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 ($P_S(t)\sim t^{-1}$) 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.

Appendix: Noise in bistable systems

This appendix is devoted to a short summary of the effect of noise in bistable systems. Our aim is not to give a full account of stochastic dynamics of bistable systems, but to provide a brief justification of why phenotype switching in the bistable population can be modelled as an activated process and, consequently, the corresponding transition rates as functions of the Arrhenius type. For a full account, we refer the reader to the the extensive literature on the subject, in particular Gardiner (1983), Maier and Stein (1996), Horsthemke and Lefever (2006), Gardiner (2009).

For concreteness, we consider the following dynamical system:

$$\begin{aligned}&\frac{dx}{dt}=\frac{k_3^{\prime }(1-x)}{J_3+(1-x)}-\frac{k_4myx}{J_4+x}\nonumber \\&\frac{dy}{dt}=k_1-(k_2^{\prime }+k_2^{\prime \prime }x)y. \end{aligned}$$

(36)

This system was proposed by Tyson and Novak (2001) as the central component of a more complex pathway regulating the G$_1$/S transition in the cell-cycle of eukaryote cells. In their original model, $x$ stands for the (normalised) concentration of active Cdh1/APC complexes, $y$ for the concentration of active CycB/CDK complexes, and $m$ for the cell size. Here, $m$ will be assumed to be the control parameter.

As $m$ increases, Eqs. (36) go through a series of bifurcations which separate different regimes (see Fig. 14): For small values of $m$ the system exhibits a single steady-state (high $x=$ Cdh1, low $y=$ CycB corresponding to G$_1$). Intermediate values of $m$ lead to a bistable regime where two stable steady-states (high Cdh1, low CycB corresponding to G$_1$ and low Cdh1 and high CycB, corresponding to S–G$_2$–M) coexist with a saddle point. Finally, as $m$ continues to increase a saddle-node bifurcation occurs which leads to the annihilation of the G$_1$-like steady-state and the saddle point. In each of the panels of Fig. 14, we show two solutions of Eqs. (36) corresponding to two different initial conditions: $y(t=0)=0.28$ and $x(t=0)=0.9$ (solid green lines) and $y(t=0)=0.3$ and $x(t=0)=0.9$ (solid purple lines). We have also plotted realisations of a stochastic system equivalent to Eqs. (36) (see Guerrero and Alarcón (2015) for details) with the same initial conditions (dashed green lines and dashed purple lines, respectively). In the two mono-stable cases shown in panels $m=0.01$ and $m=1$, we see that there are no major differences in behaviour between the mean-field (solid lines) and the stochastic (dashed lines) systems: In both cases the stochastic trajectories converge toward the mean-field fixed point, regardless of the initial condition. The behaviour in the bistable regime offers more possibilities. We have chosen the initial conditions (for both the mean-field equations and the stochastic system) so that they belong to different basins of attraction of the mean-field system: $y(t=0)=0.28$ and $x(t=0)=0.9$ (solid green lines) belongs to that of G$_1$, $y(t=0)=0.3$ and $x(t=0)=0.9$ (solid purple lines), to that of S-G$_2$-M the bistable regime ($m=0.2$). In this regime, we can see that the stochastic trajectories (dashed lines) may either behave like their mean-field counterparts and converge towards the corresponding steady-state, or, on the contrary, jump across the separatrix and converge towards the steady-state corresponding to the other (mean-field) basin of attraction.

Fig. 14

Phase spaces showing the three possible steady-state regimes associated with Eqs. (36) for different values of the control parameter $m$. In all three panels the solid blue line corresponds to the $y$-nullcline and the red line to the $x$-nullcline. The intersections between both nullclines correspond to the fixed points of Eqs. (36). For small values of $m$, e.g. $m=0.01$, the only (stable) fixed point is the $\hbox {G}_1$-like steady-state. For intermediate values of $m$, e.g. $m=0.2$, Eqs. (36) exhibit bistability: The system has two stable steady states $(\hbox {G}_1$ and $\hbox {S-G}_2-\hbox {M})$ and an unstable fixed (saddle) point. For large values of $m$, e.g. $m=1$, the only (stable) fixed point is the $\hbox {S-G}_2\hbox {-M-like steady-state}$. Parameter values: $k_1=0.04\, \hbox {min}^{-1}, \hbox {min}^{-1}$, $k_2^{\prime }=0.04$ $\hbox {min}^{-1}$, $k_2^{\prime \prime }=1$, $k_3^{\prime }=1\, \hbox {min}^{-1}$, $k_4=35\, \hbox {min}^{-1}$, $J_3=J_4=0.04$ Tyson and Novak (2001) (colour figure online)

Let us focus now on the bistable regime (e.g. $m=0.2$ in Fig. 14). If noise is ignored, then the stable steady-states have disjoint basins of attraction, being separated by a separatrix which passes through the saddle: If the initial condition is contained in the $G_1$-basin (respectively, S-G$_2$-M) then the system will evolve towards $G_1$ (respectively, S-G$_2$-M). By contrast, if noise is taken into account then the separatrix becomes a barrier that the system may cross with finite probability. This is shown in Fig. 14 where we compare the solution of Eqs. (36) with two different initial conditions, one on each side of the separatrix, with a stochastic version of the Tyson & Novak model developed in Guerrero and Alarcón (2015).

There is extensive literature Escudero and Kamenev (2009), Gardiner (1983), Gardiner (2009), Horsthemke and Lefever (2006), Maier and Stein (1996) showing that, in the limit of low noise intensity, $\sigma$, the process of switching between two stable states is an activated process, i.e. the transition rate from one steady state to the other, $W_T$, is such that $W_T\sim e^{-H}$ for $\sigma \ll 1$ where $H$ is a function of the parameters of the system.

While a detailed mathematical derivation is beyond the scope of this appendix, the physical rationale is relatively straightforward Kubo et al. (1973). In equilibrium statistical mechanics, the statistical distribution for an extensive variable, say $X$, is given by $P_e(X)=Z^{-1}e^{-\frac{{V}}{k_BT}\phi _e(x)}$, where ${V}$ is the volume of the system, $x=X/{V}$, $k_BT$ is the energy associated with thermal noise ($T$ is the temperature and $k_B$ is Boltzmann’s constant), and $\phi _e(x)$ is the equilibrium free energy per unit volume. From elementary considerations in equilibrium thermodynamics, we know that the equilibrium value of $x$ corresponds to the minimum of $\phi _e(x)$ which, in turn, corresponds to the most probable value of $x$ according to the probability distribution $P_e(X)$, with $P_e(X)$ providing the probability of deviation of $x$ from its optimal value, $x_e$. Incidentally, we remark that for low noise intensity, i.e. for $k_BT$ much smaller than a characteristic energy scale associated with $\phi _e(x)$, large deviations from $x_e$ are very unlikely as they are exponentially suppressed Touchette (2009). We note also that the larger the size of the system, ${V}$, the more insensitive the system becomes to random effects.

The essential ansatz leading to $W_T\sim e^{-H}$ is that non-equilibrium (i.e. time-evolving) states of large systems can be described by a probability distribution which is a straightforward generalisation of $P_e$, i.e.:

$$\begin{aligned} P(X(t)\vert X_0)=\mathcal{C}e^{-\frac{{V}}{\sigma }\phi (x(t)\vert x_0)} \end{aligned}$$

(37)

where $x(t)=X(t)/{V}$ is a sample path or realisation of the stochastic process, $x_0=x(0)$ denotes the initial condition and $\sigma$ is the intensity of the noise (which plays the same role as the temperature $k_BT$ in the equilibrium case). Therefore, the most likely evolution for $x(t)$ is along the path that minimises $\phi (x(t)\vert x_0)$, and $P(X(t)\vert X_0)$ can, again, be understood as the probability that a sample path deviates from the optimal (most probable) path, $x_o(t)$. For large systems, $\varOmega \gg 1$, such deviations are very unlikely (they are exponentially suppressed) and the most significant contribution comes from $x_o(t)$. Hence, provided the noise $\sigma \equiv {V}^{-1}\ll 1$, the transition probability of the system to start from $X_0$ and evolve to a state $X_1$ at time $t$, $W_T$, can be estimated by taking $\phi =\phi _0\equiv \phi (x_o(t)\vert x_0)$ in Eq. (37), namely, $W_T\sim e^{-H}$ where $H\equiv \frac{{V}}{\sigma }\phi _0$.

Although we have presented this argument in an heuristic (non-rigorous) way, it can be made mathematically rigorous within the framework of the theory of large deviations Kubo et al. (1973), Touchette (2009).