Modeling cell size control using differential scaling of cell-cycle regulators with cell size

Accurate timing of division and size homeostasis is crucial for cells. A potential mechanism for cells to decide the timing of division is the differential scaling of regulatory protein copy numbers with cell size. However, it remains unclear whether such a mechanism can lead to robust growth and division, and how the scaling behaviors of regulatory proteins affect the cell size distribution. In this study, we formulate a mathematical model combining gene expression and cell growth, in which the cell-cycle inhibitors scale sublinearly with cell size while the activators scale superlinearly. The cell divides once the ratio of their concentrations reaches a threshold value. We find that the cell can robustly grow and divide within a finite range of the threshold value. Intriguingly, the cell size at birth is proportional to the ploidy, in agreement with experiments. In a stochastic version of the model, the cell size at division is uncorrelated with that at birth. Also, the more differential the cell-size scaling of the cell-cycle regulators is, the narrower the cell-size distribution is. Finally, after the deletion of a regulator, the average cell size can change significantly while the coefficient of variation of cell size remains roughly the same, consistent with experimental observations. Author summary How cells determine the timing of cell division is a fundamental question of cell biology. It has been found that the concentration of cell-cycle activators tends to increase with cell size, while the concentration of inhibitors tends to decrease. Therefore, an attractive hypothesis is that the ratio of activators to inhibitors may trigger cell division. To investigate this hypothesis quantitatively, we formulate a model including gene expression and cell growth simultaneously. The cell divides once the activator-to-inhibitor ratio reaches a threshold. Combining theories and simulations, we analyze the conditions of robust cell cycle and the cell size distribution. Our model successfully rationalizes several experimental observations, including the relation between cell size and ploidy, the sizer behavior of cell size control, and the change of the mean and fluctuation of cell size distribution after regulator deletion.