G1/S transition and cell population dynamics

In this paper we present a model connecting the state of molecular components during the cell cycle at the individual level to the population dynamic. The complexes Cyclin E/CDK2 are good markers of the cell state in its cycle. In this paper we focus on the first transition phase of the cell cycle ($S-G_{2}-M$) where the complexe Cyclin E/CDK2 has a key role in this transition. We give a simple system of differential equations to represent the dynamic of the Cyclin E/CDK2 amount during the cell cycle, and couple it with a cell population dynamic in such way our cell population model is structured by cell age and the amount of Cyclin E/CDK2 with two compartments: cells in the G1 phase and cells in the remainder of the cell cycle ($S-G_{2}-M$). A cell transits from the G1 phase to the S phase when Cyclin E/CDK2 reaches a threshold, which allow us to take into account the variability in the timing of G1/S transition. Then the cell passes through $S-G_{2}-M$ phases and divides with the assumption of unequal division among daughter cells of the final Cyclin E/CDK2 amount. The existence and the asymptotic behavior of the solution of the model is analyzed.