Bridging time scales in cellular decision making with a stochastic bistable switch

Abstract

Background: Cellular transformations which involve a significant phenotypical change of the cell's state use bistable biochemical switches as underlying decision systems. Some of these transformations act over a very long time scale on the cell population level, up to the entire lifespan of the organism.

Results: In this work, we aim at linking cellular decisions taking place on a time scale of years to decades with the biochemical dynamics in signal transduction and gene regulation, occurring on a time scale of minutes to hours. We show that a stochastic bistable switch forms a viable biochemical mechanism to implement decision processes on long time scales. As a case study, the mechanism is applied to model the initiation of follicle growth in mammalian ovaries, where the physiological time scale of follicle pool depletion is on the order of the organism's lifespan. We construct a simple mathematical model for this process based on experimental evidence for the involved genetic mechanisms.

Conclusions: Despite the underlying stochasticity, the proposed mechanism turns out to yield reliable behavior in large populations of cells subject to the considered decision process. Our model explains how the physiological time constant may emerge from the intrinsic stochasticity of the underlying gene regulatory network. Apart from ovarian follicles, the proposed mechanism may also be of relevance for other physiological systems where cells take binary decisions over a long time scale.

Figure 1

Network schematic for the bistable switch model (1).

Figure 2

Characterisation of the phase space in the bistable switch model (1). A: Phase space diagram for the deterministic model of the bistable switch. Black lines are nullclines for the variables x and y in the deterministic switch model (1), with their intersections corresponding to equilibria of the switch. I and III are stable equilibrium points, II is an unstable one. Trajectories converge to either I or III, depending on the initial condition, as shown for the sample trajectories plotted as light blue lines. B: Schematic illustration of the configuration space for the Markov process (5) describing the cell transformation process. Circular nodes below the dashed line correspond to possible configurations (X; Y)^T of the switch, and the arrows between the nodes correspond to transitions in the configuration due to reactions. The configurations above the dashed line are collapsed into the on state, which is assumed to be irreversible due to subsequent transformation processes.

Figure 3

Steady state probability distribution for the stochastic bistable switch. 500 realizations of the stochastic reaction network model (3) were generated using the Gillespie algorithm in the stochastic simulation software Dizzy [41,42]. Each realization was for a simulated time of 300 years, and the steady state probability distribution was generated from the samples after discarding a transient phase of 50 years simulated time, using a total of about 5 · 10⁷ data points.

Figure 4

Hypothetical biochemical network for the primordial to primary transition in ovarian follicles.

Figure 5

Dynamical characteristics of the stochastic bistable switch on the single cell level and the population level. A: Probability of transformation event p_on(t) B: Population size probability distribution over time. C: Probability density function of the depletion time T_d.

Figure 6

Evolution of follicle number. Model predictions from (11) (line) vs. clinical data from [37] (crosses).