A hybrid stochastic model of the budding yeast cell cycle

The growth and division of eukaryotic cells are regulated by complex, multi-scale networks. In this process, the mechanism of controlling cell-cycle progression has to be robust against inherent noise in the system. In this paper, a hybrid stochastic model is developed to study the effects of noise on the control mechanism of the budding yeast cell cycle. The modeling approach leverages, in a single multi-scale model, the advantages of two regimes: (1) the computational efficiency of a deterministic approach, and (2) the accuracy of stochastic simulations. Our results show that this hybrid stochastic model achieves high computational efficiency while generating simulation results that match very well with published experimental measurements.

cell into mitosis (M phase). As CKIs are removed, Clb2 level rises because Clb2 activates its own transcription factor Mcm1 in an autocatalytic reaction. Rising Clb2:Cdc28 activity phosphorylates and inactivates the transcription factors SBF and MBF. In telophase, Clb2 must be degraded below a threshold so that the cell can exit mitosis and return to G1 phase. Clb2 degradation is initiated by two proteins, Cdc20 and Cdh1. Cdc20 has been kept inactive in the early stages of mitosis by the 'mitotic checkpoint complex'. When all chromosomes are properly aligned on the mitotic spindle, Cdc20 becomes active and facilitates the degradation of Clb2.
Moreover, as a yeast cell exits mitosis, Cdh1 is activated by a phosphatase, Cdc14, which has been sequestered in the nucleolus by binding to Net1. After full chromosome alignment on the metaphase plate, first Tem1 and then Cdc15 become active. Cdc15 phosphorylates Net1, which leads to the release of Cdc14. Next, Cdc14 activates Cdh1 (which takes over for Cdc20 as the primary initiator of Clb1-6 degradation in G1), and Cdc14 also activates the transcription factor, Swi5, for production of CKIs in G1 phase. In this way the scene is set for the cell to return to G1 phase, when the CKIs are abundant and all cyclins (except for Cln3) are out of the picture.

Variables, equations, reactions and parameter values
c clb2 ·mass ·Clb2+ k as,f5 ·Clb2·(AP C−AP CP) Reset rules: When the normalized concentration of Clb2 (denoted as [Clb2] n ) drops below K ez , we reset the auxiliary proteins [BUD] n and [SPN] n to zero, and divide all species in the cell, except for Cln3 and Bck2, between daughter and mother cells with a 40:60 ratio, according to observations by Di Talia et al. [1].
This ratio for Cln3 and Bck2 is set to 20:80 to match with the experimental observations in [2,3]. When The reset rules in our hybrid model are similar to the deterministic model by Chen et al. [4]. However, to prevent events from misfiring multiple times due to stochastic fluctuations in the hybrid model, we follow specific tactics [5]. In our model, there are two types of events: First, events that are associated with the states of certain species that change across predefined thresholds in only one direction (increasing or decreasing), taking BUD as an example. Second, events that are associated with the states of certain species that change across predefined thresholds in both directions (increasing and decreasing), taking Clb2 as an instance.
For the first type of events, the thresholds are set to the original predefined values at the beginning of the cell cycle. Once an event is triggered, the corresponding threshold is increased by multiplying it with a large number (e.g., 1000) to prevent misfiring.
For the second type of events, we add two dummy events to prevent misfiring. Supplementary figure S1 illustrates an example which describes the procedure we use to monitor type-two events for Clb2. The goal is to monitor the occurrence of events 2 and 4. We add events 1 and 3 to prevent misfiring of events 2 and 4 in our hybrid stochastic model. At the beginning of cell cycle, all thresholds are set to very high values except for the first event which is set to 0.5K ez . Once event i occurs, the threshold for i is set to a very large value and the threshold for event i + 1 is reset to its original predefined value. Using this procedure, we guarantee that events 2 and 4 take place in a correct manner in the presence of stochastic fluctuations.

SSA
φ → mCdc20 k s,mcdc20 · (k s,20 · c cdc20 · mass + k s,20 · c cdc20 cmcm1 · M cm1) mCdc20 → φ k d,mcdc20 · mCdc20 φ → mP ds1 k s,mpds1 ·(k s,pds ·c pds1 ·mass+k s1,pds · c pds1    Table S4.(continued) k s,mSic1 = 0.0067 k d,mSic1 = 0.14 k s,mCdc6 = 0.0083 k d,mCdc6 = 0.14 k s,mSwi5 = 0.0016 k d,mSwi5 = 0.14 k s,mCdc20 = 0.0001 k d,mCdc20 = 0.14 k s,mCdh1 = 0.97 k d,mCdh1 = 0.14 The FORTRAN code that implements the model described in Tables S1-S4 is provided in Supplementary Code. We notice that the parameters listed in Table S4 are the values that are computed after converting the concentration of species into population. In the code, however, to provide more flexibility in tuning the parameters, the original parameters from Chen's model are listed and then converted. See the README file that further elaborates on the Supplementary Code. In In In : 0 (10,000)