Host-parasite arms race in mutation modifications: indefinite escalation despite a heavy load?

If constantly changing genotypes are favorable in host and parasite coevolution, an indefinite escalation of mutation rates would result despite heavy mutational loads. We theoretically study this possibility by examining the mutation modifier dynamics of host and parasite that engage in genotype-specific epidemiological interaction. In the first model, we study the evolutionarily stable (ESS) mutation rate or switching rate if two alleles in a single locus are subjected to frequency-dependent selection favoring the rarer of the two. Mutation modifier locus is either tightly linked or unlinked to the selected locus. Sufficiently strong frequency-dependent selection may cause cycles in allele frequencies and a modifier with higher mutation rate enjoys the long-term advantage by randomizing the genotype of their offspring. Through the repeated events of invasion and replacement of mutation modifiers, the mutation rate continues to increase until the allele frequencies are stabilized. If some fraction of mutations are deleterious, there is no longer a pure ESS mutation rate: the evolutionarily stable population then consists of multiple strains concerning mutation modifier, typically one with a very high mutation rate and the other with a very low rate, stably coexisting and fighting off invasion by any other modifiers. These results are almost independent of the linkage between the selected and the modifier loci. In the second model, we consider the joint evolution of host and parasite mutation modifiers, assuming that a specific pair of host and parasite genotype densities change following the Nicholson-Bailey type model. If there is no cost of deleterious mutations, mutation rates of both species are escalated indefinitely by modifier evolution until they completely suppress the fluctuation of genotype densities. However, a small cost of deleterious mutation is enough to collapse this coevolutionary equilibrium of inflated mutations. Typical coevolutionary outcome is that the parasite mutation rate is accelerated to a high level; whereas the host mutation rate is driven to zero. Extension of our results to host-parasite coevolution of recombination modifier evolution is discussed.