A general method to derive site-site or united-residue potentials is presented. The basic principle of the method is the separation of the degrees of freedom of a system into the primary and secondary ones. The primary degrees of freedom describe the basic features of the system, while the secondary ones are averaged over when calculating the potential of mean force, which is hereafter referred to as the restricted free energy (RFE) function. The RFE can be factored into one-, two-, and multibody terms, using the cluster-cumulant expansion of Kubo. These factors can be assigned the functional forms of the corresponding lowest-order nonzero generalized cumulants, which can, in most cases, be evaluated analytically, after making some simplifying assumptions. This procedure to derive coarse-grain force fields is very valuable when applied to multibody terms, whose functional forms are hard to deduce in another way (e.g., from structural databases). After the functional forms have been derived, they can be parametrized based on the RFE surfaces of model systems obtained from all-atom models or on the statistics derived from structural databases. The approach has been applied to our united-residue force field for proteins. Analytical expressions were derived for the multibody terms pertaining to the correlation between local and electrostatic interactions within the polypeptide backbone; these expressions correspond to up to sixth-order terms in the cumulant expansion of the RFE. These expressions were subsequently parametrized by fitting to the RFEs of selected peptide fragments, calculated with the empirical conformational energy program for peptides force field. The new multibody terms enable not only the heretofore predictable α-helical segments, but also regular β-sheets, to form as the lowest-energy structures, as assessed by test calculations on a model helical protein A, as well as a model 20-residue polypeptide (betanova); the latter was not possible without introducing these new terms.

Recent improvements of a hierarchical ab initio or de novo approach for predicting both α and β structures of proteins are described. The united-residue energy function used in this procedure includes multibody interactions from a cumulant expansion of the free energy of polypeptide chains, with their relative weights determined by Z-score optimization. The critical initial stage of the hierarchical procedure involves a search of conformational space by the conformational space annealing (CSA) method, followed by optimization of an all-atom model. The procedure was assessed in a recent blind test of protein structure prediction (CASP4). The resulting lowest-energy structures of the target proteins (ranging in size from 70 to 244 residues) agreed with the experimental structures in many respects. The entire experimental structure of a cyclic α-helical protein of 70 residues was predicted to within 4.3 Å α-carbon (C ^α ) rms deviation (rmsd) whereas, for other α-helical proteins, fragments of roughly 60 residues were predicted to within 6.0 Å C ^α rmsd. Whereas β structures can now be predicted with the new procedure, the success rate for α/β- and β-proteins is lower than that for α-proteins at present. For the β portions of α/β structures, the C ^α rmsd's are less than 6.0 Å for contiguous fragments of 30–40 residues; for one target, three fragments (of length 10, 23, and 28 residues, respectively) formed a compact part of the tertiary structure with a C ^α rmsd less than 6.0 Å. Overall, these results constitute an important step toward the ab initio prediction of protein structure solely from the amino acid sequence.

A unified coarse-grained model of three major classes of biological molecules—proteins, nucleic acids, and polysaccharides—has been developed. It is based on the observations that the repeated units of biopolymers (peptide groups, nucleic acid bases, sugar rings) are highly polar and their charge distributions can be represented crudely as point multipoles. The model is an extension of the united residue (UNRES) coarse-grained model of proteins developed previously in our laboratory. The respective force fields are defined as the potentials of mean force of biomacromolecules immersed in water, where all degrees of freedom not considered in the model have been averaged out. Reducing the representation to one center per polar interaction site leads to the representation of average site–site interactions as mean-field dipole–dipole interactions. Further expansion of the potentials of mean force of biopolymer chains into Kubo’s cluster-cumulant series leads to the appearance of mean-field dipole–dipole interactions, averaged in the context of local interactions within a biopolymer unit. These mean-field interactions account for the formation of regular structures encountered in biomacromolecules, e.g., α-helices and β-sheets in proteins, double helices in nucleic acids, and helicoidally packed structures in polysaccharides, which enables us to use a greatly reduced number of interacting sites without sacrificing the ability to reproduce the correct architecture. This reduction results in an extension of the simulation timescale by more than four orders of magnitude compared to the all-atom representation. Examples of the performance of the model are presented. Components of the Unified Coarse Grained Model (UCGM) of biological macromolecules