Populations facing a nonlinear environmental gradient: Steady states and pulsating fronts

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location [Formula: see text], an individual must have a trait close to some optimal trait [Formula: see text]. Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with [Formula: see text], [Formula: see text]. We construct steady states solutions and, when [Formula: see text] is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable [Formula: see text] we take advantage of a Hilbert basis of [Formula: see text] made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable [Formula: see text] we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.

Discrete and continuum phenotype-structured models for the evolution of cancer cell populations under chemotherapy

We present a stochastic individual-based model for the phenotypic evolution of cancer cell populations under chemotherapy. In particular, we consider the case of combination cancer therapy whereby a chemotherapeutic agent is administered as the primary treatment and an epigenetic drug is used as an adjuvant treatment. The cell population is structured by the expression level of a gene that controls cell proliferation and chemoresistance. In order to obtain an analytical description of evolutionary dynamics, we formally derive a deterministic continuum counterpart of this discrete model, which is given by a nonlocal parabolic equation for the cell population density function. Integrating computational simulations of the individual-based model with analysis of the corresponding continuum model, we perform a complete exploration of the model parameter space. We show that harsher environmental conditions and higher probabilities of spontaneous epimutation can lead to more effective chemotherapy, and we demonstrate the existence of an inverse relationship between the efficacy of the epigenetic drug and the probability of spontaneous epimutation. Taken together, the outcomes of the model provide theoretical ground for the development of anticancer protocols that use lower concentrations of chemotherapeutic agents in combination with epigenetic drugs capable of promoting the re-expression of epigenetically regulated genes.

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

We consider a phenotype-structured model of evolutionary dynamics in a population of cancer cells exposed to the action of a cytotoxic drug. The model consists of a nonlocal parabolic equation governing the evolution of the cell population density function. We develop a novel method for constructing exact solutions to the model equation, which allows for a systematic investigation of the way in which the size and the phenotypic composition of the cell population change in response to variations of the drug dose and other evolutionary parameters. Moreover, we address numerical optimal control for a calibrated version of the model based on biological data from the existing literature, in order to identify the drug delivery schedule that makes it possible to minimise either the population size at the end of the treatment or the average population size during the course of treatment. The results obtained challenge the notion that traditional high-dose therapy represents a “one-fits-all solution” in anticancer therapy by showing that the continuous administration of a relatively low dose of the cytotoxic drug performs more closely to i.e. the optimal dosing regimen to minimise the average size of the cancer cell population during the course of treatment.

Dynamics of adaptation in an anisotropic phenotype-fitness landscape

We study the dynamics of adaptation of a large asexual population in a n-dimensional phenotypic space, under anisotropic mutation and selection effects. When n = 1 or under isotropy assumptions, the ‘replicator-mutator’ equation is a standard model to describe these dynamics. However, the n-dimensional anisotropic case remained largely unexplored.We prove here that the equation admits a unique solution, which is interpreted as the phenotype distribution, and we propose a new and general framework to the study of the quantitative behavior of this solution. Our method builds upon a degenerate nonlocal parabolic equation satisfied by the distribution of the ‘fitness components’, and a nonlocal transport equation satisfied by the cumulant generating function of the joint distribution of these components. This last equation can be solved analytically and we then get a general formula for the trajectory of the mean fitness and all higher cumulants of the fitness distribution, over time. Such mean fitness trajectory is the typical outcome of empirical studies of adaptation by experimental evolution, and can thus be compared to empirical data.In sharp contrast with the known results based on isotropic models, our results show that the trajectory of mean fitness may exhibit (n − 1) plateaus before it converges. It may thus appear ‘non-saturating’ for a transient but possibly long time, even though a phenotypic optimum exists. To illustrate the empirical relevance of these results, we show that the anisotropic model leads to a very good fit of Escherichia coli long-term evolution experiment, one of the most famous experimental dataset in experimental evolution. The two ‘evolutionary epochs’ that have been observed in this experiment have long puzzled the community: we propose that the pattern may simply stem form a climbing hill process, but in an anisotropic fitness landscape.