AiiiDS bsmo? PUBLICATIONS!

'"ifl'W!*'^ NisrUnited States Department of Commerce

Technology Administration

National Institute of Standards and Technology

NIST Technical Note 1509

Electrical Properties and

Dielectric Relaxation of

DNA in Solution

James Baker-Jarvis

Chriss A. Jones

Bill Riddle

QC

100

.U5753

NO. 1509

1998

k. 4

N/ST Technical Note 1509

Electrical Properties and

Dielectric Relaxation of

DNA in Solution

James Baker-Jarvis

Chriss A. Jones

Bill Riddle

Radio-Frequency Technology Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

November 1998

'^ITES O^

U.S. DEPARTMENT OF COMMERCE, William M. Daley, Secretary

TECHNOLOGY ADMINISTRATION, Gary R. Bachula, Acting Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Raymond G. Kammer, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1509, 72 pages (Novennber 1998)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1998

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
Contents

1 Introduction 2

2 Amino Acids, Peptides, and Proteins: An Overview 7

3 Structure and Electrical Properties of DNA 10

3.1 Charges and Electrical Properties of the DNA Molecule 10

3.2 Dielectric Properties of Bound Water and Polyelectrolytes Around DNA 13

3.3 Response of DNA in Electric Driving Fields 14

3.4 Dynamics of Polarization Relaxation 17

3.5 The Electric Field near the DNA Molecule 19

4 Counterions and the Helix 19

4.1 Cations 19

4.2 Hehx-Coil Transition in DNA 23

5 Modeling the Potential around the DNA Molecule 26

5.1 Phenomenological Models of DNA Dynamics 26

5.2 Poisson-Boltzmann Equation 27

5.3 Poisson-Boltzmann Apphed to Modeling the DNA Helix 29

5.4 Numerical Models for DNA Dynamics 30

6 Phenomenological Models of Dipole Moment, Relaxation Time, and

Polarizability 31

6.1 Persistence Length and Radius of Gyration 31

6.2 Polarization 31

6.3 Simple Model of Dipole Moment 33

6.4 Dipole Moment, Permittivity, and Relaxation Time 35

6.5 Statistical-Mechanical Model of Polarization for Rod-Like DNA 36

7 Charge Transfer in DNA 39

8 Measurements of DNA 42

8.1 Overview of Past Measurements 42

8.2 Permittivity Measurements and Electrode Polarization 43

111
5.3 Measurement Methods 47

8.3.1 Capacitive Measurement Techniques 47

8.3.2 Four-Probe Technique 47

8.3.3 The Open- Circuited Holder for Liquid Measurement 48

8.4 Dielectric Measurements on DNA in Solution 49

9 Discussion and Conclusions 58

10 References 59

IV
Electrical Properties and Dielectric Relaxation

of DNA in Solution

James Baker-Jarvis * Chriss A. Jones * Bill Riddle

November 20, 1998

In this report we overview and summarize the electrical properties and dielectric relaxation of solvenated DNA. We review models for counterion dynamics, dipole

moment, polarizability, and relaxation in both double- and single-stranded DNA.

We also present dielectric measurements on double- and single-stranded DNA from

herring sperm in protamine solutions. The analysis is based on polymer dynamics

and statistical mechanics. The observed dipole moment of DNA is induced through

the distortion of the counterion atmosphere by the applied electric field. Doublestranded DNA does not possess a permanent dipole moment except in the case where ligands are attached. We find evidence that denaturation proceeds in stages.

The first stage is when counterions are expelled from the molecule which causes

dissociation through strong phosphate-phosphate repulsion between strands. This

increases the conductivity of the solution. We also analyze dispersion of DNA

solutions from low to microwave frequencies. We found evidence of low-frequency

dispersion which is consistent with counterion-sheath relaxation models. We also found a dispersion at megahertz frequencies which is consistent with site-bound

counterion models.

Key words: counterions: dielectric constant; dipole moment; DNA: liquids; loss factor; microwave measurements: permittivity measurement: protamine: relaxation:

single-stranded DNA.

"HadioFreqiu'iicy Technology Division. MS S13.01. National In-tiiute of Standards and Technology,

Boulder, CO 80303-3328 einailijjarvislibouider. nist.gov
1. Introduction

This report examines the dielectric relaxation and electrical properties of deoxyribonucleic acid (DNA) in alternating electric fields [1,2].

The hterature on dielectric relaxation of DNA is copious [1-16]. There have been

a number of reviews of the polyelectrolyte properties of DNA [10-14], but none on

electrical properties and dielectric relaxation of DNA. [3-5]

Nucleic acids are high-molecular mass polymers formed of pyrimidine and purine

bases, a sugar, and phosphoric-acid backbone as shown in figure 1.1. Nucleic acids are

built up of nucleotide units which are composed of sugar, base, and phosphate groups

in helical conformation. Nucleotides are linked by three phosphates groups which are designated a, P, and 7. The phosphate groups are linked through the pyrophosphate

bond. The individual nucleotides are joined together by groups of phosphates that

form the phosphodiester bond between the 3' and 5' carbon atoms of successive sugars

(see figure 1.2). These phosphate groups are acidic. Polynucleotides have a hydroxyl

group at one end and a phosphate group on the other end. Nucleosides are subunits

of nucleotides and contain a base and a sugar. The bond between the sugar and base

is called the glycosidic bond as indicated in figure 1.3. The base can rotate only in

stericaUy permissible orientations about the glycosidic bond.

The helix is formed from two strands. The bases in adjacent strands combine by

hydrogen bonding, an electrostatic interaction, with a pyrimidine on one side and purine on the other. In DNA the purine adenine (A) pairs with the pyrimidine thymine (T)

.

The purine guanine (G) pairs with the pyrimidine cytosine (C). These are shown in

figures 1.2 and 1.3. A hydrogen bond is formed between a covalently bonded donor

hydrogen atom that is positively charged and a negatively charged acceptor atom. The

A-T base pair associates by two hydrogen bonds, whereas C-G base pairs associate by

three hydrogen bonds. The base-pair sequence is the carrier of genetic information.

The genetic code is formed of a sequence of three base pairs which determines a type of

amino acid. For example, the sequence of TTT AAA AAG GCT determines an amino

phe lyso lys ala acid sequence of: phenylalanine-lysine-lysine-alanine.

The DNA molecule has a net negative charge due to the phosphate backbone. When

dissolved in a cation solution, some of the charge of the molecule is neutralized by

cations. It is generally thought that the double-stranded DNA molecule has little in-
Figure 1.1. The right-handed hehcal conformation of DNA.

>Vci^nlr\^

^y-t^*lrt^

Orv'oc>«lci Ic bond

M -kT

3" Inycirojcyl •rtci

Figure 1.2. Single strand of DNA and the phosphodiester and glycosidic bonds.
NH,

OH OH

Figure 1.3. Glycosidic bond denoted by X.

trinsic permanent dipole moment (see figure 1.4). This is because the two strands that

compose the helix are oriented so the dipole moment of one strand cancels the other.

However, when DNA is dissolved in a solvent, such as sahne solution, an induced dipole

moment forms due to reorganization of charge into a layer around the molecule called

the counterion sheath.

Watson and Crick concluded through X-ray diffraction studies that the structure

of DNA is in the form of a double-stranded helix. In addition to x-ray structure ex- periments on DNA, information has been gleaned through nuclear magnetic resonance (NMR) experiments. Types A and B DNA are in the form of right-handed helices.

Type Z DNA is in a left-handed conformation. There is a Type B to Z transition between conformations. A transition from Type A to Type B DNA occurs when DNA is dissolved in a solvent [6] . The Watson-Crick conception of DNA as a uniform helix is an approximation. In reality DNA exists in many conformations and may contain inhomogeneities such as attached proteins. In general, double-stranded DNA is not a rigid

rod, but rather a meandering worm-like chain. Once formed, even though the individual

bonds composing DNA are weak, the molecule as a whole is very stable. The hehcal
o

o

c

(a)

O - o

(b)

OH

(c)

H^O

<d)

Figure 1.4. Permanent dipole moments of molecules. In (a) the charge in the carbon

dioxide molecule is depicted, (b) depicts the cancelation of the dipole moments in

carbon dioxide, (c) and (d) describe the dipole moment of the water molecule.
Counter

ions

Figure 1.5. Counterions and DNA molecule.

form of the DNA molecule produces major and minor grooves in the outer surface of

the molecule. There are also bound-water molecules in the grooves. Many interactions

between proteins or protons with DNA occur in these grooves. Sections 2 and 3 present an overview of the charge structure and dipole properties of DNA.

The interaction of the counterions with the DNA molecule has been a subject of

intensive research over the years. Some of the counterions bind to the phosphate backbone with a weak covalent bond. Other counterions are more loosely bound and some may penetrate into the major and minor grooves [7,8]. It is assumed that ions become

bound near charges in the DNA molecule and a double layer forms. The ions attracted

to the charged DNA molecule forms a counterion sheath that shields some of the charge

of the DNA. This is shown in figure 1.5. The counterion sheath around a DNA molecule

is composed of cations such as Na or Mg, which are attracted to the backbone negative

phosphate charges as shown in figure 1.5. These charges are somewhat mobile and oscillate about phosphate charge centers in an applied electric field. A portion of these

counterions is condensed near the surfax^e of the molecule, whereas the vast majority

are diffusely bound. The condensed counterions bind with the phosphate backbone
charges in a weakly covalent bond. In this bond, the outer s-shell of the counterion

may hybridize with the s and p shells of the phosphate oxygen [9]. Sakamoto [10] also

studied dielectric relaxation of DNA in ethanol instead of saline solution. This study

lends further proof to the counterion-dipole theory of relaxation.

The current understanding of the counterion atmosphere around the helix and the

helix-coil transition is overviewed in Section 4. An overview of relaxation time, polarization, and a statistical-mechanical theory of

relaxation is presented in Section 6. Section 7 presents an overview of charge transfer.

Section 8 discusses dielectric measurements and summarizes our measurement results on both denatured and double-stranded DNA in protamine solutions over a frequency

range of 20 Hz to 200 MHz.

2. Amino Acids, Peptides, and Proteins: An Overview

In this section we will overview dielectric relaxation of the building blocks of proteins,

amino acids, and peptides (see figure 2.1). Amino acids contain carboxyl (COOH)

groups, amide (NH2) groups, and side groups (R), as shown in figure 2.1. The side

groups and the dipole moment of the amino and carboxyl groups determine most of

the dielectric properties of the acid. Some of the side groups are polar; others are nonpolar. When ionized, the amino and carboxyl groups have positive and negative

charges respectively. This charge separation forms a permanent dipole. a amino acids

have an amino group and carboxyl group on the same carbon denoted C^. a-amino

ax^ids have a dipole moment of 15 to 17 debyes (D)(l debye equals 3.33 x 10~^° coulombmeter). P amino acids have a CH2 group in between the amino and carboxyl groups

which produces a large charge separation and therefore a dipole moment on the order

of 20 D.

Peptides are formed from condensed amino acids. A peptide consists of a small

number of amino acids cormected by peptide bonds. Peptide bonds (see figure 2.2)

provide linkages to amino acids through the CO-NH bond by using a water molecule as a bridge (see figure 3.1). The peptide unit has a dipole moment as shown in figure 2.2 on the order of 3.7 D. Chains of amino acids are called polyamino acids or polypeptides.

These are terminated by an amide group on one end and a carboxyl group on the other.

Typical dipole moments for polypeptides are on the order of 1000 D.
HgN

R

I C

1 H

(a)

— COO

H3N

H

I c -

I H

(b)

COO

CH

H,N

3 ^CHg

CH

I C — COO

I H

(c)

Figure 2.1. Amino acids: (a) standard form for amino acid where R is a side chain, (b)

glycine, and (c) vahne.

Figure 2.2. Peptide bonds and dipole moment.
Ao

^

-^ %^=Q>

'=n«t «r^^

<a) (b)

Figure 2.3. Origins of hydrogen and van der Waals bonds. In (a) the attractive force

between dipoles is depicted. In (b) the attractive force between water molecules due to charge separation is shown.

Polyamino acids can be either in the helical or random-coil phase. In the helical state,

C=0 bonds are linked by hydrogen bonds (see figure 2.3) to NH groups. The hehx can

either be right-handed or left-handed; however, the right-handed helix is more stable.

Generally, polyamino acids have permanent dipole moments and dielectric relaxation

frequencies in the kilohertz region. The dipole moment of polyamino acids has been

found to satisfy [11] < m' >= iVoMp|^, (2.1)

where Mp is molecular mass of the protein, ks is Boltzmarm's constant, T is temperature, P is the number of residues, and A^o is a constant. The brackets <> on the left

side of eq (2.1) denote ensemble average.

The origin of relaxation in proteins has been debated over the years. We know that

proteins are composed of polyamino acids with permanent dipole moments, but they

also have free and loosely bound protons. These protons bind loosely to the carboxyl htuI amino groups. It has been hypothesized that much of obser\'ed relaxation behavior of

proteins is due to movement of these nearly free protons in the applied field [12]. Strong

protonic conductivity has also been observed in DNA. At present, the consensus is that

both permanent dipoles and the proton-induced polarization contributr to dielectric
relaxation of proteins.

There are three distinct dielectric relaxations in proteins [1]. The first is in the 10

kHz to 1 MHz region and is due to rotation of the protein side chains. The second

relaxation occurs from 100 MHz to 10 GHz and is thought to be due to bound water.

The third relaxation is around 10 GHz and is due to free water. Bound water relaxes

at a frequency lower than free water due to the binding to other atoms. The relaxation

frequency in gigahertz of bound water depends on binding energy AE as [13]

JB = 18exp(-AE/Kr). (2.2)

Nandi [14] developed a theory using correlation functions that adequately predicts

dispersion in aqueous protein solutions. This theory also predicts an increase in the

static permittivity as the concentration of protein increases.

The mean-squared dipole moment of proteins has been modeled as linear in molecular mass Mp and has the form [11]

Here Aer is the dielectric increment, h is an empirical parameter, and A^ is the number

of residues. In presenting dielectric measurements frequently the difference between the

real part of the permittivity at a given frequency minus Coo ^£r = ^r ~^oo^ is called the

dielectric increment, eoo is obtained from the Cole-Cole plot. The value of the dielectric

increment varies from molecule to molecule. There are various other definitions of

dielectric increment used in the fiterature such as specific dielectric increment which

normalized to the concentration. The dielectric increment depends on cation and cation

concentration. Typical dipole moments of proteins are on the order of 500 D. In the

a-helix conformation the amide groups are aligned parallel to the axis of the hefix. This

produces a repulsive force between successive dipoles. In long a helices, dipole-dipole

interactions between the amide groups can be appreciable.

3. Structure and Electrical Properties of DNA

3.1. Charges and Electrical Properties of the DNA Molecule

Dimensional information for the DNA molecule is given in table 3.1. The dominant

charges on the DNA molecule are from oxygen atoms in the phosphate groups located

10
on the outer surfaces of the hehx. The DNA molecule also has additional 45 partially

charged atoms per base-pair site. The charge on DNA has been studied by Renugopalakrishnan and Lakshminarayanan [15] and by Pearlman.

Table 3.1. Dimensions in DNA.

Location Typical lengths (nm)

Radius of Type A DNA 09

Base-pair vertical separation 0.34

Intrastrand phosphate charge separation 0.70

counterion sheath thickness 0.30

Table 3.2. Typical bond energies.

Bond Type Typical binding energy (J/mol)

Covalent -2.8 x 10^

Ionic -1.8 X 10^

Hydrogen -1.8 x 10"^ Van der Waals -2.8 x 10^

Bases are stacked vertically by van der Waals forces, which originate in the at- traction of induced dipole moments between atoms as shown in figure 2.3. Hydrogen

bonding between bases originates from the attractive force between dipole moments and

requires the dissociation of a water molecule (see figures 1.4 and 2.3). To illustrate the

relative strength of the hydrogen bond, typical strengths of bond types are given in table

3.2. The helix strands can be either separated by heating the DNA solute followed by

quenching or by decreasing the ionic strength of the solvent. The strands also unwind

by enzymic interactions with ribonucleic acid (RNA) for self replication. The reac- tion of strand separation is called melting denaturation. The modeling of the helix-coil

transition requires a detailed analysis of polyelectrolyte theory coupled with free-energy

considerations. If the strands separate, they generally will coil. Single-stranded DNA

in a rigid, oriented state has a permanent dipole moment due to the alignment of the

11
positively charged amide groups. Single-stranded DNA has a permanent dipole moment

of approximately 20 D per base and bases are separated by 0.34 nm. However, coiling

severely decreases the net permanent dipole moment [16]. Double-stranded DNA possesses a large induced dipole moment due to the counterion atmosphere, on the order of thousands of debye. This fact is gleaned from both

dielectric relaxation studies, birefringence and dichroism experiments [17], and other

light-scattering experiments [18]. The induced dipole moment /2 in an electric field E is defined in terms of the polarizabihty jl = a- E. The polarizability a has been calculated

from statistical-mechanical means in terms of the mean-squared dipole moment < y? >

as [19] "=

dE ^ k^T <^-''

where T is the temperature and ks is Boltzmann's constant. The right-hand side of eq.

(3.1) shows that polarizability is a result of fluctuations in the dipole moment.

Since the individual strands of double-stranded DNA are antiparallel and the molecule

is symmetrical, the transverse dipole moments should cancel. However, a number of re- searchers have meeisured a small permanent dipole moment for DNA [20]. In alternating

fields, the symmetry of the molecule may be deformed slightly to produce a small permanent dipole moment [1]. Another origin of the apparent, small permanent dipole

moment is attached charged ligands such as proteins or multivalent cations [21]. These

ligands produce a net dipole moment on the DNA molecule by breaking the symmetry.

The question of how much of the relaxation of the DNA molecule is due to induced

dipole moment versus permanent moment has been studied by Hogan [17]. The response of permanent versus induced dipole moment differs in terms of field strength. The potential energy of a permanent dipole moment at an angle 6 to the

electric field \s U = —jiE cos 6, whereas the induced dipole moment in the electric field

is quadratic, U = —(Aa/2)£'^cos'^ 9, where Aa is the difference in polarizability along

anisotropy axes of the molecule. Experiments indicate that the majority of the moment

was induced rather than permanent.

Crothers have studied the effects of charge imbalance in DNA strands which they

have linked to DNA bending. This could happen for single- stranded DNA since [22,23]

DNA bends when attached proteins neutralize one side of the DNA molecule. Repulsions

between negative phosphate charges in the rest of the helix cause bending. This bending

12
is postulated to cause a permanent dipole moment since the phosphate charge balance

is changed [17,21,24].

3.2. Dielectric Properties of Bound Water and Polyelectrolytes Around DNA

The region close to the DNA molecule has a low dielectric constant and fixed charge. The

region far from the molecule has a dielectric constant close to that of water. Lamm [25]

studied the variation of dielectric constant in the grooves, near the surface, and far away

from the DNA molecule. Table 3.3 hsts predictions for dielectric constants in various

regions around the molecule. The permittivity depends on solvent concentration, dis- tance from the molecule, boundary eflFects, and dielectric field-saturation. The variation

of dielectric constant with position significantly alters the predictions for the electric

potential in the groove regions. Model predictions depend crucially on knowing the dielectric constant of water. Numerical modeling of the DNA molecule depends critically on the dielectric constant of water. When a dielectric constant of water that varies in

spax:e is used, numerical models indicate that small ions such as hydrogen can penetrate

into the minor and major grooves [8,26]. These predictions are not obtained for models

using spatially-independent dielectric constants for water.

The molecular structure of water is not simple. Besides the basic H2O triad structure

of the water molecule (see figure 1.4), there are also complicated hydrogen-bonded net- works created by dipole-dipole interactions which form hydroxyl (0H~) and hydronium

(HaO"*") ions. The dielectric constant of water is about 80, whereas biological water

contains ions which aff"ect both the real and imaginary parts of the permittivity. Water

bound in proteins and DNA has a decreased permittivity. This is due to constraints on

the movement of the molecules when they are attached to biomaterials.

DNA contains bound-water molecules. There are estimated to be 5 to 20 boundwater molecules per base pair [13]. Some of the bound water in the minor and major

grooves of DNA located on the charged nitrogenous bases and forms an aqueous bridge

or water backbone [13] (see figure 3.1). This water is tightly bound and rotational

motion is constrained in response to oscillating electric fields and therefore has a lower

dielectric constant. Water molecules are also bound to the negatively charged phosphate

groups on the exterior of the helix. In Type-B DNA the phosphate groups are far apart

and the water molecules are arranged independently around the ])]K)s])hatt's. hi FypeA DNA the water molecules form a continuous water bridge between tlu^ ])hosphate

13
Table 3.3. Real part of the permittivity around the DNA molecule as a function of

distance in nanometers from the center of the molecule and as a function of solvent

concentration in moles per liter (mol/1).

radius (nm) Location e (0 mol/1) e (0.5 mol/1)

0.07 major groove 20 20

0.07 minor groove 46 45

0.07 elsewhere 23 23

0.21 major gi-oove 31 27

0.21 minor groove 29 26

0.21 elsewhere 34 30

0.35 major groove 59 56

0.35 minor groove 41 38

0.35 elsewhere 55 52

1.3 away from molecule 77 73

sites. In Type-Z DNA, water molecules form bridges between amino groups in bases

and oxygens in the phosphate groups [13,27].

3.3. Response of DNA in Electric Driving Fields

A summary of the various dielectric relaxations in DNA is given in figure 3.2. The lowfrequency response of DNA is due primarily to longitudinal polarization of the diffuse

counterion sheath that surrounds the molecule depicted in figure 1.5. This occurs at frequencies in the range of 1 to 100 Hz. Another relaxation occurs in the megahertz

region due to movement of condensed counterions bound to individual phosphate groups.

There are many other types of motion of the DNA molecule in response to mechanical

or electrical driving fields. For example, propeller twist occurs when two adjacent bases

in a pair twist in opposite directions. Another motion is the breather mode in which

two bases oscillate in opposition as hydrogen bonds are compressed and expanded. The

Lippincott-Schroeder and Lennard-Jones potentials are commonly used for modeling

these motions. These modes resonate at wavelengths in the millimeter region; however,

relaxation damping prevents direct observation. Other static or dynamic motions of the

14
Figure 3.1. Water backbone in the minor groove of B-DNA. W denotes water molecule,

dashed lines used for hydrogen bonds in base pairing. The water molecules are around

the phosphate groups.

15
r

1 1 1 1 r I I I T T

1 \

\ ex nelcocatlon

X

S Rolaucation

x^

-3 -2 -1 1 234 6e7'8910

l_og fr^^^uoncy

Figure 3.2. Dielectric relaxations of DNA.

base pairs of the DNA molecule are roll^ twisty and slide. A number of researchers have studied dielectric relaxation of both denatured and helical conformation DNA molecules in electrolyte solutions both as a function of frequency

and applied field strength. Single-stranded DNA exhibited less dielectric relaxation than

double-stranded DNA [10,16,28]. Takashima concluded that denatured DNA tended to

coil and thereby decrease efi'ective length and therefore dipole moment. Further, a high

electric field strength affects DNA conductivity in two ways [28]. First, it promotes

increased dissociation of the molecule and thereby increases conductivity. Second, it promotes an orientation field effect where alignment of polyions increases conductivity.

Single-stranded DNA, in its stretched state, possesses a dipole moment oriented more

or less transverse to the axis. The phosphate group produces a permanent transverse

dipole moment of about 20 D per 0.34 nm base-pair section. Since the typical DNA

molecule contains thousands of base pairs, the net dipole moment can be significant.

However, as the molecule coils or the base pairs twist, the dipole moment decreases. If single strands of DNA were constrained to be rigid, relaxation would occur in the low

megahertz frequencies.

16
3.4. Dynamics of Polarization Relaxation

In order to study relaxation of DNA in solution we first consider the simplest model of a dipolar, rigid rod.

The torque on an electric moment p is N = pxE. (3.2)

For cases where the dipole moment is perpendicular to the rod axis, rotations about

the major axis can occur (see figure 3.3). The longitudinal rotation relaxation time

is [29]

^' = i^- (3-3'

where X is a constant, kB is Boltzmann's constant, and L is the length of the molecule.

The relaxation time varies as L. Major axis rotation could occur if the molecule had a

transverse dipole moment, for example, in a single strand of DNA.

When the dipole moment is parallel to the major axis, that is when there is excess charge on the dipole ends, end-over-end rotation will occur as shown in figure 3.3. The

end-over-end relaxation time for a rigid molecule of length L, viscosity 770, L/b ^ 1, 7 = 0.8, and radius b is [22,23]

' 6A;T[ln(L/6)-7]-

^'^'^^

This is the type of low- frequency relaxation that occurs with the induced dipole moment

in the counterion sheath or a permanent dipole moment parallel to the longitudinal axis

of the molecule. The relaxation time varies as L^. Since length of the molecule and

molecular mass are related, the responses for the two relaxations depend on molecular mass. Also, the model presented in this section assumed a rigid rod. In reahty DNA is not rigid, so a statistical theory of relaxation needs to be developed. We will address

these problems in a later section [29-31].

Takashima [16] and Sakamoto [10] ha\'e derived a more comprehensi\-e theor\' for counterion relaxation and found that the relaxation time \-arics in proportion to the

square of the length of the molecule |3i.32j. Most experimental evidence indicates a L"

dependence. This is in contrast to the rigid-rod model wluM'c the relaxation time varies as L^.

17
Axis of

Rotation

O-

+ + + + ^ +

+ I

Dipoie

Moment

+

+ I

I i

+ ' ^ I'M ,

,

- End over End

Axis of Rotation

^.^^^^^^..^^.^..^^^^^...^.

I +

I

I +1 '7^1 '

+ ...PJp.p!® Mp.iTi^n.t.

Figure 3.3. Torques on dipolar, rigid rod.

18
3.5. The Electric Field near the DNA Molecule

There have been a number of studies on the electric field surrounding the DNA molecule

[33,34]. Edwards et al. [33] studied a case where charge was considered to reside only on

the sugar-phosphate backbone and bases. In these studies the molecule was represented

by three concentric cylinders. The inner cylinder represented the molecule, the middle

layer represented counterions, and the outer layer represented the solvent. The charges

from the phosphate backbone were assumed to reside on the outer surface of the inner

layer. In these studies the major and minor grooves were not modeled. The skin depth

of the field, that is the distance over which the field attenuates to 1/e of its initial value, was on the order of 0.5 nm. Beyond this distance, the field approached that of a fine

charge.

4. Counterions and the Helix

4.1. Cations

The permittivity depends on concentration and type of cations [32]. As the concentration of the solvent increases, more of the phosphate charge is neutralized and the

dielectric increment decreases.

Many types of cations compounds have been used in DNA solvents, for example,

NaCl, LiCl, AgN02, CuCl2, MnCl2, MgCl2, arginines, protamine, dyes, lysine, histones,

and divalent metals such as Pb, Cd, Ni, Zn, and Hg. [10,35-37]. The simple inorganic

monovalent cations bind to the DNA molecule near the phosphate backbone to form

both a condensed and diS"use sheath. There is evidence that strong concentrations

of divalent metal cations destabilize the DNA hehx [38]. Sakamoto [36] found that

the dielectric increment decreased for divalent cations. On the other hand, histones

and protamines tightly bind in the major groove of the DNA molecule. They produce

stabifity in the double helix by neutrafizing some of the phosphate charge. Dyes can

attach to DNA, neutralize charge, and thereby decrease dielectric increment. The cation

concentration determines the dissociation of the DNA molecule. Mctague and Gibbs

showed that the Marming theory predicts that the counterion dipole moment should

increase as cation concentration increases [39,40]. In this respect the Manning theory

contradicts experiment. The theory developed by Oosawa [19] yields more realistic

predictions. In figures 4.1 through 4.3 plots of dielectric increment versus cation and

19
CO

-<1

1000 - 800 - eoo - 400 - 200 - 20 100

Figure 4.1. Effect of cations on relaxation time and dielectric increment [36].

DNA concentration are given [10,32,35,36].

Both protamine and histones are proteins. Nucleohistones and protamine-DNA

complexes occur in somatic-cell nuclei of mammals. Research on the dielectric properties

of nucleohistones and DNA has been performed by Jungner [41]. Protamine is bound

to DNA in the major groove by electrostatic interactions with phosphate charge [42]. Protamine has a total of 31 amino acid residues, with 67 percent arginine. Protamine

stabilizes DNA by neutralizing the phosphate charges in the DNA molecule and thereby

increasing the melting temperature [43,44]. The denaturing temperature of DNA in

protamine increases because of the tight binding of protamine in the major groove. The

use of protamine as a cation in DNA solutions allows a low-conducting fluid to stabihze

the molecule. This is a useful feature in applications where conductivity is important.

Expression of gene components during embryonic development is controlled by proteins, such as protamine, which interact with segments of DNA. Proteins can locate a

specific section of base pairs by recognition of specific bends and twists in the DNA

molecule. Proteins can locate short base sequences from the billions of bases in the

gene by interacting with DNA by attaching to DNA helices in the major and minor

20
^

1 c*

10=

lO^I1 o

^

N-X

1

-*-en

1

1=1

1=1

I NaCI

UIC2I TMACI

A.0NO3 • CuCIa

NilnCIa

1=1

-A- a

1 -

1 •

1 o 20 30

lOl / R

Figure 4.2. Dielectric increment versus mol/P where P is number of phosphate residues

[36].

10

10-2-

1 o

1 o

Figure 4.3. Dielectric increment and ndaxation time vs ((iTucnitration of D.\'A C,,.

21
grooves. Bound water is very important in the DNA-protein interaction since it is used

in hydrogen bonding.

An empirical fit of the melting temperature Tm of protamine-DNA solutions as a

function of salt concentration has been derived [37, 45]

Tm=^T^ + Cp\ogNa+. (4.1)

Here Na"*" is sodium molar concentration, Cp and T^ ^ 92 "C are constants. We

can compare this melting curve to that of DNA in saline solution. The hehx melting

temperature in saline solution in degrees Celsius is related to the concentration of salts by [3,46]

r^ = 100.3 +14.8 hi Cp, (4.2)

where Cp is concentration in moles per liter. This expression has been obtained by both

empirical means and derived from helix models. DNA in saline solution melts at a lower

temperature than in protamine solutions.

Bonincontro et al. [37, 45] investigated the efi"ects of protamine sulphate (clupeine)

and arginine on dielectric relaxation for herring sperm both as a function of temperature

(5°C to 40°C) and over a frequency range of 1 MHz to 10 GHz. They fitted the relaxation

data to a Cole-Cole model and then estimated the relaxation time r. Bonincontro [45] found that clupeine in safine solution showed relaxation behavior. Further, they mixed

clupeine with DNA, the relaxation properties of DNA decreased. This indicates that

clupeine decreases the induced polarization by neutralizing phosphate charges. They

also concluded that the binding of protamine to DNA reduces the number of water

molecules in the DNA complex [45]. Since protamine is a major-groove binder (see

figure 4.4) it tends to exclude bound water from the groove [45]. Bonincontro also

observed that relaxation was diminished when protamine was added to DNA solution.

They concluded this is due to the compaction of the DNA caused by the protamine and

consequential charge neutralization. The following model was used for the permittivity

' " '°° ^ (1 + (jcuri)!-) +

(1 + 3u:t^)

' ^^"''^

where Aei and /^.e^ indicate dielectric increments for DNA and water respectively,

Ti, Tyj are relaxation times, Coo is the optical limit permittivity, and a is a Cole-Cole

parameter. They obtained the activation enthalpy of the clupeine interaction from a

plot of

22
-^ = -exp{AS/R)exp{-AH/RT), (4.4)

Tl T

where AH is the enthalpy of activation energy, X is a constant, and AS is the molar

entropy of activation energy. Equation (4.4) can be used to estimate the activation

enthalpy since the slope of the plot of 1/ In (tT) versus. 1/T yields —AH/R.

4.2. Helix-Coil Transition in DNA

DNA can be either in the hehcal state or in the single-strand random-coil state. In the random-coil state, the hydrogen bonds that hold the helix together are broken.

The state that prevails depends on cation concentration and temperature of the solvent

[3,46-48]. The stability of the hehx is determined from intermolecular interactions

such as hydrogen bonds between base pairs, stacking of base pairs, interactions between

bound and free-water molecules, water in the hydrate regions of DNA, and hydrophobic

forces. The hehx stabihty depends on ion concentration since the counterions neutrahze

charge and thereby decrease repulsion between phosphate groups. Stability also depends on temperature since as kinetic energy increases, the probability that hydrogen bonds

will break increases according to an Arrhenius-type law. The presence of bound water

in the molecule also produces stabilization.

The hydrogen base-pair bonds are not the primary force that holds the DNA strands

together. It is thought that the helix structure is stabilized by water molecules [13]. Thermal denaturation destroys the ordered hydrate structure that holds the helix to- gether. Electrostatic interactions oppose base stacking, whereas nonpolar interactions

due to hydrophobic effects and van der Waals forces drive base stacking [49]. The helix-coil transition occurs at certain threshold values of pH, salt concentrar

tion, or temperature. Increasing the pH above 7 will tend to denature polynucleotides.

Denaturation readily occurs at pH's in excess of 12 or less than 2 due to base hydrogenbond ionization. The reason for this is that at higher pH the bases lose protons. A

double helix cannot be formed by deprotonated bases since the protons lost are \'ital

in the base-pairing reactions. The transition temperature depends strongly on sahne

concentration and pH [50]. Stronger concentrations yield a higher dissociation temperature. This is due to the screening of the phosphate charges by the saHne solution [46]

.

The helix-coil transition depends strongly on the free energy of the molecule together

23
Figure 4.4. Six arginine residues bound to DNA; only the phosphate groups of the DNA

molecule are shown.

24
vx > < ^x

Figure 4.5. Denaturing process modeled by Toda (T) and Lennard-Jones (LJ) potentials.

with solvent. The molecule in each of its states has different free energy. In thermal

denaturation counterions are released from the molecule as the strands separate. This

increases the conductivity of the solvent.

It is well-known that Type-B is more stable than Type-Z DNA. This is due to the fact that the phosphate charge in Type-B DNA is positioned so that it gets more counterion

screening than in Type-Z DNA [6]. A number of approaches have been used to model the denaturing of DNA. Yakushevich [51] presented a simplified model of DNA where the bonds were modeled by springs

using the sine-Gordon equation. Muto and others [52-55] modeled the melting process

by considering the hehx backbone to be composed of anharmonic Toda potentials between nucleotides and Lennard-.Iones potentials for the hydrogen crosslinks as shown m

figure 4.5. In this model the bonds were modeled by potentials and the coupled-force

problem was solved numerically. Thermalization was included in the model by introducing white noise. The analytical solutions to the resultaiit differential ecjuations are thermal solitons. This approach is ])riinitive in that the counterion atmosphere is not

adequately modeled.

Galindo [46] modeled the transition by considering the helix m two states. The first

25
state was uncoiled double-stranded DNA consisting of two parallel chains of charges; in

the other state the helix was coiled. The free energy for each of the states was calculated.

The minimum of free energy / was assumed to occur at equilibrium. It was useful to define the free energy difference between two conformations X and Y by

AfiX,Y) = J2[fiW~My)]- (4.5)

i The index i denotes the different free energies of the different conformations. The most

probable conformation was found by minimizing A/.

5. Modeling the Potential around the DNA Molecule

5.1. Phenomenological Models of DNA Dynamics

Two dissimilar materials can have different electronic affinities. When they are placed in

contact frequently a potential gradient between materials develops which allows current

to flow. As a result an electrical double layer forms at a material interface. This interface

could be between liquid and metal electrode or the layer between DNA and liquid.

Smaller electron affinity will acquire greater positive charge. The potential difference

will attrax;t ions of opposite charge to the surface and repel like charges. The potential

decreases roughly exponentially from the surface as ip{x) — ijjq exp {—x

/

\d) , where Xjj

is the Debye screening length or skin depth. The region near the electrode consists of

the Stern layer and a diffuse region which is beyond the Stern layer. A detailed analysis

of the potential requires the solution to the Poisson-Boltzmarm equation.

In the past, Manning's model, and various extensions of it, have been used to explain

the counterion interaction and dielectric relaxation of DNA [56,57]. More recently the

Poisson-Boltzmann equation has been used [58]. In the Manning model two distinct

dispersions are assumed to occur in dissolved DNA. One dispersion is at low frequencies

due to end-over-end relaxation of the sheath (see figure 3.3). The lower frequency

dispersion occurs around 1 to 100 Hz and is due to counterion fluctuations along the

sheath of the polymer. This relaxation depends on the polymer length or molecular

mass. A dispersion at higher frequencies due to relaxation of subunits of counterions

in the sheath has also been observed (see figure 3.2). This relaxation does not depend

on molecular mass. Mandel developed a model based on Manning's work where the

biopolymer was considered to be a long string of hinged, rigid rods [56,57]. Oosawa

26
m

Diffuse layer

Figure 5.1. Electrode with electrolyte

incorporated a more complicated model by solving the diffusion equation, as in Mandel's

model, but included surface diffusion of condensed counterions and two rotational modes

ofDNA [19,59].

5.2. Poisson-Boltzmann Equation

The Poisson-Boltzmaim equation is commonly used for modeling the counterion at- mosphere around a charged object [3,46,60,61]. In this section we review the timeindependent theory for counterions in an applied potential. Dukhin and Shilov [62]

present a very detailed analysis of the counterion atmosphere and dielectric relaxation

around electrodes.

We consider counterions near an electrode (see figure (5.1)). We can develop a

differential equation for the potential t/^(f^ if we substitute

E -v^. (5.1)

into

to form

V • D = p,

VV + ^ = 0,

e

(5.2)

(5.3)

27
where p is the charge density of the counterions, e is the permittivity of the solvent

and is assumed constant in space. The counterion charge density is written using the

Boltzmann distribution n^{r) — noexp{^Zcqip{r)/kBT), where r is the coordinate, Zc

is the valency per ion, q is the charge per ion, no is the ion density at infinity, kB is Boltzmarm's constant, and T is the temperature. The net charge density for a single

species of ion in one dimension is the sum of negatively and positively charged ions

p{z) = zcq{n+{z)-nr{z)) = -2noZcqsinh (^f^), (5-4)

.^.H|f. ,5.5)

The Poisson-Boltzmann eq. (5.3) in one- dimensional normalized rectangular coor- dinates can be written —^ = /c^ sinh (y)

(5.6)

where the normalized potential is

and k is the reciprocal of the Debye screening length.

The physical significance of the Debye screening length k is to produce damping.

As an example, a liquid with an ion density no = 6 x 10'^'* m""^ and e^g = 78 yields

/c ~ 2 X 10® m~^. This value of 1/k corresponds to a skin depth of 5 nm. The boundary conditions for eq. (5.6) are

,(. = 0) . ,„ = M*, (5.8) ksT

where iI)q is the potential on the boundary. At oo

y{z ^ cxd) ^ 0, (5.9)

The first integral is [63]

and the second is

^{z ^ cx)) -^ 0. (5.10)

dz

^ = -2ksmh{y/2), (5.11)

dz

y{z) = 4 tanh-i [tanh (yo/4)e^-^^)], (5.12)

28
or

Ayo/2) ^ I ^(gfao/2) _ i)g-fcz

y[z) - log

^^^^^2) ^ ;^ _ (g{yo/2) _ l)g-fcz (5.13)

where

2/0 = -r^TT- (•3-14)

The surface-charge density and differential capacitance are

a = 2v/2A;BTenosinh (|^) , (5.15)

and

Equations (5.15) and (5.16) do not yield physical results since they both increase

exponentially with ijjo. Corrections are commonly made to the Poisson-Boltzmann equation in the analysis by including an extra layer, commonly called the Stern layer, that

varies linearly with potential.

5.3. Poisson-Boltzmann Applied to Modeling the DNA Helix

It is possible to use the Poisson-Boltzmann equation to study the counterion atmosphere

around the DNA helix by assuming a charge distribution or potential on the helix

surface. A number of models have been developed to study the hehx and helix-coil

transition. For example, the model by Gahndos [46] assumes a counterion atmosphere

and a helix of infinite length with constant charge density distributed evenly on the

surface. In this study they assumed two cyUndrical layers. The first layer contained

the condensed counterions and in the second layer, the diffuse counterions. This is an approximation since most of the charge actually occurs in the phosphate groups in

the backbone. Each phosphate-sugar group was assumed to have one electron of extra

charge. The boundary conditions on the surface of the DNA at r = a are Gauss' law —d^ , . 2qi

(a) = —

,

i'o.lt)

dr a and the vanishing of the field at r ^ oo

V;(oo) = 0. (5.18)

29
qi — Cii/b, where b is the length of the cylinder over which the charge is distributed and

Cb is the Bjerum length. The solution in this case for the potential around the helix is

^[r) = ^^^In /^/ sinh^/3oln(^r), (5.19)

where e^ and ts are the dielectric constants of the bulk solvent and solute and A and

/?o are constants which are determined by the boundary conditions. It is important to

realize that the dielectric constant near the surface is different than it is far away from

the surface. The state of conformation is determined by minimizing the free energy as described in Section 4.2.

5.4. Numerical Models for DNA Dynamics

A number of extensive numerical models have been developed that solve equations for the electric field, counterion dynamics, and DNA charge sources [7,64,65]. Models such

as these are useful in predicting behavior of the DNA molecule. Saxena [66] has developed a comprehensive numerical model for dissolved DNA dynamics. In his model

equations of motion for DNA and the counterion sheath are solved numerically in the

frequency domain for the individual atoms and the counterion sheath. The model by

Saxena and Van Zandt contains a comprehensive model of site and sheath-bound counterions [9] . This study was limited to the microwave and millimeter-wave frequencies and

therefore low-frequency dispersion was not studied. Examples of equations of motion

are [9]

l|- = EK^^^ + ^^^« + ^^[^ ~ V^^]^ip5a^ , (5.20)

J-,/9 atoms

^ = -q\ls - AE. + Y.^ilqf - s/vi]5iP + 7f , (5.21)

sheath

where qf are the displacements of atom i and component a, s represents the displacement of the sheath, Dij is a force-constant matrix containing van der Waals, Coulombic,

and elastic forces, v is velocity, A and F are constants, and r} is viscosity. The last term

on the right of eq. (5.20) represents the polymer-solvent dissipation force. In these

30
equations only longitudinal motion of the sheath is considered. Specific forms for the

electric field in the various cylindrical regions are assumed.

6. Phenomenological Models of Dipole Moment, Relaxation Time, and

Polarizability

6.1. Persistence Length and Radius of Gyration

Double-stranded DNA is in the form of a meandering chain. Since the DNA molecule

is not entirely rigid, the induced dipole moment of the counterion sheath depends on

the persistence length rather than the actual length. The persistence length of a chain

of atoms is the average projection of the end-to-end distance vector on the first bond in

the chain. It is a measure of how the final direction of a chain deviates from the initial

link of the chain. The radius of gyration gives an eff'ective radius of the molecule. It

is a measure of the mean distance from the center of gravity. The radius of gyration

depends on the structural integrity of the molecule, that is, the degree of dissociation

of the molecule. Sakamoto derived the following formula for dipole moment of DNA as a function of the mean-squared radius of gyration < 5^ >, the number of phosphate

residues n and the degree of dissociation of the molecule as [32]. <//2> = naB{l - aB)q^ < S^ > . (6.1)

Sakamoto also studied protein (histone) attachment to DNA and derived an empirical formula for the dipole moment in terms of radius of gyration [36]

<fi^> =m(l--)g2<52>. (6.2)

n

Here m is the number of positively charged residues of proteins attax:hed to the DNA. In

eqs. (6.1) and (6.2), < S^ > was calculated using the Kratky-Porod equation in terms

of the persistence length Ip and contour length L for a worm-like chain [67]

/ r 21^ 2/'^ 2/"* <S2> =i^_,2 + ^_^ + ||[i_exp(-L/gi. (6.3)

6.2. Polarization

Consider a rodhke polyion surrounded by a counterion atmosphere [68]. Let S be the

instantaneous displacement between counterion and polyion charges. Then the net

31
dipole moment is // = ^iQiSi. The polarizability can be calculated from the mean

dipole moment

^ ^ ^ J ^exp{-f{^)/kBT+ j2- E/kBT)di.i

Jexp{-f{ij)/kBT + ^- E/kBT)d^i

'

where / is the free energy and E is the electric field. The polarizability is defined in

eq. (3.1). In the absence of a field, the average dipole moment < f.i. >= nq < S >=

where n is the number of charge sites. However, the fluctuations in the dipole moment

are nonzero and are the origin of polarizability.

There are a number of models for DNA polarizability. The model developed by

Mandel [56] overestimates the counterion induced polarizability since it neglects counterion repulsion. The model of Manning [69] predicts an increase in polarizability with

increasing solvent concentration, in contradiction to experiment [70]. Polarizability de- creases with increasing solvent strength [71]; probably this is due to phosphate charge

neutralization by the increased concentration of counterions. Polarization also decreases

with increasing temperature.

Oosawa developed a comprehensive time-dependent theory of polarization for bound

counterions using a diffusion equation [19,59]. This model predicts the relaxation fre- quency for the sheath. The induced dipole moment forms due to fluctuations in the

sheath density. In this theory, 5 = J2i ^i/^ is the displacement of the center of gravity

of N counterions. The coUective motion of the counterion atmosphere is given by the

simple diffusion equation

where Q is a frictional constant. Equation (6.5) has solution

5{t)=Aexp{-t/T), (6.6)

where the relaxation time is r = A^C < ^" > /kBT. In a more comprehensive theory,

Oosawa included a full-mode spectrum. This theory predicts that the relaxation time

is proportional to the square of the polyelectrolyte length and inversely proportional to temperature. These predictions are consistent with experimental results.

In another model of polarizabihty, Fixman assumed DNA to be an impenetrable

cylinder [72] and solved the Poisson-Boltzmann equation for the counterion distribution

in an external field. His model produced an expression for the induced dipole moment

32
by integrating the charge density over the length of the molecule. The Fixman model

underestimates the polarizability, but correctly predicts the solvent dependence of the

polarizability. The advantage of the Fixman model is its simplicity. The Fixman model

yields the following expression for polarizability

eLKf zi , 2tanh(t/L/2).

where L is molecule length, z/^ = 7TNACiKf/2^0a, Kj = (21n(2L/a) - 14/3)-\ e is permittivity, z\, 22 are charges on counterions and coions, A^^^ is Avogaxiro's number, c\

is the concentration of counterions, a ^ 1.2 nm is radius of the molecule, a = [l — l/Q/b',

C — q^/b'eksT, 6' = 6/(1 — p), and b is length of base pair. In a final example of polarization models. Hogan proposed a model where the dipole

moment is formed due to anisotropic counterion flow along the axis of the molecule [17].

6.3. Simple Model of Dipole Moment

The dipole moment of polymer molecules is usually modeled by a set of charges that

are connected by flexible links [32]. In this section, we review a model for the dipole

moment of a string of rigid rods [32]. The net charge (see figure 6.1) at positions fi is q5ni — q{ni— < n >), where g < n > is the mean value of charge distribution. The net

dipole moment is

/Z = ^

q{fi - fG)Sni, (6.8)

where the center of mass for N links is

^G = T7X1^^- (^-9)

The mean-squared moment is

< fi^ > =<q^Yl Yl^^i ~ ^G){rj - fG)5ni5nj > . (6.10)

i 3

This is approximately

< /x^ > = g^ 51 X](^^ ~ '^G){rj - ra) < SmSrij > . (6.11)

33
Figure 6.1. Polymer chain of dipoles of moment fXp.

34
If we apply this model to the double-stranded DNA hehx and assume there is a

probability as of an ion binding at a site, then the correlation function for the charge

distribution can be approximated by

We then have

where

< SriiSrij >= 5ijaB{l — Qb)- (6-12)

7<^2> ^ q^NaBil-as) < L2 >, (6.13)

<L'> =;^E<(r-;-rb)'>. (6.14)

z=l

This model correctly predicts the dependence of the dipole moment on the length of the

polymer, but not the temperature dependence. In the next section we will generahze

this model to get correct temperature dependence.

6.4. Dipole Moment, Permittivity, and Relaxation Time

Minakata developed a comprehensive, phenomenological model for discrete counterion

binding on rod- like polymers [73]. It includes three different types of counterion binding

sites. This model is important since it correctly predicts a number of the important

measurable quantities in the relaxation process. The mean-squared dipole moment is defined as

< n^ >= q^J^^i^J < (^^~ < ^ >){nj- < n >) >= q^ Yl^^^^^i+'^^^Jl^^iJ^^'^J^i+j')^

ij i ij (6.15)

where n is the number of bound ions at a specific site, A|n =< nf > — < rz >^, and

AimTi =< niTii^rn > — <n} > This model yields

Ae = ^^^|2^|An^ + 2EA.nl. (6.16)

Here B is the ratio of the internal to external fields, L is length of the molecule, and

N is number of polyions per unit volume. The important result of this model is that

it correctly predicts the dielectric increment to be proportional to L? and temperature

dependence to be of the form 1/T.

35
l:^

10®

10

10^ -

lOf

10«

-10-

10'

10^ 10^ 10^

l\/lolecular mass

Figure 6.2. Relaxation time and dipole moment versus molecular mass.

Another model developed by Sakamoto [32], relates dipole moment of the counterion

sheath to dielectric increment from counterion fluctuation theory and agrees well with

experiment [32] 47tN < /P >

Ac

3kBT

(6.17)

Sakamoto defined the dielectric relaxation time in terms of the maximum frequency

fm in the loss spectrum by tq = l/27rfm- For double-stranded calf thymus DNA, td was proportional to the square of the molecular mass of water Mw and the mean dipole

moment was proportional to Mw (see figure 6.2). He also found that for coiled DNA,

Td ~ 2r2, where tz = ^A2Myjr]sr}red/ RT is the Zimm viscoelastic relaxation time (see

figure 6.3). Here 77^ is solvent viscosity and iq^ed is the reduced viscosity.

6.5. Statistical-Mechanical Model of Polarization for Rod-Like DNA

In this section we consider a more rigorous model of coupled DNA and counterion dynamics [68]. We solve a generalized diffusion equation or Fokker-Planck equation for the counterion sheath dynamics in an applied electric field and we develop new mo- ment equations from the model. The model yields relaxation times and time-evolution

equations.

36
10-1

io-«

io-»

10-* TO-o 10-^ (S)

Figure 6.3. Dielectric relaxation time [td) versus Zimm relaxation time tzThe instantaneous dipole moment is yL/ = nq5^ where S is the displacement along

the major axis of the molecule from the center of charge distribution and q is electronic

charge. In addition there may be a permanent dipole moment i^ip. The polarizability can be expressed in terms of the fluctuations a =< jj? > /ksT = n^q^ < (5^ > /ksT.

The potential energy of the counterion sheath in the absence of an electric field is

V{5) = n^q^Sy2a. (6.18)

This potential-energy function is quadratic in displacement analogous to the harmonic

oscillator potential. Using this potential, the Fokker-Planck equation, which gives the

probability that a certain counterion distribution will be realized, can be written as

dp{S,t) oliM^^Pi'-m,isM. (6.19)

dt dS' dS "' 05

Here /3 = l/ksT and D is a diffusion constant. The steady-state solution is Gaussian

exp(-/3V0

If we perform the integrations, we obtain the Gaussian distribution

PeqiS)

2^2 'PiT^q ZTTa exp{-Pn^q'5y2a)

(6.20)

(6.21)

37
Since the solution is Gaussian the average dipole moment will be 0. The mean-squared

dipole moment, < //^ > = ksTa, however, is nonzero. The time-dependent solution of

eq. (6.19) is

/- N / 1 / (<^-<^"(0)exp(-t/r))\

"(*•'> =

V^ 2.rD(l -exp(-2t/r) -'-P<- 2Dr(l - exp (-2t/r))

>' <«-^^'

where r = a/DPrrq^. If we multiply eq. (6.19) by 5 and then integrate from [— oo, oo]

we obtain

—dT- - —7-' (^-^^^

with solution

<5>{t) = 5(0) exp (-t/r). (6.24)

In the hmit of large time, the mean dipole moment tends to 0. In the absence of an

electric field this behavior would be expected. The mean-squared dipole moment is <5^> it) = ^+{5^(0) - ^)exp(-t/r). (6.25)

We see that this approximation has a single relaxation time r. In the limit of large time

< 52 >^ akBT/n^q^.

In the case where there is an applied electric field on the counterions in an anisotropic

polyion atmosphere, the Fokker-Planck equation for rotational difiusion is

^^is^Ts + (^i^^^pi^^^M. (6.26)

where the potential includes efi"ects due to the counterions, permanent dipole, induced,

and intrinsic polarizability

V{5, t) = n^q^5'^/2a - nq5E{t) cos 6 - fipE{t) cos - —^—cos^ 9

counterions induced permanent ^ moment . • , • ,,.

intnnsic polarizability

(6.27)

Here the anisotropy in polarizabiUty is expressed as Aa = an — a±.

38
In the case when the polyion atmosphere is isotropic eq. (6.26) becomes

Moment equations can be developed from eq. (6.26) into a hierarchy

d<5 >

dt - <5> /t+ D(3nqE{t) < cos > , (6.29)

I ^ c2 ^ 9 << /s2 \ = 2D + 2(3DnqE{t) <5cos6> . (6.30) at T

The equations for the spatial and angular moments are \ = m{m - l)D <

6"^-^ > - + 2(5DnmqE{t) <5"^-^ cos9 > , dt T ^ ^ (6.31)

d<5'^cos'e> ^ ^^^ _^^^ sm^ecos'-^eS"^ > -2s < sin^cos^^^-^ >

dt

+QPnqE{t) <sin^cos^+^^5"^+i > +G(2sin^cos"+i ^ - ssin^ ^cos^-^ ^)

+m{m - 1)D < 5^ ^ cos^ 6 >

r +2pDnmqE{t) <

6'^-'^ cos^+^ 9 > . (6.32)

Equation (6.29) indicates that the time rate of change of the dipole moment is due

to a relaxation term plus a driving term that is hnear in electric field. This is similar to the Debye equation. Comparison to eq. (6.24) indicates that the effect of the electric

field is to add a driving term to eq. (6.29).

The model developed in this section yields an equation of motion for the counterion sheath (eq. (6.29)) and polarization. In this model the polarization has a single

relaxation time.

7. Charge Transfer in DNA

Szent-Gyorgyi concluded from experiments on proteins that biological materials have

semiconducting properties. In the ensuing years a great deal of research has been

39
performed on conduction in biopolymers. The base-pair sequences in DNA are aperiodic,

whereas the phosphate backbone is periodic. Periodicity promotes semiconductivity. In

addition, DNA contains counterions and bound water. This comphcated structure has

prevented the details of electronic conduction in DNA from being uncovered. Recently, a self-consistent field theory has been used for modeling biopolymers [74]

.

The semiconducting properties originate in internal charge-transfer mechanisms [2]

.

The semiconducting activation energies are on the order of 1 to 1.5 eV. The conduction is due to protonic and electronic activity. In the dry state, biopolymers exhibit conduction

both by electrons and protons. Risser et al. [75] and Stemp [76] have studied electron

transfer in single-stranded DNA and found it to be a good sensor of strand length with

rigidly attached donors and receptors. In DNA the periodic component of the base pairs produces band gaps. The aperiodic

component of the base pairs acts as impurities in the band gap formed by the phosphate

bax:kbone. Typical valence and conduction-band gaps are in the order of 0.3 to 0.8 eV

and energy band gaps of around 10 to 12 eV. Bloch wavefunctions can be used for modeling since the system is approximately periodic. In proteins, there are two main areas of charge transport. The charge transport is

in the main polypeptide chains and through the hydrogen-bonded crosshnks. Recent

research has concluded that conduction through the main polypeptide chains dominates

over the hydrogen-bond mechanism [74]. Beratan et al. [77] proposed a model of electron

tunneling in proteins with donor- acceptor interaction which is mediated by the covalent

bonds between amino acids.

Electron transfer in DNA takes place primarily along the axis of the molecule in

which there are four basic types. These include extra electrons, singlet and triplet tt- type molecular excited states, and holes. The electrons in 7r-orbitals are delocahzed

and do not participate in bonding. They are free to move about the carbon nuclei in a molecule. On the other hand, a"-orbitals are symmetrical about the bond axis with

localized C-C and C-H bonds. The electrons participating in these bonds are around the

line joining the two nuclei and are localized. The bases in DNA possess an electronic tt system, exhibit electron delocalization, and have low-lying 7r-type orbitals where excited

electrons may reside . [75,78-81] Since the bases are stacked close together, charge

transfer can occur from base to base. Hanlon [82] studied protonation of DNA and

found a higher density of protons in the minor groove than in the counterion solution

40
surrounding the molecule.

The electron-transfer rate between donor and acceptor has been measured with

photo-induced, excited state, and flash-quenching ground state techniques and with

transient- absorption methods. The electron transfer rate is

A; = iyexp(-(A - /\Em?/AksXT), (7.1)

where T is temperature, A is reorganization energy, and AZsm is potential difference

between sites. The predicted decay length of charge transfer is 13 nm and typical electron chargetransfer rate is 10^ s~^. This large decay with distance makes sense since long-range

transfer in DNA would not be advantageous from a molecular damage standpoint. For

example, many proteins attached to the DNA would not function normally with large

charge transfer taking place. Additionally, molecular damage becomes more probable

during times of high charge transport [79]

The eSiciency of DNA in allowing long-range electron transfer gives support to the

notion that stacked aromatic heterocycles can serve as 7r-ways. The results of research

in this area raise the possibility that DNA uses electron transfer in gene replication.

Electromagnetic fields may be able to stimulate biosynthesis in cells, particularly in

DNA. Electric fields surely interact with cell membranes, but recent evidence points to the possibifity that magnetic fields can play a role in gene activation [83]. The process

by which this could happen is through strong charge transport in DNA interacting with

applied magnetic fields. Recent experiments on charge transport have demonstrated

that current densities of at least 1x10"^ A/m^ can be achieved in DNA. The localization

of the gene segment through charge transport could be achieved either from bending or

rolfing of the base pairs or by enzyme interaction with base pairs in a segment. Electric

fields could also play a role in gene activation through bending of DNA or formation of

double layers affecting charge fiow in the molecule.

There is evidence that hydration of DNA promotes conductivity and semiconductivity [27,84]. These studies found that relaxation of double-stranded DNA in an applied

d.c. electric field of 1500 to 3000 V/cm had a 10 percent increased conductivity. Intrinsic

semi-conductivity in DNA increases exponentially with water content up to concentrations of 25 percent. Thereafter the effects are dominated by protonic activit}'. In tlie

studies of Bonincontro [27], mixtures of NaCl and DNA from herring sperm of sizes of

41
300 to 1000 nucleotide pairs were dissolved in water at a concentration of 2 mg/ml. The

samples were then dialyzed with solutions of 0.4 mol counterion solution (Na and Li)

then lyophilized (freeze drying) and ground to a powder. For many biomolecules the

conductivity follows a temperature dependence similar to inorganic semiconductors

(T^(jQe^^{-E/kBT). (7.2)

8. Measurements of DNA

8.1. Overview of Past Measurements

Measurements on conducting liquids are complicated by electrode polarization [85] and

by low-frequency dispersion (LFD) [86]. Electrode polarization is caused by the buildup of conducting ions on the capacitor plates producing an electrical double layer.

Electrode polarization influences primarily the real part of the permittivity. Since the

electrode capacitance is not a property of the material under test it must be removed

from the measurement.

In the literature three relaxations have been reported, 7, 5^ and a as indicated in

figure 3.2. The 7 relaxation occurs near 18 GHz and has been ascribed to water. The

5 relaxation is weak, occurs around 100 MHz, and is independent of molecular mass. This relaxation is due to the motion of condensed ions within a subunit of the DNA

molecule [37,87,88]. The a relaxation occurs for DNA in the range 1 to 100 Hz. The lowfrequency Oi relaxation depends on molecular mass and length and is due to fluctuations

in the counterion sheath.

Measurements on DNA in solution have been performed by many researchers over the years [32,36,40]. Good overviews of measurements of dielectric relaxation of DNA

are given in Sorriso [5], Takashima [1], and Grant [88]. The measurements can be

divided between those studies that measured DNA above 1 MHz and those studies that

measured at low frequencies (< 1000 Hz). Higher-frequency measurements were made

by a number of authors [40,89-92]. Low-frequency measurements have been made by

Takashima, Sakamoto, Hanss, and Tung [16,20,32,93]. In studies of the relaxation of

DNA in saline solutions, Takashima observed an a relaxation at around 100 Hz [1, 16],

for DNA molecular masses of 5 x 10^ D. Hanss and Sakamoto [10, 20, 32, 35, 36, 94]

performed experiments on the same length molecules and found relaxations in the 5 to 1000 Hz range. Takashima used a capacitor with platinum-blacked electrodes to

42
measure permittivity from 1 Hz to 1000 Hz. Sakamoto used a four-terminal device

and a conductivity standard to eliminate the effects of electrode polarization and d.c. conductivity [32]. He measured DNA of molecular mass 4 x 10^ D from 0.2 Hz to 30

kHz and found a relaxation near 5 Hz. The conductivity of their solution was from

2000 ^S to 13,000 iJ,S. Hanss and Bernengo [20] found a peak at 1-10 Hz. Tung [40] and Takashima [16] obtained a decrease in dielectric increment for calf-thymus DNA as concentration increased, whereas Sakamoto found the reverse trend [10]. Sakamoto [10]

found that concentration dependence of the DNA became important around 0.02 percent

and above. Sakamoto attributes this to intermolecular interaction. In figure 8.1, a ColeCole plot of dielectric increment is plotted from Sakamoto [10,32,35,36].

8.2. Permittivity Measurements and Electrode Polarization

The permittivity is can be written in the form

e* = eo[e',-j{e'; + -)]. (8.1)

UJ At low frequencies, a number of different dispersive processes occur. Direct current

conductivity where a <x l/u) produces a large dispersion in e" as cj —> 0. In this

phenomena the real part of the permittivity approaches a constant value, whereas the

imaginary part increases as l/u). Another phenomenon is low-frequency dispersion

where both the real and imaginary parts of the permittivity vary as u;"^"^ where n

is a positive number slightly greater than 0, for example n = 0.1. Maxwell-Wagner

circuit models have been used successfully to model the low-frequency behavior of highly

conductive materials. In highly conductive solutions the interaction of the electrodes with mobile charge can mask the real part of the permittivity through electrode polarization. The electrodepolarization efi'ect can be modeled by lumped circuit elements [16]

R = Rs + Rp + RsUj^R^C^, (8.3)

where C, R are the measured capacitance and resistance, Cp, Rp are the electrode

double-layer capacitance and resistance, and Cs, Rs are the sample capzicitance and

resistance.

43
O

mol/P = 0.2 ^

'—* '—

^\»

^ 1

100

eV(103)

10 15

mol/P = 0.58

-X^

r- • • -•>>

eV(103)

mol/ P = 1.5

^^^ -^

100 200 300

Figure 8.1. Dielectric increment of calf thymus for various molecular masses and phosphate residues in Cole-Cole format [36].

44
10000

1000

CO

100

"T 1 1 r-T TT T

_1 t I 1_L

"T r—r—[ T I I I

[

1 mm spacer

3 mm spacer

Accepted Value

1' I I I * ' ' '

_i I I t ' ' '

0.01 0.1 10 100 1000

Frequency (kHz)

Figure 8.2. Capacitor measurements of saline solutions of two different thicknesses.

A way to partially eliminate electrode polarization is by measuring capacitance and

resistance for two different electrode separations (see figure 8.2). It is then assumed that

Cp is the same for each measurement and Cs is then determined. The two-separation

technique works well only in cases where the electrode-polarization effect is small. Another way of minimizing the electrode-polarization effect is to coat the capacitor plates

with platinum black [95]. This lessens the influence of electrode polarization by decreasing the second term on the right side of eq. (8.2). The best approach is to bypass the

electrode-polarization problem and use a four-probe capacitor system.

45
current probe

double layer

sample

double layer

current probe

_iioltage probe

_iioltage probe

Figure 8.3. Capacitor with sample for minimizing electrode polarization.

46
8.3. Measurement Methods

8.3.1. Capacitive Measurement Techniques

The capacitor is useful when the electromagnetic wavelength in the material is much

longer than the separation between the plates. In this measurement the sample is as- sumed to be immersed in a strong electric field and weak magnetic field. A difficulty

with capacitor measurements resides in minimizing effects of the fringing field and elec- trode polarization. The fringe field is usually partially eliminated by using guards. The

capacitance for a parallel plate of separation d and area A, with no fringing of fields near edges is given by

f'A

C='-f. (8.4)

The conductance at low frequency is given by

f" A

G = uo-4-. (8.5) a The permittivity can be obtained from measurements of C and G and is given by

The * indicates the permittivity is complex.

The eflFects of electrode polarization when measuring conducting liquids at low fre- quencies must be addressed.

8.3.2. Four-Probe Technique

The main problem with capacitance techniques for highly conductive liquids is electrode

polarization. The four-probe capacitance technique overcomes electrode problems [96,

97] by measuring the voltage drop away from the plates and thereby avoiding the double

layer (see figure 8.3).

The permittivity is calculated for this measurement by

AZ' = --i-(^)2A(u;6"), (8.7)

Go (T

AZ" = -^C-^)'A{u;e'), (8.8)

Co (T

47
o- Outer Conductor o

Port 1 e shield

o Outer Conductor o

Figure 8.4. A shielded open circuit termination. The inner and outer radii of conductors

are a and b.

where uAe" is the difference in loss times angular frequency between solvent and solute

and Co is the vacuum capacitance of the cell. AZ' and AZ" are the measured differences

in real and imaginary parts of the impedance between solvent and solute.

8.3.3. The Open- Circuited Holder for Liquid Measurement

The shielded open-circuited coaxial line sample holder has been used for years (see figure

8.4) for dielectric measurements of hquids and powders in the microwave band [98]. This

fixture is composed of section of coaxial hne where the outer conductor extends beyond

the end of the inner conductor. The advantage of an open-circuited holder is the ease of

sample installation, the broad frequency capability, and the strong electric field in the

sample region. These holders can be used for measurements from 1000 Hz to 20 GHz.

The sample holder operation is based on an accurate model of a coaxial line terminated

in a shielded open circuit. The model we use is based on a full-wave solution. The full- mode model is more accurate than the commonly used capacitance expansion [99, 100].

The sample holder has been found useful in both ambient and high-temperature mea^

surements. Von Hippel [101] used an open-circuited sample holder for hquid measurements. Bussey [99] extended the open-circuited holder technique to higher frequencies

with an admittance model. Scott [102] studied the instabilities encountered in solving

the relevant nonlinear open-circuit equations. Hill [103] studied in situ measurements

of soils using open-circuited transmission lines. Jesch [104] used the shielded opencircuited holder for measurements on shale oil. Biological tissues have been measured

48
using the shielded open-circuited hne, for example by Stuchly and Stuchly [105].

8.4. Dielectric Measurements on DNA in Solution

We measured herring sperm DNA in mixtures of deionized water, protamine, and base

solution from 20 Hz to 200 MHz. Our measurements were accomplished by use of a

capacitor fixture from 20 Hz to 1 MHz and an open-circuited holder from 300 kHz to 200

MHz. We used two capacitors. The first was nickel plated; the second was platinum

blacked to minimize efi"ects from the electrode. We experienced a large decrease in

electrode efi'ects in the platinum-blacked fixture over the nickel-plated fixture. Typical

capacitor measurements of DNA solutions are given in figures 8.5 through 8.10. To

denature the DNA, samples were heated to 97 '^C for 5 min.; then the samples were

placed on ice until cooled below room temperature. The samples were then allowed

to come back to equilibrium at 22 *^C and remeasured. The measurement was taken

immediately after putting it in the fixture. The mass of the sample weis measured before

and after heating. The mass loss was less than 0.2 percent. In figures 8.5 and 8.6, the

permittivities of DNA solutions as a function of protamine concentration are displayed.

In figures 8.7 through 8.10, the permittivities of DNA solutions of both single (SS) and

double-stranded (DS) DNA as a function of protamine concentration are displayed on

a log-log scale. In figures 8.11 and 8.12, results for measurements using the shielded

open- circuited holder from 300 kHz to 200 MHz are shown.

At low frequency our results show strong electrode polarization and low-frequency

dispersion. The static values of the real part of the permittivity in figures (8.7) and

(8.9) can be estimated by drawing a straight line from the data in the 10 kHz- 100

kHz range to Hz. At very low frequencies (less than 100 Hz) we observed a drift in

capacitance with time. We consistently see higher loss and higher permittivity in the

denatured solutions than in the double-stranded solutions. These trends were also seen by Georgakilas [106]. We postulate that this is due in part to the increased conductivity

in the single-stranded solutions. As the strands separate, counterions are released and

thereby the conductivity increases. We think that the increase of the real part of the

permittivity is due to increased electrode polarization as the conductivity increases.

There are significant differences between the permittivity of the solvent and solutions

containing DNA.

The observed polarization can be interpreted as due to motion of the coimterion

49
Soln 14-DS-30 nmol - SOO bp

Soln 15-DS-100 nmol - BOO bp

Soln 14-DS-100 nmol - 400 bp

Soln 1 4-SS-30 nmol - eOO bp

Soln 1 6-SS-1 OO nmol - eOO bp

Soln 1 6-SS-1 OO nmol - 400 bp

0.01 0.1

Frequency (kHz)

Figure 8.5. Measurements of the real parts of the permittivity of double-stranded and

denatured DNA as a function of frequency. The amount of protamine cation was 30

nmol and lOOnmol. Note TBE refers to a solution of tris, boric acid, and EDTA. Also

(BP) stands for base pairs.

50
3.0

2.6 —

2.0

1.S

1.0

0.5 —

1 1 1 DS DNA-no protamine

DS DNA-30 nmol protamine

D3 D^4A-'tOO nmotprotamlriii):

SS DNA-no protamine

SS DNA-30 nmol protamine:

SS ONA-1 Oo nmol protamine

0.01 Frequency (kHz)

0.1

Figure 8.6. Capacitance measurements of the imaginary parts of the permittivity of

double-stranded (DS) and denatured (SS) DNA as a function of frequency. The amount

of protamine cation was 30 nmol and 100 nmol.

51
Er

10

10

10 ^

10

10

10

10

0.01

100 nmol DS DNA no TBE

lOOnmolSSDNAw/TBE

100 nmol no DNA w/TBE

100 nmol no DNA no TBE

100 nmol DS DNA w/TBE

Extrapolated Permittivity

L±U I I I

' 0.1 1 10 Frequency, kHz

100 1000

Figure 8.7. Capacitance measurements of the real parts of the permittivity of doublestranded (DS) and denatured (SS) DNA as a function of frequency on a log-log scale. The amount of protamine cation was 100 nmol in TBE solvent. The estimated permittivity without electrode polarization is shown by the dashed curve.

52
10

10 ^

10

10

10 .

10

10

0.01

^^— 100 nmol SS DNA no TBE

100 nmol SS DNA w/TBE

T^s— 100 nmol DS DNA no TBE

X— 100 nmol DS DNA w/TBE

B— 100 nmol no DNA w/TBE

O— 100 nmol no DNA no TBE

0.1 1 10

Frequency, kHz

100 1000

Figure 8.8. Capacitance measurements of the imaginary parts of the permittivity of

double-stranded (DS) and denatured (SS) DNA as a function of frequency on a log-log

scale in TBE solvent. The amount of protamine cation was 100 nmol.

53
10

10

10^ L

10

10"

10

10

0.01

^^— 30 nmol SS DNA no TBE

30 nmol SS DNA w/TBE

^<— 30 nmol DS DNA w/TBE

75s— 30 nmol DS DNA no TBE

€•— 30 nmol no DNA w/TBE —B— 30 nmol no DNA no TBE

Extrapolated Permittivity

0.1 1 10 100

Frequency, kHz

_i J

1 1 1 1 1 II I I1 1 1 1 1 1

1 I I 1 1 1 1 1 II I I

1 1 1 1 1 1

1 I I 1000

Figure 8.9. Capacitance measurements of the real parts of the permittivity of doublestranded (DS) and denatured (SS) DNA as a function of frequency on a log-log scale in

TBE solvent. The amount of protamine cation was 30 nmol. The estimated permittivity

without electrode polarization is shown by the dashed curve.

54
10

10 ^

10

10

10

10

10

10

-^^— 30 nmol SS DNA no TBE

30 nmol SS DNA w/TBE

X— 30 nmol DS DNA w/TBE

T^s— 30 nmol DS DNA no TBE

O— 30 nmol no DNA w/TBE

Q— 30 nmol no DNA no TBE

_1 I

'

I I I ' ' '

'

I I

' I I u

0.01 0.1 10 100 1000

Frequency, kHz

Figure 8.10. Capacitance measurements of the imaginary parts of the permittivity of

double-stranded (DS) and denatured (SS) DNA as a function of frequency on a log-log

scale in TBE solvent. The amount of protamine cation was 30 nmol.

55
85

84

83

82

81

'w" 80

79

78

77

76

"T 1 1—I—Irill 1 1 1—I—I—I T ir

75

0.0001

No DNA (08/23)

200 ng/ml DS DNA (08/23)

200 |xg/ml heat denatured DNA (0823)

200 ^ig/ml DS DNA (09/1 1

200 jig/ml heat denatured DNA (09/11)

_] _j I I

I 0.001 0.01 Frequency (GHz)

0.1

Figure 8.11. Shielded open-circuit holder measurements of the real parts of the permittivity of double-stranded DNA as a function of frequency on a log-log scale.

56
0.1

100

1.0 10

Frequency (MHz)

100

Figure 8.12. Shielded open-circuit holder measurements of the permittivity of doublestranded and denatured DNA as a function of frequency.

57
sheath ai'ound the DNA molecule. This counterion effect is an induced polarization

with the induced dipole moment parallel to the sheath major axis. The dispersion peak

is masked by the strong electrode polarization and low-frequency dispersion.

Denaturation appears to proceed in stages [106]. First counterions are expelled

from the molecule, which causes dissociation and strong phosphate-phosphate repulsion

between strands. This increases the conductivity of the solution. As temperature is increased further the repulsive forces cause hydrogen bond rupture and denaturation.

9. Discussion and Conclusions

In this Technical Note we have overviewed dielectric relaxation of DNA in solution. We

also studied the current understanding of the polarization of the DNA molecule. This

required an understanding of counterion dynamics around the molecule and polarization mechanisms. To this end we overviewed polarization theory for polymer systems

and developed an extension of the Poisson-Boltzmarm equations for the electrode-DNA

interface. We also developed a statistical-mechanical model of DNA in an ion atmosphere. We then studied past measurements of DNA and finally presented results of our measurements.

The low-frequency polarization observed in DNA is due to motion of the counterion

sheath. This polarization increases as the square of the length of the molecule and

decreases as temperature increases. An understanding of both the length and temperature dependence can be understood by models of the interaction of the counterion

atmosphere with. the phosphate backbone of the molecule. Models that represent DNA

as a string of rigid rods predict the correct temperature and length-dependent behavior.

In our measurements the dispersion peak is masked by the strong electrode-polarization

and low-frequency dispersion. We observed higher loss and higher permittivity in the

single-stranded solutions than in the double-stranded solutions. These observations were also made by Georgakilas [106]. We think that this is due in part to the in- creased conductivity in the single-stranded solutions. The increase of the real part of

the permittivity we believe is due to increased electrode polarization as the conductivity

increases.

We see evidence from conductivity measurements for the following sequence of stages

in denaturation [106]. First counterions are expelled from the molecule which causes

58
dissociation and strong phosphate-phosphate repulsion between strands. This increases

the conductivity of the solution. As temperature is increased further the repulsive forces cause hydrogen-bond rupture and denaturation.

We would like to thank Albert Young and Mark Holhs of MIT Lincoln Laboratory, Stan

Abramowitz of the Advanced Technology Program (ATP) , members of the Genosensor

Consortium, Eleanor Livingston for discussions and editing, and Claude Weil, John

Grosvenor, and Michael Janezic for measurement and design assistance.

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59
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