Protein folding is a universal process, very fast and accurate, which works consistently (as it should be) in a wide range of physiological conditions. The present work is based on three premises, namely: (i) folding reaction is a process with two consecutive and independent stages, namely the search mechanism and the overall productive stabilization; (ii) the folding kinetics results from a mechanism as fast as can be; and (iii) at nanoscale dimensions, local thermal fluctuations may have important role on the folding kinetics.Here the first stage of folding process (search mechanism) is focused exclusively. The effects and consequences of local thermal fluctuations on the configurational kinetics, treated here in the context of non extensive statistical mechanics, is analyzed in detail through the dependence of the characteristic time of folding (τ ) on the temperature T and on the nonextensive parameter q.The model used consists of effective residues forming a chain of 27 beads, which occupy different sites of a 3−D infinite lattice, representing a single protein chain in solution. The configurational evolution, treated by Monte Carlo simulation, is driven mainly by the change in free energy of transfer between consecutive configurations.We found that the kinetics of the search mechanism, at temperature T , can be equally reproduced either if configurations are relatively weighted by means of the generalized Boltzmann factor (q > 1), or by the conventional Boltzmann factor (q = 1), but in latter case with temperatures T ′ > T.However, it is also argued that the two approaches are not equivalent. Indeed, as the temperature is a critical factor for biological systems, the folding process must be optmized at a relatively small range of temperature for the set of all proteins of a given organism. That is, the problem is not longer a simple matter of renormalization of parameters. Therefore, local thermal fluctuation on systems with nanometric components, as proteins in solution, becomes a important factor affecting the configurational kinetics.As a final remark, it is argued that for a heterogeneous system with nanoscopic components, q should be treated as a variable instead of a fixed parameter.
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 p i j between configurations i → j is given by p i j = [1 + (1 − q)∆G i j /k B T ] 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 t . Additionally, for q = 1.1, t 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.
We investigate a finite chain approximation, the non-Gaussian Tsallis distribution, to the polymeric network, which gives an improvement to the Gaussian model. This distribution presents some necessary characteristics, like a cutoff to the maximum chain length and a continuous limit to the Gaussian one for a large number of monomers. It also presents a simple quadratic structure that allows to generalize the Gaussian properties such as exact-moments calculation and Wick theorem. We obtain the free-energy density in its full tensorial structure.
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