A Variational Modeling Framework for Population Genetic Dynamics

Population genetic dynamics are shaped by multiple evolutionary processes, including mutation, recombination, selection, and genetic drift. With the rapid growth of genomic data, modeling multilocus evolutionary dynamics and the resulting patterns of genetic variation has become increasingly important. Jointly modeling multiple evolutionary processes across multiple loci remains challenging, motivating the development of more flexible and extensible frameworks. Here, we develop a variational framework for population genetic dynamics based on generalized gradient-flow theory, drawing on nonequilibrium thermodynamics. In our model, mutation, recombination, and selection are represented as modular variational components, connected through a gamete-individual two-state system with active coupling. Mutation and recombination share a gamete-state free energy but use distinct dissipation potentials, while selection uses the established Shahshahani gradient structure at the individual stage. Our model represents multilocus genetic variation directly in the full haplotype-frequency space, recovers classical mutation, recombination, and selection dynamics in the corresponding limits, and provides a phenomenological stochastic extension for finite-population genetic drift. Numerical comparisons with SLiM forward simulations support consistency with allele-frequency, haplotype-frequency, and linkage-disequilibrium dynamics in the tested systems. This variational formulation provides a new perspective on modeling evolutionary dynamics and has potential for extension to more complex multilocus systems and applications in population genomics analysis.