Exactly solvable dynamics of the Eigen and the Crow-Kimura models

Abstract

We introduce a new way to study molecular evolution within well-established Hamilton-Jacobi formalism, showing that for a broad class of fitness landscapes it is possible to derive dynamics analytically within the $1/N$-accuracy, where $N$ is genome length. For smooth and monotonic fitness function this approach gives two dynamical phases: smooth dynamics, and discontinuous dynamics. The latter phase arises naturally with no explicit singular fitness function, counter-intuitively. The Hamilton-Jacobi method yields straightforward analytical results for the models that utilize fitness as a function of Hamming distance from a reference genome sequence. We also show the way in which this method gives dynamical phase structure for multi-peak fitness.