Control of synchronization ratios in clock/cell cycle coupling by growth factors and glucocorticoids

The cell cycle and the circadian clock are essential cyclic cellular processes often synchronous in healthy cells. In this work, we use previously developed mathematical models of the mammalian cell cycle and circadian cellular clock in order to investigate their dynamical interactions. Firstly, we study unidirectional cell cycle → clock coupling by proposing a mechanism of mitosis promoting factor (MPF)-controlled REV-ERBα degradation. Secondly, we analyse a bidirectional coupling configuration, where we add the CLOCK : BMAL1-mediated MPF repression via the WEE1 kinase to the first system. Our simulations reproduce ratios of clock to cell cycle period in agreement with experimental observations and give predictions of the system’s synchronization state response to a variety of control parameters. Specifically, growth factors accelerate the coupled oscillators and dexamethasone (Dex) drives the system from a 1 : 1 to a 3 : 2 synchronization state. Furthermore, simulations of a Dex pulse reveal that certain time regions of pulse application drive the system from 1 : 1 to 3 : 2 synchronization while others have no effect, revealing the existence of a responsive and an irresponsive system’s phase, a result we contextualize with observations on the segregation of Dex-treated cells into two populations.


Period Measurement Methods
Computing the period of a system in a biological context poses particular issues. In a system with a complex periodic behavior, different levels of protein peak expression may appear and it is fundamental to identify which ones would have biological meaning.
Using the mathematical period T of the solution, x(t) = x(t + T ) leads to the rejection of peaks of protein expression that may be relevant in a biological context. For example, if the clock has a complex behavior, peaks of protein expression occuring within the time interval of one mathematical period could in the real system indicate a detectable clock oscillation if high enough relatively to the protein's maximum value. Similarly, in a bidirectional coupling configuration, where MPF may also exhibit complex behavior, a peak of MPF that is slightly below the maximum may be sufficient to carry out the mitotic process, while peaks bellow a certain threshold have no effect in mitosis, which is a discrete process. Thus, it is necessary to select a threshold for the selection of relevant protein peaks.
Our algorithm is based on a numerical implementation of the first return map [1]. We consider for each protein a threshold of 50 % of their maximum value and compute the time differences between the points of the two Poincaré sections crossing the threshold (one section containing points where the signal is increasing and the other the points where the signal is decreasing), and average them. This method considers only the peaks that are above the threshold and ignores small amplitude peaks of our numerical solutions, that wouldn't be distinguishable in experimental results. Thus, in some cases of complex behavior, this period will not correspond to the mathematical period T of the solution, but will nevertheless provide a more realistic estimate of the periodlock ratios. This approach allows to better relate our work with experimental observations of clock to cell cycle period ratios observed in mammalian cells [2]. In fact, in biology normal variation among protein peaks is to be expected.
In Fig. 1 the simulation shown in Fig. 3 of the main article (bottom panel) is repeated for the thresholds of 30 and 99 % of the maximum value. One main difference between Fig. 3 and Fig. 1 A) is that the shifts between synchronization states occur for different values of GF. Nevertheless, the same values of r T appear. Moreover, Fig. 1 B) shows the synchronization state for a very high protein threshold (99 % of the maximum value), which results in computing the mathematical period, where the 6:1 ratio occurs.
Furthermore, Fig. 2 gives an example of a solution of unidirectional cell cycle → clock coupling, where the clock has a complex behavior: observe that a complete clock period occurs during the time inter-val of 6 cell cycle periods, however there are several intermediary peaks of BMAL1 and REV that are high enough relatively to their maximum value and could in the real system indicate a detectable clock oscillation. In this simulation, using either a 30 % or 50 % threshold to define the relevance of peaks allows to detect 4 peaks of BMAL1 to 6 of MPF (3:2 period-lock), but using, for instance, a 75 % threshold, would only count 3 peaks of BMAL1 as relevant in the time interval of 6 MPF oscillations (2:1 periodlock). The 99 % threshold measures the mathematical period, where a repetition of the higher peak of BMAL1 occurs for 6 peaks of MPF (6:1 period-lock) and all remaining BMAL1 peaks are ignored.
Because coupling from the cell cycle is being modeled as an action on REV, this protein tends to have a more complex behavior than the remaining clock proteins -in Fig. 2 using a threshold of 75 % would count 3 peaks of BMAL1 and only 2 of REV in the time interval of 6 MPF periods, while the 30 % and 50 % thresholds allow to count 4 peaks of each of these proteins in the same time interval. In this work, we base our computations of period-lock ratios on BMAL1 to avoid possible errors. However, using the chosen threshold (50 %) of mean protein value for each protein tends to yield similar results between REV and the other proteins.      The clock and cell cycle periods vary nonlinearly with Dex: first with a region without oscillation then decreasing and then increasing again. Fig. 7: Convergence to the 1:1 period-lock state after the application of a Dex-pulse at different circadian phases over the course of two periods. 1000 hours after the shift caused by the Dex pulse (observed in Fig. 11 of the main article) the system has returned to 1:1 synchronization: changes in periodlock caused by Dex input are transient.