Evolution of a structured cell population endowed with plasticity of traits under constraints on and between the traits

Abstract

Confronted with the biological problem of managing plasticity in cell populations, which is in particular responsible for transient and reversible drug resistance in cancer, we propose a rationale consisting of an integro-differential and a reaction-advection-diffusion equation, the properties of which are studied theoretically and numerically. By using a constructive finite volume method, we show the existence and uniqueness of a weak solution and illustrate by numerical approximations and their simulations the capacity of the model to exhibit divergence of traits. This feature may be theoretically interpreted as describing a physiological step towards multicellularity in animal evolution and, closer to present-day clinical challenges in oncology, as a possible representation of bet hedging in cancer cell populations.

Keywords: Dimorphism; Finite volume method; Plasticity; Structured population.

Fig. 1

The discrete $L^2(D_T)$ error for the semi-discrete scheme, for $T=10$ and M ranging between 2 and 128

Fig. 2

The discrete $L^2(D_T)$ error for the discrete scheme, for $T=10$ and $M_1$ ranging between 2 and 256

Fig. 3

Evolution of a population only subjected to natural selection and non-genetic epimutations

Fig. 4

Evolution of a population under the effect of the environment

Fig. 5

Evolution of two sub-populations with different levels of plasticity: initial stages

Fig. 6

Evolution of two sub-populations with different levels of plasticity: final stages

Fig. 7

Initial stages of the population density for different values of $\theta$: The differentiation process starts. At around $t=250$ (bottom left) most of the population has already concentrated around the plasticity level $\theta =0.4375$ and around $t=300$ (bottom right) we observe that the migration towards a less plastic state continues

Fig. 8

Intermediate stages of the population density for different values of $\theta$: While the differentiation process continues, we observe further loss in plasticity. Around $t=500$ (top right) most of the population has reached $\theta =0.375$ and at subsequent times the migration continues

Fig. 9

Final stages of the population density for different values of $\theta$: Around $t=900$ (bottom left) the differentiation process is over and most of the population has reached the plasticity level $\theta =0.25$. At $t=1000$ (bottom right) we observe that the population concentrated around any other level of plasticity is almost extinct, and only the one around $\theta =0.25$ survives