Coevolution fails to maintain genetic variation in a host–parasite model with constant finite population size
Introduction
There is a rich history of evolutionary theory exploring the conditions under which genetic variation is maintained or depleted in finite populations. The loss of genetic variation through drift is exacerbated, for example, by fluctuations in population size (Crow, 1970 Eq. 7.6.3.34), but variation is maintained through balancing selection in the form of overdominance (Crow, 1970 Eq. 8.6.4) or negative frequency-dependent selection (Takahata and Nei, 1990). First suggested by Haldane (1949), one process that is often posited to maintain genetic variation is coevolution between hosts and their parasites. Coevolution, it is argued, should favour pathogens that are best at infecting the most common host genotype. This in turn should favour the spread of rare host genotypes, a form of negative frequency-dependent selection (NFDS) believed to maintain genetic variation (Clarke, 1979).
Following Haldane’s initial hypotheses, balancing selection as a result of coevolution and/or overdominance was suggested as a mechanism behind the extraordinary genetic diversity found at mammalian Major Histocompatibility Complex (MHC) loci (Bodmer, 1972). These same arguments have been used recently to explain the diversity of anti-microbial peptides in Drosophila (Unckless et al., 2016, Chapman et al., 2019). MHC loci, and other immune defence genes, are notable not only for their high levels of heterozygosity ( alleles across three loci Klein and Figueroa, 1986, Zimmer and Emlen, 2013) but also for the long-term trans-specific persistence of these polymorphisms (Lawlor et al., 1988, Klein, 1987). Using a coalescent approach in a single species Takahata and Nei (1990) found that heterozygote advantage and NFDS are both capable of generating the observed levels of polymorphism. Importantly, however, the model of NFDS they used was not explicitly coevolutionary but rather explored frequency-dependent selection within a single species. Despite the long-standing interest in coevolution as a mechanism maintaining genetic variation, it remains unclear whether NFDS between species in a coevolutionary model is able to maintain more genetic variation in a finite population than expected under neutral processes alone and hence contribute to the excess genetic diversity observed at immune defence genes.
As exemplified by Takahata and Nei (1990), much of the literature on the maintenance of genetic variation alludes to yet blurs the distinction between single-species and coevolutionary NFDS (for example see Tellier et al., 2014, Otto and Michalakis, 1998, Zhao and Waxman, 2016, Llaurens et al., 2017, Ejsmond and Radwan, 2015, Rabajante et al., 2016). By the definition of NFDS in a single-species model (also called direct-NFDS, Brown and Tellier, 2011), the fitness of an allele increases as its frequency declines, which can favour the spread of rare alleles and the maintenance of genetic variation. By contrast, coevolutionary NFDS (also called indirect-NFDS), the sort that commonly arises with host–parasite coevolution, favours alleles of the focal species that correspond to ones that are rare in the interacting species. As noted by Brown and Tellier for the gene-for-gene model (2011), we show that coevolutionary NFDS in the matching-alleles model often has little if any impact on the maintenance of genetic variation relative to neutral drift.
Hints that coevolutionary NFDS does not maintain genetic variation can be found throughout the theoretical literature on matching-alleles models. Simulating coevolution in a population where genetic variation is repeatedly introduced through migration or mutation, Frank, 1991, Frank, 1993 found that the dynamics were dominated by the fixation and loss of genetic variants. He attributed this effect to the repeated population bottlenecks that occur from coevolutionary driven fluctuations in population size, but it was unclear whether this same behaviour would arise in a population of constant size. Similarly, although not explicitly discussed, many individual-based models of coevolution include mutation in either one (Agrawal and Lively, 2002) or both species (Lively, 1999, Borghans et al., 2004, Ejsmond and Radwan, 2015) in order to maintain variation and hence coevolution over the long term. Modelling coevolution in an infinite population, M’Gonigle et al. (2009) found that genetic variation is not maintained at equilibrium except when mutation is very frequent. This is echoed in simulations in finite populations, where in the absence of mutation/migration, allele fixation in either the host or pathogen is very rapid (Gokhale et al., 2013, Schenk et al., 2018). In addition to including mutation, there are several theoretical indications that, rather than being driven by NFDS, the emergent effects of coevolution are dependent on the existence of heterozygote advantage in diploids. For example, M’Gonigle and Otto (2011) showed that the evolution of parasitism depends, not solely on NFDS, but on whether the interaction induces heterozygote advantage, on average. Similarly, Nuismer and Otto (2004) showed that whether there is, on average, heterozygote advantage is the key determinant of how ploidy levels evolve in both hosts and parasites.
Despite this long and extensive history of verbal and theoretical models, it is unclear whether coevolutionary NFDS can indeed maintain genetic variation at a single locus and hence contribute to extensive diversity observed at immune defence loci. Here we compare the maintenance of genetic variation in finite coevolving populations relative to that expected under neutral drift. Previous studies of the MAM in finite populations have focused instead on the relative advantage of host versus parasite (Veller et al., 2017), the number of alleles maintained by mutation (Borghans et al., 2004, Xue and Goldenfeld, 2017), and the time to fixation/loss of alleles in either host or parasite with and without ecological feedbacks (Gokhale et al., 2013, Schenk et al., 2020). We aim to understand the effect of host–parasite interactions on the maintenance of genetic variation, relative to the neutral expectation, by examining a simple single-locus model of coevolution with constant population sizes, where some analytical progress is possible. Metaphorically, we seek to understand when the Red Queen, defined here as coevolutionary maintenance of polymorphism in both hosts and pathogens, collapses because of the loss of polymorphism in one species or the other.
Section snippets
Theoretical background
There are two classic models of coevolution involving a single locus major-effect genes in each species, the gene-for-gene model (GFGM), which was motivated by the genetic architecture of flax-rust interactions (Flor, 1956), and the matching-alleles model (MAM), a form of host–parasite specificity that may arise from lock and key molecular interactions (Dybdahl et al., 2014).
When and how genetic variation is maintained in the GFGM is relatively well understood. In the GFGM, hosts carry either a
The model
We use a continuous-time birth–death model to describe the coevolutionary dynamics between a host and a free-living pathogen, as depicted in Fig. 2. To keep the total host and pathogen population sizes constant at the same fixed value , we use a Moran model design with coupled birth–death events. Extension of the model to unequal host and pathogen population sizes is straightforward and the results do not differ qualitatively from those presented here (MacPherson et al., 2020). Both host and
Discussion
Contrary to theories that posit that host–parasite coevolution and the associated negative frequency-dependent selection (NFDS) should maintain genetic variation (Haldane, 1949, Clarke, 1979, Takahata and Nei, 1990), we use stochastic methods, both describing ensemble moments and conducting individual-based simulations, to show that genetic variation is often lost faster than in the neutral model. Although these results are consistent with stability analyses of the corresponding deterministic
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We thank Matt Pennell, Jonathan Davies, Jeff Joy, Dan Coombs, Aurélien Tellier, and two anonymous reviewers for their many helpful suggestions that improved this manuscript. This project was supported by the Godfrey-Hewitt Mobility Award, the AAUW dissertation fellowship, and a fellowship from the University of British Columbia, Canada to A.M. as well as a Natural Sciences and Engineering Research Council of Canada grant to S.P.O. (RGPIN-2016-03711).
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