Quasispecies theory for multiple-peak fitness landscapes

We use a path integral representation to solve the Eigen and Crow-Kimura molecular evolution models for the case of multiple fitness peaks with arbitrary fitness and degradation functions. In the general case, we find that the solution to these molecular evolution models can be written as the optimum of a fitness function, with constraints enforced by Lagrange multipliers and with a term accounting for the entropy of the spreading population in sequence space. The results for the Eigen model are applied to consider virus or cancer proliferation under the control of drugs or the immune system.

Figure 1

A phase diagram for the interaction of virus and drug, according to the narrow replication advantage model, Eqs. (34, 33). We set $b=3.5$ in the exponential degradation function Eq. (34), and $\gamma =1$. As the drug overlaps more with the virus, a higher viral replication advantage is required for the virus to survive. In the NS phase, the drug eliminates the virus. In the inset is shown the phase diagram for the interaction of an adaptable virus and a drug, according to the flat peak replication advantage model, Eqs. (34, 36). We use $M_0=0.9$ to represent a broad peak for the virus replication rate.

Figure 2

A phase diagram corresponding to the original antigenic sin model is shown. The degradation function (shown in the inset) is chosen to closely reproduce the binding constant behavior in [17], with a limiting degradation rate of $d_0(M_1,M_2=-1)=1$. We use $\gamma =1$. The virus replication advantage required to escape immune system control is a non-monotonic function of the overlap of the vaccine with the virus.

Figure 3

A phase diagram corresponding to the immune control of cancer is shown. We use $\gamma =1$, $B=1$, $M_0=0.9$, and $M_b=0.54$. The cancer replication advantage required to escape immune system control decreases with the overlap of the cancer with the self. Three of the four selective and one of the two nonselective phases are present for the chosen parameters.