Dielectric constant in calculations of the electrostatics of biopolymers

J. theor. Biol. (1989) 139, 143-154

Dielectric Constant in Calculations of the Electrostatics of Biopolymers L. I. KRISHTALIK

A. N. Frumkin Institute o f Electrochemistry, Academy of Sciences of the USSR, Moscow, USSR (Received and accepted 11 April 1988) The arbitrariness of the notion of the "effective", or "local", dielectric constant used in attempts to describe the heterogeneous system biopolymer/water by equations valid only for a homogeneous dielectric has been discussed. It has been shown that in solving different problems in which the atomic co-ordinates and the electron density distribution are given with a varying degree of detailing (problems of molecular dynamics, conformational analysis, theory of chemical reactions, etc.), one should use different values of the dielectric constant ofbiopolymer and surrounding water. Some particular types of problems in which e assumes the values of unity, optical (e0) or static (e~) dielectric constants as well as some intermediate cases have been considered. Different aspects of electrostatic interactions in biopolymers, proteins in particular, have been an object of intense interest, especially in the past decade (for some recent reviews see e.g. Warshel & Russell, 1984; Matthew, 1985; Matthew et al., 1985; Honig et al., 1986; Rogers, 1986). One of the most disputable problems in calculations of these interactions is the use of the notion of the dielectric constant e and estimation of its value. The subject of the present communication is an attempt at analysis of the estimates of e used in different approaches and discussion of the physical sense of the dielectric constants to be taken into consideration in solving problems of different kinds. Effective Dielectric Constant Evaluation o f the effective (sometimes termed local) dielectric constant from experimental estimates of the energies o f interaction has become popular nowadays (see e.g. Rees, 1980; Warshel & Russell, 1984; Moore et al., 1984; Wells et al., 1987; Russel et al., 1987). Such calculations are often carried out according to the following scheme: the change in p K o f a certain group when the charge of some other group is changed (by chemical modification or site-directed mutagenesis) is known, i.e. we know the energy of their electrostatic interaction. The distance between these groups is known from the X-ray analysis data and substituting these two quantities (energy and distance) into Coulomb's law, we can find the value of eel. It should be emphasized that the quantity in question is of a strictly conditional nature and cannot be considered as being a physical characteristic o f the given protein. Indeed, as has been well known since the work of Tanford & Kirkwood 143

0022-5193/89/140143 + l I $03.00/0

© 1989 Academic Press Limited

144

L. I. K R I S H T A L 1 K

(1957), the interaction of two charges, especially those localized close to the globule surface, depends significantly on e of the two media--protein and aqueous environment. It is due to the influence of the latter that the values of een found using the scheme described above greatly exceed the dielectric constant values of the protein itself. Further, at given values of e of the two media (ep and ew) and a fixed distance between the two charges, their interaction energy depends significantly on their orientation with respect to the surface of the globule ("immersion depth" of the charges). The een calculated according to the above-mentioned scheme may prove to be substantially different for different pairs of charges. This is sometimes interpreted as pointing to a dependence of the dielectric constant on the distance between the charges, but in actual fact we have to do here with the influence of dissimilar orientation of the charges with respect to the interface between the two media. It may be convenient to introduce the effective dielectric constant in the case of certain approximate comparative estimates (see e.g. Warshel, 1987; Sternberg et al., 1987), but it should be borne in mind that this quantity by itself is devoid of any physical sense and results from incorrect application to a heterogeneous system of an equation valid for a homogeneous dielectric. It is also incorrect to use, for evaluation of the solvation energy of ion in protein, Born's equation derived for a homogeneous dielectric. The dependence of the solvation energy on ep and ew and on the depth of immersion differs quantitatively from that for the interaction energy of two charges (Krishtalik, 1981; Krishtalik & Topolev, 1984). For this reason the estimates o f ee~ based on these two energies must be different. This difference is primarily due to incorrect estimates of eCn and are in no way an indication, as it was sometimes supposed (Warshel et al., 1984), of the inapplicability in general of the notion of the dielectric constant to solution of such problems. It should be noted that, for determination of the solvation energy o f ion in protein, it would be inadequate to describe it as a structureless dielectric with a certain value of ep. One should take into consideration the protein structure-primarily the existence of an intraglobular electric field which affects the ion energy materially (for review see Warshel & Russel, 1984; Krishtalik, 1986a, 1988). As it has been pointed out above, the value of een should be substantially different if the geometry of the two systems is dissimilar. From this point of view it would be worth considering the similarity of e¢~ found from the data on the effect of substitution of different amino acids for Giu-156 in subtilisine on the p K of His-64 (Russel et al., 1987)'and on the binding energy of ionic substrates (Wells et al., 1987), though the distance between the charges and the depth of immersion of the partners of Glu-156 into the globule differs greatly in the two cases. The point, however, is that the magnitude of the shift of pK is affected only by the interaction of His-64 with the 156th amino acid, whereas the binding of substrates is influenced not only by their interaction with Glu-156 (or other amino acid in this position), but also by a change in the solvation energy of ion upon its introduction into the binding pocket. Thus we have to do with two physically dissimilar effects and the similarity of the values of e,n obtained on their basis is accidental (such possibility was pointed out by Wells et al., 1987). It should be said that in the most recent papers many investigators resort to a consistent treatment of the protein-water system

DIELECTRIC CONSTANT OF BIOPOLYMERS

145

as a heterogeneous dielectric (see e.g. Honig et al., 1986; Rogers, 1986; Gilson & Honig, 1987; Sternberg et al., 1987).

Three Types of Problems of the Theory of Electrostatic Interactions in Biopolymers At this point of our consideration we shall discuss certain idealized cases which do not involve a substantially different description of the biological macromolecule and its environment. Examples of a more sophisticated treatment will be considered later as applied to more specific problems. Our present object is to discuss several limiting versions of analysis. The first conceivable case of the calculation of electrical interactions refers to a system in which we know exactly (from experiment or self-consistent calculation) the co-ordinates of all atoms and the wave functions of all electrons, i.e. we know exactly the charge density distribution. Then the electric field at any point is found by integration of the fields set up by all the volume elements with known charge density. Naturally in such calculation there is no need to introduce any dielectric constant (e = 1, field in vacuum) since all the interactions are taken into consideration explicitly. This procedure is extensively used in quantum-chemical calculations of small molecules. The molecules are considered in vacuum since it is extremely difficult to include the solvent volume into this consideration (taking account explicitly of several solvent molecules does not alter the situation radically). In view of the obvious extreme complexity of the system, in the foreseeable future this approach would be hardly feasible for biopolymers in solution. The second case entails a considerable and quite reasonable simplification of the problem. The co-ordinates o f all atoms are assumed to be known (they can be found experimentally or with account o f a certain experimental basis given as independent co-ordinates: this is the approach of conformational analysis or molecular dynamics). The electron wavefunctions are not assumed to be exactly known, but some approximations can be used which allow certain partial charges to be ascribed to the atoms of each bond (sometimes these partial charges can be localized not only at the centre of the atom but also, for example, at the centre of the electron density o f the lone pair etc.: see e.g. Naray-Szabo, 1979). We state the problem o f calculating the field set up by these partial charges. Here we have to take into account the fact that each partial charge polarizes the electron clouds o f other atoms and bonds, altering the electron density distribution. In their turn the dipole moments thus induced set up an additional field and affect the electron clouds, etc. The self-consistent change of the electron density brought about corrects our initial approximate description of the electron wave functions, bringing it closer to an exact solution. The final picture represents a superposition of the primary field of partial point charges and of the field of the self-consistent distribution of induced dipole moments. This self-consistent field can be found by an iteration procedure as it was suggested by Warshel & Levitt (1976) and after that has been intensively used in the works o f Warshel and co-workers (for review see Warshel & Russell, 1984). This basically microscopic approach raises no objections in principle, but its practical realization (especially if account is taken of the surrounding solvent simultaneously) requires

146

L.I.

KRISHTALIK

a very large volume of computer work, so that one has to resort to additional simplifications, and the quantitative aspects of the obtained results prove to be open to criticism (Krishtalik & Topolev, 1984). At the same time, it appears possible to simplify analysis of this particular type o f problem considerably. Indeed, from the continuum standpoint the self-consistent distribution of the induced dipoles of electron clouds is nothing but electronic polarization of the medium, i.e. the system can be described as a certain atomic structure carrying partial charges and immersed in a medium possessing an averaged electronic polarizability. Actually this description means introducing into electrostatic calculations the dielectric constant equal to the value of the optical dielectric constant eo (for proteins e o = 2 - 2 . 2 ) (Gilson et al., 1985). Tapia and co-workers (Tapia et al., 1978; Tapia & Johannin 1981; Tapia et al., 1985) proposed that, in quantum-chemical calculations of reacting molecules or groups, the reaction field of the surrounding medium considered as a dielectric with electronic polarizability was used. In the problems studied in these works an account o f the field o f the partial charges o f surroundings is also necessary. As a matter o f fact, this kind of question is in a sense similar to the conforrnational analysis in a heterogeneous system which will be discussed in the next section. The third type of problem is associated with the theory of chemical reactions in biopolymers. In this case we are interested not only in electrical interactions in a certain conformation but also in their changes in the course of the process. A typical situation here is when the co-ordinates of all atoms in the initial state are known with a fair degree of precision, but we do not know their positions on completion of the reactiont. An explanation is called for here. An X-ray study of native enzymes and their complexes with different ligands simulating substrates, products and transition states$ assists considerably in understanding the mechanism of the reactions and, in particular, in describing of the changes in the co-ordinates o f the atoms of the reagents themselves (and, in some cases, of the adjoining protein's groups). However, the change in the co-ordinates of most of the atoms is beyond the sensitivity of experiment. It is highly improbably that we should obtain the pattern of their m o v e m e n t (moreover, a pattern statistically averaged over a giant n u m b e r of starting configurations) by the molecular dynamics methods, which at present are capable of following the evolution of structure in time up to nanoseconds at best, whereas the characteristic times of enzymatic reactions are larger by several orders of magnitude. We must emphasize again that it is not the movements of relatively small groups of atoms that are in question, but those of all polymer atoms. At the same time there is no doubt that charge (electron, proton) transfer from one reagent to another inevitably causes a shift both o f electron density and of the centres o f gravity of the atoms carrying a partial charge. In line with the previous analysis of

tThe wave functions of electrons are described to the same approximation as in the previous problem. :]:Herethe term "transition state" is understood conditionally since not only does a certain intermediate configuration of reactants correspond to a real transition state, but so does the intermediate configuration of the surrounding medium (see e.g. review in Dogonadze & Kuznetsov, 1975a; UIstrup, 1979; Kdshtalik, 1986a).

DIELECTRIC

CONSTANT

OF BIOPOLYMERS

147

the induced electronic polarization, this self-consistent shift of atoms constitutes the correction bringing our inaccurate (for the final state) co-ordinates closer to their exact values. Just as in our discussion of the induced electronic dipoles we could, in principle, carry out an iteration procedure with the induced atomicpolarization and permanent dipoles but, in practice, this problem proves to be much more complicated and, as far as we know, there have been no real attempts in this direction. As in the previous case, however, we can introduce a certain averaged polarization associated both with a change in the lengths and valence angles of the polar bonds and with the rotation of polar groups, the bends and turns of the polypeptide chain sections. The last-mentioned effects are an analogue of the orientational polarization in low-molecular polar liquids. In other words, in the problems of the third type, a semimicroscopic approach could also be used: consideration of the atomic structure with characteristic partial charges which is immersed in a medium possessing a set of various polarization modes and described by the static dielectric constant es (for proteins es-~ 4)t. It should however, be stressed here that the analogy in using eo and es extends only to the change in the polarization of the surrounding medium upon transitition from one equilibrium conformation to another. But when we consider the kinetics of this transition we must take into account the fundamental difference between the electronic and the atomic-orientational polarization, viz. the electronic polarization is inertialess, it adjusts itself immediately to the movement o f the charge being transferred and therefore does not create an activation barrier in the reaction path. In contrast to this situation, the movement of heavy particles is fairly slow and therefore transfer of such quantum particles as electron or proton can take place only when as a result of thermal fluctuations of the co-ordinates of these heavy particles, the electron (proton) energy levels in the initial and final states become equal. In other words, charge transfer must be preceded by a certain reorganization of polar medium. Thus the orientational (and the atomic) polarization of the medium acts as a dynamic coordinate. The movement along this co-ordinate makes a significant contribution to the activation barrier value. This contribution is proportional to the interaction energy of the charge with the inertial part o f polarization, i.e. it is proportional to the Pekar's parameter C = ( l / e 0 - 1/es) (see e.g. Dogonadze & Kuznetsov, 1975a; Ulstrup, 1979; Krishtalik, 1986a). Proteins differ significantly from water and other polar liquids in that in their case eo and e~ have rather similar values and therefore the parameter C for proteins proves to be several times lower than that for water. This leads to a significantly lower reorganization energy of the medium, and this is one o f the major reasons o f the decrease in the activation energy o f enzymatic catalysis. In microscopic terms a low dielectric constant and a low reorganization energy of the medium result from fixation of the numerous dipoles of the po!ypeptide chain within the framework o f a certain structure hindering any considerable change in their orientation with tlt should be noted that Gilson & Honig (1986) have succeeded in estimating the dielectric constant of proteins by calculating possible fluctuations of dipole moments due to torsional deformations of a-helices.

148

L.I. KRISHTALIK

changing electric field. This approach to description of charge transfer processes in proteins has been developed since the middle seventies (Krishtalik, 1974, 1979, 1980; for review see Krishtalik 1986a, 1988). It should be noted that in the more advanced theory of the elementary act of charge transfer, the parameter determining the reorganization energy of the medium contains the difference ( 1 / e q - 1 / e , ) rather than ( 1 / e 0 - 1 / e s ) , where 8,t is the dielectric constant corresponding not only to the optical but also to some part o f infrared frequencies, i.e. to all frequencies much larger than k T / h . The vibrations corresponding to them (e.g. the stretching vibrations of H atom bonds) behave in a quantum manner and are not included into the classical subsystem which fluctuates slowly and determines the activation barrier (Dogonadze & Kuznetsov, 1975a; Krishtalik, 1986a). For water, an appropriate correction decreases the reorganization energy by about 20%. Unfortunately, no data are available for proteins that could be used to estimate this effect quantitatively. If we reasonably assume that we have to do with the same order of magnitude, then e o - 2 - 2 . 2 should be substituted by eq, approximately estimated as 2.5. Apart from the limiting cases considered above, some problems o f an intermediate type seem to be realistic enough. For instance, if in a simplified version of the conformational analysis we do not introduce, in the explcit form, the deformations o f valence bonds and of some angles, then we must take into account not only electronic but also atomic polarization o f the medium (or only a part o f this contribution), i.e. we also introduce a quantity of the type of the infrared dielectric constant eir (a similar approach was exploited by Gilson & Honig, 1986). In this case we have to use the value of eir for the frequency range, corresponding to vibration of the bonds not taken into account explicitly. At this point it should be noted that the situation is different in the case o f conformational analysis and molecular dynamics. In relation to the use of electronic polarization the two classes of problems may be treated in a similar manner. But when we take into account the infrared e~r the situation is essentially different. Indeed, in conformational analysis our object is to find the equilibrium energy of a definite conformation, and therefore the time required for the given equilibrium value to be reached is of no concern to us. In the molecular dynamics problem we consider the change in the atomic co-ordinates with time and use discrete time intervals of the order of 10 -t5 sec. But the period of the stretching vibrations of the highest frequency is at least an order larger. This means that infrared polarization cannot achieve relaxation in such short times, and the movement in 10 -ts sec may be treated as occurring at practically constant permanent dipole moments of the X - H bonds, i.e. we have to choose e0 as a reasonable value of the dielectric constant in molecular dynamics problems. At the same time the total time considered in problems o f this kind exceeds picoseconds and therefore the deformations o f the valence bonds X - H in this or another form should be taken into account explictly. In a rigorous consideration account should be taken of the quantum nature o f high-frequency vibrations. The difference between the problems of conformational analysis and those of molecular dynamics becomes even more apparent in heterogeneous biopolymer/water systems to be considered in the next section.

DIELECTRIC

CONSTANT

OF

BIOPOLYMERS

149

Another example of the reasonableness of an intermediate, "hybrid" approach is an analysis of kinetics of some reactions of biopolymers. Earlier we spoke about the dielectric continuum formalism in describing the reorganization of the inertial polarization of the medium. We emphasized that such a description makes it unnecessary to calculate the fluctuational changes in the co-ordinates of an enormous number of atoms, for which, by the way, X-ray analysis reveals no significant shifts of their equilibrium positions. In some cases, however, in the course of the reaction, individual side groups of protein undergo more substantial shifts detected experimentally. This phenomenon seems to be observed, in particular, in the oxidation of cytochrome C. For this protein Churg et al. (1983) have performed a very interesting analysis which revealed a direct dependence of the energy of the system on the co-ordinates of the labile groups, and the resulting formation of a corresponding activation barrier. We think that with systems of this kind it would be advisable to combine analysis of the reorganization of labile groups (e.g. on the same lines as the analysis in the work cited above) with the estimation of the reorganization energy of the rest of protein in terms of dielectric formalism. This approach is similar to that used widely in the theory of homogeneous chemical reactions, e.g. redox reactions of metal complexes when the energy of the "inner sphere" reorganization is calculated for a discrete model, taking into account explicitly the lengths of the corresponding bonds, the frequency of vibrations, etc, and the "outer sphere" reorganization of the solvent, is described in terms of the continuum dielectric formalism (see e.g. Ulstrup, 1979; German, 1983). Similarly, it may be expedient to explicitly consider the induced dipole of some easily polarizable group situated in the vicinity of the charged reactant while treating the rest of the protein as a dielectric continuum.

Heterogeneous System Biopolymer-Water As we stressed in the first section of this paper, the electric field of any charge localized in biopolymer or near it brings about polarization both of the macromolecule proper and of the surrounding medium, the resulting field being a superposition of the primary field of the charge and of the field set up by the polarization of both media. In this connection it is necessary to consider in what manner the dielectric properties of the surroudning aqueous phase should be taken into account in solving problems of various kinds. The situation is simplest in the case of problems of the third type--the static dielectric constant should be used both for polymer and aqueous environments. Here arise some complications in connection with the presence of ions in the aqueous phase. In calculating the reorganization energy, account should be taken both of the reorganization of water as a dielectric (contribution proportional to C) and the reorganization of the ionic atmosphere (Dogonadze & Kuznetsov, 1975 b ). The latter, however, makes a small contribution (German & Kuznetsov, 1987) and for most problems not related specially to investigation of the influence of the ionic strength this may be ignored. In calculating electrostatic fields, it would be sound practice to introduce the Debye screening in solution as it was done in Tanford-Kirkwood's

150

L.I. KRISHTALIK

theory (1957). The field inside the polymer can be calculated assuming for the surrounding medium es--oo (quasimetal approximation). In this case the reaction field differs from the exact value at es = 80 by several percent. With allowance for the screening by the ionic atmosphere and depending on the conditions, this difference may practically disappear*. The quasimetal approximation is, of course, unsuitable for calculating the field on the surface of polymer and around it. The situation is more complicated for problems of the second type. In conformational analysis .and molecular dynamics studies the estimate of e depending on the distance between charges is often used as well as the approximation e = R. This approximation is convenient for calculations (the root extraction operation is eliminated) but it has no physical sense*. The increase of e with rising value of R is usually substantiated by qualitative considerations of two kinds. The first version assumes that at small distances molecules interact "almost as in vacuum", and at large distances polarization of the environment takes effect and e approaches es. This reasoning, however, does not take into account the fact that the charges of two interacting molecules polarize the environment regardless of the distance between them. Further, if, as in the problems under consideration, the position of all atoms is given, ep is determined, as shown in the preceding section, by electronic polarization, i.e. ep =eop. The second possible substantiation of the e ( R ) dependence lies in the fact that with increasing R one of the charges may approach the surface of the macromolecule and this will increase its interaction with the surrounding water. It should be taken into consideration, however, that an effect of this kind depends significantly on the localization of charges inside the globule. For example, for two charges localized at the distance R = 4 ~, on the same diameter on both sides of the centre of the globule with the radius of 2 0 ~ the interaction energy is equal to q ~ q 2 / l ' 2 8 , epR, and for the same two charges one of which is at 2 A distance from the globule surface, the corresponding coefficient in the denominator is equal to 2.18. When the charge approaches the surface 1 ~, further this coefficient increases to 3.33. For a similar geometry but at R = 6 .~ the relevant coefficients are 1.41, 2.98 and 4.68§. We see thus that the effective dielectric constant depends more on the position of the charges with respect to the interface than on R and cannot be described by a single interpolation formula. In other words one has to solve a particular electrostatic problem.

tFor certain problems an exact solution for the reaction field of the charge q can be represented as the image field of the point charge q(e r - ew)/(el, + e,,.) and of a certain distributed charge density making a small additional contribution (Finkel'shtein, 1977; see also Linse, 1986). :[:Itshould be noted that the e = R dependence is generally used in the range from R = 3 ,~ to R = 10 A.. At larger R, electrostatic interactions are not taken into account. Such cutting of Coulomb forces is justified in solutions, but in proteins, due to their fixed structuure, of importance are interactions at much larger distances as has been shown by our calculations for a -chymotripsin (Krishtalik, 1977; Krishtalik & Topolev, 1983; Topolev & Krishtalik, 1983). §The calculations have been carded out by the image method.

DIELECTRIC

CONSTANT

OF BIOPOLYMERS

151

As has been described in the previous section, in solving problems of conformational analysis and molecular dynamics one should use the optical dielectric constant of polymer (in some cases a somewhat larger value which takes account of a part of infrared polarization). The description of the solvent surrounding the macromolecule, however, should be significantly different in the two cases. Indeed, in the case of conformational analysis we are primarily interested in finding the equilibrium conformations corresponding to the energy minima. This means that in our anlysis we must allow the sytem to attain equilibrium in each conformation. Consequently, when passing from one conformation to another, there must be enough time for the polarization of the surrounding medium to relax to a new equilibrium value. In other words the dielectric behaviour of the bulk of the solvent (except its molecules treated explicitly) is described by its static dielectric constant esw. Thus in a conformational analysis we have to do with a heterogeneous system of two dielectrics Cop and esw. In molecular dynamics problems we are interested in processes occurring during short times: the usual step is of the order of femtoseconds. During this time only electronic polarization is able to respond and therefore the surrounding medium should be ascribed the value of Cow. Since the values of e0p and Cow do not differ very significantly, the problem can be greatly simplified if we consider a homogeneous system with e0 = Cop (usually e0w is somewhat less than e0p but this difference may be neglected especially if account is taken of some increase in the effective e~ due to the involvement of infrared vibrations of the highest frequency). The total duration of the process analyzed by the molecular dynamics methods is usually no less than picoseconds. Not only electronic but also atomic polarization of the medium undergoes a change during these times and this manifests itself in the value of the dielectric constant of the order of eir (for the part of the solvent whose molecules are not included explicitly into calculation). The simplest way of taking into account the changes in the polarization of this surrounding medium is to recalculate at regular intervals (e.g. after 100 steps) the reaction field corresponding to infrared polarization, keeping this field constant during the subsequent cycle. It would be also advisable to use a similar approach to orientational polarization (here the static constant es~ is included). The field set up by this polarization may be assumed to be constant for the time intervals of the order of the water dielectric relaxation times, i.e. up to 10 psec. We can suggest the following calculation scheme. The whole period of time of interest to us is broken into cycles, e.g. of 10 psec, the starting conformation of each cycle corresponding to the reaction field calculated for a heterogeneous system with e0p and e~w, and this field is constant during the entire cycle. The calculation of the energetics of this starting conformation coincides with that in conformational analysis, i.e. the two methods merge at this point. The cycle is broken into subcycles, e.g. of 0.1 psec. Each of the subcycles corresponds to a change of the reaction field of atomic polarization only (0.1 psec in its order of magnitude corresponds to the characteristic times of vibrations of covalent bond). Each single step, however, is

152

L.I. KRISHTALIK

calculated in the approximation of a homogeneous system with Cop subjected to a permanent field of inertial polarizationt. Dielectric Constant and Discretenes of the Medium The dielectric constant is a quantity characteristic for a continuum description of the polarization of a dielectric. At the same time, when one considers short-range interactions it becomes clear that polarization of the medium changes with distance not continuously but discretely: the value and orientation of the dipole moment are constant in bounds of one molecule (polar group) and change jumpwise when passing to a neighbouring molecule. These stepwise changes should be averaged. This averaging is carried out in the so-called nonlocal electrostatics or the theory of media with spatial dispersion of the dielectric constant (see, e.g. Kornyshev, 1985). In this theory the continuum description is also used but the dielectric response of the medium proves to be significantly dependent on the correlation length A - - a parameter characteristic for every kind o f polarization. For orientational polarization Aor is supposed to be of the order of the radius of a polar molecule or of a strong associate o f molecules moving as a single whole. For atomic and electronic polarizations the corresponding values o f A are much less. When the field is considered at the distances r >>Aor, orientational polarization manifests itself completely and the medium behaves as a dielectric with a macroscopic dielectric constant es. If, however, r<< Aor, the influence of orientational polarization becomes nil and the effective dielectric constant approaches cir. It should be stressed that this decrease o f the dielectric constant is due precisely to the discreteness o f the medium rather than to dielectric saturation, and at small distances it occurs even in weak fields. In the transition region the correction in the Coulomb field can be described by the exponential function of ; t / r . Decrease of the dielectric constant at small distances manifests itself in different phenomena, in the properties o f the electric double layer at the metal/electrolyte interface, in particular. The value of the electrical capacitance o f this molecular capacitor and the data on its structure suggest that at this interface a water layer 3 A thick (monomolecular layer) behaves as a dielectric with e = 6, i.e. a value close to ei~,w= 5 (see e.g. Bockris & Reddy, 1970). Taking this fact into account in calculating the electrostatic interaction of ionogenic groups located on the protein surface, one can obtain a reasonable agreement with experiment for the shift of p K on the basis of X-ray data, alone, without introducing empirical corrections not sutficiently justified physically (Krishtalik et aL, 1988). It is to be thought that taking

tBreaking of calculations into cycles was suggested by Gilson et al. (1985) on the basis of different considerations, namely a slow change in the protein geometry. They, however, did not separate the polarization of the surrounding medium into the infrared and static components, which is fundamental for this problem. The simplification of calculations with the use of the image method (surrounding medium--quasimetal) proposed by the above authors may be used in the version with e~,,. but is quite unsuitable for calculation with ei,.,,, since the condition eir.w >> Cop necessary in this case is not fulfilled (in actual fact for water e~r.,,.= 2 to 3cop, i.e. the characteristic coefficient (cop- e~r.,.)/(eop + e~r.,,.)= -1/3 to -1/2 and hence is far from the value -1 corresponding to quasimetal).

DIELECTRIC CONSTANT OF BIOPOLYMERS

153

a c c o u n t o f t h e d e c r e a s e o f t h e d i e l e c t r i c c o n s t a n t n e a r t h e i n t e r f a c e , i.e. t r a n s i t i o n to a t h r e e - p h a s e h e t e r o g e n e o u s m o d e l (ep, elf.w, esw) is n e c e s s a r y in all c a s e s w h e r e interactions with charges localized near the surface of a macromolecule are concerned. As r e g a r d s t h e s t a t i c d i e l e c t r i c c o n s t a n t o f b i o p o l y m e r itself, in this c a s e t h e problem of the influence of spatial dispersion has practically not been elucidated. It m a y b e s u p p o s e d t h a t d u e to t h e p r e s e n c e o f a d e v e l o p e d s p a t i a l s t r u c t u r e , t h e o r i e n t a t i o n s o f d i p o l e s a r e c o r r e l a t e d at r a t h e r l a r g e d i s t a n c e s , so t h a t in c a l c u l a t i o n s o f i n t r a g l o b u l a r fields ep p r o v e s to b e s o m e w h a t l o w e r t h a n t h e m a c r o s c o p i c v a l u e o f es, a p p r o a c h i n g t h a t o f eir ( K r i s h t a l i k , 1986b). S i n c e f o r p r o t e i n s , h o w e v e r , t h e s e v a l u e s a r e s i m i l a r e n o u g h ( t h e y a r e e s t i m a t e d as es.p -~ 4 a n d e~r.p ~- 3 - 3 " 5 ) a n e x i s t i n g u n c e r t a i n t y in this m a t t e r s h o u l d n o t h i n d e r e l e c t r o s t a t i c c a l c u l a t i o n s .

REFERENCES BOCKRIS, J. O'M. & REDDY, A. K. N. (1970). Modern Electrochemistry. New York: Plenum Press. CHURG, A. K., WEISS, R. M., WARSHEL, A. & TAKANO T. (1983). J. Phys. Chem. 87, 1683. DOGONADZE, R. R. & KUZNETSOV, A. M. (1975a). Progr. Surf Sci. 6, 1. DOGONADZE, R. R. & KUZNETSOV, A. M. (1975b). J. Electroanal. Chem. 65, 545. FINKEL'SHTEIN, A. V. (1977). Molekulyarnaya Biologiya (Moscow) 11,811. GERMAN, E. D. (1983). Rev. Inorganic Chem. 5, 123. GERMAN, E. D. & KUZNETSOV, A. M. (1987). Elektrokhimiya 23, 1671. GILSON, M. K. & HONIG, B. H. (1986). Biopolymers 25, 2097. GILSON, M. K. & HONIG, B. H. (1987) Nature, Lond. 33tl, 84. GILSON, M. K., RASHIN, A., FINE, R. & HONIG, B. (1985). J. molec. Biol. 184, 503. HONIG, B. n., HUBBELL, W. L. & FLEWELLING, R. F. (1986). A. Rev. Biophys. Biophys. Chem. 15, 163. KORNYSHEV, A. A. (1985). In: The Chemical Physics of Solvation. Part A. (Dogonadze, R. R., Kalman, E., Kornyshev, A. A. & Ulstrup, J., eds) p. 77. Amsterdam: Elsevier. KRlSHTALIK, L. I. (1974). Molekulyarnaya Biologiya (Moscow) 8, 91. KRISHTALIK, L. I. (1977). IFIAS Workshop on physico-chemical aspects of electron transfer processes in enzyme systems (HEDI~N, G. ed.) p. 76. Stockholm: IFIAS. KR1SHTALIK, L. I. (1979). Molekulyarnaya Biologiya (Moscow) 13, 577. KR1SHTALIK, L. I. (1980). J. theor. Biol. 86, 757. KRISHTALIK, L. 1. (1981). Molekulyarnaya Biologiya (Moscow) 15, 290. KRISHTALIK, L. I. (1986a). Charge transfer reactions in electrochemical and chemical processes. New York; London: Consultants Bureau. KRISHTALIK, L. I. (1986b). J. Electroanalyt. Chem. 2114, 245. KRlSHTAUK, L. I. (1988). In: The Chemical Physics of Solvation. Part C. (Dogonadze, R. R., Kalman, E., Kornyshev, A. A. & Ulstrup, J., eds) p. 707. Amsterdam: Elsevier. KRISHTALIK, L. I. & TOPOLEV, V. V. (1983). Molekulyarnaya Biologiya (Moscow) 17, 1034. KRISHTALtK, L. I. & TOPOLEV, V. V. (1983). Molekulyarnaya Biologiya (Moscow) 18, 712, 892. KRISHTALIK, L. I., TOPOLEV, V. V. & KHARKATS, YU. I. (1988). Biofizika (in press). LINSE, P. (1986). J. phys. Chem. 90, 6821. MATTHEW, J. B. (1985). A. Reu. Biophys. Biophys. Chem. 14, 387. MATTHEW, J. B., GURD, F. R. N., GARCIA-MORENO, B., FLANAGAN, M. A., MARCH, K. L. & SHIRE, S. J. (1985). Crit. Rev. Biochem. 18, 91. MOORE, G. R., HARRIS, D. E., LEITCH, L. A. & PE'VI'IGREW, G. W. (1984). Biochim. Biophys. Acta 764, 331. NARAY-SzABO, G. (1979). Int. J. Quantum. Chem. 16, 265. REES, D. C. (1980). J. molec. BioL 141, 323. ROGERS, N. K. (1986). Progr. Biophys. MoL Biol. 48, 37. RUSSEL, A. J., THOMAS, P. G. & FERSHT, A. R. (1987). J. molec. Biol. 193, 803. STERNBERG, M. J. E., HAYES, F. R. T., RUSSEL, A. J., THOMAS, P. G. & FERSHT, A. R. (1987). Nature, Lond. 330, 86.

154

D. WINNE

TANFORD, C. & KIRKWOOD, J. G. (1957). J. Am. Chem. Soc. 79, 5333. TAPIA, O. & JOHANNIN, G. (1981). £ Chem. Phys. 75, 3624. TAPIA, O., STAMATO, F. i . L. G. & SMEYERS, Y. G. (1985). J. molec. Struct. (Teochem.) 123, 67. TAPIA, O., SUSSMAN, F. & POULAIN, E. (1978)..I. theor. Biol. 71, 4~. TOPOLEV, V. V. & KRISHTALIK, L. I. (1983). Molekulyarnaya Biologiya (Moscow) 17, 1177. ULSTRUP, J. (1979). Charge Transfer Processes in Condensed Media, In: Lecture Notes in Chemistry. Vol. 10. Berlin: Springer-Verlag. WARSFIEL,A. (1987). Nature, Lond. 330, 15. WARSHEL,A. & LEVITT, M. (1976). J. molec. Biol. 103, 227. WARSHEL, A. & RUSSELL, S. T. (1984). Q. Rev. Biophys. 17, 283. WARSHEL, A., RUSSELL, S. T. & CHURG, A. K. (1984). Proc. natn. Acad. Sci. U.S.A. 81, 4785. WELLS, J. A., POWERS, D. B., BOTT, R. R., GRAYCOV, T. P. & ESTELL, D. A. (1987). Proc. natn. Acad. ScL U.S.A. 84, 1219.