Phase Diagrams of Quasispecies Theory with Recombination and Horizontal Gene Transfer

We consider how transfer of genetic information between individuals influences the phase diagram and mean fitness of both the Eigen and the parallel, or Crow-Kimura, models of evolution. In the absence of genetic transfer, these physical models of evolution consider the replication and point mutation of the genomes of independent individuals in a large population. A phase transition occurs, such that below a critical mutation rate an identifiable quasispecies forms. We show how transfer of genetic information changes the phase diagram and mean fitness and introduces metastability in quasispecies theory, via an analytic field theoretic mapping.

Figure 1

The selected phase, in which a finite fraction of the population has a nonzero magnetization, is shown for the parallel model with recombination. The rate of genetic exchange is $\nu$, and the rate of mutation is $\mu$. The phase diagram is shown for the replication rate $f(u)=A{\delta }_{u,1}$ (solid line, with dotted line the metastable limit) and the replication rate $f(u)=k{u}^2/2$ (dashed line). Also shown is the phase diagram for the Eigen model for $f(u)=(A-A_0){\delta }_{u,1}+A_0$ (solid line, with dotted line the metastable limit) and $f(u)=k{u}^2/2+k_0$ (long-dashed line) [ordinate coordinates in brackets].