Figure 1.

Modeling single gene expression. (a) mRNA molecules are synthesized at rate k_R from the template DNA strand. Proteins are translated at a rate k_P off each mRNA molecule. Proteins and mRNA degrade at rates γ_P and γ_R, respectively. Degradation into constituents is denoted by φ. All reactions are assumed to be Poisson, so that the probability of a reaction with rate k happening in a time dt is given by kdt, and the waiting times between successive reactions are exponentially distributed. (b) A simplified timecourse illustrates the intuition behind the result of Eq. 1. Transcription initiation events occur with average frequency k_R and are indicated by arrows; each mRNA transcript releases a burst of proteins of average size b, and proteins decay between bursts. (c) Controlling mean and variance of protein number by varying system parameters for single gene expression. The Fano factor ratio (δp²/<p> ≡ variance/mean) is plotted versus the mean. The mRNA half-life is fixed at 2 min. The base case corresponds to a burst size b = 20, a transcript initiation rate k_R = 0.01 s⁻¹ and a protein half-life ln(2)/γ_P = 1 h. The three curves are produced by varying one of these parameters while keeping the other two fixed. b is varied between 5 and 40 (circles); k_R is varied between 0.0025 s⁻¹ and 0.02 s⁻¹ (triangles); protein half-life is varied from 15 min to 2 h (squares). The Poisson value of δp²/<p> = 1 is marked for comparison. Monte Carlo results (symbols) match analytic values given by Eq. 1 exactly (solid lines).