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A finite locus effect diffusion model for the evolution of a quantitative trait.

J R Miller¹, M C Pugh, M B Hamilton

¹Department of Mathematics, Georgetown University, Washington, DC 20057-1233, USA. jrm32@georgetown.edu

Journal of Mathematical Biology

|February 8, 2006

Summary

This study models quantitative trait evolution under selection in large populations. It finds that trait response to selection is largely unaffected by variations in genetic effect sizes when mutation is minimal.

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Area of Science:

Population Genetics
Quantitative Genetics
Evolutionary Biology

Background:

Additive quantitative traits are influenced by multiple genetic loci.
Understanding the evolutionary response to selection requires modeling allele frequencies and effect sizes.
Previous models often simplify the distribution of genetic effects.

Purpose of the Study:

To develop a diffusion model for the joint distribution of locus effects and allele frequencies.
To approximate a discrete model in the limit of infinite population size and loci.
To analyze the long-time response of quantitative traits to selection.

Main Methods:

Construction of a diffusion model for joint distributions.
Application of multiple-timescale asymptotics for constant effect sizes.

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Numerical solutions for two distinct effect sizes.

Main Results:

The diffusion model accurately approximates discrete models under specific limits.
Asymptotic analysis predicts long-time trait mean response to selection.
Numerical results show trait response is insensitive to effect size variability when mutation is negligible.

Conclusions:

The developed diffusion model provides a powerful tool for studying quantitative trait evolution.
Variability in genetic effect sizes has minimal impact on selection response in large populations without significant mutation.
This work advances theoretical understanding of adaptation and evolution.