A mathematical model of marine bacteriophage evolution

To explore how particularities of a host cell–virus system, and in particular host cell replication, affect viral evolution, in this paper we formulate a mathematical model of marine bacteriophage evolution. The intrinsic simplicity of real-life phage–bacteria systems, and in particular aquatic systems, for which the assumption of homogeneous mixing is well justified, allows for a reasonably simple model. The model constructed in this paper is based upon the Beretta–Kuang model of bacteria–phage interaction in an aquatic environment (Beretta & Kuang 1998 Math. Biosci. 149, 57–76. (doi:10.1016/S0025-5564(97)10015-3)). Compared to the original Beretta–Kuang model, the model assumes the existence of a multitude of viral variants which correspond to continuously distributed phenotypes. It is noteworthy that the model is mechanistic (at least as far as the Beretta–Kuang model is mechanistic). Moreover, this model does not include any explicit law or mechanism of evolution; instead it is assumed, in agreement with the principles of Darwinian evolution, that evolution in this system can occur as a result of random mutations and natural selection. Simulations with a simplistic linear fitness landscape (which is chosen for the convenience of demonstration only and is not related to any real-life system) show that a pulse-type travelling wave moving towards increasing Darwinian fitness appears in the phenotype space. This implies that the overall fitness of a viral quasi-species steadily increases with time. That is, the simulations demonstrate that for an uneven fitness landscape random mutations combined with a mechanism of natural selection (for this particular system this is given by the conspecific competition for the resource) lead to the Darwinian evolution. It is noteworthy that in this system the speed of propagation of this wave (and hence the rate of evolution) is not constant but varies, depending on the current viral fitness and the abundance of susceptible bacteria. A specific feature of the original Beretta–Kuang model is that this model exhibits a supercritical Hopf bifurcation, leading to the loss of stability and the rise of self-sustained oscillations in the system. This phenomenon corresponds to the paradox of enrichment in the system. It is remarkable that under the conditions that ensure the bifurcation in the Beretta-Kuang model, the viral evolution model formulated in this paper also exhibits a rise in self-sustained oscillations of the abundance of all interacting populations. The propagation of the travelling wave, however, remains stable under these conditions. The only visible impact of the oscillations on viral evolution is a lower speed of the evolution.