Combinatorial modeling of protein folding kinetics: free energy profiles and rates

A combinatorial approach has been used to determine the optimal assumptions and robustness of simple statistical mechanical models for protein folding. By combining alternative plausible assumptions and different enumeration schemes for constructing partition functions, 76 closely related Ising-like models were generated. These include various contiguous sequence approximations, the possibility of forming a disordered loop between ordered segments, the use of an atomstic or coarse-grained representation of the protein structure, and the choice of ordered residues or native contacts as a reaction coordinate. We also consider all 2(N) conformations of an N-residue protein (e.g., 10(30) conformations for a 100-residue protein). Although the number of configurations is much too large to list, a free energy profile can be calculated using a build-up procedure by accumulation and recombination of partial partition functions. The predictions of the 76 models were tested against two kinds of experimental data - folding rates and the determination of two-state behavior for 25 proteins. Two-state behavior was judged by the relative magnitude of the dominant relaxation calculated from the rate matrix for motion on the free energy profile. The relative performance of each assumption was evaluated using rank sum statistics which show, with the exception of the single sequence approximation, that this class of models is not sensitive to alternative assumptions. Two-state free energy profiles are calculated for almost all but the a-helical proteins, and surprisingly accurate rates are predicted in both the absence and presence of denaturant. The combinatorics also show that an a-carbon representation of the protein structure does nearly as well as atomistic descriptions, possibly reflecting partial interatomic interactions between native residues in structures of the transition state ensemble. With its coarse-grained description of both the energy and entropy, and only 3 or 4 adjustable parameters, the a-carbon version may be regarded as the simplest possible analytical model of protein folding capable of predicting experimental properties of specific proteins. (C) 2004 Elsevier B.V. All rights reserved.