Analytical Expressions and Physics for Single-Cell mRNA Distributions of the lac Operon of E. coli

Mechanistic models of stochastic gene expression are of considerable interest, but their complexity often precludes tractable analytical expressions for messenger RNA (mRNA) and protein distributions. The lac operon of Escherichia coli is a model system with regulatory elements such as multiple operators and DNA looping that are shared by many operons. Although this system is complex, intuition suggests that fast DNA looping may simplify it by causing the repressor-bound states of the operon to equilibrate rapidly, thus ensuring that the subsequent dynamics are governed by slow transitions between the repressor-free and the equilibrated repressor-bound states. Here, we show that this intuition is correct by applying singular perturbation theory to a mechanistic model of lac transcription with the scaled time constant of DNA looping as the perturbation parameter. We find that at steady state, the repressor-bound states satisfy detailed balance and are dominated by the looped states; moreover, the interaction between the repressor-free and the equilibrated repressor-bound states is described by an extension of the Peccoud-Ycart two-state model in which both (repressor-free and repressor-bound) states support transcription. The solution of this extended two-state model reveals that the steady-state mRNA distribution is a mixture of the Poisson and negative hypergeometric distributions, which reflects mRNAs obtained by transcription from the repressor-bound and repressor-free states. Finally, we show that the physics revealed by perturbation theory makes it easy to derive the extended two-state model equations for complex regulatory architectures.