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. Existing approaches often face challenges in jointly describing multiple evolutionary processes and modeling multilocus systems, 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, with different life-cycle stages connected through a gamete-individual two-state system. Mutation and recombination act on the gamete state, selection acts on the individual state, and finite-population genetic drift is represented by a stochastic extension. 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 natural stochastic extension for finite populations. Numerical experiments and SLiM forward simulations show that our model captures the dynamics of allele frequencies, haplotype frequencies, and linkage disequilibrium in multilocus systems. The variational formulation opens avenues for future extensions to additional evolutionary processes and more complex multilocus systems, as well as for developing differentiable computational methods for gradient-based parameter inference and scalable genomic modeling.