A detailed mathematical model for the fission yeast mitotic cycle is developed based on positive and negative feedback loops by which Cdc13͞Cdc2 kinase activates and inactivates itself. Positive feedbacks are created by Cdc13͞Cdc2-dependent phosphorylation of specific substrates: inactivating its negative regulators (Rum1, Ste9 and Wee1͞Mik1) and activating its positive regulator (Cdc25). A slow negative feedback loop is turned on during mitosis by activation of Slp1͞anaphase-promoting complex (APC), which indirectly re-activates the negative regulators, leading to a drop in Cdc13͞Cdc2 activity and exit from mitosis. The model explains how fission yeast cells can exit mitosis in the absence of Ste9 (Cdc13 degradation) and Rum1 (an inhibitor of Cdc13͞Cdc2). We also show that, if the positive feedback loops accelerating the G2͞M transition (through Wee1 and Cdc25) are weak, then cells can reset back to G2 from early stages of mitosis by premature activation of the negative feedback loop. This resetting can happen more than once, resulting in a quantized distribution of cycle times, as observed experimentally in wee1 ؊ cdc25⌬ mutant cells. Our quantitative description of these quantized cycles demonstrates the utility of mathematical modeling, because these cycles cannot be understood by intuitive arguments alone.

Background: There is considerable controversy concerning the exact growth profile of size parameters during the cell cycle. Linear, exponential and bilinear models are commonly considered, and the same model may not apply for all species. Selection of the most adequate model to describe a given data-set requires the use of quantitative model selection criteria, such as the partial (sequential) F-test, the Akaike information criterion and the Schwarz Bayesian information criterion, which are suitable for comparing differently parameterized models in terms of the quality and robustness of the fit but have not yet been used in cell growth-profile studies.

An analysis was made of cell length and cycle time in time-lapse films of the fission yeast Schizosaccharomyces pombe using wild-type (WT) cells and those of various mutants. The more important conclusions about ‘size controls’ are: (1) there is a marker in G2 in WT cells provided by a rate change point (RCP) where the linear rate of length growth increases by approximately 30%. The period before this RCP is dependent on size and can be called a ‘sizer’. The period after the RCP is nearly independent of size and can be called a ‘timer’. The achievement of a critical threshold size is at or near the RCP which is on average at about 0.3 of the cycle (halfway through G2). This is much earlier than was previously believed. (2) The RCP is at about the time when H1 histone kinase activity and the B type cyclin cdc13 start to rise in preparation for mitosis. The RCP is also associated with other metabolic changes. (3) In wee1 mutants, the mitotic size control is replaced by a G1/S size control which is as strong as the mitotic control. As in WT cells, there is a sizer which precedes the RCP followed by a timer but the RCP is at about the G1/S boundary and has a larger increase (approximately 100%) in rate. (4) cdc25 is not an essential part of the size control at mitosis or at the G1/S boundary. (5) Three further situations have been examined in which the mitotic size control has been abolished. First, induction synchronisation by block and release of cdc2 and cdc10. In the largest oversize-cells which are produced, the RCP is pushed back to the beginning of the cycle. There is no sizer period but only a timer. Second, when both the antagonists wee1 and cdc25 are absent in the double mutant wee1-50 cdc25 delta. In this interesting situation there is apparently no mitotic size control and the cycle times are quantised. Third, in rum1 delta wee1-50 where the normal long G1 in wee1 is much reduced, there is probably no size control either in G1 or in G2 causing a continuous shortening of division length from cycle to cycle.

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