Chemical reaction networks involving molecular species at low copy numbers lead to stochasticity in protein levels in gene expression at the single-cell level. Mathematical modelling of this stochastic phenomenon enables us to elucidate the underlying molecular mechanisms quantitatively. Here we present a two-stage stochastic gene expression model that extends the standard model by an mRNA inactivation loop. The extended model exhibits smaller protein noise than the original two-stage model. Interestingly, the fractional reduction of noise is a non-monotonous function of protein stability, and can be substantial especially if the inactivated mRNA is stable. We complement the noise study by an extensive mathematical analysis of the joint steady-state distribution of active and inactive mRNA and protein species. We determine its generating function and derive a recursive formula for the protein distribution. The results of the analytical formula are cross-validated by kinetic Monte-Carlo simulation.
2020
LNSB
Stationary distributions and metastable behaviour for self-regulating proteins with general lifetime distributions
Candan Çelik, Pavol Bokes, and Abhyudai Singh
In International Conference on Computational Methods in Systems Biology , 2020
Regulatory molecules such as transcription factors are often present at relatively small copy numbers in living cells. The copy number of a particular molecule fluctuates in time due to the random occurrence of production and degradation reactions. Here we consider a stochastic model for a self-regulating transcription factor whose lifespan (or time till degradation) follows a general distribution modelled as per a multi-dimensional phase-type process. We show that at steady state the protein copy-number distribution is the same as in a one-dimensional model with exponentially distributed lifetimes. This invariance result holds only if molecules are produced one at a time: we provide explicit counterexamples in the bursty production regime. Additionally, we consider the case of a bistable genetic switch constituted by a positively autoregulating transcription factor. The switch alternately resides in states of up- and downregulation and generates bimodal protein distributions. In the context of our invariance result, we investigate how the choice of lifetime distribution affects the rates of metastable transitions between the two modes of the distribution. The phase-type model, being non-linear and multi-dimensional whilst possessing an explicit stationary distribution, provides a valuable test example for exploring dynamics in complex biological systems.
