Why do protein architectures have boltzmann‐like statistics?

Reports on progress in physics. Physical Society (Great Britain), 2018

In the course of evolution, proteins undergo important changes in their amino acid sequences, while their three-dimensional folded structure and their biological function remain remarkably conserved. Thanks to modern sequencing techniques, sequence data accumulate at unprecedented pace. This provides large sets of so-called homologous, i.e. evolutionarily related protein sequences, to which methods of inverse statistical physics can be applied. Using sequence data as the basis for the inference of Boltzmann distributions from samples of microscopic configurations or observables, it is possible to extract information about evolutionary constraints and thus protein function and structure. Here we give an overview over some biologically important questions, and how statistical-mechanics inspired modeling approaches can help to answer them. Finally, we discuss some open questions, which we expect to be addressed over the next years.

Annual Review of Physical Chemistry, 1997

The energy landscape theory of protein folding is a statistical description of a protein's potential surface. It assumes that folding occurs through organizing an ensemble of structures rather than through only a few uniquely defined structural intermediates. It suggests that the most realistic model of a protein is a minimally frustrated heteropolymer with a rugged funnel-like landscape biased toward the native structure. This statistical description has been developed using tools from the statistical mechanics of disordered systems, polymers, and phase transitions of finite systems. We review here its analytical background and contrast the phenomena in homopolymers, random heteropolymers, and protein-like heteropolymers that are kinetically and thermodynamically capable of folding. The connection between these statistical concepts and the results of minimalist models used in computer simulations is discussed. The review concludes with a brief discussion of how the theory helps in the interpretation of results from fast folding experiments and in the practical task of protein structure prediction.

Brazilian Journal of Physics, 2009

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 -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.

Physical Review E, 2011

Random heteropolymers do not display the typical equilibrium properties of globular proteins, but are the starting point to understand the physics of proteins and, in particular, to describe their non-native states. So far, they have been studied only with mean-field models in the thermodynamic limit, or with computer simulations of very small chains on lattice. After describing a self-adjusting parallel-tempering technique to sample efficiently the low-energy states of frustrated systems without the need of tuning the system-dependent parameters of the algorithm, we apply it to random heteropolymers moving in continuous space. We show that if the mean interaction between monomers is negative, the usual description through the random energy model is nearly correct, provided that it is extended to account for non-compact conformations. If the mean interaction is positive, such a simple description breaks out and the system behaves in a way more similar to Ising spin glasses. The former case is a model for the denatured state of globular proteins, the latter of naturally-unfolded proteins, whose equilibrium properties thus result qualitatively different.

Journal of Chemical Physics, 2019

In this paper we show that ensembles of well-structured and unstructured proteins can be distinguished by borrowing concepts from non-equilibrium statistical mechanics. For this purpose, we represent proteins by two different polymer models and interpret the resulting polymer configurations as random walks of a diffusing particle in space. The first model is the trace of the Cα-atoms along the protein main chain and the second their projections onto the protein axis. The resulting trajectories are subsequently analyzed using the theory of the Generalized Langevin Equation. Velocities are replaced by displacements relating consecutive points on the discrete protein axes and equilibrium ensemble averages by averages over appropriate protein structure ensembles. The resulting displacement autocorrelation functions resemble those of velocity autocorrelation functions of simple liquids and display a minimum, which can be related to the lengths of secondary structure elements. This minimum is clearly more pronounced for well-structured proteins than for unstructured ones and the corresponding memory function displays a slower decay, indicating a stronger "folding memory".

The native state structures of globular proteins are stable and well-packed indicating that self-interactions are favored over protein-solvent interactions under folding conditions. We use this as a guiding principle to derive the geometry of the building blocks of protein structures, alpha-helices and strands assembled into beta-sheets, with no adjustable parameters, no amino acid sequence information, and no chemistry. There is an almost perfect fit between the dictates of mathematics and physics and the rules of quantum chemistry. Our theory establishes an energy landscape that channels protein evolution by providing sequence-independent platforms for elaborating sequence-dependent functional diversity. Our work highlights the vital role of discreteness in life and has implications for the creation of artificial life and on the nature of life elsewhere in the cosmos.

Protein Folding Protocols

The high packing density inside proteins leads to certain geometric regularities and also is one of the most important contributors to the high extent of cooperativity manifested by proteins in their cohesive domain motions. The orientations between neighboring non-bonded residues in proteins substantially follow the similar geometric regularities, regardless of whether the residues are on the surface or buried-a direct result of hydrophobicity forces. These orientations are relatively fixed and correspond closely to small deformations from those of the face-centered cubic lattice, which is the way in which identical spheres pack at the highest density. Packing density also is related to the extent of conservation of residues, and we show this relationship for residue packing densities by averaging over a large sample or residue packings. There are three regimes: 1) over a broad range of packing densities the relationship between sequence entropy and inverse packing density is nearly linear, 2) over a limited range of low packing densities the sequence entropy is nearly constant, and 3) at extremely low packing densities the sequence entropy is highly variable. These packing results provide important justification for the simple elastic network models that have been shown for a large number of proteins to represent protein dynamics so successfully, even when the models are extremely coarse-grained. Elastic network models for polymeric chains are simple and could be combined with these protein elastic networks to represent partially denatured parts of proteins. Finally, we show results of applications of the elastic network model to study the functional motions of the ribosome, based on its known structure. These results indicate expected correlations among its components for the step-wise processing steps in protein synthesis, and suggest ways to use these elastic network models to develop more detailed mechanisms-an important possibility, since most experiments yield only static structures.

Biochemistry, 1985

Using lattice statistical mechanics, we develop theory to account for the folding of a heteropolymer molecule such as a protein to the globular and soluble state. Folding is assumed to be driven by the association of solvophobic monomers to avoid solvent and opposed by the chain configurational entropy. Theory predicts a phase transition as a function of temperature or solvent character. Molecules that are too short or too long or that have too few solvophobic residues are predicted not to fold. Globular molecules should have

2004

In the past decade, a large amount of biological data has been generated, enabling new quantitative approaches in biology. In this thesis, we focus on two biological questions by using techniques from statistical physics: hydrophobicity patterns in proteins and their impact on the designability of protein structures and regulatory motif finding in DNA sequences. Proteins fold into specific structures to perform their functions. Hydrophobicity is the main force of folding; protein sequences try to lower the ground state energy of the folded structure by burying hydrophobic monomers in the core. This results in patterns, or correlations, in the hydrophobic profiles of proteins. In this thesis, we study the designability phenomena: the vast majority of proteins adopt only a small number of distinct folded structures. In Chapter 2, we use principal component analysis to characterize the distribution of solvent accessibility profiles in an appropriate high-dimensional vector space and sh...

2011

The series of events that drive a polypeptide chain into its stable native structure is not yet fully understood. Protein systems involve so numerous complex interactions and remarkable properties that they continuously require new experiments, as well as theoretical and computational approaches [1–5].