The rise of the neutral theory of molecular evolution seems to have aroused a renewed interest in mathematical population genetics among biologists, who are primarily experimenters rather than theoreticians. This has encouraged me to set out the mathematics of the evolutionary process in a manner that, I hope, will be comprehensible to those with only a basic knowledge of calculus and matrix algebra. I must acknowledge from the start my great debt to my students. Equipped initially with rather limited mathematics, they have pursued the subject with much enthusiasm and success. This has enabled…mehr
The rise of the neutral theory of molecular evolution seems to have aroused a renewed interest in mathematical population genetics among biologists, who are primarily experimenters rather than theoreticians. This has encouraged me to set out the mathematics of the evolutionary process in a manner that, I hope, will be comprehensible to those with only a basic knowledge of calculus and matrix algebra. I must acknowledge from the start my great debt to my students. Equipped initially with rather limited mathematics, they have pursued the subject with much enthusiasm and success. This has enabled me to try a number of different approaches over the years. I was particularly grateful to Dr L. J. Eaves and Professor W. E. Nance for the opportunity to give a one-semester course at the Medical College of Virginia, and I would like to thank them, their colleagues and their students for the many kindnesses shown to me during my visit. I have concentrated almost entirely on stochastic topics, since these cause the greatest problems for non-mathematicians. The latter are particularly concerned with the range of validity of formulae. A sense of confidence in applying these formulae is, almost certainly, best gained by following their derivation. I have set out proofs in fair detail, since, in my experience, minor points of algebraic manipulation occasionally cause problems. To avoid loss of continuity, I have sometimes put material in notes at the end of chapters.
1 Introduction.- Notes and exercises.- 2 Wright-Fisher, Moran and other models.- On simple models.- Stochastic models: deterministic models.- Random genetic drift: general approach.- Distribution of allele frequency under drift.- Change of allele frequency in one generation.- The mean in any generation.- Change of allele frequency in one generation: the variance.- Is the binomial distribution of allele number relevant?.- Effective population size.- Variance of allele frequency in any generation.- Inbreeding effective size.- Populations under systematic 'pressure'.- A continuous approximation to the conditional distribution.- Diffusion methods.- Moran's model.- Notes and exercises.- 3 On the description of changes in allele frequency.- A bewildering abundance of descriptions.- Probability distribution of allele frequency.- A numerical example.- Probability of fixation.- Dominant latent root (maximum non-unit eigenvalue).- Modified process.- Probability distribution of absorption time.- Mean absorption time.- Variance of absorption time.- Mean sojourn times.- Variance of sojourn time.- Probability of ever reaching a given frequency.- A matrix representation of mean sojourn time.- Notes and exercises.- 4 Survival of new mutations: branching processes.- On special methods.- Probability of survival.- The need for accuracy.- Are diffusion methods appropriate?.- A fundamental simplification.- Probability generating functions.- Independent propagation: branching processes.- Utility of the branching process approach.- Fundamental equations for branching processes.- Calculating the probability of survival.- The case Ne < N.- A problem in genetic conservation.- Probability of survival when t is fairly large.- Pattern of survival in natural populations.- A caution.- Genieselection.- Probability of ultimate fixation.- Calculating the probability of fixation.- Population subdivision.- Cyclical variation in population size.- A note on epistasis and linkage.- Spread of a fungal pathogen.- Notes and exercises.- 5 Probability of fixation: the more general case.- Fundamental equation.- Uniqueness of the solution.- Probability of fixation for Moran's model.- Moran's method for the Wright-Fisher haploid model.- Population subdivision.- The case Ne < N.- Effect of selection on fixation probability.- Kimura's method for Anding the probability of fixation.- Population subdivision reconsidered.- Fluctuating viabilities.- Rate of evolution.- Notes and exercises.- 6 Some notes on continuous approximations.- The problem stated.- Probability of reaching a given allele frequency.- Events at the boundaries.- The problem of representing the initial frequency.- Ad hoc method.- Objections to our ad hoc method.- Dirac's delta function.- Notes and exercises.- 7 Mean sojourn, absorption and fixation times.- Retrospect.- Fundamental equations.- Uniqueness of solutions.- Mean times for Moran's model.- A tentative approach for the Wright-Fisher model.- A generating function for mean sojourn times.- Calculating mean sojourn times for the Wright-Fisher model.- Mean fixation times.- Continuous approximations.- Calculating mean sojourn and absorption times.- Interpreting the results.- Mean sojourn times in the modified process.- Mean fixation time.- Effect of selection.- Mean times under genie selection in the modified process.- Effect of selection: some results.- Calculation of mean times under selection: a numerical example.- Mean fixation time under selection.- Variance of sojourn, absorption and fixation times.- Concerted evolution of multigenefamilies.- Notes and exercises.- 8 Introduction to probability distributions: probability flux.- On the need to calculate probability distributions.- Continuous approximation of the allele frequency.- Approximation of zero and unit frequencies.- Continuous time approximation.- Probability flux.- Components of flux.- Calculating the probability flux.- Probability flux for discontinuous frequency.- Expressions for M and V.- Values of P(x, f) when x takes limiting values.- Notes and exercises.- 9 Stationary distributions: frequency spectra.- Long-term distributions.- Wright's formula for the stationary distribution.- Stationary distributions (case of no selection): mean and variance.- Variation at nucleotide sites: effective number of bases.- Finding the stationary distribution in the case of no selection.- Stationary distribution (continued): allele frequencies zero and unity.- Accuracy of the results.- Nucleotide sites (resumed): probability of monomorphism.- Multiple alleles: infinite alleles model.- Mean number of alleles present.- Mean number of alleles in a given frequency range.- The frequency spectrum.- Infinite sites models.- Frequency spectrum for transposable elements.- Stationary distributions under selection.- Harmful recessives.- Notes and exercises.- 10 Diffusion methods.- Aims: notation.- Forward equation.- Backward equation.- Generality of the backward equation.- Kimura-Ohta equation.- Initial, terminal and boundary conditions.- An alternative approach.- Finding ?(p, x, t) in the neutral, no-mutation case.- Finding v(p, t) and u(p, t) in the neutral, no-mutation case.- Voronka-Keller formulae.- Moments of the distribution.- Fisher's view on the reliability of diffusion methods.- Two-way mutation: no selection.- Selection.- Notes and exercises.- 11General comments and conclusions.- A note on methods.- General summary and conclusions.- References.
1 Introduction.- Notes and exercises.- 2 Wright-Fisher, Moran and other models.- On simple models.- Stochastic models: deterministic models.- Random genetic drift: general approach.- Distribution of allele frequency under drift.- Change of allele frequency in one generation.- The mean in any generation.- Change of allele frequency in one generation: the variance.- Is the binomial distribution of allele number relevant?.- Effective population size.- Variance of allele frequency in any generation.- Inbreeding effective size.- Populations under systematic 'pressure'.- A continuous approximation to the conditional distribution.- Diffusion methods.- Moran's model.- Notes and exercises.- 3 On the description of changes in allele frequency.- A bewildering abundance of descriptions.- Probability distribution of allele frequency.- A numerical example.- Probability of fixation.- Dominant latent root (maximum non-unit eigenvalue).- Modified process.- Probability distribution of absorption time.- Mean absorption time.- Variance of absorption time.- Mean sojourn times.- Variance of sojourn time.- Probability of ever reaching a given frequency.- A matrix representation of mean sojourn time.- Notes and exercises.- 4 Survival of new mutations: branching processes.- On special methods.- Probability of survival.- The need for accuracy.- Are diffusion methods appropriate?.- A fundamental simplification.- Probability generating functions.- Independent propagation: branching processes.- Utility of the branching process approach.- Fundamental equations for branching processes.- Calculating the probability of survival.- The case Ne < N.- A problem in genetic conservation.- Probability of survival when t is fairly large.- Pattern of survival in natural populations.- A caution.- Genieselection.- Probability of ultimate fixation.- Calculating the probability of fixation.- Population subdivision.- Cyclical variation in population size.- A note on epistasis and linkage.- Spread of a fungal pathogen.- Notes and exercises.- 5 Probability of fixation: the more general case.- Fundamental equation.- Uniqueness of the solution.- Probability of fixation for Moran's model.- Moran's method for the Wright-Fisher haploid model.- Population subdivision.- The case Ne < N.- Effect of selection on fixation probability.- Kimura's method for Anding the probability of fixation.- Population subdivision reconsidered.- Fluctuating viabilities.- Rate of evolution.- Notes and exercises.- 6 Some notes on continuous approximations.- The problem stated.- Probability of reaching a given allele frequency.- Events at the boundaries.- The problem of representing the initial frequency.- Ad hoc method.- Objections to our ad hoc method.- Dirac's delta function.- Notes and exercises.- 7 Mean sojourn, absorption and fixation times.- Retrospect.- Fundamental equations.- Uniqueness of solutions.- Mean times for Moran's model.- A tentative approach for the Wright-Fisher model.- A generating function for mean sojourn times.- Calculating mean sojourn times for the Wright-Fisher model.- Mean fixation times.- Continuous approximations.- Calculating mean sojourn and absorption times.- Interpreting the results.- Mean sojourn times in the modified process.- Mean fixation time.- Effect of selection.- Mean times under genie selection in the modified process.- Effect of selection: some results.- Calculation of mean times under selection: a numerical example.- Mean fixation time under selection.- Variance of sojourn, absorption and fixation times.- Concerted evolution of multigenefamilies.- Notes and exercises.- 8 Introduction to probability distributions: probability flux.- On the need to calculate probability distributions.- Continuous approximation of the allele frequency.- Approximation of zero and unit frequencies.- Continuous time approximation.- Probability flux.- Components of flux.- Calculating the probability flux.- Probability flux for discontinuous frequency.- Expressions for M and V.- Values of P(x, f) when x takes limiting values.- Notes and exercises.- 9 Stationary distributions: frequency spectra.- Long-term distributions.- Wright's formula for the stationary distribution.- Stationary distributions (case of no selection): mean and variance.- Variation at nucleotide sites: effective number of bases.- Finding the stationary distribution in the case of no selection.- Stationary distribution (continued): allele frequencies zero and unity.- Accuracy of the results.- Nucleotide sites (resumed): probability of monomorphism.- Multiple alleles: infinite alleles model.- Mean number of alleles present.- Mean number of alleles in a given frequency range.- The frequency spectrum.- Infinite sites models.- Frequency spectrum for transposable elements.- Stationary distributions under selection.- Harmful recessives.- Notes and exercises.- 10 Diffusion methods.- Aims: notation.- Forward equation.- Backward equation.- Generality of the backward equation.- Kimura-Ohta equation.- Initial, terminal and boundary conditions.- An alternative approach.- Finding ?(p, x, t) in the neutral, no-mutation case.- Finding v(p, t) and u(p, t) in the neutral, no-mutation case.- Voronka-Keller formulae.- Moments of the distribution.- Fisher's view on the reliability of diffusion methods.- Two-way mutation: no selection.- Selection.- Notes and exercises.- 11General comments and conclusions.- A note on methods.- General summary and conclusions.- References.
Rezensionen
"The author of this book sets out to accomplish a very difficult task...In my view the author is successful....I recommend the book to all biologists..." (Theoretical Population Genetics)
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