Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences

Abstract

Abstract. Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logical testing by proof-of-concept mathematical models. Here, we present a stochastic agent-based model and a related deterministic integrodifferential equation model for the evolution of a phenotype-structured population composed of asexually-reproducing and competing organisms which are exposed to novel environmental conditions. This setting has relevance to the study of biological systems where colonising asexual populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene expression can change the reproductive capacity of individuals within the population in the new environment. Simulations and analyses of our models clarify the conditions under which certain evolutionary paths are possible, and illustrate that whilst epigenetic mechanisms may facilitate adaptation in asexual species faced with environmental change, they can also lead to a type of "epigenetic load" and contribute to extinction. Moreover, our results offer a formal basis for the claim that constant environments favour individuals with low rates of stochastic phenotypic variation. Finally, our model provides a "proof of concept" of the verbal hypothesis that phenotypic stability is a key driver in rescuing the adaptive potential of an asexual lineage, and supports the notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic instability and biased epimutations, and steer an asexual population back from the brink of an evolutionary dead end.

Replicator-mutator equations with quadratic fitness

“…First, very recently, replicator-mutator equations (or related problems) with quadratic fitness have attracted a lot of attention: let us mention the works [10], [3], [5], [11], [9], [18] and the references therein. In particular, Chisholm et al [5] study -among other things -the long time behaviour of the nonlocal term of an equation very close to (1.1), in the case f (x) = −x 2 , with compactly supported initial data. In Section 2 we completely solve (1.1) for any initial data, and can therefore study the long time behaviour not only of the nonlocal term f (t) but also of the full profile u(t, x).…”

Dynamical Patterns of Coexisting Strategies in a Hybrid Discrete-continuum Spatial Evolutionary Game Model

“…In conclusion, we would like to remark that experience in a variety of contexts has demonstrated the value of relating individual-based stochastic models to deterministic continuum models [5,7,8,10,31,32,38], which make it possible to complement numerical simulations with rigorous analytical results, to achieve conclusions with broad structural stability under parameter changes. In this regard, a fundamental problem is what method of proof can be employed to derive deterministic mesoscopic models from the type of stochastic individual-based models which are defined in our framework.…”

Evolution of cancer cell populations under cytotoxic therapy and treatment optimisation: insight from a phenotype-structured model

“…which can be obtained from stochastic individual-based models that track the dynamics of single cells via meanfield approximation [12], through probabilistic methods in the limit of large cell numbers [9,10] or by using the principle of mass conservation [63]. In equation (2.2), the diffusion term models the effects of spontaneous epimutations, which occur at rate β ∈ R >0 [3,46].…”

“…The parameter d ∈ R >0 models the rate of cell death due to intrapopulation competition. Based on the ideas proposed by Chisholm et al [12] and Lorenzi et al [38,37], we define the functions p and k as…”