Noise-induced multistability in chemical systems: Discrete versus continuum modeling

The noisy dynamics of chemical systems is commonly studied using either the chemical master equation (CME) or the chemical Fokker-Planck equation (CFPE). The latter is a continuum approximation of the discrete CME approach. It has recently been shown that for a particular system, the CFPE captures noise-induced multistability predicted by the CME. This phenomenon involves the CME's marginal probability distribution changing from unimodal to multimodal as the system size decreases below a critical value. We here show that the CFPE does not always capture noise-induced multistability. In particular we find simple chemical systems for which the CME predicts noise-induced multistability, whereas the CFPE predicts monostability for all system sizes.

Figure 1

Comparison of the stationary distribution of protein concentration obtained from SSA simulations with the QSA solution $\pi \left({\varphi }_P\right)$ obtained by Eq. (12), the numerical solution of the CFPE given by Eq. (3) and the CCLE given by Eq. (13) for different values of volume. The volumes used are $\Omega =1,10,50,100$ for (a)–(d), respectively; since the gene concentration ${\varphi }_N$ is fixed to 1, this implies that the gene copy numbers $N$ are the same as the volumes. Parameters (see text) are such that timescale separation is enforced. The QSA and SSA predict multimodality below a certain volume whereas the CFPE predicts a unimodal distribution for all volumes.

Figure 2

Plot of the number of modes of $\pi \left({\varphi }_P\right)$ as a function of the volume $\Omega$, for different values of $\delta =\sqrt{q_2}-\sqrt{q_1}$. The larger is the difference between the production rates $q_1,q_2$, the larger is the volume range over which the stationary distribution of the CME is multimodal; in contrast, the solution of the CFPE is unimodal for all volumes.

Figure 3

Comparison of the stationary distribution of protein concentration obtained from SSA simulations, the numerical solution of the CFPE for different values of volume, $\Omega$, and simulations of the CCLE for the genetic feedback loop Eq. (14). See text for parameter values. Note that as for the gene feedback loop with no feedback (Fig. 1), the CFPE does not capture the onset of multimodality as the volume (and the number of gene molecules $N)$ is decreased below a critical value.