Fluctuations about the native conformation of proteins have proven to be suitably reproduced with a simple elastic network model, which has shown excellent agreement with a number of different properties for a wide variety of proteins. This scalar model simply investigates the magnitudes of motion of individual residues in the structure. To use the elastic model approach further for developing the details of protein mechanisms, it becomes essential to expand this model to include the added details of the directions of individual residue fluctuations. In this paper a new tool is presented for this purpose and applied to the retinol-binding protein, which indicates enhanced flexibility in the region of entry to the ligand binding site and for the portion of the protein binding to its carrier protein.

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 reflect 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, i.e., 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 Á, 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.

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