Statistical Mechanics Perspective Research Articles

How large should proteins be? The minimal size of a good structure seeker

This work deals with the determination of the minimum length of a protein chain required for expeditious and robust folding into a specific target structural motif. Our goal requires that we explore the relationship between the size of a protein and its ability to fold from the statistical mechanics perspective. A coarse partition of the soft-mode conformation space in basins of attraction of the potential energy surface (PES) associated to different contact patterns (CPs) is assumed. This coarse description is consistent with low-temperature experimental probes, revealing the fine resolution of each PES basin, a feature known to be washed out at typical renaturation temperatures. This experimental observation provides the signature of an entrainment of fast-relaxing variables by a coarse CP-resolution of the dynamics. Our statistical analysis is thus focused on the size-dependent energy dispersion within basins and the observation that large energy-level dispersions prevent the entrainment of long-time torsional dynamics by CP transitions due to the breakdown of relaxation timescale separation. The entrainment by coarse dynamics is required by a good structure seeker, since it renders adiabatically the search in conformation space, a regime warranted by a relatively fast equilibration within each CP-basin. This drastic reduction in the number of dominant variables, probed for real proteins at low unphysiological temperatures, is reasonably assumed to be responsible for the robustness and expediency of the folding process, which is thus no longer dependent on the fine details of the torsional dynamics. Finding the minimal size of a protein chain fulfilling such requirements is precisely the goal of this work. If the chain is not long enough, the large relative dispersion of energy levels within a basin introduces correlations between substates of different basins and leads to a breakdown of the dynamic entrainment, rendering the folding process ineffective, and considerably decreasing its expediency. This is so since energy fluctuations within a CP-basin relax on timescales which are no longer incommensurably shorter than those of CP-transitions and intra-CP-basin relaxation interferes with the folding process as viewed from the coarser perspective.

MDL and the statistical mechanics of protein potentials.

The combination of a wealth of structural data and impressive computational power provides detailed information pertaining to the structure and dynamics of biomacromolecules. A natural inclination is to incorporate this information into models to gain added predictive power on protein folding and stability. There has been considerable recent interest in developing "knowledge-based" potentials to describe internal interactions in proteins. In these approaches, probability distribution functions are inferred from existing knowledge. A common assumption has been the "quasi-chemical approximation" or "Boltzmann device". This method relates statistical mechanical probabilities to observed frequencies. The validity of this approach is discussed in detail from a statistical mechanics perspective. Because statistical mechanics is a form of statistical inference based on a lack of knowledge of the system, the "Boltzmann device" does not have a rigorous theoretical justification. In the present work, a statistical mechanics based on partial knowledge of the system is employed. This statistical mechanical scheme uses the minimum description length (MDL) of phase space as its main tool. With this approach, "knowledge-based" potentials can be derived in a rigorous fashion. In practical calculations, these potentials are best obtained using Bayesian inference methods similar to those used in image reconstruction.

An analysis of the statistical model adsorption isotherm

AbstractThe statistical model adsorption isotherm developed by Ruthven and coworkers and used in the prediction and correlation of adsorption in microporous materials especially zeolites is critically examined from the perspective of statistical mechanics. This is done by applying the method to a class of molecular models for which the statistical thermodynamics may be solved analytically without approximation. The models considered are finite length single component and binary one‐dimensional systems in which the molecules interact via square well potentials. These are among the simplest realistic yet analytically solvable molecular models of adsorption in porous materials. Our analysis clarifies the theoretical status of the Ruthven approach as well as revealing some of the weaknesses in it.

Functional Random-Walk Model of the Many-Particle System

By Fourier-transforming the author's recently proposed state functional formalism for the BBGKY hierarchy, a new perspective of nonequilibrium statistical mechanics is given: the basic equation is formally very close to the Fokker-Planck equation and may readily be modified to a universal master equation (with irreversibility) by a slight change. Hence, the problem reduces to one of a generalized random-walk such that the stochastic quantity to be considered is the particle-number density in the 1-body phase space. A general solution is formulated for the weak interaction case.