BIOPOLYMERS
VOL. 16, 2653-2669 (1977)
Influence of Temperature and Ionic Strength on the Low-Frequency Dielectric Dispersion of DNA Solutions MING SUNG TUNG,* ROBERT J. MOLINARI, ROBERT H. COLE, and JULIAN H. GIBBS, Brown University, Department of Chemistry, Providence, Rhode Island 02912
Synopsis The dielectric properties of DNA solutions a t low frequencies (5 Hz to 2 kHz) have been measured by means of a four-terminal bridge method utilized to minimize electrode polarization errors. At 24OC native salt-free DNA has a very large specific dielectric increment, At/c = 9.8 X lo6 l/mol and a very low frequency relaxation centered a t 18 Hz. Both the dielectric increment and the relaxation time are greatly decreased by partial heat denaturation a t temperatures above 6OoCor by addition of salt, the effects being much larger for divalent anions. These results are shown to be in qualitative agreement with theoretical treatments of counterion fluctuation polarization by McTague and Gibbs for the equilibrium case and by Mandel for relaxation. The ratio of the relaxation time for the low-frequencyprocess to that previously observed a t much higher frequencies suggests that these relaxations result from counterion fluctuations along the longitudinal and transverse axes of the molecule, respectively.
INTRODUCTION The dielectric properties of DNA solutions have been the subject of several experimental and theoretical inve~tigations,l-'~ but the evidence has been incomplete and interpreted in a variety of ways. Allgen and Junger2 first observed a dispersion, which they attributed to a transverse permanent dipole moment (i.e., one at right angles to the helix axis), in the vicinity of 100 kHz. A symmetrical rise and decay of birefringence on the application and removal of a biasing field, observed by Benoit20and confirmed by Haltner,21 was attributed by them to an induced dipole. Recently, Hornick and Wei11,22Colson et al.,23and Greve and DeHeij2*have arrived at a similar conclusion from consideration of the birefringence of sonicated DNA. In dielectric measurements at frequencies above 50 Hz T a k a ~ h i m a ~ - ~ found a low-frequency relaxation with an amplitude and a characteristic frequency strongly dependent on molecular weight. He eventually interpreted his results in terms of induced ionic polarization. On the other hand, Hanss and Berengog concluded that the dna polarizability, measured as a conductivity dispersion in the very low frequency range (0.5 Hz to 1kHz), * Present address: National Bureau of Standards, Washington, D.C. 20234. 2653 0 1977 by John Wiley & Sons, Inc.
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could be explained only by a permanent ionic dipole. McTague and Gibbs,l3 treating the polyion as a linear array of counterion adsorption sites, were able to qualitatively describe the low-frequency dielectric behavior by means of a counterion polarization model. Mandel and van der TouwlO showed that two separate dielectric dispersion regions exist in solutions containing linear polyelectrolytes such as DNA. They attributed the higher-frequency dispersion, in which both the mean relaxation time and the dielectric increment are independent of molecular weight, to fluctuations in the distribution of bound counterions along short rigid sections of the molecules joined to each other by loosely jointed flaws in the helix structure. Recently, Sakamoto et al.l9 assumed that counterion fluctuations along the DNA helix were slower than the overall rotational diffusion of the DNA. They then concluded that the low-frequency dielectric relaxation corresponded to a rotational relaxation of an ionic dipole, the origin of which they attributed to counterion fluctuations that were “permanent” on the time scale of the faster rotational diffusion. Those authors ignored, however, any explanation of the high-freqency dispersion in their model. These and other studies (for a general review see S c h w a r ~provide ~~) a variety of evidence and conclusions. It seems clear, however, that two distinct polarizations and corresponding dispersion regions exist in DNA solutions. The one at radio frequencies is better characterized experimentally than the apparently much larger one at audio or subaudio frequencies. Neither a longitudinal nor a transverse permanent moment in the DNA molecule is a likely explanation, in particular for the largeamplitude, low-frequency process. (The antiparallel arrangement of the nucleotide chains in the helix would cause cancellation of most permanent longitudinal moments.) Alternative explanations utilizing induced counterion polarization effects have not been fully established. In the interest of better characterizationof the low-frequencydispersion, we have made studies of aqueous solutions of DNA under various conditions of concentration and added salt to frequenciesas low as 5 Hz. In addition, the temperature dependence df the real part of the dielectric dispersion in solutions of native salt-,fFee DNA was measured. These studies were made possible by use of a special four-terminal, low-frequency bridge, developed in this lab by Berberian and Cole,26which greatly reduced the errors introduced by electrode polarization effects. The only other study of DNA made with a four-terminal bridge technique, that of Sakamoto et al.,19 utilized a bridge design which was virtually identical to the original Cole-Berberian four-terminal bridge26but which employed a much more sophisticated detection system on the bridge output. Whereas their study was limited to only one species of added salt and two concentrations of of DNA, we report here more extensive results covering the effects on dielectric behavior of degree of helicity, species, valency, and concentration of added salt, temperature, and a wide range of salt-free DNA concentrations. The results of both studies are consistent with predictions of the
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static model of McTague and Gibbs,13 which includes counterion nearest-neighbor interactions, and a modified relaxation model of van der Touw and Mandel.18
EXPERIMENTAL Three samples of calf-thymus DNA (as the sodium salt) were used in the work. The first (NaDNA-S3),from Sigma Chemical Co., was stated to have an average molecular weight 1.3-1.5 X lo6 as supplied. The second (DNA-M1, M4, M5), from Miles Laboratory, Inc., had a relative viscosity of 1.88 at a concentration 0.5 mg/ml and at 25OC, corresponding to an average molecular weight of 2 X lo6. The third, 2-001, from Cal-biochem, had a weight-average molecular weight of 9 X lo6 as supplied. Doubly-distilled water with specific conductance less than 8 X mho cm-l and freed of COa by boiling for 5 min was used for the preparation of DNA solutions of concentration 3 mg/ml (by weight of dried sample) and of pH about 6.7. The concentration expressed as molarity is referred to phosphorus. Two samples M-4 and M-5 were dialyzed against COz freedistilled water for 72 hr at 4°C for the reduction of salt concentration. The stock solutions were diluted just before the measurements, which were completed in a few hours. Denaturation was accomplished by the heating of solutions for 15-17 min a t temperatures from 64 to 97OC, followed by a rapid cooling to 4OC. Measurements were made at 24°C. Most of the admittance measurements were made with a special lowfrequency bridge developed by Berberian and Cole26for determination of parallel capacitance and conductance a t frequencies from 2 kHz to 1Hz. An essential feature of the instrument for the present work was its ability to make four-terminal guarded electrode measurements largely free of electrode polarization error^.^^,^^ The measuring cell for the guarded four-terminal measurements was the same as in Refs. 26 and 27, except that the probe cell constant, Cg, equaled 0.0529 pf. A different cell (C, = 0.0438 pf) was used for the temperaturedependence studies. The dielectric increment, Ad, of a DNA solution was obtained as the difference of capacitances of DNA and matching KCl solutions at a given frequency, divided by the geometric cell capacitance C, in pf. The dielectric loss, At”, was obtained from the relation At” = (G - Go)/27rfCg, where G and Go are the conductance of the solution at a frequency, f , and the dc conductances, respectively. The measured conductances were in the range 8-30 pmho, making measurements of capacitanbes on the order of pf inaccurate below 5 Hz as a consequence of the enormously larger conduction currents. Conductances were measured to 1Hz, and the 1 Hz value was taken as the dc value, Go, for calculation of A&’. Some measurements to determine the absence of a noticeable dispersion
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Fig. 1. Dielectric increment dispersion 24'C of aqueous NaDNA-S3 solution at conc = 2.94 10-4M. (1) Native: ( 0 )24"C, f = 25 Hz,T = 6.37 msec; (2) heated to 64°C for 15min: (v) :.64OC, f = 28 Hz,T = 5.7; (3) heated to 84% for 16 min: (X) 84OC, f = 50 Hz,T = 3.18; (4) 97"C, f = 800 Hz,7 = 0.2. heated to 97OC for 17 min: (0) X
in the range 1-100 kHz were made with a conventional three-terminal cell and a transformer bridge developed in this laboratory for direct reading of parallel capacitances and conductances.
RESULTS Effect of the Thermally Induced Helix-Coil Transition Dispersion curves of A d vs frequency at 24°C are shown in Fig. 1for a solution of native NaDNA-S3 and for solutions partly denatured by maintenance for 15-17 min a t temperatures of 64,84, and 97°C. It is evident from these results that the dielectric increment is still frequency dependent even a t 5 Hz for all but the completely denatured samples. Complex plane plots of At" vs At' in Fig. 2 are circular arcs within experimental error. When extrapolated to zero frequency they give estimates of the static dielectric increment, AcO, and of the relaxation time, 7 , corresponding to the midpoint of the dispersion. The quantities r and ACOare plotted in Fig. 3 against the temperature to which the samples were heated. Salt Effects Effects of added mono- and divalent salts on the dielectric increment are shown in Fig. 4 (for samples M4 and M5). Marked decreases in relaxation times, 7 , and, as is better shown by the complex plane plots in Fig. 5, static dielectric increments, AEO,were observed with an increase in salt concentration. The changes in the dielectric properties with concentration of added mono- and divalent salts are shown in Fig. 6 and Table I. The
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Fig. 2. Cole-Cole plots for NaDNA-S3 (24OC) heated to different temperatures: temperature, 24OC, ( 0 )heated a t 84"C, (u) heated a t 64"C, (A)heated a t 97OC. 1100 1000
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effect of divalent ions was much stronger than the effect of monovalent ions; the specific dielectric increment (DNA conc. = 4 X 10-4N) decreased from almost lo7,when no salt was added, to very near zero when the normality of Mg was 1.5 X 10-N. While the charge and concentration of the counterion species strongly affected the dielectric behavior, the size of the counterion had little or no effect (Fig. 4).
++
Temperature Dependence Figure 10 shows the dispersion curve of a DNA solution 2-001 (DNA concentration = 6.2 X lOV4N)a t a number of different temperatures (t =
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(b) Fig. 4. (a) Dielectric increment dispersion of DNA-M4 (conc. = 3.75 X 10-4M) with added monovalent salt. Curve 1: no salt added; X: CsCl = 4 X lO-5N; 0: NaCl = 4 X 10-5N. Curve 2: CsCl = 1.6 X lO-4N. (b) Dielectric increment dispersion of DNA-M5 with added salts. Curve 1: DNA-MI (conc. = 3.75 X lo-%) MgClz (conc. = 1.5 X 0: DNA conc. = 4.69 X 10e4N,no salt added; X: MgClz conc. = 0.75 X W 4 N ;0 : BaClz conc. = 0.75 X 10-4N, A: NaCl conc. = 5 X 10-4N.
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5,10,15,20, and 25OC). The dielectric constant of the solution increased with increases in temperature. Figure 11 shows that the temperature dependence of the DNA dielectric constant did indeed extend throughout the entire dispersion region and was not, for example, a simple displacement to higher frequencies of a dispersion curve with constant or decreasing amplitude a t higher temperatures. Since no data were available on the temperature dependence of the various electrode-blocking polarizations at such low frequencies, the well-known temperature dependences of the dielectric properties of water and matched-conductance KC1 solutions were studied in order to mimic the electric properties of the polyelectrolyte solution in the absence of a dispersion. Figures 12 and 13 show the measured
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(b) Fig. 5. Typical Cole-Cole plot for salt effect. (a) DNA-M4 (conc. = 3.75 X 10-4M) with monovalent salt: ( 0 )no salt added, ( X ) NaCl = 4 X 10-5N. (b) DNA-M5 (conc. = 4.69 x 10-4M) with divalent salt: ( 0 )no salt added, ( X ) BaClz (conc. 0.75 X lO-4N).
temperature dependence of the low-frequency dielectric behavior of KC1 solutions and pure water over the frequency range of interest. The static dielectric constants thus obtained are compared in Table I1 with the expected values from the literature.28 Both the direction and magnitude of the published temperature dependences of the static dielectric constant have been reproduced by our measurements. TABLE I Dielectric Properties of DNA with and without Salts Sample
DNA-M4
M, x 10-6 Conc. x lo4 (molfl) Salt conc.
0
3500 8.8 Conductivity (mho cm-l) 14.05 (24°C) Root-mean-square endto-end length ACL
T L (msec)
DNA-M5
NaDNA-S3
2 2 3.75 4.69 1.6 x 1 0 - 4 ~1.5 x 1 0 - 4 ~ 5 x 1 0 - 4 ~ CSCl MgClz NaCl 1300 60 610 3.2 0.23 1 29.54 24.26
1040 6.1 26.03
3430 8,
2830A
3430A
1.4 2.94
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NORMALITY OF A W E D SALT/ NORMALITY OF DNA
Fig. 6. Dielectric increment and relaxationtime vs the ratio of the concentration of added mono- and divalent salts to that of DNA. ( 0 )monovalent ion, ( X ) divalent ion.
CONC. x
lo4
Fig. 7. Specific dielectric increment and charge fraction vs DNA concentration.
DISCUSSION AND CONCLUSIONS The results of our measurements to frequencies of 5 Hz show that DNA solutions have huge dielectric increments and very long relaxation times, with values that are very dependent on the DNA dimensions and the chemical environment. For native salt-free DNA a t a concentration of 0.934 X 10-3M, the static dielectric increment is 9200 and the relaxation frequency is 18 Hz. The accepted molecular structure of DNA, that is, two oppositely directed backbones with the base pairs perpendicular to a helix axis, cannot possess a permanent dipole moment. Therefore, the large observed increases in dielectric constant at low frequency must be explained in some
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Fig. 8. Dielectric increment of diluted DNA (conc. = 3.75 X 10-5M) vs time.
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Fig. 9. Dielectric increment dispersion of DNA with and without dialysis. Curve 1: DNA-M5 (conc. = 4.69 X lO-*M) dialysis for 72 hr. Curve 2: DNA-M1 (conc. = 8.48 X 10-4M) without dialysis.
other way. It should also be pointed out that neither the strong effect of added salt on the dielectric constant nor the temperature dependence can be explained by the assumption of a permanent dipole moment. It is therefore necessary to consider interpretations in which the low-frequency dielectric dispersion of DNA is attributed to induced ionic polarization. Several different models have been proposed for explanations of the dielectric dispersion as an effect of ionic p o l a r i z a t i ~ n . ~ ~O’Konski15 ~J~-~~,~~ formulated a model with a frequency-independent surface conductivity. His model predicted a much smaller dielectric increment and a much shorter relaxation time than our experiments indicate and was open to the objection that effects of chemical potential gradients of ion concentrations were not treated. The theories of Schwarz16and Takashimas considered the properties of the inner counterion layer in terms of a complex surface conductivity, but failed to take into account the interaction between
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Fig. 11. DNA dielectric increment vs temperatureat frequencies spanning the entire relaxation region.
nearest-neighbor counterions and thus were unable to explain the observations by Mande130and Minakata3l that the dielectric increment of certain polyelectrolytes went through a maximum on addition of salt. Since experiments have shown that the dielectric properties are strongly influenced by counterions, we shall discuss our results in the framework of those theories in which the distribution and interaction of the counterions are emphasized, i.e., the theory of McTague and Gibbsl3 (hereafter M-G) for static polarization and the relaxation theory due to Mandel.17 In these theories three important factors which determine the dielectric dispersion are: the mean-square end-bend length of the polyelectrolyte,the amount of salt added, and the number and valency of the bound counterions.13J7
.
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Fig. 13. Temperature dependence of KCl solution (2 X IW4N)dielectric constant.
Oosawal* has also derived a dynamic theory with a relaxation equation similar to that of Mandel. The following discussion summarizes along with other phenomena experimentally observed effects of variations of these three quantities on the dielectric properties.
Magnitude of Dielectric Increments Assuming that DNA with an average molecular weight of 2 X lo6 has a root-mean-square end-to-end length of 3430 A (a contour length of 10,400 A and a persistence length of 600 A was used1'), and assuming random orientation of the DNA rod, then the M-G theory gives a value of 3.8 X lo6 l/mol for Adc, in fair agreement with the measured value of 9.8 X 106 l/mol for M-5 a t a concentration of 0.93 X 10-3M.
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TABLE I1 Calibrations of Temperature Dependence t("C)
co (meas.)
co (lit.)
G (conductance)
84.1 80.4 76.7
10.84 15.35 16.19
KC1 Temperature Dependence 10 20 30
83.8 80.6 16.4
HzO Temperature Dependence 10 20 30
84. I 80.4 76.7
82.0 79.4 76.9
Effect of the Thermally Induced Helix-Coil Transition The dielectric increment and relaxation time of a DNA solution are decreased by heating of the sample as shown in Fig. 3. The behavior is similar to that of the degree of helicity as a function of maximum heating temperature, in low-ionic-strength solutions, as determined from intrinsic viscosity measurements.32 Thus, dielectric dispersion can be used for the detection of the helix-coil transition in DNA solutions without the application of physical forces that may degrade the DNA. The mean-square end-to-end lengths of DNA molecules decrease in the transition from the rigid helical rod to the flexible random coil, with a resultant decrease in the dielectric increment and associated relaxation time. The results reported here are similar to those obtained by Takashima? but our dielectric increments and relaxation times are considerably larger. This discrepancy is believed to be due to the difference in the method used for measurement of the dielectric properties. Takashima was limited to measurements above a frequency of 50 Hz by his reliance on the technique of electrode spacing variation to correct for electrode polarization effects. He reported that the profile of the dielectric dispersion at an intermediate stage of the helix-coil transition was different from that of both helical DNA and randomly coiled DNA (i.e., the low-frequency dispersion appeared to be split in two). He attributed this split to the presence of an intermediate form of DNA. Our study shows no such peculiarity in the profile of the dielectric properties of partly denatured DNA, but rather a smooth change from helix properties to coil properties. Salt Effects Figure 6 (plotted as A& vs the ratio of normalities of added salt to that of DNA) shows that the dielectric increment decreases with the addition of salt. These experimental effects are in qualitative agreement with Sakamoto et al.19 Smaller end-to-end lengths, due to the decrease in electrostatic repulsion between DNA segments which accompanies increases of ionic strength, can cause dielectric changes in the observed di-
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rection. Sakamato et al.19 assert that the salt effect is explained primarily by this electrostatic chain contraction. Two phenomena, however, support our belief that the dramatic decrease in the dielectric increment with the addition of salt must be attributed mostly to changes in the magnitude of the polarization induced among the bound counterions. The first is that the observed decreases in the dielectric constant are much larger than those which would be expected from the decrease in end-to-end lengths derived from electrostatic considerations. The hydrodynamic length of DNA was reported to decrease by 25% when the concentration of sodium chloride was varied from to 10-3M,7yet the relaxation time decreased by a factor of 8.8 when the concentration of sodium chloride was increased by 5 X lO-*M. The M-G theory predicts such a dramatic decrease in the counterion polarization if the fraction of occupied counterion sites on the polyion is greater than one-half. Sakamoto et al.19 attribute their observed decreases in specific dielectric increment with increases in ionic strength to the above-mentioned chain contraction. They derive from their own results, however, that the fraction of sites occupied on the polyion is approximately two-thirds, i.e., exactly the case where the counterion polarization would be expected to decrease without invoking chain contraction. The second phenomenon is that divalent ions yield a much larger effect than monovalent ones, a circumstance readily attributable to a much stronger bonding of the divalent ions to the DNA.
Concentration Effects (Salt-Free DNA) Figure 7 shows that the specific dielectric increment increases as the concentration of salt-free DNA is reduced. The increase is interpreted as due to the decrease in the number of bound ions, i.e., the concomitant increase in the fraction of sites unoccupied upon dilution of salt-free DNA. This concentration effect and the salt effect discussed above indicate that a decrease in the number of bound counterions increases the dielectric increment. The above is an effect yielded theoretically by the M-G theory which also predicts that the dielectric increment will pass through a maximum on further decrease in the number of bound ions and drop toward zero as the number of bound ions approaches zero (a charge fraction of 1). The fact that DNA denatures more readily when the charge fraction is increased prevents the detection of this maximum in DNA solutions by preventing the study of salt-free DNA solutions at low concentrations. The denaturation of salt-free DNA on dilution has been reported12 and was indeed observed a t the lowest concentrations in this study in the time dependence of the dielectric increment (see Fig. 8). The lowest concentrations of DNA used in this study represent the low concentration limit for which DNA can be stable in salt-free solutions, and are in qualitative agreement with room-temperature values of Gruenwedel and Lu33for that limit. Much lower concentrations have been reached in solutions stabilized by added
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salts; see, for example, Sakamoto et al.,19 but the added salt itself drastically changes the ion environment of the DNA in solution and hence the dielectric properties. Extrapolation of the inverse specific dielectric increment to zero concentration (suggested by van der Touw and MandeP4) for various synthetic aqueous polyelectrolytes did not give a straight line for our salt-free solutions of native DNA, as shown in Fig. 7.
Temperature Effects Figure 10 shows that the dielectric increment of DNA solutions increases with temperature, and Fig. 11shows that it does so proportionately. It is important to make two observations. The first is that this behavior of DNA solutions is opposite to the temperature dependences of the static dielectric constant of the solvent (water) and of various matching-conductance aqueous solutions containing electrolytes but no DNA. The second is that the increase in dielectric constant with temperature is consistent with the M-G static model of counterion polarization. This model predicts a maximum in the dielectric increment, at a charge fraction equal to one-half, with the dielectric increment going to zero for a charge fraction of both one (no bound ions) and zero (all counterion sites bound). Since denaturation12 at high charge fractions prevents observation of this maximum in DNA solutions (see Concentration Effects), we are limited in our experimental observations to charge fractions somewhere between zero and presumably one-half. An increase in temperature must give rise to an equilibrium state in which fewer counterions are bound to the polyion. In the case of a charge fraction between zero and one-half, this decrease in the number of bound counterions associated with an increase in temperature will, according to the M-G model, cause the counterion polarization and dielectric increment to increase as observed. The observed temperature dependence would contradict any mechanism of relaxation through rotation of permanent dipoles.
Dielectric Relaxation Time The low-frequency relaxation of DNA in aqueous solutions differs considerably in all the cases we have studied from the Debye behavior predicted by various simple models. This difference is seen in the complex plane plots, Figs. 2 and 5, as large deviations from a semicircular locus of E’ vs 6’’. Within the uncertainties of the data, these plots are fitted by the circular arc function proposed by Cole and Cole to describe relaxations of a variety of dielectrics, with the parameter a (equal to zero for the Debye behavior) of order 0.36-0.44 for native and partially denatured DNA solutions and of order 0.5 for solutions with added salts. These values imply broad distributions of relaxation times if the behavior is described in such terms. We believe, however, that the deviations are preferably taken as indicative
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of cooperative interaction effects, and not necessarily or even desirably as alternatively described. None of the interpretations of the low-frequency relaxation so far proposed provide, to the extent they have been developed, an explanation of the observed form of the relaxation. Their possible validity can only be considered qualitatively on other grounds. The modified Maxwell-Wagner heterogeneous polarization model proposed by Pollacks has been shown by Takashima5 to be unsatisfactory for DNA solutions. Another explanation by Hanss and Bernengog attributes the effects to the orientation and transport of ionic dipoles on the basis that their observed relaxation times for conductance dispersion are of the same order of magnitude as the expected values for molecular reorientations. Although their observations are not inconsistent with ours, extensive r e s ~ l t s on ~ ~electric - ~ ~ birefringence of DNA solutions show that induced rather than permanent dipole forces are associated with the molecular reorientation. We therefore consider a third, alternative, interpretation in terms of induced counterion polarizations. Models based on mobile counterions bound to negative charge sites have been considered by several a ~ t h o r s ~ ~ and J ~ result J ~ J ~in a longitudinal relaxation time, 7,proportional to the mean-square length, L2,and the reciprocal of the counterion mobility, u. After neglecting counterioncounterion interactions, Mandel17derived the formula: 7
= L2/12ukt = al/ur(ze)2
(1)
where a1 is the polarizability of the polyion, r is the linear counterion density, and ze is the charge on the ion. In the absence of an independent value of u along the polyelectrolyte surface, absolute values of 7 for comparison with experiment cannot be calculated. The prediction of Eq. (1) that T should be proportional to a1 and hence to At/c is consistent with the observed increase of T with Adc. This dependence could, however, easily result from a variety of models (see, for example, Ref. 25). A further difficulty in attempting to identify observed relaxation times with ones calculated on the basis of a mobile counterion model is that the observed relaxation can result from a combination of this process, relaxation by reorientation of the polyion molecular axis (i.e., rotational diffusion), and relaxation by exchange of free and bound counterions. Mandel and van der TouwlO have shown that if these processes are assumed to operate independently, with an exponential decay of correlation for each, the observed rate of decay (the reciprocal of the relaxation time) would be the sum of the rates for the separate processes. Only if one of these is considered faster than the others can it be identified with the observed rate of decay. More generally, the experimental dipole correlation function is the product of the functions for each mechanism of charge displacement, if these mechanisms operate independently. Mandel and van der TouwlO commented on various possibilities for the relative rates of these mechanisms and showed that the success of their model in predicting two relaxations
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for DNA depended on the counterion fluctuation along small subunits of the polyion being faster than rotation of the molecule. Sakomoto et al.I9 noted an order of magnitude agreement between relaxation times calculated from viscosity measurements and the observed dielectric relaxation times. They then assumed that the low-frequency dielectric relaxation corresponded to a rotational relaxation that was faster than counterion fluctuations, although they noted that experimental and theoretical studies of counterion exchange rates either did not agree or were inconclusive. We see no method of theoretically determining in an unambiguous way which of these mechanisms would dominate. We have already suggested that the large derivations of the observed relaxation from ideal Debye behavior (with an exponential decay of correlations) may well be the result of counterion interaction effects. In that case, further progress in interpretation of the low-frequencyrelaxation will require a proper dynamical treatment of such effects, as for example, by a generalization of the equilibrium theory of McTague and Gibbs.13 One further point of interest can be made: namely, that the ratio of the lowfrequency relaxation time, ca. 8.8 X sec, to the high-frequency relaxsec, reported by Mandel and van der TouwlO is 7.3 ation time, 1.2 X X lo4. The ratio of the mean-square end-to-end length (3430 to the square of the hydrated diameter (26 for our DNA samples is 1.7 X lo4. This agreement suggests that the two relaxation processes and times are associated with counterion fluctuations along the longitudinal and transverse axes of the DNA polyion, respectively. The fact that the relaxation times and specific dielectric increments of the higher-frequency relaxation have been found to be independent of molecular weightlo is also consistent with this view. %1)2935
This work was supported by Research Grant GM 10906 from the United States Public Health Service.
References 1. Allgen, L. G. (1950) Acta Physiol. Scand. 22, Suppl. 76. 2. Jungner, G., Jungner, I. & Allgen, L. G. (1949) Nature 63,849450. 3. Takashima, S. (1963) J. Mol. Biol. 7,455-467. 4. Takashima, S. (1966) J. Phys. Chem. 70,1372-1380. 5. Takashima, S. (1967) Adu. Chem. Ser. 63,232-252. 6. Takashima, S. (1966) Biopolymers 4,663-676. 7. Takashima, S. (1967) Biopolymers 5,899-913. 8. Pollack, M. (1965) J. Chem. Phys. 43,908-909. 9. Hanss, M. & Bernengo, J. B. (1973) Biopolymers 12,2151-2159. 10. Mandel, M. & van der Touw, F. (1974) in Polyelectrolytes, Selengy, E., Ed., Reidel Dordrecht, pp. 285-300. 11. Eisenberg, H. (1975) J. Polym. Sci. Symp. 51,5748. 12. Inman, R. B. & Jordon, D. 0.(1960) Biochim. Biophys. Acta 42,421-434. 13. McTague, J. P. & Gibbs, J. H. (1967) J. Chem. Phys. 44,4295-4301. 14. Oosawa, F. (1970) Biopolymers 9,677-688. 15. O’Konski, C. T. (1960) J. Phys. Chem. 64,605-619. 16. Schwarz, G. (1962) J. Phys. Chem. 66,2636-2642.
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17. Mandel, M. (1961) Mol. Phys. 4,489-496. 18. Van der Touw, F. & Mandel, M. (1974) Biophys. Chem. 2,21&230. 19. Sakamoto, M., Kanda, H., Hayakawa, R. & Wada, Y. (1976) Biopolymers 15, 879892. 20. Benoit, H. (1951) Ann. Phys. 6,561-609. 21. Haltner, A. H., Jr. (1955) Thesis, University of California, Berkeley, California. 22. Hornick, C. & Weill, G. (1971) Biopolymers, 10,2345-2358. 23. Colson, P., Houssier, C. & Fredericq, E. (1974) Biochim. Biophys. Acta 340, 244261. 24. Grever, J. & de Heij, M. E. (1975) Biopolymers 14,2441-2443. 25. Schwarz, G. (1972) in Dielectric and Related Molecular Processes, Vol. 1, The Chemical Society, London. 26. Berberian, J. G. & Cole, R. H. (1969) Reo. Sci. Instrum. 40 ( 6 ) ,811-817. 27. Tung, M. S. (1973) Thesis, Brown University, Providence, R.I. 28. Hasted, J. B. (1973) in Aqueous Dielectrics, Chapmann and Hall, London, p. 39. 29. Kirkwood, J. G. & Shumaker, J. B. (1952) Proc. Natl. Acad. Sci. U S A 38,855-862. 30. Mandel, M. & Jenard, A. (1963) Trans. Faraday SOC.59,2158-2170. 31. Minakata, A. (1972) Biopolymers 11,1567-1582. 32. Doty, P., Boedtker, H., Fresco, J., Haskelkorn, R. & Litt, M. (1959) Proc. Natl. Acad. Sci. U S A 45,482-499. 33. Gruenwedel, D. W., Hsu, C-H. & Lu, D. S. (1971) Biopolymers 10,47-68. 34. Van der Touw, F. & Mandel, M. (1974) Biophys. Chem. 2,231-241. 35. Gray, H. B., Bloomfield, V. A. & Hearst, J. E. (1967) J. Chem. Phys. 46,1493-1498.
Received December 9,1976 Accepted April 2,1977
VOL. 16, 2653-2669 (1977)
Influence of Temperature and Ionic Strength on the Low-Frequency Dielectric Dispersion of DNA Solutions MING SUNG TUNG,* ROBERT J. MOLINARI, ROBERT H. COLE, and JULIAN H. GIBBS, Brown University, Department of Chemistry, Providence, Rhode Island 02912
Synopsis The dielectric properties of DNA solutions a t low frequencies (5 Hz to 2 kHz) have been measured by means of a four-terminal bridge method utilized to minimize electrode polarization errors. At 24OC native salt-free DNA has a very large specific dielectric increment, At/c = 9.8 X lo6 l/mol and a very low frequency relaxation centered a t 18 Hz. Both the dielectric increment and the relaxation time are greatly decreased by partial heat denaturation a t temperatures above 6OoCor by addition of salt, the effects being much larger for divalent anions. These results are shown to be in qualitative agreement with theoretical treatments of counterion fluctuation polarization by McTague and Gibbs for the equilibrium case and by Mandel for relaxation. The ratio of the relaxation time for the low-frequencyprocess to that previously observed a t much higher frequencies suggests that these relaxations result from counterion fluctuations along the longitudinal and transverse axes of the molecule, respectively.
INTRODUCTION The dielectric properties of DNA solutions have been the subject of several experimental and theoretical inve~tigations,l-'~ but the evidence has been incomplete and interpreted in a variety of ways. Allgen and Junger2 first observed a dispersion, which they attributed to a transverse permanent dipole moment (i.e., one at right angles to the helix axis), in the vicinity of 100 kHz. A symmetrical rise and decay of birefringence on the application and removal of a biasing field, observed by Benoit20and confirmed by Haltner,21 was attributed by them to an induced dipole. Recently, Hornick and Wei11,22Colson et al.,23and Greve and DeHeij2*have arrived at a similar conclusion from consideration of the birefringence of sonicated DNA. In dielectric measurements at frequencies above 50 Hz T a k a ~ h i m a ~ - ~ found a low-frequency relaxation with an amplitude and a characteristic frequency strongly dependent on molecular weight. He eventually interpreted his results in terms of induced ionic polarization. On the other hand, Hanss and Berengog concluded that the dna polarizability, measured as a conductivity dispersion in the very low frequency range (0.5 Hz to 1kHz), * Present address: National Bureau of Standards, Washington, D.C. 20234. 2653 0 1977 by John Wiley & Sons, Inc.
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SUNG TUNG ET AL.
could be explained only by a permanent ionic dipole. McTague and Gibbs,l3 treating the polyion as a linear array of counterion adsorption sites, were able to qualitatively describe the low-frequency dielectric behavior by means of a counterion polarization model. Mandel and van der TouwlO showed that two separate dielectric dispersion regions exist in solutions containing linear polyelectrolytes such as DNA. They attributed the higher-frequency dispersion, in which both the mean relaxation time and the dielectric increment are independent of molecular weight, to fluctuations in the distribution of bound counterions along short rigid sections of the molecules joined to each other by loosely jointed flaws in the helix structure. Recently, Sakamoto et al.l9 assumed that counterion fluctuations along the DNA helix were slower than the overall rotational diffusion of the DNA. They then concluded that the low-frequency dielectric relaxation corresponded to a rotational relaxation of an ionic dipole, the origin of which they attributed to counterion fluctuations that were “permanent” on the time scale of the faster rotational diffusion. Those authors ignored, however, any explanation of the high-freqency dispersion in their model. These and other studies (for a general review see S c h w a r ~provide ~~) a variety of evidence and conclusions. It seems clear, however, that two distinct polarizations and corresponding dispersion regions exist in DNA solutions. The one at radio frequencies is better characterized experimentally than the apparently much larger one at audio or subaudio frequencies. Neither a longitudinal nor a transverse permanent moment in the DNA molecule is a likely explanation, in particular for the largeamplitude, low-frequency process. (The antiparallel arrangement of the nucleotide chains in the helix would cause cancellation of most permanent longitudinal moments.) Alternative explanations utilizing induced counterion polarization effects have not been fully established. In the interest of better characterizationof the low-frequencydispersion, we have made studies of aqueous solutions of DNA under various conditions of concentration and added salt to frequenciesas low as 5 Hz. In addition, the temperature dependence df the real part of the dielectric dispersion in solutions of native salt-,fFee DNA was measured. These studies were made possible by use of a special four-terminal, low-frequency bridge, developed in this lab by Berberian and Cole,26which greatly reduced the errors introduced by electrode polarization effects. The only other study of DNA made with a four-terminal bridge technique, that of Sakamoto et al.,19 utilized a bridge design which was virtually identical to the original Cole-Berberian four-terminal bridge26but which employed a much more sophisticated detection system on the bridge output. Whereas their study was limited to only one species of added salt and two concentrations of of DNA, we report here more extensive results covering the effects on dielectric behavior of degree of helicity, species, valency, and concentration of added salt, temperature, and a wide range of salt-free DNA concentrations. The results of both studies are consistent with predictions of the
DIELECTRIC DISPERSION OF DNA SOLUTIONS
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static model of McTague and Gibbs,13 which includes counterion nearest-neighbor interactions, and a modified relaxation model of van der Touw and Mandel.18
EXPERIMENTAL Three samples of calf-thymus DNA (as the sodium salt) were used in the work. The first (NaDNA-S3),from Sigma Chemical Co., was stated to have an average molecular weight 1.3-1.5 X lo6 as supplied. The second (DNA-M1, M4, M5), from Miles Laboratory, Inc., had a relative viscosity of 1.88 at a concentration 0.5 mg/ml and at 25OC, corresponding to an average molecular weight of 2 X lo6. The third, 2-001, from Cal-biochem, had a weight-average molecular weight of 9 X lo6 as supplied. Doubly-distilled water with specific conductance less than 8 X mho cm-l and freed of COa by boiling for 5 min was used for the preparation of DNA solutions of concentration 3 mg/ml (by weight of dried sample) and of pH about 6.7. The concentration expressed as molarity is referred to phosphorus. Two samples M-4 and M-5 were dialyzed against COz freedistilled water for 72 hr at 4°C for the reduction of salt concentration. The stock solutions were diluted just before the measurements, which were completed in a few hours. Denaturation was accomplished by the heating of solutions for 15-17 min a t temperatures from 64 to 97OC, followed by a rapid cooling to 4OC. Measurements were made at 24°C. Most of the admittance measurements were made with a special lowfrequency bridge developed by Berberian and Cole26for determination of parallel capacitance and conductance a t frequencies from 2 kHz to 1Hz. An essential feature of the instrument for the present work was its ability to make four-terminal guarded electrode measurements largely free of electrode polarization error^.^^,^^ The measuring cell for the guarded four-terminal measurements was the same as in Refs. 26 and 27, except that the probe cell constant, Cg, equaled 0.0529 pf. A different cell (C, = 0.0438 pf) was used for the temperaturedependence studies. The dielectric increment, Ad, of a DNA solution was obtained as the difference of capacitances of DNA and matching KCl solutions at a given frequency, divided by the geometric cell capacitance C, in pf. The dielectric loss, At”, was obtained from the relation At” = (G - Go)/27rfCg, where G and Go are the conductance of the solution at a frequency, f , and the dc conductances, respectively. The measured conductances were in the range 8-30 pmho, making measurements of capacitanbes on the order of pf inaccurate below 5 Hz as a consequence of the enormously larger conduction currents. Conductances were measured to 1Hz, and the 1 Hz value was taken as the dc value, Go, for calculation of A&’. Some measurements to determine the absence of a noticeable dispersion
SUNG TUNG ET AL.
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Fig. 1. Dielectric increment dispersion 24'C of aqueous NaDNA-S3 solution at conc = 2.94 10-4M. (1) Native: ( 0 )24"C, f = 25 Hz,T = 6.37 msec; (2) heated to 64°C for 15min: (v) :.64OC, f = 28 Hz,T = 5.7; (3) heated to 84% for 16 min: (X) 84OC, f = 50 Hz,T = 3.18; (4) 97"C, f = 800 Hz,7 = 0.2. heated to 97OC for 17 min: (0) X
in the range 1-100 kHz were made with a conventional three-terminal cell and a transformer bridge developed in this laboratory for direct reading of parallel capacitances and conductances.
RESULTS Effect of the Thermally Induced Helix-Coil Transition Dispersion curves of A d vs frequency at 24°C are shown in Fig. 1for a solution of native NaDNA-S3 and for solutions partly denatured by maintenance for 15-17 min a t temperatures of 64,84, and 97°C. It is evident from these results that the dielectric increment is still frequency dependent even a t 5 Hz for all but the completely denatured samples. Complex plane plots of At" vs At' in Fig. 2 are circular arcs within experimental error. When extrapolated to zero frequency they give estimates of the static dielectric increment, AcO, and of the relaxation time, 7 , corresponding to the midpoint of the dispersion. The quantities r and ACOare plotted in Fig. 3 against the temperature to which the samples were heated. Salt Effects Effects of added mono- and divalent salts on the dielectric increment are shown in Fig. 4 (for samples M4 and M5). Marked decreases in relaxation times, 7 , and, as is better shown by the complex plane plots in Fig. 5, static dielectric increments, AEO,were observed with an increase in salt concentration. The changes in the dielectric properties with concentration of added mono- and divalent salts are shown in Fig. 6 and Table I. The
DIELECTRIC DISPERSION OF DNA SOLUTIONS 300
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Fig. 2. Cole-Cole plots for NaDNA-S3 (24OC) heated to different temperatures: temperature, 24OC, ( 0 )heated a t 84"C, (u) heated a t 64"C, (A)heated a t 97OC. 1100 1000
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t-c Fig. 3. Dielectric increment and relaxation vs temperature of heating for NaDNA-S3. (-) = dielectric increment, (- - -) = relaxation time.
effect of divalent ions was much stronger than the effect of monovalent ions; the specific dielectric increment (DNA conc. = 4 X 10-4N) decreased from almost lo7,when no salt was added, to very near zero when the normality of Mg was 1.5 X 10-N. While the charge and concentration of the counterion species strongly affected the dielectric behavior, the size of the counterion had little or no effect (Fig. 4).
++
Temperature Dependence Figure 10 shows the dispersion curve of a DNA solution 2-001 (DNA concentration = 6.2 X lOV4N)a t a number of different temperatures (t =
SUNG TUNG ET AL.
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(b) Fig. 4. (a) Dielectric increment dispersion of DNA-M4 (conc. = 3.75 X 10-4M) with added monovalent salt. Curve 1: no salt added; X: CsCl = 4 X lO-5N; 0: NaCl = 4 X 10-5N. Curve 2: CsCl = 1.6 X lO-4N. (b) Dielectric increment dispersion of DNA-M5 with added salts. Curve 1: DNA-MI (conc. = 3.75 X lo-%) MgClz (conc. = 1.5 X 0: DNA conc. = 4.69 X 10e4N,no salt added; X: MgClz conc. = 0.75 X W 4 N ;0 : BaClz conc. = 0.75 X 10-4N, A: NaCl conc. = 5 X 10-4N.
+
5,10,15,20, and 25OC). The dielectric constant of the solution increased with increases in temperature. Figure 11 shows that the temperature dependence of the DNA dielectric constant did indeed extend throughout the entire dispersion region and was not, for example, a simple displacement to higher frequencies of a dispersion curve with constant or decreasing amplitude a t higher temperatures. Since no data were available on the temperature dependence of the various electrode-blocking polarizations at such low frequencies, the well-known temperature dependences of the dielectric properties of water and matched-conductance KC1 solutions were studied in order to mimic the electric properties of the polyelectrolyte solution in the absence of a dispersion. Figures 12 and 13 show the measured
DIELECTRIC DISPERSION OF DNA SOLUTIONS
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(b) Fig. 5. Typical Cole-Cole plot for salt effect. (a) DNA-M4 (conc. = 3.75 X 10-4M) with monovalent salt: ( 0 )no salt added, ( X ) NaCl = 4 X 10-5N. (b) DNA-M5 (conc. = 4.69 x 10-4M) with divalent salt: ( 0 )no salt added, ( X ) BaClz (conc. 0.75 X lO-4N).
temperature dependence of the low-frequency dielectric behavior of KC1 solutions and pure water over the frequency range of interest. The static dielectric constants thus obtained are compared in Table I1 with the expected values from the literature.28 Both the direction and magnitude of the published temperature dependences of the static dielectric constant have been reproduced by our measurements. TABLE I Dielectric Properties of DNA with and without Salts Sample
DNA-M4
M, x 10-6 Conc. x lo4 (molfl) Salt conc.
0
3500 8.8 Conductivity (mho cm-l) 14.05 (24°C) Root-mean-square endto-end length ACL
T L (msec)
DNA-M5
NaDNA-S3
2 2 3.75 4.69 1.6 x 1 0 - 4 ~1.5 x 1 0 - 4 ~ 5 x 1 0 - 4 ~ CSCl MgClz NaCl 1300 60 610 3.2 0.23 1 29.54 24.26
1040 6.1 26.03
3430 8,
2830A
3430A
1.4 2.94
2660
v)
SUNG TUNG ET AL.
NORMALITY OF A W E D SALT/ NORMALITY OF DNA
Fig. 6. Dielectric increment and relaxationtime vs the ratio of the concentration of added mono- and divalent salts to that of DNA. ( 0 )monovalent ion, ( X ) divalent ion.
CONC. x
lo4
Fig. 7. Specific dielectric increment and charge fraction vs DNA concentration.
DISCUSSION AND CONCLUSIONS The results of our measurements to frequencies of 5 Hz show that DNA solutions have huge dielectric increments and very long relaxation times, with values that are very dependent on the DNA dimensions and the chemical environment. For native salt-free DNA a t a concentration of 0.934 X 10-3M, the static dielectric increment is 9200 and the relaxation frequency is 18 Hz. The accepted molecular structure of DNA, that is, two oppositely directed backbones with the base pairs perpendicular to a helix axis, cannot possess a permanent dipole moment. Therefore, the large observed increases in dielectric constant at low frequency must be explained in some
DIELECTRIC DISPERSION OF DNA SOLUTIONS
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Fig. 8. Dielectric increment of diluted DNA (conc. = 3.75 X 10-5M) vs time.
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Fig. 9. Dielectric increment dispersion of DNA with and without dialysis. Curve 1: DNA-M5 (conc. = 4.69 X lO-*M) dialysis for 72 hr. Curve 2: DNA-M1 (conc. = 8.48 X 10-4M) without dialysis.
other way. It should also be pointed out that neither the strong effect of added salt on the dielectric constant nor the temperature dependence can be explained by the assumption of a permanent dipole moment. It is therefore necessary to consider interpretations in which the low-frequency dielectric dispersion of DNA is attributed to induced ionic polarization. Several different models have been proposed for explanations of the dielectric dispersion as an effect of ionic p o l a r i z a t i ~ n . ~ ~O’Konski15 ~J~-~~,~~ formulated a model with a frequency-independent surface conductivity. His model predicted a much smaller dielectric increment and a much shorter relaxation time than our experiments indicate and was open to the objection that effects of chemical potential gradients of ion concentrations were not treated. The theories of Schwarz16and Takashimas considered the properties of the inner counterion layer in terms of a complex surface conductivity, but failed to take into account the interaction between
SUNG TUNG ET AL.
2662 1000
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Fig. 11. DNA dielectric increment vs temperatureat frequencies spanning the entire relaxation region.
nearest-neighbor counterions and thus were unable to explain the observations by Mande130and Minakata3l that the dielectric increment of certain polyelectrolytes went through a maximum on addition of salt. Since experiments have shown that the dielectric properties are strongly influenced by counterions, we shall discuss our results in the framework of those theories in which the distribution and interaction of the counterions are emphasized, i.e., the theory of McTague and Gibbsl3 (hereafter M-G) for static polarization and the relaxation theory due to Mandel.17 In these theories three important factors which determine the dielectric dispersion are: the mean-square end-bend length of the polyelectrolyte,the amount of salt added, and the number and valency of the bound counterions.13J7
.
DIELECTRIC DISPERSION OF DNA SOLUTIONS
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Fig. 12. Temperature dependence of H20 dielectric constant. 120
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Fig. 13. Temperature dependence of KCl solution (2 X IW4N)dielectric constant.
Oosawal* has also derived a dynamic theory with a relaxation equation similar to that of Mandel. The following discussion summarizes along with other phenomena experimentally observed effects of variations of these three quantities on the dielectric properties.
Magnitude of Dielectric Increments Assuming that DNA with an average molecular weight of 2 X lo6 has a root-mean-square end-to-end length of 3430 A (a contour length of 10,400 A and a persistence length of 600 A was used1'), and assuming random orientation of the DNA rod, then the M-G theory gives a value of 3.8 X lo6 l/mol for Adc, in fair agreement with the measured value of 9.8 X 106 l/mol for M-5 a t a concentration of 0.93 X 10-3M.
SUNG TUNG ET AL.
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TABLE I1 Calibrations of Temperature Dependence t("C)
co (meas.)
co (lit.)
G (conductance)
84.1 80.4 76.7
10.84 15.35 16.19
KC1 Temperature Dependence 10 20 30
83.8 80.6 16.4
HzO Temperature Dependence 10 20 30
84. I 80.4 76.7
82.0 79.4 76.9
Effect of the Thermally Induced Helix-Coil Transition The dielectric increment and relaxation time of a DNA solution are decreased by heating of the sample as shown in Fig. 3. The behavior is similar to that of the degree of helicity as a function of maximum heating temperature, in low-ionic-strength solutions, as determined from intrinsic viscosity measurements.32 Thus, dielectric dispersion can be used for the detection of the helix-coil transition in DNA solutions without the application of physical forces that may degrade the DNA. The mean-square end-to-end lengths of DNA molecules decrease in the transition from the rigid helical rod to the flexible random coil, with a resultant decrease in the dielectric increment and associated relaxation time. The results reported here are similar to those obtained by Takashima? but our dielectric increments and relaxation times are considerably larger. This discrepancy is believed to be due to the difference in the method used for measurement of the dielectric properties. Takashima was limited to measurements above a frequency of 50 Hz by his reliance on the technique of electrode spacing variation to correct for electrode polarization effects. He reported that the profile of the dielectric dispersion at an intermediate stage of the helix-coil transition was different from that of both helical DNA and randomly coiled DNA (i.e., the low-frequency dispersion appeared to be split in two). He attributed this split to the presence of an intermediate form of DNA. Our study shows no such peculiarity in the profile of the dielectric properties of partly denatured DNA, but rather a smooth change from helix properties to coil properties. Salt Effects Figure 6 (plotted as A& vs the ratio of normalities of added salt to that of DNA) shows that the dielectric increment decreases with the addition of salt. These experimental effects are in qualitative agreement with Sakamoto et al.19 Smaller end-to-end lengths, due to the decrease in electrostatic repulsion between DNA segments which accompanies increases of ionic strength, can cause dielectric changes in the observed di-
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rection. Sakamato et al.19 assert that the salt effect is explained primarily by this electrostatic chain contraction. Two phenomena, however, support our belief that the dramatic decrease in the dielectric increment with the addition of salt must be attributed mostly to changes in the magnitude of the polarization induced among the bound counterions. The first is that the observed decreases in the dielectric constant are much larger than those which would be expected from the decrease in end-to-end lengths derived from electrostatic considerations. The hydrodynamic length of DNA was reported to decrease by 25% when the concentration of sodium chloride was varied from to 10-3M,7yet the relaxation time decreased by a factor of 8.8 when the concentration of sodium chloride was increased by 5 X lO-*M. The M-G theory predicts such a dramatic decrease in the counterion polarization if the fraction of occupied counterion sites on the polyion is greater than one-half. Sakamoto et al.19 attribute their observed decreases in specific dielectric increment with increases in ionic strength to the above-mentioned chain contraction. They derive from their own results, however, that the fraction of sites occupied on the polyion is approximately two-thirds, i.e., exactly the case where the counterion polarization would be expected to decrease without invoking chain contraction. The second phenomenon is that divalent ions yield a much larger effect than monovalent ones, a circumstance readily attributable to a much stronger bonding of the divalent ions to the DNA.
Concentration Effects (Salt-Free DNA) Figure 7 shows that the specific dielectric increment increases as the concentration of salt-free DNA is reduced. The increase is interpreted as due to the decrease in the number of bound ions, i.e., the concomitant increase in the fraction of sites unoccupied upon dilution of salt-free DNA. This concentration effect and the salt effect discussed above indicate that a decrease in the number of bound counterions increases the dielectric increment. The above is an effect yielded theoretically by the M-G theory which also predicts that the dielectric increment will pass through a maximum on further decrease in the number of bound ions and drop toward zero as the number of bound ions approaches zero (a charge fraction of 1). The fact that DNA denatures more readily when the charge fraction is increased prevents the detection of this maximum in DNA solutions by preventing the study of salt-free DNA solutions at low concentrations. The denaturation of salt-free DNA on dilution has been reported12 and was indeed observed a t the lowest concentrations in this study in the time dependence of the dielectric increment (see Fig. 8). The lowest concentrations of DNA used in this study represent the low concentration limit for which DNA can be stable in salt-free solutions, and are in qualitative agreement with room-temperature values of Gruenwedel and Lu33for that limit. Much lower concentrations have been reached in solutions stabilized by added
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SUNG TUNG ET AL.
salts; see, for example, Sakamoto et al.,19 but the added salt itself drastically changes the ion environment of the DNA in solution and hence the dielectric properties. Extrapolation of the inverse specific dielectric increment to zero concentration (suggested by van der Touw and MandeP4) for various synthetic aqueous polyelectrolytes did not give a straight line for our salt-free solutions of native DNA, as shown in Fig. 7.
Temperature Effects Figure 10 shows that the dielectric increment of DNA solutions increases with temperature, and Fig. 11shows that it does so proportionately. It is important to make two observations. The first is that this behavior of DNA solutions is opposite to the temperature dependences of the static dielectric constant of the solvent (water) and of various matching-conductance aqueous solutions containing electrolytes but no DNA. The second is that the increase in dielectric constant with temperature is consistent with the M-G static model of counterion polarization. This model predicts a maximum in the dielectric increment, at a charge fraction equal to one-half, with the dielectric increment going to zero for a charge fraction of both one (no bound ions) and zero (all counterion sites bound). Since denaturation12 at high charge fractions prevents observation of this maximum in DNA solutions (see Concentration Effects), we are limited in our experimental observations to charge fractions somewhere between zero and presumably one-half. An increase in temperature must give rise to an equilibrium state in which fewer counterions are bound to the polyion. In the case of a charge fraction between zero and one-half, this decrease in the number of bound counterions associated with an increase in temperature will, according to the M-G model, cause the counterion polarization and dielectric increment to increase as observed. The observed temperature dependence would contradict any mechanism of relaxation through rotation of permanent dipoles.
Dielectric Relaxation Time The low-frequency relaxation of DNA in aqueous solutions differs considerably in all the cases we have studied from the Debye behavior predicted by various simple models. This difference is seen in the complex plane plots, Figs. 2 and 5, as large deviations from a semicircular locus of E’ vs 6’’. Within the uncertainties of the data, these plots are fitted by the circular arc function proposed by Cole and Cole to describe relaxations of a variety of dielectrics, with the parameter a (equal to zero for the Debye behavior) of order 0.36-0.44 for native and partially denatured DNA solutions and of order 0.5 for solutions with added salts. These values imply broad distributions of relaxation times if the behavior is described in such terms. We believe, however, that the deviations are preferably taken as indicative
DIELECTRIC DISPERSION OF DNA SOLUTIONS
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of cooperative interaction effects, and not necessarily or even desirably as alternatively described. None of the interpretations of the low-frequency relaxation so far proposed provide, to the extent they have been developed, an explanation of the observed form of the relaxation. Their possible validity can only be considered qualitatively on other grounds. The modified Maxwell-Wagner heterogeneous polarization model proposed by Pollacks has been shown by Takashima5 to be unsatisfactory for DNA solutions. Another explanation by Hanss and Bernengog attributes the effects to the orientation and transport of ionic dipoles on the basis that their observed relaxation times for conductance dispersion are of the same order of magnitude as the expected values for molecular reorientations. Although their observations are not inconsistent with ours, extensive r e s ~ l t s on ~ ~electric - ~ ~ birefringence of DNA solutions show that induced rather than permanent dipole forces are associated with the molecular reorientation. We therefore consider a third, alternative, interpretation in terms of induced counterion polarizations. Models based on mobile counterions bound to negative charge sites have been considered by several a ~ t h o r s ~ ~ and J ~ result J ~ J ~in a longitudinal relaxation time, 7,proportional to the mean-square length, L2,and the reciprocal of the counterion mobility, u. After neglecting counterioncounterion interactions, Mandel17derived the formula: 7
= L2/12ukt = al/ur(ze)2
(1)
where a1 is the polarizability of the polyion, r is the linear counterion density, and ze is the charge on the ion. In the absence of an independent value of u along the polyelectrolyte surface, absolute values of 7 for comparison with experiment cannot be calculated. The prediction of Eq. (1) that T should be proportional to a1 and hence to At/c is consistent with the observed increase of T with Adc. This dependence could, however, easily result from a variety of models (see, for example, Ref. 25). A further difficulty in attempting to identify observed relaxation times with ones calculated on the basis of a mobile counterion model is that the observed relaxation can result from a combination of this process, relaxation by reorientation of the polyion molecular axis (i.e., rotational diffusion), and relaxation by exchange of free and bound counterions. Mandel and van der TouwlO have shown that if these processes are assumed to operate independently, with an exponential decay of correlation for each, the observed rate of decay (the reciprocal of the relaxation time) would be the sum of the rates for the separate processes. Only if one of these is considered faster than the others can it be identified with the observed rate of decay. More generally, the experimental dipole correlation function is the product of the functions for each mechanism of charge displacement, if these mechanisms operate independently. Mandel and van der TouwlO commented on various possibilities for the relative rates of these mechanisms and showed that the success of their model in predicting two relaxations
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for DNA depended on the counterion fluctuation along small subunits of the polyion being faster than rotation of the molecule. Sakomoto et al.I9 noted an order of magnitude agreement between relaxation times calculated from viscosity measurements and the observed dielectric relaxation times. They then assumed that the low-frequency dielectric relaxation corresponded to a rotational relaxation that was faster than counterion fluctuations, although they noted that experimental and theoretical studies of counterion exchange rates either did not agree or were inconclusive. We see no method of theoretically determining in an unambiguous way which of these mechanisms would dominate. We have already suggested that the large derivations of the observed relaxation from ideal Debye behavior (with an exponential decay of correlations) may well be the result of counterion interaction effects. In that case, further progress in interpretation of the low-frequencyrelaxation will require a proper dynamical treatment of such effects, as for example, by a generalization of the equilibrium theory of McTague and Gibbs.13 One further point of interest can be made: namely, that the ratio of the lowfrequency relaxation time, ca. 8.8 X sec, to the high-frequency relaxsec, reported by Mandel and van der TouwlO is 7.3 ation time, 1.2 X X lo4. The ratio of the mean-square end-to-end length (3430 to the square of the hydrated diameter (26 for our DNA samples is 1.7 X lo4. This agreement suggests that the two relaxation processes and times are associated with counterion fluctuations along the longitudinal and transverse axes of the DNA polyion, respectively. The fact that the relaxation times and specific dielectric increments of the higher-frequency relaxation have been found to be independent of molecular weightlo is also consistent with this view. %1)2935
This work was supported by Research Grant GM 10906 from the United States Public Health Service.
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17. Mandel, M. (1961) Mol. Phys. 4,489-496. 18. Van der Touw, F. & Mandel, M. (1974) Biophys. Chem. 2,21&230. 19. Sakamoto, M., Kanda, H., Hayakawa, R. & Wada, Y. (1976) Biopolymers 15, 879892. 20. Benoit, H. (1951) Ann. Phys. 6,561-609. 21. Haltner, A. H., Jr. (1955) Thesis, University of California, Berkeley, California. 22. Hornick, C. & Weill, G. (1971) Biopolymers, 10,2345-2358. 23. Colson, P., Houssier, C. & Fredericq, E. (1974) Biochim. Biophys. Acta 340, 244261. 24. Grever, J. & de Heij, M. E. (1975) Biopolymers 14,2441-2443. 25. Schwarz, G. (1972) in Dielectric and Related Molecular Processes, Vol. 1, The Chemical Society, London. 26. Berberian, J. G. & Cole, R. H. (1969) Reo. Sci. Instrum. 40 ( 6 ) ,811-817. 27. Tung, M. S. (1973) Thesis, Brown University, Providence, R.I. 28. Hasted, J. B. (1973) in Aqueous Dielectrics, Chapmann and Hall, London, p. 39. 29. Kirkwood, J. G. & Shumaker, J. B. (1952) Proc. Natl. Acad. Sci. U S A 38,855-862. 30. Mandel, M. & Jenard, A. (1963) Trans. Faraday SOC.59,2158-2170. 31. Minakata, A. (1972) Biopolymers 11,1567-1582. 32. Doty, P., Boedtker, H., Fresco, J., Haskelkorn, R. & Litt, M. (1959) Proc. Natl. Acad. Sci. U S A 45,482-499. 33. Gruenwedel, D. W., Hsu, C-H. & Lu, D. S. (1971) Biopolymers 10,47-68. 34. Van der Touw, F. & Mandel, M. (1974) Biophys. Chem. 2,231-241. 35. Gray, H. B., Bloomfield, V. A. & Hearst, J. E. (1967) J. Chem. Phys. 46,1493-1498.
Received December 9,1976 Accepted April 2,1977
