J. Phys. Chem. 1994, 98, 10881-10887
10881
Brownian Dynamics of the Polarization of Rodlike Polyelectrolytes Tomasz Grycuk and Jan Antosiewicz Department of Biophysics, Warsaw University, 02-089 Warsaw, Poland
Dietmar Porschke* Max Planck Institut jiir biophysikalische Chemie, 37077 Gottingen, Germany Received: May 16, 1994; In Final Form: August 2, 1994@
The effect of external electric field pulses on the ion atmosphere around linear polyelectrolytes is characterized by Brownian dynamics simulations. The polymer is fixed at the center of a cube and is surrounded by mobile counterions and byions; electrostatic interactions beyond the cube are evaluated using periodic boundary conditions and the Ewald summation technique; hydrodynamic interactions are not included in the simulation. The polymers are constructed in analogy to DNA double helices with a constant charge spacing of 0.17 nm. Electric fields applied parallel to the long axis of the polymer induce a dipole moment parallel to this axis with time constants around 10 ns (at ion concentrations around 1 mM). The dipole of a polymer with 40 charged residues approaches saturation at field strengths in the range around 50 kV/cm. Parallel to the dipole rise, there is a dissociation of counterions from the polymer reflected by an increase of the root mean square distance (s,2)ll2 of the counterions from the center of the cube. The ( s > ) ~ / ~value increases linearly with the field strength E up to E rz 100 kV/cm; this effect and its dependence on E are in close analogy to the “dissociation field effect”. The dipole moment at a given field strength and ion concentration does not increase with more than the square of the chain length under the conditions of our present simulations. A relatively small decrease of the dipole moment is observed when the ion concentration is increased.
Introduction The effects of external electric fields on polyelectrolytes have been investigated extensively by various experimental and theoretical procedure^.^-^' The polarization of the ion atmosphere around polyelectrolytes is known to be the basis of most processes induced by electric fields in biopolymers, but because of the complexity of polyelectrolytes, it has been very difficult to describe the polarization quantitatively. Among the various polyelectrolytes, DNA double helices have been a preferred object for investigations, partly because of its biological relevance but also because the structure of DNA double helices is well-defined and relatively rigid. A major advantage results from the possibility to prepare DNA double helices of various well-defined uniform chain lengths. The effects of electric fields on “DNA restriction fragments” have been mainly analyzed by electrooptical procedures, because electrooptical measurements can be conducted with very small amounts of material. Results from various laboratories demonstrated that the polarizability at low field strengths increases with the square of the chain length,8~10~12 but in one casell an increase of the polarizability with the cube of the chain length has also been reported. At high field strengths and/or at high chain lengths, the induced dipole tends to saturate;8s10s12under these conditions, the experimental data appear as if DNA bears a permanent dipole moment. The polarization has been mainly characterized by electrooptical measurements of the stationary degree of orientation using the linear dichroism or the birefringence, which reflects the polarization process itself rather indirectly. A more direct experimental approach to the polarization has been found during a detailed quantitative analysis of the dynamics of the molecular alignment induced by application of external field pulses. Measurements of dichroism rise curves at a very high time @
Abstract published in Advance ACS Abstracts, September 15, 1994.
0022-365419412098-10881$04.50/0
r e s ~ l u t i o nrevealed ~~ that the rise of the dichroism is a convolution product of polarization and rotational diffusion; deconvolution showed that polarization proceeds with time constants in the range around 10 ns. The dependence of the polarization time constant on the electric field strength and on the salt concentration could be described by a simple consistent kinetic m0de1.l~ Subsequently, a more detailed model on the coupling of field-induced polarization and orientation has been developed by Szabo et al.15 The theoretical interpretation of polyelectrolyte polarization has been tried by various models. S ~ h w a r z l -has ~ analyzed the anisotropy of the ion conductivity of polyelectrolytes; according to his model the polarizability increases with the third power of the chain length. Mande14proposed a cylinder model with charges regularly distributed over its length and with counterions, which are mobile but cannot leave the region of the ion cloud around the polymer. According to Mandel’s model, the dipole moment is expected to increase with the third power of the chain length. Oosawa5calculated the mean square electric moment due to thermal fluctuations of the bound counterions. Rau and Chamey6 developed a model based on the counterion condensation theory of Manning7 and concluded that the contribution of the Debye Hiickel ion atmosphere should lead to an increase of the dipole moment with the square of the chain length. Hogan et aL8 proposed a model based on anisotropic ion flow and predicted an increase of the dipole moment with the square of the chain length. “Convective polarization” due to the relative electrophoretic motion has been analyzed by Fixman and Jaga~mathan;~ according to their calculations, the polarizability is expected to increase approximately with the square of the chain length. The literature on the theory of polyelectrolyte polarization is quite extensive and cannot be reviewed completely in this context. Obviously, polarization of polyelectrolytes is a very complex process, which 0 1994 American Chemical Society
10882 J. Phys. Chem., Vol. 98, No. 42, 1994
can hardly be described by simple models. Thus, it should be useful to analyze polyelectrolyte polarization by numerical simulations. A first analysis of polyelectrolyte polarization by numerical procedures has been published by Watanabe and co-workers18-20 who used a Metropolis Monte Carlo procedure. In our present investigation we have used a Brownian dynamics simulation, which provides a more detailed view of the polarization process and of its dynamics. In the first stage of the development of the project we have neglected hydrodynamic interactions (cf. ref 9 and references cited therein). Preliminary simulations have shown that consideration of hydrodynamic interactions requires much more computer time and indicated that hydrodynamic interactions are not dominant under the conditions used for our present simulations.
Simulation Procedure
Our simulation procedure is based on an algorithm published by E ~ m a k .We ~ ~analyze a model consisting of a single rigid cylindrical polyion with a given number of counterions and byions in a cube with a side length L. This cube is surrounded by identical images using periodical boundary conditions. The ions are assumed to be in a liquid solvent; we used a relative dielectric constant cr = 80.2 corresponding to an aqueous solution at 20 "C. We want to compare the results of our simulations with experimental data obtained for DNA, but we did not try to model the DNA double helix in detail. For example, we have not considered the low dielectric constant inside the DNA: in our simulations we have used the same dielectric constant inside and outside the cylinder representing DNA. The DNA is modeled as a line of N negative charges with a constant spacing parameter b; the charges are located on the symmetry axis of a cylinder with the radius a. The length of the cylinder is 1 = ( N - l)b 2a. For the simulations described below we used b = 0.17 nm and a = 0.5 nm. The separation of charges along the axis corresponds to that of B-DNA; the radius of the cylinder corresponds to that used in the simulations of Watanabe et aLZo The radius is lower than that of double-helical DNA; a relatively low radius had to be used for an approximately correct description of the electrostatic interactions, because the charges of the DNA are located at the center of the cylinder. The DNA model is fixed to the center of the cube, and its symmetry axis is identical with the z-axis of our coordinate system. Counterions and byions are mobile with a translational diffusion coefficient 1.16 x m2 s-l. This value has been calculated for 20 "C from the one reported by YoshidaZ2(cf. ref 23) for 25 "C by the standard viscosity/ temperature conversion factor [D2/D1 = (~717'2)/(~72T1)1.For simplicity we have used the same diffusion coefficient for the mobile anions. This coefficient for translational diffusion corresponds to a hydrodynamic radius of 0.185 nm; thus, counterions and byions are modeled as spheres with a radius of 0.185 nm. Our program is designed to include motion of the DNA model. In our present simulations the DNA model is kept in the center of the cube and, thus, the diffusion coefficients of the small ions should be somewhat modified. However, the diffusion coefficient of DNA is at least 1 order of magnitude smaller than that of the small ions and, thus, the correction has been neglected. For comparison we note that Ermak21 used a translational diffusion coefficient of 2 x m2 s-l at 20 "C and an ion radius of 0.107 nm; Watanabe and co-workersZ0used an ion radius of 0.15 nm. The counterions and the byions are first distributed randomly within the cube but are not allowed to enter the cylinder representing the DNA. The ions are then subjected to electro-
+
Grycuk et al. static forces-and to random displacement: the overall electrostatic force Fi results from interactions with the other charges and from any external slectric field that may be applied; the random displacement &(At) results from interactions with solvent molecules. The motion within a small time step At is described by Ti(t
+ At) = Ti(t) + iT,(At) + DoFfi(t)Aht
(1)
where Do is the diffusion coefficient (in the absence of effects resulting from electrostatics); kT is the thermal energy; the random displacement &(At) is described by a Gaussian probability distribution function with an average value of 0 and a mean square value of each component of 2DoAt. In our present simulations the interactions between counterions and byions at small distances is described by the model of "soft" spheres. The potential of interaction is given by
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Hydrodynamic interactions between moving ions as well as between ions and the DNA model are neglected. The time period At = 4 x lo-" s used in our simulations is equivalent to that selected by Ermak.*l The cube described above is surrounded by other cubes, which are identical with the central one. The electrostatic interactions between charges extend beyond the central cube, Le., each ion interacts with a given other ion in the cube and with all its images in surrounding cubes. These interactions are calculated by the Ewald summation t e ~ h n i q u e . ~ According ~,~~,~~ to ,this ~~ technique, the positions of ions are monitored only in the central cube, because the system is assumed to be periodic. If a particle travels out of the cube at one side, an image enters the cube through the opposite side; the number of particles in the cube (and also in 211 the surrounding cubes) remains constant. The force Fv on the ion i due to the ion j is calculated as the gradient of the modified pair potential @g
-F . .= -VQ... 11
11
where
x = 7q/L, Ui/l) is the isotropic term of the modified potential,
and Ui1(2) is the anisotropic term: 1 U i l ) ( x )= ( ~ r f c ( & ) - 1 - E,)
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2
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We have used a polynomial expansion for the calculation of
Polarization of Polyelectrolytes
J. Phys. Chem., Vol. 98,No. 42, 1994 10883
the anisotropic term25U V ( ~ The ) . analytic form of the modified pair force is directly obtained from the gradient of the potential
a
I
0j j . During movement of the ions according to eq 1, different parameters characterizing the basic cube are calculated and recorded. In our present communication we use the dipole moment of the basic cube ji and the root of mean square distance of counterions from the center of the basic cube ( s , ~ ) ~ ~ ~ :
I
1500 Clz
.N.-l ji = CqiFi
500
i= 1
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where NI is the number of ions, including all ions with both positive and negative charge and NC is the number of counterions, corresponding to the ions with positive charge. Results from several tests provide evidence that the algorithm used for our simulations is correct. First, it has been shown that the ion distribution around a spherical polyion in the absence of an external electric field corresponds exactly to the one simulated by Ermak.21 Then, the ion distribution around a linear polyelectrolyte, again in the absence of an extemal electric field, proved to be equivalent to that reported by various authors (LeBret and Z i n ~ mFuoss, ; ~ ~ Katchalsky and Lif~on;~O Alfrey, Berg, and Morawetz3I). It is virtually impossible to check the validity of data simulated for polyelectrolytes in the presence of an external electric field because sufficiently accurate predictions are not available. However, we have checked that our procedure does not lead to a field-induced dipole moment of the cube with ions when there is no fixed electrolyte. Trajectories have been collected according to the following procedure: after random assembly of the ions in the cube, a time period was simulated in the absence of an external electric field for equilibration of the ions around the polymer. At the end of this period the configuration of ions was stored, the electric field was turned on, and a trajectory was simulated until a stationary state. For the next trajectory the ion configuration at the end of the (last) preequilibration was restored and used to start another preequilibration period, in order to get a new independent starting configuration. At the end of this preequilibration, the configuration was stored again, before the extemal electric field was applied and the next trajectory was simulated. This procedure was repeated and the information extracted from the individual trajectories, e.g., the dipole moment and the root mean square distance, was accumulated. Accumulation of data included the preequilibration periods; the preequilibration data have been used for the evaluation of the ( s > ) ~ /data: ~ the average level obtained from this period was introduced as the starting level at t = 0 for the exponential fit of the ( s , ~ ) ~ /rise * curve.
Results Application of an electric field parallel to the long axis of the polymer (i.e., in the z direction) induces a dipole moment along this axis. At low electric field strengths, the rise of the dipole moment may be described by a single exponential (cf. Figure la), whereas two exponentials with amplitudes of opposite sign are required at higher field strengths (cf. Figure lb). All the rise curves found at higher field strengths exhibit a distinct maximum of the dipole moment during the initial period, which is due to the following effect: application of an electric field induces dissociation of counterions, which have been associated with the polymer in the absence of the electric
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Figure 1. Rise curve of the dipole moment simulated for a linear polymer with 80 charges at a distance of 0.17 nm between these charges with an equivalent number of counterions in a cube of 403nm3.Electric field strength: (a) 20 kV/cm; (b) 50 kV/cm (average of 5 12 trajectories; the lines without noise represent least squares fits with the following parameters: (a) r = 18.6 ns, pz = 1678 D; (b) tl = 19.2 ns, pzl= 37067 D; t2 = 21.3 ns, p> = -34084 D).
field; these counterions move away from the polymer in the z direction and, thus, provide a special contribution to the dipole moment; these dissociated counterions are not only subject to electrophoretic motion but also to Brownian motion and, thus, the special contribution to the dipole moment usually disappears after a short time (cf. Figure lb). Another way to look at the process(es) induced by application of the electric field has been used by recording of the root mean square distance ( s , ~ ) ~of/ ~the counterions from the center of the cube. As shown in Figure 2, the electric field induces an increase of the ( S C Z ) ~ / value. ~ This increase reflects dissociation of counterions from the polymer. Whenever the signal to noise ratio was sufficiently high, two exponentials with amplitudes of opposite sign were required for a satisfactory fit of the (sc2)1/2
10884 J. Phys. Chem., Vol. 98, No. 42, 1994
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Figure 2. Root-mean-square distance ( s , ~ ) ~of~ the ~ counterions simulated for a polymer with 80 charges at a distance of 0.17 nm between these charges with an equivalent number of counterions in a cube of 403nm3 after application of an electric field of 50 kV/cm; the line without noise represents a least squares fit by two exponentials; the lower panel shows the residuals (tl = 17.5 ns, t2 = 20.4 ns, A((s?)~/~)I= -0.152, A ( ( S , ~ ) ~ = / ~ )0.167; ~ ( s , ~ ) values ~ ~ ~ are given relative to the cube side length and, thus, are without dimension; average of 512 trajectories). rise curves. At the given accuracy of the available data, it is not clear yet whether the time constants derived from the (s:)lI2 rise curves are identical with the time constants of the dipole rise curves. Usually the time constants are rather close to each other, but in some cases we have observed differences, which require further investigation. It is likely that the response of the polymer with its many counterions to electric fields cannot be described by one or two simple normal modes of reaction. Probably there is a relatively broad spectrum of processes with different time constants; the dipole moment and the root mean square distance may represent different averages of the spectrum of individual processes. The example given in Figure 2 shows that the (s?)l/* value increases with some time delay after application of the field pulse. This effect has been found in most of our simulations. Evaluation of such data by standard exponential fitting routines leads to two exponentials with amplitudes of opposite sign. It is possible that the delay reflects a convolution product of two processes (cf. ref 14). As should be expected, the dipole moment increases with increasing field strength. However, the stationary values of the dipole moments tend to saturate already at relatively low electric field strengths. Simulations at low electric field strengths, Le., E 5 20 kVIcm, require particularly large numbers of trajectories corresponding to extensive computer time for a reasonable signal to noise ratio and, thus, the range of constant polarizability has hardly been documented in the present investigation. In the case of a polymer with 40 residues (cf. Figure 3a), the dipole moment is saturated already at a field strength of E 50 kV1 cm. In this case the dipole moment and its field strength dependence are hardly affected by addition of 10 salt particles into the cube. The addition of 10 salt particles to the polymer with 40 charged residues including counterions represents an increase of the counterion concentration of 25% (from 1.04 to 1.3 mM).
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Figure 3. (a) Dipole moments in z direction pz as a function of the external electric field strength E for a polymer with 40 residues in a cube of 403nm3:(a) without added salt (x) and (b) with 10 salt particles (+). (b) Increase of the root mean square distance A(S?)’~induced by an external electric field of strength E (parameters as in a). The stationary values of the root mean square distances for a given system increase with increasing field strengths. As shown in Figure 3b, the A(sC2)l/*values are a linear function of the electric field strength. This result is in close analogy with the “2. Wien effect” observed for simple weak electrolyte^.^^ The analogy is also found with respect to the “threshold” in the electric field strength, which must be exceeded before the linear relation is ~ a l i d . ~A~theoretical ,~~ analysis of this “dissociation field effect” has been given by O n ~ a g e r . ~ ~ . ~ ~ Among the various dependences, which may be simulated, the dependence on the chain length is of particular interest. Simulations on linear polymers with 20,40, and 80 residues in a cube of 403nm3demonstrate an increase of the dipole moment with the chain length N , but a double logarithmic plot (cf. Figure 4) shows that the dipole moment increases only with N1.58. Various described in the literature predict an increase with fl or with N3. Thus, we have looked for factors affecting the magnitude of our simulated dipole moments. Application of an external electric field induces a deformation of the ion atmosphere around the polymer. We have to check how far the deformed ion atmosphere extends into space. The time required for a given system to arrive at its stationary state increases with the size of the cube and, thus, simulations have been restricted to relatively small cubes wherever possible. However, as shown in Figure 5 , the dipole moment of a polymer with 60 residues increases considerably with the size of the cube-even in a range of cube sizes, where the distance between the terminal charges of the polymer and the boundary of the cube is about 10 nm. This result demonstrates that the polarized ion atmosphere extends to a considerable distance from the
Polarization of Polyelectrolytes
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J. Phys. Chem., Vol. 98, No. 42, I994 10885 0 0 0
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Figure 5. Dipole moment in z direction pL induced in a polymer with 60 residues by an electric field E = 50 kV/cm as a function of the cube side length L (without byions).
polymer. It should be mentioned that variation of the cube size around a given polymer implies changes of the counterion concentration. Thus, a major part of the effect shown in Figure 5 may be due to changes of the counterion concentration. The dependence of the dipole moment on the cube size may affect the chain length dependence of the dipole moment given in Figure 4 because the chain length dependence has been simulated in a cube of constant dimensions. Due to limits in the available computer time, the cube size has not been increased until the magnitude of the dipole moment arrives at its maximal value and the cube size does not affect the result anymore. Nevertheless, it should be possible to get a chain length dependence of dipole moments, which is not perturbed by this effect, if the distance between the ends of the polymers and the boundary of the cube is kept constant. For this reason data have been simulated where the chain length and the size of the cube has been increased simultaneously, such that the distance between the terminal charges of the polymer and the cube boundary was at a constant value of 16.6 nm. These simulations (cf. Figure 4) performed in the range of chains with 40-80 residues show an increase of the dipole moment with N1,53.
The dipole moments for different chain lengths discussed above were calculated in the absence of added salt. Under these conditions, variation of the chain length leads to some variation of the counterion concentration, which may affect the magnitude of the calculated dipole moments. In order to compensate for any distortion of the chain length dependence resulting from this effect, data were simulated under conditions, where not only the distance between the polymer ends and the cube boundary but also the concentration of counterions was kept at a constant value. As shown by the data included in Figure 4, there is some influence of the salt concentration: according to the two data points for the chain lengths 40 and 80, the dipole moment at constant counterion concentration and at a constant distance of the polymer ends to the cube boundary increases with N1,68. For comparison, the chain length dependence of the induced dipole moment has been studied in the range of very low chain lengths-starting at the limit, where a single charged residue is fixed at the center of the cube and is surrounded by a single counterion. Although the dipoles generated by external electric fields are rather small in this range of chain lengths, the magnitude of these dipoles can be recorded at a reasonable signal to noise ratio. In the range from one to four residues, the dipole increases with the square of the chain length (cf. Figure 6). The increase of the dipole moment with the chain length levels off already in the range of 8-16 residues. Some simulations have been performed on the dipole moment at different salt concentrations for a polyelectrolyte with 40 charged residues in a cube of 403 nm3. Due to limitations in the available computer time, the simulations had to be restricted to the range of relatively small salt concentrations. Addition of 60 salt particles corresponding to a 1.56 mM concentration of added salt resulted in a decrease of the dipole moment by about 15%. The time constants of the field-induced processes are not presented because their accuracy does not appear to be sufficient yet. Nevertheless, some preliminary conclusions should be mentioned. Simulations on a polymer with a given number of residues in the absence of added salt show that the time constants decrease with increasing cube size. This effect is attributed to the decrease of the counterion concentration with increasing cube dimensions. The dependence of the time constants on the chain length in the range from 40 to 80 residues appears to be rather small, whereas a clear decrease of the time constants is found at lower chain lengths. These results appear to be equivalent to experimental polarization time constants derived from dichroism rise curves.14 Currently our simulations are extended to higher numbers of trajectories for a more accurate assignment of these effects.
10886 J. Phys. Chem., Vol. 98, No. 42, 1994 Discussion
Grycuk et al.
Brownian dynamics simulations indicate that the influence of this effect remains relatively small when standard values are Although theoretical models on the effect of external electric used for the diffusion coefficients. However, final conclusions fields in polyelectrolytes have been developed by many different on the effect of hydrodynamic interactions require more author^,'-^.^-^ it has not been possible yet to arrive at a consistent extensive computations. Our present simulations were also description of these effects. Because of the complexity of restricted to the longitudinal dipole generated by electric fields. polyelectrolytes, it proved to be difficult to include all the For a more accurate comparison with experimental data we have various effects induced by external electric fields. Under these to calculate the difference between the longitudinal and the conditions, numerical simulations are required for an assignment perpendicular dipole moment. Another problem for further of the effects resulting from electric field pulses. The simulation investigations is nature of the dipole moment. In our present approach avoids simplifications, which have to be introduced simulations we have calculated the dipole moment by simple in analytical or semianalytical procedures. Nevertheless, we summation of the contributions from all charges in the cube. A have to check whether any of the numerical procedures used in more direct measure of the dipole moment-at least more closely our approach may introduce some error. The most important corresponding to the one observed in experiments-is the torque approximation in our approach is the use of periodic boundary on the polyelectrolyte induced by the external electric field. conditions together with the Ewald summation t e c h n i q ~ e ? ’ * ~ * ~ Future ~ ~ ~ ~ simulations of this parameter at various orientations of In the literature it has been demonstrated by other authors and the polymer axis to the field vector may help to characterize it has been checked independently for our computer program the field-induced dipole in more detail. that the ion distributions calculated according to this procedure More information may also be expected from simulations of are correct. Thus, it may be expected that correct ion distributhe time constants with an increased accuracy. The prediction tions are provided by this procedure also in the presence of of these time constants is one of the major advantages of the external electric fields. Brownian dynamics approach. The time constants obtained by Part of our results are similar to the ones reported by our present simulations are consistent with the available Watanabe and co-workers.18-20 However, some important experimental data. In the past it has been difficult to assign results of our simulations, in particular the chain length experimental time constants because of the complexity of dependence of the induced dipole moment, are clearly different. polyelectrolytes. The results of simulations, including, for Watanabe and co-workers report an increase of the dipole example, the root mean square distance of counterions (cf. moment with the cube of the chain length, whereas the dipole above), are clearly useful as a reference for comparison with moment does not increase with more than the square of the experimental data, which have been performed to characterize chain length according to our simulations. Our results are the state of ions associated with polyelectrolytes. Time consistent with the experimental results reported by Hogan et constants may be used, for example, to distinguish “outer-sphere’’ al.,sStellwagen,’O Diekmann et al.,12 and a later independent from “inner-sphere’’ c o m p l e ~ e s , ~and ~ - thus, ~ ~ the results of investigation by one of the authors.13 A different result has simulations may also contribute to the identification of structures been reported by Elias and Eden;” according to their data the in solution. dipole moment increases with the cube of the chain length. It may be concluded, however, that there is clearly more evidence, Acknowledgment. The facilities of the Gesellschaft fur both from theory and experiment, for an increase of the wissenschaftliche Datenverarbeitung mbH Gottingen were used polarizability with not more than the square of the chain length. for part of our calculations. Our project was supported by a The absolute values of the dipole moments obtained by our grant from the “Sonderprogramm zur Forderung der Zusamsimulations are in the correct order of magnitude, but the menarbeit mit Mittel- und Osteuropa” of the Max Planck calculated dipole moments are larger than experimental ones Gesellschaft. by factors in the range 2-3. For example, the polarizability 1.05 x C m2 V-’ found for a DNA restriction fragment References and Notes with 43 bp (cf. ref 12; in a buffer with an ionic strength ~ 2 . 4 4 (1) Schwarz, G. Z . Phys. 1956, 145, 563. mM at 20 “C) corresponds to a dipole moment of 630 D at 20 (2) Schwarz, G. 2.Phys. Chem. N. F. 1959, 19, 286. kV/cm; the dipole moment simulated at this field strength for (3) Schwarz, G. J. Phys. Chem. 1962, 66, 2636. (4) Mandel, M. Mol. Phys. 1961, 4, 489. a linear arrangement of 80 charges at a spacing of 0.17 nm is (5) Oosawa, F. Biopolymers 1970, 9, 677. e1700 D (in the absence of added byions). The difference (6) Rau, D. C.; Charney, E. Biophys. Chem. 1981, 14, 1. appears to be partly due to the fact that the concentration of (7) , . Mannine. G. 0. Rev. Bioohvs. 1978. 11. 179. (8) Hogan, k;Dzttagupta, N.; Crothers; D.’M. Proc. Natl. Acad. Sci. byions in our simulations was smaller than that used in the U S A . 1978, 75, 195. experimental investigation. Another part of the difference is (9) Fixman, M.; Jagannathan, S. J . Chem. Phys. 1981, 75, 4048. expected to come from the simple DNA model used in our (IO) Stellwagen, N. Biopolymers 1981, 20,399. (11) Elias, J. G.; Eden, D. Macromolecules 1981, 14, 410. simulations. Although the results are somewhat different with (12) Diekmann, S.; Hillen, W.; Jung, M.; Wells, R. D.; Porschke, D. respect to details, the main features of the field-induced effects Biophys. Chem. 1982, 15, 157. observed in experiments are reproduced by the simulated data. (13) Porschke, D. Biopolymers 1989,28, 1383. For example, saturation of the polarizability,8~10~12 polarization (14) Porschke, D. Biophys. Chem. 1985, 22, 237. (15) Szabo, A,; Haleem, M.; Eden, D. J. Chem. Phys. 1986, 85, 7472. time constants in the range around 10 ns (ref 14), and field(16) Mandel, M.; Odijk, T. Annu. Rev. Phys. Chem. 1984, 35, 75. induced d i s s ~ c i a t i o nof~ ~ions are quite similar. Because we (17) Charney, E. Q. Rev. Biophys. 1988, 21, 1. have used a rather simplified model of the DNA double helix (18) Kikuchi, K.; Yoshida, M.; Maekawa, T.; Watanabe, H. Colloid and Molecular Electro-Optics 1991; Jennings, B. R., Stoylov, S. P., Eds.; in our present simulation, it should not be a major problem to Institute of Physics Publ.: Bristol, 1992; p 7. get a better approximation of the experimental data by fine (19) Watanabe, H.; Yoshida, M.; Kikuchi, K.; Maekawa, T. Colloid and adjustment of model parameters. Molecular Electro-Optics 1991; Jennings, B. R., Stoylov, S. P., Eds.; Institute of Physics Publ.: Bristol, 1992; p 61. One of the problems, which should be analyzed in further (20) Yoshida, M.; Kikuchi, K.; Maekawa, T.; Watanabe, H. J . Phys. detail, is the effect of hydrodynamic interactions (also denoted Chem. 1992, 96, 2365. as “convective transport”; cf. ref 9 and references cited therein) (21) Ermak, D. L. J . Chem. Phys. 1975, 62, 4189. on the magnitude of the dipole moment. Some preliminary (22) Yoshida, N. J . Chem. Phys. 1978, 69, 4867.
Polarization of Polyelectrolytes (23) (24) (25) (26) 2102. (27) (28)
Guldbrand, L. Mol. Phys. 1989, 67, 217. Ewald, P. P. Ann. Phys. ( k i p z i g ) 1921, 64, 253. Nijboer, B. R. A.; de Wette, F. W. Physica 1957, 23, 309. Brush, S. G.; Sahlin, H. L.; Teller, E. J . Chem. Phys. 1966, 45, Hansen, J. P. Phys. Rev. A 1973, 8, 3096. Wood, M. D.; Friedman, H. L. Z. Phys. Chem. N . F. 1987, 155,
121
(29) LeBret, M.; Zimm, B. H. Biopolymers 1984, 23, 271. (30) Fuoss, R. M.; Katchalsky, A.; Lifson, S. Proc. Nail. Acad. Sci. U S A . 1951, 37, 579.
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