Dominant Pathways in Protein Folding

Abstract

We present a method to investigate the kinetics of protein folding on a long time-scale and the dynamics underlying the formation of secondary and tertiary structures during the entire reaction. The approach is based on the formal analogy between thermal and quantum diffusion: by writing the solution of the Fokker-Planck equation for the time-evolution of a protein in a viscous heat-bath in terms of a path integral, we derive a Hamilton-Jacobi variational principle from which we are able to compute the most probable pathway of folding. The method is applied to the folding of the Villin Headpiece Subdomain, in the framework of a Go-model. We have found that, in this model, the transition occurs through an initial collapsing phase driven by the starting coil configuration and a later rearrangement phase, in which secondary structures are formed and all computed paths display strong similarities. This method is completely general, does not require the prior knowledge of any reaction coordinate and represents an efficient tool to perfom ab-initio simulations of the entire folding process with available computers.Understanding the kinetics of protein folding and the dynamical mechanisms involved in the formation of their structures in an all-atom approach involves simulating a statistically significant ensemble of folding trajectories for a system of ∼ 10 4 degrees of freedom. Unfortunately, the existence of a huge gap between the microscopic time-scale of the rotational degrees of freedom ∼ 10 −12 s and the macroscopic time scales of the full folding process ∼ 10 −6 − 10 1 s makes it extremely computationally challenging to follow the evolution of a typical ∼ 100-residue protein for a time interval longer than few tens of nanoseconds.Several approaches have been proposed to overcome such computational difficulties and address the problem of identifying the relevant pathways of the folding reaction [2]. Unfortunately these methods are either affected by uncontrolled systematic errors associated to ad-hoc approximations, or can only be applied to small proteins with typical folding time of the order of few nanoseconds (fast folders). In this Letter we present a novel approach to overcome these difficulties: we adopt the Langevin approach and devise a method to rigorously define and practically compute the most statistically relevant protein folding pathway. As a first exploratory application, we have studied the folding transition of the 36-monomer Villin Headpiece Subdomain (PDB code 1VII). This molecule has been extensively studied in the literature because it is the smallest polypeptide that has all of the properties of a single domain protein and in addition, it is one of the fastest folders [3]. The ribbon representation of this system is shown in Fig.1. We analyze the transition from different random selfavoiding coil states to the native state, whose structure was obtained from the Brookhaven Protein Data Bank.Our study is based on the analogy between Langevin diffusion and quantum propagation. Previous studies...

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