Based on the dipole model of peptide groups developed in our earlier work [Liwo et al., Prot. Sci., 2, 1697 (1993)], a cumulant expansion of the average free energy of the system of freely rotating peptide‐group dipoles tethered to a fixed α‐carbon trace is derived. A graphical approach is presented to find all nonvanishing terms in the cumulants. In particular, analytical expressions for three‐ and four‐body (correlation) terms in the averaged interaction potential of united peptide groups are derived. These expressions are similar to the cooperative forces in hydrogen bonding introduced by Koliński and Skolnick [J. Chem. Phys., 97, 9412 (1992)]. The cooperativity arises here naturally from the higher order terms in the power‐series expansion (in the inverse of the temperature) for the average energy. Test calculations have shown that addition of the derived four‐body term to the statistical united‐residue potential of our earlier work [Liwo et al., J. Comput. Chem., 18, 849, 874 (1997)] greatly improves its performance in folding poly‐l‐alanine into an α‐helix. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 259–276, 1998

Recent improvements of a hierarchical ab initio or de novo approach for predicting both ␣ and ␤ 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 ␣-helical protein of 70 residues was predicted to within 4.3 Å ␣-carbon (C ␣ ) rms deviation (rmsd) whereas, for other ␣-helical proteins, fragments of roughly 60 residues were predicted to within 6.0 Å C ␣ rmsd. Whereas ␤ structures can now be predicted with the new procedure, the success rate for ␣͞␤-and ␤-proteins is lower than that for ␣-proteins at present. For the ␤ portions of ␣͞␤ structures, the C ␣ 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 ␣ 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. I mportant progress has been made in recent years toward the physics-based computation of protein structure based solely on knowledge of the amino acid sequence. This approach, commonly referred to as an ab initio or de novo method (1-3), is based on the thermodynamic hypothesis formulated by Anfinsen (4), according to which the native structure of a protein corresponds to the global minimum of its free energy under given conditions. Protein structure prediction by using ab initio methods is accomplished by a search for a conformation corresponding to the global-minimum of an appropriate potential energy function without use of secondary structure prediction, homology modeling, threading, etc.Until recently, ab initio protein structure prediction based solely on the thermodynamic hypothesis was considered unfeasible (5-7) mainly because of the inaccuracy of the potential functions used to describe protein conformational energy and the lack of powerful global optimization methods for exploring the energy landscapes represented by those functions. Other types of knowledge-based methodologies, such as homology modeling (8-13) or threading methods (9,12,14) have been considered to be the most successful approaches. However, the success of these methods depends on the presence of sequentially or structurally homologous proteins in the databases. Furthermore, they do not provide a general understanding of the...

Recent improvements in the protein-structure prediction method developed in our laboratory, based on the thermodynamic hypothesis, are described. The conformational space is searched extensively at the united-residue level by using our physics-based UNRES energy function and the conformational space annealing method of global optimization. The lowest-energy coarse-grained structures are then converted to an all-atom representation and energyminimized with the ECEPP͞3 force field. The procedure was assessed in two recent blind tests of protein-structure prediction. During the first blind test, we predicted large fragments of ␣ and ␣؉␤ proteins [60 -70 residues with C ␣ rms deviation (rmsd) <6 Å]. However, for ␣؉␤ proteins, significant topological errors occurred despite low rmsd values. In the second exercise, we predicted whole structures of five proteins (two ␣ and three ␣؉␤, with sizes of 53-235 residues) with remarkably good accuracy. In particular, for the genomic target TM0487 (a 102-residue ␣؉␤ protein from Thermotoga maritima), we predicted the complete, topologically correct structure with 7.3-Å C ␣ rmsd. So far this protein is the largest ␣؉␤ protein predicted based solely on the amino acid sequence and a physics-based potential-energy function and search procedure. For target T0198, a phosphate transport system regulator PhoU from T. maritima (a 235-residue mainly ␣-helical protein), we predicted the topology of the whole six-helix bundle correctly within 8 Å rmsd, except the 32 C-terminal residues, most of which form a ␤-hairpin. These and other examples described in this work demonstrate significant progress in physics-based protein-structure prediction.global optimization ͉ thermodynamic hypothesis T o date, the great majority of successful algorithms for proteinstructure prediction are knowledge-based approaches; they make explicit use of homology modeling (1, 2) or fold recognition methods (2-6). This feature even pertains to most of the methods considered as ab initio (7,8), which, in theory, should not make explicit use of structural databases. However, in-depth understanding of the physical principles of formation of protein structure requires the development of physics-based methods for proteinstructure prediction (9). Moreover, such methods will be independent of structural databases used in the training of knowledgebased methods. Furthermore, physics-based methods will enable us to study the structures of proteins that seem to possess a degenerate native state, such as the prion proteins, to simulate protein-folding pathways, to understand the mechanisms of protein folding, and to study interactions of proteins with other biomacromolecules and their assemblies (e.g., nucleic acids, polysaccharides, lipids, etc.). The underlying principle of physics-based methods for proteinstructure prediction is Anfinsen's thermodynamic hypothesis (10), according to which protein molecules adopt the conformations that are the global minima of their potential-energy surfaces. The methods based on this hypoth...