Modeling the correlation functions of conformational motions in proteins

Non-Markovian noise mediated through anomalous diffusion within ion channels

It is evident from a wide range of experimental findings that ion channel gating is inherently stochastic. The issue of "memory effects" (diffusional retardation due to local changes in water viscosity) in ionic flow has been recently addressed using Brownian dynamics simulations. The results presented indicate such memory effects are negligible, unless the diffusional barrier is much higher than that of free solute. In this paper using differential stochastic methods we conclude that the Markovian property of exponential dwell times gives rise to a high barrier, resulting in diffusional memory effects that cannot be ignored in determining ionic flow through channels. We have addressed this question using a generalized Langevin equation that contains a combination of Markovian and non-Markovian processes with different time scales. This approach afforded the development of an algorithm that describes an oscillatory ionic diffusional sequence. The resulting oscillatory function behavior, with exponential decay, was obtained at the weak non-Markovian limit with two distinct time scales corresponding to the processes of ionic diffusion and drift. This will be analyzed further in future studies using molecular dynamics simulations. We propose that the rise of time scales and memory effects is related to differences of shear viscosity in the cytoplasm and extracellular matrix.

Modeling the "glass" transition in proteins

A model of a protein as a disordered system of identical spherical particles (which imitate protein side chains) interacting with each other via a repulsive soft sphere potential U(r) infinity r(-beta) is constructed. The particles undergo the conformational motion (CM) within their own harmonic conformational potentials around some mean equilibrium positions ascribed by the tertiary structure of the protein. A first principles calculation of the positional correlation functions for the CM is carried out. The general analysis is exemplified by the case in which the mean equilibrium positions of the particles form a cubic tightly-packed (face- centered) lattice (each particle has 12 nearest neighbors) with the step b(hydr) =6.6 A (the average distance between the centers of mass of hydrated protein subunits). The model yields dramatic slowing down of the relaxation with the decrease of temperature followed by a sharp glass transition at some crossover temperature T(c) < 200 K. At the transition the liquid-like dynamic behavior (the correlation functions tend to zero with time) is altered by the glass-like one (the correlation functions tend with time to some non-zero limit). In the liquid-like region above the crossover temperature the relaxation exhibits distinct alpha-process following the beta-one. The glass transition results from the interaction of the particles. Thus the model suggests that namely direct interactions of the fragments of protein structure rather than protein-solvent interactions are the origin of the phenomenon of the glass transition. The known increase of T(c) up to 300 K at dehydration of the protein is attributed to the known concomitant compression of the globule upon drying by about 4-6% so that positions of individual atoms displace by about 0.6 A (modeled by the decrease of the step of the lattice b by 0.6 A so that b(dehydr)=6 A). The model suggests that the solvent influences the phenomenon of the glass transition indirectly determining the tertiary structure of the protein rather than via own freezing. In the model the transition from the liquid-like dynamic behavior to the glass-like one can be obtained even in a cluster containing a few particles. Thus the results of the model can be considered as an argument in favor of the point of view that the transition to the glassy behavior can take place for a very small domains of the protein comprising only several constituting fragments of its structure. The model predicts that for the dehydrated protein the alpha-relaxation process is strongly repressed.