Turbulent phenomena in protein folding

Abstract

Protein folding and hydrodynamic turbulence are two long-standing challenges, in molecular biophysics and fluid dynamics, respectively. The theories of these phenomena have been developed independently and used different formalisms. Here we show that the protein folding flows can be surprisingly similar to turbulent fluid flows. Studying a benchmark model protein (an SH3 domain), we have found that the flows for the slow folding trajectories of the protein, in which a partly formed N- and C-terminal β sheet hinders the RT loop from attaching to the protein core, have many properties of turbulent flows of a fluid. The flows are analyzed in a three-dimensional (3D) space of collective variables, which are the numbers of native contacts between the terminal β strands, between the RT loop and the protein core, and the rest of the native contacts. We have found that the flows have fractal nature and are filled with 3D eddies; the latter contain strange attractors, at which the tracer flow paths behave as saddle trajectories. Two regions of the space increment have been observed, in which the flux variations are self-similar with the scaling exponent h=1/3, in surprising agreement with the Kolmogorov inertial range theory of turbulence. In one region, the cascade of protein rearrangements is directed from larger to smaller scales (net folding), and in the other, it is oppositely directed (net unfolding). Folding flows for the fast trajectories are essentially "laminar" and do not have the property of self-similarity. Based on the results of our study, we infer, and support this inference by simulations, that the origin of the similarity between the protein folding and turbulent motion of a fluid is in a cascade mechanism of structural transformations in the systems that underlies these phenomena.