Estimation of effective interresidue contact energies from protein crystal structures: quasi-chemical approximation

Effective interresidue contact energies for proteins in solution are estimated from the numbers of residue-residue contacts observed in crystal structures of globular proteins by means of the quasi-chemical approximation with an approximate treatment of the effects of chain connectivity. Employing a lattice model, each residue of a protein is assumed to occupy a site in a lattice and vacant sites are regarded to be occupied by an effective solvent molecule whose size is equal to the average size of a residue. A basic assumption is that the average characteristics of residue-residue contacts formed in a large number of protein crystal structures reflcct actual differences of interactions among residues, as if there were no significant contribution from the specific amino acid sequence in each protein as well as intraresidue and short-range interactions. Then, taking account of the effects of the chain connectivity only as imposing a limit to the size of the system, Le. , the number of lattice sites or the number of effective solvent molecules in the system, the system is regarded to be the mixture of unconnected residues and effective solvent molecules. The quasi-chemical approximation, that contact pair formation resembles a chemical reaction, is applied to this system to obtain formulas that relate the statistical averages of the numbers of contacts to the contact energies. The number of effective solvent molecules for each protein is chosen to yield the total number of residue-residue contacts equal to its expected value for the hypothetical case of hard sphere interactions among residues and effective solvent molecules; the expected number of residue-residue contacts at this condition has been crudely estimated by means of a freely jointed chain distribution and an expansion originating in hard sphere interactions. Each residue is represented by the center of its side chain atom positions, and contacts among residues and effective solvent molecules are defined to be those pairs within 6.5 A, a distance that has been chosen on the basis of the observed radial distribution of residues; nearest-neighbor pairs along a chain are explicitly excluded in counting contacts. Coordination numbers, for each type of residue as well as for solvent molecules, are estimated from the mean volume of each type of residue and used to evaluate the numbers of residue-solvent and solvent-solvent contacts from the numbers of residue-residue contacts. The estimated values of contact energies have reasonable residue-type dependences, reflecting residue distributions in protein crystals; nonpolar-residue-in and polar-residue-out are seen as well as the segregation of those residue groups. In addition, there is a linear relationship between the average contact energies for nonpolar residues and their hydrophobicities reported by Nozaki and Tanford; however, the magnitudes on average are about twice as large. The relevance of results to protein folding and other applications are discussed. Introduction A complete treatment of protein conformations in solution requires inclusion of solvent effects. Solvent molecules interact with atoms in proteins not only in shortrange interactions such as hydrogen-bond formation and van der Waals interactions but modify electrostatic interactions between protein atoms. Also the entropy of water molecules around protein molecules differs from that of bulk water by forming more ordered cagelike structures or binding to specific sites. As originally pointed out by Kauzmann,' hydrophobic interactions, which would occur explicitly because of the nonspecific solvent effects, might be a principal force in leading to a collapsed protein molecule. Hydrophobic energies have been evaluated, among other ways, as the free energy changes of transfer of amino acids from ethanol or dioxane to water2 and of liquid, hydrocarbons into water.3* Chothia7-" evaluated the contributions of hydrophobic energy to the formation of secondary, tertiary, and quaternary structures by employing the estimates in the reference2 quoted above for values of the hydrophobic energy of interfacial areas exposed to water. His and others12 estimates indicate that the hydrophobic energies, or the solvent effects, are a major contributor to the energetics of protein folding, essentially because large surface areas of protein molecules become buried in the interior upon folding. However, there is the fundamental question of whether liquid hydrocarbons and the organic solvents can completely represent a protein in t e r i~ r . ' ~ Lee14J5 has pointed out on the basis of a scaled particle theory that thermodynamic properties such as the partial molecular volume of the solute in dilute binary solution^'^ and the change in the Ben-Naim local standard chemical potential of a solute molecule upon transferring i t from the gas phase to a liquid phase15 depend significantly on both the packing density of pure solvent and the ratio of the size of the solvent molecule to that of the solute molecule. Then, he has claimed that an obvious major difference of the high packing density and solidlike rigidity of protein interiors from small nonpolar solvents and even simple polymers makes i t difficult to justify using the transfer data generally in quantitative studies of protein folding. Thus, estimates of hydrophobic interactions which are specific to protein molecules would be desirable. Protein folding processes include a wide range of protein conformations from denatured to native states. The conformational freedom of a protein is vast. This makes it difficult to simulate the whole process of protein folding, if all atoms of a protein and solvent molecules are to be included in a detailed energy calculation. The geometry of molecules and interaction potentials require some simplification. The principal purpose of the present work is to include solvent effects into effective interresidue contact energies, which can then provide a crude estimate of the long-range component of conformational energies. Tanaka and Scheraga16 estimated contact energies by a method which may appear to be similar but ignores solvent and is different in essence from the present one; incidentally, their method yielded extremely large magnitudes for contact energies. Here the effective contact energies between residues in proteins will be estimated directly from the numbers of residue-residue contacts observed in protein crystal structures by regarding them as statistical averages in the quasi-chemical with an approximate treatment of the effects of chain connectivity. Estimated contact energies will be compared with experimental values This article not subject to U.S. Copyright. Published 1985 by the American Chemical Society Macromolecules, Vol. 18, No. 3, 1985 of hydrophobic energies. Also, the relevance of results to protein folding and other applications will be discussed.