Variation approach to error threshold in generic fitness landscape

Error threshold is the upper bound of the mutation rate such that the genetic information can pass on during reproduction. In Eigen's original proposal, the fitness landscape consists of a single peak along with other background sequences. However, realistic fitness landscapes are far more complicated and thus obstruct deeper understanding from either analytic or numerical approaches. Here we propose a variational approach to derive the error thresholds in generic fitness landscapes. When applying to the Eigen model, it delivers the correct error threshold and reveals the scaling relations between relevant parameters. For the fitness landscape with a peak-mesa profile, the two-dimensional phase diagram in the mesa fitness and the mutation rate can be computed with minimal cost. When the mesa fitness is larger than some critical value, upon increasing the mutation rate, the quasispecies changes from the peak to the mesa and then dissolves into the random background. The phase boundaries can be computed analytically within the effective-fitness approximation. At the end, the universal phase diagram in terms of the rescaled variables is explained in detail.