Sanzo Miyazawa

How effective for fold recognition is a potential of mean force that includes relative orientations between contacting residues in proteins

We estimate the statistical distribution of relative orientations between contacting residues from a database of protein structures and evaluate the potential of mean force for relative orientations between contacting residues. Polar angles and Euler angles are used to specify two degrees of directional freedom and three degrees of rotational freedom for the orientation of one residue relative to another in contacting residues, respectively. A local coordinate system affixed to each residue based only on main chain atoms is defined for fold recognition. The number of contacting residue pairs in the database will severely limit the resolution of the statistical distribution of relative orientations, if it is estimated by dividing space into cells and counting samples observed in each cell. To overcome such problems and to evaluate the fully anisotropic distributions of relative orientations as a function of polar and Euler angles, we choose a method in which the observed distribution is represented as a sum of delta functions each of which represents the observed orientation of a contacting residue, and is evaluated as a series expansion of spherical harmonics functions. The sample size limits the frequencies of modes whose expansion coefficients can be reliably estimated. High frequency modes are statistically less reliable than low frequency modes. Each expansion coefficient is separately corrected for the sample size according to suggestions from a Bayesian statistical analysis. As a result, many expansion terms can be utilized to evaluate orientational distributions. Also, unlike other orientational potentials, the uniform distribution is used for a reference distribution in evaluating a potential of mean force for each type of contacting residue pair from its orientational distribution, so that residue-residue orientations can be fully evaluated. It is shown by using decoy sets that the discrimination power of the orientational potential in fold recognition increases by taking account of the Euler angle dependencies and becomes comparable to that of a simple contact potential, and that the total energy potential taken as a simple sum of contact, orientation, and (phi,psi) potentials performs well to identify the native folds.

Long- and short-range interactions in native protein structures are consistent/minimally frustrated in sequence space.

We show that long‐ and short‐range interactions in almost all protein native structures are actually consistent with each other for coarse‐grained energy scales; specifically we mean the long‐range inter‐residue contact energies and the short‐range secondary structure energies based on peptide dihedral angles, which are potentials of mean force evaluated from residue distributions observed in protein native structures. This consistency is observed at equilibrium in sequence space rather than in conformational space. Statistical ensembles of sequences are generated by exchanging residues for each of 797 protein native structures with the Metropolis method. It is shown that adding the other category of interaction to either the short‐ or long‐range interactions decreases the means and variances of those energies for essentially all protein native structures, indicating that both interactions consistently work by more‐or‐less restricting sequence spaces available to one of the interactions. In addition to this consistency, independence by these interaction classes is also indicated by the fact that there are almost no correlations between them when equilibrated using both interactions and significant but small, positive correlations at equilibrium using only one of the interactions. Evidence is provided that protein native sequences can be regarded approximately as samples from the statistical ensembles of sequences with these energy scales and that all proteins have the same effective conformational temperature. Designing protein structures and sequences to be consistent and minimally frustrated among the various interactions is a most effective way to increase protein stability and foldability. Proteins 2003;50:35–43.

Prediction of Contact Residue Pairs Based on Co-Substitution between Sites in Protein 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 in varying degrees 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 phylogenetic correlations and indirect dependences through other residues within a protein or even through other molecules. Rapid growth of protein families with unknown folds requires an accurate de novo prediction method for protein structure. Recent attempts of disentangling direct from indirect dependences of amino acid types between residue positions in multiple sequence alignments have revealed that inferred residue-residue proximities can be sufficient information to predict a protein fold without the use of known three-dimensional structures. Here, we propose an alternative method of inferring coevolving site pairs from concurrent and compensatory substitutions between sites in each branch of a phylogenetic tree. Substitution probability and physico-chemical changes (volume, charge, hydrogen-bonding capability, and others) accompanied by substitutions at each site in each branch of a phylogenetic tree are estimated with the likelihood of each substitution, and their direct correlations between sites are used to detect concurrent and compensatory substitutions. In order to extract direct dependences between sites, partial correlation coefficients of the characteristic changes along branches between sites, in which linear multiple dependences on feature vectors at other sites are removed, are calculated and used to rank coevolving site pairs. Accuracy of contact prediction based on the present coevolution 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.

Advantages of a Mechanistic Codon Substitution Model for Evolutionary Analysis of Protein-Coding Sequences

Background A mechanistic codon substitution model, in which each codon substitution rate is proportional to the product of a codon mutation rate and the average fixation probability depending on the type of amino acid replacement, has advantages over nucleotide, amino acid, and empirical codon substitution models in evolutionary analysis of protein-coding sequences. It can approximate a wide range of codon substitution processes. If no selection pressure on amino acids is taken into account, it will become equivalent to a nucleotide substitution model. If mutation rates are assumed not to depend on the codon type, then it will become essentially equivalent to an amino acid substitution model. Mutation at the nucleotide level and selection at the amino acid level can be separately evaluated. Results The present scheme for single nucleotide mutations is equivalent to the general time-reversible model, but multiple nucleotide changes in infinitesimal time are allowed. Selective constraints on the respective types of amino acid replacements are tailored to each gene in a linear function of a given estimate of selective constraints. Their good estimates are those calculated by maximizing the respective likelihoods of empirical amino acid or codon substitution frequency matrices. Akaike and Bayesian information criteria indicate that the present model performs far better than the other substitution models for all five phylogenetic trees of highly-divergent to highly-homologous sequences of chloroplast, mitochondrial, and nuclear genes. It is also shown that multiple nucleotide changes in infinitesimal time are significant in long branches, although they may be caused by compensatory substitutions or other mechanisms. The variation of selective constraint over sites fits the datasets significantly better than variable mutation rates, except for 10 slow-evolving nuclear genes of 10 mammals. An critical finding for phylogenetic analysis is that assuming variable mutation rates over sites lead to the overestimation of branch lengths.

Selective constraints on amino acids estimated by a mechanistic codon substitution model with multiple nucleotide changes.

Background 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. Results 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. Conclusions/Significance 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.

On the optimal contact potential of proteins

Abstract 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 G o ¯ 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.

Properties of contact matrices induced by pairwise interactions in proteins.

The properties of contact matrices ( C matrices) needed for native proteins to be the lowest-energy conformations are considered in relation to a contact energy matrix ( E matrix). 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 the C matrix) and a sequence-dependent energy parameter (an element of the 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, where the E matrix is expanded in a series of eigenspaces with an additional constant term. The additional constant term in the spectral expansion of the E matrix is indicated by the lowest bound of the total energy function to correspond to a threshold of contact energy that approximately separates native contacts from non-native ones. Inner products between the principal eigenvector of the C matrix, that of the E matrix, and a contact number vector have been examined for 182 proteins, each of which is a representative from each family of the SCOP database [Murzin, J. Mol. Biol. 247, 536 (1995)], and the results indicate the parallel tendencies between those vectors. A statistical contact potential [S. Miyazawa and R. L. Jernigan, Proteins 34, 49 (1999); S. Miyazawa and R. L. Jernigan, Proteins50, 35 (2003)] estimated from protein crystal structures was used to evaluate pairwise residue-residue interactions in the proteins. In addition, the spectral representation of C and E matrices reveals that pairwise residue-residue interactions, which depend 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 unique lowest-energy conformations.

Monte Carlo calculation of the quantum J1−J2 model on the square lattice

Abstract We used the modified decoupled cell method (mDCM) of the quantum Monte Carlo simulation to calculate the thermodynamic properties and spin configurations of the frustrated J 1 − J 2 model on a square lattice by increasing the frustration parameter α= J 2 J 1 from 0 to 1. The size N of the system used in the Monte Carlo simulation in this study is 32 × 32. For small values of α the Neel state is a ground state spin configuration of this model system, and for α > 0.6 a collinear state is the ground state configuration instead of the Neel state.

Numerical Experiments on Frustrated Quantum Spin Systems

Thermodynamic properties of the antiferromagnetic Heisenberg model with frustration on the square lattice are investigated using quantum simulation based on the decoupled cell Monte Carlo method. The effect of frustration is introduced through an additional next-nearest neighbor antiferromagnetic interaction J 2. With increasing J 2, certain spin ordering was observed at low temperatures.