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Register a new userA comparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environments
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Added by
Aleksandra Arda\\v{s}eva
Publication date
2020
fields
Biology
and research's language is
English
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Deterministic continuum models formulated in terms of non-local partial differential equations for the evolutionary dynamics of populations structured by phenotypic traits have been used recently to address open questions concerning the adaptation of asexual species to periodically fluctuating environmental conditions. These deterministic continuum models are usually defined on the basis of population-scale phenomenological assumptions and cannot capture adaptive phenomena that are driven by stochastic variability in the evolutionary paths of single individuals. In this paper, we develop a stochastic individual-based model for the coevolution between two competing phenotype-structured cell populations that are exposed to time-varying nutrient levels and undergo spontaneous, heritable phenotypic variations with different probabilities. The evolution of every cell is described by a set of rules that result in a discrete-time branching random walk on the space of phenotypic states. We formally show that the deterministic continuum counterpart of this model comprises a system of non-local partial differential equations for the cell population density functions coupled with an ordinary differential equation for the nutrient concentration. We compare the individual-based model and its continuum analogue, focussing on scenarios whereby the predictions of the two models differ. Our results clarify the conditions under which significant differences between the two models can emerge due to stochastic effects associated with small population levels. These differences arise in the presence of low probabilities of phenotypic variation, and become more apparent when the two populations are characterised by less fit initial mean phenotypes and smaller initial levels of phenotypic heterogeneity.
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We derive the full kinetic equations describing the evolution of the probability density distribution for a structured population such as cells distributed according to their ages and sizes. The kinetic equations for such a sizer-timer model incorporates both demographic and individual cell growth rate stochasticities. Averages taken over the densities obeying the kinetic equations can be used to generate a second order PDE that incorporates the growth rate stochasticity. On the other hand, marginalizing over the densities yields a modified birth-death process that shows how age and size influence demographic stochasticity. Our kinetic framework is thus a more complete model that subsumes both the deterministic PDE and birth-death master equation representations for structured populations.
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Microbial populations often have complex spatial structures, with homogeneous competition holding only at a local scale. Population structure can strongly impact evolution, in particular by affecting the fixation probability of mutants. Here, we propose a model of structured microbial populations on graphs, where each node of the graph contains a well-mixed deme whose size can fluctuate, and where migrations are independent from birth and death events. We study analytically and numerically the mutant fixation probabilities in different structures, in the rare migration regime. In particular, we demonstrate that the star graph continuously transitions between amplifying and suppressing natural selection as migration rate asymmetry is varied. This elucidates the apparent paradox in existing constant-size models on graphs, where the star is an amplifier or a suppressor depending on the details of the dynamics or update rule chosen, e.g. whether each birth event precedes or follows a death event. The celebrated amplification property of the star graph for large populations is preserved in our model, for specific migration asymmetry. We further demonstrate a general mapping between our model and constant-size models on graphs, under a constraint on migration rates, which directly stems from assuming constant size. By lifting this constraint, our model reconciles and generalizes previous results, showing that migration rate asymmetry is key to determining whether a given population structure amplifies or suppresses natural selection.
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