In this paper, we introduce an approach to the protein folding problem from the point of view of statistical physics. Protein folding is a stochastic process by which a polypeptide folds into its characteristic and functional 3D structure from random coil. The process involves an intricate interplay between global geometry and local structure, and each protein seems to present special problems. We introduce CSAW (conditioned self-avoiding walk), a model of protein folding that combines the features of self-avoiding walk (SAW) and the Monte Carlo method. In this model, the unfolded protein chain is treated as a random coil described by SAW. Folding is induced by hydrophobic forces and other interactions, such as hydrogen bonding, which can be taken into account by imposing conditions on SAW. Conceptually, the mathematical basis is a generalized Langevin equation. To illustrate the flexibility and capabilities of the model, we consider several examples, including helix formation, elastic properties, and the transition in the folding of myoglobin. From the CSAW simulation and physical arguments, we find a universal elastic energy for proteins, which depends only on the radius of gyration $R_{g}$ and the residue number $N$. The elastic energy gives rise to scaling laws $R_{g}\sim N^ν$ in different regions with exponents $ν=3/5,3/7,2/5$, consistent with the observed unfolded stage, pre-globule, and molten globule, respectively. These results indicate that CSAW can serve as a theoretical laboratory to study universal principles in protein folding.
The process of protein folding from an unfolded state to a biologically active, folded conformation is governed by many parameters e.g the sequence of amino acids, intermolecular interactions, the solvent, temperature and chaperon molecules. Our study, based on random matrix modeling of the interactions, shows however that the evolution of the statistical measures e.g Gibbs free energy, heat capacity, entropy is single parametric. The information can explain the selection of specific folding pathways from an infinite number of possible ways as well as other folding characteristics observed in …
When described by a low-dimensional reaction coordinate, the rates of protein folding are determined by a subtle interplay between free-energy barriers and friction. While it is commonplace to extract free-energy profiles from molecular trajectories, a direct evaluation of friction is far more elusive, and one typically evaluates it indirectly via memoryless reaction rate theories. Here, using memory-kernel extraction methods founded on a generalised Langevin equation (GLE) formalism, we directly calculate the memory-dependent friction for eight fast-folding proteins, taken from a published se…
The statistical properties of protein folding within the ϕ^4 model are investigated. The calculation is performed using statistical mechanics and path integral method. In particular, the evolution of heat capacity in term of temperature is given for various levels of the nonlinearity of source and the strength of interaction between protein backbone and nonlinear source. It is found that the nonlinear source contributes constructively to the specific heat especially at higher temperature when it is weakly interacting with the protein backbone. This indicates increasing energy absorption as the…
We present a novel statistical mechanics formalism for the theoretical description of the process of protein folding$\leftrightarrow$unfolding transition in water environment. The formalism is based on the construction of the partition function of a protein obeying two-stage-like folding kinetics. Using the statistical mechanics model of solvation of hydrophobic hydrocarbons we obtain the partition function of infinitely diluted solution of proteins in water environment. The calculated dependencies of the protein heat capacities upon temperature are compared with the corresponding results of e…
Energy landscape theory describes how a full-length protein can attain its native fold by sampling only a tiny fraction of all possible structures. Although protein folding is now understood to be concomitant with synthesis on the ribosome, there have been few attempts to modify energy landscape theory by accounting for cotranslational folding. Here we provide a model for cotranslational folding that leads to a natural definition of a nested energy landscape. By applying concepts drawn from submanifold differential geometry, the physics of protein folding on the ribosome can be explored in a q…
Repeat proteins are made with tandem copies of similar amino acid stretches that fold into elongated architectures. Due to their symmetry, these proteins constitute excellent model systems to investigate how evolution relates to structure, folding and function. Here, we propose a scheme to map evolutionary information at the sequence level to a coarse-grained model for repeat-protein folding and use it to investigate the folding of thousands of repeat-proteins. We model the energetics by a combination of an inverse Potts model scheme with an explicit mechanistic model of duplications and delet…
In the traditional random-conformational-search model, various hypotheses with a series of meta-stable intermediate states were often proposed to resolve the Levinthal paradox. Here we introduce a quantum strategy to formulate protein folding as a quantum walk on a definite graph, which provides us a general framework without making hypotheses. Evaluating it by the mean of first passage time, we find that the folding time via our quantum approach is much shorter than the one obtained via classical random walks. This idea is expected to evoke more insights for future studies.
A simple statistical mechanical model proposed by Wako and Sait$\hat{\rm o}$ has explained the aspects of protein folding surprisingly well. This model was systematically applied to multiple proteins by Mu$\tilde{\rm n}$oz and Eaton and has since been referred to as the Wako-Sait$\hat{\rm o}$-Mu$\tilde{\rm n}$oz-Eaton (WSME) model. The success of the WSME model in explaining the folding of many proteins has verified the hypothesis that the folding is dominated by native interactions, which makes the energy landscape globally biased toward native conformation. Using the WSME and other related m…
Contact Order Dependent Protein Folding Rates: Kinetic Consequences of a Cooperative Interplay Between Favorable Nonlocal Interactions and Local Conformational Preferences
Topological and energetic factors: what determines the structural details of the transition state ensemble and "on-route" intermediates for protein folding? An investigation for small globular proteins
