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PROTEINS: Structure, Function, and Genetics 28:174–182 (1997)

Electrostatics of Proteins: Description in Terms of Two Constants Simultaneously

L.I. Krishtalik,* A.M. Kuznetsov, and E.L. Mertz A.N. Frumkin Institute of , Russian Academy of Sciences, Moscow, Russia

ABSTRACT In the semi-continuum treat- constant being about 4 (see, e.g., ref. 5). Thus pro- ment of the energetics of charge formation (or teins can be defined as highly polar low-dielectric transfer) inside a protein, two components of media, a combination that is impossible for low- the energy are inevitably present: the energy molecular-weight solvents. of interaction of the with the pre-existing In low-molecular-weight liquids, the electric field intraprotein electric field, and the energy due set up by their at any point fluctuates around to polarization of the medium by the newly zero. In the presence of a permanent electric field, formed charge. The pre-existing field is set up e.g., upon immersion of an ion into the solvent, a by charges (partial or full) of the protein reorganization of the medium takes place, some fixed in a definite structure. The calculation of average permanent orientation of dipoles appears, this field involves only the electronic polariza- and their field acquires a non-zero value. On the tion (the optical dielectric constant eo)ofthe other hand, in proteins, the permanent component of protein because the polarization due to shifts the average dipoles’ field at any point is non-zero, the of heavy atoms has already been accounted for spatial distribution of the field being determined by by their equilibrium coordinates. At the same the . The intraprotein electric field time, the aqueous surroundings should be de- exists before introduction of any ion into the macro- scribed by the static constant esw, as the posi- , and therefore proteins can be defined as tions of water are not fixed. The ‘‘preorganized media’’ as opposed to the usual sol- formation of a new charge, absent in the equi- vents.6 Some fluctuations of the dipoles’ field do take librium X-ray structure, results in shifts of place in proteins too, but the amplitude of these and polar atoms, i.e., it involves all fluctuations is relatively low (low dielectric constant). kinds of medium polarization described by the An ion appearing inside the protein molecule, e.g., static dielectric constant of protein es. Thus, in at electrolytic dissociation of some side chain, is calculations of the total energy, two different subject to the action of the permanent electric field dielectric constants of the protein are opera- existing at the corresponding point of this preorga- tive simultaneously. This differs from a widely nized medium. This field substantially affects the used algorithm employing one effective dielec- ion’s energy. On the other hand, the ion’s own field tric constant for both components of the polarizes the surroundings, resulting in both elec- ion’s energy. Proteins: 28:174–182, 1997. tronic and atomic polarization (a part of the latter is r 1997 Wiley-Liss, Inc. in a sense an analog of orientational polarization). We will consider here the atomic polarization due to Key words: proteins as preorganized media; small shifts of atoms and small-angle turns of pro- intraprotein electric field; ion tein polar groups not accompanied by a major change charging; ion formation energy; op- in the protein conformation. The case of a total tical and static dielectric con- restructuring of the macromolecule upon stants; reorganization energy of some group calls for separate consideration, which can hardly be done in the framework of dielectric INTRODUCTION formalism only. In the case of a substantial shift of It is widely recognized that electrostatic interac- only the closest ion’s neighbors, a seemingly promis- tions are of primary importance in the properties ing approach combines the molecular level of model- and function of proteins (for some recent detailed ing of the nearest surroundings with the dielectric reviews, see refs. 1–4). From the point of view of formalism for the rest of the system (similar to , proteins are quite specific media. They ‘‘inner-sphere’’ and ‘‘outer-sphere’’ reorganization for possess a high concentration of strongly polar group- chemical reactions in solutions). The same is true for ings (peptide bonds first of all). However, these dipoles are fixed inside a definite structure, and hence their mobility is severely restricted. There- *Correspondence to: L.I. Krishtalik, A.N. Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninskii fore, the dielectric response of proteins to an external prosp. 31, 177071 Moscow, Russia. electric field is rather weak, their static dielectric Received 1 April 1996; Accepted 9 September 1996 r 1997 WILEY-LISS, INC. TWO DIELECTRIC CONSTANTS OF PROTEINS 175 formation (or a substantial change upon ionization) knowledge of the protein’s electronic structure is in- of covalent bonds between the ion and its ligands; accessible, and therefore we must restrict ourselves this problem should be treated quantum-chemically. to an approximate description of the distribution of Thus the energy of ion interaction with a protein- , e.g., ascribing to each (or each aceous medium consists of two major components: bond) a definite partial charge. The values of these the ion energy in the pre-existing intraprotein field charges are usually assumed on the basis of experi- and its charging (Bornian) energy due to the polariza- mental data (e.g., the moments) and theoreti- tion of the medium by the ion. Both these compo- cal (quantum-chemical) calculations for rather small nents were considered many times in a continuum model molecules, like free amides or short peptides. dielectric formalism. Such an analysis needs a proper (We are leaving here aside the problem of the choice definition of the dielectric constant. The notion of the of a definite set of partial charges. Different systems dielectric constant of proteins in the framework of of these charges adjusted for different purposes continuum formalism has been discussed many appear in the literature. Most suitable for the prob- times.7–10 The main conclusion of these consider- lem considered in this paper are the sets evaluated ations is that the effective dielectric constant that in view of calculations in a semi-continuum approxi- should be used in calculations depends on the prob- mation. A more detailed discussion of the problem is lem, namely, on the approximations used to describe given in the Appendix.) Now, using these charges, we the protein structure. Considering explicitly coordi- can, in principle, find the electric field distribution. nates and motions of only some part of the constitu- However, in these calculations, we should keep in ents of the system, we must describe the behavior of mind that our description of the charge distribution the rest of the particles as an averaged dielectric (e.g., partial charges fixed at the centers of atoms) is response, making use of, say, optical, , or approximate, and we should take into account the static dielectric constant. For example, if the coordi- effect of atom charges on the electronic clouds of nates of all nuclei are assumed to be known (or other atoms, that is, the effect of their electronic calculated), and the electronic density distribution is polarization. Averaging this effect, we come to a approximately reflected by ascribing to each atom description of the system as a structure formed by some partial charge, then we can use, in microscopic fixed charges embedded in a medium with an optical analysis, atomic or, in a semi- (electronic) dielectric constant eo. For proteins, this continuum approach, the optical dielectric constant. constant can be estimated as about 2.1. (The usual With the movement of all the nuclei being unspeci- value for aliphatic amides is close to 2.0; here we fied, the static dielectric constant is suitable to have introduced some additional correction taking describe the total response. The different character- into account that for aromatics, heteroaromatics, istic times of the various modes of polarization and disulfides eo varies between 2.3 and 2.6.) should result in different effective es used in the It should be noted that the mutual interaction of analysis of processes occurring in different time partial charges of course influences their equilib- scales, etc.11,12 rium positions, but this effect should not be ac- The aim of the present is to consider how to counted for as some quasi-orientational polarization apply correctly the ideas of refs. 7–10 described because we start with known equilibrium coordi- above to calculations of the energetics of ion forma- nates of an already folded protein. tion. We will show that, in contrast to the protocol Let us now consider the formation of an ion inside usually employed, in calculations of two components the protein molecule, for instance, by trans- of ion energy, viz., the effect of the pre-existing field fer to, or by electrolytic dissociation of, one of the and the charging energy, two different effective dielec- protein groups. On the one hand, this ion appears at tric constants should be used. We start our analysis a definite place in the structure, and at this point with a general description of the problem, and we some electric field has existed before formation of the then discuss the technical questions of the applica- ion. The energy of the newly formed charge is tion of these results to proteins. influenced by this field, and the field, as discussed A preliminary account of this study has been above, can be expressed with the help of the optical 13 reported. dielectric constant eo. On the other hand, the new ion not only disturbs electronic shells of the atoms of the ELECTROSTATICS OF PROTEINS AS medium, but also produces some shifts of nuclei, STRUCTURED DIELECTRICA resulting in a new equilibrium structure, but with We will consider a protein or other dielectric with a unknown new atomic coordinates. The situation can well-defined structure, i.e., with known coordinates be described as a superposition of a set of induced of all its atoms. If we knew, in addition, the exact dipoles on the original network of partial charges. distribution of electron density, we could calculate That means that the new charge interacts with all the electric field at any point of the system consider- kinds of medium polarization, and hence the energy ing the charge density distribution in , i.e., at of this interaction depends on the total value of the dielectric constant e51. However, such a complete static dielectric constant es. 176 L.I. KRISHTALIK ET AL.

We can also describe the same situation as an Using the standard procedure, one can transform imaginary two-step process. First, the ion is formed Eq. 3 to an equivalent form: at a definite point under the condition that all modes of the protein and water polarization are frozen. In 1 1 W 5 E D dV 1 E D dV. (4) this step, the ion interacts only with the pre-existing 4p eV 0 i 8p eV i i field, the latter being determined through the optical dielectric constant of the medium, . Then, in the eo Here Di 5eSEi is the vector of the electric displace- step, the full polarization of the protein and ment produced by charge density ri. In Eqs. 3 and 4, its surrounding is released, the electronic density wi, Ei and Di refer to the charge density ri. distribution and atomic positions adjust to the field In Eq. 3, the first two integrals (or the first term in of the newly formed ion, and this results in an Eq. 4) represent the energy of new in the ion-medium interaction energy corresponding to the pre-existing field of partial charges organized in total static polarization es. some spatial structure. This expression has the Our task now is to consider quantitatively the usual form for the electrostatic interaction of a set of result of superposition of the two effects discussed: charges ‘‘i’’ with the potential w0, created by charges the pre-existing field, formed with only the electronic ‘‘0,’’ the value of w0 depending on the dielectric polarization operative; and the static polarization of constant eo. The second part of Eqs. 3 or 4 gives the the medium by the newly formed ion. charging energy of new ion(s) in the medium with We will proceed from the general formula defining the static dielectric constant es. (This charging en- the energy of creation of some charge(s): ergy includes, in the case of formation of two or more ions, their mutual interactions.) r(1) The two components of the free energy called W 5 dV w(l)dr(l) eV er(0) ‘‘static’’ and ‘‘’’ terms (which, in the present s(1) notations, are responsible for the effects of the 1 dS w(l)ds(l). (1) eS es(0) pre-existing field and of induced medium polariza- tion) were considered in the recent papers by Simon- 14,15 Here w(l) is potential; r(l) and s(l) are the volume son et al. However, they have not analyzed the and densities, all these values being problem of the dielectric constant for the continuum functions of the degree of charging l. In the course of calculations of the pre-existing field effect. A more detailed semi-continuum formalism and charging, r(l) and s(l) increase from zero to ri and some problems of its application are described in the si, and dr(l) 5ridland ds(l) 5sidl. The total field E(l) can be written as a superposition of the original Appendix. pre-existing field (the subscript 0) and the field of the newly formed ion, i.e., its own field plus the field of ELECTROSTATIC CALCULATIONS medium polarization produced by this ion (the sub- FOR PROTEINS script i) First of all, some refinement of the parameter eo should be made here. Strictly speaking, in the experi- 2=w(l) 5 E(l) 5 E0 1 Ei(l) 5 2=w0 2=w(l) (2a) mental X-ray structures only the coordinates of heavy atoms are defined, while the coordinates of which is equivalent, to within a constant, to are found from some model construction based on the bond lengths and angles for small molecules. In this sense, the situation here is some- w(l) 5w0 1wi(l) (2b) what similar to our description of electronic clouds. The indefinite constant in Eq. 2b is omitted, because, Therefore, some correction for the shift of H atoms in the absence of partial and newly formed charges, from their idealized positions (small changes in the length and orientation of N-H or O-H dipoles, etc.) we assume both w0 and wi equal to zero. In the integration of Eq. 1 with condition (2), we should be introduced. The effective eo now becomes not the purely electronic one, but one that includes should take into account that w0 is the potential of the pre-existing field, and hence it is independent of some part of the infrared polarization. It is difficult to define this part exactly. The correction to eo we are the parameter of charging degree l, while wi(l), the potential due to the new charges, is proportional to l interested in originates from the stretchings and deformations of the ’s polar bonds having (wi(l) 5lwi). rather high characteristic . Therefore, it should be close to the effective ‘‘quantum’’ boundary W 5 w r dV 1 w s dS eV 0 i eS 0 i eq of proteins (the value corresponding to frequencies 16 "v $ 4 kT). For water, this effect increases eq up to 1 1 2.1 (compared with e 5 1.8); for proteins, the total 1 w r dV 1 w s dS (3) o 2 eV i i 2 eS i i concentration of H atoms is of the same order of TWO DIELECTRIC CONSTANTS OF PROTEINS 177 magnitude as in water, but only about 15% of them ous distribution, depending on the system geom- belong to highly polar N-H or H bonds. Therefore, we etry). Each image charge depends, besides geometric estimate for proteins eq < 2.2. On the other hand, if parameters, on the ratio of the dielectric constants of the H atom positions are defined more precisely, only the protein and its surroundings (usually, water), optical value eo should be used. e.g., in the case of a planar boundary, on the param- We have derived Eqs. 3 and 4 in a general form, eter K 5 (eo 2esw)/(eo 1esw). Thus the solution and hence they are valid for both homogeneous and cannot be expressed as a function of a single eo of heterogeneous systems. Proteins are often sur- protein, but involves some combination of constants. rounded by another medium, in particular water. In It should be stressed here that, in considering the this case, the electric fields should be calculated pre-existing field, one should include in qms not only taking due account of the heterogeneity. This means the partial charges of the neutral multipoles of the that, in calculation of the pre-existing field, we protein (peptide groups, etc.) but also the charges of should use optical eo for protein but static esw for ions that have existed in an ionized form at the pH of water because water is not preorganized. The inte- the solution used to crystallize the protein (see, e.g., grals in these equations are taken over the total ref. 8). Indeed, ‘‘orientational’’ polarization, i.e., the volume of the system, including both protein and its shift of heavy atoms created by these ions has aqueous surroundings. In calculations of w0 (or corre- already been accounted for by the real positions of sponding E0) in this heterogeneous system, we sub- the protein atoms in a given protein structure. The stitute eo for the proteinaceous part and esw for the situation changes when one considers the possibility aqueous surroundings. Calculations of wi (and/or Ei, of neutralization of these ions at some quite different Di) employ es and esw, correspondingly. pH. Practically, this is equivalent to the charging of In this respect, calculation of the equilibrium the particle by the charge of an opposite sign. In this energy of ions in a protein globule differs from the case, one should use eo, as described above, for the calculation of such an essentially non-equilibrium effect of the pre-existing field of all other charges, but value as the reorganization energy that is relevant to the whole static dielectric constant es in calculation the kinetics of the charge transfer inside the protein. of its charging energy because upon neutralization While considering the last problem, one should bear the pre-existing ‘‘orientational’’ polarization disap- in mind that the pre-existing field, by definition, does pears in a normal way. not change in the course of reaction, and hence The last terms of Eq. 3 give the usual Bornian affects only the difference of equilibrium energies of charging energies of each newly formed ion and the the charge in its initial and final positions. Indeed, energy of their mutual interaction. For the latter, the the reorganization energy depends on the difference reasoning similar to that given above leads to Cou- of Ds (and/or Es) in initial and final states, and hence lomb law (q1q2/es R). Of course, the assumption that e the constant value of E0 cancels out, only the differ- inside the ion’s body is equal to the surrounding es is ence of Dis(Eis) being important. The reorganiza- very rough, and hence, in the case of a complex shape tion energy describes the medium repolarization, of the ion, one could expect deviation from the simple and so it is connected only with its response to the Coulombic interaction of two point charges. How- redistribution of free charges. To calculate this quan- ever, if the charge density distribution inside the ion tity, we have to find the difference of the recharging has a spherical symmetry, then the field has the energies in ‘‘slow’’ and ‘‘fast’’ dielectric media, the character of the central field, irrespective of the ‘‘fast’’ medium being described by optical ratio of the inner and outer dielectric constants. only, for both protein and water, and the ‘‘slow’’ by es Therefore, the interaction energy retains its simple and esw. Coulombic form. In calculating the energy of ion(s) in the pre- The charging energy of each separate charge (two existing field (first two terms of Eq. 3), we can use the last integrals in Eq. 3) may be considered, for a reasonable approximation that eo in the body of an heterogeneous system, to be composed of two contri- ion and in its surroundings is practically the same. butions: the charging in an infinite medium with the Then, we can consider w0 as a sum of potentials of properties of the protein, and the interaction of the central , i.e., Coulombic potentials set up by emerging charge with its own images. The second each partial charge m. In this case, the energy is part can be represented as these integrals with a equal to the product of the total charge and the potential of image charges wim; in the simple case of a 1 potential in the center of this charge: spherical symmetric charge, it equals ⁄2 o wimqi. The charging energy proper also depends on the charac- ter of the charge density distribution in the ion W 5 qmqi/eoRm. (5) om under consideration. In the case of a charge evenly distributed over the surface of a spherical ion of 2 In the case of a heterogeneous system, it is often radius a, the energy is qi /2aes. (This seems to be a convenient to present the result in the same - good model for many ions because the excessive like form of Eq. 5, adding to the real partial charges charge is concentrated mainly in the outer valence qm their images qm8 (discrete images or their continu- orbital.) As will be described in more detail else- 178 L.I. KRISHTALIK ET AL.

2 where, this expression (qi /2aes) is valid for the ever, this question was not treated quantitatively external medium-dependent part of charging energy there.12) The correct protocol of the semi-continuum at any spherically symmetric charge density distribu- analysis of the energetics of charge transfer pro- tion inside an ion of radius a. cesses in proteins should make use of different dielectric constants for computation of the two com- CONCLUDING REMARKS ponents of the total reaction energy. The mutual interaction of the atomic charges of This will result in substantial quantitative protein determines the definite equilibrium configu- changes. It is well known that the pre-existing ration of the atoms in the folded macromolecule. intraprotein electric field brings a marked contribu- Therefore, in calculations of the electric field gener- tion to reaction energy, usually compensating for a ated by this structure, only the electronic polariza- large part of the loss of ion solvation energy (see, e.g., tion of the medium should be employed because all the reviews quoted at the beginning of this paper). possible shifts of heavy atoms are already accounted With eo used for the calculation of this field, its effect for in their equilibrium coordinates. On the contrary, will increase about 1.5–2 times. the formation of a new charge, absent in the equilib- rium X-ray structure, results in small shifts of atom ACKNOWLEDGMENTS positions, revealing itself as an atomic (quasi- This work was supported by ISF grant MD 9300 orientational) polarization additional to the elec- and by NIH Fogarty Russian-USA Exchange grant tronic polarization. TW00063. E.L.M. was supported by an A.N. Frumkin These two cases were discussed, in application to a fellowship. semi-continuum description of protein electrostatics, in the papers quoted in the introduction. The princi- APPENDIX pal task of the present paper is to apply consistently In this appendix we give more details on con- the notions elaborated in these papers to the calcula- tinuum treatment and discuss some aspects of its tions of the energetics of charging or recharging of a application. particle inside the protein. The energy of the process is determined by the sum of two physically different Introduction of the Effective Dielectric Media components: the energy of the ion interaction with For a description of recharging processes in protein- the pre-existing intraprotein field, and the energy of water medium we formally introduce three effective charging of this newly formed ion in the polarizable dielectric media. Generally, the dielectric constants medium. Therefore, in semi-continuum calculations of these effective media inside both water and pro- of the total energy, two different dielectric constants tein parts of the system may be functions of coordi- are operative simultaneously: optical (electronic) eo nates (for some related data and considerations see, for the effect of the intraprotein field and static (es) e.g., refs. 11, 12, 14, 15, and 17–21). for the charging of the newly formed ion. In the real, The first effective medium is required to describe heterogeneous systems, one should use e of protein o the pre-existing field E0. This field is produced by the and static esw of aqueous surroundings for computa- pre-existing fixed charges and by the electronic tions of the intraprotein field, and es and esw in the polarization of protein (characterized by the optical charging energy calculations. dielectric constant, eo(r)) and by the total polariza- 17 Warshel et al. have criticized the widely em- tion of water (characterized by the static dielectric ployed approach by which the static dielectric con- constants, esw(r)). We may formally introduce the stant was used in calculating the intraprotein field: pre-existing dielectric displacement D0, which is thus, the effect of the permanent dipoles was in- related to the electric field by the effective dielectric cluded twice, as a source of the intraprotein field and constant of the first effective medium, e1(r). The as a source of dielectric polarization. In the present source of the fields E0 and D0 is the pre-existing paper, this contradiction has been eliminated. The charge distribution, r0(r). In the first effective me- field of permanent dipoles is screened dielectrically dium the pre-existing charges r0(r) are always at the only by the electronic polarization, and their contri- fixed equilibrium positions. The solution of the elec- bution to the static polarization is taken into account trostatic problem for charge distribution r0(r)inthe only for the field of external (in respect to these first medium is given by the potential w0. For this dipoles) charges. effective medium common relations for usual dielec- We have shown that, for calculation of the energy tric media are valid. of ions in protein, one should employ two dielectric constants of the protein simultaneously, optical and E0 52grad(w0); divD0 524p·r0(r); static. This differs from the protocol applied in many works (including our previous ones), in which a esw(r) D0 ; e1(r)E0; e1(r) 5 . (A1) single value of e was chosen. (As far as we know, only 5eo(r) in our recent paper was the intraprotein field consid- ered as independent of the static dielectric constant, As discussed in the main text of this paper, the the latter being expressed as evolving in time; how- protein dielectric constant depends on a particular TWO DIELECTRIC CONSTANTS OF PROTEINS 179 way of describing protein charge distribution. There- The general expression for the electrostatic work fore, unlike the pre-existing electric field, the pre- of a recharging process is as follows: existing dielectric displacement depends on the method of description (see Eq. A1). That is why D0 is 1 dr(l) a rather formal quantity. W 5 e e w(l)· dV dl. (A3) l50 1 V dl 2 We introduce the second effective dielectric medium to describe the changes in the system upon a slow Here the integral over l represents the integral recharging process. This effective medium is charac- along arbitrary trajectory connecting the initial and terized by the effective static dielectric constant, final states of the process. Choosing a particular path of the process, we write esw(r) e2(r) 5 . 5es(r) E( ) E ·E; D( ) D ·D. (A4) l 5 0 1l i l 5 0 1l i

Finally, the third effective dielectric medium de- From equations E(l) ; 2gradw(l), Eq. A4, and scribes the changes in the system upon the fast relationships between the electric fields and poten- recharging process. This effective medium is charac- tials (Eqs. A1 and A2), we derive 2gradw(l) 5 terized by the effective dielectric constant account- 2gradw0 2l·gradwi. Hence, with an indefinite ing for the inertialess polarization of the medium, constant equal to zero,

eow(r) w(l) 5wo 1l·wi. (A5) e3(r) 5 . 5eo(r) Similarly, from equations divD(l) ; 24p · r(l) For the third medium the dielectric constant of the and Eq. A4, and relationships between the dielectric protein depends on the purpose of the current formal- displacements and charge distributions (Eqs. A1 and ism. For calculation of the total reorganization en- A2), we derive ergy of electron transfer, e3(r) is the optical dielectric constant (eo(r) 5 2.1 and eow(r) 5 1.8), while for pro- r(l) 5r0(r)1l·Dr(r). (A6) ton transfer e3(r) coincides with the ‘‘quantum’’ dielec- tric constant (eo(r) < 2.2 and eow(r) < 2.1). For calcu- Substituting Eqs. A5 and A6 in Eq. A3, we obtain lating the activation energy of reaction, keep in mind the work that only classic polarization modes participate in formation of the activation barrier. Therefore, for 1 both and electron transfer the same ‘‘quan- Wi 5 e w0 · Dr(r)dV 1 e wi · Dr(r)dV. (A7) tum’’ dielectric constant is operative. V 2 V Let us consider an explicit change of charge distri- bution function Dr(r) proceeding in the second or We here that the relation A7 is valid for an third effective media. In these cases the common arbitrary dielectric provided that the changes pro- relations are valid duced by explicit charge redistribution can be de- scribed with the aid of a continuum approach using some effective dielectric constant. Ei 52grad(wi); Substituting for i the subscript 2 in Eq. A7, we divDi 524p·Dr(r); Di 5ei(r)·Ei. (A2) immediately obtain the work of the slow recharging process. Here the subscript i 5 2, 3 numerates the effective From Eq. A7 we infer two important consequences. medium. First, the energy of interaction of the charge with the pre-existing field is affected directly only by the General Relations for the Electrostatic Work values of electric field or potential before the process. of Recharging Process This means that for calculation of work it is of no Let us assume that the electric field inherent to a importance how the pre-existing electric field or the medium before a recharging process (the pre- potential have been calculated. Generally, one may existing electric field), E0, and the corresponding chose any procedures of the calculation of the , w0, are known. Suppose that the change of field before the process, e.g., the quantum chemical the explicit charge distribution function in protein- calculations for the protein dissolved in water or water system produces a change in the fields Ei and semi-continuum calculations with protein partial Di (see Eq. A2) so that the fields change from E0 and charges fixed in the first effective medium. D0 to E0 1 Ei and D0 1 Di. This is equivalent to the Second, the work of the recharging process (Eq. statement that changes in all fields (electric field, A7) is equivalent to the work of charging of effective dielectric displacement, and polarization) may be charge Dr(r). Thus, the recharging process may be described using a continuum approach. reduced to the charging one with the effective charge. 180 L.I. KRISHTALIK ET AL.

During a fast recharging process all inertial polar- This equation coincides with that for the reorgani- ization is frozen. Fast process induces only the zation of a dielectric medium derived by Liu and change in the electronic polarization. This change in Newton22 and Marcus.23 Thus the expression for polarization, along with the change in the electric reorganization free energy for the preorganized pro- field and dielectric displacement, can be described, tein medium turns out to be the same as for the in continuum formalism, with the use of the effective usual dielectric medium. charge Dr(r) in the electronic (third) effective me- According to Eq. A9, the reorganization energy is dium. Hence, the work of fast process may be ob- independent of the potential of the pre-existing field, tained from Eq. A7 by formally substituting the i by and one may chose any reference structure, whether the subscript 3. it is known or not. It may be helpful to have relationships for the potential after slow and fast recharging processes, The Relation Between Protein Dielectric Constants and Microscopic Changes of the w and w . During the recharging process the slow fast Protein Structure potential is given by Eq. A5. At the final stage of the process l51, and we get An external source of electric field induces protein atomic shifts from the equilibrium reference posi-

wslow 5w0 1w2; wfast 5w0 1w3 (A8) tions. At the same time, these shifts could be de- scribed using the continuum approach of this paper. where w2 and w3 are the potentials obtained from the Therefore it would be possible in principle to get solution of the electrostatic problems (Eqs. A2) for from the experiment the relation between the two the effective charge Dr(r) in the second and third dielectric constants of protein and the atomic shifts, effective media. if it would be possible to obtain these shifts from Generally speaking, one can operate with any structural studies of the protein in the presence and reference state (the state for which the protein absence of the field source. Here, any source of the structure is available) because the total work of the external field could be considered, e.g., the oxidized recharging process depends directly only on the or reduced group of the protein, the external perma- potential before the process under consideration. nent electric field applied to the protein, and so on. The procedure of the calculation of, e.g., the charging For simplicity, we consider the case of a spherical energy of charge q2 in the presence of charge q1 that ion formation in an infinite ‘‘uniform’’ protein that is was absent in the reference equilibrium protein in equilibrium reference structure. We assume the structure (with the potential of the reference struc- ion charge distribution function, q(r), to be such that ture being w0), is as follows. The potential after the the total charge q is uniformly distributed inside a charging by q1 (determined through Eq. A8) gives the thin surface layer of an ion of radius a. potential of the pre-existing field for the second The work of such charging is given by Eq. A7 in the charging process (by q2): w01 5wslow 5w0 1w21, with second medium, with the effective charge Dr(r) 5 w21 being the potential of charge q1 in the second q(r): effective medium. The required charging energy of charge q2 is derived through Eq. A7 with effective 1 W 5 wr · q(r)dV 1 w (q)·q(r)dV (A10) charge Dr(r) 5 q2 and the potential w0 5w01. eV 0 2 eV 2 Reorganization Free Energy where To obtain reorganization free energy for the fast recharging process, Er, we should use the standard 1 qk definition, namely, reorganization energy is the dif- wr 5 0 o r ference between the energies of the final states eo k 0r 2 rk 0 obtained at fast and slow charge transfer from the same initial state. This difference may be calculated is the potential of the pre-existing field at point r as a difference in work of the fast and slow recharg- produced by all the pre-existing charges qk in the r ing process beginning from the same initial reference equilibrium positions rk of the reference structure, state. and Thus, for the reorganization energy for redistribu- tion of the explicit charges (with effective charge 1 q(r8) w2(q) 5 e dr8 Dr(r)): es 0r 2 r80

1 is the potential created by the charge distribution E 5 W 2 W 5 w · Dr(r)dV r 3 2 2 eV 3 q(r) in the static medium. The ion charging results in new equilibrium posi- 1 tions, r , of the protein pre-existing charges. Let us 2 w · Dr(r)dV. (A9) k 2 eV 2 discharge the same ion and calculate the discharging TWO DIELECTRIC CONSTANTS OF PROTEINS 181 work, W8, using the same formula (Eq. A7). Now the Some Remarks on the Choice sign of effective charge is the opposite, Dr(r) 52q(r), of Partial Charges and the ion appearing in the new equilibrium struc- It is worthwhile discussing the problem of the ture must be considered a pre-existing charge. We choice and treatment of partial charges. First we have recall that partial charges are not directly observ- able quantities, and their values depend on the purpose for which they are used. For calculation of W8 52ew0 ·q(r)dV V the pre-existing field, it is hardly possible to perform 1 calculations of electronic density distribution of the 1 w (2q)·(2q(r))dV (A11) 2 eV 2 entire protein molecule. Therefore, we must use the data on electronic densities of separate protein frag- with the potential of the pre-existing charges in the ments. The electric field due to a protein fragment new equilibrium structure may be approximated by the field of model charge distribution. It is usually convenient to describe this model distribution by a set of point charges at 1 q 1 q(r8) k certain positions. For our purpose, the most appropri- w0 5 o 1 e dr8 eo k 0r 2 rk 0 eo 0r 2 r80 ate set would be a set of point charges that reproduce accurately enough the potential around the separate and the potential of the charge distribution 2q(r)in fragment. Generally speaking, this set is not unique. the static medium w2(2q). It is well known that polarizable surroundings Then we must demand that the slow recharging affect the electronic density distribution of solute. process is reversible. This means that the calculated Therefore, the values of partial charges approximat- charging and discharging energies must be the same ing the solute charge density are also affected by the in value but opposite in sign: W 52W8. surrounding reaction field, which is formed by all So, taking into account that w2(2q)·(2q(r)) 5 kinds of polarization. Thus the most suitable partial w2(q)·q(r), we derive from Eqs. A10 and A11 charges are those related to the separate fragment

placed in dielectric with e5es. 2 1 1 q 1 qqk 1 qqk Different sets of partial charges are available in 2 · 5 2 o o r the literature. Along with other parameters, they are 1es eo2 a eo k 0R 2 rk 0 eo k 0R 2 r 0 k designed to reproduce the pair-wise atom-atom where R is the radius vector of the center of the ion. interactions and particularly the electrostatic inter- This relation can be simplified assuming that the actions. The latter are screened by the electronic polarization that reduces, in continuum description, atomic shifts from the reference positions, Drk 5 rk 2 rr, are small: the interaction energy by e5e0. To avoid the explicit k allowance for the screening effect, the partial charges

r should be scaled in an ideal case, so that the atom- eo q Drk ·(R2rk) 1 2 · > 2 q . (A12) atom interaction of scaled atomic charges will look o k r 3 1 es2 a k R2r 0 k0 like the interaction in vacuum, i.e., at e51. This scaling procedure can be applied quite rigorously Here the summation is taken over all pre-existing only to the case of charges incorporated in an infinite charges qk (except the ion q). uniform medium. The application of such sets to the The electric field of the ion q placed in vacuum present formalism deserves separate consideration. (with charge distribution function introduced above) Let us consider the Poisson equation for a number inside the ion body is absent. Therefore, the forma- of arbitrary disposed charges (both partial and full tion of such an ion in protein results in shifts of ones) r0(r) in some dielectrically inhomogeneous partial charges only outside the ion. Thus the radius medium with constant e(r): a is involved in Eq. A12 as a radius of a protein region where the shifts of partial charges upon the div(e(r)gradw) 5 4pr0(r). ion formation are absent. For a general case of finite protein surrounded by We may introduce fictitious charges, , and water, Eq. A12 is more complicated and includes the potential, r0(r) ; r0(r)/Œc, e(r) ; e(r)/c, and w ; wŒc, dielectric constant of water by the effect of the respectively (where c is an arbitrary positive con- images of pre-existing charges. stant), for which the Poisson equation is also valid Evaluated from Eq. A12, the relation between the protein dielectric constants may be compared with div(e(r)gradw) 5 4pr0(r). independent estimates of these constants, and thus the self-consistency of the semi-continuum approach The electrostatic interaction energy is generally may be studied. represented by a product of potential owing to some 182 L.I. KRISHTALIK ET AL. charge and the value of another charge (more ex- considered as performed for protein placed in a actly, the spatial integral of this product). Hence, the uniform medium with dielectric constant eo. product of a fictitious charge and a fictitious poten- REFERENCES tial gives the true interaction energy: wr5wr. For the case of the above-mentioned sets of partial 1. Harvey, S.C. Treatment of electrostatic effects in macromo- lecular modeling. Proteins 5:78–92, 1989. charges, the form of the Coulombic interaction be- 2. Davis, M.E., McCammon, J.A. Electrostatics in biomolecu- tween two partial charges in an infinite uniform lar structure and dynamics. Chem. Rev. 90:509–521, 1990. 3. Sharp, K.A., Honig, B.H. Electrostatic interactions in medium is qkqj/rkj (here rkj is the distance between macromolecules: Theory and applications. Annu. Rev. Bio- scaled partial charges qk and qj), while the true form phys. Biophys. Chem. 19:301–332, 1990. 4. Warshel, A., Åqvist, J. 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