We study folding of Trp-cage miniprotein in the conditions when the native state of the protein is stable and unfolding events are improbable, which corresponds to physiological conditions. Using molecular dynamics simulations with an implicit solvent model, an ensemble of folding trajectories from unfolded (practically extended) states of the protein to the native state was generated. To get insight into the folding kinetics, the free energy surface and kinetic network projected on this surface were constructed. This, "conventional" analysis of the folding reaction was followed by a recently proposed hydrodynamic description of protein folding (Chekmarev et al. in Phys Rev Lett 100(1):018107, 2008), in which the process of the first-passage folding is viewed as a stationary flow of a folding "fluid" from the unfolded to native state. This approach is conceptually different from the previously used approaches and thus allows an alternative view of the folding dynamics and kinetics of Trp-cage, the conclusions about which are very diverse. In agreement with most previous studies, we observed two characteristic folding pathways: in one pathway (I), the collapse of the hydrophobic core precedes the formation of the [Formula: see text]-helix, and in the other pathway (II), these events occur in the reverse order. We found that although pathway II is complicated by a repeated partial protein unfolding, it contributes to the total folding flow as little as ≈10%, so that the folding kinetics remain essentially single-exponential.
Recently, when studying folding of a SH3 domain, we discovered that the flows of transitions between protein states can be surprisingly similar to turbulent fluid flows. This similarity was not restricted by a vortex pattern of the flow fields but extended to a spatial correlation of flow fluctuations, resulting, in particular, in the structure functions such as in the Kolmogorov theory of homogeneous and isotropic turbulence. Here, we undertake a detailed analysis of spatial distribution of folding flows and their similarity to turbulent fluid flows. Using molecular dynamics simulations, we study folding of another benchmark system—Trp-cage miniprotein, which has different content of secondary structure elements and mechanism of folding. Calculating the probability fluxes of transitions in a three-dimensional space of collective variables, we have found that similar to the SH3 domain, the structure functions of the second and third orders correspond to the Kolmogorov functions. The spatial distributions of the probability fluxes are self-similar with a fractal dimension, and the fractal index decreases toward the native state, indicating that the flow becomes more turbulent as the native state is approached. We also show that the process of folding can be viewed as Brownian diffusion in the space of probability fluxes. The diffusion coefficient plays a role of the key parameter that defines the structures functions, similar to the rate of dissipation of kinetic energy in hydrodynamic turbulence. The obtained results, first, show that the very complex dynamics of protein folding allows a simple characterization in terms of scaling and diffusion of probability fluxes, and, secondly, they suggest that the turbulence phenomena similar to hydrodynamic turbulence are not specific of folding of a particular protein but are common to protein folding.
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