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.

The implementation of molecular dynamics (MD) with our physics-based protein united-residue (UNRES) force field, described in the accompanying paper, was extended to Langevin dynamics. The equations of motion are integrated by using a simplified stochastic velocity Verlet algorithm. To compare the results to those with all-atom simulations with implicit solvent in which no explicit stochastic and friction forces are present, we alternatively introduced the Berendsen thermostat. Test simulations on the Ala(10) polypeptide demonstrated that the average kinetic energy is stable with about a 5 fs time step. To determine the correspondence between the UNRES time step and the time step of all-atom molecular dynamics, all-atom simulations with the AMBER 99 force field and explicit solvent and also with implicit solvent taken into account within the framework of the generalized Born/surface area (GBSA) model were carried out on the unblocked Ala(10) polypeptide. We found that the UNRES time scale is 4 times longer than that of all-atom MD simulations because the degrees of freedom corresponding to the fastest motions in UNRES are averaged out. When the reduction of the computational cost for evaluation of the UNRES energy function is also taken into account, UNRES (with hydration included implicitly in the side chain-side chain interaction potential) offers about at least a 4000-fold speed up of computations relative to all-atom simulations with explicit solvent and at least a 65-fold speed up relative to all-atom simulations with implicit solvent. To carry out an initial full-blown test of the UNRES/MD approach, we ran Berendsen-bath and Langevin dynamics simulations of the 46-residue B-domain of staphylococcal protein A. We were able to determine the folding temperature at which all trajectories converged to nativelike structures with both approaches. For comparison, we carried out ab initio folding simulations of this protein at the AMBER 99/GBSA level. The average CPU time for folding protein A by UNRES molecular dynamics was 30 min with a single Alpha processor, compared to about 152 h for all-atom simulations with implicit solvent. It can be concluded that the UNRES/MD approach will enable us to carry out microsecond and, possibly, millisecond simulations of protein folding and, consequently, of the folding process of proteins in real time.

We report the application of Langevin dynamics to the physicsbased united-residue (UNRES) force field developed in our laboratory. Ten trajectories were run on seven proteins [PDB ID codes 1BDD (␣; 46 residues), 1GAB (␣; 47 residues), 1LQ7 (␣; 67 residues), 1CLB (␣; 75 residues), 1E0L (␤; 28 residues), and 1E0G (␣؉␤; 48 residues), and 1IGD (␣؉␤; 61 residues)] with the UNRES force field parameterized by using our recently developed method for obtaining a hierarchical structure of the energy landscape. All ␣-helical proteins and 1E0G folded to the native-like structures, whereas 1IGD and 1E0L yielded mostly nonnative ␣-helical folds although the native-like structures are lowest in energy for these two proteins, which can be attributed to neglecting the entropy factor in the current parameterization of UNRES. Average folding times for successful folding simulations were of the order of nanoseconds, whereas even the ultrafast-folding proteins fold only in microseconds, which implies that the UNRES time scale is approximately three orders of magnitude larger than the experimental time scale because the fast motions of the secondary degrees of freedom are averaged out. Folding with Langevin dynamics required 2-10 h of CPU time on average with a single AMD Athlon MP 2800؉ processor depending on the size of the protein. With the advantage of parallel processing, this process leads to the possibility to explore thousands of folding pathways and to predict not only the native structure but also the folding scenario of a protein together with its quantitative kinetic and thermodynamic characteristics.Langevin dynamics ͉ mesoscopic models ͉ restricted free energy T here are two protein-folding problems in contemporary computational biology. The first problem is to predict protein structure from sequence, and the second one is to predict protein-folding pathways. There are many approximate methods to attack the folding problem, which belong to two broad categories of physics and knowledge-based methods (1-3). Molecular dynamics (MD) is the only computational method that provides a time-dependent analysis of a system in molecular biology and, consequently, can be implemented to solve the second protein-folding problem.Ideally, both the protein and the surrounding solvent should be represented at the all-atom level (4) because this approach is the closest to experiment. However, there are two severe limitations to such a treatment, namely the multidimensionality of the system (typically, Ͼ10 4 degrees of freedom with explicit solvent) and the small values of the time step in integrating the equations of motion (of the order of femtoseconds). Because of these two limitations, explicit-solvent all-atom MD algorithms can simulate events in the range of 10 Ϫ9 to 10 Ϫ8 s for typical proteins and 10 Ϫ6 s for very small proteins (4-6). These time scales are at least one order of magnitude smaller than the folding times of proteins (4). Consequently, all-atom simulations of real-size proteins are usually limited to unfolding the nati...