O n the Mechanism of Dielectric Relaxation in Aqueous DNA Solutions 6. SAIF, R.
K. MOHR, C. J. MONTROSE, and T. A. LlTOVlTZ
Vitreow %ate Laboratory, Catholic University of America, Washington, DC 20064
SYNOPSIS
The complex dielectric response of calf thymus DNA in aqueous saline solutions has been measured from 1 MHz to 1 GHz. The results are presented in terms of the relaxation of the incremental contributions to the permittivity and conductivity from the condensed counterions surrounding the DNA molecules. Measurements of the low-frequency conductivity of the samples also lends support to the condensed counterion int,erpretation.
INTRODUCTION Understanding the mechanism of the coupling of electromagnetic energy to DNA molecules, as well as the effect of this coupling on the biochemical functioning of DNA in living systems, is of great interest and considerable biological importance. The dielectric response of DNA in solution is characterized by relaxations in a t least three frequency regions. One relaxation (called the a relaxation) occurs in the low-frequency region below a few kilohertz. It is characterized by a large dielectric increment whose magnitude is dependent on the size of the DNA molecule. A second relaxation ( t h e 6 relaxation) characterized by a much smaller, molecular weight independent, dielectric increment lies in the frequency region roughly from 1 MHz to 1 GHz. I t is the main subject of this paper. The last ( t h e y relaxation) occurs a t high frequencies (above 1 G H z ) and is attributed to the reorientation of dipolar water molecules. DNA molecules in aqueous salt solutions have negatively charged sites a t the phosphate groups. These sites are neutralized by positive counterions ( Na ' , M g2+ , etc.) in the solution, ensuring overall charge neutrality. The nature of the binding between these positive ions and the negative sites on the DNA molecule has been a subject of debate, but in recent years the idea of nonlocalized binding' of these ions
to the negative sites on DNA has been generally accepted. Nonlocalized binding implies that the positive ions can move relatively freely on the surface of the DNA molecule and in the surrounding hydration shell. These ions generally do not leave the immediate vicinity of the DNA molecules, but since they are fully hydrated, their mobility can reasonably be expected to be about the same as that for free ions. A reasonable model for the configuration of a DNA molecule in solutions of low to moderate salinity (which includes the 0-90-mN range of the present study) and near neutral p H is that of a worm-like chain,'-3 that is, a chain made of a sequence of mechanically rigid sections (subunits) that are connected a t fixed angles. Such a chain will exhibit a stiffness that is intermediate between those of a rigid rod and a freely joined chain. It is accepted that DNA molecules possess no permanent dipoles moments4; however, when a n external field is applied to charged molecules such as DNA a moment is induced, arising from the migration of ions over individual subunits and over the length of the molecule. The resulting induced dipole experiences a torque that tends to align it with the external field. The dipole formation and alignment processes require a finite time to occur. and thus give rise to the observed dielectric relaxat ions. Mandel2I5has experimentally studied both the cy and 6 relaxations of DNA solutions. He has proposed that both relaxations have their origin in the movement of ions around the DNA molecule. He argued that, since the a relaxation is dependent on molecular weight (i.e., the length of the molecule), this 1171
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SAIF ET AL.
relaxation reflects the migration of counterions over the entire dimension of the DNA molecule; conversely, since the 6 relaxation is independent of the molecular weight, it is reasonable to suppose that it arises from the migration of ions over a n identifiable segment of the DNA molecule. Natural candidates for the segments are the subunits associated with the wormlike chain model of DNA. Although Mandel reported this work some time ago, the origin of the 6 relaxation is still the subject of debate. In addition to the various condensed counterion models that have been proposed, there are several other possible explanations of the relaxation. Recently, Grosse' suggested that, since DNA molecules in a saline solution present a heterogeneous mixture, a Maxwell-Wagner type polarization model could account for the 6 relaxation. In this approach the relaxation of the bulk dielectric susceptibility arises from the charging of the interfaces within the material. Grant' has speculated that the 6 relaxation is due to the motion of "bound" water molecules, which are strongly held by the DNA. The dynamical response of these to a n external electric field can be expected to be slower than unbound water molecules for which the relaxation occurs above 10 GHz. No theoretical calculations have been offered to quantify this hypothesis, so that it remains simply a suggestion. T o investigate the nature of the 6 relaxation in sufficient detail so that the reasonableness of the various models can be assessed, the dielectric properties of DNA solutions were measured in the frequency range 1 MHz t o 1 GHz. The results were analyzed in terms of the measured permittivity and conductivity increments relative to the DNA-free solvent.
METHODS Sample Preparation
Calf thymus DNA was purchased from Sigma Chemical Company. Ultrapure type D 4764 or preparations purified by standard phenol extraction8 from type D 4522 were used. Both are supplied as highly polymerized DNA. The type D 4522 DNA on which most of our measurements were made was sheared prior t o purification to reduce the average molecular weight and lower the solution viscosity. Although a wide range of DNA concentrations were examined in preliminary measurements, the results we report here are for concentrations near 6 mg/ mL as determined by uv absorption. This concen-
tration is relatively high but is necessary to obtain dielectric and conductivity increments large enough to measure with high accuracy. The NaCl concentrations of the solutions were fixed by dialysis for two to four days against known concentrations of salt. The salt concentrations were zero, 30 m M , and 90 m M added salt. The added salt does not include counterions bound to the DNA and not removed by dialysis against deionized water. Dielectric Measurements
Characterizing the 6 relaxation for calf thymus DNA requires measurement of its complex dielectric response over the frequency range of 1MHz to 1 GHz. In addition, the low-frequency ( 100-Hz ) conductivity of the samples was measured. The measurements were all made a t 25 k 0.1"C using a circulating water bath to control the temperature of the sample and measuring cell. T h e 100-Hz conductivity measurements were made with a commercial conductivity probe (Beckman CEL-G50-Y184) capable of measuring less than 150 pL of sample. The measurements up to 40 MHz were made with a parallel plate capacitance cell having platinum black electrodes similar in design to cells used by K. Foster (University of Pennsylvania, private communication). Above 45 MHz open-ended coaxial probes as described by Stuchlyg were employed. The measurements require the use of two instruments. The low-frequency range ( 100 Hz to 40 MHz) was covered by a Hewlett-Packard 4194A impedance analyzer. T h e high-frequency range (45 MHz to 1 GHz) was covered by a Hewlett-Packard 8510B network analyzer. Standard calibration methods were employed for each instrument. The effects of DNA on the dielectric behavior of aqueous salt solutions are relatively small in the 6 relaxation frequency regime. In order to isolate the effects of DNA and its associated counterions, we assumed that the various contributions to the relative permittivity t and conductivity u were simply additive as explained in more detail below. We determined increments for c and CJ by measuring the properties of the bulk solvent with and without DNA, and taking the difference. This procedure has the advantage of making the effects of DNA visible, and also of partially correcting certain calibration errors and neglected stray impedances.13 Since the errors are nearly the same in both the sample solution and the solvent, they are reduced by the subtraction. Although the two instruments used employ very different methods to extract the dielectric behavior from the measured parameters, there is rel-
MECHANISM OF DIELECTRIC RELAXATION
atively little discrepancy in the data in going from one instrument to the other. This is rather remarkable since the dielectric and conductivity increments are relatively small compared to the solvent values. The agreement between the two instruments increases our confidence in the results.
RESULTS low-Frequency Conductivity Measurements
A set of experiments was performed on DNA solutions having a DNA concentration of 1 g / L and dialyzed against a series of saline solutions. The first in the series was a DNA solution dialyzed against deionized water (typical conductivity less than 1 X 10 .‘s/m) for seven days. After this period (during which the dialysis solution was changed frequently), it was expected that the concentration of free bulk ions in the solution would be negligible; any remaining cations should be bound to the DNA. Nevertheless, a conductivity of 0.005 S / m (equivalent to that for a NaCl concentration of 0.5 m N ) was measured. T o complete the series the DNA was dialyzed against saline solutions of 0.0015,0.005, and 0.015 normalities, respectively. After each dialysis, conductivities of the DNA and saline solutions were measured (see Table I ) . It was found that in each case the conductivity of the DNA solution increased to a value equal t o that of the dialysis solution plus 0.005 k 0.001 S / m . As is shown below, this excess conductivity is a consequence of the counterion condensation. Dielectric Relaxation Data
Dielectric relaxation measurements were made on three different DNA solutions over a frequency range of 1 MHz to 1 GHz. These solutions contained calf thymus DNA, NaC1, and water. The solutions differed in the concentrations of both the DNA and
the salt. The solutions were the following: (1)a 5.7 g / L DNA solution dialyzed against deionized water for 4 days, yielding a nominal bulk normality of 0 mN; ( 2 ) a 6.1 g / L DNA solution dialyzed for 2 days against a 30 mN NaCl salt solution, yielding a nominal bulk normality of 30 mN; and ( 3 ) a 3.9 g / L DNA solution dialyzed for 2 days against a 90 mN NaCl salt solution, yielding a bulk normality of 90 mN. To analyze the relaxation data we assumed that the conductivity contribution of each entity in the solution is additive. Thus, the total conductivity of the DNA solution a t angular frequency w [ a( o)]is given by
where u D , ac, g B , and uDNA are the conductivity contributions of the diffuse counterions, condensed counterions, bulk (i.e., added salt) ions, and the DNA polyions, respectively; uWD is the dipolar conductivity contribution of the water molecules. If, as the data below show, the relaxing part of the total conductivity A a ( w ) is related t o motion of the condensed ions (i.e., frequency-dependent condensed ion conductivity), then Aa( w ) = ( T ~ To . extract A a ( o)from the measured values of a( o),values of UB UD were obtained by assuming that a t 1 MHz the condensed counterions and water dipoles polarize but do not conduct, and thus CT(, and uWD can be neglected:
+
The values of uB, UD , and CTDNAcan be taken as constant over the range 1MHz to 1 GHz. A t frequencies above 1 MHz, A g ( w ) was obtained by subtracting from a( w ) the conductivity, a( 1 M H z ) , and the measured values of u W D (of course corrected for the excluded volume caused by the presence of the D N A ) . The expression
Ao(w) Table I Conductivity of Various Solutions at 100 Hz (Units are Siemens per Meter)
Conductivity of Dialyzing Solutions
Conductivity of Dialyzed Solutions
Excess 100 Hz Conductivity
0 0.01 5 0.050 0.150
0.005 0.020 0.054 0.156
0.005 0.005 0.004 0.006
1173
= U(U)
-
~ (M lH z ) - (1 - V ) ( T ~ D ( 3 )
was used to obtain the relaxing part of the ion conductivity. T h e quantity V is the volume fraction of the sample occupied by the DNA. This equation presumes that the dipolar contribution of the water is not affected by the presence of the DNA molecules in solution. Using Eq. ( 3 ) we obtained values of Aa( o)for each of the three DNA solutions. These are plotted in Figure 1. Following a line of logic similar to that in the conductivity discussion above, an expression for
1174
SAIF ET AL.
At( w ) ,the dielectric increment of the DNA solution (i.e., the relaxing component of the real part of the dielectric permittivity), can be written a s
and At = 0 a t this frequency. The dielectric increment of the DNA solutions in the range from 1MHz to 1 GHz can then be written as At( W
where t is the permittivity of the DNA solutions and tWL) is the dielectric permittivity of water in the presence of DNA and salt. Since the relaxation frequency of water in DNA solutions is still of the order of 10 GHz or higher, t & D is constant over the range of our measurements. The assumption above that the dipolar water contribution to the conductivity is not much affected by the presence of the DNA turned out to be quite reasonable. However, the data a t 1 GHz clearly showed that DNA lowers the real part of the dielectric permittivity of the aqueous salt solution. The value o f t a t 1 GHz is less than that of a salt water solution of the same salinity (even after correction for the excluded volume due to the presence of the D N A ) . This is consistent with the notion that some of the water is affected by the DNA.7i'4 T o obtain a n estimate of the value of tW1) in the presence of the DNA, we assumed (after analyzing the frequency-dependent conductivity data) that by 1 GHz the condensed ion relaxation had effectively reached its limiting values (i.e., 07 % 1). Thus t a t 1 GHz is given by t ( 1 GHz)
=
(1 -
V)twl)
(5)
FREQUENCY (Hz)
Figure 1. Frequency dependence of the conductivity increment for calf thymus DNA solutions containing 0 ( 0 ), 3 0 ( A ) , and 90 (0) m M added NaCI. The 30 and 90 m M curves have been shifted by 0.01 and 0.02 S / m respectively for clarity. The data points shown are typical, and represent a fraction of the data taken and used to obtain least-squares fits to Eq. (8).The fits are represented by the solid curves.
)
= t ( 0)
- t(
1GHz)
(6)
T h e conductivity data were fit to several commonly used functions. First, an attempt was made to describe the data with a single Debye relaxation process. T h e resulting fit was rather poor as was that resulting from a superposition of two Debye processes. However a superposition of three Debye processes yielded an excellent fit. The relevant formulas are
and
where the At, and 7 , ( i = 1, 2, 3 ) refer to the ith relaxation strength and relaxation time, respectively. As is evident from Figure 1, the least-squares fit of the A u data to Eq. ( 8 )is excellent. T h e relaxation parameters determined by the fitting procedure were then used to calculate the dielectric increment over the same frequency range using Eq. ( 7 ) . The comparison of the calculated curve and experimental data are shown in Figure 2. Above 8 MHz the fit is quite good. Below 8 MHz significant departures from the predictions of Eq. ( 8 ) occur. These departures are consistent with the existence of the low-frequency N relaxation reported by Mandel' and Takashima,lSamong others. The effects of a tail of the a relaxation are to be expected in both the Au and A6 data. However, these effects are negligible in the A u data. Since Au'? = 2.rrf,,toAt,, ( t h e subscript N refers to the dielectric parameters associated with the cy relaxation). Using fee = 500 Hz and At,, = 60,000, gives Au,, x S / m , which is several orders of magnitude less than either ffl,u)k or .la6 measured in this work. Since A u ( w ) data should be insensitive to the tail of the N relaxation, it was used for the fit. Table I1 presents the fitting parameters determined for each sample. T h e results have been normalized to the values corresponding to a sample concentration of 1 g / L DNA. It can be argued that the good fit obtained is simply a reflection of the
MECHANISM OF DIELECTRIC RELAXATION
-
and
3
2
1175
tt
28
-Oo 00
where At is the total dielectric increment for the 6 relaxation and G ( l n T ) is the relaxation time distribution function. I n this case the log normal distribution given by G(ln7) 1o7
106
FREQUENCY
1o9
108
(Hz)
Figure 2. Frequency dependence of the dielectric increment for calf thymus DNA solutions containing 0 (0), 30 (A) , and 90 (0)m M added NaCl. The 0 and 30 m M curves have been shifted by 16 and 8 units, respectively, for clarity. The data points shown are typical and represent a fraction of the points taken. The solid curves are generated using Eq. (7) and the fitting parameters obtained in the fits of the conductivity data to Eq. (8).
relatively large number of adjustable parameters used. For this reason we have also fitted the data using a distribution of relaxation times. In this case the equations take the form
=
a -exp
G
has been employed where the parameter ro is the most probable relaxation time, and the quantity a determines the width of the distribution of relaxation times and is related to the standard deviation of the distribution C by the relation a = ( \/2c) Comparisons of the data with the predictions of Eqs. ( 9 ) - ( 11) are virtually indistinguishable from those for three Debyes and are not shown here. The parameters giving the best fit and their standard deviations are listed in Table 111. Again the parameters have been normalized for a concentration of 1 g / L DNA.
DISCUSSION Several models have been proposed to explain the 6 relaxation. These include a modified MaxwellWagner model,6 a number of models based on the
Table I1 Summary of Parameters Used in Fitting Relaxation Data to a Sum of Three Debye Relaxation Functions*
f2
(MHz) (MHz)
f3
(MHz)
fi
All
At? A13
Au, X lo4 (S/m) Au2 x i04 (S/m) Au3 X lo4 (S/m) At Au x 104 (S/m) fa", (MHz)
0 mN Na-DNA (5.7 g/L)
30 mN Na-DNA (6.1 g/L)
90 mN Na-DNA (3.9 g/L)
7 i 2.4 33 i 4.6 200 1- 8.9 0.88 f 0.12 0.64 f 0.12 0.19 i 0.01 3.4 i 1.2 11.7 f 2.7 20.8 i 1.8 1.94 i 0.17 39.9 3.5 33.3 t 2.0
7 f 1.1 54 f 7 300 i 13.5 1.70 f 0.14 0.45 -t 0.05 0.21 f 0.01 6.6 f 1.1 13.4 f 2.3 35.5 -t 2.9 2.55 ? 0.15 58.9 f 3.9 39.2 t 1.8
7 t 2.1 56 f 6.8 400 f 12.5 1.20 k 0.21 0.53 f 0.07 0.22 i 0.01 4.7 f 1.6 16.6 f 3.0 48.2 k 2.7 2.23 t 0.22 74.1 i 4.4 55.9 f 2.3
*
* The values of Ac and A u have been normlaized to 1 g/L. The uncertainties represent one standard deviation for the least-squares fit parameters. Thee favK is the average of the three frequencies weighted by the Ac associated with each.
1176
SAIF ET AL.
Table I11 Summary of Parameters Used in Fitting Relaxation Data to a Log Normal Distribution of Relaxation Times"
DNA Sample Ae fo
(MW
a
x lo8 ( s ) A m x lo4 (S/m) To
0 mN (5.7 g/L)
30 mN (6.1 g/L)
1.95 k 0.05 12.3 f 0.9 0.47 k 0.01 1.28 f 0.09 40 f 4
2.23 0.15 12.8 k 2.6 0.41 0.02 1.24 i 0.24 65 i 11
+
*
90 mN (3.9 g/L) 1.79 21 0.41 0.76 85
k 0.15 2 5.8 f 0.03 f 0.21 2 10
a Uncertainties represent one standard deviation of the least squares fit parameters.
concept of condensed counterions, and models relating to bound water. As will be shown below, the condensed counterion model is the most successful in predicting the basic features of the observed 6 relaxation. The Maxwell-Wagner approach is most directly evaluated by using the calculation due to Grosse.' He treats the DNA molecules as long rigid cylinders that have a thin (compared to their radius) conductive layer around them. Using the MaxwellWagner mixture formulas, he derives expressions for the conductivity increment and the dielectric increment of the 6 relaxation in the limit u1 % C J , and ern & tp. Here u1 and u, are the conductivities of the conducting counterion layer and the bulk medium, respectively, and t, and tp are the dielectric permittivities of medium and DNA molecule, respectively. These limits are applicable t o the aqueous DNA systems reported here. With these assumptions Grosse predicts a single relaxation with amplitude
close to that in the bulk. This leads to a conductivity of 6.2 S / m . It can readily be seen that the Grosse model yields values in rather poor agreement with experiment. In searching for possible reasons for this it was noted that Grosse assumed the thickness of the counterion layer is small compared with the radius of the DNA molecule. This in fact is not the case: the radius of the double-stranded DNA is 10 A and the thickness of the layer around DNA is 7 A.' Consequently, we modified his model taking into account the finite thickness of the layer. T h e modified equations are
'35~17
At
8 Vt, 3
=-
and a relaxation frequency f c given by
fc
Using these modified equations, the parameters were recalculated. Comparing the modified values (also shown in Table IV) with the measured ones, one finds that the values for Au are comparable, but that the values for At and fc are still rather different. The reasonable agreement of the Au values appears fortuitous: Au is proportional to the product of At and f c , the former of which comes out much too high while fc is much too low and the errors tend to compensate. We conclude that a Maxwell-Wagner mechanism is not the explanation for the observed 6 relaxation. If one wishes to consider a counterion relaxation mechanism, many models are available, including one due to Mande15 and another to Manning.'Il6 The models have similarities but Manning's model includes a n expression that can be used to calculate the relaxation amplitude and its ionic strength dependence. Mandel's model was interesting because it suggested three possible mechanisms that could contribute to the 6 relaxation and that might correspond to the fit to three Debye processes. For these reasons aspects of both models are used in the analysis. Table IV Maxwell-Wagner Theory Predictions: Comparison of Theory and Experiment
ff1
=-
Zxtot,
In Table IV we compare Grosse's predicted values with experimental results for the 5.7 g / L DNA solution. The value of ul used in the calculations was obtained from the reported' concentration of Na' ions in the counterion layer ( 1 . 2 M ) and using the assumption that the ion mobility in the layer was
Au X
lo4 (S/m)
A6
fc ( G W
Grosse Theory
Grosse Modified
Measured Values
165 .21 1.4
27 0.035 1.4
1.9 80"
40
a This is the frequency a t which Ao(w) has reached f its limiting value.
MECHANISM OF DIELECTRIC RELAXATION
Manning assumes that a polyelectrolyte molecule in an ionic solution is a linear array of discrete charge sites that are surrounded by a cloud of positive ( N a ) counterions. Above a certain critical value of the linear charge density of the polyion, the model predicts that a fraction of the counterions will condense into a thin layer surrounding the DNA. The number of condensed N a + ions is less than the number of charge sites. This "counterion condensation" is related to the existence of an electrostatic potential that decreases logarithmically as one moves outward from the surface of the polyion. The critical value of the charge density of the polyion for condensation is determined in Manning's model by the quantity (, the ratio of the electrostatic energy (due to interaction of the polyion sites and the condensed counterions) to the thermal energy. The condition for condensation is +
[=-
'1
t,kTb
21
where q is the protonic charge, t , is the bulk dielectric constant for the solvent, k is Boltzmann's constant, 7'is the temperature, and b is the distance between the charge sites. The fraction of the counterions that condense 4 is given by the expression
where Z is the valence of the counterion. For BDNA in a n aqueous solution ( a s in the case here), ( = 4.8. When the solution contains only one species of' small cation such as Na', Eq. (18) predicts a value of 4 equal to 0.76. This value has been confirmed by nmr measurements.' In the Manning model the ions in an aqueous solution of DNA can be classified into three groups: ( 1) condensed counterions, ( 2 ) diffuse counterions, and ( 3 ) bulk ions. The condensed counterions neutralize a fraction of the negatively charged phosphate groups in the DNA backbone. They are hydrated and have a mobility similar to that in the bulk but their movement is on average restricted to a cylindrical volume around the DNA. This type of binding of condensed counterions and phosphate groups in DNA is called delocalized binding. The diffuse counterions are responsible for neutralizing the rest of the DNA charge. T h e number density of the diffuse counterions decreases exponentially with the distance from the DNA axis. The bulk ions (sometimes called the added salt) behave as ions in ordinary aqueous solution.
11 77
Using the condensed ion model, Manning has derived a n expression for the contribution of condensed ions to the electrostatic polarizability of polyions such as DNA.'' H e first writes the free energy of the condensed ion system, taking into account free energies due to (1) the electrostatic interaction of charge sites gelin the presence of excess simple salt, ( 2 ) mixing of condensed counterions with the rest of the ions in the solution g,, and ( 3 ) the interaction of the external field with the condensed counterions gex.The last term in the free energy expression for the system is given by
where E is the external electric field and 1 is a distance coordinate along the polyion axis with respect to a n arbitrary origin. A t equilibrium, the total free energy, gtotal= gel g, g,, , is a minimum and thus its derivative with respect to 1 must vanish. From this condition, Manning obtains the charge distribution in the presence of the external electric field which leads to a n expression for the polarizability of the system. Manning's expression for N , ,,the polarizability of the condensed counterion layer along a direction parallel to the DNA axis, is
+ +
=
AZ 'q'nL' 12kT
where L is the length of the DNA parallel to the external field and n is the number of counterions condensed around the polyion, the value of which is given by
a n d the factor A , which is related to the ionic strength of the solution, is
where K, = (0.329 k ' )l 1.I 2 where 1 is the ionic strength of the solution. We now compare the predictions of the Manning model with the experimental data. In order to account for the dielectric increment due to the 6 relaxation, we follow Mande15 and model the DNA molecule as a sequence of rigid rods, the length of which is taken t,o be 300 -t 50 A.This value is the
1178
SAIF ET AL.
segment length for the wormlike chain model of DNA and is related to the persistence length.’*”Using Eq. ( 2 0 ) and the relation
where N is the number of subunits/m3, At, the dielectric increment due to the polarizability of the condensed counterions along the subunits (rigid rods), was calculated. The factor accounts for the random orientation of the rigid rods. T h e conductivity increment was calculated assuming the mobility p of the condensed ions was the same as that of bulk ions:
where pc is the number of condensed counterions per unit volume and again a factor o f f appears due to the random orientation of the rods with respect to the field. One must remember that by using Eq. ( 2 3 ) to calculate the dielectric increment, the assumption is made that the interaction between DNA molecules is negligible. In Table V we see that the measured and calculated dielectric and conductivity increments agree within a factor of 2. This is quite good if one considers the approximations made in the Manning model and the uncertainties in the experimental data. Further, the motion of the ions along the rods in only one of the possible mechanisms that could contribute to the 6 relaxation. T h e ionic strength dependence of the relaxation parameters predicted by Eqs. ( 2 3 ) and ( 2 4 ) is not evident in the experimental data. For example, the Manning approach would predict that a n increase in the bulk ionic strength should cause a n increase in the polarizability and thus At increasing the ionic strength increases the strength of the Debye shielding. This weakens the electrostatic interactions between the fixed negative charge sites and the condensed counterions, and also between the condensed counterions themselves. Therefore a t the higher ionic strengths it is easier for a n external electric field to displace the condensed counterions over distances comparable to a subunit. In Table V the measured and calculated polarizabilities are listed for various ionic strengths. It is evident that, although there is reasonable agreement between the calculated and measured polarizabilities (within a factor of 2 ) , the predicted trends are not obvious. Again we must note that over the range of
salinities occurring in these measurements the predicted change in (Y is only a factor of 2 -t 0.6. The uncertainties in the measured values preclude any serious discussion of the ionic strength dependence. The values of Aucalc do not increase with salinity because it is assumed that neither n’ nor p changes with increased ionic strength. The values of Auexp agree well with the Ancalc at the lowest salinity. However, contrary to the calculations, the experimental values of 0 rise with increasing salinity. Because other experimental data’ show that n’ stays relatively constant over this range of bulk ion concentrations, these results suggest that the mobility of the condensed ions might be increasing with increased salinity. This would not be inconsistent with increased Debye shielding of the condensed ions from the effects of the negative sites on the DNA. Although there are some difficulties, we believe the agreement that does exist between the experimental and theoretical values to be strong evidence that the 6 relaxation is due to condensed counterions. As a further test of the condensed counterion model, we used the concept of condensed and diffuse ions to account for the “excess” dc conductivity measured in the 1g / L calf thymus DNA sample. A conductivity of 0.005 f 0.001 S / m was measured a t 100 Hz after the DNA solution had been dialyzed against deionized water for a week to eliminate excess bulk ions from the sample. From Manning’s model, it is known that the DNA molecules are neutralized by both condensed counterions ( 76 2 10) % and diffuse ions. At frequencies below the 6 relaxation the condensed counterions, because of their restricted motion, do not contribute to conductivity but only to polarization. However, the diffuse counterions and the DNA polyions can contribute to the 100-Hz conductivity. In order to estimate these contributions, we have calculated the normality of the 1 g / L calf thymus DNA (2.86 m N ) and the fraction of this normality, which is neutralized by diffuse ions (0.69 m N ) . This normality corresponds equally t o the diffuse ionic charge concentration and the net concentration of charge from the DNA polyions, i.e., the number of charges per unit volume p is the same for both. T h e resulting Conductivity is
where p N a and p D N A are the mobilities of Na ions and DNA polyions, respectively. In the case of Na ions we have assumed that the mobility is the same as in the bulk. T h e values for DNA are taken from electrophoresis measurements l 7 on very dilute solutions and are of the same order of magnitude a s
MECHANISM OF DIELECTRIC RELAXATION
1179
Table V Dielectric Parameters from the Manning Model: Comparison of Experiment and Theory
k x ,
Atcalc
allelp X 10"" (cou12 m/N) allca~c X (cod2m / N ) Anexl'X lo4 (S/m) Auc,lc x lo4 (S/m)
0 m N Na-DNA (5.7 g/L)
30 mN Na-DNA (6.1 g/L)
90 mN Na-DNA (3.9 g/L)
1.94 i 0.17 1.0 i 0.3 5.3 +- 0.5 2.7 +- 0.9 39.9 i 3.5 38 -t 4
2.55 k 0.15 1.5 i 0.5 6.9 i 0.4 3.9 i 1.4 58.9 3.9 38 i- 4
2.23 i 0.22 1.9 i 0.6 6.1 t 0.6 5.1 t 1.8 74.1 i 4.4 38 i 4
*
In calculating t h e quantities in this table t h e following parameters were used: length of rigid subunit = 300 50 A; number of subunits/(g. m3) = 3.8 X 10"; Na ion mobility = 5.4 X lo-*m'/(V. s). T h e ionic strength was t h a t of th e hulk ions.
*
the sodium mobility. The actual DNA mobilities in our experiments may be lower since the DNA concentrations were relatively high, increasing the viscosity and lowering the mobility. The measured mobilities, however, allow a n estimate of the upper limit of the contribution of DNA to the low-frequency conductivity. Note that here we do not use the factor f since the ions are free to move in the direction of the field. If we assume further that no bulk ions are left after one week of dialysis, the d c ( o r 100 H z ) conductivity is due only to the diffuse ions and DNA, and using Eq. ( 2 5 ) is found to be 0.007 +- 0.001 S / m for a DNA concentration of 1 g/L. This is in excellent agreement with the measured value. Similarly, the excess conductivity seen in the remaining solutions in Table I is explained by the diffuse ions and DNA polyions. It was mentioned above that several mechanisms5 have been suggested that could account for the 6 relaxation. These mechanisms include the ( 1) orientational motion of a subunit, ( 2 ) migration of condensed ions over a subunit, ( 3 ) motion of condensed ions perpendicular to a subunit, '' and ( 4 ) fluctuations in the concentration of condensed ions with the bulk and diffuse ions. While some or all of these probably play a role in the relaxation, neither the models nor the data are sufficiently detailed to allow u s to make more than general conclusions. In the vase of the migration of ions over a subunit, we have calculated the permittivity and conductivity increments, and found that this mechanism could explain most of what we observe. For this case, as suggested by Mandel,* one can estimate the magnitude of the expected relaxation time by calculating the diffusion time or the associated relaxation frequency for a counterion along a subunit of the DNA,
-
Using the values for the parameters given previously yields f R 2 MHz, which is a t about the same order of magnitude as that predicted for the largest relaxation found in the three Debye fit. Considering the crudeness of the model, this is about as good as one can expect to do. It is probably not fruitful to speculate further as to relaxation times or contributions to the magnitude of the relaxation from other possible mechanisms. As Mandel and Odijk I" concluded, it will require a significant amount of both theoretical and experimental work to explain fully the kinetics of the dielectric behavior of large polyelectrolytes such as DNA.
SUMMARY AND CONCLUSIONS T h e goal of this research was to explore the mechanism of the dielectric dispersion in solutions of DNA. Dielectric measurements were made over the range from 100 Hz to 1GHz. The data were analyzed by assuming that DNA has two types of counterions, diffuse and condensed. Analysis of the data using the Manning model of condensation has shown that the diffuse ions and the DNA polyions account for the low-frequency conductivity of D h A solutions (i.e., with zero added salt or the conductivity in excess of that caused by added s a l t ) . At high frequencies (1 MHz to 1 G H z ) , dielectric dispersion was observed. Using Manning's model of polarizability of condensed ions, we were able to reasonably account for the observed conductivity and dielectric increments of the high-frequency relaxation ( e.g., within a factor of two). We believe this work shows that the high-frequency dispersion observed in DNA solutions is related to the displacement of condensed ions over a subunit of length of the DNA molecule. The relaxation frequency is not addressed by the
1180
SAIF ET AL.
Manning model and thus we have not been able to account for its magnitude.
This work was supported in part by contract number DAMD17-86-6260 with the Walter Reed Army Institute of Research United States Army Medical Command. That support is gratefully acknowledged by the authors.
REFERENCES 1. Manning, G. S. (1978) Quarto Rev. Biophys. 11,179246. 2. Mandel, M. (1977) Ann. N Y Acad. Sci. 303, 74-87. 3. Bloomfield, V. A., Crothers, D. M. & Tinoco, Jr., I., (1974) Physical Chemistry of Nucleic Acids, Harper & Row, New York. 4. Pethig, R. & Kell, B. K. Phys. Med. Biol. (1987) 32, 933-970. 5 . van der Touw, F. & Mandel, M. ( 1974) Biophys. Chem. 2, 218-230. 6. Grosse, C. (1989) Alta Frequenza 58, 365. 7. Grant, E. H., Sheppard, R. J . & South, G. P. (1978) Dielectric Behaviour of Biological Molecules in Solution, Oxford University Press, Oxford.
8. Davis, L. G., Dibner, M. D. & Battey, J. F. (1986) Molecular Biology, Elsevier, New York. 9. Athey, T. W., Stuchly, M. A. & Stuchly, S. S. (1982) IEEE Tran. Microwave Th. Tech. MTT-30, 82. 10. Kraszewski, A., Stuchly, M. A. & Stuchly, S. S. (1983) IEEE Trans. Instrum. Meas. IM-32,385-386. 11. Hewlett-Packard Company. (1985) H P 8510 Network Analyzer System Operating and Programming Manual, Santa Rosa, CA. 12. Yokagawa-Hewlett-Packard, Ltd. ( 1986) Model 4194A Impedance GainlPhase Analyzer Operation Manual, Tokyo. 13. Misra, D., Chabbra, M., Epstein, B. R., Mirotznik, M. & Foster, K. ( 1990) IEEE Tran. Microwave Th. Tech. 38,8. 14. Takashima, S., Gabriel, C., Sheppard, R. J. & Grant, E. H. (1984) Biophys. J . 46, 29-34. 15. Takashima, S. (1967) Biopolymers 5 , 899-913. 16. Manning, G. S. (1978) Biophys. Chem. 9, 65-70. 17. Manning, G. S . (1981) J . Phys. Chem. 85,1506-1515. 18. de Xammar Oro, J. R. & Grigera, J. R. (1984) Biopolymers 23, 1457-1463. 19. Mandel, M. & Odijk, T. (1984) Ann. Rev. Phys. Chem. 35, 75-108.
Received September 11, 1990 Accepted June 10, 1991
K. MOHR, C. J. MONTROSE, and T. A. LlTOVlTZ
Vitreow %ate Laboratory, Catholic University of America, Washington, DC 20064
SYNOPSIS
The complex dielectric response of calf thymus DNA in aqueous saline solutions has been measured from 1 MHz to 1 GHz. The results are presented in terms of the relaxation of the incremental contributions to the permittivity and conductivity from the condensed counterions surrounding the DNA molecules. Measurements of the low-frequency conductivity of the samples also lends support to the condensed counterion int,erpretation.
INTRODUCTION Understanding the mechanism of the coupling of electromagnetic energy to DNA molecules, as well as the effect of this coupling on the biochemical functioning of DNA in living systems, is of great interest and considerable biological importance. The dielectric response of DNA in solution is characterized by relaxations in a t least three frequency regions. One relaxation (called the a relaxation) occurs in the low-frequency region below a few kilohertz. It is characterized by a large dielectric increment whose magnitude is dependent on the size of the DNA molecule. A second relaxation ( t h e 6 relaxation) characterized by a much smaller, molecular weight independent, dielectric increment lies in the frequency region roughly from 1 MHz to 1 GHz. I t is the main subject of this paper. The last ( t h e y relaxation) occurs a t high frequencies (above 1 G H z ) and is attributed to the reorientation of dipolar water molecules. DNA molecules in aqueous salt solutions have negatively charged sites a t the phosphate groups. These sites are neutralized by positive counterions ( Na ' , M g2+ , etc.) in the solution, ensuring overall charge neutrality. The nature of the binding between these positive ions and the negative sites on the DNA molecule has been a subject of debate, but in recent years the idea of nonlocalized binding' of these ions
to the negative sites on DNA has been generally accepted. Nonlocalized binding implies that the positive ions can move relatively freely on the surface of the DNA molecule and in the surrounding hydration shell. These ions generally do not leave the immediate vicinity of the DNA molecules, but since they are fully hydrated, their mobility can reasonably be expected to be about the same as that for free ions. A reasonable model for the configuration of a DNA molecule in solutions of low to moderate salinity (which includes the 0-90-mN range of the present study) and near neutral p H is that of a worm-like chain,'-3 that is, a chain made of a sequence of mechanically rigid sections (subunits) that are connected a t fixed angles. Such a chain will exhibit a stiffness that is intermediate between those of a rigid rod and a freely joined chain. It is accepted that DNA molecules possess no permanent dipoles moments4; however, when a n external field is applied to charged molecules such as DNA a moment is induced, arising from the migration of ions over individual subunits and over the length of the molecule. The resulting induced dipole experiences a torque that tends to align it with the external field. The dipole formation and alignment processes require a finite time to occur. and thus give rise to the observed dielectric relaxat ions. Mandel2I5has experimentally studied both the cy and 6 relaxations of DNA solutions. He has proposed that both relaxations have their origin in the movement of ions around the DNA molecule. He argued that, since the a relaxation is dependent on molecular weight (i.e., the length of the molecule), this 1171
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SAIF ET AL.
relaxation reflects the migration of counterions over the entire dimension of the DNA molecule; conversely, since the 6 relaxation is independent of the molecular weight, it is reasonable to suppose that it arises from the migration of ions over a n identifiable segment of the DNA molecule. Natural candidates for the segments are the subunits associated with the wormlike chain model of DNA. Although Mandel reported this work some time ago, the origin of the 6 relaxation is still the subject of debate. In addition to the various condensed counterion models that have been proposed, there are several other possible explanations of the relaxation. Recently, Grosse' suggested that, since DNA molecules in a saline solution present a heterogeneous mixture, a Maxwell-Wagner type polarization model could account for the 6 relaxation. In this approach the relaxation of the bulk dielectric susceptibility arises from the charging of the interfaces within the material. Grant' has speculated that the 6 relaxation is due to the motion of "bound" water molecules, which are strongly held by the DNA. The dynamical response of these to a n external electric field can be expected to be slower than unbound water molecules for which the relaxation occurs above 10 GHz. No theoretical calculations have been offered to quantify this hypothesis, so that it remains simply a suggestion. T o investigate the nature of the 6 relaxation in sufficient detail so that the reasonableness of the various models can be assessed, the dielectric properties of DNA solutions were measured in the frequency range 1 MHz t o 1 GHz. The results were analyzed in terms of the measured permittivity and conductivity increments relative to the DNA-free solvent.
METHODS Sample Preparation
Calf thymus DNA was purchased from Sigma Chemical Company. Ultrapure type D 4764 or preparations purified by standard phenol extraction8 from type D 4522 were used. Both are supplied as highly polymerized DNA. The type D 4522 DNA on which most of our measurements were made was sheared prior t o purification to reduce the average molecular weight and lower the solution viscosity. Although a wide range of DNA concentrations were examined in preliminary measurements, the results we report here are for concentrations near 6 mg/ mL as determined by uv absorption. This concen-
tration is relatively high but is necessary to obtain dielectric and conductivity increments large enough to measure with high accuracy. The NaCl concentrations of the solutions were fixed by dialysis for two to four days against known concentrations of salt. The salt concentrations were zero, 30 m M , and 90 m M added salt. The added salt does not include counterions bound to the DNA and not removed by dialysis against deionized water. Dielectric Measurements
Characterizing the 6 relaxation for calf thymus DNA requires measurement of its complex dielectric response over the frequency range of 1MHz to 1 GHz. In addition, the low-frequency ( 100-Hz ) conductivity of the samples was measured. The measurements were all made a t 25 k 0.1"C using a circulating water bath to control the temperature of the sample and measuring cell. T h e 100-Hz conductivity measurements were made with a commercial conductivity probe (Beckman CEL-G50-Y184) capable of measuring less than 150 pL of sample. The measurements up to 40 MHz were made with a parallel plate capacitance cell having platinum black electrodes similar in design to cells used by K. Foster (University of Pennsylvania, private communication). Above 45 MHz open-ended coaxial probes as described by Stuchlyg were employed. The measurements require the use of two instruments. The low-frequency range ( 100 Hz to 40 MHz) was covered by a Hewlett-Packard 4194A impedance analyzer. T h e high-frequency range (45 MHz to 1 GHz) was covered by a Hewlett-Packard 8510B network analyzer. Standard calibration methods were employed for each instrument. The effects of DNA on the dielectric behavior of aqueous salt solutions are relatively small in the 6 relaxation frequency regime. In order to isolate the effects of DNA and its associated counterions, we assumed that the various contributions to the relative permittivity t and conductivity u were simply additive as explained in more detail below. We determined increments for c and CJ by measuring the properties of the bulk solvent with and without DNA, and taking the difference. This procedure has the advantage of making the effects of DNA visible, and also of partially correcting certain calibration errors and neglected stray impedances.13 Since the errors are nearly the same in both the sample solution and the solvent, they are reduced by the subtraction. Although the two instruments used employ very different methods to extract the dielectric behavior from the measured parameters, there is rel-
MECHANISM OF DIELECTRIC RELAXATION
atively little discrepancy in the data in going from one instrument to the other. This is rather remarkable since the dielectric and conductivity increments are relatively small compared to the solvent values. The agreement between the two instruments increases our confidence in the results.
RESULTS low-Frequency Conductivity Measurements
A set of experiments was performed on DNA solutions having a DNA concentration of 1 g / L and dialyzed against a series of saline solutions. The first in the series was a DNA solution dialyzed against deionized water (typical conductivity less than 1 X 10 .‘s/m) for seven days. After this period (during which the dialysis solution was changed frequently), it was expected that the concentration of free bulk ions in the solution would be negligible; any remaining cations should be bound to the DNA. Nevertheless, a conductivity of 0.005 S / m (equivalent to that for a NaCl concentration of 0.5 m N ) was measured. T o complete the series the DNA was dialyzed against saline solutions of 0.0015,0.005, and 0.015 normalities, respectively. After each dialysis, conductivities of the DNA and saline solutions were measured (see Table I ) . It was found that in each case the conductivity of the DNA solution increased to a value equal t o that of the dialysis solution plus 0.005 k 0.001 S / m . As is shown below, this excess conductivity is a consequence of the counterion condensation. Dielectric Relaxation Data
Dielectric relaxation measurements were made on three different DNA solutions over a frequency range of 1 MHz to 1 GHz. These solutions contained calf thymus DNA, NaC1, and water. The solutions differed in the concentrations of both the DNA and
the salt. The solutions were the following: (1)a 5.7 g / L DNA solution dialyzed against deionized water for 4 days, yielding a nominal bulk normality of 0 mN; ( 2 ) a 6.1 g / L DNA solution dialyzed for 2 days against a 30 mN NaCl salt solution, yielding a nominal bulk normality of 30 mN; and ( 3 ) a 3.9 g / L DNA solution dialyzed for 2 days against a 90 mN NaCl salt solution, yielding a bulk normality of 90 mN. To analyze the relaxation data we assumed that the conductivity contribution of each entity in the solution is additive. Thus, the total conductivity of the DNA solution a t angular frequency w [ a( o)]is given by
where u D , ac, g B , and uDNA are the conductivity contributions of the diffuse counterions, condensed counterions, bulk (i.e., added salt) ions, and the DNA polyions, respectively; uWD is the dipolar conductivity contribution of the water molecules. If, as the data below show, the relaxing part of the total conductivity A a ( w ) is related t o motion of the condensed ions (i.e., frequency-dependent condensed ion conductivity), then Aa( w ) = ( T ~ To . extract A a ( o)from the measured values of a( o),values of UB UD were obtained by assuming that a t 1 MHz the condensed counterions and water dipoles polarize but do not conduct, and thus CT(, and uWD can be neglected:
+
The values of uB, UD , and CTDNAcan be taken as constant over the range 1MHz to 1 GHz. A t frequencies above 1 MHz, A g ( w ) was obtained by subtracting from a( w ) the conductivity, a( 1 M H z ) , and the measured values of u W D (of course corrected for the excluded volume caused by the presence of the D N A ) . The expression
Ao(w) Table I Conductivity of Various Solutions at 100 Hz (Units are Siemens per Meter)
Conductivity of Dialyzing Solutions
Conductivity of Dialyzed Solutions
Excess 100 Hz Conductivity
0 0.01 5 0.050 0.150
0.005 0.020 0.054 0.156
0.005 0.005 0.004 0.006
1173
= U(U)
-
~ (M lH z ) - (1 - V ) ( T ~ D ( 3 )
was used to obtain the relaxing part of the ion conductivity. T h e quantity V is the volume fraction of the sample occupied by the DNA. This equation presumes that the dipolar contribution of the water is not affected by the presence of the DNA molecules in solution. Using Eq. ( 3 ) we obtained values of Aa( o)for each of the three DNA solutions. These are plotted in Figure 1. Following a line of logic similar to that in the conductivity discussion above, an expression for
1174
SAIF ET AL.
At( w ) ,the dielectric increment of the DNA solution (i.e., the relaxing component of the real part of the dielectric permittivity), can be written a s
and At = 0 a t this frequency. The dielectric increment of the DNA solutions in the range from 1MHz to 1 GHz can then be written as At( W
where t is the permittivity of the DNA solutions and tWL) is the dielectric permittivity of water in the presence of DNA and salt. Since the relaxation frequency of water in DNA solutions is still of the order of 10 GHz or higher, t & D is constant over the range of our measurements. The assumption above that the dipolar water contribution to the conductivity is not much affected by the presence of the DNA turned out to be quite reasonable. However, the data a t 1 GHz clearly showed that DNA lowers the real part of the dielectric permittivity of the aqueous salt solution. The value o f t a t 1 GHz is less than that of a salt water solution of the same salinity (even after correction for the excluded volume due to the presence of the D N A ) . This is consistent with the notion that some of the water is affected by the DNA.7i'4 T o obtain a n estimate of the value of tW1) in the presence of the DNA, we assumed (after analyzing the frequency-dependent conductivity data) that by 1 GHz the condensed ion relaxation had effectively reached its limiting values (i.e., 07 % 1). Thus t a t 1 GHz is given by t ( 1 GHz)
=
(1 -
V)twl)
(5)
FREQUENCY (Hz)
Figure 1. Frequency dependence of the conductivity increment for calf thymus DNA solutions containing 0 ( 0 ), 3 0 ( A ) , and 90 (0) m M added NaCI. The 30 and 90 m M curves have been shifted by 0.01 and 0.02 S / m respectively for clarity. The data points shown are typical, and represent a fraction of the data taken and used to obtain least-squares fits to Eq. (8).The fits are represented by the solid curves.
)
= t ( 0)
- t(
1GHz)
(6)
T h e conductivity data were fit to several commonly used functions. First, an attempt was made to describe the data with a single Debye relaxation process. T h e resulting fit was rather poor as was that resulting from a superposition of two Debye processes. However a superposition of three Debye processes yielded an excellent fit. The relevant formulas are
and
where the At, and 7 , ( i = 1, 2, 3 ) refer to the ith relaxation strength and relaxation time, respectively. As is evident from Figure 1, the least-squares fit of the A u data to Eq. ( 8 )is excellent. T h e relaxation parameters determined by the fitting procedure were then used to calculate the dielectric increment over the same frequency range using Eq. ( 7 ) . The comparison of the calculated curve and experimental data are shown in Figure 2. Above 8 MHz the fit is quite good. Below 8 MHz significant departures from the predictions of Eq. ( 8 ) occur. These departures are consistent with the existence of the low-frequency N relaxation reported by Mandel' and Takashima,lSamong others. The effects of a tail of the a relaxation are to be expected in both the Au and A6 data. However, these effects are negligible in the A u data. Since Au'? = 2.rrf,,toAt,, ( t h e subscript N refers to the dielectric parameters associated with the cy relaxation). Using fee = 500 Hz and At,, = 60,000, gives Au,, x S / m , which is several orders of magnitude less than either ffl,u)k or .la6 measured in this work. Since A u ( w ) data should be insensitive to the tail of the N relaxation, it was used for the fit. Table I1 presents the fitting parameters determined for each sample. T h e results have been normalized to the values corresponding to a sample concentration of 1 g / L DNA. It can be argued that the good fit obtained is simply a reflection of the
MECHANISM OF DIELECTRIC RELAXATION
-
and
3
2
1175
tt
28
-Oo 00
where At is the total dielectric increment for the 6 relaxation and G ( l n T ) is the relaxation time distribution function. I n this case the log normal distribution given by G(ln7) 1o7
106
FREQUENCY
1o9
108
(Hz)
Figure 2. Frequency dependence of the dielectric increment for calf thymus DNA solutions containing 0 (0), 30 (A) , and 90 (0)m M added NaCl. The 0 and 30 m M curves have been shifted by 16 and 8 units, respectively, for clarity. The data points shown are typical and represent a fraction of the points taken. The solid curves are generated using Eq. (7) and the fitting parameters obtained in the fits of the conductivity data to Eq. (8).
relatively large number of adjustable parameters used. For this reason we have also fitted the data using a distribution of relaxation times. In this case the equations take the form
=
a -exp
G
has been employed where the parameter ro is the most probable relaxation time, and the quantity a determines the width of the distribution of relaxation times and is related to the standard deviation of the distribution C by the relation a = ( \/2c) Comparisons of the data with the predictions of Eqs. ( 9 ) - ( 11) are virtually indistinguishable from those for three Debyes and are not shown here. The parameters giving the best fit and their standard deviations are listed in Table 111. Again the parameters have been normalized for a concentration of 1 g / L DNA.
DISCUSSION Several models have been proposed to explain the 6 relaxation. These include a modified MaxwellWagner model,6 a number of models based on the
Table I1 Summary of Parameters Used in Fitting Relaxation Data to a Sum of Three Debye Relaxation Functions*
f2
(MHz) (MHz)
f3
(MHz)
fi
All
At? A13
Au, X lo4 (S/m) Au2 x i04 (S/m) Au3 X lo4 (S/m) At Au x 104 (S/m) fa", (MHz)
0 mN Na-DNA (5.7 g/L)
30 mN Na-DNA (6.1 g/L)
90 mN Na-DNA (3.9 g/L)
7 i 2.4 33 i 4.6 200 1- 8.9 0.88 f 0.12 0.64 f 0.12 0.19 i 0.01 3.4 i 1.2 11.7 f 2.7 20.8 i 1.8 1.94 i 0.17 39.9 3.5 33.3 t 2.0
7 f 1.1 54 f 7 300 i 13.5 1.70 f 0.14 0.45 -t 0.05 0.21 f 0.01 6.6 f 1.1 13.4 f 2.3 35.5 -t 2.9 2.55 ? 0.15 58.9 f 3.9 39.2 t 1.8
7 t 2.1 56 f 6.8 400 f 12.5 1.20 k 0.21 0.53 f 0.07 0.22 i 0.01 4.7 f 1.6 16.6 f 3.0 48.2 k 2.7 2.23 t 0.22 74.1 i 4.4 55.9 f 2.3
*
* The values of Ac and A u have been normlaized to 1 g/L. The uncertainties represent one standard deviation for the least-squares fit parameters. Thee favK is the average of the three frequencies weighted by the Ac associated with each.
1176
SAIF ET AL.
Table I11 Summary of Parameters Used in Fitting Relaxation Data to a Log Normal Distribution of Relaxation Times"
DNA Sample Ae fo
(MW
a
x lo8 ( s ) A m x lo4 (S/m) To
0 mN (5.7 g/L)
30 mN (6.1 g/L)
1.95 k 0.05 12.3 f 0.9 0.47 k 0.01 1.28 f 0.09 40 f 4
2.23 0.15 12.8 k 2.6 0.41 0.02 1.24 i 0.24 65 i 11
+
*
90 mN (3.9 g/L) 1.79 21 0.41 0.76 85
k 0.15 2 5.8 f 0.03 f 0.21 2 10
a Uncertainties represent one standard deviation of the least squares fit parameters.
concept of condensed counterions, and models relating to bound water. As will be shown below, the condensed counterion model is the most successful in predicting the basic features of the observed 6 relaxation. The Maxwell-Wagner approach is most directly evaluated by using the calculation due to Grosse.' He treats the DNA molecules as long rigid cylinders that have a thin (compared to their radius) conductive layer around them. Using the MaxwellWagner mixture formulas, he derives expressions for the conductivity increment and the dielectric increment of the 6 relaxation in the limit u1 % C J , and ern & tp. Here u1 and u, are the conductivities of the conducting counterion layer and the bulk medium, respectively, and t, and tp are the dielectric permittivities of medium and DNA molecule, respectively. These limits are applicable t o the aqueous DNA systems reported here. With these assumptions Grosse predicts a single relaxation with amplitude
close to that in the bulk. This leads to a conductivity of 6.2 S / m . It can readily be seen that the Grosse model yields values in rather poor agreement with experiment. In searching for possible reasons for this it was noted that Grosse assumed the thickness of the counterion layer is small compared with the radius of the DNA molecule. This in fact is not the case: the radius of the double-stranded DNA is 10 A and the thickness of the layer around DNA is 7 A.' Consequently, we modified his model taking into account the finite thickness of the layer. T h e modified equations are
'35~17
At
8 Vt, 3
=-
and a relaxation frequency f c given by
fc
Using these modified equations, the parameters were recalculated. Comparing the modified values (also shown in Table IV) with the measured ones, one finds that the values for Au are comparable, but that the values for At and fc are still rather different. The reasonable agreement of the Au values appears fortuitous: Au is proportional to the product of At and f c , the former of which comes out much too high while fc is much too low and the errors tend to compensate. We conclude that a Maxwell-Wagner mechanism is not the explanation for the observed 6 relaxation. If one wishes to consider a counterion relaxation mechanism, many models are available, including one due to Mande15 and another to Manning.'Il6 The models have similarities but Manning's model includes a n expression that can be used to calculate the relaxation amplitude and its ionic strength dependence. Mandel's model was interesting because it suggested three possible mechanisms that could contribute to the 6 relaxation and that might correspond to the fit to three Debye processes. For these reasons aspects of both models are used in the analysis. Table IV Maxwell-Wagner Theory Predictions: Comparison of Theory and Experiment
ff1
=-
Zxtot,
In Table IV we compare Grosse's predicted values with experimental results for the 5.7 g / L DNA solution. The value of ul used in the calculations was obtained from the reported' concentration of Na' ions in the counterion layer ( 1 . 2 M ) and using the assumption that the ion mobility in the layer was
Au X
lo4 (S/m)
A6
fc ( G W
Grosse Theory
Grosse Modified
Measured Values
165 .21 1.4
27 0.035 1.4
1.9 80"
40
a This is the frequency a t which Ao(w) has reached f its limiting value.
MECHANISM OF DIELECTRIC RELAXATION
Manning assumes that a polyelectrolyte molecule in an ionic solution is a linear array of discrete charge sites that are surrounded by a cloud of positive ( N a ) counterions. Above a certain critical value of the linear charge density of the polyion, the model predicts that a fraction of the counterions will condense into a thin layer surrounding the DNA. The number of condensed N a + ions is less than the number of charge sites. This "counterion condensation" is related to the existence of an electrostatic potential that decreases logarithmically as one moves outward from the surface of the polyion. The critical value of the charge density of the polyion for condensation is determined in Manning's model by the quantity (, the ratio of the electrostatic energy (due to interaction of the polyion sites and the condensed counterions) to the thermal energy. The condition for condensation is +
[=-
'1
t,kTb
21
where q is the protonic charge, t , is the bulk dielectric constant for the solvent, k is Boltzmann's constant, 7'is the temperature, and b is the distance between the charge sites. The fraction of the counterions that condense 4 is given by the expression
where Z is the valence of the counterion. For BDNA in a n aqueous solution ( a s in the case here), ( = 4.8. When the solution contains only one species of' small cation such as Na', Eq. (18) predicts a value of 4 equal to 0.76. This value has been confirmed by nmr measurements.' In the Manning model the ions in an aqueous solution of DNA can be classified into three groups: ( 1) condensed counterions, ( 2 ) diffuse counterions, and ( 3 ) bulk ions. The condensed counterions neutralize a fraction of the negatively charged phosphate groups in the DNA backbone. They are hydrated and have a mobility similar to that in the bulk but their movement is on average restricted to a cylindrical volume around the DNA. This type of binding of condensed counterions and phosphate groups in DNA is called delocalized binding. The diffuse counterions are responsible for neutralizing the rest of the DNA charge. T h e number density of the diffuse counterions decreases exponentially with the distance from the DNA axis. The bulk ions (sometimes called the added salt) behave as ions in ordinary aqueous solution.
11 77
Using the condensed ion model, Manning has derived a n expression for the contribution of condensed ions to the electrostatic polarizability of polyions such as DNA.'' H e first writes the free energy of the condensed ion system, taking into account free energies due to (1) the electrostatic interaction of charge sites gelin the presence of excess simple salt, ( 2 ) mixing of condensed counterions with the rest of the ions in the solution g,, and ( 3 ) the interaction of the external field with the condensed counterions gex.The last term in the free energy expression for the system is given by
where E is the external electric field and 1 is a distance coordinate along the polyion axis with respect to a n arbitrary origin. A t equilibrium, the total free energy, gtotal= gel g, g,, , is a minimum and thus its derivative with respect to 1 must vanish. From this condition, Manning obtains the charge distribution in the presence of the external electric field which leads to a n expression for the polarizability of the system. Manning's expression for N , ,,the polarizability of the condensed counterion layer along a direction parallel to the DNA axis, is
+ +
=
AZ 'q'nL' 12kT
where L is the length of the DNA parallel to the external field and n is the number of counterions condensed around the polyion, the value of which is given by
a n d the factor A , which is related to the ionic strength of the solution, is
where K, = (0.329 k ' )l 1.I 2 where 1 is the ionic strength of the solution. We now compare the predictions of the Manning model with the experimental data. In order to account for the dielectric increment due to the 6 relaxation, we follow Mande15 and model the DNA molecule as a sequence of rigid rods, the length of which is taken t,o be 300 -t 50 A.This value is the
1178
SAIF ET AL.
segment length for the wormlike chain model of DNA and is related to the persistence length.’*”Using Eq. ( 2 0 ) and the relation
where N is the number of subunits/m3, At, the dielectric increment due to the polarizability of the condensed counterions along the subunits (rigid rods), was calculated. The factor accounts for the random orientation of the rigid rods. T h e conductivity increment was calculated assuming the mobility p of the condensed ions was the same as that of bulk ions:
where pc is the number of condensed counterions per unit volume and again a factor o f f appears due to the random orientation of the rods with respect to the field. One must remember that by using Eq. ( 2 3 ) to calculate the dielectric increment, the assumption is made that the interaction between DNA molecules is negligible. In Table V we see that the measured and calculated dielectric and conductivity increments agree within a factor of 2. This is quite good if one considers the approximations made in the Manning model and the uncertainties in the experimental data. Further, the motion of the ions along the rods in only one of the possible mechanisms that could contribute to the 6 relaxation. T h e ionic strength dependence of the relaxation parameters predicted by Eqs. ( 2 3 ) and ( 2 4 ) is not evident in the experimental data. For example, the Manning approach would predict that a n increase in the bulk ionic strength should cause a n increase in the polarizability and thus At increasing the ionic strength increases the strength of the Debye shielding. This weakens the electrostatic interactions between the fixed negative charge sites and the condensed counterions, and also between the condensed counterions themselves. Therefore a t the higher ionic strengths it is easier for a n external electric field to displace the condensed counterions over distances comparable to a subunit. In Table V the measured and calculated polarizabilities are listed for various ionic strengths. It is evident that, although there is reasonable agreement between the calculated and measured polarizabilities (within a factor of 2 ) , the predicted trends are not obvious. Again we must note that over the range of
salinities occurring in these measurements the predicted change in (Y is only a factor of 2 -t 0.6. The uncertainties in the measured values preclude any serious discussion of the ionic strength dependence. The values of Aucalc do not increase with salinity because it is assumed that neither n’ nor p changes with increased ionic strength. The values of Auexp agree well with the Ancalc at the lowest salinity. However, contrary to the calculations, the experimental values of 0 rise with increasing salinity. Because other experimental data’ show that n’ stays relatively constant over this range of bulk ion concentrations, these results suggest that the mobility of the condensed ions might be increasing with increased salinity. This would not be inconsistent with increased Debye shielding of the condensed ions from the effects of the negative sites on the DNA. Although there are some difficulties, we believe the agreement that does exist between the experimental and theoretical values to be strong evidence that the 6 relaxation is due to condensed counterions. As a further test of the condensed counterion model, we used the concept of condensed and diffuse ions to account for the “excess” dc conductivity measured in the 1g / L calf thymus DNA sample. A conductivity of 0.005 f 0.001 S / m was measured a t 100 Hz after the DNA solution had been dialyzed against deionized water for a week to eliminate excess bulk ions from the sample. From Manning’s model, it is known that the DNA molecules are neutralized by both condensed counterions ( 76 2 10) % and diffuse ions. At frequencies below the 6 relaxation the condensed counterions, because of their restricted motion, do not contribute to conductivity but only to polarization. However, the diffuse counterions and the DNA polyions can contribute to the 100-Hz conductivity. In order to estimate these contributions, we have calculated the normality of the 1 g / L calf thymus DNA (2.86 m N ) and the fraction of this normality, which is neutralized by diffuse ions (0.69 m N ) . This normality corresponds equally t o the diffuse ionic charge concentration and the net concentration of charge from the DNA polyions, i.e., the number of charges per unit volume p is the same for both. T h e resulting Conductivity is
where p N a and p D N A are the mobilities of Na ions and DNA polyions, respectively. In the case of Na ions we have assumed that the mobility is the same as in the bulk. T h e values for DNA are taken from electrophoresis measurements l 7 on very dilute solutions and are of the same order of magnitude a s
MECHANISM OF DIELECTRIC RELAXATION
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Table V Dielectric Parameters from the Manning Model: Comparison of Experiment and Theory
k x ,
Atcalc
allelp X 10"" (cou12 m/N) allca~c X (cod2m / N ) Anexl'X lo4 (S/m) Auc,lc x lo4 (S/m)
0 m N Na-DNA (5.7 g/L)
30 mN Na-DNA (6.1 g/L)
90 mN Na-DNA (3.9 g/L)
1.94 i 0.17 1.0 i 0.3 5.3 +- 0.5 2.7 +- 0.9 39.9 i 3.5 38 -t 4
2.55 k 0.15 1.5 i 0.5 6.9 i 0.4 3.9 i 1.4 58.9 3.9 38 i- 4
2.23 i 0.22 1.9 i 0.6 6.1 t 0.6 5.1 t 1.8 74.1 i 4.4 38 i 4
*
In calculating t h e quantities in this table t h e following parameters were used: length of rigid subunit = 300 50 A; number of subunits/(g. m3) = 3.8 X 10"; Na ion mobility = 5.4 X lo-*m'/(V. s). T h e ionic strength was t h a t of th e hulk ions.
*
the sodium mobility. The actual DNA mobilities in our experiments may be lower since the DNA concentrations were relatively high, increasing the viscosity and lowering the mobility. The measured mobilities, however, allow a n estimate of the upper limit of the contribution of DNA to the low-frequency conductivity. Note that here we do not use the factor f since the ions are free to move in the direction of the field. If we assume further that no bulk ions are left after one week of dialysis, the d c ( o r 100 H z ) conductivity is due only to the diffuse ions and DNA, and using Eq. ( 2 5 ) is found to be 0.007 +- 0.001 S / m for a DNA concentration of 1 g/L. This is in excellent agreement with the measured value. Similarly, the excess conductivity seen in the remaining solutions in Table I is explained by the diffuse ions and DNA polyions. It was mentioned above that several mechanisms5 have been suggested that could account for the 6 relaxation. These mechanisms include the ( 1) orientational motion of a subunit, ( 2 ) migration of condensed ions over a subunit, ( 3 ) motion of condensed ions perpendicular to a subunit, '' and ( 4 ) fluctuations in the concentration of condensed ions with the bulk and diffuse ions. While some or all of these probably play a role in the relaxation, neither the models nor the data are sufficiently detailed to allow u s to make more than general conclusions. In the vase of the migration of ions over a subunit, we have calculated the permittivity and conductivity increments, and found that this mechanism could explain most of what we observe. For this case, as suggested by Mandel,* one can estimate the magnitude of the expected relaxation time by calculating the diffusion time or the associated relaxation frequency for a counterion along a subunit of the DNA,
-
Using the values for the parameters given previously yields f R 2 MHz, which is a t about the same order of magnitude as that predicted for the largest relaxation found in the three Debye fit. Considering the crudeness of the model, this is about as good as one can expect to do. It is probably not fruitful to speculate further as to relaxation times or contributions to the magnitude of the relaxation from other possible mechanisms. As Mandel and Odijk I" concluded, it will require a significant amount of both theoretical and experimental work to explain fully the kinetics of the dielectric behavior of large polyelectrolytes such as DNA.
SUMMARY AND CONCLUSIONS T h e goal of this research was to explore the mechanism of the dielectric dispersion in solutions of DNA. Dielectric measurements were made over the range from 100 Hz to 1GHz. The data were analyzed by assuming that DNA has two types of counterions, diffuse and condensed. Analysis of the data using the Manning model of condensation has shown that the diffuse ions and the DNA polyions account for the low-frequency conductivity of D h A solutions (i.e., with zero added salt or the conductivity in excess of that caused by added s a l t ) . At high frequencies (1 MHz to 1 G H z ) , dielectric dispersion was observed. Using Manning's model of polarizability of condensed ions, we were able to reasonably account for the observed conductivity and dielectric increments of the high-frequency relaxation ( e.g., within a factor of two). We believe this work shows that the high-frequency dispersion observed in DNA solutions is related to the displacement of condensed ions over a subunit of length of the DNA molecule. The relaxation frequency is not addressed by the
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SAIF ET AL.
Manning model and thus we have not been able to account for its magnitude.
This work was supported in part by contract number DAMD17-86-6260 with the Walter Reed Army Institute of Research United States Army Medical Command. That support is gratefully acknowledged by the authors.
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8. Davis, L. G., Dibner, M. D. & Battey, J. F. (1986) Molecular Biology, Elsevier, New York. 9. Athey, T. W., Stuchly, M. A. & Stuchly, S. S. (1982) IEEE Tran. Microwave Th. Tech. MTT-30, 82. 10. Kraszewski, A., Stuchly, M. A. & Stuchly, S. S. (1983) IEEE Trans. Instrum. Meas. IM-32,385-386. 11. Hewlett-Packard Company. (1985) H P 8510 Network Analyzer System Operating and Programming Manual, Santa Rosa, CA. 12. Yokagawa-Hewlett-Packard, Ltd. ( 1986) Model 4194A Impedance GainlPhase Analyzer Operation Manual, Tokyo. 13. Misra, D., Chabbra, M., Epstein, B. R., Mirotznik, M. & Foster, K. ( 1990) IEEE Tran. Microwave Th. Tech. 38,8. 14. Takashima, S., Gabriel, C., Sheppard, R. J. & Grant, E. H. (1984) Biophys. J . 46, 29-34. 15. Takashima, S. (1967) Biopolymers 5 , 899-913. 16. Manning, G. S. (1978) Biophys. Chem. 9, 65-70. 17. Manning, G. S . (1981) J . Phys. Chem. 85,1506-1515. 18. de Xammar Oro, J. R. & Grigera, J. R. (1984) Biopolymers 23, 1457-1463. 19. Mandel, M. & Odijk, T. (1984) Ann. Rev. Phys. Chem. 35, 75-108.
Received September 11, 1990 Accepted June 10, 1991
