J. Phys. Chem. 1991,95, 5983-5988 The same experiment was repeated with a 0.8 mM solution of
GM2 and a 31 mM solution of Triton X100, a synthetic nonionic surfactant. N o time evolution was observed as shown in Figure 2. The open dot at zero time represents the intensity value predicted for unmixed micelles. In this case the mixed micelle has a molecular weight that is between the molecular weights of the independent micelles of ganglioside and Triton.I3 A similar experiment performed with sodium cholate and GM2 showed the same behavior as shown in Figure 2, that is, no time evolution. Discussion The large difference in mixing times between the two systems GMZ-GTI b and GM2-Triton is easily justified within the framework of our simple kinetic model. The inverse characteristic time of the mixing process, eq 10, is a bilinear function of the escape constants of the two amphiphiles. Therefore, the larger of the two constants will dominate. For the case of the two gangliosides both constants are small, k-M = lo-" s-' and k-T = 1.2 X lo" s-l. Therefore L = -0.96 X lo" s-I and the evolution time is long enough to be measured directly. Due to its high cmc, the escape constant in Triton is much larger than k-M,so that k-M can be neglected in eq 10 and the overall evolution time is determined by the fast Triton dissolution process, which is undetectable on the time scale of our experiment. With sodium cholate the situation is exactly the same as for Triton.
5983
The order of magnitude of the escape constants of gangliosides is obtained from eq 3 with reasonable values of the collision frequencies u, in the range 107-109 s-l and the cmc of gangliosides, about 10-8 M.Is The collision frequency is roughly taken to be the inverse rotational diffusion time of a molecule inside a micelles2 The quality of the fit of Figure 1 is quite good, indicating that our simple model is effective. In conclusion, this work, besides giving a clean verification of simple self-assembly models, can also be of help in biochemistry research, where gangliosides are mixed with different amphiphiles. If they are mixed with low-cmc molecules such as phospholipids, mixing times can be quite long, while they may be quite different for other, less hydrophobic, amphiphiles.
Acknowledgment. Thanks are due to V. Degiorgio, S.Sonnino, and G. Tettamanti for useful discussions. L.C. thanks Fidia Spa and P. Salina Eni Ricerche for financial support. Work was partially supported by CNR Progetto Finalizzato Chimica Fine 11. Registry No. Ganglioside Gm2,19600-01-2; ganglioside GTI6,5924713-1. (15) Corti, M.; Cantti, L.; Sonnino, S.;Tettamanti, G. In New Trends in Ganglioside Research; Liviana Press: Padova, Italy, 1988; F.R.S.Vol 14, p
79.
Electric Moments of Rodlike Molecules Due to Asymmetry of Ligand Binding Induced by Electric Fields Jan Antosiewicz Department of Biophysics, Warsaw University, 02-089 Warsaw, Poland
and Dietmar Porschke* Max PIanck Institut f u r biophysikalische Chemie, 3400 Gottingen, Germany (Received: November 14, 1990; In Final Form: February 28, 1991)
Ligand binding to rigid linear and helical arrays of charges is simulated by a Monte Carlo procedure, in order to obtain information on electric moments due to asymmetry of site binding induced by an external electric field. Ligand binding is described by a free energy of site binding and by "internal" electrostatic interactions between sites. The effect of an external electric field is included by the energy of the dipole, which results from any asymmetry of ligand binding, in this field. Due to the internal electrostatic interactions the average degree of ligand binding is much larger than that expected according to the site binding constant; this effect corresponds to Yon condensation". The root-mean-squaredipole moment due to ligand fluctuations in the absence of an external electric field is decreased considerably by the internal electrostatic interactions. External electric fields induce an anisotropy of the ion binding corresponding to a mean dipole moment, which increases with the electric field. Optical anisotropies due to this effect, simulated for a wide range of field strengths, can be fitted with high accuracy by an induced dipole orientation function. The influence of the ion atmosphere due to Debye-HUckel screening on the resulting polarizability a remains relatively small, except for high salt concentrations. A rearrangement of the charges from a linear to a helical array, with charge densities corresponding to that of double helical DNA, leads to an increase of the a values by factors in the range 2-3. The polarizability a increases with the chain length n; for DNA fragments in the range from 43 to 95 base pairs the a values are linear with n2.68for the linear array and with n2.56(n2.'I) for helical arrays without (with) Debye-HUckel screening. The polarizability decreases with increasing salt concentration in the absence of screening but increases with increasing salt in the presence of screening. Some decrease of the polarizability is observed with increasing electric field strengths, but the saturation of the simulated polarizability is not as distinct as found experimentally for DNA fragments. The results obtained from the simulations for salt concentrations of a few millimoles per liter are surprisingly close to experimental data, in spite of the approximations used in these simulations.
Introduction Electric moments of macromolecules have been a challenge to physical chemistry because of their unusually large magnitude and because of considerable difficulties in interpretation. The difficulties mainly result from the fact that macromolecules usually are associated with many charged residues leading to strong electrostatic interactions, which are rather complex already in the absence of external electric fields. Various approaches to the 0022-3654/91/2095-5983$02.50/0
problem have been described. In most cases rodlike molecules have been used as models and the polarization has been attributed to motions of counterions along the long axis, to counterion fluctuations or also to convective polarization due to relative electrophoretic motion.'-" Very recently Plum and BloomfieldI3 (1) (2)
Mandel, M.Mol. fhys. 1961, I , 489. McTague, J. P.;Gibbs, J. H.J . Chcm. fhys. 1966, 44, 4295.
Q 1991 American Chemical Society
5984 The Journal of Physical Chemistry, Vol. 95, No. 15, 1991
have used Monte Carlo simulations of ligand binding to DNA double helices to support the view that asymmetries of ligand binding existing in the absence of an external electric field provides a sizeable contribution to the apparent permanent dipole moment observed for DNA. Detailed measurements of dichroism rise curves for rodlike DNA restriction fragments with high time resolution have demonstrated, however, that the dipole causing alignment of DNA parallel to the field vector does not preexist but is generated by the electric fie1d.l‘ By these measurements it has been possible for the first time to characterize the dynamics of polarization itself. The experimental data indicate that a major contribution to polarization comes from field-indud dissociation of ions from DNA molecules. The pulse reversal experiments quoted by Plum and B1oomfieldI3 as a support of their view are not conclusive, because all the pulse reversal experiments given in the literature are on relatively long DNA fragments, which show a strong contribution to optical anisotropy from internal m0bi1ity.l~ In this case electrooptical transients recorded upon pulse reversal cannot be used to demonstrate permanent dipole contributions, because the change of the field direction by the standard generators takes a considerable amount of time relative to the time constants of the internal dynamics. Thus, it is not very likely that asymmetries of ligand binding, which exist a t zero field strength, provide a major contribution to the electric parameters of molecules such as DNA. Another argument against the interpretation suggested by Plum and Bloomfield is the very fast relaxation of ion bindingI4 with time constants around 10 ns. Thus, fluctuations have a very short lifetime and cannot simulate a permanent dipole moment in electrooptical transients-except for the case of small objects with very high rotational diffusion coefficients. This argument holds for the whole range of electric field strengths. None of the claims in the literature for observation of permanent moments cited by Plum and Bloomfield is for sufficiently short DNA fragments. Electrooptical investigations of a short DNA fragment with 43 base pairs demonstrateI6 the existence of a standard induced dipole moment for the whole range of electric field strengths up to 80 kV/cm. In our present investigation we analyze the fluctuations of ion binding to a rodlike polyelectrolyte by Monte Carlo procedures and characterize two contributions, which have to be included for any description of field-induced effects in polyelectrolytes. We use a linear and a helical array of charges as a model of a rodlike polyelectrolyte. The spacing of the charges is defined corresponding to that of DNA, in order to facilitate comparison with experimental data. Most of the charges are compensated by ligand binding, which is partly driven by a free energy term of site binding. We account for the free energy of interaction between the charges on the rod, which gives rise to a particularly strong driving force for counterion binding. Finally, the influence of an external electric field is described by the free energy of interaction of dipole moments, which result from asymmetry of ligand binding, with the external electric field. (3) Oosawa, F. Biopolymers 1970, 9, 677. (4) Neumann, E.; Katchalsky, A. Proc. Natl. Acad. Sci. U.S.A. 1972, 69, 993. (5) Hogan, M.; Dattagupta, N.; Crothers, D. M.Proc. Natl. Acad. Sci. U.S.A. 1978, 73, 195. (6) Manning, G . S. Blophys. Chem. 1978, 9,65. (7) Charney, E.; Yamaoka, K.; Manning, G. S. Biophys. Chem. 198O,II, 167. (8) Rau, D. C.; Charncy. E. Biophys. Chem. 1981, 14, 1. (9) van Dijk, W.; van der Touw, F.; Mandel, M.Macromolecules 1981, 14, 192. (10) van Dijk, W.; van der T o w , F.; Mandel, M. Macromolecules 1981, 14, 1554. ( I I ) Morita, A.; Watanabe, H. Macromolecules 1984, 17, 1545. (12) Mandel, M.;Odijk, T.Annu. Rev. Phys. Chem. 1984, 35.75. (13) Plum,G. E.; Bloomfield, V. A. Biopolymers 1990, 29, 1137. (14) Porschke, D. Biophys. Chem. 1985, 22, 237. (1 5) Porschke, D. Biopolymers 1989, 28, 1383. (16) Diekmann, S.; Hillen, W.; Jung, M.; Wells, R. D.; Porschke, D. Biophys. Chem. 1982. I S , 157.
Antosiewicz and Porschke Description of the Monte Carlo Procedure
Definition of the Models. Simulations are presented for two different arrays of charges. The first array is a rigid, linear lattice of n equally spaced binding sites with a distance between subsequent sites of 1.7 A. In the second array the coordinates of the charged sites closely reflect the three-dimensional spacing of phosphate residues of double-helical D N A the binding sites were arranged on two helical lines, symmetrical around the z axis. The helical lines were calculated according to equations given by Bronstein and Semendjajew” with a helix-radius of 9.5 A and a pitch of 34 A. The increase of the helix length for each pair of binding sites is 3.37 A. Each binding site is characterized by an intrinsic binding constant K. Thus, at a molar concentration of counterions c, the average degree of binding for noninteracting sites is given by Kc l =1 + Kc The charge of each “free” binding site is q, = -1.0 (in units of elementary charge). The charge of counterions is qc = +1.0, and thus occupied sites are not charged. The state of our polymer is given by the pattern of counterion binding and by the orientation of the polymer with respect to the direction of an external electric field. For the mathematical description we use an n-element vector (x,, i = 1, ..., n) with xl = 0 or xi = 1, when the ith site is free or occupied, respectively. The orientation with respect to the field vector is described by the angle 6. The probability of a given state is determined by its energy. There are three types of energy considered within the present model, which contribute to the total energy of a given state E1. The first is the sum of free energies for the binding of ions to isolated sites given by
where kT is the thermal energy. The second type of energy describes the electrostatic interactions between charges on the polymer model. This energy reads E,, =
1 cc 4TCO j - 2 i l l n
i-1
(4s + x p c ) ( q , + X W C ) Eljhj
(2)
where eo is the dielectric permittivity of vacuum, cij is the relative dielectric constant of the medium between the sites i and j ; r, is the distance between the sites. We have also used a modified version of the electrostatic energy, which accounts for ionic screening in the form of a Debye screening exponential. In this case the electrostatic energy is given by
eX(*ru) (4s + Xjqc)(q, + x ~ c )
1 e,=c c-1 + 4TE0 n j-1
j - 2 i-1
KO
(24 ClfU
where K is the inverse Debye screening length and a is the effective radius of the binding site (cf. textbooks of physical chemistry). We have used a = 2 A. The third energy term is also of electrostatic nature and is called here the orientational energy, because it gives the energy of interaction of the dipole moment p of a given state with the external electric field. This term is given by (3)
where E is the strength of the electric field, and 0 is the angle between the vectors of the dipole and the electric field. We note that the entropy of mixing occupied and empty sites, which is regained upon translation of bound ions on the polymer lattice, is automatically taken care of by the Monte Carlo procedure. (1 7) Bronstein, 1. N.; Semendjajew, K . A. Taschcnbuch der Mathcmatik; H. Deutsch: Frankfurt, 1979.
The Journal of Physical Chemistry, Vol. 95, No. 15, 1991 5985
Electric Moments of Rodlike Molecules
The Monte Carla Pro". The Monte Carlo algorithm used in the present work is based on the procedure published by Metropolis et al.'* Three versions of the program were used: the first one for independent sites ( M a ) is analogous to that presented by Plum and Bloomfield;I3the second one includes the three types of energy described above, but the model polymer is always kept in the direction of the field (MCl); in the third version the orientation of the model is not fixed (MC2). We present a full description of the algorithm for the third version of the program below; the other versions are reduced as indicated above. First the initial state is defined by n + 1 random numbers, which are sampled from an uniform [0,1] distribution by the DURAND subroutine of the IMSL library.19 Each of the first n random numbers is compared with the binding probability { for independent sites; if the random number is less than or equal to 3; then the site is considered to be occupied. The last random number represents the cosine of the angle 0 according to cos 0 = 2fn+, - 1
(4)
Now the energy of the initial state is calculated according to the eqs 1-3. For the next state, n + 2 random numbers are sampled. If a given number among the first n numbers is less than or equal to 0.96, then the state of the respective site remains unchanged; if the random number is larger than 0.98, then the site is considered occupied, and otherwise it is empty. The numerical values used for selection may be varied. We used the values 0.96 and 0.98, in order to limit the number of sites, which change their state in each Monte Carlo step, to a few percent. This limitation proved to be necessary in order to decrease the number of "nonproductive" Monte Carlo steps and to get a final result with a satisfactory standard deviation below a few percent after a sufficiently low total number of Monte Carlo steps. The random number r,,+l serves, as previously, to determine the orientation. The random number r,,+2is used when the energy of the next state is larger than the energy of the previous one. In this case exp(-hE,/kr) is calculated and compared with r,,+2. If the random number is less than exp(-AE,/kT), then the new state is accepted; if not, the state accepted in the previous step is added to the distribution and the next Monte Carlo step is started. For each accepted state its dipole moment P is calculated according to the standard equation
where 7, is the one-dimensional position vector of the site i on the model referred to its center of symmetry. As output data we calculate the mean dipole moment, the root-mean-square dipole moment, the mean occupancy of sites, and the mean value of the function (3 cos2 0 - 1)/2
(6)
This quantity represents the stationary degree of the molecular orientation at a given field strength.20 Technical Details. Most of the calculations were done for a model with 168 sites representing a DNA double helix with 84 base pairs (bp). Some data were also simulated for models representing DNA with 43,69, and 95 bp. The results of these simulations may be compared to experimental data.I5 For the simulations in the absence of ionic screening the individual magnitudes of the binding constant and of the ion concentration do not matter, but only their product Kc; this product was varied over a broad range from 0.002 to I . For simulations with ionic screening we used a binding constant 1 M-'. According to the the equilibrium constant for the association of, e.g., (18) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J . Chem. Phys. 1953, 21, 1087. ( 19) IMSL-Fortran Subroutines for Mathematical Applicarions; IMSL Inc.: Houston, TX, 1987. (20) Frederiq, E.; Houssier. C. Electric Dichroism and Electric Birefringencr; Clarendon Press: Oxford, 1973.
yi
iY
800.
r 3
300.
1
-200.
-700.
10.
100
1000.
l.E+04
l.E+05
1 E+06
Figure 1. Dependence of mean di le moment ( p ) (circles) and rootmean-square dipole moment ( p z ) 'p"(squares) on the number of Monte Carlo steps used during simulation (Kc = 1, 168 sites 84 bp).
Na+ ions to a phosphate residue with a single negative net charge is expected to be around 1 M-I. The concentration of monovalent ions was varied between 2 mM and 1 M. The temperature was 293.1 K. We used three different values for the dielectric constant. Two of them, 80.0 and 5.0, were constants as usual, whereas the third one was designed as distance dependent according to cij = 80.0 - 79.0 exp(-rij/ro) (7) with ro = 10.0 A. This distance dependence is similar to that recommended by Hingerty et aLZ4 The three dielectric constants were selected to check the dependence of our results on the choice; we do not want to present any detailed discussion of dielectric effects in biopolymers. Results First we describe the results of simulations for the "simple" linear array of charges, which comprise all essential phenomena. Then, we proceed to the modulation of the resulting parameters by Debye-Hiickel screening and by the geometry of the array of charges. Independent Sites. When only the free energy of site binding is taken into account, Le., the sites are independent, the results obtained by our Monte Carlo program are identical with those reported by Plum and Bloomfield.13 Our algorithm is less efficient than that used by Plum and Bloomfield, but their algorithm cannot include electrostatic interactions between sites and/or interactions with an external electric field. A comparison of the efficiency of the algorithms may be based on the number of Monte Carlo steps required for a reasonably small mean dipole moment ( p ) of the model at zero field strength. Because of symmetry, the mean dipole moment of the model in the absence of an external electric field should be zero for independent sites as well as for interacting sites. For the model representing a DNA with 84 bp and for Kc = 1, the algorithm used by Plum and Bloomfield provides for lo4 Monte Carlo step (randomly chosen starting points for subroutine DURAND) a mean dipole moment ( p ) of the order f 5 O D and a root-mean-square dipole moment ( p 2 ) ' I 2of the order 2600 D. When the algorithm designed for the present investigation is used for a corresponding calculation (independent sites), the same range of variations for (1)is found after 3.4 X lo5 Monte Carlo steps. Interacting Sites. As a first step of our simulations we have calculated the dependence of ( p ) and ( p 2 ) ' / z for the model representing a DNA with 84 bp (1168 sites) on the number of Monte Carlo steps. For all calculations we used the same starting number for the random number generator. As shown in Figure (21) (22) (23) (24)
Eigen, M. Z. Phys. Chrm. 1954, NFI. 176. Fuoss, R. J . Am. Chem. Soc. 1958,80,5059. Davies, C. W. Ion Association; Butterworths: London, 1960; p 39. Hingerty, B. E.; Ritchie, R. H.; Ferrell, T. L.; Turner, J. E. B b polymers 1905, 24, 427.
5986 The Journal of Physical Chemistry, Vol. 95, No. IS, 1991
Antosiewicz and Porschke 1 .o
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