Substitution (logic)

Abstract: DNA sequence evolution through nucleotide substitution may be assimilated to a stationary Markov process. The fundamental equations of the general model, with 12 independent substitution parameters, are used to obtain a formula which corrects the effect of multiple and parallel substitutions on the measure of evolutionary divergence between two homologous sequences. We show that only reversible models, with six independent parameters, allow the calculation of the substitution rates. Simulation experiments on DNA sequence evolution through nucleotide substitution call into question the effectiveness of the general model (and of any other more detailed description); nevertheless, the general model results are slightly superior to any of its particular cases.

Abstract: Comparison of relative fixation rates of synonymous (silent) and nonsynonymous (amino acid-altering) mutations provides a means for understanding the mechanisms of molecular sequence evolution. The nonsynonymous/synonymous rate ratio (omega = d(N)d(S)) is an important indicator of selective pressure at the protein level, with omega = 1 meaning neutral mutations, omega < 1 purifying selection, and omega > 1 diversifying positive selection. Amino acid sites in a protein are expected to be under different selective pressures and have different underlying omega ratios. We develop models that account for heterogeneous omega ratios among amino acid sites and apply them to phylogenetic analyses of protein-coding DNA sequences. These models are useful for testing for adaptive molecular evolution and identifying amino acid sites under diversifying selection. Ten data sets of genes from nuclear, mitochondrial, and viral genomes are analyzed to estimate the distributions of omega among sites. In all data sets analyzed, the selective pressure indicated by the omega ratio is found to be highly heterogeneous among sites. Previously unsuspected Darwinian selection is detected in several genes in which the average omega ratio across sites is <1, but in which some sites are clearly under diversifying selection with omega > 1. Genes undergoing positive selection include the beta-globin gene from vertebrates, mitochondrial protein-coding genes from hominoids, the hemagglutinin (HA) gene from human influenza virus A, and HIV-1 env, vif, and pol genes. Tests for the presence of positively selected sites and their subsequent identification appear quite robust to the specific distributional form assumed for omega and can be achieved using any of several models we implement. However, we encountered difficulties in estimating the precise distribution of omega among sites from real data sets.

Abstract: This chapter analyzes the benefits and shortcomings of inferring a temporal trend from a study of different aged sites. This technique, called space-for-time substitution, assumes that spatial and temporal variation are equivalent. Although this assumption has been challenged, studies continue to rely on space-for-time substitution due to necessity or convenience.

Abstract: Most current models of sequence evolution assume that all sites of a protein evolve under the same substitution process, characterized by a 20 x 20 substitution matrix. Here, we propose to relax this assumption by developing a Bayesian mixture model that allows the amino-acid replacement pattern at different sites of a protein alignment to be described by distinct substitution processes. Our model, named CAT, assumes the existence of distinct processes (or classes) differing by their equilibrium frequencies over the 20 residues. Through the use of a Dirichlet process prior, the total number of classes and their respective amino-acid profiles, as well as the affiliations of each site to a given class, are all free variables of the model. In this way, the CAT model is able to adapt to the complexity actually present in the data, and it yields an estimate of the substitutional heterogeneity through the posterior mean number of classes. We show that a significant level of heterogeneity is present in the substitution patterns of proteins, and that the standard one-matrix model fails to account for this heterogeneity. By evaluating the Bayes factor, we demonstrate that the standard model is outperformed by CAT on all of the data sets which we analyzed. Altogether, these results suggest that the complexity of the pattern of substitution of real sequences is better captured by the CAT model, offering the possibility of studying its impact on phylogenetic reconstruction and its connections with structure-function determinants.

Abstract: Foundations.- Getting started with Isabelle.- Advanced methods.- Basic use of Isabelle.- Proof management: The subgoal module.- Tactics.- Tacticals.- Theorems and forward proof.- Theories, terms and types.- Defining logics.- Syntax transformations.- Substitution tactics.- Simplification.- The classical reasoner.- Basic concepts.- First-order logic.- Zermelo-Fraenkel set theory.- Higher-order logic.- First-order sequent calculus.- Constructive Type Theory.- Syntax of Isabelle Theories.