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Variational principles in evolution

N Behera1

  • 1Development Biology and Genetics Laboratory, Indian Institute of Science, Bangalore, India. behera@dbgl.iisc.ernet.in

Bulletin of Mathematical Biology
|January 1, 1996
PubMed
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This study extends variational principles for population genetics to multiple loci. While simple cases allow extension, complex models with linkage and epistasis present challenges, requiring a Riemannian geometry approach.

Area of Science:

  • Population Genetics
  • Mathematical Biology
  • Evolutionary Dynamics

Background:

  • Svirezhev's integral variational principle for one-locus selection models.
  • Lagrangian formulation for population undergoing selection.
  • Need for extending principles to multi-locus genetic systems.

Purpose of the Study:

  • To investigate the extension of variational principles to multi-locus selection models.
  • To explore the applicability of the Lagrangian approach in complex genetic systems.
  • To develop a general method for constructing population trajectories in metric spaces.

Main Methods:

  • Generalization of the one-locus Lagrangian to multiple loci.
  • Formulation of population trajectories as steepest ascent paths in Riemannian metric spaces.

Related Experiment Videos

  • Determination of metric tensors and surfaces for gene frequency variations.

    • Main Results:

    • The variational principle can be extended to multiple loci in simple cases.
    • Extension is challenging with linkage and epistasis in two-locus or more general models.
    • Population trajectories lie on specific surfaces within Riemannian metric spaces, adhering to a local optimality principle.

    Conclusions:

    • A general method using Riemannian geometry is established for multi-locus population dynamics.
    • The rate of change of mean fitness is maximized along these trajectories.
    • In specific cases (zero linkage disequilibria), trajectories move on product spaces of spheres; for two loci, this surface is a hyper-torus.