Analysis of transcription-factor binding-site evolution by using the Hamilton-Jacobi equations

We investigate a quasi-species mutation-selection model of transcription-factor binding-site evolution. By considering the mesa and the crater fitness landscapes designed to describe these binding sites and point mutations, we derive an evolution equation for the population distribution of binding sequences. In the long-length limit, the evolution equation is replaced by a Hamilton-Jacobi equation which we solve for the stationary state solution. From the stationary solution, we derive the population distributions and find that an error threshold, separating populations in which the binding site does or does not evolve, only exists for certain values of the fitness parameters. A phase diagram in this parameter space is derived and shows a critical line below which no error threshold exists. We also investigate the evolution of multiple binding sites for the same transcription factor. For two binding sites, we perform an analysis similar to that for a single site and determine a phase diagram showing different phases with both, one, or no binding sites selected. In the phase diagram, the phase boundary between the one-or-two selected site phases is qualitatively different for the mesa and the crater fitness landscapes. As fitness benefits for a second bound transcription factor tend to zero, the minimum mutation rate at which the two-site phase occurs diverges in the mesa landscape whereas the mutation rate at the phase boundary tends to a finite value for the crater landscape.