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

Abstract

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.

Coupling gene expression dynamics to cell size dynamics and cell cycle events: exact and approximate solutions of the extended telegraph model

“…(shown by the yellow dashed line) and when replication occurs halfway through the cell cycle (w = 0.5) [48]. Note that the region where D is minimized is exactly around the yellow dashed line.…”

“…Recently, numerous efforts have been made to extend the conventional one-state and two-state models to include some description of these processes and yet retain analytical tractability. Some studies focused on the moment statistics (mean and variance) of mRNA and protein numbers [40][41][42][43][44], while other studies additionally obtain the analytical distributions of molecule numbers [22,[45][46][47][48][49]. Please refer to Table 1 for a summary of exactly solvable extensions of the one-state and two-state models that explicitly capture cell birth, growth, and division.…”

“…[49] Refs. [22,48] Two-state models gene replication not considered gene replication considered volume-independent transcription × × volume-dependent transcription Ref. [49] × Table 1: Exactly solvable gene expression models that explicitly describe cell birth, growth, and division.…”

“…Due to mathematical complexity, most previous work is limited to the effective one-state model with the gene product (mRNA or protein) produced in a constitutive or bursty manner [22,[45][46][47][48]. Some of these models incorporate the scaling of transcription activity with cell volume [22,48], while the rest do not.…”

“…Due to mathematical complexity, most previous work is limited to the effective one-state model with the gene product (mRNA or protein) produced in a constitutive or bursty manner [22,[45][46][47][48]. Some of these models incorporate the scaling of transcription activity with cell volume [22,48], while the rest do not. We note that the latter case is not to be seen as unphysical since while the scaling of transcription with volume is commonly observed, it is by no means a universal phenomenon (in both prokaryotic [50,51] and eukaryotic cells [52][53][54][55] there are examples where there is no such scaling).…”