Fig. 1

Stochastic model for an enzymatic reaction. a The model integrates reversible Michaelis–Menten kinetics with the three-stage model for gene expression^24,27. The model includes consumption of the metabolite by downstream pathways, degradation of mRNA transcripts, and dilution of all chemical species by cell growth (not shown in the diagram); rate constants are shown in the figure and model reactions are shown in Eqs. (R1)–(R9), Methods. The inset shows a typical simulation for a realistic parameter set shown in Table 1. b Construction of the Poisson Mixture Model (PMM) for the number of metabolite molecules (n_p). This approximation is valid under a separation of timescales between enzyme expression and enzyme catalysis. The mixture model, shown in Eq. (1), comprises Poisson distributions weighted by the distribution of enzyme expression P(n_etot). The Poisson parameter λ(n_etot) depends on enzyme kinetics via the nonlinear relation in Eq. (4). In the irreversible case (k_rev = 0), the λ(n_etot) parameter scales linearly and produces equi-spaced Poisson modes. The first mode, Poisson (n_p, 0), is highlighted as a bar. c The PMM provides an accurate approximation of the stationary distributions. Insets show distributions for enzyme and metabolite, computed via Gillespie simulations and the PMM approximation for fixed λ_∞ = 1080 molecules, K = 8 molecules, and three different promoter switching parameters, shown in Table 1