Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments
Introduction
Evolution can be thought of as a complex and dynamic interplay between hereditary phenotypic modifications, environmental change and natural selection. In this framework, it is largely an open question in evolutionary biology how individuals and populations adapt to fluctuating environments.
Previous theoretical and experimental work involving asexual populations has shed some light on the way phenotypic diversity can evolve in the presence of environmental fluctuations (Acar et al., 2008, Avery, 2006, Casadesus and Low, 2006, Dubnau and Losick, 2006, Gander et al., 2007, Kussell et al., 2005, Kussell and Leibler, 2005, Lachmann and Jablonka, 1996, Thattai and van Oudenaarden, 2004, Wolf et al., 2005). With the aim of dissecting the relative contributions of phenotypic variation, environmental oscillations and natural selection as drivers of phenotypic adaptation, here we formulate and analyse an integro-differential model of adaptive dynamics in a phenotype-structured population embedded in a changing environment. Models of this type can be derived from stochastic individual-based models in the limit of large numbers of individuals (Champagnat et al., 2006, Champagnat et al., 2001), and they have been proven to constitute a suitable conceptual apparatus to study evolutionary processes in population dynamics (Bouin and Calvez, 2014, Bouin et al., 2012, Chisholm et al., 2015a, Delitala et al., 2013, Delitala and Lorenzi, 2012, Lavi et al., 2014, Lorz et al., 2013, Stiehl et al., 2014).
We focus on the ecological scenario where a population has a fitness landscape with one single peak, the location of which undergoes periodic oscillations in time. Due to random epimutation events (which change the way genes are expressed), individuals within this population undergo stochastic variation in phenotype (Brock et al., 2009, Sharma et al., 2010, Gupta et al., 2011, Pisco et al., 2013). We assume that small (large) epimutations correspond to small (large) phenotypic changes, and noting that small epimutations occur at a much higher frequency than large epimutations (Becker et al., 2011), we model the effects of heritable variations in gene expression by means of a diffusion operator, along the lines of Lorz et al. (2011), Mirrahimi et al. (2015), and Perthame and Barles (2008). Moreover, in order to take into account the fact that epimutations can be inherently biased towards particular variants (Arthur and Farrow, 1999, Donoghue and Ree, 2000, Laland et al., 2014, Wallace, 2002), we follow the modelling strategy presented in Chisholm et al., 2015a, Chisholm et al., 2015b and include a drift operator in our model.
From the mathematical point of view, our work follows earlier papers on the analysis of integro-differential equations that arise in models of adaptive evolution of phenotype-structured populations (Lorz et al., 2011, Mirrahimi et al., 2015, Perthame and Barles, 2008, Calsina et al., 2013, Desvillettes et al., 2008, Raoul, 2011). These papers are devoted to the study of solutions of such equations when the rate of diffusion across the phenotypic space is small or tends to zero. The main novelty of our work is that we do not impose any smallness assumptions on the diffusion rate. We also allow the presence of a drift term in the governing equation. In this setting, we are able to establish the existence of periodic solutions with a Gaussian profile, without any specific assumptions concerning the nature of the periodic variation in the trait associated with the maximum of the fitness landscape.
Exploiting the analytical tractability of the model, we perform a systematic investigation of the ways in which the presence of a time-varying environment, the evolution of the epigenetic state, the level of phenotypic diversity and the size of the population are shaped by the rate of epimutations, the degree of bias in the generation of novel phenotypic variants, the strength of natural selection, and the frequency of environmental oscillations. The generality of this model makes the results of our study applicable to a broad range of asexual populations evolving in fluctuating environments.
Section snippets
The model
We study evolutionary dynamics in a well-mixed population that is structured by a phenotypic trait . Individuals inside the population proliferate through asexual reproduction, die due to competition for limited resources, and undergo epimutations. To reduce biological complexity to its essence, we make the prima facie assumption that stochastic variations in gene expression yield infinitesimally small phenotypic modifications. Moreover, we let the environment evolve independently of the
Analysis of the model
Subject to a single condition below [the inequality (3.5)], there is a solution of the problem (2.1), (2.2), (2.3), (2.4), where
- (i)
is periodic with period T;
- (ii)
has a Gaussian profile,where and are periodic (this ensures that the mean phenotype is also the most prevalent one);
- (iii)
the instantaneous most prevalent phenotype is
Numerical solutions
In order to illustrate the analytical results established in the previous section, here we present the results of numerical solutions for the mathematical problem (2.1), (2.2), (2.3), (2.4) with and Further technical details of the numerical solution method are provided in Appendix B, but we mention here several important general points. The model that we have analysed in full mathematical detail is defined with , but the finite-interval numerical
Discussion and conclusions
Recently, Serviedio and co-workers observed that an important purpose of mathematical models in evolutionary research is “to act as ‘proof-of-concept׳ tests of the logic in verbal explanations, paralleling the way in which empirical data are used to test hypotheses׳ (Servedio et al., 2014). In this spirit, our goal here is to contribute to a systematic identification of the relative contributions of heritable variations in gene expression, environmental changes and natural selection as drivers
Acknowledgements
This work was supported in part by the French National Research Agency through the “ANR blanche” project Kibord [ANR-13-BS01-0004] and by the Australian Research Council [DP110100795 and DP140100339]. T.L. was also supported by the Hadamard Mathematics Labex, backed by the Fondation Mathématique Jacques Hadamard, through a grant overseen by the French National Research Agency [ANR-11-LABX-0056-LMH].
References (42)
- et al.
The pattern of variation in centipede segment number as an example of developmental constraint in evolution
J. Theor. Biol.
(1999) - et al.
Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration
C. R. Math.
(2012) - et al.
Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models
Theor. Popul. Biol.
(2006) - et al.
A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions
J. Theor. Biol.
(2012) - et al.
A mathematical model for immune and autoimmune response mediated by T-cells
Comput. Math. Appl.
(2013) - et al.
Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells
Cell
(2011) - et al.
The inheritance of phenotypes: an adaptation to fluctuating environments
J. Theor. Biol.
(1996) - et al.
Simplifying the complexity of resistance heterogeneity in metastatic cancer
Trends Mol. Med.
(2014) - et al.
Time fluctuations in a population model of adaptative dynamics
Ann. I. H. Poincaré-AN
(2015) - et al.
A chromatin-mediated reversible drug-tolerant state in cancer cell subpopulations
Cell
(2010)
Diversity in times of adversity: probabilistic strategies in microbial survival games
J. Theor. Biol.
Stochastic switching as a survival strategy in fluctuating environments
Nat. Genet.
Fixation in finite populations evolving in fluctuating environments
J. R. Soc. Interface
Microbial cell individuality and the underlying sources of heterogeneity
Nat. Rev. Microbiol.
Spontaneous epigenetic variation in the arabidopsis thaliana methylome
Nature
Travelling waves for the cane toads equation with bounded traits
Nonlinearity
Non-genetic heterogeneity–a mutation-independent driving force for the somatic evolution of tumours
Nat. Rev. Genet.
Asymptotics of steady states of a selection-mutation equation for small mutation rate
Soc. Edinb. A
Epigenetic gene regulation in the bacterial world
Microbiol. Mol. Biol. Rev.
The canonical equation of adaptive dynamics: a mathematical view
Selection
Phenotypic plasticity in evolutionary rescue experiments
Philos. Trans. R. Soc. B Biol. Sci.
Cited by (0)
- 1
The authors contributed equally to this paper.