arXiv subjects

Sanzo Miyazawa

Publications and source records attributed to Sanzo Miyazawa.

10 recordsLinked to original sources

arXiv2026

Boltzmann Machine Learning with a Parallel, Persistent Markov chain Monte Carlo method for Estimating Evolutionary Fields and Couplings from a Protein Multiple Sequence Alignment

The inverse Potts problem for estimating evolutionary single-site fields and pairwise couplings in homologous protein sequences from their single-site and pairwise amino acid frequencies observed in their multiple sequence alignment would be still one of useful methods in the studies of protein structure and evolution. Since the reproducibility of fields and couplings are the most important, the Boltzmann machine method is employed here, although it is computationally intensive. In order to reduce computational time required for the Boltzmann machine, parallel, persistent Markov chain Monte Carlo method is employed to estimate the single-site and pairwise marginal distributions in each learning step. Also, stochastic gradient descent methods are used to reduce computational time for each learning. Another problem is how to adjust the values of hyperparameters; there are two regularization parameters for evolutionary fields and couplings. The precision of contact residue pair prediction is often used to adjust the hyperparameters. However, it is not sensitive to these regularization parameters. Here, they are adjusted for the fields and couplings to satisfy a specific condition that is appropriate for protein conformations. This method has been applied to eight protein families.

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arXiv2024

Selection originating from protein stability/foldability: Relationships between protein folding free energy, sequence ensemble, and fitness

Assuming that mutation and fixation processes are reversible Markov processes, we prove that the equilibrium ensemble of sequences obeys a Boltzmann distribution with $\exp(4N_e m(1 - 1/(2N)))$, where $m$ is Malthusian fitness and $N_e$ and $N$ are effective and actual population sizes. On the other hand, 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 $\exp(-ψ_N)$, where $ψ_N$ is represented as the sum of one body and pairwise potentials. A protein folding theory indicates that homologous sequences obey a canonical ensemble characterized by $\exp(-ΔG_{ND}/k_B T_s)$ or by $\exp(- G_{N}/k_B T_s)$ if an amino acid composition is kept constant, where $ΔG_{ND} \equiv G_N - G_D$, $G_N$ and $G_D$ are the native and denatured free energies, and $T_s$ is selective temperature. Thus, $4N_e m (1 - 1/(2N))$, $-Δψ_{ND}$, and $-ΔG_{ND}/k_B T_s$ must be equivalent to each other. Based on the analysis of the changes ($Δψ_N$) of $ψ_N$ due to single nucleotide nonsynonymous substitutions, $T_s$, and then glass transition temperature $T_g$, and $ΔG_{ND}$ are estimated with reasonable values for 14 protein domains. In addition, approximating the probability density function (PDF) of $Δψ_N$ by a log-normal distribution, PDFs of $Δψ_N$ and $K_a/K_s$, which is the ratio of nonsynonymous to synonymous substitution rate per site, in all and in fixed mutants are estimated. It is confirmed that $T_s$ negatively correlates with the mean of $K_a/K_s$. Stabilizing mutations are significantly fixed by positive selection, and balance with destabilizing mutations fixed by random drift. Supporting the nearly neutral theory, neutral selection is not significant.

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arXiv2024

Boltzmann machine learning and regularization methods for inferring evolutionary fields and couplings from a multiple sequence alignment

The inverse Potts problem to infer a Boltzmann distribution for homologous protein sequences from their single-site and pairwise amino acid frequencies recently attracts a great deal of attention in the studies of protein structure and evolution. We study regularization and learning methods and how to tune regularization parameters to correctly infer interactions in Boltzmann machine learning. Using $L_2$ regularization for fields, group $L_1$ for couplings is shown to be very effective for sparse couplings in comparison with $L_2$ and $L_1$. Two regularization parameters are tuned to yield equal values for both the sample and ensemble averages of evolutionary energy. Both averages smoothly change and converge, but their learning profiles are very different between learning methods. The Adam method is modified to make stepsize proportional to the gradient for sparse couplings. It is shown by first inferring interactions from protein sequences and then from Monte Carlo samples that the fields and couplings can be well recovered, but that recovering the pairwise correlations in the resolution of a total energy is harder for the natural proteins than for the protein-like sequences. Selective temperature for folding/structural constrains in protein evolution is also estimated.

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arXiv2018

Prediction of Structures and Interactions from Genome Information

Predicting three dimensional residue-residue contacts from evolutionary information in protein sequences was attempted already in the early 1990s. However, contact prediction accuracies of methods evaluated in CASP experiments before CASP11 remained quite low, typically with $<20$% true positives. Recently, contact prediction has been significantly improved to the level that an accurate three dimensional model of a large protein can be generated on the basis of predicted contacts. This improvement was attained by disentangling direct from indirect correlations in amino acid covariations or cosubstitutions between sites in protein evolution. Here, we review statistical methods for extracting causative correlations and various approaches to describe protein structure, complex, and flexibility based on predicted contacts.

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arXiv2017

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 $\exp(-ψ_N)$, where the total interaction $ψ_N$ 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 $\exp(-ΔG_{ND}/k_B T_s)$ or by $\exp(- G_{N}/k_B T_s)$ if an amino acid composition is kept constant, meaning $ψ_N = ΔG_{ND}/k_B T_s +$ constant, where $ΔG_{ND} \equiv G_N - G_D$, $G_N$ 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 ($Δψ_N$) due to single nucleotide nonsynonymous mutations, and have found that the variance of their $Δψ_N$ over all sites hardly depends on the $ψ_N$ of each homologous sequence, indicating that the variance of $ΔG_N (= k_B T_s Δψ_N)$ is nearly constant irrespective of protein families. As a result, $T_s$ is estimated from the ratio of the variance of $Δψ_N$ to that of a reference protein, which is determined by a direct comparison between $ΔΔψ_{ND} (\simeq Δψ_N)$ and experimental $ΔΔG_{ND}$. Based on the REM, glass transition temperature $T_g$ and $ΔG_{ND}$ are estimated from $T_s$ and experimental melting temperatures ($T_m$) for 14 protein domains. The estimates of $ΔG_{ND}$ 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 $ΔΔG_{ND}$ than previous methods.

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arXiv2015

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 $s = κ\exp(βΔG) (1 - \exp(βΔΔG))$, where $s$ and $ΔΔG$ are selective advantage and stability change of a mutant protein, $ΔG$ is the folding free energy of the wild-type protein, and $κ$ represents protein abundance and indispensability. The distribution of $ΔΔG$ 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 $ΔG$. The evolution of this gene has an equilibrium ($ΔG_e$) 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 $ΔG_e > -2.5$ 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 $ΔΔG$ through decreasing $ΔG_e$, and by strong structural constraints, which directly make $ΔΔG$ more positive.

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arXiv2012

Inference of Co-Evolving Site Pairs: an Excellent Predictor of Contact Residue Pairs in Protein 3D structures

Residue-residue interactions that fold a protein into a unique three-dimensional structure and make it play a specific function impose structural and functional constraints on each residue site. Selective constraints on residue sites are recorded in amino acid orders in homologous sequences and also in the evolutionary trace of amino acid substitutions. A challenge is to extract direct dependences between residue sites by removing indirect dependences through other residues within a protein or even through other molecules. Recent attempts of disentangling direct from indirect dependences of amino acid types between residue positions in multiple sequence alignments have revealed that the strength of inferred residue pair couplings is an excellent predictor of residue-residue proximity in folded structures. Here, we report an alternative attempt of inferring co-evolving site pairs from concurrent and compensatory substitutions between sites in each branch of a phylogenetic tree. First, branch lengths of a phylogenetic tree inferred by the neighbor-joining method are optimized as well as other parameters by maximizing a likelihood of the tree in a mechanistic codon substitution model. Mean changes of quantities, which are characteristic of concurrent and compensatory substitutions, accompanied by substitutions at each site in each branch of the tree are estimated with the likelihood of each substitution. Partial correlation coefficients of the characteristic changes along branches between sites are calculated and used to rank co-evolving site pairs. Accuracy of contact prediction based on the present co-evolution score is comparable to that achieved by a maximum entropy model of protein sequences for 15 protein families taken from the Pfam release 26.0. Besides, this excellent accuracy indicates that compensatory substitutions are significant in protein evolution.

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arXiv2011

Properties of contact matrices induced by pairwise interactions in proteins

The total conformational energy is assumed to consist of pairwise interaction energies between atoms or residues, each of which is expressed as a product of a conformation-dependent function (an element of a contact matrix, C-matrix) and a sequence-dependent energy parameter (an element of a contact energy matrix, E-matrix). Such pairwise interactions in proteins force native C-matrices to be in a relationship as if the interactions are a Go-like potential [N. Go, Annu. Rev. Biophys. Bioeng. 12. 183 (1983)] for the native C-matrix, because the lowest bound of the total energy function is equal to the total energy of the native conformation interacting in a Go-like pairwise potential. This relationship between C- and E-matrices corresponds to (a) a parallel relationship between the eigenvectors of the C- and E-matrices and a linear relationship between their eigenvalues, and (b) a parallel relationship between a contact number vector and the principal eigenvectors of the C- and E-matrices; the E-matrix is expanded in a series of eigenspaces with an additional constant term, which corresponds to a threshold of contact energy that approximately separates native contacts from non-native ones. These relationships are confirmed in 182 representatives from each family of the SCOP database by examining inner products between the principal eigenvector of the C-matrix, that of the E-matrix evaluated with a statistical contact potential, and a contact number vector. In addition, the spectral representation of C- and E-matrices reveals that pairwise residue-residue interactions, which depends only on the types of interacting amino acids but not on other residues in a protein, are insufficient and other interactions including residue connectivities and steric hindrance are needed to make native structures the unique lowest energy conformations.

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arXiv2011

Selective Constraints on Amino Acids Estimated by a Mechanistic Codon Substitution Model with Multiple Nucleotide Changes

Empirical substitution matrices represent the average tendencies of substitutions over various protein families by sacrificing gene-level resolution. We develop a codon-based model, in which mutational tendencies of codon, a genetic code, and the strength of selective constraints against amino acid replacements can be tailored to a given gene. First, selective constraints averaged over proteins are estimated by maximizing the likelihood of each 1-PAM matrix of empirical amino acid (JTT, WAG, and LG) and codon (KHG) substitution matrices. Then, selective constraints specific to given proteins are approximated as a linear function of those estimated from the empirical substitution matrices. Akaike information criterion (AIC) values indicate that a model allowing multiple nucleotide changes fits the empirical substitution matrices significantly better. Also, the ML estimates of transition-transversion bias obtained from these empirical matrices are not so large as previously estimated. The selective constraints are characteristic of proteins rather than species. However, their relative strengths among amino acid pairs can be approximated not to depend very much on protein families but amino acid pairs, because the present model, in which selective constraints are approximated to be a linear function of those estimated from the JTT/WAG/LG/KHG matrices, can provide a good fit to other empirical substitution matrices including cpREV for chloroplast proteins and mtREV for vertebrate mitochondrial proteins. The present codon-based model with the ML estimates of selective constraints and with adjustable mutation rates of nucleotide would be useful as a simple substitution model in ML and Bayesian inferences of molecular phylogenetic trees, and enables us to obtain biologically meaningful information at both nucleotide and amino acid levels from codon and protein sequences.

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arXiv2008

On the optimal contact potential of proteins

We analytically derive the lower bound of the total conformational energy of a protein structure by assuming that the total conformational energy is well approximated by the sum of sequence-dependent pairwise contact energies. The condition for the native structure achieving the lower bound leads to the contact energy matrix that is a scalar multiple of the native contact matrix, i.e., the so-called Go potential. We also derive spectral relations between contact matrix and energy matrix, and approximations related to one-dimensional protein structures. Implications for protein structure prediction are discussed.

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