Geometric phase in the Crow-Kimura model of molecular evolution on dynamic environments

When taking place under fluctuating environments, some classical results of evolutionary dynamics in fitness landscapes need to be reconsidered. Under such nonequilibrium conditions, the properties of adaptive evolution might escape from the expectations grounded in equilibrium systems. Here, an important contribution to this nonequilibrium dynamics results from the presence of a geometry (Berry) phase in the Crow-Kimura model of molecular evolution with asymmetric mutations. By considering changes in fitness alone as well as changes in both fitness and mutation, analytical expressions for the Berry phase are derived, showing strong singularities at bulk transition points. Periodically varying parameters are also analyed for the two-dimensional case. The potential implications for evolutionary and prebiotic scenarios are discussed.