicts the final biophysical assay implemented todetect TBP bending DNA. On the right, sample stabilization is performed on a streptavidincoated bead fixed to the surface by biotinylated-PEG. The position (e.g. xf id , zf id ) is measured by a separate laser and actively stabilized using the piezo stage. On the left, TBP
bending DNA is measured by axial detection of a DNA-tethered bead trapped directly above
the tethering point. The bead is tethered by biotinylated DNA attached to the surface bystreptavidin bound to covalently attached biotinylated-PEG molecule. When TBP bends
the DNA, the bead is pulled down in the trap.

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4.2

MATERIALS AND METHODS
The instrument and sample preparation were as described in Chapter 2. The actively

stabilized optical-trapping microscope ensured the stability essential for measuring the small,infrequent events of TBP bending DNA. Beam steering and lateral stiffness were calibrated as
described. Samples surfaces were coated with covalently attached PEG, which was essential
for consistent, reproducible data acquisition. These surfaces incorporated mPEG (LaysanBio) and 1-3% biotin PEG (Rapp Polymere) for specific attachment of DNA tethers to
the surface, as well as for anchoring 860 nm dia. streptavidin beads to the surface to use as
fiducials for sample stabilization. Tethers were 92 nm DNA, with biotin on one end to attach
to biotin-PEG anchored streptavidin, and digoxigenin on the other end to attach to 330 nm
dia. antidigoxigenin-modified beads. The short DNA was essential for single, TATA-specificextension changes, as was a carefully engineered DNA sequence with one or no TATA boxes
and no other TATA-like sequences.
Experiments were exceptionally sensitive to buffer conditions. For all experiments, weused TBP buffer (10 mM Tris-Cl - pH 7.9, 10 mM Hepes - pH 7.9, 150 mM KCl, 1mM
DTT, 0.5 mM MgCl2 , 0.2% tween-20, 0.4 mg/ml BSA). TBP buffer was actually a 1:1
combination of two different buffers, mixed just before experiments. One buffer was DB300- 20 mM Tris-Cl, 300 mM KCl, 1 mM DTT, 0.8 mg/ml BSA, the other buffer was RM 20 mM Hepes, 1mM MgCl2 , 1 mM DTT, 0.4% tween-20. The reason for mixing the buffers
in this way was so that dilutions of TBP, which was in a buffer almost identical to theDB300, could be made in such a way as to not change the final concentrations of the buffer
components. The storage buffer for TBP was DB100 (20 mM Tris - pH 7.9, 100 mM KCl,
1mM DTT, 20% glycerol, 50 µg/ml BSA), the same as DB300 but with lower KCl, slightly
higher BSA, and 20% glycerol. Transcription experiments have previously shown sensitivity
to BSA solutions (personal communication from Jim Goodrich), and our TBP experiments
also showed different behavior with different BSA solutions, failing when we used Sigma,

140
Bovine Albumin Fraction V A-3059. For this reason, we used the same BSA solution usedin transcription experiments in the Goodrich lab (Roche BSA, Cat. No. 10 711 454 001).
Sensitivity of TBP to other buffer conditions is discussed in the text.
Recombinant, human TBP was prepped as described in [27], and stored at -80◦ C,
warmed in fingers and diluted into TBP buffer just before experiments. The TBP dilution
was flowed into samples and data acquisition started ∼3 min. afterwards. Leftover TBP wasstored on ice and new TBP (from leftover on ice) was added after ∼30 min. After 1 hourfrom removal from 80◦ C, any leftover TBP was discarded.
Data acquisition was as described in Chapter 2 and Chapter 3. Lateral sample andlaser positioning and displacement calibrations were as described in Chapter 2, and axial
positioning and calibrations were based on the description in Chapter 3. Axial displacement
calibrations were 7th-order polynomial fits to Pralle reconstructions of the axial detection
signal of each bead to be measured. Axial stiffness vs. power and vs. height were basedon the PSD method estimates (see Chapter 2 and 3). The first approximation of axial
trap equilibrium during acquisition was by the peak offset corresponding to the equilibrium
voltage at the measurement height, as described in Chapter 3, and first approximation of trap
height during acquisition was also determined by an offset from the voltage peak determinedby the height dependent lateral drag measurement and stiffness dependent peak corrections.
The final reported axial force and extension were calculated in analysis as described, basedon the PSD of control 92-nm DNA-tether data, the height-dependent axial drag, and thetheoretical dF /dExt curve. Axial detection was essential for measurements on short DNA.
For data acquisition on each tether, we pre-tensioned the tether for screening andpre-centering over the tetherpoint. If the tether remained attached and had a (highly preliminary) persistence length measurement of P > 7 nm and contour length measurement 200
nm < L < 265 nm, we continued acquisition. We then positioned the sample and lasers andcalibrated detection of x, y, z using automated LabVIEW. In detail, after the pretensioning,
we set the sample height based on the axial detection peak, calibrated x, y detection sensi-

141
tivity of the trapped bead, moving the trap and stage simultaneously for calibration of theshort DNA. We then centered over the tetherpoint via the stage and DNA centering curve,
set the height again, and centered the fiducial laser over the fixed 860 nm bead by the PZT
mirror. Finally, we performed the final x, y, z detection calibration of both detection lasers
by moving the stage and the 1064 PZT mirror simultaneously.
For extension vs. time measurements, we acquired x, y, and z bead displacement
vs. time for both the trapped bead and fiducial. We acquired 4000 data points/s, with 2
kHz antialiasing, stabilizing the sample at 100 Hz. In these measurements, changes in theaxial trapped bead displacement were equivalent to changes in DNA extension to within an
uncertainty estimate of relative bead position of 20% (based on the estimates in the previous
chapter). For final display, we converted to absolute extension by adding the force-dependent
extension calculated in the previous chapter to the bead position at that force minus theaverage bead position at that force.
Extension vs. time measurements were acquired with several variables. One variable
was force: varying 0.3, 0.5, 0.8 pN, typically 100-300 s at each force. The force was set bychanging the trap stiffness and moving the sample to change the bead displacement fromthe trap center to the desired displacement. The other two variables provided the main
measurement vs. the controls: the main measurement was TBP + TATA box DNA. The
controls were 1) TATA box DNA with no TBP present and 2) TBP + no TATA DNA.
For data analysis, extension changes, equivalent to axial displacement changes, wereanalyzed by three methods. The first method was a qualitative analaysis of extension change
events (see text below for details). The second method was an adapted hidden Markov
modeling (HMM) algorithm developed for FRET data [34]. For this analysis we scaled
displacement to between 0 and 1, and analyzed using the Hammy 4.0 GUI (available online
at TJ Ha’s website). The third method of analysis was a more sophisticated HMM algorithm,
modified from the Hammy algorithm to introduce Fourier modes to correct for residual drift
in the data.

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4.3

TBP AND TATA-DEPENDENT EXTENSION CHANGES IN DNA
The first step in studying TBP bending of DNA was establishing TBP dependent

extension changes of the DNA in a DNA tethering experiment. Our first dramatically TBPdependent results on DNA tethers were achieved on a DNA construct containing 48 repeats ofa consensus TATA box on 502 nm DNA. This construct was designed to maximize extension
changes by TBP, creating compaction of the DNA that would be obvious by visual inspection of video microscopy, thus allowing faster screening of conditions. With this construct,
we observed TBP dependent partial DNA compaction. As designed, this compaction was
obvious by visual inspection of the CCD microscopy image of the DNA tethered beads, andit was also observed by stretching the DNA and observing discontinuities in the stretching
curves. In addition, during extension vs. time data, we observed multiply-sized extension
changes.
These experiments suggested that the DNA compaction did not occur from the construct sticking to the surface. In the stretching experiments, the centerpoint of repeatedstretching experiments with discontinuities did not change within a few nm. The extension vs. time experiments also suggested the effects were not surface sticking because thelateral position of the bead in the laser was stable during the axial extension changes. However, without establishing TATA dependence, these experiments could not rule out sticking
between DNA-TBP and the trapped bead. These experiments did demonstrate that applied load was able to disrupt some of the compaction, but often the compaction seemed tostabilize over time to become increasingly resistant to disruption.
We next set out to establish TATA box dependence. On similar length DNA to themulti-TATA construct but without engineered TATA boxes, we still observed similar partial
DNA compaction. We decreased the DNA length to 285 nm and a sequence with no consensus
TATA boxes and observed fewer discontinuities in stretching curves and fewer extension
changes during extension vs. time data. We then further decreased the length of the DNA

143
tethers to 92 nm DNA. The 92 nm DNA improved both the TATA box dependence andalso decreased Brownian noise due to the increased stiffness of the shorter DNA molecules,
improving signal/noise.
Simultaneously to the experiments on different lengths of DNA, we performed experiments with varying KCl, MgCl2 , competitor GC DNA, and TBP concentration in attempts
to find optimum conditions for further experiments. We also varied the DNA among fourconstructs: 92 nm DNA with one TATA box and no TATA box and ∼300 nm DNA with
one TATA box and no TATA box. Testing was generally preliminary in the sense that weonly tested conditions on 2-6 samples, trying to quickly screen for the best conditions. The
goals were to maximize interpretability and statistics, i.e. multiple consistently sized events
over 100 s, with maximum specificity for the TATA box. Multiple events over 100 s would
allow efficient gathering of statistically significant numbers of extension changes and dwell
times in different states. Consistently sized events would suggest we had determined thebase level extension change. Specificity to TATA leads to easier interpretability, because
we would only be studying bending at one DNA sequence, reducing variables. The final
optimized conditions were 150 mM KCl, 0.5 mM MgCl2 , no GC DNA, and 10-20 nm TBP.
Interestingly, this KCl and MgCl2 concentration mirror in vivo conditions [50].
4.3.1

Data reduction

Once DNA length, [TBP], [KCl], and [MgCl2 ] were optimized, we obtained statistically significant amounts of data at identical conditions with controlled variables to morequantitatively show TBP and TATA box dependence. We obtained DNA extension vs. time
data on 92 nm TATA DNA in the presence and absence of 20 nM TBP, and on 92 nm no
TATA DNA in the presence and absence of 20 nM TBP, all under an applied load of 0.5
pN. In addition, we obtained 92 nm TATA ± TBP under applied loads of 0.3 and 0.8 pN.
Under these consistent conditions, we obtained extension vs. time traces on a total of 30
individual samples. At the tether level, we attempted to acquire data on some 300 tethers.

144
Of these 300, 70 stuck or detached before we could acquire >100 s of data. We chose 100 s
as a cutoff in an attempt to ensure multiple events per typical TBP + TATA trace. Of the
230 remaining tethers, we cut 31 more due to poor persistence or contour length fit values.
We cut 4 tethers due to qualitatively strange force vs. extension curves, one because of an
unstable attachment to a fiducial, and 8 because they were acquired after TBP had beenpresent in the sample without being refreshed for >30 min. We were left with 186 remaining
tethers, 62% of the original 300.
Before performing further analysis on the extension vs. time data, the x, y, and z
vs. time traces of both beads were visually inspected at both full bandwidth and 5 Hzsmoothing and decimating. Regions of the traces representing different applied force wereseparated, resulting in 285 total traces. Further cuts were made here, at the trace level.
Specifically regions of traces were cut if the tether detached (the axial bead displacement
abruptly moved to ∼0). Regions were also cut if the 0.2 Hz smoothed bead displacement
moved > ±10 nm from the mean displacement. Regions were cut with spurious x or y tether
and/or x, y, or z fiducial motion. Segments of traces adjacent to the removed regions wereconsidered independent traces. If <100 s of data remained in a trace, the trace was discarded
from the remaining analysis. These cuts resulted in discarding 34 full traces (12%) leaving
a remaining 251 traces for further analysis.
From the initial visual inspection, distinct TBP and TATA dependent behavior was
clear. Fig. 4.2 shows the typical observed behavior. Almost all traces in the absence of
TBP showed a steady extension with a 1-2 nm noise level and some slow drift. Almost allno TATA + TBP data traces were similarly ”quiet” with possibly a slight increase in eventswith abrupt extension changes. Almost all TATA + TBP traces showed multiple abruptchanges in DNA extension of ∼3 nm. We interpret the dips as states where TBP is bound toand bending the DNA. To further quantify the TBP and TATA dependence, we performeda more rigorous visual inspection, counting events on the different types of traces.
On the final data set, examining the data at 5Hz smoothed/decimated, we observed

145

A

B

C

Figure 4.2: TBP and TATA dependent extension changes in DNA. DNA extension vs. time
traces demonstrate clear TBP and TATA dependent extension changes. (A) DNA extension
measured with no TBP present demonstrates typical noise levels for our experiment. (B) A
trace on DNA containing no TATA box shows no step-like extension changes. (C) A trace
on on DNA containing a consensus TATA box shows several step-like changes.

146

A

B

C

Figure 4.3: Common events in DNA extension vs. time traces. (A) Dips in extension fromthe state with the majority dwell time were the most common event on TBP+TATA traces.
(B) Downward spikes in extension were the 2nd most common event. (C) Upward hops inextension from the state with the majority dwell time were more rare, but common enoughto classify as an event.

147
three distinct classes of events: dips, spikes and hops. Fig. 4.3. We defined dips as >3 data
points 2-5 nm below the extension with the majority dwell time. We defined spikes as 1-3
data points at >5 nm from main level of trace, and we defined hops as >3 data points 2+
nm above the extension with the majority dwell time. We then quantified numbers of eachof these events by visual inspection for the data at 0.5 pN. For traces in the absence of TBP,
on 51 traces we observed an average of 0.3 dips per trace, 0 spikes, and 0.04 hops/trace. The
average trace length was ∼200 s for each of these data sets. For no TATA + TBP traces,on 36 traces we observed an average of 0.3 dips/trace, 1.14 spikes/trace, and 0.2 hops/trace.
For the TATA + TBP traces, on 88 traces we observed 3.8 dips/trace, 1.8 spikes/trace, and
0.2 hops/trace. While a more quantitative analysis than visual inspection could provide
more precise results, the dramatic difference in the number of dips/trace firmly establishes
the TBP and TATA dependence of these step-like DNA extension changes.
We also further analyzed the spike class of events in an attempt to elucidate their
origin. Note that we observed 0 spikes in the no TBP data, 1.14/trace in the no TATA +TBP data, and 1.8/trace in the TATA + TBP data. The similar frequency of spikes in theno TATA + TBP and TATA + TBP traces suggests the spikes are TBP, but not TATA
box, dependent. In addition, we quantified the presence of spikes for the TBP + TATA
data at 0.3 and 0.8 pN. At 0.3 pN, we saw a dramatic increase in the frequency of spikes
to an average of 5 spikes/trace. In contrast, at 0.8 pN, we saw only 0.49 spikes/trace. The
spikes display a dramatic force dependence, qualitatively consistent with force increasing
the energetic cost for larger DNA bends and decreasing the probability for longer range
interactions under increased load. Finally, we quantified by visual inspection the incidenceof spikes with origins stemming from dips vs. the upper level of traces that typically had
the majority dwell time. In the No TATA data, spikes did not stem from dips. In the TATA
+ TBP data, across all forces, > 83% of spikes originated from dips, with a small force
dependence (0.3 pN: 83%, 0.5 pN: 90%, 0.8 pN: 94%).
The origin of these spikes is currently unclear. They appear TBP dependent but not148entirely TATA dependent. They seem to vary in size of extension change, though determining
the size of such fast events is difficult due to Brownian motion noise limits. The duration of
the extension change associated with the spikes varies; examining the 4 kHz data leads toduration estimates of ∼0.1-0.5 s. The spikes occur from dips far more frequently than not.
Our interpretation that dips represent states where TBP is bound then suggests spikes may
be an interaction of bound TBP with something else, another part of the DNA, another TBP
bound to another part of the DNA, or possibly a brief interaction with the bead. The spikes
are not consistent with interactions with the surface because no x or y motion accompanies
them. One attractive hypothesis credits spikes to interactions with another TBP boundtransiently to the DNA. Though untested, this hypothesis would have interesting implicationsin that the other bound TBP would have to be bound in an unbent, and probably highly
unstable state.
4.4

EXTENSION CHANGES VS FORCE
For more quantitative analysis of extension changes and rates, we performed hidden

Markov modeling analysis of our data traces. After initially applying the analysis to the 5
Hz smoothed data, evaluation of the extension change to standard deviation ratio (∆Ext/σ)
showed the average ∆Ext/σ barely passed the threshold of 2 established in robustness measurements of the Hammy algorithm. For the low force data, ∆Ext/σ was <2 for over half
the individual molecules. This evaluation led to re-analysis of the data at 1.67 Hz smoothing.
Furthermore, we rejected traces that still had ∆Ext/σ ratios <2 at 1.67 Hz.
We fit the data to the simplest model, assuming 2 states, bent and unbent. We processed traces using the Hammy 4.0 GUI. We rejected data after processing if no transitions
were present, if it was unclear which 2 states shared the majority dwell time, or if the Hammy
algorithm could not fit the 2 states with the majority dwell times. We tested the influenceof fitting to a 2-state model in the presence of the spikes. When fitting traces with many
spikes, the difference in the measured extension change was <10% between fits including a

149
3rd state for spikes vs. fits with 2 states. The data and the HMM analysis show consistently
sized extension changes with variable dwell times in the upper and lower extension states,
see Fig. 4.4 for a trace and HMM fit at 0.5 pN. We interpret the lower state to represent
bent DNA, and the upper state to represent unbent DNA. The 0.5 pN trace also shows longand short dwell times, with an expanded view to show excellent correspondence between
short dwell times in the fit and the data. Traces and fits for 0.3 pN and 0.8 traces are shownin Fig. 4.5, showing similar extension changes and varying dwell times to the 0.5 pN data.
Note also the spikes so frequently present in the 0.3 pN data.
The similarity of extension changes for the three forces is shown in Fig. 4.6. The histograms show distributions all peaked near 3 nm. Statistical analysis yields mean extension
changes for the three forces all within statistical error of one another (2.8 ± 0.2 nm for 0.3
pN, 3.1 ± 0.1 nm for 0.5 pN, and 2.9 ± 0.2 for 0.8 pN). This lack of force dependence did not
follow the trend expected from theoretical predictions of extension changes for kinked DNA
under varying load, shown in Fig. 4.7 [9], though the magnitude of the extension changes issimilar to the theoretical prediction.
4.5

KINETICS
HMM analysis yielded two separate measures for the transition rates between the twostates. In one measure, the transition probabilities computed during the fit to each trace
are converted into transition rates (rate = probability*data acquisition rate). Histograms ofthese transition rates are shown in Fig. 4.8. The statistics are low, but on inspection, thehistograms do not appear to represent a normal distribution, especially the rates to unbend.
In the Hammy analysis paper on which our HMM analysis is based, the transition rate histograms also appear to be non-normal distributions [34]. The authors convert to the naturallog of the rates to yield normal distributions, then measure the mean and exponentiate it toobtain an expectation value of the rates. Conversion to natural logarithms of the transition
rates shown here does not result in a normal distribution. Despite the skewed distributions,

150
A

B

Figure 4.4: Extension changes show long and short dwell times. We analyzed the step-like
extension changes in our data traces with a hidden Markov model (HMM). The fit is shownin black. (A) Long and short dwell times are evident in the data and well fit by the model.
(B) A closer look at short dwell times and the HMM fit.

A

B

Figure 4.5: Exploring force dependence. Extension vs. time traces at 0.3 pN (A) and 0.8
pN (B) show TBP-dependent, step-like changes of similar size to extension changes for 0.5
pN traces.

A

151

B

C

Figure 4.6: Histograms of extension changes for three forces. The extension changes from
HMM analysis are histogrammed for three applied forces: 0.3 (A), 0.5 (B) and 0.8 (C) pN.
The mean ± std. error for the three forces respectively were 2.8 ± 0.2 nm, 3.1 ± 0.1 nm,
and 2.9 ± 0.2 nm, all within statistical error of each other.

we report mean rates ± standard errors in Table 4.1.
As the 2nd method of rate analysis yielded by HMM, we accumulated the dwell times ineach state into dwell time histograms. A simple reaction should yield dwell time histograms
well-fit by single exponentials, with the decay rate giving the rate of the process. For ourmodel, we would expect fits to dwell times in the unbent state to yield bending rates, and fits
to dwell times in the bent state to yield unbending rates. Fits to the dwell time histogramsfor our data were dependent on histogram binning and details of the fit. We optimized thebinning and fitting using simulations (see next section) and determined 6 s time bins andChange in extension (nm)

A

B

152
20

α=80
90

o

o

100 o

15
10
5
0 −2
10

−1

10
10
Force F (pN)

0

10

1

Figure 4.7: Force vs. extension change, theory and experiment. (A) A theoretical estimation of the expected extension change for TBP bending short DNA shows a strong force
dependence of the extension change [9]. (Figure credit: figure by Prof. Meredith Betterton.)(B) The measured extension changes are similar in magnitude to the theoretical predictions,but do not follow the trend for different forces. Extension changes for each force (2.8 ± 0.2
nm for 0.3 pN, 3.1 ± 0.1 nm, for 0.5 pN and 2.9 ± 0.2 nm for 0.8 pN) are shown with thestandard error.

linear fits to ln(N ) provided the most accurate measure of rates for simulations based on thedata values and a 2 state model with single exponentially distributed steps. Fig. 4.9 showsln(N ) and the linear fits for the bent and unbent dwell times, for each of the applied forcesin our experiment. The obtained rates are listed in Table 4.2. The unbent state (bendingrate) seems like a reasonable fit by this single exponential model. The bent state (unbendingrate) is fit poorly by a single exponential, which also correlates with poor agreement between

153
A

B

C

D

E

F

Figure 4.8: Transition rates calculated from transition probabilities. Transition probabilities for bending and unbending were calculated for each trace via HMM analysis. These
probabilities were converted to transition rates (rate = probability * data acquisition rate),and histogrammed to show the distributions. Rates to bend are shown on the left; Rates tounbend on the right. Histograms of the rates are shown for three forces, 0.3 pN (A and B),
0.5 pN (C and D) and 0.8 pN (E and F).

Table 4.1: Transition rates from HMM transition probabilities.
Force
0.3 pN
0.5 pN0.8 pNrate to bend (s−1 )0.018 ± 0.002
0.029 ± 0.003
0.025 ± 0.003

rate to unbend (s−1 )0.048 ± 0.009
0.12 ± 0.02
0.09 ± 0.01

154
the unbending rates determined by transition rates vs. dwell time fits.
Table 4.2: Transition rates from dwell time fits.
Force
0.3 pN
0.5 pN0.8 pN

4.6

rate to bend (s−1 )0.016 ± 0.003
0.03 ± 0.003
0.025 ± 0.003

rate to unbend (s−1 )0.018 ± 0.005
0.034 ± 0.006
0.016 ± 0.006

SIMULATIONS TO VALIDATE ANALYSIS
Before interpreting the values of these rates, we performed simulations to test thevalidity of our analysis methods on a simulation set with similar numbers and lengths oftraces, similar extension changes, similar noise, and similar rates. We performed simulationsof the sets of traces for each force, with the smallest set being the low force data with 17
traces. We then evaluated the percent difference between the input and output values forthe extension change and the rates. The simulation for the smallest data set is shown in Fig.
4.10. The input, output and percent difference values are listed in Table 4.3. Though thepercent difference values in the table show a maximum of 20%, repeated simulations actually
showed percent differences in rates up to ±50% measured relative to the input rates. Thislarge variation could actually be a factor of 2 times the measured rate, if the measured rate
is 50% smaller than the input rate. The simulations suggest that even for an ideal systemwith the exact 2 state model we posit, with single exponential bending and unbending rates,
the measured rates may only be estimates within 2x for our data sets.
4.7

DRIFT CORRECTING HIDDEN MARKOV ANALYSIS
Inspection of our long data traces reveals residual drift, despite active stabilization of

our instrument. To quantify the impact of that drift on our TBP traces and potentially
improve our measurements, a drift-correcting HMM algorithm was developed that incorporates Fourier modes to account for this drift (unpublished work by George Emanuel, Tatiana

155

A

B

C

D

E

F

Figure 4.9: Dwell time histograms for three forces. Histograms of dwell times, plotted asln(N), and linear fits to determine the bending rate and unbending rate at each force. The
single-exponential rates are equivalent to the slopes, k, of the linear fits to ln(N ) vs. time.
The bending rate is the slope of the fit to the dwell times in the unbent state (left). The
unbending rate is the slope of the fit to the dwell times in the bent state (right). Plots andfits are shown for three forces, 0.3 pN (A and B), 0.5 pN (C and D), and 0.8 pN (E and F).
The unbending rates are poorly fit by the single-exponential model, indicating that a more
complex model is needed to explain the data.

156

A

B

C

Figure 4.10: Simulations to verify analysis methods. Data analysis was performed with ahidden Markov modeling program. To verify this analysis, we simulated traces at a noiselevel (σ = 1 nm at 1.67 Hz) and trace length (∼300 s)similar to our data set. We added
exponentially distributed steps equal to the measured extension changes (∼3 nm) with rates
determined from the transition probabilities in our data analysis (bending rate = 0.02/s,
unbending rate = 0.05/s). A simulation trace (A) and dwell time histograms of time spentin the bent (B) and unbent (C) states are shown. With the number of traces equivalent toour smallest data set (N = 17 at 0.3 pN) and single exponential fits to the dwell times, werecover our input values to within 20% (see table).

157
Table 4.3: TBP trace simulations.
∆Ext
Rates from transition probabilities
rate to bend
rate to unbend
Rates from dwell time analysis
rate to bend
rate to unbend

Input
3

Output3.04% difference
1.3

0.02
0.05

0.024
0.052

20
4.9

0.02
0.05

0.023
0.044

15
-12

Kuriabova, and Meredith Betterton). As a first application of this method, analysis was
performed, again with the simplest 2 state model, allowing for 1 Fourier mode to accountfor drift. A trace with obvious drift and the improved fit by the drift correcting algorithm
is shown in Fig. 4.11. The entire data set was process with this algorithm, and extension
changes and rates were quantified. Comparing those values and calculating percent differences demonstrates a < 4% difference in the measured extension change and differences inrates varying from -44% to + 56%. Though this rate variation is substantial, it is well within
the 2x uncertainty suggested by simulations. Nonetheless, this drift-correcting HMM algorithm shows excellent potential for improving the precision of measurements on long traces
with drift.
4.8

CONCLUSIONSTBP bound to TATA box DNA recruits the transcription machinery to initiate site-

specific transcription and is not ordinarily thought to be a transient complex. Nevertheless,our results indicate step-like dynamic bending of DNA by TBP on timescales of tens ofseconds. These dynamics imply that TBP-TATA complexes must be caught and stabilized
in a bent state during recruitment of the transcription machinery, in contrast to the textbook model of TBP stably bending TATA box DNA for hundreds of seconds and forminga virtually static site-specific scaffold for the transcription machinery. We anticipate that

158

Figure 4.11: Drift correcting Hidden Markov Modelling analysis. Traces were analyzed witha more sophisticated HMM algorithm that accounted for steps and allowed Fourier modes
to account for drift.

the developed biophysical assay will allow not only further characterizations of TBP-DNA
interactions, but also direct measurements of the architectural changes that occur duringassembly of the human transcription preinitiation complex.

Chapter 5
CONCLUSIONS AND FUTURE DIRECTIONS

5.1

CONCLUSIONSOne of the major results in this thesis is the development of a biophysical assay capableof performing stable, precision optical trapping measurements on DNA-binding proteinsinteracting with very short DNA molecules. The key developments for this assay included an
actively stabilized optical trapping instrument (based on previous work [10, 11, 12]), a novel
axial detection technique for measuring changes in DNA extension and determining DNA
extension and force, and custom PEG surfaces to prevent non-specific adhesion of proteinsto the sample surfaces. These innovations allowed measurements to be performed on short
DNA molecules (<100 nm), which not only expands the range of biological phenomenon thatcan potentially be studied by optical trapping, but also, due to the increase in DNA stiffness
for shorter molecules, improves the spatio-temporal resolution of these measurements.
The techniques described in this thesis allowed direct, single-molecule measurementsof human TBP bending DNA. These measurements found a step-like TBP-induced DNA
extension change of ∼3 nm, constant over a force range of 0.3-0.8 pN. The magnitude of theextension change was similar to the theoretical predictions at these forces [9] for the 100◦
bend expected for TBP, but smaller by a factor of ∼2-4x, and without the predicted force
dependence. Over this force range the measurements showed TBP bending and unbending
DNA on a timescale of tens of seconds. The measured unbending rates were much fasterthan dissociation rates in the literature measured at lower KCl concentrations, but similar

160
to a more recently published measurement of TBP dissociation rates under a similar, more
physiological KCl concentration. The faster dissociation rates observed at more physiologicalmonovalent salt concentrations could have interesting implications for models describing therole of TBP in transcription initiation and regulation.
5.2

FUTURE DIRECTIONS

5.2.1

Technique and instrumentation improvements

The axial measurement techniques described in this thesis were adept at measuring
axial trap stiffness and DNA extension changes with low uncertainty (<20%), as well as estimating an average applied force and DNA extension. However, these techniques fell shortwhen attempts were made at obtaining a simple force vs. extension curve on DNA and atdynamically measuring force and absolute bead displacement from trap equilbrium on individual molecules. This failure was mostly credited to an inability to determine the positionof the axial trap equilibrium with accuracy. The biggest impediment to determining thetrap equilibrium position was the voltage oscillations that occur in the detection signal withchanges in depth from the sample surface. A technique for removing these voltage oscillations from the detection signal has shown promise (personal communication from Allison
Churnside), and this oscillation removal technique could dramatically decrease the uncertainty in measurements of the axial trap equilibrium position. With this improvement, theaxial measurement technique could potentially measure not only DNA extension changes andaverage force and extension, but also dynamically measure the force, DNA extension, andabsolute bead displacement from trap equilibrium for individual molecules with a precisionsimilar to lateral measurement techniques.

161
5.2.2

Exploring TBP-induced spikes in DNA extension

During the measurements of TBP bending DNA, larger, faster extension decreases
also occurred, appearing in the time-averaged data as downward spikes in extension. The
spikes were a TBP-dependent phenomenon but not completely dependent on the presenceof a TATA box. The origin of these spikes is currently unclear, but the observation that amajority of spikes originated from the bent state (∼90%) strongly suggests they originate
from a bound TBP interacting with another region of the DNA or possibly with the bead.
Further analysis of the existing data could examine the size and duration of these spikes,
determining a range of interaction zones, and/or determining the likelihood that spikes
originate from an interaction with the bead. If the TBP-induced spikes in extension prove
an interesting phenomenon (i.e. not likely resulting from an interaction with the bead),future experiments could explore the origin of this interaction.
Part of the interest in this spike phenomenon is based on the hypothesis that the spikes
may originate from an interaction with the N-terminal tail of TBP. The N-terminal tail of
TBP is made up of a varying number of glutamine repeats, typically between 29 and 42[2]. Polyglutamine tracts, like those in the tail of TBP, are associated with protein aggregation [51], which fuels the hypothesis that the TBP tail may be the source of the interaction
causing these TBP-dependent spikes in DNA extension. Further fueling this hypothesis is
the extreme TBP concentration dependence observed during data acquired while optimizing
conditions for our experiments. When the TBP concentration was increased by as little as
10%, the observed behavior changed dramatically. Rather than observing step-like extension
changes, we observed highly variable extension changes, with increasing stability over time,
resulting in apparently compacted DNA after only a few minutes. Such dramatic concentration dependence is often associated with cooperative processes, which supports a hypothesisthat larger extension changes may be caused by multiple TBP molecules binding to the
DNA and interacting with one another. Driving home the relevance of studying interactions

162
caused by the tail of TBP, expansion of the number of glutamine repeats in the human TBP
tail to >42 causes nuclear aggregates of TBP in cells and the neurodegenerative diseasespinocerebellar ataxia type 17 (SCA17) [52]. The possible relevance of these interactions
to a disease-causing mechanism solidifies the potential importance of understanding theseinteractions.
To elucidate the origin of the spikes, two experiments may prove useful. If the interactions are between TBP and other regions of the DNA, introducing non-specific competitor
GC DNA could decrease the frequency of the spikes. Addressing the question from anotherangle, if the interactions are caused by interactions with the TBP tail, performing experiments with truncated TBP containing only the C-terminal DNA-binding region should eliminate the spikes. Additionally, if the extremely concentration dependent DNA compaction
originates from interactions with the N-terminal tail, the truncated TBP would not cause
this DNA compaction. If proven true, an added benefit would be the ability to measure
TBP-DNA interactions at an increased TBP concentration, which should also accelerate thebinding rate, allowing faster and concentration-dependent measurements to be made of TBP
bending DNA. These types of measurements are currently impeded by the compaction that
occurs when the concentration of TBP is increased.

5.2.3

Angle vs. stability changes for bending of non-consensus TATA boxes

TBP preferentially binds TATA-box sequences of DNA, but to function in promoters
without a TATA box, TBP must also bind non-TATA DNA sequences. This requirement
begs the question of what differences may exist between TBP binding TATA box vs. nonTATA box DNA, and how these differences may influence the regulation of genes with TATA
promoters vs. genes without TATA promoters. FRET studies have begun to answer thesequestions [53, 54, 55], finding a significant decrease in apparent bend angle for non-TATA box
DNA sequences. These studies also found a broadening of the distribution of bend angles for
non-TATA sequences, indicating a more heterogenous population of bending angles. Thus

163
the question remains unanswered whether the apparent bend angle decrease was causedby an actual decrease in bend angle for the individual molecules, vs. a decrease in thestability of the bending, which would be time-averaged over the population of molecules inthese measurements. Our single-molecule optical trapping experiments, performed on DNA
with non-consensus TATA box sequences, could potentially differentiate between these two
possibilities, either showing a decreased bend angle for individual molecules, a decrease instability of bending, or even a combination of the two.

5.2.4

Geometry and stability changes with additional PIC proteins

TBP bending of DNA is but one of the conformational changes of DNA thought tooccur during transcription initiation. FRET studies have shown bending angles are different
for TBP-DNA complexes alone vs. TBP-DNA complexes containing TFIIA [5]. Crosslinking
and electron microscopy studies have suggested wrapping of the DNA around members of
the preinitiation complex (PIC) [8, 56, 57]. In addition, TBP can bind DNA both alone
and as a subunit of the general transcription factor TFIID, and the structure of TFIIDDNA complexes remains an open question [58]. This structural question, once answered,
could greatly influence the interpretation of the role of bending and DNA architecture intranscription initiation and regulation.
The assay established in this thesis could potentially elucidate how the geometry andstability of protein-DNA complexes change as different transcription factors are introduced.
The optical trapping experiments described here studied dynamic bending of DNA by TBP
on timescales of tens of seconds, an ideal timescale for these extension vs. time measurements.
Adding additional transcription factors often increases the stability of the protein-DNA complexes, which could create complexes with dynamics too slow to study reasonably by simply
measuring extension vs. time. However, with an optical trap, the force can be increased,
which should increase the dissociation/unbending rates of the DNA-protein complexes. Preliminary studies we performed showed increasing the force dynamically to 5-20 pN may even

164
cause abrupt unwrapping of DNA-PIC complexes, and these complexes may rewrap as theforce on the DNA is relaxed.
These type of single-molecule optical trapping studies of various complexes of DNA
with PIC components could not only measure the changing geometry and stability of thesecomplexes, but potentially even observe the order these complexes may assemble on the DNA.
This order of assembly remains another open question in transcription, with contrastingmodels, one suggesting an ordered assembly of proteins on the DNA vs. another suggestingassembly of proteins in solution before binding to the DNA [2]. Elucidating the architectureof PIC complexes and the order of assembly of these complexes is an exciting possibility that
could have an impact on our understanding of transcription initiation and regulation of gene
expression.

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