Proteins fold by subdiffusion of the order parameter

It is shown that the folding of a C(alpha) model of chymotyprsin inhibitor (CI2) protein cannot be described by either diffusion (Smoluchowski equation, SE) or a normal-diffusion continuous time random walk of a single order parameter under the influence of the thermodynamic force. The reason for these failures is that the order parameter follows subdiffusion. A theory is proposed based on the idea that an ordinary SE holds along a contour representative of the folding pathways, and that displacements along the contour obey a fractal relationship to, and are longer than, those along the reaction coordinate defined by the order parameter. With a new, constraint-free method to determine the order-parameter-dependent diffusion constant, and statistical temperature molecular dynamics (STMD) enhanced sampling of the free energy, the fractal SE theory is completely characterized by short-time simulations, and its predictions are in quantitative agreement with simulated long-time folding dynamics. Thus, the fractal SE may serve as an accelerated algorithm to study the folding of proteins too slow to be simulated directly.