An inactive sequence of DNA and a transcription factor bind to produce an active gene G. This produces mRNA, denoted by M at a rate α₁, and in turn the mRNA produces protein at rate α₂. Eventually, the transcription factor will unbind (at rate β₁), and the gene will become inactive again. Each copy of mRNA produced will also be degraded (at rate β₂).

Here, G represents an active gene, G* an active gene with a bound RNA polymerase, M an mRNA molecule, M* an mRNA molecule bound to a ribosome, P a protein, and S₀ states which correspond to transcription factor unbinding and mRNA transcript decay.

In both cases the blue line shows the best fit of the model to the data, obtained using the method of maximum likelihood giving Â₂ = 3.57 for Tsr-Venus and Â₂ = 20.96 for β-gal. The error bars show the upper and lower bounds of the 95% confidence interval for the fitted parameter.

For A we have α₁ = 0.018 and β₁ = 0.086 and for B we have α₁ = 0.009 and β₁ = 0.043, resulting in the same Â₂ and hence identical burst size distributions. Other parameters were α₀ = 0.012, α₂ = 0.013, β₂ = 0.0039, and β₃ = 0.0007, based on previous simulation studies [22]. The distributions shown are for a run of 10,000 seconds using the Stocks implementation of Gillespie's method [23], after an initial transient of 10,000 seconds. Previous studies have indicated that the steady state is in fact attained in under 1000 seconds.

The results of 1000 runs of the Nelder-Mead maximisation of the log-likelihood for the parameters α₀, α₁, β₁, and β₃, with α₂ determined by the relationship in Equation 6, and with the mRNA degradation rate β₂ set to 7.2×10⁻³, corresponding to a half-life for β-gal mRNA of 1.6 mins [24]. The panels in column A show the estimates of the values of the parameters and the percentage of times the Nelder-Mead algorithm converged to those values. The panels in column B are scattergrams of the values of the parameter estimates against the value of the log-likelihood. Each simulation is run 10,000 times to simulate a population of 10,000 cells, and each simulation is run for 5000 reaction steps. The starting values for the optimisation routine are: α₀ = 0.01 s⁻¹, α₁ = 0.02 s⁻¹, β₁ = 0.1 s⁻¹, and β₃ = 0.0007 s⁻¹, and are based on previous simulation studies [16].

The results of 10,000 runs of the Nelder-Mead maximisation of the log-likelihood for the parameters α₀, α₁, β₁, and β₂, with α₂ determined by the relationship in Equation 6, and with the protein degradation rate set, β₃ set to 2.77×10⁻⁴, consistent with a half-life for β-gal 60 mins. The panels in column A show the estimates of the values of the parameters and the percentage of times the Nelder-Mead algorithm converged to those values. The panels in column B are scattergrams of the values of the parameter estimates against the value of the log-likelihood. Each simulation is run 10,000 times to simulate a population of 10,000 cells, and each simulation is run for 5000 reaction steps. The starting values for the optimisation routine are: α₀ = 0.01 s⁻¹, α₁ = 0.02 s⁻¹, β₁ = 0.1 s⁻¹, and β₂ = 0.007 s⁻¹, and are based on previous simulation studies [16].