Selection originating from protein foldability: I. A new method to estimate selection temperature

The probability distribution of sequences with maximum entropy that satisfies a given amino acid composition at each site and a given pairwise amino acid frequency at each site pair is a Boltzmann distribution with $exp(-psi_N)$, where the total interaction $psi_N$ is represented as the sum of one body and pairwise interactions. A protein folding theory based on the random energy model (REM) indicates that the equilibrium ensemble of natural protein sequences is a canonical ensemble characterized by $exp(-Delta G_{ND}/k_B T_s)$ or by $exp(- G_{N}/k_B T_s)$ if an amino acid composition is kept constant, meaning $psi_N = Delta G_{ND}/k_B T_s +$ constant, where $Delta G_{ND} equiv G_N - G_D$, $G_N$ and $G_D$ are the native and denatured free energies, and $T_s$ is the effective temperature of natural selection. Here, we examine interaction changes ($Delta psi_N$) due to single nucleotide nonsynonymous mutations, and have found that the variance of their $Delta psi_N$ over all sites hardly depends on the $psi_N$ of each homologous sequence, indicating that the variance of $Delta G_N (= k_B T_s Delta psi_N)$ is nearly constant irrespective of protein families. As a result, $T_s$ is estimated from the ratio of the variance of $Delta psi_N$ to that of a reference protein, which is determined by a direct comparison between $DeltaDelta psi_{ND} (simeq Delta psi_N)$ and experimental $DeltaDelta G_{ND}$. Based on the REM, glass transition temperature $T_g$ and $Delta G_{ND}$ are estimated from $T_s$ and experimental melting temperatures ($T_m$) for 14 protein domains. The estimates of $Delta G_{ND}$ agree well with their experimental values for 5 proteins, and those of $T_s$ and $T_g$ are all within a reasonable range. This method is coarse-grained but much simpler in estimating $T_s$, $T_g$ and $DeltaDelta G_{ND}$ than previous methods.

Selection maintaining protein stability at equilibrium

The common understanding of protein evolution has been that neutral or slightly deleterious mutations are fixed by random drift, and evolutionary rate is determined primarily by the proportion of neutral mutations. However, recent studies have revealed that highly expressed genes evolve slowly because of fitness costs due to misfolded proteins. Here we study selection maintaining protein stability. Protein fitness is taken to be $s = kappa exp(betaDelta G) (1 - exp(betaDeltaDelta G))$, where $s$ and $DeltaDelta G$ are selective advantage and stability change of a mutant protein, $Delta G$ is the folding free energy of the wild-type protein, and $kappa$ represents protein abundance and indispensability. The distribution of $DeltaDelta G$ is approximated to be a bi-Gaussian function, which represents structurally slightly- or highly-constrained sites. Also, the mean of the distribution is negatively proportional to $Delta G$. The evolution of this gene has an equilibrium ($Delta G_e$) of protein stability, the range of which is consistent with experimental values. The probability distribution of $K_a/K_s$, the ratio of nonsynonymous to synonymous substitution rate per site, over fixed mutants in the vicinity of the equilibrium shows that nearly neutral selection is predominant only in low-abundant, non-essential proteins of $Delta G_e > -2.5$ kcal/mol. In the other proteins, positive selection on stabilizing mutations is significant to maintain protein stability at equilibrium as well as random drift on slightly negative mutations, although the average $langle K_a/K_s rangle$ is less than 1. Slow evolutionary rates can be caused by high protein abundance/indispensability, which produces positive shifts of $DeltaDelta G$ through decreasing $Delta G_e$, and by strong structural constraints, which directly make $DeltaDelta G$ more positive.

Protein Folding and Machine Learning: Fundamentals

In spite of decades of research, much remains to be discovered about folding: the detailed structure of the initial (unfolded) state, vestigial folding instructions remaining only in the unfolded state, the interaction of the molecule with the solvent, instantaneous power at each point within the molecule during folding, the fact that the process is stable in spite of myriad possible disturbances, potential stabilization of trajectory by chaos, and, of course, the exact physical mechanism (code or instructions) by which the folding process is specified in the amino acid sequence. Simulations based upon microscopic physics have had some spectacular successes and continue to improve, particularly as super-computer capabilities increase. The simulations, exciting as they are, are still too slow and expensive to deal with the enormous number of molecules of interest. In this paper, we introduce an approximate model based upon physics, empirics, and information science which is proposed for use in machine learning applications in which very large numbers of sub-simulations must be made. In particular, we focus upon machine learning applications in the learning phase and argue that our model is sufficiently close to the physics that, in spite of its approximate nature, can facilitate stepping through machine learning solutions to explore the mechanics of folding mentioned above. We particularly emphasize the exploration of energy flow (power) within the molecule during folding, the possibility of energy scale invariance (above a threshold), vestigial information in the unfolded state as attractive targets for such machine language analysis, and statistical analysis of an ensemble of folding micro-steps.

Free-Energy Calculations in Protein Folding by Generalized-Ensemble Algorithms

We review uses of the generalized-ensemble algorithms for free-energy calculations in protein folding. Two of the well-known methods are multicanonical algorithm and replica-exchange method; the latter is also referred to as parallel tempering. We present a new generalized-ensemble algorithm that combines the merits of the two methods; it is referred to as the replica-exchange multicanonical algorithm. We also give a multidimensional extension of the replica-exchange method. Its realization as an umbrella sampling method, which we refer to as the replica-exchange umbrella sampling, is a powerful algorithm that can give free energy in wide reaction coordinate space.

The Principle of Stationary Action in Biophysics: Stability in Protein Folding

Processes that proceed reliably from a variety of initial conditions to a unique final form, regardless of moderately changing conditions, are of obvious importance in biophysics. Protein folding is a case in point. We show that the action principle can be applied directly to study the stability of biological processes. The action principle in classical physics starts with the first variation of the action and leads immediately to the equations of motion. The second variation of the action leads in a natural way to powerful theorems that provide quantitative treatment of stability and focusing and also explain how some very complex processes can behave as though some seemingly important forces drop out. We first apply these ideas to the non-equilibrium states involved in two-state folding. We treat torsional waves and use the action principle to talk about critical points in the dynamics. For some proteins the theory resembles TST. We reach several quantitative and qualitative conclusions. Besides giving an explanation of why TST often works in folding, we find that the apparent smoothness of the energy funnel is a natural consequence of the putative critical points in the dynamics. These ideas also explain why biological proteins fold to unique states and random polymers do not. The insensitivity to perturbations which follows from the presence of critical points explains how folding to a unique shape occurs in the presence of dilute denaturing agents in spite of the fact that those agents disrupt the folded structure of the native state. This paper contributes to the theoretical armamentarium by directing attention to the logical progression from first physical principles to the stability theorems related to catastrophe theory as applied to folding. This can potentially have the same success in biophysics as it has enjoyed in optics.