Protein Folding and Heteropolymers

Abstract

We present a statistical mechanics approach to the protein folding problem. We first review some of the basic properties of proteins, and introduce some physical models to describe their thermodynamics. These models rely on a random heteropolymeric description of these non random biomolecules. Various kinds of randomness are investigated, and the connection with disordered systems is discussed. We conclude by a brief study of the dynamics of proteins.Natural proteins have the property of folding into an (almost) unique compact native structure, which is of biological interest 1,2 . The compactness of this unique native state is largely due to the existence of an optimal amount of hydrophobic amino-acid residues 3 , since these biological objects are usually designed to work in water. The task of predicting the conformation of the three-dimensional structure from the linear primary sequence is often referred to as the protein folding problem. Already at this stage, a rigorous analytical theory appears difficult, since it amounts to study a mesoscopic system (most proteins have between 100 and 500 residues), notwithstanding the solvent's properties.This mesoscopic system is of a classical nature; the quantum mechanical valence electrons of the atoms induce interactions between the heavy "nuclei" which can then be treated as classical objects interacting through classical many-body interactions. This observation forms the basis of Molecular Dynamics, Monte Carlo, or Statistical Mechanical models.To further complicate the matter, both the compactness and the chemical heterogeneity of a given protein tend to slow down dynamical processes: the question of the kinetic control of the folding (as opposed to a thermodynamic control) is therefore periodically asked. This remark suggests that the folding process may have something in common with the physics of glassy systems, where competing interactions (frustration) and/or disorder lead to a very rugged phase space, resulting in slow dynamical processes. In this review, we will assume that the folded state of proteins is thermodynamically stable (the folded state is the state of minimal free energy).There has been a considerable amount of numerical simulations of proteins, using molecular dynamics or Monte Carlo calculations (for a review see 2,3,4 ).This promising approach will not be discussed in this review. We will use only analytical approaches throughout. Similarly, models emphasizing the micro-crystalline character of the folded proteins will not be addressed here 5 .The outline of this review is the following. In the first section, we introduce at an elementary level, some notions of the physics, chemistry and biology of proteins. In the second section, we study (using statistical physics methods) some heteropolymer models which are possibly relevant to the protein folding problem. The phase diagrams of these models bear some qualitative resemblance to the real systems.In the last section, we tackle the issues of dynamics. In view of the complexity of 1

Early stages of homopolymer collapse

“…Note that τ p is independent of N and p as expected for a local process. A short relaxation time scaling as N o is also predicted by Langevin type theories of collapse [8,10].…”

“…It is interesting however to note that both the theory of Garel at al. [10] and of Kuzentsov et al [8] distinguish between two regimes in the early collapse: first, a very fast process, whose characteristic time is independent of N and is thus interpreted as local rearrangement; and second, a much slower process, reflecting large scale reorganization, with a characteristic time that increases with N.…”

“…Theoretical studies have utilized a variety of approaches in focusing on different aspects of the problem. Langevin models [8][9][10], phenomenological models [11][12][13] and computer simulations [14][15][16][17][18][19][20] have been used to consider the kinetics of collapse in the absence of topological constraints. Other studies have focused on the role of internal entanglements [21,22] or the competition with chain-chain aggregation [23,24].…”

A Polymer Model with Epigenetic Recolouring Reveals a Pathway for thede novoEstablishment and 3D organisation of Chromatin Domains

“…5). We argue that the physics underlying the emergence of a frozen network of intra-chain interactions might be reminiscent of the physics of spin glasses with quenched disorder [56,70,83] SI Fig. S10).…”

Coupled dynamics of sequence selection and compactification in mean-field hetero-polymers

“…The main complications in solving protein models are the chain constraint and the presence of disorder, embodied in the amino-acid sequences (see, e.g., [7,8]). Natural aminoacid sequences have evolved genetically, driven mainly by the two (competing) demands of structure reproducibility and functionality.…”