Nonextensive statistical mechanics applied to protein folding problem: Kinetics aspects

Abstract

A reduced (stereo-chemical) model is employed to study kinetic aspects of globular protein folding process, by Monte Carlo simulation. Nonextensive statistical approach is used: transition probability pij between configurations i → j is given by pij = [1+(1-q)ΔGij/kBT]1/(1-q), where q is the nonextensive (Tsallis) parameter. The system model consists of a chain of 27 beads immerse in its solvent; the beads represent the sequence of amino acids along the chain by means of a 10-letter stereo-chemical alphabet; a syntax (rule) to design the amino acid sequence for any given 3D structure is embedded in the model. The study focuses mainly kinetic aspects of the folding problem related with the protein folding time, represented in this work by the concept of first passage time (FPT). Many distinct proteins, whose native structures are represented here by compact self avoiding (CSA) configurations, were employed in our analysis, although our results are presented exclusively for one representative protein, for which a rich statistics was achieved. Our results reveal that there is a specific combinations of value for the nonextensive parameter q and temperature T, which gives the smallest estimated folding characteristic time : Additionally, for q = 1:1, stays almost invariable in the range 0:9 ≤ T ≤ 1:3, slightly oscillating about its average value (t) = 27 ±σ where σ = 2 is the standard deviation. This behavior is explained by comparing the distribution of the folding times for the Boltzmann statistics (q → 1), with respect to the nonextensive statistics for q = 1:1, which shows that the effect of the nonextensive parameter q is to cut off the larger folding times present in the original (q → 1) distribution. The distribution of natural logarithm of the folding times for Boltzmann statistics is a triple peaked Gaussian, while, for q = 1:1 (Tsallis), it is a double peaked Gaussian, suggesting that a log-normal process with two characteristic times replaced the original process with three characteristic times. Finally we comment on the physical meaning of the present results, as well its significance in the near future works.