Finding the optimal solution to a complex optimization problem is of great importance in many fields, ranging from protein structure prediction to the design of microprocessor circuitry. Some recent progress in finding the global minima of potential energy functions is described, focusing on applications of the simple "basin-hopping" approach to atomic and molecular clusters and more complicated hypersurface deformation techniques for crystals and biomolecules. These methods have produced promising results and should enable larger and more complex systems to be treated in the future.

A Monte Carlo-minimization method has been developed to overcome the multiple-minima problem. The Metropolis Monte Carlo sampling, assisted by energy minimization, surmounts intervening barriers in moving through successive discrete local minima in the multidimensional energy surface. The method has located the lowest-energy minimum thus far reported for the brain pentapeptide [Met5]enkephalin in the absence of water. Presumably it is the global minimumenergy structure. This supports the concept that protein folding may be a Markov process. In the presence of water, the molecules appear to exist as an ensemble of different conformations.Optimization procedures are required for an ultimate understanding as to how interatomic interactions lead to the folded, most-stable conformation of a protein from a linear polypeptide chain. A major problem in locating the global minimum of the empirical potential function that describes the conformations of a protein arises from the existence of many local minima in the multidimensional energy surface: the multipleminima problem (1). This problem exists even for a system as small as a terminally blocked amino acid and becomes aggravated as the size of the system increases. Whereas algorithms are available for minimizing a function of many variables, none exist for passing from one local minimum, over an intervening barrier, to the next local minimum-and ultimately to the global minimum-in a many-dimensional space (1, 2). Several procedures have been developed to overcome this problem (1); these include the "buildup" method (3), optimization of electrostatics (4), relaxation of dimensionality (5, 6), adaptive importance sampling Monte Carlo (7-10), pattern recognition based on factor analysis of protein data (11, 12), use of distance constraints (13), and use of short-, medium-, and long-range interactions (14). Most of these procedures have been tested so far on short oligopeptides (up to 20 residues, in some cases), and their possible extension to proteins containing of the order of 100 residues would be of great interest. In our continual search for procedures to overcome this problem, we have developed an approach that appears to work very efficiently on the pentapeptide [Met5]enkephalin (H-Tyr-Gly-Gly-Phe-Met-OH) and hopefully can be extended to larger structures. The application of this procedure to enkephalin is reported here.The multiple-minima problem is not unique to protein folding but arises in many other fields of biology, chemistry, and physics whenever complexity appears (e.g., for intrinsically heterogeneous systems with a large number of strongly coupled degrees of freedom). A protein, composed of chemically distinct amino acids in a unique sequence, is a heterogeneous system that is fundamentally different from a homopolymer, and its many degrees offreedom contribute to the formidable difficulty of the multiple-minima problem. From a computational point of view, the multiple-minima problem is reminiscent of the NP (nondeterministic polynomial time) pro...

The difference In temperature between the initial and final states is not important, because in this theory the relevant energies (e.g., E^,,) are taken to be temperature Independent.

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.

A method is described for the inclusion of the effects of hydration in empirical conformational energy computations on polypeptides. The free energy of hydration is composed of additive contributions of various functional groups. The hydration of each group is assumed to be proportional to the accessible surface area of the group. The constants of proportionality, representing the free energy of hydration per unit area of accessible surface, have been evaluated for seven classes of groups (occurring in peptides) by least-squares fitting to experimental free energies of solution of small monofunctional aliphatic and aromatic molecules. The same method has also been applied to the modeling of the enthalpy and heat capacity ofhydration, each of which is computed from the accessible surface area.The free energy of folding of a protein consists of the sum of contributions from the energy of its intramolecular interactions (1, 2) and from the free energy of interaction of the molecule with the surrounding solvent water. Exact computation of the latter contribution still poses problems (3). As a practical approach, hydration-shell models have been used. In these models, the free energy of interaction of water molecules with the solute is expressed in the form of an averaged effective potential of interaction of atoms (and functional groups) of a solute molecule with a layer of solvent around each atom (4-10)-i.e., in terms ofa potential ofmean force (3). An empirical free energy of hydration is assigned to every atom and group. When the conformation of the protein changes, some water is eliminated from the hydration shell whenever groups on the protein approach each other. The free energy change accompanying this process depends on the total free energy of hydration of the groups and on the amount of water being eliminated from the hydration shells. This amount, in turn, depends on the size and distance of separation of the groups that approach each other, and it can be computed by geometrical methods from the volumes of overlapping spheres (4-6, 10, 11).The hydration-shell model contains several approximations, which may be sources of error and also reduce the speed of computer-based numerical computations (8), such as the thickness of the shell, the apportioning of the free energy between overlapping hydration shells of covalently connected atoms, and the calculation of the volume of overlap of three or more hydration spheres that belong to nearby atoms. The latter problem can be overcome, however, by modifying the computing procedures (10, 11).We have initiated an alternative approach, in order to avoid these problems. We assume that the extent of interaction of any functional group i of a solute with the solvent is proportional to the solvent-accessible surface area Ai of group i (12-14) because the group can interact directly only with the water molecules that are in contact with the group at this surface. Thus, the total free energy of hydration of a solute molecule is given by Eq. 1: AGh = E Ai [1] where...

Some of the parameters that are used in the computer program ECEPP (Empirical Conformational Energy Program for Peptides) to describe the geometry of amino acid residues and the potential energy of interactions have been updated. The changes are based on recently available experimental information. The most signifcant changes improve the geometry and the interactions of prolyl and hydroxyprolyl residues, on the basis of crystallographic structural data. The structure of the pyrrolidine ring has been revised to correspond to the experimentally determined extent of out-of-plane puckering of the five-membered ring. The geometry of the peptide group preceding a Pro residue has also been altered. The parameters for nonbonded interactions involving the C6 and H* atoms of Pro and Hyp have been modified. Use of the revised parameters provides improvements in the computed minimum-energy conformations of peptides containing the Pro-Pro and Ala-Pro sequences. In particular, it is demonstrated that an a-helix-like conformation of a residue preceding Pro is now only of moderately high energy, and thus it is an accessible state. This result corroborates the observed occurrence of Pro residues in kinked a-helices in globular proteins. The structure of the poly(G1y-PrePro) triple helix, a computational model for collagen structure, has been recomputed. The validity of previous computations for this model structure has been confirmed. The refinement of the computed interactions has provided a new general model structure to be used for future computations on collagen-like polypeptides.

Conformational energy calculations using ECEPP (Empirical Conformational Energy Program for Peptides) were carried out on the N-acetyl-N'-methylamides of the 20 naturally occurring amino acids. Minimum-energy conformations were located, and the relative conformational energy, librational entropy, and free energy each minimum were calculated. The effects of intrinsic torsional potentials, intramolecular hydrogen bonds, and librational entropy on relative conformational energies and locations of minima are discussed. The results are categorized most easily by use of a new conformational letter code that is introduced here.

We report the modification and parameterization of the united-residue (UNRES) force field for energy-based protein-structure prediction and protein-folding simulations. We tested the approach on three training proteins separately: 1E0L (β), 1GAB (α), and 1E0G (α + β). Heretofore, the UNRES force field had been designed and parameterized to locate native-like structures of proteins as global minima of their effective potential-energy surfaces, which largely neglected the conformational entropy because decoys composed of only lowest-energy conformations were used to optimize the force field. Recently, we developed a mesoscopic dynamics procedure for UNRES, and applied it with success to simulate protein folding pathways. How ever, the force field turned out to be largely biased towards α-helical structures in canonical simulations because the conformational entropy had been neglected in the parameterization. We applied the hierarchical optimization method developed in our earlier work to optimize the force field, in which the conformational space of a training protein is divided into levels each corresponding to a certain degree of native-likeness. The levels are ordered according to increasing native-likeness; level 0 corresponds to structures with no native-like elements and the highest level corresponds to the fully native-like structures. The aim of optimization is to achieve the order of the free energies of levels, decreasing as their native-likeness increases. The procedure is iterative, and decoys of the training protein(s) generated with the energy-function parameters of the preceding iteration are used to optimize the force field in a current iteration. We applied the multiplexing replica exchange molecular dynamics (MREMD) method, recently implemented in UNRES, to generate decoys; with this modification, conformational entropy is taken into account. Moreover, we optimized the free-energy gaps between levels at temperatures corresponding to a predominance of folded or unfolded structures, as well as to structures at the putative folding-transition temperature, changing the sign of the gaps at the transition temperature. This enabled us to obtain force fields characterized by a single peak in the heat capacity at the transition temperature. Furthermore, we introduced temperature dependence to the UNRES force field; this is consistent with the fact that it is a free-energy and not a potential-energy function.