VOL. 15, 879-892 (1976)
BIOPOLYMERS
Dielectric Relaxation of DNA in Aqueous Solutions MASANORI SAKAMOTO, HIROSHI KANDA,* REINOSUKE HAYAKAWA, and YASAKU WADA, Department of Applied Physics, Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo, Japan
Synopsis Using a four-electrode cell and a new electronic system for direct detection of the frequency difference spectrum of solution impedance, the complex dielectric constant of calf thymus DNA ( M r = 4 X lo6)in aqueous NaCl a t 10°C is measured a t frequencies ranging from 0.2 Hz to 30 kHz. The DNA concentrations are C, = 0.01% and 0.05%, and the NaCl M . A single relaxation region is concentrations are varied from C, = lop4 M to found in this frequency range, the relaxation frequency being 10 Hz a t C, = 0.01% and C , = M . At C, = 0.05% it is evidenced that the DNA chains have appreciable intermolecular interactions. The dielectric relaxation time Td a t C, = 0.01% agrees well with the rotational relaxation time estimated from the reduced viscosity on the assumption that the DNA is not representable as a rigid rod but a coiled chain, I t is concluded that the dielectric relaxation is ascribed to the rotation of the molecule. Observed values of dielectric increment and other experimental findings are reasonably explained by assuming that the dipole moment of DNA results from the slow counterion fluctuation which has a longer relaxation time than r d .
INTRODUCTION The dielectric relaxation of DNA solutions has been studied by several workers. 1-13 Takashima* among them measured the dielectric constant and conductivity of DNA in aqueous NaCl in the frequency range 50 Hz to 200 kHz and found a relaxation around 100 Hz to 1 kHz for DNA with various molecular weights. He found that the rotational relaxation time obtained from the flow birefringence has a molecular weight dependence different from the dielectric one. Furthermore, he observed'O that the salt concentration and the salt species affect considerably the dielectric relaxation time while they have a minor effect on the rotational one. From these facts he concluded8J0 that the dielectric relaxation does not arise from the rotation of molecules but from an induced polarization of counterions around the molecules. Since Takashima employed a conventional two-electrode method for measuring dielectric * Present address: Central Research Laboratory, Hitachi Ltd., Kokubunji-shi, Tokyo, Japan. a79 0 1976 by John Wiley & Sons, Inc.
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SAKAMOTO E T AL.
constant and conductivity of the solution, his results may be somewhat obscured in particular a t low frequencies and high conductances where the d.c. conductance is very much larger than the dielectric contribution. More recently, Hanss and Bernengo12 observed the conductivity dispersion of DNA in aqueous NaCl by use of a four-electrode technique14J5 and differential measurements with two cells, one being filled with a solution of DNA and the other with the solvent whose d.c. conductivity was adjusted to the solution conductivity. With these techniques, they extended the frequency range down to 0.1 Hz and found a dielectric loss peak around 1Hz to 10 Hz, which agreed in order of magnitude with the rotational relaxation time obtained from the electric birefringence decay. Their measurements, however, were limited to a single concentration of added salt M NaCl) and to a relatively high concentration of DNA (0.05%in weight) where considerable intermolecular interactions might be present. We have developed a new method16 for measuring the complex dielectric constant of a highly conductive material by simultaneous employment of the four-electrode technique and the direct detection system. This method enabled us to extend the frequency range down to an ultralow range even for solutions of a considerably high d.c. conductivity. We felt it would be interesting to apply this method to dilute solutions of DNA with various salt concentrations and to examine the inconsistency between the data by Takashima and Hanss and Bernengo. This paper describes experimental results of dielectric relaxation of DNA in aqueous NaCl a t frequencies from 0.2 Hz to 30 kHz a t two polymer concentrations, C , = 0.01 and 0.05%, with varying NaCl concentrations C,. The results indicate that there exists a single relaxation region in the covered frequency range and both dielectric increment and relaxation time increase remarkably with decreasing salt concentration. For solutions of C , = 0.01%, the rotational relaxation time agrees fairly well with the dielectric one over a whole range of salt concentration, and thus i t is concluded that the dielectric relaxation of DNA in solution arises from the rotation of molecules. The origin of dipole moment in DNA is discussed on the basis of experimental findings as well as theoretical considerations and it is suggested that the counterion fluctuation with relaxation times longer than the rotational one is more probable as the origin of dipole moment than the dipole moment model due to a difference in the ionic composition of the two molecular ends, such as differences in residual bound amino acids or proteins.
EXPERIMENTAL The sodium salt of calf thymus DNA purchased from Sigma Chemical Co. (lot D-1501) was dissolved in aqueous NaCl with concentrations C , = lo-* M to M . The average molecular weight of DNA was deter-
DNA DIELECTRIC RELAXATION
881
mined from the intrinsic viscosity in 0.2 M NaCl as 4 X lo6 through an empirical relation by Reinert et al.17 The NaCl solutions were made of a stock solution of M by dilution to the prescribed concentration with freshly deionized water, the conductivity of which is less than 0.2 pmho cm-l. The sample solutions were left at 3°C for 40 hr till complete dissolution of DNA. To prevent CO2 gas in the atmosphere from dissolving into sample solutions, the air in the sample flask was replaced by nitrogen gas. The concentration of DNA was chosefi as C, = 0.01% in weight which is suitable for reducing the effect of intermolecular interactions and the data were compared with those at C, = 0.05% where appreciable intermolecular interactions are expected. Both dielectric and viscosity measurements were carried out at 10°C, which has been proved to be sufficiently lower than the denaturation temperature of DNA at C, higher than 3 x lop4 M.18 We have confirmed the stability of DNA at this temperature even a t C, = lop4M by the uv absorption a t 260 nm. The conductivity in the solution changed from 20 pmho cm-I to 130 pmho cm-l with variations of C, and C,. The conductivity was always found to obey the additivity law.19 Difficulties in measuring the complex dielectric constant, E* = t' id', of a highly conductive solution like DNA in aqueous NaCl arise from effects of the electrode polarization and the fluctuation of solution impedance owing to the drift of d.c. conductivity with time. Essential points of our new method which satisfactorily overcomes these two difficulties have been already reported elsewhere.16 Briefly summarized, the method consists of the four-electrode technique14J5 (two current electrodes and two potential ones) which is useful for avoiding the effect of electrode polarization and the direct measuring technique of the frequency difference spectrum of solution impedance between measuring frequency w and reference one wo. The four-electrode cell used in this study is illustrated in Figure 1. The size of the cylindrical cell made of Pyrex glass was 14 cm3 in volume, 8 cm in length, and 1.5 cm in diameter. A pair of current electrodes made of platinum black disks (El and E4) were set at the both ends of the cylinder. Two potential electrodes (E2 and E3) made of platinum black thin needles, 0.7 mm in diameter, were set 1.75 cm apart from the cylindrical axis to minimize the size effect of the e1e~trodes.l~ The separation of the potential electrodes was 4 cm. The cell was immersed in a silicone bath regulated at 10°C within *O.l"C. The second technique employed here is a direct measurement of the difference between the real parts of solution impedances a t two frequencies, AZ'(w) = Z ' ( w ) - Z'(oo). This technique has an advantage to eliminate the frequency-independent d.c. conductivity which overwhelmingly contributes to 2' and may drift with time. In the present technique, the sum of two sinusoidal currents of equal amplitude, I cos w t and I cos w o t , was supplied to the solution through
882
SAKAMOTO ET AL. I
t-
I
20 A 20
-
20 --20
80
I
-
..
Fig. 1. Dimensions in mm of the four-electrode cell; El and Ed-current electrodes, E2 and Es-potential electrodes, and I-solution inlet.
the current electrodes. The voltage across the potential electrodes was picked up by a differential amplifier with a very high input impedance. The most part of the voltage arises from the d.c. conductance of the solution. After elimination of this part by an automatic bridge balance, the amplified signal was fed to two demodulators. In one of them, the signal was demodulated by a reference wave A cos w t in a half switching cycle and by A cos wot in the other half. The output of the demodulator therefore gives the frequency difference spectrum of the real part of solution impedance, AZ’(w). The other demodulator with reference signal A sin w t yields Z ” ( w ) . In the case of highly conductive solution where t’ and t’’ are much smaller than alwto ( a denoting the d.c. conductivity and eo the vacuum permittivity), 1
A Z ’ ( U ) = - - (:)2
A(wt”)
CO
1
(?) CO
Z”(w) = -
2 UE’
where COis the vacuum capacitance between two cross-sectional planes a t the potential electrodes and A ( w t ” ) denotes w t ” ( w ) - wot”(w0). The real part t’ can be directly obtained from Z ” ( w ) by the use of Eqs. (1). On the other hand, we’’ at the low-frequency limit of the relaxation region should be proportional to w2. This limiting law enables us to get absolute value of t N if the measuring frequency is extended to a sufficiently low frequency. In the present case of DNA, the lowest frequency in the present experiment (0.2 Hz) was found to satisfy this requirement. In the present study, f = w / 2 x was varied from 0.2 Hz to 30 kHz, f o = w o / 2 a was fixed a t 7.63 Hz, and the switching frequency was chosen as 6.045 Hz. Measurements for DNA in aqueous NaCl were made relatively to aqueous NaC1, the concentration of which was adjusted to have the same conductivity as the DNA solution within fl%. The dielectric
DNA DIELECTRIC RELAXATION
883
SHEAR RATE ( sec-1)
Fig. 2. Shear stress plotted against shear rate for DNA (C, = 0.01%) in aqueous NaCl measured by,a rotating cylinder viscometer a t 10°C. Dashed line represents the data for pure water for comparison.
constant d obtained in this way is, therefore, the increment over that of the aqueous NaC1, which is almost equal to the dielectric constant of pure water because the decrease in dielectric constant of water due to NaCl is only 5.5 X low3at C, = lop3M.20 The apparent viscosity of DNA solution depends strongly on the shear rate i/. Therefore we employed a low-shear rotating cylinder viscometer in which the data is reliable down to y = 0.1 sec-l and the relation between shear stress and i/ was recorded on an x-Y recorder as illustrated in Figure 2. The viscosity 17 was obtained from the initial slope of the curve.
RESULTS Figures 3 and 4 indicate the frequency dependence of t' and t" at C, = 0.01% and 0.05%, respectively, each for three NaCl concentrations. A single loss peak is found in the covered frequency range in all the cases. Figure 5 illustrates Cole-Cole plots of the data in Figures 3 and 4. The plots deviate considerably from the semicircular locus, indicating a distribution of relaxation times. In Table I are summarized the dielectric relaxation time T d , which is obtained from the loss peak frequency f m with 2 i r f r f m r d = 1, and the dielectric increment At obtained from the Cole-Cole plots, together with the reduced viscosity, vred = (7 - ~ , ) / v , C , (17, denoting the solvent viscosity). Both At and 7d increase with decreasing salt concentration. In Table I, we find that 7 d at a particular salt concentration is considerably longer at C, = 0.05% than at C, = 0.01%. This implies that, a t C, = 0.05% for which the dielectric measurement has been carried out by Hanss and Bernengo," there may exist to a considerable extent intermolecular interactions like entanglements of DNA chains. In the following, therefore, we shall discuss the data at C, = 0.01% alone.
SAKAMOTO E T AL.
884
4 2
-
0
0
-
0 16 w
0
0 40 20 0 F R E Q U E N C Y (.Hz 1
Fig. 3. Real ( 0 )and imaginary (0) parts of complex dielectric constant of DNA at C , = 0.05% for three NaCl concentrations (lO°C). Real part represents the increment over water.
FREQUENCY
( Hz)
Fig. 4. Real ( 0 )and imaginary (0) parts of complex dielectric constant of DNA at C , = 0.01% for three NaCl concentrations (lO°C). Real part represents the increment over water.
DNA DIELECTRIC RELAXATION
2K1 C p = 0 . 0 5 "1;
cp=o.o 1 % m;02&]i v
.
4,
lr*ypg)F] 0
-
-?
885
02
04
&'
06
E'
lo3)
G
4
2
0
(
103)
v 5
u
C 0 . a
bm; loE
0
&'
m0 -
.
2
v
2
1
0
E'
( lo3)
4,
Q
0
2
4
E'
5
(
6
10
15
(
lo3)
20
t
0
20
10
E'
lo3 )
(
30
lo3)
Fig. 5. Cole-Cole plots of the data in Figures 3 and 4 for DNA at 10°C.
DISCUSSION A t C, = 0.01%, the solution volume per a DNA molecule is 6.6 X cm3, which is equivalent to the volume of a cube with the side of ca. 0.4 p. The contour length of DNA of 4 X lo6 in molecular weight is ca. 2 p , whereas Kuhn's statistical segment length of the DNA was estimated by Triebel e t a1.21 as 900 A from hydrodynamic data a t C, = 0.2 M . This implies that the DNA chain consists of 22 statistical segments and thus TABLE I Results o n DNA a t 10°C for Various NaCl Concentrations
c, A€
(103)
'rd
sec)
( l o 2dl/g) 2 r Z (10-3 ~ ) a set)' 3'rKA ( p 2 ) 1 ' 2( l o 6D)b
%ed
?&
0.05 0.01 0.05 0.01
0.01 0.01 0.01 0.01
10-3~ 5.2 0.62 45 16 1.2 22 100 0.52
3x 10-4~
18 2.4 85 34 2.3 43 192 1.02
acalculated from Eqs. (2) and ( 3 ) , respectively, using '+d b Calculated from Eq. (6) by use of A€.
10-
4
33 6.8 120 69 5.5 103 458 1.72
in place of [q].
~
SAKAMOTO E T AL.
886
the rms end to end length, (h2)l12,is equal to 0.42 /I a t C, = 0.2 M . We observed the intrinsic viscosity [q] of the DNA as 30 dl/g a t C, = 0.2 M M . Since [q] is proportional to (h2)3/2,22 and 70 dl/g a t C, = ( h 2 ) at C, = M is estimated as 0.56 /I. From these facts, we may assume that the DNA chain may be representable as a coiled one rather than a rigid rod. This is not surprising in view of the fact that the DNA is locally stiff but, on account of its large length, distances between sequentially widely separated groups are obtainable from random chain formulas. Furthermore, the fact that ( h 2 )112is comparable with the average distance between DNA molecules indicates that the DNA a t C, = M or lower may exhibit intermolecular interactions to some extent even a t C, = 0.01%, but the interactions are greatly diminished compared with the DNA a t C, = 0.05%. For calf thymus DNA (Mr = 5 X lo6), Takashimas found a loss peak a t 100 Hz for C, = 0.01% and C, lower than A4 NaC1. Hanss and Bernengo,12 on the other hand, observed a loss peak for the calf thymus DNA (Mr = 5 X lo6) a t 5 Hz for C, = 0.05% and C, = M NaC1. These results might suggest two separate relaxation mechanisms for the DNA, but the present results a t similar experimental conditions evidence a single relaxation region over the covered frequency range. The present data are rather closer to those by Hanss and Bernengo than those by Takashima when compared a t similar experimental conditions. The rotational relaxation time rrot of DNA in solution has been measured by electric dichroism by Ding et al.,23flow dichroism by Callis and D a ~ i d s o n , ~and * dynamic light scattering by Schmitz and S ~ h u r r . ~ ~ The data are summarized in Figure 6 as a function of molecular weight, where values are reduced to 10°C, assuming that rrotis proportional to the solvent viscosity. An alternative way for obtaining rrotis to use the intrinsic viscosity [q]. For a bead-spring model with a dominant hydrodynamic interaction among beads (Zimm the viscoelastic relaxation time rz related to the rotation of end to end vector of the chain (the longest relaxation time in Zimm’s theory) is given as 72
= 0-42Mrqs[q1
RT
(coiled chain)
where M , is the molecular weight, R is the gas constant, and T is the absolute temperature. For a rigid rod, on the other hand, KirkwoodAuer’s theory27gives the viscoelastic rotational relaxation time T K A as TKA =
5Mrqs[q1
4RT
(rigid rod)
(3)
Values of T Z and ?KA as functions of Mr were calculated from Eqs. (2) and (3) along with an empirical relation between [q] and M , given by Reinert, et al.17 Results a t 10°C are drawn in Figure 6 by solid lines.
DNA DIELECTRIC RELAXATION
887
4
-2 I
5
6
7
0
9
LOG M Fig. 6. Doubly logarithmic plot of rotational relaxation times (lO°C) against molecular electric d i c h r o i ~ m (O), ; ~ ~ flow d i c h r o i ~ m (a), ; ~ ~ dynamic light scatweight of DNA: (a), and solid lines; T Z and T K A calculated from Eqs. (2) and (3), respectively.
At low molecular weights where DNA is rodlike, T K A agrees with Trot, while, at high molecular weights where DNA may be representable as a coiled chain, T Z agrees with Trot. In any case, these results indicate that we can estimate Trot from the viscosity data if the shape of the chain is appropriately assumed. If we assume that, first, the dielectric relaxation arises from the rotation of DNA molecules and secondly, the dipole moment vector and the end to end vector of the molecule have the same direction and are proportional in magnitude to each other, Td should be equal to the rotational relaxation time of end to end vector and thus 272
Td
N
7d
=3
(coiled chain)
or 7
~ (rigid ~ rod)
Equation (4)was proved analytically by Fujimori et a1.28 for a Rouse model, which is a bead-spring model without hydrodynamic interaction. Equation (4)holds for the Zimm model with a minor error. In Table I, 272 and 3 T K A are calculated from Eqs. (2) and (3), respectively. In this calculation we used the reduced viscosity Vred at C , = 0.01% in place of [ v ] , because 7d was also obtained at a finite concentration, C , = 0.01%. The intermolecular interactions, if any, should affect Td and Vred by the same factor. In Table I, T d compares rather well with 272 for three salt concentrations. This evidences that the DNA in the present case forms a coiled chain rather than a rigid rod and the dielec-
SAKAMOTO ET AL.
888
tric relaxation arises from the rotation of the molecule. It has been evidenced from the electric dichroismZ3and electric birefringence measurements12 that the DNA behaves as if i t might possess a permanent dipole. The permanent dipole means here the dipole which may change more slowly than the rotation of the DNA. Ding et al.23 suggested that the permanent dipole might arise from the saturation effect of induced polarization of counterions. The reversed pulse experiment on the electric birefringence by Hanss et a1.,I2 however, seems to evidence a mechanism of permanent dipole different from this saturation effect. In fact, the dielectric relaxation which occurs in a low electric field requires the existence of a permanent dipole even at the zero electric field. The increases in both Ac and 7d with decreasing salt concentration (see Table I) may be primarily explained by the expansion of chain. Salt ions shield the electrostatic repulsion between charges on DNA and reduce the chain expansion. In Table I, the rms dipole moment of DNA is calculated from At at C , = 0.01% by Kirkwood's formula:29 (p2)1/2N
3.3
(C,:mole DNAh)
(6)
Our values of ( p 2 ) ' l 2 (or simply denoted p ) = 5.2 X lo5 D and 7d = 16 msec ( M , = 4 x 106, C , = 10-3 M , C , = 0.01%) are in general agreement with those by Hanss and Bernengo,12p = 7.2 X lo5 D and rd = 18 msec ( M , = 5 X lo6) (note that p1 for sample 1 in Table I in their paper should be read as 7.2 X lo5 D). Values by T a k a ~ h i m ap, ~= 3.2 X lo4 D and 7 d = 0.09 msec (M,= 5 X lo6) are much lower than ours probably because the frequency range in his experiment covered only the highfrequency part of the relaxation region. Analogously, Jennings and Plummer30 who observed light-scattering in a.c. field a t frequencies above 100 Hz reported lower values than ours, p = 7.4 X lo4 D and Td = 0.4 msec (M, = 3 X lo6), probably because the frequency was not extended to sufficiently low values. In Figure 6, the closed circles represent maximum relaxation times obtained from the electric dichroism measurement by Ding et al.23 Two circles for M , = 6.0 X lo6 and 6.7 X lo6 are located somewhat lower than both 7KA and TZ probably because the duration of exciting d.c. pulse in their experiment, ca. 40 psec, was too short to observe the slow relaxation mechanism. The slow mechanisms are not excited by a short pulse and thus do not appear in the subsequent decay. Accordingly, the value of dipole moment estimated from the field dependence of dichroism reported in their paper is p = 1.0 X lo4 D ( M , = 6.7 X lo6) which is much lower than our value. In fact, Hanss and Bernengo12 who used a d.c. pulse of 20 msec for excitation in their electrical birefringence measurement found two relaxation times: the longer one of them agrees with rd estimated from the dielectric loss peak. For DNA with a much lower molecular weight, however, the pulse duration employed by Ding et al.23seems satisfactory to observe the whole relaxa-
DNA DIELECTRIC RELAXATION
889
tional behavior. Actually the circles for M , < lo6 in Figure 6 fall on the curve of 7 K A and p estimated from the field dependence of dichroism ( p = 0.7 x lo4 D for M , = 0.32 X lo6) is close to that for M , = 6.7 X lo6 despite of the large difference in M,. T o summarize, observation of the whole relaxational behavior of DNA with a wide distribution of relaxation times requires a wide frequency range or a long duration of exciting d.c. pulse. On account of a double-stranded structure of antiparallel helixes, the DNA molecule by itself does not possess a dipole moment along the chain.31 Two mechanisms have been proposed by Hanss and Bernengo12 for the origin of dipole moment in DNA: ( I ) a difference in the ionic composition of the two molecular ends, such as differences in residual bound amino acids or proteins (end charge mechanism), and (2) counterion fluctuation with a relaxation time longer than Trot (counterion mechanism). The end charge mechanism, however, seems unprobable because the magnitude of (p2)l12is too large to be accounted for by this mechanism. As has been described in the first part of this section, the rms end to end length ( h 2 )112a t C , = lov3A4 is estimated as 0.56 1.1. For explaining the value of ( p 2 )112 = 5.2 X lo5 D by this mechanism, we must assume the end charges of f 1 9 e (e being the elementary charge), which are unreasonably large. In the following we shall discuss on the counterion mechanism. We assume two kinds of counterions diss’ociated from the polymer:19 ( I ) counterions which are bound near the polyion (site-bound ions), and (2) counterions which are located in the environment apart from the polyion and contribute to the conductivity of the solution (free counterions). We consider the charge distribution on the polyion including site-bound ions and calculate the rms dipole moment of polyion plus site-bound ions, which will be denoted simply “polyion” in the following. Let us consider a polyion consisting of N monomeric units, each having a single monovalent binding site (phosphate group in the case of DNA). The dipole moment vector p of the polyion is defined as N
p
=
1 6nie(ri - rc) i= 1
(7)
where ri and r G = (1/N) ZE1 ri are position vectors of the i-th site and the center of gravity, respectively, and Gnie is defined by 6n;e = (n; - no)e
(8)
Here nie is the charge on the i-th site (ni = 0 for a site which is either undissociated or occupied by a site-bound ion, n; = -1 for an unoccupied acid site) and noe is the mean value of nie. From Eq. (7) we obtain the mean-square value of p as
SAKAMOTO E T AL.
890
where the angular brackets represent double averaging processes; an average over different conformations of the polyion and an average over different charge distributions on the sites of the polyion. It is assumed here that the conformational change (including rotation) of the polyion is fast enough to attain an equilibrium and, on the other hand, the change in the charge distribution is slow enough compared with the time scale in which we are considering (p2) in Eq (9). In other words, the first average above mentioned may be regarded as a time average and the second as a spatial one, i.e., an average over different polyions. Since these two averaging processes can be treated as approximately independent of each other, we may obtain from Eq. (9) N
(p2) =
N
C C (6ni6nj)e2((r;- r d + j - r d ) i=l j=1
(10)
Furthermore, if we assume for simplicity that there is no correlation between ion bindings at different sites and the probability of ion binding at each site is 1 - a (adenoting the fraction of free counterions), then we have (6ni6nj)
=0
(6ni6nj)
= a ( 1 - a)
(11)
Substituting Eqs. (11) into Eq. (lo),we have the final result
where ( S2)is the mean-square radius of gyration of the polyion defined by
Incidentally, this result is valid also for the case where the change in the charge distribution on the polyion is fast enough to realize a charge distribution equilibrium because the time average of the charge distribution appearing in this case is formally equivalent to the spatial average in deriving Eq. (12). Recently, Van der Touw and Mande132 have developed a theory on the dielectric relaxation of polyelectrolytes. They took into account two mechanisms of the polarization, the local counterion fluctuation within the subunit of the polyion and the counterion fluctuation over the whole molecule. The latter is essentially the same as the counterion mechanism in this paper and Eq. (12) is analogous to Eq. (31) in their paper except that we considered a discrete distribution of bound ions along the polyion whereas they assumed a continuous distribution of them. In the present case of DNA, quantities appearing on the righthand
DNA DIELECTRIC RELAXATION
891
side of Eq. (12) can be estimated as follows: The number of monomeric units in the double-stranded DNA, N , is calculated as 1.3 x lo4 from the molecular weight. As mentioned earlier, the d.c. conductivity data always satisfied the additivity law, and one-third of Na+ ions originating from Na-DNA was found to contribute to the conductivity. Thus, we may take a = %,. The mean-square radius of gyration ( S 2 )is ap; proximately related with ( h 2 )as ( S 2 ) N ( h 2 ) / 6 . 2 2 In the case of C, = M , we obtain ( S 2 ) l I 2= 0.23 I.L from ( h 2 ) l I 2= 0.56 p. Substituting these values into Eq. (12), we have ( p 2 ) l I 2 = 5.9 X lo5 D, which is in a good agreement with 5.2 X lo5 D in Table I. The fluctuation of charge distribution on the polyion is induced by the transfers of counterions between polyion and environment and/or along the polyion. The counterion mechanism presented in this paper assumes that the relaxation time of this transfer, 7t, is longer than the dielectric relaxation time of the polyion, 7d. Estimation of T~ is still obscure a t present. G ~ t t l i e bdemonstrated ~~ by a radioactive tracer experiment that the transfer time of counterions from polyion to environment is less than one second for Na-polyacrylate of molecular weight 9.4 X lo5 and C, = 0.1-0.45%. Lifson and Jackson34predicted from a theoretical point of view that the transfer time should depend on the electrostatic potential of the polyion and a value less than one millisecond may be probable. In any case, we might not expect 7t longer than 7d as far as we consider only a free diffusion of counterions between polyion and environment or along the polyion because the diffusion length is one micron in order of magnitude in the present case of DNA. We must consider, therefore, some transfer mechanism of counterions different from the free diffusion for explaining 7t longer than 7d. Relative magnitudes of 7d and 7t were discussed by Van der Touw et a1.,32735,36but no conclusion has been derived by them. Results on the dielectric relaxation of DNA in the present study as well as those of partially neutralized polyacrylic acid and Na-polystyrene sulfonate by Van der Touw and Mande1,35 both having been reasonably explained by the counterion mechanism, seem to suggest that 7t is longer than 7d. The shielding effect on the polyion charge due to surrounding salt ions may be stronger in the end charge mechanism that in the counterion one because, in the former, the charges on the polyion are localized on the ends of chain. Since the radius of ion atmosphere in the DebyeHuckel theory is ca. 100 8, a t C, = M , being much smaller than ( h 2 ) I l 2the , end charges, if any, would be considerably shielded by salt ions. It is to be noted in this connection that, as has been calculated by Hall,37 the relaxation time of formation of ion atmosphere is very much shorter than the rotational relaxation time of DNA. In the present stage, the reason for 7t longer than 7d is still obscure. A direct measurement of 7t is desirable to answer this question. Dielectric measurements for DNA with lower molecular weights, purified DNA without residual proteins, and DNA in various kinds of salt solution may be also useful to clarify the origin of dipole moment.
892
SAKAMOTO E T AL.
This research was supported by a Grant-in-Aid for Fundamental Scientific Research, Japan. The authors would like to express their gratitude to Dr. E. Fukada and Dr. M. Kaibara a t the Institute for Physical and Chemical Research for making available a lowshear viscometer. Thanks are also due to Miss N. Fujinawa for her help in preparing the manuscript.
References 1. Jungner, G. (1945) Acta Physiol. Scand. 10, Suppl. 32. 2. Jungner, G., Jungner, I. & Allgen, L. G. (1949) Nature, 163,849-850. 3. Jungner, I. (1950) Acta Physiol. Scand., 20, Suppl. 69. 4. Allgen, L. G. (1950) Acta Physiol. Scand., 22, Suppl. 76. 5. Jacobson, B. (1953) Reo. Sci. Znstr., 24,949-954. 6. Jerrard, H. G. & Simmons, B. A. W. (1959) Nature, 184,1715-1716. 7. Takashima, S. (1963) J.Mol. Biol., 7.455-467. 8. Takashima, S. (1966) J. Phys. Chem., 70,1372-1380. 9. Takashima, S. (1967) Adu. Chem. Ser., 63,232-252. 10. Takashima, S. (1967) Biopolymers, 5,899-913. 11. Takashirna, S. (1973) Biopolymers, 12,145-155. 12. Hanss, M. & Bernengo, J. C. (1973) Biopolymers, 12,2151-2159. 13. Goswami, D. N. & Das Gupta, N. N. (1974) Biopolymers, 13,1549-1556. 14. Schwan, H. P. (1968) Ann. N. Y. Acad. Sci.,148, Art. I, 191-209. 15. Schwan, H. P. (1963) Physical Techniques in Biological Research, Vol. 6, Academic Press, New York, Chap. 6. 16. Hayakawa, R., Kanda, H., Sakamoto, M. & Wada, Y. (1975) Japan. J. Appl. Phys., 14.2039-2052. 17. Reinert, K. E., Strassburger, J. & Triebel, H. (1971) Biopolyrners, 10,285-307. 18. Schildkraut, C. & Lifson, S. (1965) Biopolymers, 3,195-208. 19. Oosawa, F. (1971) Polyelectrolytes, Marcel Dekker, Inc., New York. 20. Cole, R. H. (1955) J. Amer. Chem. SOC.,77,2012-2013. 21. Triebel, H., Reinert, K. E. & Strassburger, J. (1971) Biopolymers, 10,2619-2621. 22. Flory, P. J. (1953) Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, N.Y. 23. Ding, D.-W., Rill, R. & Van Holde, K. E. (1972) Biopolymers, 11,2109-2124. 24. Callis, P. R. & Davidson, N. (1969) Biopolymers, 8.379-390. 25. Schmitz, K. S. & Schurr, J. M. (1973) Biopolymers, 12, 1543-1564. 26. Zimm, B. H. (1956) J. Chem. Phys., 24,269-278. 27. Kirkwood, J. G. & Auer, P. L. (1951) J. Chem. Phys., 19,281-283. 28. Fujimori, S., Doi, M., Nakajima, H. & Wada, Y. (1975) J. Polym. Sci.-Polym. Phys. Ed., 13,2135-2153. 29. Kirkwood, J. G. (1943) Proteins, Amino Acids and Peptides as Ions and Dipolar Ions, Cohn, E. J. & Edsall, J. T., Eds., Reinhold, New York, p. 296. 30. Jennings, B. R. & Plumrner, H. (1970) Biopolymers, 9,1361-1372. 31. Schwarz, G. (1972) Chem. SOC.Specialist Periodical Rep., Dielectric and Related Molecular Processes, 1,163-191. 32. Van der Touw, F. & Mandel, M. (1974) Biophys. Chem. (Amsterdam),2,21%230. 33. Gottlieb, M. H. (1971) J. Phys. Chem., 75.1990-1993. 34. Lifson, S. &Jackson, J. L. (1962) J. Chem. Phys., 36,2410-2414. 35. Van der Touw, F. & Mandel, M. (1974) Biophys. Chem. (Amsterdam),2,231-241. 36. Muller, G., Van der Touw, F., Zwolle, S. & Mandel, M. (1974) Biophys. Chem. (Amsterdam), 2,242-254. 37. Hall, L. (1952) J. Acoust. SOC.Amer., 24,704-708.
Received July 8,1975 Accepted November 26,1975
BIOPOLYMERS
Dielectric Relaxation of DNA in Aqueous Solutions MASANORI SAKAMOTO, HIROSHI KANDA,* REINOSUKE HAYAKAWA, and YASAKU WADA, Department of Applied Physics, Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo, Japan
Synopsis Using a four-electrode cell and a new electronic system for direct detection of the frequency difference spectrum of solution impedance, the complex dielectric constant of calf thymus DNA ( M r = 4 X lo6)in aqueous NaCl a t 10°C is measured a t frequencies ranging from 0.2 Hz to 30 kHz. The DNA concentrations are C, = 0.01% and 0.05%, and the NaCl M . A single relaxation region is concentrations are varied from C, = lop4 M to found in this frequency range, the relaxation frequency being 10 Hz a t C, = 0.01% and C , = M . At C, = 0.05% it is evidenced that the DNA chains have appreciable intermolecular interactions. The dielectric relaxation time Td a t C, = 0.01% agrees well with the rotational relaxation time estimated from the reduced viscosity on the assumption that the DNA is not representable as a rigid rod but a coiled chain, I t is concluded that the dielectric relaxation is ascribed to the rotation of the molecule. Observed values of dielectric increment and other experimental findings are reasonably explained by assuming that the dipole moment of DNA results from the slow counterion fluctuation which has a longer relaxation time than r d .
INTRODUCTION The dielectric relaxation of DNA solutions has been studied by several workers. 1-13 Takashima* among them measured the dielectric constant and conductivity of DNA in aqueous NaCl in the frequency range 50 Hz to 200 kHz and found a relaxation around 100 Hz to 1 kHz for DNA with various molecular weights. He found that the rotational relaxation time obtained from the flow birefringence has a molecular weight dependence different from the dielectric one. Furthermore, he observed'O that the salt concentration and the salt species affect considerably the dielectric relaxation time while they have a minor effect on the rotational one. From these facts he concluded8J0 that the dielectric relaxation does not arise from the rotation of molecules but from an induced polarization of counterions around the molecules. Since Takashima employed a conventional two-electrode method for measuring dielectric * Present address: Central Research Laboratory, Hitachi Ltd., Kokubunji-shi, Tokyo, Japan. a79 0 1976 by John Wiley & Sons, Inc.
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SAKAMOTO E T AL.
constant and conductivity of the solution, his results may be somewhat obscured in particular a t low frequencies and high conductances where the d.c. conductance is very much larger than the dielectric contribution. More recently, Hanss and Bernengo12 observed the conductivity dispersion of DNA in aqueous NaCl by use of a four-electrode technique14J5 and differential measurements with two cells, one being filled with a solution of DNA and the other with the solvent whose d.c. conductivity was adjusted to the solution conductivity. With these techniques, they extended the frequency range down to 0.1 Hz and found a dielectric loss peak around 1Hz to 10 Hz, which agreed in order of magnitude with the rotational relaxation time obtained from the electric birefringence decay. Their measurements, however, were limited to a single concentration of added salt M NaCl) and to a relatively high concentration of DNA (0.05%in weight) where considerable intermolecular interactions might be present. We have developed a new method16 for measuring the complex dielectric constant of a highly conductive material by simultaneous employment of the four-electrode technique and the direct detection system. This method enabled us to extend the frequency range down to an ultralow range even for solutions of a considerably high d.c. conductivity. We felt it would be interesting to apply this method to dilute solutions of DNA with various salt concentrations and to examine the inconsistency between the data by Takashima and Hanss and Bernengo. This paper describes experimental results of dielectric relaxation of DNA in aqueous NaCl a t frequencies from 0.2 Hz to 30 kHz a t two polymer concentrations, C , = 0.01 and 0.05%, with varying NaCl concentrations C,. The results indicate that there exists a single relaxation region in the covered frequency range and both dielectric increment and relaxation time increase remarkably with decreasing salt concentration. For solutions of C , = 0.01%, the rotational relaxation time agrees fairly well with the dielectric one over a whole range of salt concentration, and thus i t is concluded that the dielectric relaxation of DNA in solution arises from the rotation of molecules. The origin of dipole moment in DNA is discussed on the basis of experimental findings as well as theoretical considerations and it is suggested that the counterion fluctuation with relaxation times longer than the rotational one is more probable as the origin of dipole moment than the dipole moment model due to a difference in the ionic composition of the two molecular ends, such as differences in residual bound amino acids or proteins.
EXPERIMENTAL The sodium salt of calf thymus DNA purchased from Sigma Chemical Co. (lot D-1501) was dissolved in aqueous NaCl with concentrations C , = lo-* M to M . The average molecular weight of DNA was deter-
DNA DIELECTRIC RELAXATION
881
mined from the intrinsic viscosity in 0.2 M NaCl as 4 X lo6 through an empirical relation by Reinert et al.17 The NaCl solutions were made of a stock solution of M by dilution to the prescribed concentration with freshly deionized water, the conductivity of which is less than 0.2 pmho cm-l. The sample solutions were left at 3°C for 40 hr till complete dissolution of DNA. To prevent CO2 gas in the atmosphere from dissolving into sample solutions, the air in the sample flask was replaced by nitrogen gas. The concentration of DNA was chosefi as C, = 0.01% in weight which is suitable for reducing the effect of intermolecular interactions and the data were compared with those at C, = 0.05% where appreciable intermolecular interactions are expected. Both dielectric and viscosity measurements were carried out at 10°C, which has been proved to be sufficiently lower than the denaturation temperature of DNA at C, higher than 3 x lop4 M.18 We have confirmed the stability of DNA at this temperature even a t C, = lop4M by the uv absorption a t 260 nm. The conductivity in the solution changed from 20 pmho cm-I to 130 pmho cm-l with variations of C, and C,. The conductivity was always found to obey the additivity law.19 Difficulties in measuring the complex dielectric constant, E* = t' id', of a highly conductive solution like DNA in aqueous NaCl arise from effects of the electrode polarization and the fluctuation of solution impedance owing to the drift of d.c. conductivity with time. Essential points of our new method which satisfactorily overcomes these two difficulties have been already reported elsewhere.16 Briefly summarized, the method consists of the four-electrode technique14J5 (two current electrodes and two potential ones) which is useful for avoiding the effect of electrode polarization and the direct measuring technique of the frequency difference spectrum of solution impedance between measuring frequency w and reference one wo. The four-electrode cell used in this study is illustrated in Figure 1. The size of the cylindrical cell made of Pyrex glass was 14 cm3 in volume, 8 cm in length, and 1.5 cm in diameter. A pair of current electrodes made of platinum black disks (El and E4) were set at the both ends of the cylinder. Two potential electrodes (E2 and E3) made of platinum black thin needles, 0.7 mm in diameter, were set 1.75 cm apart from the cylindrical axis to minimize the size effect of the e1e~trodes.l~ The separation of the potential electrodes was 4 cm. The cell was immersed in a silicone bath regulated at 10°C within *O.l"C. The second technique employed here is a direct measurement of the difference between the real parts of solution impedances a t two frequencies, AZ'(w) = Z ' ( w ) - Z'(oo). This technique has an advantage to eliminate the frequency-independent d.c. conductivity which overwhelmingly contributes to 2' and may drift with time. In the present technique, the sum of two sinusoidal currents of equal amplitude, I cos w t and I cos w o t , was supplied to the solution through
882
SAKAMOTO ET AL. I
t-
I
20 A 20
-
20 --20
80
I
-
..
Fig. 1. Dimensions in mm of the four-electrode cell; El and Ed-current electrodes, E2 and Es-potential electrodes, and I-solution inlet.
the current electrodes. The voltage across the potential electrodes was picked up by a differential amplifier with a very high input impedance. The most part of the voltage arises from the d.c. conductance of the solution. After elimination of this part by an automatic bridge balance, the amplified signal was fed to two demodulators. In one of them, the signal was demodulated by a reference wave A cos w t in a half switching cycle and by A cos wot in the other half. The output of the demodulator therefore gives the frequency difference spectrum of the real part of solution impedance, AZ’(w). The other demodulator with reference signal A sin w t yields Z ” ( w ) . In the case of highly conductive solution where t’ and t’’ are much smaller than alwto ( a denoting the d.c. conductivity and eo the vacuum permittivity), 1
A Z ’ ( U ) = - - (:)2
A(wt”)
CO
1
(?) CO
Z”(w) = -
2 UE’
where COis the vacuum capacitance between two cross-sectional planes a t the potential electrodes and A ( w t ” ) denotes w t ” ( w ) - wot”(w0). The real part t’ can be directly obtained from Z ” ( w ) by the use of Eqs. (1). On the other hand, we’’ at the low-frequency limit of the relaxation region should be proportional to w2. This limiting law enables us to get absolute value of t N if the measuring frequency is extended to a sufficiently low frequency. In the present case of DNA, the lowest frequency in the present experiment (0.2 Hz) was found to satisfy this requirement. In the present study, f = w / 2 x was varied from 0.2 Hz to 30 kHz, f o = w o / 2 a was fixed a t 7.63 Hz, and the switching frequency was chosen as 6.045 Hz. Measurements for DNA in aqueous NaCl were made relatively to aqueous NaC1, the concentration of which was adjusted to have the same conductivity as the DNA solution within fl%. The dielectric
DNA DIELECTRIC RELAXATION
883
SHEAR RATE ( sec-1)
Fig. 2. Shear stress plotted against shear rate for DNA (C, = 0.01%) in aqueous NaCl measured by,a rotating cylinder viscometer a t 10°C. Dashed line represents the data for pure water for comparison.
constant d obtained in this way is, therefore, the increment over that of the aqueous NaC1, which is almost equal to the dielectric constant of pure water because the decrease in dielectric constant of water due to NaCl is only 5.5 X low3at C, = lop3M.20 The apparent viscosity of DNA solution depends strongly on the shear rate i/. Therefore we employed a low-shear rotating cylinder viscometer in which the data is reliable down to y = 0.1 sec-l and the relation between shear stress and i/ was recorded on an x-Y recorder as illustrated in Figure 2. The viscosity 17 was obtained from the initial slope of the curve.
RESULTS Figures 3 and 4 indicate the frequency dependence of t' and t" at C, = 0.01% and 0.05%, respectively, each for three NaCl concentrations. A single loss peak is found in the covered frequency range in all the cases. Figure 5 illustrates Cole-Cole plots of the data in Figures 3 and 4. The plots deviate considerably from the semicircular locus, indicating a distribution of relaxation times. In Table I are summarized the dielectric relaxation time T d , which is obtained from the loss peak frequency f m with 2 i r f r f m r d = 1, and the dielectric increment At obtained from the Cole-Cole plots, together with the reduced viscosity, vred = (7 - ~ , ) / v , C , (17, denoting the solvent viscosity). Both At and 7d increase with decreasing salt concentration. In Table I, we find that 7 d at a particular salt concentration is considerably longer at C, = 0.05% than at C, = 0.01%. This implies that, a t C, = 0.05% for which the dielectric measurement has been carried out by Hanss and Bernengo," there may exist to a considerable extent intermolecular interactions like entanglements of DNA chains. In the following, therefore, we shall discuss the data at C, = 0.01% alone.
SAKAMOTO E T AL.
884
4 2
-
0
0
-
0 16 w
0
0 40 20 0 F R E Q U E N C Y (.Hz 1
Fig. 3. Real ( 0 )and imaginary (0) parts of complex dielectric constant of DNA at C , = 0.05% for three NaCl concentrations (lO°C). Real part represents the increment over water.
FREQUENCY
( Hz)
Fig. 4. Real ( 0 )and imaginary (0) parts of complex dielectric constant of DNA at C , = 0.01% for three NaCl concentrations (lO°C). Real part represents the increment over water.
DNA DIELECTRIC RELAXATION
2K1 C p = 0 . 0 5 "1;
cp=o.o 1 % m;02&]i v
.
4,
lr*ypg)F] 0
-
-?
885
02
04
&'
06
E'
lo3)
G
4
2
0
(
103)
v 5
u
C 0 . a
bm; loE
0
&'
m0 -
.
2
v
2
1
0
E'
( lo3)
4,
Q
0
2
4
E'
5
(
6
10
15
(
lo3)
20
t
0
20
10
E'
lo3 )
(
30
lo3)
Fig. 5. Cole-Cole plots of the data in Figures 3 and 4 for DNA at 10°C.
DISCUSSION A t C, = 0.01%, the solution volume per a DNA molecule is 6.6 X cm3, which is equivalent to the volume of a cube with the side of ca. 0.4 p. The contour length of DNA of 4 X lo6 in molecular weight is ca. 2 p , whereas Kuhn's statistical segment length of the DNA was estimated by Triebel e t a1.21 as 900 A from hydrodynamic data a t C, = 0.2 M . This implies that the DNA chain consists of 22 statistical segments and thus TABLE I Results o n DNA a t 10°C for Various NaCl Concentrations
c, A€
(103)
'rd
sec)
( l o 2dl/g) 2 r Z (10-3 ~ ) a set)' 3'rKA ( p 2 ) 1 ' 2( l o 6D)b
%ed
?&
0.05 0.01 0.05 0.01
0.01 0.01 0.01 0.01
10-3~ 5.2 0.62 45 16 1.2 22 100 0.52
3x 10-4~
18 2.4 85 34 2.3 43 192 1.02
acalculated from Eqs. (2) and ( 3 ) , respectively, using '+d b Calculated from Eq. (6) by use of A€.
10-
4
33 6.8 120 69 5.5 103 458 1.72
in place of [q].
~
SAKAMOTO E T AL.
886
the rms end to end length, (h2)l12,is equal to 0.42 /I a t C, = 0.2 M . We observed the intrinsic viscosity [q] of the DNA as 30 dl/g a t C, = 0.2 M M . Since [q] is proportional to (h2)3/2,22 and 70 dl/g a t C, = ( h 2 ) at C, = M is estimated as 0.56 /I. From these facts, we may assume that the DNA chain may be representable as a coiled one rather than a rigid rod. This is not surprising in view of the fact that the DNA is locally stiff but, on account of its large length, distances between sequentially widely separated groups are obtainable from random chain formulas. Furthermore, the fact that ( h 2 )112is comparable with the average distance between DNA molecules indicates that the DNA a t C, = M or lower may exhibit intermolecular interactions to some extent even a t C, = 0.01%, but the interactions are greatly diminished compared with the DNA a t C, = 0.05%. For calf thymus DNA (Mr = 5 X lo6), Takashimas found a loss peak a t 100 Hz for C, = 0.01% and C, lower than A4 NaC1. Hanss and Bernengo,12 on the other hand, observed a loss peak for the calf thymus DNA (Mr = 5 X lo6) a t 5 Hz for C, = 0.05% and C, = M NaC1. These results might suggest two separate relaxation mechanisms for the DNA, but the present results a t similar experimental conditions evidence a single relaxation region over the covered frequency range. The present data are rather closer to those by Hanss and Bernengo than those by Takashima when compared a t similar experimental conditions. The rotational relaxation time rrot of DNA in solution has been measured by electric dichroism by Ding et al.,23flow dichroism by Callis and D a ~ i d s o n , ~and * dynamic light scattering by Schmitz and S ~ h u r r . ~ ~ The data are summarized in Figure 6 as a function of molecular weight, where values are reduced to 10°C, assuming that rrotis proportional to the solvent viscosity. An alternative way for obtaining rrotis to use the intrinsic viscosity [q]. For a bead-spring model with a dominant hydrodynamic interaction among beads (Zimm the viscoelastic relaxation time rz related to the rotation of end to end vector of the chain (the longest relaxation time in Zimm’s theory) is given as 72
= 0-42Mrqs[q1
RT
(coiled chain)
where M , is the molecular weight, R is the gas constant, and T is the absolute temperature. For a rigid rod, on the other hand, KirkwoodAuer’s theory27gives the viscoelastic rotational relaxation time T K A as TKA =
5Mrqs[q1
4RT
(rigid rod)
(3)
Values of T Z and ?KA as functions of Mr were calculated from Eqs. (2) and (3) along with an empirical relation between [q] and M , given by Reinert, et al.17 Results a t 10°C are drawn in Figure 6 by solid lines.
DNA DIELECTRIC RELAXATION
887
4
-2 I
5
6
7
0
9
LOG M Fig. 6. Doubly logarithmic plot of rotational relaxation times (lO°C) against molecular electric d i c h r o i ~ m (O), ; ~ ~ flow d i c h r o i ~ m (a), ; ~ ~ dynamic light scatweight of DNA: (a), and solid lines; T Z and T K A calculated from Eqs. (2) and (3), respectively.
At low molecular weights where DNA is rodlike, T K A agrees with Trot, while, at high molecular weights where DNA may be representable as a coiled chain, T Z agrees with Trot. In any case, these results indicate that we can estimate Trot from the viscosity data if the shape of the chain is appropriately assumed. If we assume that, first, the dielectric relaxation arises from the rotation of DNA molecules and secondly, the dipole moment vector and the end to end vector of the molecule have the same direction and are proportional in magnitude to each other, Td should be equal to the rotational relaxation time of end to end vector and thus 272
Td
N
7d
=3
(coiled chain)
or 7
~ (rigid ~ rod)
Equation (4)was proved analytically by Fujimori et a1.28 for a Rouse model, which is a bead-spring model without hydrodynamic interaction. Equation (4)holds for the Zimm model with a minor error. In Table I, 272 and 3 T K A are calculated from Eqs. (2) and (3), respectively. In this calculation we used the reduced viscosity Vred at C , = 0.01% in place of [ v ] , because 7d was also obtained at a finite concentration, C , = 0.01%. The intermolecular interactions, if any, should affect Td and Vred by the same factor. In Table I, T d compares rather well with 272 for three salt concentrations. This evidences that the DNA in the present case forms a coiled chain rather than a rigid rod and the dielec-
SAKAMOTO ET AL.
888
tric relaxation arises from the rotation of the molecule. It has been evidenced from the electric dichroismZ3and electric birefringence measurements12 that the DNA behaves as if i t might possess a permanent dipole. The permanent dipole means here the dipole which may change more slowly than the rotation of the DNA. Ding et al.23 suggested that the permanent dipole might arise from the saturation effect of induced polarization of counterions. The reversed pulse experiment on the electric birefringence by Hanss et a1.,I2 however, seems to evidence a mechanism of permanent dipole different from this saturation effect. In fact, the dielectric relaxation which occurs in a low electric field requires the existence of a permanent dipole even at the zero electric field. The increases in both Ac and 7d with decreasing salt concentration (see Table I) may be primarily explained by the expansion of chain. Salt ions shield the electrostatic repulsion between charges on DNA and reduce the chain expansion. In Table I, the rms dipole moment of DNA is calculated from At at C , = 0.01% by Kirkwood's formula:29 (p2)1/2N
3.3
(C,:mole DNAh)
(6)
Our values of ( p 2 ) ' l 2 (or simply denoted p ) = 5.2 X lo5 D and 7d = 16 msec ( M , = 4 x 106, C , = 10-3 M , C , = 0.01%) are in general agreement with those by Hanss and Bernengo,12p = 7.2 X lo5 D and rd = 18 msec ( M , = 5 X lo6) (note that p1 for sample 1 in Table I in their paper should be read as 7.2 X lo5 D). Values by T a k a ~ h i m ap, ~= 3.2 X lo4 D and 7 d = 0.09 msec (M,= 5 X lo6) are much lower than ours probably because the frequency range in his experiment covered only the highfrequency part of the relaxation region. Analogously, Jennings and Plummer30 who observed light-scattering in a.c. field a t frequencies above 100 Hz reported lower values than ours, p = 7.4 X lo4 D and Td = 0.4 msec (M, = 3 X lo6), probably because the frequency was not extended to sufficiently low values. In Figure 6, the closed circles represent maximum relaxation times obtained from the electric dichroism measurement by Ding et al.23 Two circles for M , = 6.0 X lo6 and 6.7 X lo6 are located somewhat lower than both 7KA and TZ probably because the duration of exciting d.c. pulse in their experiment, ca. 40 psec, was too short to observe the slow relaxation mechanism. The slow mechanisms are not excited by a short pulse and thus do not appear in the subsequent decay. Accordingly, the value of dipole moment estimated from the field dependence of dichroism reported in their paper is p = 1.0 X lo4 D ( M , = 6.7 X lo6) which is much lower than our value. In fact, Hanss and Bernengo12 who used a d.c. pulse of 20 msec for excitation in their electrical birefringence measurement found two relaxation times: the longer one of them agrees with rd estimated from the dielectric loss peak. For DNA with a much lower molecular weight, however, the pulse duration employed by Ding et al.23seems satisfactory to observe the whole relaxa-
DNA DIELECTRIC RELAXATION
889
tional behavior. Actually the circles for M , < lo6 in Figure 6 fall on the curve of 7 K A and p estimated from the field dependence of dichroism ( p = 0.7 x lo4 D for M , = 0.32 X lo6) is close to that for M , = 6.7 X lo6 despite of the large difference in M,. T o summarize, observation of the whole relaxational behavior of DNA with a wide distribution of relaxation times requires a wide frequency range or a long duration of exciting d.c. pulse. On account of a double-stranded structure of antiparallel helixes, the DNA molecule by itself does not possess a dipole moment along the chain.31 Two mechanisms have been proposed by Hanss and Bernengo12 for the origin of dipole moment in DNA: ( I ) a difference in the ionic composition of the two molecular ends, such as differences in residual bound amino acids or proteins (end charge mechanism), and (2) counterion fluctuation with a relaxation time longer than Trot (counterion mechanism). The end charge mechanism, however, seems unprobable because the magnitude of (p2)l12is too large to be accounted for by this mechanism. As has been described in the first part of this section, the rms end to end length ( h 2 )112a t C , = lov3A4 is estimated as 0.56 1.1. For explaining the value of ( p 2 )112 = 5.2 X lo5 D by this mechanism, we must assume the end charges of f 1 9 e (e being the elementary charge), which are unreasonably large. In the following we shall discuss on the counterion mechanism. We assume two kinds of counterions diss’ociated from the polymer:19 ( I ) counterions which are bound near the polyion (site-bound ions), and (2) counterions which are located in the environment apart from the polyion and contribute to the conductivity of the solution (free counterions). We consider the charge distribution on the polyion including site-bound ions and calculate the rms dipole moment of polyion plus site-bound ions, which will be denoted simply “polyion” in the following. Let us consider a polyion consisting of N monomeric units, each having a single monovalent binding site (phosphate group in the case of DNA). The dipole moment vector p of the polyion is defined as N
p
=
1 6nie(ri - rc) i= 1
(7)
where ri and r G = (1/N) ZE1 ri are position vectors of the i-th site and the center of gravity, respectively, and Gnie is defined by 6n;e = (n; - no)e
(8)
Here nie is the charge on the i-th site (ni = 0 for a site which is either undissociated or occupied by a site-bound ion, n; = -1 for an unoccupied acid site) and noe is the mean value of nie. From Eq. (7) we obtain the mean-square value of p as
SAKAMOTO E T AL.
890
where the angular brackets represent double averaging processes; an average over different conformations of the polyion and an average over different charge distributions on the sites of the polyion. It is assumed here that the conformational change (including rotation) of the polyion is fast enough to attain an equilibrium and, on the other hand, the change in the charge distribution is slow enough compared with the time scale in which we are considering (p2) in Eq (9). In other words, the first average above mentioned may be regarded as a time average and the second as a spatial one, i.e., an average over different polyions. Since these two averaging processes can be treated as approximately independent of each other, we may obtain from Eq. (9) N
(p2) =
N
C C (6ni6nj)e2((r;- r d + j - r d ) i=l j=1
(10)
Furthermore, if we assume for simplicity that there is no correlation between ion bindings at different sites and the probability of ion binding at each site is 1 - a (adenoting the fraction of free counterions), then we have (6ni6nj)
=0
(6ni6nj)
= a ( 1 - a)
(11)
Substituting Eqs. (11) into Eq. (lo),we have the final result
where ( S2)is the mean-square radius of gyration of the polyion defined by
Incidentally, this result is valid also for the case where the change in the charge distribution on the polyion is fast enough to realize a charge distribution equilibrium because the time average of the charge distribution appearing in this case is formally equivalent to the spatial average in deriving Eq. (12). Recently, Van der Touw and Mande132 have developed a theory on the dielectric relaxation of polyelectrolytes. They took into account two mechanisms of the polarization, the local counterion fluctuation within the subunit of the polyion and the counterion fluctuation over the whole molecule. The latter is essentially the same as the counterion mechanism in this paper and Eq. (12) is analogous to Eq. (31) in their paper except that we considered a discrete distribution of bound ions along the polyion whereas they assumed a continuous distribution of them. In the present case of DNA, quantities appearing on the righthand
DNA DIELECTRIC RELAXATION
891
side of Eq. (12) can be estimated as follows: The number of monomeric units in the double-stranded DNA, N , is calculated as 1.3 x lo4 from the molecular weight. As mentioned earlier, the d.c. conductivity data always satisfied the additivity law, and one-third of Na+ ions originating from Na-DNA was found to contribute to the conductivity. Thus, we may take a = %,. The mean-square radius of gyration ( S 2 )is ap; proximately related with ( h 2 )as ( S 2 ) N ( h 2 ) / 6 . 2 2 In the case of C, = M , we obtain ( S 2 ) l I 2= 0.23 I.L from ( h 2 ) l I 2= 0.56 p. Substituting these values into Eq. (12), we have ( p 2 ) l I 2 = 5.9 X lo5 D, which is in a good agreement with 5.2 X lo5 D in Table I. The fluctuation of charge distribution on the polyion is induced by the transfers of counterions between polyion and environment and/or along the polyion. The counterion mechanism presented in this paper assumes that the relaxation time of this transfer, 7t, is longer than the dielectric relaxation time of the polyion, 7d. Estimation of T~ is still obscure a t present. G ~ t t l i e bdemonstrated ~~ by a radioactive tracer experiment that the transfer time of counterions from polyion to environment is less than one second for Na-polyacrylate of molecular weight 9.4 X lo5 and C, = 0.1-0.45%. Lifson and Jackson34predicted from a theoretical point of view that the transfer time should depend on the electrostatic potential of the polyion and a value less than one millisecond may be probable. In any case, we might not expect 7t longer than 7d as far as we consider only a free diffusion of counterions between polyion and environment or along the polyion because the diffusion length is one micron in order of magnitude in the present case of DNA. We must consider, therefore, some transfer mechanism of counterions different from the free diffusion for explaining 7t longer than 7d. Relative magnitudes of 7d and 7t were discussed by Van der Touw et a1.,32735,36but no conclusion has been derived by them. Results on the dielectric relaxation of DNA in the present study as well as those of partially neutralized polyacrylic acid and Na-polystyrene sulfonate by Van der Touw and Mande1,35 both having been reasonably explained by the counterion mechanism, seem to suggest that 7t is longer than 7d. The shielding effect on the polyion charge due to surrounding salt ions may be stronger in the end charge mechanism that in the counterion one because, in the former, the charges on the polyion are localized on the ends of chain. Since the radius of ion atmosphere in the DebyeHuckel theory is ca. 100 8, a t C, = M , being much smaller than ( h 2 ) I l 2the , end charges, if any, would be considerably shielded by salt ions. It is to be noted in this connection that, as has been calculated by Hall,37 the relaxation time of formation of ion atmosphere is very much shorter than the rotational relaxation time of DNA. In the present stage, the reason for 7t longer than 7d is still obscure. A direct measurement of 7t is desirable to answer this question. Dielectric measurements for DNA with lower molecular weights, purified DNA without residual proteins, and DNA in various kinds of salt solution may be also useful to clarify the origin of dipole moment.
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SAKAMOTO E T AL.
This research was supported by a Grant-in-Aid for Fundamental Scientific Research, Japan. The authors would like to express their gratitude to Dr. E. Fukada and Dr. M. Kaibara a t the Institute for Physical and Chemical Research for making available a lowshear viscometer. Thanks are also due to Miss N. Fujinawa for her help in preparing the manuscript.
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Received July 8,1975 Accepted November 26,1975
