Figure 1
Stochastic corrections to rates of enzymatic reactions and efficiency of transcriptional gene-regulation.
(A) Normalised rate v0 of a single-enzyme reaction as a function of time for bursty substrate production. The system is initially void of substrate-molecules. They are produced in batches of size 40 with the production times determined by a Poisson process and there are on average 117.6 molecules present in the stationary state. The variance-corrected MME (red solid line) coincides almost exactly with the simulation results (circles). The uppermost, blue solid line is the Michaelis-Menten upper bound based on the mean density alone and the lowermost, green solid line shows the optimal lower bound. The figure also shows the third (dashed) and fourth (dash-dot) order Taylor approximation to 〈v0〉. Squares are the solution of Stefanini et al.14 for the system with batch size 1. Inset: Predictions by the usual MME and optimal lower bound deviate linearly from the variance-corrected MME and simulation results with increasing batch size. (B) The occupation probability p0 of the TF-Operator complex as a function of time in a non-stationary gene regulation experiment. The uppermost, blue solid line is the Michaelis-Menten upper bound based on the mean density alone and the lowermost, green solid line shows the optimal lower bound. The variance-corrected MME (red solid line) yields an excellent approximaxxtion to the results (12)–(18) of the analytical model (dashed line) and the simulation results (circles). The pink dash-dotted line is the Poissonian approximation given by Eq. (19). Inset: The density (blue solid line) and population variance (purple dashed line) of the number of TF molecules in a single cell as a function of time.
