Gene Expression Variability with Feedback Regulation Implemented via Protein-Dependent Cell Growth

Variability in gene expression among single cells and growing cell populations can arise from the stochastic nature of protein synthesis, which often occurs in random bursts. This study investigates the variability in the expression of a growth-sustaining protein, whose concentration is regulated by a negative feedback loop due to cell growth-induced dilution. We model the distribution of protein concentration using a Chapman-Kolmogorov equation for single cells and a population balance equation for growing cell populations. For single cells, we derive an explicit solution for the protein concentration distribution in state space and represent it as a Bessel function in Laplace space. For growing populations, we find that the distribution satisfies a Heun differential equation with singular boundary conditions. By addressing the central connection problem for the Heun equation, we quantify the population-level protein distribution and determine the Mathusian parameter, which characterizes population growth. This work provides a comprehensive analytical framework for understanding how stochastic protein synthesis impacts gene expression variability and population dynamics.