From invasion to latency: Intracellular noise and cell motility as key controls of the competition between resource-limited cellular populations

Abstract

In this paper we analyse stochastic models of the competition between two resource-limited cell populations which differ in their response to nutrient availability: the resident population exhibits a switch-like response behaviour while the invading population exhibits a bistable response. We investigate how noise in the intracellular regulatory pathways and cell motility influence the fate of the incumbent and invading populations. We focus initially on a spatially homogeneous system and study in detail the role of intracellular noise. We show that in such well-mixed systems, two distinct regimes exist: In the low (intracellular) noise limit, the invader has the ability to invade the resident population, whereas in the high noise regime competition between the two populations is found to be neutral and, in accordance with neutral evolution theory, invasion is a random event. Careful examination of the system dynamics leads us to conclude that (i) even if the invader is unable to invade, the distribution of survival times, P_S(t), has a fat-tail behaviour (P_S(t) ~ t⁻¹) which implies that small colonies of mutants can coexist with the resident population for arbitrarily long times, and (ii) the bistable structure of the invading population increases the stability of the latent population, thus increasing their long-term likelihood of survival, by decreasing the intensity of the noise at the population level. We also examine the effects of spatial inhomogeneity. In the low noise limit we find that cell motility is positively correlated with the aggressiveness of the invader as defined by the time the invader takes to invade the resident population: the faster the invasion, the more aggressive the invader.

1. Introduction

The aim of this paper is to analyse competition between two cell populations which employ different strategies to exploit a shared resource, i.e. they differ in how they transform the available resource into offspring. In practice, these strategies are controlled by complex signalling pathways which control gene expression and protein synthesis in response to varying resource availability (see, for example [2, 7], where models of how oxygen concentration regulates cell-cycle progression are proposed). Cell activity can then be described in terms of so-called “response curves”. These curves represent the level of activation of a gene or the concentration of a protein driving a particular cellular response as a function of the concentration of a substance (in our case, the resource, e.g. oxygen) or the level of some stimulus. We consider the case in which the response curve represents the steady-state level of the corresponding output as a function of the stimulus [4]. Two typical response curves are shown in Fig. 1 and described below.

Fig. 1.

Response curves corresponding to two competing populations: (a) switch-like response and (b) bistable response. Plot (a) depicts a situation where the probability of triggering a particular type of cellular response (say, proliferation) increases monotonically with the intensity of the corresponding stimulus or signal (say, abundance of a resource). Plot (b) corresponds to a system where a particular cell can be in one of two phenotypes (corresponding to the upper and lower branches) with sharp, abrupt transitions between them. For the bistable response as shown in plot (b), we distinguish between two distinct situations: the low intracellular noise regime where transitions between phenotypes occur only when the signal concentration falls below (exceeds) c₁ (c₂), (solid blue arrows in panel (b)), and the high intracellular noise level, where noise-driven transitions can also occur for signal concentrations in the interval c₁ ≤ c ≤ c₂ (dashed red arrows in panel (b)). Plots (c) and (d) show how the spatially-dependent resource profile relates to the response functions shown in plots (a) and (b), respectively, and also how resource levels give rise to (local) population limitation. Both plots (c) and (d) show a spatial array of three compartments occupied by switch-like and bistable cells, respectively. Each compartment, x_i−1, x_i, and x_i+1 is depicted to have a different resource concentration. The level of resource in each compartment is such that c(x_i−1) > c₂, c(x_i) ≃ c₀, c(x_i+1) > c₁, respectively. Consider a population of switch-like cells (plot (c)). Solid hexagons represent cells whose birth rate exceeds their death rate, spotted hexagons correspond to cells whose birth and death rates are equal, and striped hexagons represent cells whose death rate exceeds their birth rate. In compartment x_i−1 where c(x_i−1) > c₂, most cells are proliferating. Cells in x_i (c(x_i) ≃ c₀) are as likely to proliferate as to die. Most cells in x_i+1 (c(x_i+1) ≃ c₁) are undergoing apoptosis (see Table 1). Bistable cells (plot (d)) exhibit two types of behaviour: proliferation (solid hexagons) and quiescence (striped hexagons). In compartment x_i−1 the former are more abundant than the latter since c(x_i−1) > c₂. In x_i+1, by contrast, quiescent cells are more abundant than proliferative cells since c(x_i+1) < c₁. In compartment x_i we have a mixture of proliferating and quiescent cells, their relative proportion being determined by the noise-to-activation barrier ratio (see Table 1 and Eq. (4)).

As stated above, we analyse the competition between two populations whose cells exhibit different response curves. We consider a resident population of cells with a switch-like response curve similar to that represented in Fig. 1(a) and we interpret this curve in the following way: when the concentration of the resource is below a threshold value (c₀, in Fig. 1(a)), proliferation cannot be activated. Such cells remain in a quiescent state and will eventually die. If the resource concentration exceeds c₀, proliferation can occur (i.e. the cell-cycle is activated) and the cell will divide.

We consider that this population of switch-like cells is in competition with an invading or mutant population of cells characterised by a bistable response curve (see Fig. 1(b)) whose interpretation is more involved. When the resource concentration falls below c₁ (see Fig. 1(b)), proliferation is assumed to be inactive and the corresponding cell is in a quiescent state which will eventually lead to its death. If the resource levels exceed c₂, a threshold value in Fig. 1(b), then the cell-cycle is active and the cell divides. The feature which distinguishes these cells from the resident ones is the existence of an intermediate region of resource concentrations, c ∈ [c₁, c₂], for which cells can be in either phenotype: quiescent or proliferating.

Both types of response curves are known to correspond to well-characterised biological situations [49]. For example, Tyson et al. [49] have shown that a sigmoidal response curve (see Fig. 1(a)) can arise if the cellular response depends on a protein whose activity is controlled by phosphorylation and dephosphorylation reactions governed by Michaelis-Menten kinetics. This mechanism yields a very steep response. Similar behaviour can be generated by multistep processes involving a cascade of phosphorylation-dephosphorylation reactions [20], as occurs, for example, in the MAP kinase cascade [28].

Regarding the bistable behaviour (Fig. 1(b)), several biochemical mechanisms including mutual inhibition (see [49,48] and Appendix A) could give rise to such a cellular response. Alternatively, double phosphorylation of a protein can generate bistable behaviour provided additional constraints are satisfied [42]. From the mathematical viewpoint, such a response curve originates from a subcritical pitchfork bifurcation [45] of the non-linear system regulating cell proliferation, with resource levels serving as the control parameter.

The literature is replete with examples of biological systems in which a regulatory system switches between a graded, monostable response (i.e. our switch-like response curve) and a bistable response curve. For example, Kelemen et al. [27] have proposed a gene regulatory circuit which can exhibit switch-like and bistable behaviour, depending on the spatial distribution of epigenetic repressors: if the repressors nucleate at two segments that flank the gene, they create two interacting repressor gradients (one upstream and the other downstream of the gene) and the corresponding response is bistable. Alternatively, if repressors localise on a single segment, then the response is monostable [27]. Similarly, analysis of mathematical models has revealed that bistable activation responses in pathways involved in apoptosis [35], cell survival [36], and cell-cycle progression [15] depend upon the presence of positive feedback loops. If such positive feedbacks are rendered inactive, the corresponding response curves are monotonic (switch-like) instead of bistable. Our model is thus relevant to situations in which a mixed population of cells (i.e. one population in which the feedbacks are active and a second population in which they are inactive) compete to determine which of them has the ability to invade or persist. This scenario enables us to investigate which of these response mechanisms is robust in a given environment in terms of the outcome of our competition model.

An example of a situation to which our results may be applicable is tumour growth. Once the results regarding robustness of the different response mechanisms have been established, a natural avenue for further investigation is to ascertain which factors may lead to the emergence of malignant phenotypes which are able to thrive under a variety of conditions, including the hostile environments created by treatment with chemotheropeutic drugs [32, 46]. In this paper we investigate whether the intracellular noise associated with a bistable cellular response and cell motility is a feasible candidate to play such a role.

Our aim is to understand the role of intracellular noise (i.e. randomness affecting the dynamics of the underlying regulatory pathways) and cell motility on the outcome of the competition between the two populations, switch-like and bistable. In particular, the question we address here is which of these strategies has the evolutionary advantage of being able to withstand invasion by the other cell type. For the resident, switch-like population noise blurs the threshold c₀ (see Fig. 1(a)): cells with resource concentration below c₀ have a small, but finite, probability of dividing; conversely cells experiencing resource concentration above c₀ have a small, but finite, probability of becoming quiescent. The effect of noise on bistable cells is more complex. As mentioned above, the bistable response curve Fig. 1(b) can be understood as the result of a bifurcation of the underlying cell-cycle control system which means that two attractors coexist for a range of values of the resource concentration, in our case c ∈ [c₁, c₂]. If the level of (intrinsic) noise present in the proliferation pathway is low, then the phenotype of a particular bistable cell for resource level c, such that c ∈ [c₁, c₂], is determined by the basins of attraction of each of the attractors represented by the upper and lower branches. Transitions between states are only possible if c falls below the lower threshold, c₁, or exceeds the upper threshold, c₂ [45]. As the intracellular noise increases in intensity, stochastic transitions between both basins of attraction can occur [17, 39] and there is a finite probability that for c ∈ [c₁, c₂] there are noise-induced phenotype changes in the bistable cells.

In this paper, we investigate how noise in the intracellular regulatory pathways and cell motility determine the fate of the competition between a switch-like incumbent and a bistable invader. We show that two distinct regimes exist, namely, the low (intracellular) noise limit, in which the invader has the ability to invade the resident population, and the high noise regime, in which the competition between the two populations is found to be neutral and, in accordance with neutral evolution theory, invasion is a random event with probability equal to the proportion of invasive cells initially present in the environment [29, 11]. Further examination of the dynamics in the high-noise limit leads us to conclude that, even if the invader is unable to invade, the distribution of survival times, P_S(t), has a fat-tail behaviour: P_S(t) ~ t⁻¹, which implies that small colonies of mutants can coexist with the resident population for arbitrarily long times. Furthermore, the bistable structure of the invading population increases the stability of the latent populations and their long-term likelihood of survival, by decreasing the noise intensity at the population level. This behaviour suggests that our system exhibits a transition to latency or dormancy when the intracellular noise of the bistable mutants increases. In the context of this work, latency refers to the state in which an invader, which is unable to colonise the resident population, persists as a small, residual population incapable of growing but not completely eliminated [5]. Biologically relevant examples of such latent states include tumour dormancy [51] and latent infection of HIV-1-infected patients undergoing potent anti-retroviral therapy [44]. Moreover, we analyse the effects of spatial inhomogeneity and find that cell motility is positively correlated with the aggressiveness of the invader, aggressiveness being assumed to correlate with the time the invader cells take to invade the resident population. The faster the invasion, the more aggressive the invader. Thus, our model can be useful for studying the emergence of latency and also the role of cell motility in invasion.

The remainder of this paper is organised as follows. In section 2 we introduce the model formulation and the numerical methodology that is used throughout the paper. In Section 3, we give a detailed description of the particular problem we are studying. Sections 4 and 5 contain extended discussions of our results for well-mixed and spatially-inhomogeneous systems, respectively. Finally, Section 6 is devoted to a discussion of our results as well as future work.

2. Model formulation and methodology

In this section, we introduce our general approach, model formulation and numerical methodology. We start by describing a general model of competition between cell populations that rely upon a common resource for survival. Our model is formulated in terms of a master equation for the dynamics of the cell populations coupled to a differential equation that describes the dynamics of the resource. The model is formulated in general terms, including its application to a spatially-extended system. We then specialise the model to describe competition between a resident population of switch-like cells and an invasive population of bistable cells. The section concludes with a description of our numerical methodology, namely, a hybrid Gillespie algorithm.

2.1. Model formulation

In order to account for spatial structure, we consider a compartmental model, where cells in each compartment are assumed to concentrate on the regularly-spaced vertices, x_i ∈ ℝ^d, of a d-dimensional lattice, ℒ^d, embedded in ℝ^d. In each compartment, we consider N_T types of competing cells. We introduce the vector ${\chi }^{\left(k\right)}\left(t\right)\equiv \left({\chi }^{\left(k\right)}\left(x_1,t\right),\dots {\chi }^{\left(k\right)}\left(x_N,t\right)\right)\equiv ({\chi }_1^{\left(k\right)}\left(t\right),\dots {\chi }_N^{\left(k\right)}\left(t\right)),k=1,\dots ,N_T,$ whose components correspond to the number of cells of type k at site x_i ∈ ℝ^d and time t.

Our model comprises a simple birth-and-death process with random migration between nearest neighbours, and a variable resource (for example, oxygen) whose concentration controls the rates of cell birth and death. We assume that the spatial distribution, c(x, t), of this resource evolves as a result of diffusion and consumption by the cell populations. In more detail, we assume that the spatio-temporal evolution of the resource (for example, oxygen) is determined by the following reaction-diffusion equation:

$$\frac{\partial c}{\partial t}=D{\nabla }^{2}c-\overline{k}c{\displaystyle \sum _{k=1}^{N_T}{\displaystyle \sum _{x_i\in {ℒ}^d}{\chi }^{(k)}}}(t)\delta (x-x_i)$$ (1)

where D is the corresponding diffusivity and $\overline{k}$ is the per-cell resource depletion rate. The resource is assumed to enter the system through the boundaries of the domain. In particular, in one dimension we have:

$$D{\frac{\partial c}{\partial x}|}_{x=0}=h_0,D{\frac{\partial c}{\partial x}|}_{x=L}=-h_L$$ (2)

where h₀ and h_L are the nutrient fluxes at the ends of the domain. We fix $h_0-h_L=\Omega L\overline{k}{c}_{0,}$ where L is the length of our one-dimensional domain where c₀ is a typical oxygen concentration, chosen to ensure that sufficient oxygen is supplied to maintain a total population (resident plus mutant) of O(Ω) per unit length. Thus, Ω measures the average per-site population that our resource can sustain. A schematic is presented in Figs. 1(c) and (d) showing how the spatially-dependent resource profile relates to the response function.

The model we propose is a stochastic, spatially extended, birth-death process with random motility between nearest neighbours, which can be described by a master equation for the probability distribution P(χ^(k), t), i.e. the probability that the population be χ^(k) at time t. The master equation is formulated in terms of transition rates, which describe the probability that each of the different (random) events associated with the population dynamics occurs per unit of time. We assume that the birth and death rates, $\overline{b}\left(c\right)\text{and}\overline{\mu }\left(c\right),$ respectively, depend on the local oxygen concentration. At each site of ℒ^d, cells of each type die, change phenotype or migrate at the following rates:

• –Proliferate $({\chi }_i^{\left(k\right)}\to {\chi }_i^{\left(k\right)}+1),$ with rate $T_{{\chi }_i^{\left(k\right)}+1|{\chi }_i^{\left(k\right)}}={\overline{b}}_k\left(c\right){\chi }_i^{\left(k\right)},$ where ${\overline{b}}_k\left(c\right)$ is the oxygen-dependent birth rate for cells of type k;
• –Die $({\chi }_i^{\left(k\right)}\to {\chi }_i^{\left(k\right)}-1),$ with rate $T_{{\chi }_i^{\left(k\right)}-1|{\chi }_i^{\left(k\right)}}={\overline{\mu }}_k\left(c\right){\chi }_i^{\left(k\right)},$ where ${\overline{\mu }}_k\left(c\right)$ is the oxygen-dependent death rate for cells of type k;
• –Switch phenotype from type k to type $l({\chi }_i^{\left(k\right)},{\chi }_i^{\left(l\right)}\to {\chi }_i^{\left(k\right)}-1,{\chi }_i^{\left(l\right)}+1),$ with rate $T_{{\chi }_i^{\left(k\right)}-1{\chi }_i^{\left(l\right)}+1|{\chi }_i^{\left(k\right)}{\chi }_i^{\left(l\right)}}={\overline{w}}_0{e}^{-H_{kl}\left(c\right)}{\chi }_i^{\left(k\right)},$ where w₀ is the inverse of the characteristic time scale for phenotypic switching and P_kl(c) = e^−H_kl(c) is the oxygen-dependent probability of a switch from phenotype k to type phenotype l. (More details about P_kl(c) are in Section 3);
• –Diffuse to neighbouring compartments $({\chi }_i^{\left(k\right)},{\chi }_j^{\left(k\right)}\to {\chi }_i^{\left(k\right)}-1,{\chi }_j^{\left(k\right)}+1),$ with rate $T_{{\chi }_i^{\left(k\right)}-1{\chi }_j^{\left(k\right)}+1|{\chi }_i^{\left(k\right)}{\chi }_j^{\left(k\right)}}=\frac{{\overline{\nu }}_k}{h^2}{\chi }_i^{\left(k\right)},$ where $j\in \langle i\rangle ,{\overline{\nu }}_k\ge 0$ is the diffusion coefficient for cells of type k, and h is the lattice spacing. Regarding cell motility, we ignore excluded volume interactions [9], i.e. competition for space, assuming instead that the carrying capacity of each compartment is solely determined by resource availability.

Having defined the various transition rates, it is straightforward to state the master equation for this spatially-extended, resource-dependent birth-and-death process [38]:

$$\begin{array}{c}\frac{\partial }{\partial t}P({\chi }^{\left(1\right)},\dots ,{\chi }^{(N_T)},t)={\displaystyle \sum _{k=1}^{N_T}{\displaystyle \sum _{x_i\in {ℒ}^d}[\left({\epsilon }_{{\chi }_i^{\left(k\right)}}^{-}-1\right)T_{{\chi }_i^{\left(k\right)}+1|{\chi }_i^{\left(k\right)}}P({\chi }^{\left(1\right)},\dots ,{\chi }^{(N_T)},t)+}}\hfill \\ \left({\epsilon }_{{\chi }_i^{\left(k\right)}}^{+}-1\right)T_{{\chi }_i^{\left(k\right)}-1|{\chi }_i^{\left(k\right)}}P({\chi }^{\left(1\right)},\dots ,{\chi }^{\left(N_T\right)},t)]+\hfill \\ {\displaystyle \sum _{k=1}^{N_T}{\displaystyle \sum _{j\ne k}{\displaystyle \sum _{x_i\in {ℒ}^d}[\left({\epsilon }_{{\chi }_i^{\left(k\right)}}^{-}{\epsilon }_{{\chi }_i^{\left(j\right)}}^{+}-1\right)T_{{\chi }_i^{\left(k\right)}+1{\chi }_i^{\left(j\right)}-1|{\chi }_i^{\left(k\right)}{\chi }_i^{\left(j\right)}}P({\chi }^{\left(1\right)},\dots ,{\chi }^{(N_T)},t)+}}}\hfill \\ \left({\epsilon }_{{\chi }_i^{\left(k\right)}}^{+}{\epsilon }_{{\chi }_i^{\left(j\right)}}^{-}-1\right)T_{{\chi }_i^{\left(k\right)}-1{\chi }_i^{\left(j\right)}+1|{\chi }_i^{\left(k\right)}{\chi }_i^{\left(j\right)}}P({\chi }^{\left(1\right)},\dots ,{\chi }^{\left(N_T\right)},t)]+\hfill \\ {\displaystyle \sum _{k=1}^{N_T}{\displaystyle \sum _{x_i\in {ℒ}^d}{\displaystyle \sum _{x_j\in \langle x_i\rangle }[\left({\epsilon }_{{\chi }_i^{\left(k\right)}}^{-}{\epsilon }_{{\chi }_j^{\left(k\right)}}^{+}-1\right)T_{{\chi }_i^{\left(k\right)}+1{\chi }_j^{\left(k\right)}-1|{\chi }_i^{\left(k\right)}{\chi }_j^{\left(k\right)}}P({\chi }^{\left(1\right)},\dots ,{\chi }^{\left(N_T\right)},t)+}}}\hfill \\ \left({\epsilon }_{{\chi }_i^{\left(k\right)}}^{+}{\epsilon }_{{\chi }_j^{\left(k\right)}}^{-}-1\right)T_{{\chi }_i^{\left(k\right)}-1{\chi }_j^{\left(k\right)}+1|{\chi }_i^{\left(k\right)}{\chi }_j^{\left(k\right)}}P({\chi }^{\left(1\right)},\dots ,{\chi }^{\left(N_T\right)},t)]\hfill \end{array}$$ (3)

where ${\epsilon }_{{\chi }_i^{\left(k\right)}}^{\pm }f({\chi }_i^{\left(k\right)})=f({\chi }_i^{\left(k\right)}\pm 1).$

2.2. Numerical methodology

We use a straightforward generalisation of the stochastic simulation algorithm (SSA) proposed by Gillespie [18,19] to produce numerical results. The SSA enables us to generate sample paths or realisations of the Master Equation (3) and involves two basic steps: (i) generate the waiting time between two random events, τ_w, and (ii) determine which of the elementary events (i.e. birth, death, migration, phenotype switch) occurs and update the populations accordingly. Since our Master Equation is coupled to a reaction-diffusion equation (see Eq. (1)), we must account for this coupling in our numerical method. By definition, during the waiting time between two events, i.e. during the time interval [t, t + τ_w), the cell populations remain constant. We thus solve Eq. (1) in the interval [t, t + τ_w) with constant χ^(k) = χ^(k)(t). In summary, our version of the SSA involves three steps:

1. Generate the waiting time between two random events, τ_w;

2. Solve Eq. (1) in the interval [t, t + τ_w) with constant χ^(k) = χ^(k)(t);

3. Choose which elementary event occurs and update the populations accordingly.

These three steps are repeated until a stopping condition is satisfied. Step 2 involves solving an ordinary diffirential equation (ODE) or a partial differential equation (PDE), depending on whether we are considering a well-mixed system or a spatially-inhomogeneous situation, respectively. The corresponding solvers and boundary and/or initial conditions are specified in detail in Sections 4 and 5.

3. Competition between populations with switch-like and bistable responses

The model we study in this section considers a resident population of cells endowed with a switch-like response curve (see Fig. 1(a)) and an invading population of cells exhibiting a bistable response curve (see Fig. 1(b)). For high resource concentrations the invading cells have a proliferative phenotype and divide with probability one whilst, for low resource levels, they exhibit a quiescent phenotype and undergo apoptosis with probability one. However, there is a regime of intermediate values of resource concentration for which transitions between the two phenotypes occur with finite probability (see Fig. A1, Supplementary Material). The question we address here is which of these strategies (i.e. switch-like or bistable response), has the evolutionary advantage of being able to withstand invasion by the other cell type.

Our model is defined by the following quantities and parameters. There are three distinct cell types (N_T = 3) corresponding to resident (switch-like) cells, χ⁽¹⁾, proliferating mutant (bistable) cells, χ⁽²⁾, and quiescent mutant (bistable) cells χ⁽³⁾. The evolution of the resource concentration is given by Eq. (1). The dynamics of the cell populations is determined by the transition rates of the Master Equation (Eq. (3)) stated in Table 1.

Table 1.

Transition rates for our stochastic model of competition between switch-like and bistable populations.

Transition rate	Event
$T_{{\chi }_i^{(1)}+1|{\chi }_i^{\left(1\right)}}={\tau }_p^{-1}\frac{e^{{\overline{\alpha }}_0(c-c_0)}}{e^{{\overline{\alpha }}_0\left(c-c_0\right)}+e^{-{\overline{\alpha }}_0(c-c_0)}}{\chi }_i^{(1)}$	Division of switch-like cell
$T_{{\chi }_i^{(1)}-1|{\chi }_i^{(1)}}={\tau }_p^{-1}\frac{e^{-{\overline{\alpha }}_0(c-c_0)}}{e^{{\overline{\alpha }}_0(c-c_0)}+e^{-{\overline{\alpha }}_0(c-c_0)}}{\chi }_i^{(1)}$	Switch-like cell death
$T_{{\chi }_i^{(2)}+1|{\chi }_i^{(2)}}={\tau }_p^{-1}{\chi }_i^{(2)}$	Proliferation of bistable cell
$T_{{\chi }_i^{(3)}+1,{\chi }_i^{(2)}-1|{\chi }_i^{(3)},{\chi }_i^{(2)}}={\overline{w}}_0{e}^{-H_{PA}(c)}{\chi }_i^{(2)}$	Bistable cell phenotype change (proliferating to quiescent)
$T_{{\chi }_i^{(2)}+1,{\chi }_i^{(3)}-1|{\chi }_i^{(2)},{\chi }_i^{(3)}}={\overline{w}}_0{e}^{-H_{AP}(c)}{\chi }_i^{(3)}$	Bistable cell phenotype change (quiescent to proliferating)
$T_{{\chi }_i^{(3)}-1|{\chi }_i^{(3)}}={\tau }_p^{-1}{\chi }_i^{(3)}$	Quiescent cell death
$T_{{\chi }_i^{(k)}-1{\chi }_j^{(k)}+1|{\chi }_i^{(k)}{\chi }_j^{(k)}}=\frac{{\overline{\nu }}_k}{h^2}{\chi }_i^{(k)}$	Cell migration ∀ k and ∀ j ∈< i >

We further assume that random changes of phenotype in the bistable population are the result of an activated process which can be modelled by rates of Arrhenius type, i.e. P_AP(c) = e^−H_AP(c) and P_PA(c) = e^−H_PA(c), where P_AP(c) and P_PA(c) are the probabilities of switching between the quiescent and proliferative phenotypes. The motivation for this modelling assumption is discussed in detail in Appendix A. The functions H_AP(c) and H_PA(c) depend on the height of the activation barrier relative to the noise intensity and the concentration of resources [16,17,23] in the following way:

$$\begin{array}{c}{H}_{PA}(c)=\{\begin{array}{l}{\overline{\delta }}_2{(c-c_2)}^n+{\overline{\alpha }}_2(c_2-c_1)\text{for}c>c_2\hfill \\ {\overline{\alpha }}_2(c-c_1)\text{for}{c}_1<c\le c_2\hfill \\ 0\text{for}c\le c_1.\hfill \end{array}\\ H_{AP}(c)=\{\begin{array}{l}0\text{for}c\ge c_2\hfill \\ {\overline{\alpha }}_1(c_2-c)\text{for}{c}_1\le c<c_2\hfill \\ {\overline{\delta }}_1{(c_1-c)}^n+{\overline{\alpha }}_1(c_2-c_1)\text{for}c<c_1\hfill \end{array}\end{array}$$ (4)

In (4) the constants ${\overline{\alpha }}_1\text{and}{\overline{\alpha }}_2$ represent the ratio between the height of the activation barrier and the noise intensity, while c₁ and c₂ are defined in Fig. 1. The rationale for these choices is as follows. For concreteness, consider H_PA(c). From the discussion in Appendix A, we know that, in the bistable regime (i.e. c₁ < c ≤ c₂), H_PA(c) must be proportional to the ratio between the effective energy barrier and the intensity of the noise. This ratio is given by the parameter ${\overline{\alpha }}_2.$ Moreover, we know that P_PA(c), the probability of switching from a proliferating to a quiescent phenotype, must be such that P_PA(c) → 0 when c ≥ c₂ and P_PA(c) → 1 when c ≤ c₁. The former condition implies that H_PA(c) must increase rapidly when c ≥ c₂, and so we assume algebraic dependence on (c − c₂), with exponent n ≫ 1. The latter condition means that H_PA(c) → 0 when c → c₁ and grows as c approaches c₂. The linear dependence on c − c₁ in Eq. (4) is the simplest functional form that satisfies these conditions. A similar argument justifies the functional form used for H_AP(c) (see Appendix A for a more detailed discussion). The transition probabilities associated with Eq. (4) are plotted in Fig. A1, Supplementary Material, for two different sets of values of ${\overline{\alpha }}_1\text{and}{\overline{\alpha }}_2:$ low noise intensity (green lines in Fig. A1, Supplementary Material) and high noise intensity (red lines in Fig. A1, Supplementary Material).

In the context of the definitions of ${\overline{\alpha }}_1\text{and}{\overline{\alpha }}_2,$ fluctuations are assumed to be produced by intrinsic noise in the intracellular regulatory pathway which generates the bistable response. Intrinsic noise is attributed to finite size effects associated with the regulatory reactions within the cell. Therefore, according to the theory of large deviations (see Appendix A and [33]) ${\overline{\alpha }}_i\propto V,$ where V is the characteristic scale associated with the volume of the cell.

We allow for the general situation in which random transitions between states in the bistable regime are asymmetric, i.e. systems for which transitions in one direction are more likely than in the other. In our model, Eq. (4), this is accounted for by assuming ${\overline{\alpha }}_1\ne {\overline{\alpha }}_2.$ If this condition is satisfied, transitions will occur more frequently out of the state with smaller barrier-to-noise ratio. An example of such an asymmetric bistable system has been reported in [26], where the evolution of the GAL network in yeast was studied. In this system no transitions were observed from the active to the inactive state, in contrast to inactive-to-active transitions.

3.1. Non-dimensionalisation

We first define the dimensionless time and space variables as $\tau ={\tau }_p^{-1}t$ and x → h⁻¹x, respectively, where τ_p is the characteristic time for a cell of either type to proliferate or die and h corresponds to the average cell diameter. We further define τ_D = h²/D as the characteristic time for oxygen diffusion. With this choice of dimensionless scales, the definition of the dimensionless parameter grouping shown in Table 2 follows naturally.

Table 2.

Dimensionless variables and parameters.

Dimensionless variables	Definition
$c\to c/c_0$	Dimensionless resource concentration
Dimensionless parameter	Definition
$ϵ={\tau }_D/{\tau }_p$	Ratio between the diffusion time and the proliferation time
$w_0={\tau }_p{\overline{w}}_0$	Dimensionless characteristic frequency for phenotype switch
$\nu ={\tau }_p{h}^{-2}\overline{\nu }$	Dimensionless cellular diffuusion coefficient
$\kappa ={\tau }_D\overline{\kappa }$	Dimensionless resource depletion rate
${\alpha }_i={\overline{\alpha }}_0{c}_i(\text{i}=0,2,3)$	See Fig. 1

The values of the parameters used in this paper are given in Table 3.

Table 3.

Parameter values used in our simulations. The values of w₀, α₁, α₂, ${\overline{\nu }}_2,$ and ${\overline{\nu }}_3,$ which are not given in this table, are specified in the captions of specific simulation results. In all simulations we fix $n=4,{\overline{\delta }}_1={\overline{\delta }}_2=1.$

Parameter	Value	Units
ϵ	10−4	Dimensionless
h	10	μm
D	103	μm2/s
$\overline{\kappa }$	10−3	s−1

For simplicity, we further assume that α₁ = α₂, c₁ = 1/2 and c₂ = 3/2. Additionally, we rescale time in the following way: t → ϵτ. In these dimensionless units, the resource equation becomes:

$$\frac{\partial c}{\partial \tau }={\nabla }^{2}c-\kappa c{\displaystyle \sum _{k=1}^{N_T}{\chi }^{\left(k\right)}}$$ (5)

with boundary conditions given by Eq. (2) and initial conditions c(x, τ = 0) = 1.

For a well-mixed system (with no spatial variation), Eq. (5) is superseded by:

$$\frac{dc}{dt}=-\kappa c{\displaystyle \sum _{k=1}^{N_T}{\chi }^{(k)}}+S$$ (6)

where S = κΩ, so that Eq. (5) admits a spatially-homogeneous, steady-state solution with c(x) = 1 and ${\displaystyle {\sum }_{k=1}^{N_T}{\chi }^{(k)}}=\Omega ,$ i.e. Ω measures the total population that resource levels equivalent to c₀ can sustain. The corresponding dimensionless transition rates are given in Table 4.

Table 4.

Dimensionless transition rates for our stochastic model of competition between switch-like and bistable populations. These are obtained by rescaling the transition rates in Table 1.

Transition rate	Event
$T_{{\chi }_i^{(1)}+1|{\chi }_i^{(1)}}=ϵ\frac{e^{{\alpha }_0(c-1)}}{e^{{\alpha }_0(c-1)}+e^{-{\alpha }_0(c-1)}}{\chi }_i^{(1)}$	Division of switch-like cell
$T_{{\chi }_i^{(1)}-1|{\chi }_i^{(1)}}=ϵ\frac{e^{-{\alpha }_0(c-1)}}{e^{{\alpha }_0(c-1)}+e^{-{\alpha }_0(c-1)}}{\chi }_i^{(1)}$	Switch-like cell death
$T_{{\chi }_i^{(2)}+1|{\chi }_i^{(2)}}=ϵ{\chi }_i^{(2)}$	Proliferation of bistable division
$T_{{\chi }_i^{(3)}+1{\chi }_i^{(2)}-1|{\chi }_i^{(3)}{\chi }_i^{(2)}}=ϵw_0{e}^{-H_{PA}(c)}{\chi }_i^{(2)}$	Bistable cell phenotype change: proliferating to quiescent
$T_{{\chi }_i^{(2)}+1,{\chi }_i^{(3)}-1|{\chi }_i^{(2)},{\chi }_i^{(3)}}=ϵw_0{e}^{-H_{AP}(c)}{\chi }_i^{(3)}$	Bistable cell phenotype change: quiescent to proliferating
$T_{{\chi }_i^{(3)}-1|{\chi }_i^{(3)}}=ϵ{\chi }_i^{(3)}$	Quiescent cell death
$T_{{\chi }_i^{(k)}-1,{\chi }_j^{(k)}+1|{\chi }_i^{(k)},{\chi }_j^{(k)}}=ϵ{\nu }_k{\chi }_i^{(k)}$	Cell migration ∀ k and ∀ j ∈ < i >

4. Well-mixed systems

We start by analysing the well-mixed, spatially-homogeneous version of our model for which the analysis simplifies greatly. Indeed, we find that there exist two asymptotic limits in which it is possible to obtain analytical results.

4.1. High barrier-to-noise ratio: Invasion in well-mixed systems

In order to determine whether a mutant bistable population out-competes the resident switch-like population, we adopt a standard two-step procedure [31, 11]: (i) the incumbent population is allowed to evolve in the absence of the invader until it reaches its steady-state population, N_I, and then, (ii) a small number of invasive cells is introduced into the system which continues to evolve until either the invaders out-compete the residents or the invasive species becomes extinct. In the stochastic setting, coexistence is not possible as the system has two absorbing states where either resident or invader becomes extinct. From Eq. (5), it is straight-forward to verify that in the absence of invading cells (i.e. χ⁽²⁾ = 0 and χ⁽³⁾ = 0) and when spatial effects are neglected (i.e. no diffusion), the steady-state of the incumbent population is given by N_I = Ω and c = 1.

We have conducted numerical simulations in which a population N_M = yN_I = yΩ of mutants are introduced into a resident population with size N_I = (1 − y)Ω, where y is the initial mutant-to-resident ratio and with c(t = 0) = 1 as the initial resource levels. The system evolve until either the resident population becomes extinct or until a time τ_m has elapsed. In all our simulations we have fixed τ_m = 10⁶ which in dimensional terms corresponds to τ_m = 100τ_p, where τ_p is the cell doubling time. This value of τ_m has been set by running preliminary simulations with different values of τ_m and choosing the smallest value for which statistics were not altered.

We assess which population is the fittest by comparing the probability of invasion, P_inv ≡ P(χ⁽¹⁾ = 0, τ ≤ τ_m|χ⁽²⁾ + χ⁽³⁾ > 0), with the results obtained in neutral theory [11]. If the mutant were neutral, i.e. it was as fit as the resident, then P_inv = y [29, 8]. Further if P_inv > y then the invader is fitter than the resident [11].

In Fig. 2 we present simulation results showing how the fixation probability P_inv = P(χ⁽¹⁾ = 0, τ ≤ τ_m|χ⁽²⁾ + χ⁽³⁾ > 0) changes as we vary the barrier-to-noise ratio, α. We observe that for low noise intensity the invasion probability is well above that of the neutral case. In fact, for a wide range of values of α, we find that P(χ⁽¹⁾ = 0, τ ≤ τ_m|χ⁽²⁾ + χ⁽³⁾ > 0) ≃ 1. For low barrier-to-noise ratio (that is, as the intensity of the intracellular noise increases), the invasion probability decreases until it falls below the neutral case. However, for high barrier-to-noise ratios, i.e. α₁ = α₂ ≫ 1, the bistable phenotype is fitter than its switch-like counterpart.

Fig. 2.

Simulation results for the fixation probability of the invader (bistable) cell population as a function of the barrier-to-noise ratio α = α₁ = α₂. Parameter values: The initial mutant-to-incumbent ratio y = 1/500. w₀ = 10².

4.1.1. Mean-field theory

In order to strengthen our quantitative understanding of the competition between the bistable and switch-like populations, we develop a mean-field theory to describe their interactions. It is straightforward to show that the dimensionless mean-field equations for this process are:

$$\frac{dc}{d\tau }=\kappa \Omega -\kappa \left(n_s+n_p+n_q\right)c,$$ (7)

$$\frac{d{n}_s}{d\tau }=ϵ\text{tanh}\left({\alpha }_0\left(c-1\right)\right)n_s,$$ (8)

$$\frac{d{n}_p}{d\tau }=ϵn_p-ϵw_0{P}_{PA}\left(c\right)n_p+ϵw_0{P}_{AP}\left(c\right)n_q,$$ (9)

$$\frac{d{n}_q}{d\tau }=-ϵn_q+ϵw_0{P}_{PA}\left(c\right)n_p-ϵw_0{P}_{AP}\left(c\right)n_q,$$ (10)

where c is the resource concentration, n_s is the number of resident, switch-like cells, n_p and n_q are the number of proliferating and quiescent bistable cells and ϵ ≪ 1 is defined in Table 2. The dimensionless parameters in Eqs. (7)-(10) have the same meaning as for their stochastic counterparts. The associated initial conditions are:

$$c\left(\tau =0\right)=1,n_s\left(\tau =0\right)=\left(1-y\right)\Omega ,n_p\left(\tau =0\right)=n_q\left(\tau =0\right)=y\Omega /2.$$ (11)

Note that in the absence of the mutant population, i.e. n_p = n_q = 0, the corresponding stable steady-state is given by c = 1 and n_s = Ω.

One may ask whether a mean-field theory can accurately predict the behaviour of the stochastic system. To address this question, we have solved numerically Eqs. (7), (9) and (10) with Ω = 100 (see Fig. A2(a), Supplementary Material) and Ω = 1000 (see Fig. A2(b), Supplementary Material) and compared the results to those generated from our stochastic model. The results in Fig. A2, Supplementary Material, show that the mean-field theory provides a reasonable approximation to the full stochastic process, even for system sizes as small as Ω = 100.

Usually, invasion criteria in deterministic systems are based on Malthusian theory where the governing equations are linearised about the relevant attractors and the dominant eigenvalue provides an estimate of the growth rate of the invading population. If the corresponding growth rate is positive then the invading population will eventually displace the resident one. Otherwise, if the growth rate is negative then the incumbent population will resist invasion [40,43,31]. According to this formulation, the likelihood of invasion by a small mutant population is determined by linearising the dynamics of the invader around the invader-free steady state of the system. In this limit, the equations for the resource levels and the cell populations decouple at O(ϵ) and we may focus on the linearised equations for n_p and n_q alone. In our case this corresponds to the steady state of Eqs. (7) and (8) with n_p = n_q = 0, c = 1 and n_s = Ω. The eigenvalue problem corresponding to the linearised system can be written as

$$\frac{d{n}_p}{d\tau }=ϵn_p-ϵw_0{e}^{-\alpha /2}\left(n_p-n_q\right),$$ (12)

$$\frac{d{n}_q}{d\tau }=-ϵn_q+ϵw_0{e}^{-\alpha /2}\left(n_p-n_q\right),$$ (13)

where, for simplicity, we have assumed that α = α₁ = α₂ (i.e. the barriers that bistable cells have to overcome to switch between proliferative and quiescent phenotypes have the same height) determines the ability of the mutant population to invade. Eqs. (12)-(13) have two eigenvalues given by

$${\lambda }_{\pm }=ϵ\left(-w_0{e}^{-\alpha /2}\pm {\left(1+w_0{e}^{-\alpha }\right)}^{1/2}\right).$$

In Fig. 3 we present results showing how λ = λ_±/ϵ, i.e. the rescaled dominant eigenvalue (the largest positive eigenvalue) associated to the linearised system Eqs. (12)-(13) varies with, α, the barrier-to-noise ratio. We observe an abrupt transition between two regimes: a low noise, large barrier-to-noise ratio (α) regime where λ ≃ 1 and a high noise, small barrier-to-noise ratio regime where λ ≃ 0. The former region corresponds to an invasive mutant which almost certainly takes over the resident population and the latter to a neutral mutant whose invasion likelihood is, according to the neutral theory [29, 11], equal to the initial mutant-to-resident ratio (which, in realistic situations, is very small). This analytical result is consistent with simulation results obtained using the SSA [18,19] and presented in Fig. 2.

Fig. 3.

Series of curves showing how, for different values of w₀, the rescaled dominant eigenvalue, λ, of Eqs. (12)-(13) changes as the barrier-to-noise ration, α, is varied. Key: w₀ = 10² (dashed blue line), w₀ = 10³ (dash-dotted green line), and w₀ = 10⁴ (solid red line).

Fig. 3 also shows how, as w₀ increases (i.e. as the characteristic time for a bistable cell to switch phenotype decreases), the invading regime loses ground to the neutral regime. This is to be expected, as increasing w₀ amplifies the effect of intracellular noise on the bistable population.

In the context of the linear analysis, we have also investigated the behaviour of the bistable population in terms of its ability to invade the switch-like population when the parameters c₁ and c₂ of the bistable cells (see Fig. 1) vary with respect to c₀, the parameter characterising the switch-like cells. More specifically, fixing c₀ = 1 for switch-like cells we determined the eigenvalue λ for a range of values of c₁ and c₂. The results are shown in Fig. 4. If c₁, c₂ < c₀ (Fig. 4, blue line), we observe that λ > 0 for all α. This is natural since, according to the scheme in Fig. 1, if c₁, c₂ < c₀ then all bistable cells have the proliferative phenotype which gives the invading population an advantage over the resident one. If, on the contrary, c₀ > c₁, c₂ (Fig. 4, green line) then λ < 0 for all α. In this case the invader cells are unable to invade regardless of the value of α. This behaviour can also be understood in terms of Fig. 1: if all bistable cells carry the quiescent phenotype then the resident population has an advantage over the invader one. The third case (c₁ < c₀ < c₂) has already been discussed (see discussion regarding Fig. 3).

Fig. 4.

Series of curves showing how the rescaled dominant eigenvalue, λ, of Eqs. (12)-(13) changes as α ≡ α₁ = α₂ varies. We plot λ for different values of c₁ and c₂. Key: c₁ = 3/2 and c₂ = 6/3 (dash-dotted green line), c₁ = 2/6 and c₂ = 2/3 (dashed blue line), and c₁ = 1/2 and c₂ = 3/2 (solid red line). w₀ = 10⁴.

4.2. Low barrier-to-noise ratio: Dormancy in well-mixed systems

In this section, we analyse our competition model in the low barrier-to-noise ratio limit where α₁ = α₂ ≲ 1. As suggested by the simulation results shown in Fig. 2, as the level of intracellular noise in the bistable population increases, the barrier-to-noise ratio falls and the bistable population loses its competitive advantage with respect to the switch-like population.

4.2.1. Embedded branching process

In order to understand how this change of behaviour arises, we first look at the numerical simulation of our model and realise that the oxygen concentration remains constant at c ≃ 1 (see Fig. A3, Supplementary Material, for a representative realisation of the dynamics of our model in the regime of high intracellular noise). This observation motivates a significant model simplification which enables us to make analytical progress.

If c ≃ 1 then the incumbent and mutant populations are effectively decoupled and we can study the (sub)process (χ⁽²⁾, χ⁽³⁾) independently of χ⁽¹⁾ and with constant coefficients (see Table 5).

Table 5.

Dimensionless transition rates for our stochastic model for the birth-and-death process of the bistable population. These rates are the low barrier-to-noise ratio of the rates given in Table 1. We have also considered the oxygen level to be constant: c ≃ 1 (see Fig. A3, Supplementary Materials, for numerical evidence supporting this assumption).

Transition rate	Event
Tχ(2)+1|χ(2) = ϵχ(2)	Proliferation of bistable cell
Tχ(3)+1χ(2) −1|χ(3)χ(2) = ϵw0e−α2/2χ(2)	Bistable cell phenotype change (proliferating to quiescent)
Tχ(2)+1χ(3) −1|χ(2)χ(3) = ϵw0e−α1/2χ(3)	Bistable cell phenotype change (quiescent to proliferating)
Tχ(3) −1|χ(3) = ϵχ(3)	Quiescent cell death

We formulate our analytical theory by considering the embedded branching process equivalent to the birth-death process associated with the transition rates shown in Table 5 [22]. The process of embedding involves coarse-graining time in the following way. After birth, each individual (of type j) lives for a length of time which is exponentially distributed with characteristic time ${\tau }_c^{\left(j\right)}={\left({ϵ}^2(1+w_0{e}^{-\alpha /2})\right)}^{-1}$ where α ≡ α₁ = α₂. At the end of their life-span, each individual produces off-spring according to the corresponding generating functions of the per-cell offspring probabilities [30]:

$$G_P(x,y)=\frac{w_0{e}^{-\alpha /2}}{1+w_0{e}^{-\alpha /2}}y+\frac{1}{1+w_0{e}^{-\alpha /2}}{x}^2$$ (14)

$$G_Q(x,y)=\frac{1}{1+w_0{e}^{-\alpha /2}}+\frac{w_0{e}^{-\alpha /2}}{1+w_0{e}^{-\alpha /2}}x.$$ (15)

For full specification of the age-dependent, embedded branching process, we need to specify the age distribution $f\left(\tau \right)={\tau }_c^{-1}{e}^{-\tau /{\tau }_c}.$ By construction, the derivatives of the generating function of the per-cell offspring probability specified in Eqs. (14)-(15) can be used to determine P_IJ (I, J = P, Q), the probability that a cell of type J will produce ξ_IJ descendants of type I, $P_{IJ}=\frac{1}{{\xi }_{IJ}!}{\partial }_I^{{\xi }_{IJ}}{G}_{J|x=y=0·}$

Eqs. (14) and (15) allow us to determine the matrix M, whose entries, m_i,j, correspond to the average offspring of type i produced by each individual of type j, simply by computing the derivatives of the generating functions:

$$\text{M}=\left(\begin{array}{ll}{m}_{PP}\hfill & m_{PQ}\hfill \\ m_{QP}\hfill & m_{QQ}\hfill \end{array}\right)=\left(\begin{array}{ll}{\partial }_x{G}_P\left(1,1\right)\hfill & {\partial }_y{G}_P\left(1,1\right)\hfill \\ {\partial }_x{G}_Q\left(1,1\right)\hfill & {\partial }_y{G}_Q\left(1,1\right)\hfill \end{array}\right)=\left(\begin{array}{ll}\frac{2}{1+\omega }\hfill & \frac{\omega }{1+\omega }\hfill \\ \frac{\omega }{1+\omega }\hfill & \hfill 0\hfill \end{array}\right)$$ (16)

where ω ≡ w₀e^−α/2. From elementary considerations in branching process theory [30], we know that, λ₁, the dominant eigenvalue of M, is equal to the growth rate and contains valuable information about the long-term behaviour of our system, including, P_E, the probability of eventual extinction. If λ₁ ≤ 1 then P_E = 1 whereas if λ₁ > 1 then P_E < 1 and the system has a finite probability of long-term survival. With M defined by Eq. (16) we can explicitly calculate λ₁ as:

$${\text{λ}}_1=\frac{1+{\left(1+{\omega }^2\right)}^{1/2}}{1+\omega }$$ (17)

which is plotted in Fig. 5 as a function of the parameter ω = w₀e^−α/2. We can see that, as ω increases, our embedded branching process switches from supercritical (λ₁ > 1) to critical (λ₁ = 1), i.e. the population transitions from having a positive growth rate to having a zero growth rate. In terms of our competition problem this means that for small ω the bistable population can thrive in the presence of the switch-like population and will therefore eventually invade it. However, as ω increases, the switch-like and the bistable populations engage in a neutral competition scenario, where invasion is a random event whose probability is equal to the initial mutant-to-resident ratio [8, 11]. Since, in realistic situations, this ratio is very small, the resulting invasion probability is negligible. This analytic result is in good agreement with the simulation results presented in Fig. 6, where we have plotted the probability of fixation of the invading population P(χ⁽¹⁾ = 0, τ ≤ τ_m|χ⁽²⁾ + χ⁽³⁾ > 0) as a function of w₀. We can see that as w₀ increases, invasion becomes less likely.

Fig. 5.

Dominant eigenvalue, λ₁, of the embedded branching process defined by the generating functions Eqs. (14) and (15) as a function of the parameter ω = w₀e^−α/2 with α ≡ α₁ = α₂.

Fig. 6.

Invasion probability as a function of ω for α ≡ α₁ = α₂= 1.

4.2.2. Extinction dynamics of the invading population in the critical case: Emergence of latency in well-mixed systems

When studying the competition between two populations it is common practice to look at the growth rate or, correspondingly, the dominant eigenvalue of the growth operator and then to draw conclusions about the invasion probability [31,40,43]. We would like to extend our argument and analyse the dynamics of the extinction of the invader (bistable) population when they are critical (i.e. when ω > 1). Results from the theory of critical branching processes [24,30] show that the long-time asymptotics of the survival probability, P_S(τ) ≡ P(χ⁽²⁾(τ) + χ⁽³⁾(τ) > 0), exhibits a fat-tail behaviour: P_S(τ) ~ Aτ⁻¹ where the coefficient A is proportional to the inverse of the variance (covariance matrix, for multi-type processes [24]) of the per-cell offspring probabilities. Accordingly, the probability distribution of the time to extinction, T, has a long-time asymptotic behaviour of the form f(T) ~ AT⁻². Further, the average extinction time diverges logarithmically, 𝔼[T] ~ log(τ_m) as τ_m → ∞, where τ_m corresponds to the simulation time, and 𝔼[T²] → ∞. Guided by this asymptotic behaviour we expect that the invading population should be able to survive for arbitrarily long times, with a non-vanishing, albeit small, probability.

The value of the exponent of the survival probability, P_S(τ) ~ Aτ^−β with β = 1, persists for a wide range of critical branching processes [24]. As a result, there is no scope for its control. However, the coefficient A is model-dependent and, therefore, sensitive to changes in the model parameters. Thus, although we cannot change the long-time asymptotics (i.e. the exponent β), we can alter the survival probability at time τ by varying A. In order to explore this issue further, we need to establish a random benchmark population against which to compare the process defined by Eqs. (14) and (15). We choose as a random benchmark population a single-cell type process for which the probability of proliferation, P(ξ_R = 2), and the probability of death, P(ξ_R = 0), are identical and therefore the growth process is a maximally random, unbiased process. The corresponding generating function, G_R(x), is given by:

$$G_R(x)=\frac{1}{2}(1+x^2)$$ (18)

with $f\left(\tau \right)={\tau }_c^{-1}{e}^{-\tau /{\tau }_c}$ where τ_c = ϵ⁻². The aim of this comparison is to ascertain if bistability has any effect on the long time properties of our system in the critical case (ω > 1). It is trivial to verify that 𝔼[ξ_R] = 1 and that Var[ξ_R] = 1. The equivalent birth-and-death process and corresponding Master Equation are determined by the transition rates shown in Table 6.

Table 6.

Dimensionless transition rates for our stochastic model for the birth-and-death process of the random population.

Transition rate	Event
Tχ(R)+1|χ(R) = ϵχ(R)	Random-population cell division
Tχ(R)−1|χ(R) = ϵχ(R)	Random-population cell death

Notice that the entries of the matrix M specified in Eq. (16) can be written as M = (ξ_IJ P_IJ) where I, J = P, Q, and ξ_IJ is a random variable representing the number of offspring of type I produced by a parent of type J, and $P_{IJ}=(1/{\xi }_{IJ}!){\partial }_I^{{\xi }_{IJ}}{G}_J{|}_{x=y=0}$ is the corresponding probability. We can thus define a matrix σ² = ((ξ_IJ − 〈ξ_IJ〉)²P_IJ) for I, J = P, Q where:

$${\sigma }^2=\left(\begin{array}{l}{\left(2-\frac{2}{1+\omega }\right)}^2\frac{1}{1+\omega }{\left(1-\frac{\omega }{1+\omega }\right)}^2\frac{\omega }{1+\omega }\hfill \\ {\left(1-\frac{\omega }{1+\omega }\right)}^2\frac{\omega }{1+\omega }0\hfill \end{array}\right).$$ (19)

and λ_σ², the dominant eigenvalue of σ², is given by:

$${\lambda }_{{\sigma }^2}=\frac{2{\omega }^2+w{(1+4{\omega }^2)}^{1/2}}{{(1+\omega )}^3}.$$ (20)

In Fig. 7 we use Eq. (20) to show how λ_σ² varies with ω.

Fig. 7.

Dominant eigenvalue, λ_σ², of the matrix σ² Eq. (20) as a function of ω.

Since λ_σ² < Var[ξ_R] for ω > 0, for those values of ω for which λ_σ² ≃ 1, we expect that the bistable population, whose population dynamics are determined by the generating functions Eqs. (14) and (15), will exhibit smaller fluctuations and, therefore, be more stable than the random random benchmark population specified in Eq. (18). Further, the coexistence probability at τ = τ_m for the bistable population competing against the switch-like population should be bigger than that corresponding to the competition between the random and the switch-like populations. In order to verify these predictions we use the SSA to perform direct simulations.

To check for stability, consider the interval ℐ = [0, 100]. In Fig. 8, we plot R_S, the ratio of the probabilities that the bistable and random populations stay in ℐ for all τ ≤ τ_m:

$$R_S=\frac{P({\chi }^{(1)}>0,{\chi }^{(2)}+{\chi }^{(3)}\in ℐ)}{P({\chi }^{(1)}>0,{\chi }^{(R)}\in ℐ)}.$$ (21)

If the bistable population is more stable than the random benchmark population, we anticipate R_S > 1. This is confirmed by the simulation results presented in Fig. 8 where we plot the ratio R_S as a function of ω for α = α₁ = α₂ = 1.

Fig. 8.

Ratio R_S, as defined by Eq. (21), as a function of the parameter ω = w₀e^−α/2 with α = α₁ = α₂ = 1. The initial mutant population in these simulations, for the competition between the switch-like and random populations, is χ^(R)(τ = 0) = 10, whilst for the competition between the switch-like and bistable populations, it is χ⁽²⁾(τ = 0) = χ⁽³⁾(τ = 0) = 5.

To test whether the bistable population is longer-lived than its random counterpart we have computed, for each mutant population, the probability of coexistence between the invader and resident populations after a time τ_m has elapsed, i.e. P(χ⁽¹⁾(τ_m) > 0, χ⁽²⁾(τ_m) + χ⁽³⁾(τ_m) > 0 and P(χ⁽¹⁾(τ_m) > 0, χ^(R)(τ_m) > 0), respectively. We define the ratio R_C as:

$$R_C=\frac{P({\chi }^{(1)}({\tau }_m)>0,{\chi }^{(2)}({\tau }_m)+{\chi }^{(3)}({\tau }_m)>0)}{P({\chi }^{(1)}({\tau }_m)>0,{\chi }^{(R)}({\tau }_m)>0)},$$ (22)

and deduce that if R_C > 1 then the probability of the bistable mutant surviving (without invading the resident population) beyond τ_m is greater than that corresponding to the random mutant. In such cases if the bistable mutant population is unable to invade it is likely to linger for longer times than the random benchmark population. This is to be expected since λ_σ² <Var[ξ_R] and is confirmed by the numerical results presented in Fig. 9.

Fig. 9.

Ratio R_C, as defined by Eq. (22), as a function of the parameter ω = w₀e^−α/2 with α = α₁ = α₂ = 1. The initial mutant population in these simulations, for the competition between the switch-like and random populations, is χ^(R)(τ = 0) = 10, whilst for the competition between the switch-like and bistable populations, it is χ⁽²⁾(τ = 0) = χ⁽³⁾(τ = 0) = 5.

In Fig. 9 we see that R_C exhibits a non-monotonic dependence on ω, behaviour which is consistent with the non-monotonic behaviour of λ_σ² (see Fig. 7). According to our argument regarding the survival probability of a critical random walk [24,30] whereby the smaller the variance the longer-lived the population is expected to be, Fig. 7 shows that as ω increases, λ_σ² exhibits first a minimum, then a maximum, and, after that, a monotonic decrease, which, at the level of the survival probability, should translate into first a maximum, then a minimum, and then a monotonic increase. This scenario is consistent with the behaviour of R_C presented in Fig. 9 and suggests that λ_σ² provides a reasonable description of the variance of the embedded branching process defined by Eqs. (14) and (15).

5. Spatially-inhomogeneous systems

We now study the impact of cell movement and spatial variation on the dynamics of the two cell populations. For simplicity, we focus on the one-dimensional case and the high barrier-to-noise ratio. In particular, we are interested in the effect that cell motility has on the ability of the invading population to invade the resident population.

We start by analysing the low noise regime using a mean-field theory, the natural generalisation of Eqs. (7)-(10) to the spatially-extended case:

$$\frac{\partial c}{\partial \tau }=\frac{{\partial }^{2}c}{\partial x^2}-\kappa (n_s+n_p+n_q)c,$$ (23)

$$\frac{\partial n_s}{\partial \tau }=ϵ{\nu }_1\frac{{\partial }^2{n}_s}{\partial x^2}+ϵ\text{tanh}({\alpha }_0(c-1))n_s,$$ (24)

$$\frac{\partial n_p}{\partial \tau }=ϵ{\nu }_2\frac{{\partial }^2{n}_p}{\partial x^2}+ϵn_p-ϵw_0{P}_{PA}(c)n_p+ϵw_0{P}_{AP}(c)n_q,$$ (25)

$$\frac{\partial n_q}{\partial \tau }=ϵ{\nu }_3\frac{{\partial }^2{n}_q}{\partial x^2}-ϵn_q+ϵw_0{P}_{PA}(c)n_p-ϵw_0{P}_{AP}(c)n_q,$$ (26)

Spatial heterogeneity is induced via flux boundary conditions for the resource concentration,

$$\frac{\partial c}{\partial x}{{|}_{x=0}=h_0,\frac{\partial c}{\partial x}|}_{x=L}=-h_L.$$ (27)

whereas no-flux boundary conditions are imposed for the cell populations n_s, n_p and n_q,

$$\frac{\partial n_s}{\partial x}{{|}_{x=0}=\frac{\partial n_s}{\partial x}|}_{x=L}=0,\frac{\partial n_p}{\partial x}{{|}_{x=0}=\frac{\partial n_p}{\partial x}|}_{x=L}=0,\frac{\partial n_q}{\partial x}{{|}_{x=0}=\frac{\partial n_q}{\partial x}|}_{x=L}=0.$$ (28)

The initial conditions used to close Eqs. (23)-(26) are:

$$\begin{array}{l}c\left(x,\tau =0\right)=1\hfill \\ n_s\left(x,\tau =0\right)=\Omega /L-2\delta \left(x-L/2\right)\hfill \\ n_p\left(x,\tau =0\right)=n_q\left(x,\tau =0\right)=\delta \left(x-L/2\right).\hfill \end{array}$$ (29)

where δ(x) is the Dirac delta distribution i.e., we introduce mutant bistable cells at x = L/2. We assume that the resident cells are not motile (ν₁ = 0) whereas the invading population, as part of their evolution towards malignancy, possess a degree of mobility (i.e. ν₂ ≠ 0 and ν₃ ≠ 0). We solve Eqs. (23)-(26) numerically using the method of lines, performing a finite difference discretisation in space and integrating the resulting ODE system in Matlab via its stiff solver ode23s.

As Fig. 10 reveals, the outcome of the competition between the two populations in a spatial environment is that the bistable population invades the resident one. At early times, the invader cells remain localised in a neighbourhood of the site at which they were seeded. They evolve locally via dynamics similar to those shown in Fig. A2, Supplementary Material. When a sufficiently large number of mutant cells have accumulated, they form an advancing front of proliferative cells which moves up the oxygen gradient. This process continues until the invading population reaches the oxygen-rich region (in our set-up, the region of the domain closer to x = 0), where the resident population concentrates. The resident and mutant cells then engage in a competition process similar to that described in Section 4.1, whereupon the resident population becomes extinct.

Fig. 10.

Four snapshots of the time evolution of Eqs. (23)-(26). Panels (a), (b), (c), and (d) correspond to τ = 2.5 × 10⁴, τ = 5 × 10⁴, τ = 10⁵, and τ = 1.5 × 10⁵, respectively. Colour code: green lines correspond to c(x, t), red lines to n_s(x, t), blue lines to n_p(x, t), and orange lines to n_q(x, t). Parameter values: ν₂ = ν₃ = 10⁻¹ μm²s⁻¹. w₀ = 10⁴.

We now investigate how the outcome of the competition between the switch-like resident and the bistable mutant populations changes as ν₂ and ν₃ vary. We assume that ν₂ = ν₃ > 0 (all mutant cells are equally motile, regardless of their phenotype) and determine whether the resident population becomes extinct by computing the total population size of each cell type in the domain:

$$N_s(t){\displaystyle ={\int }_0^L{n}_s(x,t)dx,N_p(t)={\displaystyle {\int }_0^L{n}_p(x,t)dx,N_q(t)={\displaystyle {\int }_0^L{n}_q(x,t)dx}}}$$ (30)

Typical results are presented in Figs. 11 and 12. Fig. 11 shows that the invader cells become more invasive as their motility increases: the four panels in Fig. 11 reveal that the time taken for the invading population to eradicate the resident population increases monotonically as ν₂ = ν₃ decreases.

Fig. 11.

Series of results obtained from numerical simulations of Eqs. (23)-(29) showing how the time to extinction of the switch-like resident population changes as the motility rate ${\overline{nu}}_2={\overline{nu}}_3$ of the invading bistable cells varies. Key: N_s(t) (dash-dotted red lines), N_p(t) (dashed blue lines), N_q(t) (solid orange lines) (see Eq. (30)). ${\overline{\nu }}_1$ = 0. Parameter values: cell diffusion coefficients are given in μm²s⁻¹, ${\overline{\nu }}_1$ = 0 and w₀ = 10⁴.

Fig. 12.

This plot shows the emergence of coexistence between a motile switch-like resident and a non-motile bistable mutant (ν₂ = ν₃ = 0). Dash-dotted red lines correspond to N_s, dashed blue lines to N_p, and solid orange lines to N_q. Cell diffusion coefficients are given in μm²s⁻¹. w₀ = 10⁴.

Fig. 12 shows results for ν₂ = ν₃ = 0. We see that a non-motile mutant population of mutant cells is unable to invade the resident population in an inhomogeneous environment (generated by an initially heterogeneous distribution of oxygen): the two populations coexist via oscillations, with periods of stasis of the resident population and virtual (although not complete) elimination of the invading population interrupted by bursts of growth and decay.

Taken together, these results suggest that up-regulated cell motility increases the aggressiveness of the bistable mutant, as the time it takes to clear the resident population is inversely proportional to the cell motility coefficient. Our results also indicate that cell motility may be necessary for invasion into an inhomogeneous environment: if the invader cells are immobile, then both resident and mutant populations coexist in an oscillatory state.

We now proceed to determine whether the increases in invasiveness are induced by cell motility or the bistable structure (i.e. could a mutant cell with switch-like population dynamics and increased motility invade a switch-like, non-motile resident?). We address this question by considering a system in which a non-motile, switch-like population is invaded by a motile, switch-like mutant. In this case, the mean-field dynamics are given by the following system of PDEs:

$$\frac{\partial c}{\partial \tau }=\frac{{\partial }^{2}c}{\partial x^2}-\kappa (n_s+n_m)c$$ (31)

$$\frac{\partial n_s}{\partial \tau }=ϵ\text{tanh}({\alpha }_0(c-1))n_s$$ (32)

$$\frac{\partial n_m}{\partial \tau }=ϵ{\nu }_m\frac{{\partial }^2{n}_m}{\partial x^2}+ϵ\text{tanh}({\alpha }_0(c-1))n_m,$$ (33)

where n_s is the number of (resident) non-motile, switch-like cells and n_m is the number of (mutant) motile, switch-like cells. We prescribe no-flux boundary conditions for Eq. (33). The boundary conditions for Eq. (31) are given by Eq. (27) with $h_0-h_L=L{h}^{-1}\overline{\kappa }\Omega c_0.$ The initial conditions are:

$$\begin{array}{l}c(x,\tau =0)=1\hfill \\ n_s(x,\tau =0)=\Omega /L-\delta (x-L/2)\hfill \\ n_m(x,\tau =0)=\delta (x-L/2).\hfill \end{array}$$ (34)

Simulation results showing how the quantities

$$N_s(t)={\displaystyle {\int }_0^L{n}_s(x,t)dx\text{and}{N}_m(t)={\displaystyle {\int }_0^L{n}_m(x,t)dx}}$$ (35)

evolve over time are presented in Fig. 13 and suggest that increased motility is not sufficient for a mutant population to invade the switch-like population. Rather, that another mutation prior to increased motility (e.g. a mutation leading to bistable cells) is needed in order for the invader cells to invade.

Fig. 13.

This plot illustrates that up-regulation of cell motility alone is insufficient to enable a switch-like mutant to invade a resident switch-like population. Dash-dotted red lines correspond to N_s and solid blue lines to N_m, as defined in Eq. (31). Cell diffusion coefficients are given in μm²s⁻¹. w₀ = 10⁴.

6. Conclusions

In this paper we have presented a model that describes competition between two cell populations for a common (and limited) resource. The populations differ in the strategies that they use to exploit the resource and produce offspring. These alternative strategies are incorporated via response functions, which relate the level of activation of the reproductive pathway (i.e. intracellular levels of cell-cycle-activating proteins) to the amount of resource available (see Fig. 1): whereas the resident population exhibits a sigmoidal, switch-like response curve, the invading population is characterised by a bistable response curve. We used a variety of mathematical approaches (stochastic simulations, mean-field deterministic differential equations and branching processes theory), to investigate how the ability of the bistable population to invade the switch-like population depends on two factors, the levels of intracellular noise in the bistable cells and their motility. From our analysis, we conclude that:

• –For a well-mixed system, the quantity ω = w₀e^−α/2, which measures the effect of noise on the kinetics of phenotype change in bistable cells, plays a key role in controlling the ability of the bistable mutant population to invade the switch-like resident cells: when ω ≪ 1, intracellular noise levels are low (see Section 4.1), the bistable population has a competitive advantage and invades the resident switch-like population. If, however, ω ≳ 1, and intracellular noise levels are high (see Section 4.2), the bistable mutant population loses its competitive advantage and the competition becomes neutral (invasion is a random event, with probability equal to the initial proportion of mutant cells [29, 11]). Physically, ω represents the average frequency with which bistable cells change phenotype (from proliferative to quiescent and vice versa). This parameter can be altered by changing either the barrier-to-noise ratio, α, or the characteristic frequency ω₀
• –The behaviour of the system in the well-mixed limit when ω ⩾ 1 exhibits a number of features, in particular in its relation to the emergence of latency. In this regime, the dynamics of the bistable mutant population becomes critical in the sense that its net average growth rate is zero and its dynamics become dominated by fluctuations. However, the structure of the bistable population endows it with additional stability, enabling it to be longer-lived than a random population whose individuals do not possess such additional structure. This fact provides a mechanism for latency where small, long-lived mutant colonies do not invade the resident population but persist with non-zero probability until an additional growth-promoting event (e.g. gene mutation) occurs. This scenario could be relevant to a number of long-lived, latent infections, such as HIV-1 under anti-retroviral therapy [44], and also in tumour dormancy [1,51,13,50].
• –Cell motility seems to be a major contributing factor when spatial effects are taken into account. In the low noise limit, we observe that (i) the invader (bistable) cells must be motile if they are to invade, (ii) the aggressiveness of the invading population, measured in terms of the extinction time of the resident population, is positively correlated with the motility of the invader cells, and (iii) a non-motile bistable mutant cannot invade the switch-like resident in spatially inhomogeneous systems. Instead, both resident and mutant populations coexist via a quasi-steady oscillation, with periods of stasis, where the resident population remains constant and the invading population is reduced to very low levels, interrupted by bursts of rapid growth of the bistable mutant, followed by fast decay. Although we have shown that motility increases the likelihood of invasion of our bistable mutant, we have also shown that up-regulation of cell motility alone with no previous transformation is unable to produce an invasive mutant. In our case, this suggests that a sequence of mutations from switch-like to bistable to motile bistable may produce an aggressive invasive phenotype.

An important issue concerning the applicability of our model to biological situations is noise control at the level of intracellular pathways and gene regulatory networks. Recent studies have concluded that there are a number of mechanisms, including (but not limited to) negative feed-backs and oligomerisation, which act to reduce intracellular noise [6,12,4, 34]. Therefore, in principle, control of the level of intracellular noise is attainable through regulation of such negative feedbacks. Related to control of the levels of intracellular noise, we should also consider whether there are limits on noise suppression [37].

We have introduced the hypothesis that all cell types consume oxygen at the same rate. However, it is conceivable that cells using different strategies to utilise resources have different metabolic needs, and, therefore, that we need to consider species-specific oxygen consumption rates [10]. The impact of these factors on the mechanisms that regulate the transition from invasion to latency proposed in this paper is left as an open problem for future work.

Biological applications of our model centre around recent results where positive feed-back loops have been identified as necessary for bistable activation responses in pathways regulating apoptosis [35], cell survival [36], and cell-cycle progression [15]. Therefore, one can consider mixed populations in which the different cell types are characterised according to whether these positive feed-backs are functional: cells for which positive feed-back loops are active can be viewed as bistable while cells for which positive feed-backs are inactive, resemble switch-like ones. Such mixed populations, under environmental conditions of limited resources/signalling cues, undergo competition. Our model enables us to investigate which of these response mechanisms is robust in a given environment in terms of the outcome of such competition.