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Error estimate for the diffusion approximation of the Wright--Fisher model.

S N Ethier, M F Norman

    Proceedings of the National Academy of Sciences of the United States of America
    |November 1, 1977
    PubMed
    Summary
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    Diffusion approximations of the two-locus Wright-Fisher model.

    Journal of mathematical biology·1989

    The Wright-Fisher model and its diffusion approximation differ in expected gene frequency values. This difference, without selection, depends on mutation rate and population size, offering insights into population genetics models.

    Area of Science:

    • Population Genetics
    • Mathematical Biology
    • Evolutionary Dynamics

    Background:

    • The Wright-Fisher model is a fundamental model in population genetics.
    • Diffusion approximations are often used to simplify complex genetic models.
    • Understanding the accuracy of these approximations is crucial for genetic research.

    Purpose of the Study:

    • To compare the Wright-Fisher model with its diffusion approximation.
    • To analyze the expected value of a function of gene frequency in both models.
    • To quantify the difference between the two models under specific conditions.

    Main Methods:

    • Mathematical comparison of the Wright-Fisher model and its diffusion approximation.
    • Analysis of the expected value of an arbitrary smooth function of nth-generation gene frequency.

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  • Derivation of the difference between the two models in the absence of selection.

    • Main Results:

    • The expected value of a smooth function of gene frequency differs between the two models.
    • In the absence of selection, this difference is bounded.
    • The difference is a linear combination of mutation rate and reciprocal population size.

    Conclusions:

    • The diffusion approximation introduces a quantifiable error compared to the Wright-Fisher model.
    • The magnitude of this error is influenced by mutation rate and population size.
    • This finding is important for selecting appropriate models in population genetics studies.