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| Wright-Fisher model | |
|---|---|
| Name | Wright–Fisher model |
| Field | Population genetics |
| Introduced | 1931 |
| Authors | Sewall Wright; Ronald A. Fisher |
Wright-Fisher model The Wright–Fisher model is a foundational stochastic model in population genetics describing allele frequency dynamics in a finite, diploid population with nonoverlapping generations. Developed in the early 20th century by Sewall Wright and Ronald Fisher, the model formalizes genetic drift and provides a baseline against which selection, mutation, and migration processes are compared. It underpins theoretical work by later figures such as Motoo Kimura, John Maynard Smith, J. B. S. Haldane, and informs modern computational methods by groups at institutions like Cold Spring Harbor Laboratory, University of Chicago, and University of Cambridge.
The model represents a population of constant size N where alleles are sampled each generation according to multinomial sampling influenced by forces studied by Sewall Wright and Ronald Fisher. It contrasts with the deterministic models developed in the 19th century by figures like Gregor Mendel and later theoreticians such as Thomas Hunt Morgan and Hermann Joseph Muller. The Wright–Fisher framework motivated key theoretical advances by Motoo Kimura on neutral theory, by George R. Price on covariance, and by William Hamilton in kin selection contexts.
The canonical formulation considers 2N gene copies in a hermaphroditic or dioecious population, with allele counts evolving via binomial or multinomial sampling. Parameters include population size N, mutation rates as in Kimura's models, selection coefficients reminiscent of work by J. B. S. Haldane and John Maynard Smith, and migration terms comparable to treatments by Wright in his island model. The model is often presented in discrete time with transition probabilities given by the binomial distribution and is tied to limiting diffusion approximations developed by Motoo Kimura and Feller.
Key mathematical results include fixation probabilities, time to fixation or loss, and sampling distributions. Fixation probabilities under neutrality follow from martingale properties and were elaborated by Kimura; with selection, approximate formulas trace to diffusion theory by Kolmogorov and Feller. The model connects to coalescent theory introduced by John Kingman, which gives genealogical duality and links to genealogical processes studied by Hudson and Tavaré. Spectral decompositions, eigenfunctions, and orthogonal polynomial techniques used in analysis echo methods from Sturm–Liouville theory and contributions by S. Karlin and H. Taylor.
Extensions include overlapping generations, age-structured models by researchers at Imperial College London and University of Oxford, metapopulation formulations akin to Wright's island model, and models incorporating recombination as in work by Felsenstein and Bruce Walsh. Continuous-time limits yield the Moran model and diffusion processes treated by Ito calculus; spatial extensions relate to stepping-stone models studied by Kimura and Weiss. Polygenic selection frameworks and models with dominance, epistasis, and linkage disequilibrium draw on methods from Sewall Wright's adaptive landscape and quantitative genetics literature by Ronald Fisher and Lande.
The model serves as a null hypothesis for tests of neutrality used in studies by groups at Max Planck Institute for Evolutionary Anthropology, Wellcome Sanger Institute, Broad Institute, and field teams working on organisms such as Drosophila melanogaster, Homo sapiens, Arabidopsis thaliana, and E. coli. It underlies inference methods for demographic history used in analyses by Li and Durbin style pairwise sequential Markovian coalescent, site frequency spectrum approaches by T. L. Marth and Gutenkunst, and likelihood frameworks employed by Peter Donnelly and Jonathan K. Pritchard. Conservation genetics applications appear in work on endangered taxa by ICUN committees and field programs like those at San Diego Zoo and Smithsonian Institution.
Simulations implement Wright–Fisher dynamics in software packages developed by communities around Bioconductor, PLINK, SLiM, msprime, and fastsimcoal. Forward-time simulators by groups at University of British Columbia and University College London enable modeling selection, recombination, and demography; coalescent simulators by Hudson and optimized engines like msprime exploit Kingman duality. Numerical methods for diffusion approximations use finite-difference schemes and spectral methods favored in computational work at Los Alamos National Laboratory and academic centers such as Princeton University and Stanford University.
Critiques emphasize unrealistic assumptions: constant population size unlike demography studied by Lewontin and E. O. Wilson; nonoverlapping generations unlike many natural populations documented by David Lack and G. Evelyn Hutchinson; panmixia versus population structure analyzed by Wright and Slatkin; and the omission of ecological interactions highlighted by Robert MacArthur and Edward O. Wilson. Empiricalists at Harvard University, Yale University, and University of California, Berkeley have demonstrated departures from model assumptions in genomic datasets, motivating richer models and careful model checking.
Category:Population genetics models