Concentration fluctuations in growing and dividing cells: Insights into the emergence of concentration homeostasis

Intracellular reaction rates depend on concentrations and hence their levels are often regulated. However classical models of stochastic gene expression lack a cell size description and cannot be used to predict noise in concentrations. Here, we construct a model of gene product dynamics that includes a description of cell growth, cell division, size-dependent gene expression, gene dosage compensation, and size control mechanisms that can vary with the cell cycle phase. We obtain expressions for the approximate distributions and power spectra of concentration fluctuations which lead to insight into the emergence of concentration homeostasis. We find that (i) the conditions necessary to suppress cell division-induced concentration oscillations are difficult to achieve; (ii) mRNA concentration and number distributions can have different number of modes; (iii) two-layer size control strategies such as sizer-timer or adder-timer are ideal because they maintain constant mean concentrations whilst minimising concentration noise; (iv) accurate concentration homeostasis requires a fine tuning of dosage compensation, replication timing, and size-dependent gene expression; (v) deviations from perfect concentration homeostasis show up as deviations of the concentration distribution from a gamma distribution. Some of these predictions are confirmed using data for E. coli, fission yeast, and budding yeast.Author summary: Experiments show that often the number of mRNA or protein in a cell is proportional to its volume, i.e. as a cell grows, the mRNA or protein concentration remains approximately constant. This suggests that the maintenance of a constant concentration, i.e. concentration homeostasis, is important to proper cellular function and that there are mechanisms responsible behind this regulation. However most mathematical models of gene expression do not describe the coupling of transcription and cell size; the few models that do include such a description, ignore many salient aspects of cell dynamics and hence it is presently difficult to understand how concentration homeostasis emerges. In this article, we develop a mathematical model of gene expression that overcomes the aforementioned difficulties; it includes a detailed description of cellular biology such as cell growth, cell division, size-dependent gene expression, DNA replication, and cell size control mechanisms that can vary with the cell cycle phase. We devise a method to approximately solve this complex model to obtain expressions for the distributions and power spectra of concentration fluctuations. This allows us to deduce that accurate concentration homeostasis results from a complex tradeoff between many mechanisms. We confirm our theoretical predictions using data for bacteria and yeast.