Effect of local thermal fluctuations on folding kinetics: A study from the perspective of nonextensive statistical mechanics

Abstract

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.

Entropic formulation for the protein folding process: Hydrophobic stability correlates with folding rates

“…Besides in-depth support for some results already pointed out in our previous works [10,11,13,[15][16][17], one of our new results shows the existence of a strong correlation between the hydrophobic component of the protein stability and its folding rate.…”

“…The increase in T depends on each specific target; it has to do with topological features of this specific target. In other words, when one uses the Boltzmann weight, a quite similar sampling can be achieved by increasing the original reservoir temperature T, however the specific increment depends on the particular chosen target [15]. Surely this is not what is desired, because for all proteins of the same organism, the folding process must work fine at the same temperature, or at least at the same narrow temperature interval [15].…”

“…In other words, when one uses the Boltzmann weight, a quite similar sampling can be achieved by increasing the original reservoir temperature T, however the specific increment depends on the particular chosen target [15]. Surely this is not what is desired, because for all proteins of the same organism, the folding process must work fine at the same temperature, or at least at the same narrow temperature interval [15]. The net effect of variable q over the chain's configurational evolution is overcome energetic/steric traps easiest than with the standard Boltzmann weight.…”

“…here is the instantaneous radius of gyration of the chain, and max = 4/3, for more details see reference [15]. Given the instantaneous spatial position of the molecule, the entropic index q reflects the effect of local thermal fluctuations acting on the chain.…”

“…In order to take the thermal noise into account, instead of apply the usual Boltzmann weight, we adopt Tsallis weight to sample chain configurations in our MC simulations [14,15].…”