Dielectric relaxation of a DNA fragment

117

Dielectric Relaxation of a DNA Fragment R. B. Dowda and W. E. Vaughana aDepartment of Clzemisrry. University University Ave., Madison, Wi’ 53706

of Wisconsin-Madison,

f 101

1 ABSTRACT The real part of the dielectric permittivity in the frequency range 4-50 MHz of a monodisperse DNA fragment in aqueous solution at 23°C as a function of concentration is determined by time domain reflectometry. The fragment is short enough to be considered a rigid rod. The data are interpreted in light of a microscopic model based on a forced diffusion equation. The observed dielectric dispersion derives from counterion motion perpendicular to the rod axis. Using parameters derived from analysis of transient electric birefringence using the same model, the model predicts a limit than the extrapolated slower response in the zero concentration The discrepancy presumably lies with the experimental data exhibit. extrapolation that is hindered by lack of data at very low concentrations and the sensitivity of polyelectrolyte solution properties to even small salt concentrations. 2 INTRODUCTION

A number of determinations of the frequency dependence of +he .!ielectric permittivity (real part) appear in the literature. These investigations are on long polydisperse DNA fragments. Often the data are interpreted in terms of phenomenological models, yielding derived parameters of uncertain microscopic significance. In this work we examine aqueous solutions of a monodisperse DNA fragment short enough to be modeled as a rigid rod. Further, a microscopic model with well defined dynamical parameters is available to interpret the dielectric behavior. The temperature is fixed (23OC) and the study is over a limited range of concentration and frequency.

0167-7322 /93/sos.00

Q 1993 - Elswier Science Publishers B.V. AU rights reserved

3 EXPERIMENTAL The determination of the dielectric permittivity E* = E’ - is” of aqueous solutions of polyelectrolytes from low to radio frequencies is very difficult. The dielectric loss &’ is obscured by the contribution of ionic conduction and only the real part of the permittivity E’ is used to characterize the motion of the polyelectrolyte and the distortion of its ion atmosphere. A survey of methods and results for various polyelectrolyte solutions has been reported by Altig [ 11. For this study on aqueous solutions of DNA oligomers, we choose the method of time domain reflectometry. The instrument is an upgraded version [I] of an apparatus constructed by Meyer [2]. The instrument accesses the frequency window 4-50 MHz. If the DNA samples are sufficiently short, they can be regarded as rigid rods, and the frequency window will reach both the “high” and “low” frequency dispersions commonly observed for polydisperse samples. A voltage pulse is applied to a coaxial line that feeds through a sampling oscilloscope, The pulse reflects off a dielectric filled section of the line terminated with a matched impedance, and the returning pulse is probed by sample-and-hold circuitry_ The entire operation is under computer control and the raw data is a discrete set of voltages as a function of time. These data are (discrete) Fourier transformed and the permittivity determined by deconvolution of the successive influence of the empty line. connector, sample filled line, and termination on the reflected waveform. Details are found in reference 1. The apparatus was tested on water and simple salt solutions. Even water, which has not yet approached its dispersion at 50 MHz, exhibits reproducible oscillations between 78 and SO units of relative permittivity (dielectric constant) as a result of uncharacterized systematic error in the instrument. Parallel oscillations appear in the salt and polyelectrolyte data and our data are smoothed by subtracting the solvent contribution and readding the static permittivity of water (23OC) to the data. Double helical DNA, isolated from calf thymus. consisted of 163 f 5 base pairs and was a gift of M. Thomas Record. The DNA was in aqueous solution (23°C) at pH 7.75 in 0.05 mM EDTA, 1.5 mM NaN3 and 1 mM Na2HP04. This solution was used neat and diluted with water to produce lo*ver concentrations_ 163 base pairs corresponds to a rod length of 5.54( IO)-* m. Three concentrations were used: i2.3, 17.4 and 26.2 mM in PO43-. 4 THEORY The derived data, dielectric pexmittivity (real part) E’ as a function of frequency, are interpreted in terms of a microscopic model for the motion of

119

rodlike polyions and the associated counterion complement in external electric fields [34]. The model describes the bulk diffusion of the counterions, the rotational diffusion of the rod, the interaction of the rod and counterion charges with the external field, and the CouIombic interactions of the counterion charges with each other and with the fixed charges on the rod. The counterion motion takes place in a cylindrical shell with length equal to the rod length, inner radius corresponding to the DNA radius, and an outer radius sufficiently large so that the number density of counterions is virtually zero in that region of configuration space. rhe model presumes no added salt. The dynamical equation to be solved (a forced diffusion equation) is of the Master equation form and the result for the dipolar correlation function y is a sum of exponentials.

(m(O)- m(t))

’ = (m(0) - m(O))

-

y gives the (normalized) correlation of the system dipole at time t given some value at time 0 for the field free decay of po!arizat ion. The - hk are the reIaxation rates and are the eigenvalues of the free diffusion opeator.

-hk

=

L(L + l)D,

+ m’(D,l-

Dl> + D&

-

)I I

(2)

DI and Dir are the components of the rod rotational diffusion tensor, D3 is the bulk diffusion coefE:ient of a counter-ion, L and m, mi. and Oniki are labels for the eigenfunctions describing respectively orientation of the rod in space, location of counter-ion i in the cylindrical z-direction, and location of counterion i in the r- and $-directions. There are n** counterions and the second occurrence of L in equation 2 is the rod length. The Ck are determined by equilibrium averages [3]. A simplifying feature is that the model predicts that only two relaxation modes have large amplitudes - these can be interpreted roughly as the first mode involving counterion motion in the zdirection and the first mode involving motion in the r-direction (a longitudinal and a transverse mode). y is related to the complex dielectric permittivity E* by [5]

120

I

dr exp(imt)dt - dt

=

Eow - E,:(~E* + E*(Q - E,)(~Q

l)( 1 + B(E+E,,A~))

+ 1)(1 -t B(E*,E,.A~))

(3)

0

EJJ ar,d b are low and high frequency limits of E* and Al is a shape factor. B is a known function [5]. The DNA oligomers are “needle shaped” so to good accuracy

r

J

- %xp( dt

C-E,

iwt)dt

=

&O-%0

0

(4)

Thus the model predicts that the real part of the permittivity is represented by the superposition of two Debye relaxations. We fit the data to find &o. the relative amplitudes and corresponding relaxation rates - ki.

6 =

E,

+

1+ 5 RESULTS

(1 -Xl)

XI

(&(-y-E,)

(o/hi)’ + 1 + (o/h*)*

(3

AND CONCLUSIONS

The amplitudes AE~ = (~g - Em)Xi and relaxation times Ti (- hi)-* found by fitting the data to a superposition of two Debye reIaxations are shown in Table 1. & = 79.2 (the dielectric constant of water at 23°C). Table

I. Dielectric

Cont.

mM PO43-

Dispersion

Parameters

(aqueous DNA 23°C)

‘El ns

A=1

Qns

k2

12.3

31.8&l-7

17.8k1.4

3-l-11.4

2.1HI.4

17.4

255&l

21.x1.8

3.9&l -3

2.7s.4

26.2

33.Sfl.7

35.3*2-o

4.1%3.0

1.3HI.5

.8

We find the amplitude of the lower frequency proportional to the concentration so the specific

dispersion increment

to ‘be roughly* is constant as

shown by Figure 1. The specific increment for the high frequency is less well determined but decreases with increasing concentration. Figure f. Dielectric Increments versus Concentration dispersion), Hollow squares (fast dispersion)

y =

0.63592

dispersion

- Filled squares

(slow

+ 1295.4~

30-

DNA M POjThe relaxation times are linear in concentratron (within experimental error) as shown by Figure 2. Since the DNA solutions. which contain added salt, were diluted with water, the intercepts of the plots of Figure 2 are for infinite dilution with s added salt, and the microscopic theory should be directly applicable. To show the quality of fit of equation 5 to *he data, the data and backfit arz shown in Figure 3. Some oscillations remain In the experi‘mental data as a result of incomplete compensation for SySiZliiC& eri-or in the instrument. About 90% of the amplitude comes from the slower dispersion. The relaxation time (for the least concentrated solution) of 21-s ns corresponds to

122

a critical frequency of 7_3( 10)6 Hz and an inflection point on the E*- Iogv at :ogv = 6.86. This feature is seen in the backfit and indicates that the instrument has accessed only the upper frequency half of the siower dispersion (and all of the dispersion at higher frequency). Figure 2. ReIaxation Times versus Concentration. dispersion), Hollow sqttares (fast dispersion).

=

Filled

squares (SIOW

11.010+ 867.09x

y = 2.4601 + 66.541x

0

-*------*

0.000

. . 0.010

I

.

0.020

.

.

.

.

.

.

.

.

0.030

DNA M POjThe extrapolated relaxation times can be used to estimate the dynamical parameters of the model. Using equation 2, the relaxation rate (l/cc) of the first longitudinal mode is

Similarly, the rate of the first transverse mode is (7)

12.3

Figure 3. Dielectric Permittivity as a Function of Frequency (least concentrated sohrtion). Line with oscillations = experimental data. Smooth line = backfit.

IQ0

I-

s-b .= ._> =

em

E

90

8

6.6

6.8

7.0

7.2

7.4

7.6

7-g

Log(Frequency) However. it is easy to see that the slower rate (9.0( 10)’ s-*) does not correspond to equation 6 and the faster raie (4_ 1(IO)8 s-l) to equation 7. Rather it appears that the slower rate is for the dispersion associated with the first transverse mode and the faster rate with the second transverse mode. The longitudinal dispersion appears at frequencies Mow the range accessed by the instrument. Our model was fit previously r3-4) to the transient electric birefringence of a solution containing short (42 nm) DNA fragments [6]_ The field off decay of biref-ingence yields DI and the value of DI found was in good agreement with values calculated from rheological models [7-8]. For our fragment, WC calculate DI = S-99( !0)4 s-t. DII was found by fitting the fieId on growth of birefringence. The value found for DIIcan be transferred to our fragment by presuming that DIIis inversely proportional to the rod length [3]. We obtain DII

= 2.60( IO)6 s-1. Finally D3 can be taken to be the diffusicn constant of a sodium ion (23°C). Our estimate is D3 = 1.27(10)-g m%-* . These numbers when inserted into equations 6 and 7 (011 = 3.536(10)7 m-l) yield - Xl = 4.1(10)6s-1 and - hz= 4.2( lo)6 which are much less than either observed rate. Our experiments can be interpreted in light of diefectric measurements on aqueous DNA solutions in the literature [9-IO]. However, this earlier work [9] is m on monodisperse samples (albeit with a known length distribution), nor were the samples short enough to be considered rigid rods. The aqueous sample of Vreugdenil et al. [9J had solute with 194 nm length whereas our DNA has 9 length of 55 nm. The persistence length for DNA is about 50 nm so even our sample is somewhat flexible Our concentrations were higher than those of reference 9. Added salt is necessary to prevent denatuntion of the DNA. Nevertheless the data of reference 9 are useful in resolving the discrepancy Vreugdenil et al. obtained data to low between theory and experiment. frequencies and observed two dispersion regions. In the spirit of our model, we wouId assign the high frequency dispersion to transverse modes. Since the amplitude of this mode depends rn the number of base monomers (phosphate concentration) but not on ien&;. we extrapolated the reported dielectric increments of the high frequency dispersion to our lowest concentration of 12.3 mM and predict AE~ = 17.2 close to our observed value (17.8). The estimate of - Xl from the model seems firm, so we conclude that our lower frequency dispersion is the ftrst transverse mode and the higher frequency dispersion the much smaller amplitude second transverse mode. Th,e longitudinal modes occur below the frequency range of our instrument. Even so, the (extrapolated) rates are much larger than the model prediction, However. the extrapolation of the relaxation times is open to question. Figure 2 shows a decrease of relaxation time with decreasing concentration. The high frequency dispersion of reference 9 shows an increase of relaxation time by more than a factor of ten when *Lheconcentration went from 2.88 mM to 0.1 mM. We might expect simiIar behavior for odr system with the T-concentration curve taking a sharp upturn at low concentrations thereby greatly reducin g the extrapolated relaxation rates and producing better agreement with the mode1 prediction. One should not be surprised by large effects caused by mM salt concentrations in poiyelectrolyte soWions_ Fixman and Jagannathan’s [I I] phenomenological model for the (counter-ion) polarizability of polyelectrolyte sohttions predicts a reversal from a 2:1 transverseflongitudinai polarization ratio at 0.1 mM added salt to a 1:3 transverse/longitudinal polarization ratio at 2 mM addzd sak

6 REFERENCES 1

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2

P. I. Meyer. Thesis University of Wisconsin-Madison,

3.

W. E. Sormen, Thesis University of Wisconsin-Madison,

4.

W. E. Sonnen, G. E. Wesenberg. W. E. Vaughan, manuscript submitted to Biophysical Chemistry.

5.

D. 0.

6.

A. Szabo, M. Haleem. D. Eden, J. Chem. Phys., 85 (1986) 7472-7479.

7.

P. J. Hagerman and B. H. Zimm, Biopolymers, 20 (1981) 1481-1502.

8.

M. M. Tirade, C. L. Martinez, J. G. de la Terre, J. Chem. Phys.. 8 1 (1984) 2047-2052.

9.

Th. Vreugdenhil, F. van der Touw, M. Mandel, Biophys. Chem.. 10 ( 1979) 67-80.

10.

M. S. Tung, R. J. Molinari. R. H. Cole. J. H. Gibbs. Biopolymers, 16 (1977) 2653-2669.

11.

M. Fixman and S. Jagannathan. J. Cbem. Phys., 75 (1981) 4048-+059.

Klug

(1989). (1982). (1991).

and W. E. Vaughan, J. Chem. Phys., 56 (1972) 5005-5007.