Quantifying the potential and flux landscapes of multi-locus evolution

Exploration of multi-locus evolution is critically important for understanding evolutionary dynamics. Recombination and epistasis lead to complex evolutionary dynamics. Quantifying the stability and function of such multi-locus evolutionary systems globally is a long-standing challenge for evolutionary biologists. The conventional Wright, Fisher and quasi-linkage equilibrium (QLE) theories can only be applied to highly restricted, simplified and special evolutionary scenarios. In this study, we developed a non-equilibrium potential and flux landscape theory to explore the multi-locus evolution beyond Wright, Fisher and the QLE. We found that the intrinsic potential landscape as a Lyapunov function under the zero noise limit can be used to describe the global stability of a deterministic multi-locus system. We identified and quantified two driving forces responsible for multi-locus evolution: the underlying landscape and the curl flux. We studied the evolution of different cases under recombination, epistasis and mutation. Recombination, frequency-dependent selection and mutation can give rise to non-zero curl flux. In particular, we investigated the dynamics of a simple example of a two-locus system. We explored the underlying potential landscape which shows stable basins attracting the system down to valleys and rotating curl flux, which aids the global communication under recombination, epistasis and mutation. Discontinuities in the first derivative of the non-equilibrium free energy functional shows the non-equilibrium phase transition as the recombination rate increases. We also explored the effect of epistasis on mono-stability and bi-stability evolution. Mutation may drive the system far from equilibrium and be another source of non-zero probability flux. Entropy production rate can quantify energy consumption or dissipation. We explained the origin of the Red Queen hypothesis for endless evolution using the curl flux. Our landscape and flux framework can be applied more generally to multi-locus evolutionary systems experiencing recombination and epistasis.