The polymorphic evolution sequence for populations with phenotypic plasticity

Abstract

In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth-and death rates as well as the competition kernel, c(x, y) which describes the induced death rate that an individual of type x experiences due to the presence of an individual or type y. When a new individual is born, with a small probability a mutation occurs, i.e. the offspring has different genotype as the parent. The novel aspect of the models we study is that an individual with a given genotype may express a certain set of different phenotypes, and during its lifetime it may switch between different phenotypes, with rates that are much larger then the mutation rates and that, moreover, may depend on the state of the entire population. The evolution of the population is described by a continuous-time, measure-valued Markov process. In [4], such a model was proposed to describe tumor evolution under immunotherapy. In the present paper we consider a large class of models which comprises the example studied in [4] and analyse their scaling limits as the population size tends to infinity and the mutation rate tends to zero. Under suitable assumptions, we prove convergence to a Markov jump process that is a generalisation of the polymorphic evolution sequence (PES) as analysed in [8,10].

Experimental and stochastic models of melanoma T-cell therapy define impact of subclone fitness on selection of antigen loss variants

“…To study the evolution of melanomas under the improved ACT METi therapy, we use an individualbased continuous-time Markov process that is an extension of the model in (Baar et al, 2016). This model itself is based on an individual-based model of adaptive dynamics, introduced in Pacala, 1997, 1999;Dieckmann and Law, 1996;Law and Dieckmann, 1999;Fournier and Méléard, 2004) and further developed over the last years (Champagnat, 2006;Champagnat and Méléard, 2011;Champagnat et al, 2008), in particular for the scenario of phenotypic switching (Baar and Bovier, 2018). The tumor is composed of a finite number of different cells and cytokines.…”

Individual-based models under various time-scales

“…I refer to [18] for the exact definition of the sets Q α,β 1/2 (R), where the value R > 0 defines a bound on the Hölder regularity of respectively order α in x and β in y. The "1/2" refers to the property (1). Given such a regularity, the authors propose to adjust the order of the estimation kernel and explain how to choose the associated window sizes given |U n | (for some large n ≥ 1).…”

“…• It is not difficult to generalize the model to include frequency dependent effects of selection at individual level. We would then replace s > 0 by some smooth function (s(x)) x∈ [0,1] . Generalizations of σ as a function are also not a mathematical issue.…”

“…A large family of extensions has emerged, notably to include coexistence of traits (cf. [11], [1]), or more complicated interactions, for instance with horizontal transfer [6] or aging [21]. Since they deal with two additional time-scales, we rather focus on [7], where events of invasion affect the dynamics of some marker,…”

A Stochastic Adaptive Dynamics Model for Bacterial Populations with Mutation, Dormancy and Transfer

“…The stochastic individual based modelling and analysis of the dynamics and evolution of bacterial populations has attracted significant interest in recent years (see e.g. [Cha06, FM04, BCF + 16, BCF + 18, LFL17,BB18]). This can on the one hand be motivated externally by the relevance of bacterial population dynamics in biology, medicine and industry, and on the other hand internally by the presence of interesting and distinctive features which invite new modelling approaches and lead to new patterns and results.…”