PROTEINS: Structure, Function, and Genetics 28:174–182 (1997) Electrostatics of Proteins: Description in Terms of Two Dielectric Constants Simultaneously L.I. Krishtalik,* A.M. Kuznetsov, and E.L. Mertz A.N. Frumkin Institute of Electrochemistry, 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 ion with the pre-existing In low-molecular-weight liquids, the electric field intraprotein electric field, and the energy due set up by their dipoles 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 atoms 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 protein structure. 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- molecule, and therefore proteins can be defined as tions of water molecules 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 electrons 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 ionization 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 electrostatics, 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- charge density, e.g., ascribing to each atom (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 dipole 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, infrared, 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 polarizabilities 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 work 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 electron 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.