Cells of the cyanobacterium Synechococcus elongatus possess a circadian clock in the form of three core clock proteins (the Kai proteins) whose concentrations and phosphorylation states oscillate with daily periodicity under constant conditions. The circadian clock regulates the cell cycle such that the timing of cell divisions is biased towards certain times during the circadian period, but the mechanism underlying how the clock regulates division timing remains unclear. Here, we propose a mechanism in which a protein limiting for division accumulates at a rate proportional to cell volume growth and modulated by the clock. This "modulated rates" model, in which the clock signal is integrated over time to affect division timing, differs fundamentally from the previously proposed "gating" concept, in which the clock is assumed to suppress divisions during a specific time window. We found that while both models can capture the single-cell statistics of division timing in S. elongatus , only the modulated rates model robustly places divisions away from darkness during changes in the environment. Moreover, within the framework of the modulated rates model, existing experiments on S. elongatus are consistent with the simple mechanism that division timing is regulated by the accumulation of a division limiting protein in phase with genes whose activity peak at dusk.
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