Quantitative Protein Dynamics from Dominant Folding Pathways

Abstract

We develop a theoretical approach to the protein folding problem based on out-of-equilibrium stochastic dynamics. Within this framework, the computational difficulties related to the existence of large time scale gaps in the protein folding problem are removed and simulating the entire reaction in atomistic details using existing computers becomes feasible. In addition, this formalism provides a natural framework to investigate the relationships between thermodynamical and kinetic aspects of the folding. For example, it is possible to show that, in order to have a large probability to remain unchanged under Langevin diffusion, the native state has to be characterized by a small conformational entropy. We discuss how to determine the most probable folding pathway, to identify configurations representative of the transition state and to compute the most probable transition time. We perform an illustrative application of these ideas, studying the conformational evolution of alanine di-peptide, within an all-atom model based on the empiric GROMOS96 force field.A critical part of the protein-folding problem is to understand its kinetics and the underlying physical processes. To this aim, several different theoretical methods have been recently developed, spanning from analytical approaches[1, 2, 3] to detailed computer simulations [4,5,6]. A major problem in simulating the folding process using standard molecular dynamics (MD) is the huge gap between the time scale of "elementary moves", of the order of 10-100 ps, and that of the entire folding process, which ranges from a few microseconds for fast-folders [7], up to several seconds or even minutes for more complex proteins. This peculiarity of the folding process makes the brute-force molecular dynamics approach too demanding, and a substantial part of the efforts in the field of protein folding simulation aims at bridging this gap.In a recent paper [8] we have presented a novel theoretical framework for investigating the folding dynamics, named hereafter Dominant Folding Pathways (DFP), which is based on a reformulation in terms of path integrals of the dynamics described by the Langevin equation. The DFP analysis allows to compute rigorously (i.e. without any assumptions other than the validity of the underlying Langevin equation) the most probable conformational pathway connecting an arbitrary initial conformation to an arbitrary final conformation. The major advantage of the method is the possibility of bypassing the computational difficulties associated with the existence of different time scales in the problem, while retaining the ability to recover information on the time evolution of the system. As we shall see, the resulting computational simplification is dramatic and makes it feasible to study the formation pattern of conformational structures along the entire folding process using realistic all-atom force fields, on available computers.In this Letter we further develop our formalism and we present the first DFP simulation performed in full atomistic deta...

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