Computational Molecular Evolution (Oxford Series in Ecology and Evolution)

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Computational Molecular Evolution | Ziheng Yang | | Oxford University Press Canada

To purchase, visit your preferred ebook provider. Oxford Scholarship Online This book is available as part of Oxford Scholarship Online - view abstracts and keywords at book and chapter level. Computational Molecular Evolution Ziheng Yang Oxford Series in Ecology and Evolution Authored by a world-renowned specialist in the field Adopts a statistical approach to phylogenetics Web-based example data sets used to clarify the theory Emphasises the models and methods designed for understanding the evolutionary process of genes and genomes Ideal graduate seminar course material.

Molecular Evolution Ziheng Yang. Aboveground-Belowground Linkages Richard D. Bardgett and David A.

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Ships from the UK. Great condition for a used book! Your purchase also supports literacy charities. Ergodebooks , Texas, United States Seller rating: Oxford University Press, Ships with Tracking Number! May not contain Access Codes or Supplements. Buy with confidence, excellent customer service! We're sorry - this copy is no longer available. In general, the degree of each state node i is given by the number minus one of nonzero elements of the i th column in the associated substitution matrix.

We can investigate the effect of sparseness of substitution matrices on the dispersion index with the formalism developed above. We compare the case of fully connected graphs F where any amino acid can replace any other one with the case where only amino acids one nucleotide mutation apart can replace each other nonsynonymous substitution, NS. As before, we generate 10 7 random matrices in each class and compute their statistical properties.

We observe again fig. For the fully connected matrix, any amino acid can be replaced by any other. For the NS matrices, only amino acids one nucleotide mutation apart can replace each other. For the NS matrices, transitions are weighted by the number of nucleotide substitutions that can lead from one amino acid to another: There are, for example, 6-nt mutations that transform a phenylalanine into a leucine, but only one mutation that transforms a lysine into an isoleucine. The substitution process of nucleotides, amino acids, etc. One of the most fundamental tasks in molecular evolutionary investigation is to characterize the random variable n from molecular data.

In molecular evolution, the main observable is the probability p d t that two different sequences are different at a given site.

Statistical and Mathematical Challenges in Molecular Evolution

This quantity however is very different from the intrinsic variance of the substitution number. The next crucial step is to evaluate the variance V of this number. What we have achieved in this article is to find a simple expression for V. In particular, we have shown that for both short and long time scales, the variance V can be easily deduced from Q. For short times, only the diagonal elements of Q are required to compute V [relation 31 ]. One possibility to produce a higher dispersion index is sparse matrices, where the ensemble of possible transitions has been reduced.

The authors are grateful to Erik Geissler, Olivier Rivoire, and Ivan Junier for fruitful discussions and critical reading of the manuscript. The corresponding balance equations are captured in equations 4 and 5. Obtaining equations of the moments such as 12 and 13 from the Master equation is a standard procedure of stochastic processes Gardiner ; Houchmandzadeh We give here the outline of the derivation.

Multiplying each row by n and summing over all n leads, in vectorial form, to.

Computational Molecular Evolution. Oxford Series in Ecology and Evolution.

For the second term, we have. The equation for the second moment 13 is obtained by the same procedure where each row of equation 42 is multiplied by n 2. Higher moments and the probability generating function equations can be obtained by similar computations. We give here the proof for GTR matrices. These matrices can be factorized into.

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F , where F is a symmetric matrix we stress again that in our notation, the substitution matrix is the transpose of that used in most of the literature. The advantage of the factorization 44 is that except for one zero eigenvalue, all other eigenvalues of S are negative. S can be therefore be written as. The pseudoinverse of S is defined as. The general solution of the undetermined equation 47 is therefore.

To see this, we can expand relation For all the specific models used in the literature, the variance at all times can also be determined explicitly through relation As an example, consider the equal input model F81 which we studied in the main text. For this model, the reduced matrix is simply. Relation 50 then becomes.

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We have previously shown eq. However, the time can be expressed as a function of mean the substitution number. National Center for Biotechnology Information , U. Published online Apr 6. For commercial re-use, please contact journals. Abstract The number of substitutions of nucleotides, amino acids, etc. Introduction Evolution at the molecular level is the process by which random mutations change the content of some sites of a given sequence of nucleotides, amino acids, etc.

Materials and Methods Background and Definitions The problem we investigate in this article is mainly that of counting transitions of a random variable fig. Open in a separate window.

Problem Formulation To count the number of substitutions fig. Solution of the Equation for the Moments One standard way of solving equation 9 would be to express the matrix Q in its eigenbasis; equation 9 is then diagonalized and can be formally solved. Rate Heterogeneity The substitution models we have considered here can be applied to sequences where the Q matrix is identical for all sites.

Let us define n S as the substitution number for the sequence: Results Application to Specific Nucleotides Substitution Models Nucleotide substitution models are widely used in molecular evolution Graur and Li ; Yang , for example, to deduce distances between sequences. Statistical Investigation of the Dispersion Index and the Influence of Sparseness The relation 28 can be solved explicitly for general substitution matrices. Discussion and Conclusion The substitution process of nucleotides, amino acids, etc. Click here to view. Acknowledgment The authors are grateful to Erik Geissler, Olivier Rivoire, and Ivan Junier for fruitful discussions and critical reading of the manuscript.

Lack of self-averaging in neutral evolution of proteins. A Fresh Approach to Numerical Computing.

Thermodynamics of neutral protein evolution. The modern molecular clock. The index of dispersion of molecular evolution: Evolutionary trees from DNA sequences: Handbook of stochastic methods: Exact stochastic simulation of coupled chemical reactions. Lineage effects and the index of dispersion of molecular evolution. Fundamentals of molecular evolution.

Dating of the human-ape splitting by a molecular clock of mitochondrial DNA. Molecular-clock methods for estimating evolutionary rates and timescales. Theory of neutral clustering for growing populations. A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences.

The neutral theory of molecular evolution.

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