A two-stage procedure for the determination of a united-residue potential designed for protein simulations is outlined. In the first stage, the long-range and local-interaction energy terms of the total energy of a polypeptide chain are determined by analyzing protein᎐crystal data and averaging the Ž all-atom energy surfaces. In the second stage described in the accompanying . article , the relative weights of the energy terms are optimized so as to locate the native structures of selected test proteins as the lowest energy structures. The

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

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Based on the concept that hydrophobic interactions cause a polypeptide chain to adopt a compact structure, a method is proposed to predict the structure of a protein. The procedure is carried out in four stages: (1) use of a virtual-bond united-residue approximation with the side chains represented by spheres to search conformational space extensively using specially designed interactions to lead to a collapsed structure, (2) conversion of the lowest-energy virtual-bond united-residue chain to one with a real polypeptide backbone, with optimization of the hydrogen-bond network among the backbone groups, (3) perturbation of the latter structure by the electrostatically driven Monte Carlo (EDMC) procedure, and (4) conversion of the spherical representation of the side chains to real groups and perturbation of the whole molecule by the EDMC procedure using the empirical conformational energy program for peptides (ECEPP/2) energy function plus hydration. Application of this procedure to the 36-residue avian pancreatic polypeptide led to a structure that resembled the one determined by X-ray crystallography; it had an a-helix starting at residue 13, with the N-terminal portion of the chain in an extended conformation packed against the a-helix. Similar structures with slightly higher energies, but looser packing, were also obtained.Keywords: compact conformations; conversion from a united-residue representation to an all-atom chain; hydrophobic-residue packing; Monte Carlo methods; multiple-minima problem; potential of mean force; protein folding; united-residue representation of a polypeptide chain In our continuing effort to surmount the multiple-minima problem (Scheraga, 1989;Kostrowicki & Scheraga, 1992;Olszewski et al., 1992) in computing the structure of a protein, we have developed a procedure that takes advantage of the fact that the protein core tends to consist of tightly packed nonpolar residues, with the polar ones located on the surface (Kauzmann, 1959;Rackovsky & Scheraga, 1977;Richards, 1977;Wertz & Scheraga, 1978;Chan & Dill, 1990;. Such an example of selforganization is reminiscent of early work by Onsager (1949) on the packing of tobacco mosaic virus particles and by Flory (1956) on the packing of rodlike polymers (including a-helices). These workers showed that, as the solution concentration increases, the system separates into two phases, an organized ordered one and a more dilute isotropic one. The self-organization of the moreconcentrated ordered phase arises solely from entropic effects; i.e., in a concentrated solution it is easier to pack rods in an ordered anisotropic array than in a disordered isotropic one. Of course, while entropy alone can account for this phenomenon, possible attractive interactions between the ordered rods can also contribute to this selforganization. We thus consider the possibility that, if the available conformational space of a polypeptide is confined, organized structure will be promoted.

A new optimization method is presented to search for the global minimum-energy conformations of polypeptides. The method combines essential aspects of the build-up procedure and the genetic algorithm, and it introduces the important concept of ''conformational space annealing.'' Instead of considering a single conformation, attention is focused on a population of conformations while new conformations are obtained by modifying a ''seed conformation.'' The annealing is carried out by introducing a distance cutoff, D , which is defined in the conformational space; D effectively divides the cut cut whole conformational space of local minima into subdivisions. The value of D cut is set to a large number at the beginning of the algorithm to cover the whole conformational space, and annealing is achieved by slowly reducing it. Many distinct local minima designed to be distributed as far apart as possible in conformational space are investigated simultaneously. Therefore, the new method finds not only the global minimum-energy conformation but also many other distinct local minima as by-products. The method is tested on Metenkephalin, a 24-dihedral angle problem. For all 100 independent runs, the accepted global minimum-energy conformation was obtained after about 2600 minimizations on average.

An algorithm is proposed for the conversion of a virtual-bond polypeptide chain (connected C" atoms) to an allatom backbone, based on determining the most extensive hydrogen-bond network between the peptide groups of the backbone, while maintaining all of the backbone atoms in energetically feasible conformations. Hydrogen bonding is represented by aligning the peptide-group dipoles. These peptide groups are not contiguous in the amino acid sequence. The first dipoles to be aligned are those that are both sufficiently close in space to be arranged in approximately linear arrays termed dipole paths. The criteria used in the construction of dipole paths are: to assure good alignment of the greatest possible number of dipoles that are close in space; to optimize the electrostatic interactions between the dipoles that belong to different paths close in space; and to avoid locally unfavorable amino acid residue conformations. The equations for dipole alignment are solved separately for each path, and then the remaining single dipoles are aligned optimally with the electrostatic field from the dipoles that belong to the dipole-path network. A least-squares minimizer is used to keep the geometry of the a-carbon trace of the resulting backbone close to that of the input virtual-bond chain. This procedure is sufficient to convert the virtual-bond chain to a real chain; in applications to real systems, however, the final structure is obtained by minimizing the total ECEPP/2 (empirical conformational energy program for peptides) energy of the system, starting from the geometry resulting from the solution of the alignment equations. When applied to model a-helical and &sheet structures, the algorithm, followed by the ECEPP/2 energy minimization, resulted in an energy and backbone geometry characteristic of these a-helical and P-sheet structures. Application to the a-carbon trace of the backbone of the crystallographic SPTI structure of bovine pancreatic trypsin inhibitor, followed by ECEPP/2 energy minimization with C"-distance constraints, led to a structure with almost as low energy and root mean square deviation as the ECEPP/2 geometry analog of SPTI, the best agreement between the crystal and reconstructed backbone being observed for the residues involved in the dipole-path network.

The conformation of the 29-residue rat galanin neuropeptide was studied using the Monte Carlo with energy minimization (MCM) and electrostatically driven Monte Carlo (EDMC) methods. According to a previously elaborated procedure, the polypeptide chain was first treated in a united-residue approximation, in order to enable extensive exploration of the conformational space to be carried out (with the use of MCM). Then the low-energy united-residue conformations were converted to the all-atom representations, and EDMC simulations were carried out for the all-atom polypeptide chains, using the ECEPP/3 force field with hydration included. In order to estimate the effect of environment on galanin conformation, the low-energy conformations obtained as a result of these simulations were taken as starting structures for further EDMC runs that did not include hydration. The lowest-energy conformation obtained in aqueous solution calculations had a nonhelical N-terminal part packed against the nonpolar face of a residual helix that extended from Pro13 toward the C-terminus. One next lowest-energy structure was a nearly-all-helical conformation, but with a markedly higher energy. In contrast, all of the low-energy conformations in the absence of water were all-helical differing only by the extent to which the helix was kinked around Pro13. These results are in qualitative agreement with the available NMR and CD data of galanin in aqueous and nonaqueous solvents.

The performance of the physics-based protocol, whose main component is the United Residue (UNRES) physics-based coarse-grained force field, developed in our laboratory for the prediction of protein structure from amino acid sequence, is illustrated. Candidate models are selected, based on probabilities of the conformational families determined by multiplexed replica-exchange simulations, from the 10th Community Wide Experiment on the Critical Assessment of Techniques for Protein Structure Prediction (CASP10). For target T0663, classified as a new fold, which consists of two α + β domains homologous to those of known proteins, UNRES predicted the correct symmetry of packing, in which the domains are rotated with respect to each other by 180°in the experimental structure. By contrast, models obtained by knowledge-based methods, in which each domain is modeled very accurately but not rotated, resulted in incorrect packing. Two UNRES models of this target were featured by the assessors. Correct domain packing was also predicted by UNRES for the homologous target T0644, which has a similar structure to that of T0663, except that the two domains are not rotated. Predictions for two other targets, T0668 and T0684_D2, are among the best ones by global distance test score. These results suggest that our physicsbased method has substantial predictive power. In particular, it has the ability to predict domain-domain orientations, which is a significant advance in the state of the art.protein folding | structure symmetry | multi-domain packing P rediction of protein structures from amino acid sequence still remains an unsolved problem of computational biology. Although, since the famous experiments by Anfinsen (1), it is known that a protein adopts the structure which is the (kinetically reachable) global minimum of the free energy of a system, it is not straightforward to implement this physical principle in practice because of the inaccuracy of existing force fields and because of the enormous difficulty to search the conformational space of the system. Therefore, the most effective methods for protein-structure prediction nowadays are knowledge-based approaches, in which database information is incorporated explicitly into the procedure (2). These methods can be divided into three categories, namely, comparative (homology) modeling (3-5), in which the target sequence is compared with the sequences for which experimental structures are known and those structures are usually selected as candidate models for which the greatest similarity is observed; threading (6-8), in which the target sequence is superposed on structures from a database, and those which give the highest score (lowest pseudoenergy) are selected as candidate predictions; and, finally, the fragment-assembly or minithreading method developed by David Baker and colleagues (9, 10), in which the predicted structure is assembled from nine-residue fragments extracted from a protein-structure database, and knowledge-and physicsbased filters are applied at each asse...