Elucidating effects of reaction rates on dynamics of the lac circuit in Escherichia coli

Gene expression is regulated by a complex transcriptional network. It is of interest to quantify uncertainty of not knowing accurately reaction rates of underlying biochemical reactions, and to understand how they affect gene expression. Assuming a kinetic model of the lac circuit in Escherichia coli, regardless of how many reactions are involved in transcription regulation, transcription rate is shown to be the most important parameter affecting steady state production of mRNA and protein in the cell. In particular, doubling the transcription rate approximately doubles the number of mRNA synthesized at steady state for any rates of transcription inhibition and activation. On the other hand, increasing the rate of transcription inhibition by 10% reduces the average steady state count of mRNA by about 7%, whereas changes in the rate of transcription activation appear to have no such effect. Furthermore, for wide range of reaction rates in the kinetic model of the lac genetic switch considered, protein production was observed to always reach a maximum before the degradation reduces its count to zero, and this maximum was found to be always at least 27 protein molecules. Such value appears to be a fundamental structural property of genetic circuits making it very robust against changes in the internal and external conditions.