Comparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environments

Aleksandra Ardaševa, Alexander R. A. Anderson, Robert A. Gatenby, Helen M. Byrne, Philip K. Maini, and Tommaso Lorenzi
Phys. Rev. E 102, 042404 – Published 8 October 2020

Abstract

Deterministic continuum models formulated as nonlocal 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 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 light of these considerations, in this paper we develop a stochastic individual-based model for the coevolution of two competing phenotype-structured cell populations that are exposed to time-varying nutrient levels and undergo spontaneous, heritable phenotypic changes with different probabilities. Here, 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, and nutrient levels are governed by a difference equation in which a sink term models nutrient consumption by the cells. We formally show that the deterministic continuum counterpart of this model comprises a system of nonlocal 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 analog, focusing on scenarios whereby the predictions of the two models differ. The results obtained clarify the conditions under which significant differences between the two models can emerge due to bottleneck effects that bring about both lower regularity of the density functions of the two populations and more pronounced demographic stochasticity. In particular, bottleneck effects emerge in the presence of lower probabilities of phenotypic variation and are more apparent when the two populations are characterized by lower fitness initial mean phenotypes and smaller initial levels of phenotypic heterogeneity. The emergence of these effects, and thus the agreement between the two modeling approaches, is also dependent on the initial proportions of the two populations. As an illustrative example, we demonstrate the implications of these results in the context of the mathematical modeling of the early stage of metastatic colonization of distant organs.

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  • Received 2 April 2020
  • Revised 17 August 2020
  • Accepted 14 September 2020

DOI:https://doi.org/10.1103/PhysRevE.102.042404

©2020 American Physical Society

Physics Subject Headings (PhySH)

Physics of Living Systems

Authors & Affiliations

Aleksandra Ardaševa1,*, Alexander R. A. Anderson2, Robert A. Gatenby2, Helen M. Byrne1, Philip K. Maini1, and Tommaso Lorenzi3,†

  • 1Wolfson Centre for Mathematical Biology, University of Oxford, Oxford OX2 6GG, United Kingdom
  • 2Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center, Tampa, Florida 33612, USA
  • 3Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, Italy

  • *aleksandra.ardaseva@maths.ox.ac.uk
  • tommaso.lorenzi@polito.it

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Vol. 102, Iss. 4 — October 2020

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