From START to FINISH: The Influence of Osmotic Stress on the Cell Cycle

The cell cycle is a sequence of biochemical events that are controlled by complex but robust molecular machinery. This enables cells to achieve accurate self-reproduction under a broad range of different conditions. Environmental changes are transmitted by molecular signalling networks, which coordinate their action with the cell cycle. The cell cycle process and its responses to environmental stresses arise from intertwined nonlinear interactions among large numbers of simpler components. Yet, understanding of how these pieces fit together into a coherent whole requires a systems biology approach. Here, we present a novel mathematical model that describes the influence of osmotic stress on the entire cell cycle of S. cerevisiae for the first time. Our model incorporates all recently known and several proposed interactions between the osmotic stress response pathway and the cell cycle. This model unveils the mechanisms that emerge as a consequence of the interaction between the cell cycle and stress response networks. Furthermore, it characterises the role of individual components. Moreover, it predicts different phenotypical responses for cells depending on the phase of cells at the onset of the stress. The key predictions of the model are: (i) exposure of cells to osmotic stress during the late S and the early G2/M phase can induce DNA re-replication before cell division occurs, (ii) cells stressed at the late G2/M phase display accelerated exit from mitosis and arrest in the next cell cycle, (iii) osmotic stress delays the G1-to-S and G2-to-M transitions in a dose dependent manner, whereas it accelerates the M-to-G1 transition independently of the stress dose and (iv) the Hog MAPK network compensates the role of the MEN network during cell division of MEN mutant cells. These model predictions are supported by independent experiments in S. cerevisiae and, moreover, have recently been observed in other eukaryotes.

Here we provide the details of our approach to build the mathematical model, which describes the influence of osmotic stress on the cell cycle.
The next cell cycle transition, G2-to-M, is mainly governed by the activity of Cdc28-Clb2 [1], which in turn is regulated by several mechanisms. On the one hand, the protein kinase Swe1 inhibits Cdc28-Clb2 activity by tyrosine phosphorylation of Cdc28 (link sr10 in Figure S1) [5]. Swe1 is part of the morphogenesis check-point, and it prevents the cell entering into the G2 phase when aspects of bud formation are defective [5,7]. Swe1 is quickly degraded, mediated by the Hsl1-Hsl7 complex, as well as Cdc28-Clb2 (links sr11, sr12 and sr22 in Figure S1) . It is worth mentioning that the phosphatase Mih1 reverses the tyrosine phosphorylation of Cdc28 (link sr9 in Figure S1) [5]. On the other hand, the availability of Clb2 is transcriptionally controlled; Cdc28-Clb2 activates Mcm1 (link sr20 in Figure S1) [3], which is a transcription factor of CLB2, thereby establishing a positive feedback loop (links sr20 and sr21 in Figure S1). Active Cdc28-Clb2, therefore, transfers the cell to the M phase, during which the replicated chromosomes are segregated and nuclear division takes place.
Exit from mitosis is achieved mainly by the inactivation of Cdc28-Clb2 [24]. This is mediated by the degradation of Clb2, performed by two major APC (Anaphase Promoting Complex) components, Cdc20 and Cdh1 (links sr13 and sr14 in Figure S1) [16,17]. First, Cdh1 is absent and Clb2 is primarily degraded by a Cdc20 dependent mechanism. Cdc20 also triggers a pathway that results in the activation of Cdh1 and Cdc14 [16,17,24]. The remaining fraction of the Clb2 is degraded by Cdh1 in the exit from mitosis [17]. The return to the G1 phase prompted by the phosphatase Cdc14 occurs in multiple steps: (i) Cdc14 dephosphorylates and activates Cdh1 (link sr24 in Figure S1); (ii) Cdc14 activates Swi5 (link sr25 in Figure S1), which is a transcription factor of Sic1 and Cdc6 (Sic1 and Cdc6 are shown by CKIs in Figure S1) [6], and (iii) Cdc14 also dephosphorylates and activates Sic1 and Cdc6 (link sr26 in Figure S1). Therefore, the activity of CKIs (Sic1 and Cdc6) and Cdh1 ensures Cdc28-Clb2 inactivation (see Figure S1).

Osmotic stress response
Osmoregulation is a homeostatic process, highly conserved across species, which regulates internal turgor pressure, as well as the water content and the volume of the cell. Osmotic stress is sensed by various receptors and causes the activation of different signalling pathways [29]. The High-Osmolarity Glycerol (HOG) MAPK signalling plays a key role in osmoregulation [30,31]. The HOG MAP kinase pathway, like any other MAP kinase pathway, has three categories of protein kinases: a MAP kinase, a MAP kinase kinase (MAPKK), and a MAP kinase kinase kinase (MAPKKK) [29,32]. For S. cerevisiae, Sln1, Msb2, Hkr1 and Sho1 [33][34][35][36] have been reported as osmosensors of three upstream branches which monitor changes in the turgor pressure and independently regulate three MAPKKKs (Ste11, Ssk2 and Ssk22) [29,37]. Consequently, MAPKKKs phosphorylate and activate the MAPKK (Pbs2) [29]. Activation of Pbs2 results in the activation of Hog1 via phosphorylation [29]. Dually phosphorylated Hog1 (Hog1PP) then accumulates in the nucleus and activates gene expression of proteins involved in glycerol production (see Figure S2). Thereby, glycerol production increases to compensate the turgor pressure loss [32].
As we will see in the next section, phosphorylated Hog1, a result of osmotic stress, is the main player in the interactions between the osmotic stress response and cell cycle networks.

Interaction between cell cycle and osmotic stress networks
The kinase Hog1PP interacts with different cell cycle regulated proteins [18][19][20][21][22]. The phase of the cell cycle during which the osmotic stress is applied dictates the mechanisms of interaction of Hog1PP with the cell cycle machinery.
The G1-to-S transition is halted upon activation of Hog1PP. Activity of Hog1PP blocks this transition by a dual mechanism [18,20]: (i) Hog1PP downregulates the transcription of the G1 cyclins [18,20] and (ii) Hog1PP phosphorylates Sic1 directly on a specific site [20]. This reduces the binding of Sic1 to the Cdc4 complex [20]. This complex formation is required to initiate the ubiquitin-mediated proteolysis pathway responsible for Sic1 degradation [38]. Therefore, the presence of Hog1PP stabilised the level of Sic1 [20]. Hence, G1 arrest as response to osmotic stress is the consequence of Sic1 accumulation, which is caused by direct phosphorylation of Sic1 and downregulation of the G1 cyclins by Hog1PP [20].
Application of osmotic stress after the transition to the S phase delays the cell cycle due to three main reported mechanisms: (i) direct downregulation of CLB5 transcription by Hog1PP [22], (ii) accumulation of the protein kinase Swe1, and (iii) downregulation of the M phase cyclin CLB2. The protein kinase Swe1 is cell cycle regulated [7,11]. Swe1 synthesis is mediated by the transcription factor SBF during late G1 phase and early S phase [5]. Rapid degradation of Swe1 is regulated by sequential phosphorylations, which are caused by the activity of the Hsl1-Hsl7 complex, as well as by the activity of Cdc28-Clb2 [5,7]. However, in the presence of osmotic stress, Hog1PP phosphorylates Hsl1. Phosphorylated Hsl1 then cannot bind to Hsl7 and hence the Hsl1-Hsl7 complex is not formed. As a consequence, Swe1 is not degraded [21]. Hence, osmotic stress applied at the onset of the G2 phase inhibits Cdc28-Clb2 activity due to two mechanisms: (i) stabilisation of Swe1 [21], and (ii) direct downregulation of CLB2 [19,21]. Inactivation of Cdc28-Clb2 blocks the G2-to-M transition.
These are the reported interactions between the osmotic stress response and the cell cycle networks. Despite the existence of all these experimental data, some questions still remain open. For example, there is experimental evidence that Hog1PP phosphorylates Sic1 [20], but it is unknown whether Hog1PP phosphorylates Sic1 when the latter is in a complex with Cdc28-Clb5 or with Cdc28-Clb2. Hence, we made a series of assumptions, which are stated in the main text section and summarised in Figure 1 of the main text (indicated by orange).

Model description
The dynamics of the concentration [C] of each component is described by an ordinary differential equation in which the rate of change of [C] depends on the sum of its production/activation rates v p/a , and the sum of its degradation/inhibition rates v d/i : where v p/a and v d/i depend on the kinetics of the corresponding interactions. The molecular interactions are modelled by mass-action kinetics, Michaelis-Menten kinetics, and Hill functions. For components which are either active or inactive during the cell cycle, such as transcription factors SBF, MBF and Mcm1, the Goldbeter-Koshland switch-like function is used [39].
As mentioned in Materials and Methods, our model starts from two basic modules: the cell cycle module and the osmotic stress response module [40,41]. First, in order to model the molecular interactions between the osmotic stress response and the cell cycle network components, we have to introduce the dynamics of a substantial number of new cell cycle components and their complexes not considered before [40]. Then the dynamics of these new components has to be incorporated into the cell cycle network model such that the new model of the cell cycle reproduces the phenotypical behaviour of the wild-type and mutated cell [5, 7, 11-15, 23, 42]. Finally, we take into account the interactions of Hog1PP with the cell cycle regulated components such that the model mimics the experimental observations of the reaction of wild-type and mutated cell to osmotic stress [18][19][20][21][22].
In the following subsections we present in detail the substantial extensions of the cell cycle module and the modelling of the key interactions between the cell cycle and the osmotic stress pathway which make possible the coupling between both modules. We start with the modelling of the morphogenesis checkpoint.

Modelling the morphogenesis checkpoint
The morphogenesis checkpoint refers to the control mechanisms that hinder the progression through the G2 phase in unfavourable conditions. These mechanisms are achieved through the control of Cdc28-Clb2 activity [5,7]. This checkpoint is inactive under optimal growth conditions, but it becomes active under a malformation of bud or osmotic stress, halting cell cycle progression.
Swe1 inhibits the activity of Cdc28-Clb2 during the late G1 and S phase. In turn, the activity of Swe1 is inhibited via the Hsl1-Hsl7 complex and Cdc28-Clb2 [5,7] (for details of Cdc28-Clb2 regulation and Swe1 regulation see Sections 1.1 and 2.1 of Supplementary Information).
We therefore developed a model for the morphogenesis checkpoint which includes the dynamics of Hsl1-Hsl7 regulation explicitly, based on the recently unveiled molecular mechanisms of the morphogenesis checkpoint [12][13][14][15]. This is in contrast to the model of Ciliberto et al., where the dynamics of Hsl1-Hsl7 was not considered, and instead, reduced to a fixed parameter [43]. We first assumed that Swe1 can exist in four different forms, namely, unphosphorylated (denoted by Swe1), phosphorylated via Hsl1-Hsl7 (denoted by Swe1M), phosphorylated via Cdc28-Clb2 (denoted by Swe1P), and doubly phosphorylated (denoted by Swe1MP), in accordance with biological studies [5,7,[11][12][13]23,42] (see Figure S3). Note that the last form is expected to be highly unstable [44]. We furthermore included the interactions of all forms of Swe1 with Mih1, Hsl1, Hsl7, Cdc28-Clb2 and, additionally, we incorporated the interactions that link the morphogenesis checkpoint to the osmotic stress response module. Note that the mechanisms responsible for the morphogenesis checkpoint alone are highly complex and, in our model, we do not include components such as Gin4, Elm1, Cla4, which have been implicated in the morphogenesis checkpoint, but do not interact with Hog1PP. As such, we keep the model as simple as possible, while preserving the main biological mechanisms. We then integrated this model into the cell cycle module by considering the interaction of the morphogenesis checkpoint elements with the rest of the cell cycle components. We validated this new cell cycle model by reproducing the phenotypical observation of the cell cycle of the wild-type and various mutated cells [5, 7, 11-13, 23, 42].

Modelling the regulation of Swe1
Swe1 is not a crucial component for cells growing in optimal conditions, since swe1∆ cells exhibit a normal cell cycle in controlled environments [23]. However, Swe1 becomes pivotal in cells exposed to osmotic stress. We assumed that Swe1 can be present in four different forms during the cell cycle, in accordance with biological studies [5, 7, 11-13, 23, 42]. Activity of Swe1, is regulated by several mechanisms: (i) Synthesis of Swe1 takes place via transcription factor SBF [5]. Hence, in wild type cells, accumulation of Swe1 begins in late G1 and peaks in S phase or early G2 phase. The production of Swe1 is described by: where k sswe represents the synthesis rate of Swe1 by SBF and k ssweC denotes the basal production rate of Swe1.
(ii) Swe1 is tagged for rapid degradation during G2 and M phase via a pathway which contains the Hsl1-Hsl7 complex and Cdc28-Clb2 [7,11,12,44]. When the Hsl1-Hsl7 complex is present, the kinase Swe1, which has been accumulated in the nucleus until then, is moved to the bud neck [12,13,44]. We denote the product of post-translational modification of Swe1 by Hsl1-Hsl7 by Swe1M, Eq.(6). We modelled the inhibition of Swe1 by the Hsl1-Hsl7 complex with a kinetic law similar to a Hill function: where k hsl1 represents the rate of formation of Swe1M and J iwee is the inverse of the inflection point. The formation of the Hsl1-Hsl7 complex is explained in Section 2.2.
(iii) Also, Cdc28-Clb2 phosphorylates Swe1 [44]. We denote this intermediate product by Swe1P, Eq.(7), which is targeted for degradation [7]. We modelled the phosphorylation of Swe1 by Cdc28-Clb2 also by a Hill function since they are the antagonistic: Therefore, the dynamics of Swe1 was modelled by the following equation: where v rSwe1M describes the backward reaction from Swe1M to Swe1, v rSwe1P explains the reverse reaction from Swe1P to Swe1 and v dSwe1 represents the natural degradation of kinase Swe1.
Using similar arguments, we derived the following mathematical equations describing the dynamics of Swe1M, Swe1P and Swe1MP (see Figure S3): The phosphorylation of Swe1 by Cdc28-Clb2 decreases the activity of the kinase Swe1. Both Swe1M and Swe1P are stable, but the product Swe1MP is highly unstable [43]. Swe1MP is a post-translational modification of Swe1 for which Hsl1-Hsl7 and Cdc28-Clb2 are needed. Hence, in accordance with the reported experiments we assumed that for degradation of Swe1, both Hsl1-Hsl7 and Cdc28-Clb2 are required [7,11,44,45].

Modelling the regulation of Hsl1-Hsl7 complex under osmotic stress
Formation of the Hsl1-Hsl7 complex is necessary for Swe1 degradation [45]. Therefore we included it in our model. This complex is located on the bud neck [45], but spatial modelling of Hsl1 and Hsl7 localisation on the bud neck is beyond the scope of this paper. Instead, we built a temporal model for the regulation of this complex. Since Hsl1 is the dominant component in the formation of the Hsl1-Hsl7 complex [13], we can abstract the dynamics of the formation of the Hsl1-Hsl7 complex by Hsl1. Hsl1 activity correlates with bud emergence and remains stable up to nuclear division [13]. The localisation of Hsl1 and Hsl7 to the septin cortex takes place exactly after bud formation [13]. This supports our choice of Hsl1 as a representative for the Hsl1-Hsl7 complex.
In the presence of osmotic stress, Hog1PP interacts with Hsl1 and hinders the formation of the Hsl1-Hsl7 complex. As a consequence, Swe1 is not degraded and Hsl7 is delocalised from the bud neck [21]. Therefore, the formation of the Hsl1-Hsl7 complex was modelled by: where BUD denotes a mathematical function which describes the observation of bud formation and depends on Cln2, Cln3 and Clb5 (Eq.(70) of Section 4) [40]. Moreover, Hsl1 is stable until nuclear division. As well as Clb2, Hsl1 is a substrate of APC [14,15]. Therefore, following this experimental evidence [14,15] we assumed that its degradation follows a similar APC dependent mechanism to the one of Clb2. Moreover, we assumed that it also has a natural selfdegradation. Hence, the degradation terms of Hsl1-Hsl7 are given by: where represents the APC dependent degradation of the Hsl1-Hsl7 complex and the parameter kkd Hsl1Hsl7 captures the rate of self-degradation. As a result, considering Eqs. (9) and (10), the regulation of the Hsl1-Hsl7 complex is described by:

Modelling the regulation of Mih1
The protein phosphatase Mih1 reverses the phosphorylation of Cdc28-Clb2 which is caused by activity of Swe1 (see link sr9 in Figure S1) [5]. We assumed that there is a positive feedback between Cdc28-Clb2 and Mih1 (Cdc28-Clb2 activates Mih1) [43]. Since details of Mih1 regulations are still unknown, we used the following Hill functions to describe its activation by Clb2 and its self-inhibition [43] where [M ih1 T ] is the total concentration of Mih1 in the cell, which is assumed to be constant. Note that

Modelling the regulation of Clb2
The pair of mitotic cyclins Clb1 and Clb2 are represented by Clb2 in our model. Clb2 is crucial for successful mitosis and its mutation causes G2 arrest [1]. The activity of Cdc28-Clb2 is regulated by several mechanisms. The activity of Hog1PP blocks the G2-to-M transition by influencing the activity of Cdc28-Clb2. The protein kinase Swe1, accumulated upon activation of Hog1PP, inhibits Cdc28-Clb2 by tyrosine phosphorylation of Cdc28 [21]. Also, Hog1PP downregulates the transcriptional activity of CLB2 [19,21]. Hence, we need to take the interaction of Clb2 with Swe1 and also Hog1PP into consideration: (i) The availability of Clb2 is transcriptionally controlled. Cdc28-Clb2 activates Mcm1, which is the transcription factor of CLB2, thereby establishing a positive feedback loop [3]. Also the presence of an osmotic stress influences the transcription of the M phase cyclin [19,21]. Hence, the transcriptional production of Clb2 is described by: where the parameter kk sb2 captures the basal transcription of CLB2, and kkk sb2 describes the induced expression of CLB2 by Mcm1. Activity of Hog1PP also changes the transcription of CLB2 (see Section 2.6). Note that we use Goldbeter-Koshland function to model the Mcm1 activity, Eq.(114) of Section 4.

Modelling cell growth
Since osmotic stress arrests cells at different stages, a pure exponential model for the mass would lead to cells reaching unrealistic sizes. We therefore modelled cell growth such that cell growth is limited. Cell size is proportional to its mass [40,46] and also upon osmotic stress, cell volume decreases [47]. Hence, cell growth under osmotic stress is slower compared with an untreated cell. Hence, we describe cell growth under osmotic stress as: where k g is the growth rate in untreated conditions, M mass,max is the maximum reported mass for the cell, and k dHog1mass represents the influence of Hog1PP on the cell growth.

Modelling the regulation of Sic1 under osmotic stress
The activity of Sic1 in the G1 phase controls the G1-to-S transition [9]. Several mechanisms are involved in the regulation of Sic1: (i) The availability of Sic1 is transcriptionally controlled. The transcription factor Swi5 activates the transcription of SIC1 at the M/G1 phase boundary and in the G1 phase [49] v where the first term represents the basal production of Sic1, and the second term represents the transcriptional regulation of SIC1 by Swi5. Regulation of Swi5 is described by Eq.(53) of Section 4.
(ii) Sic1 binds to the B-type cyclin complexes (Cdc28-Clb5 and Cdc28-Clb2) to inhibit their activity in the G1 phase. We modelled these associations by mass-action kinetics: where k asb5 and k asb2 are the rates of Sic1 binding to Cdc28-Clb5 and Cdc28-Clb2 to build C5 and C2 complexes, respectively. We also assumed that Sic1 binds to Cdc28P-Clb2 denoted by PClb2, using mass-action kinetics to model this reaction: where k pasb2 captures the association rate of Sic1 to PClb2.
(iii) To enable the G1-to-S transition, Sic1 has to be inactivated. Cdc28 phosphorylates and consequently inactivates Sic1 [9]. We modelled the phosphorylation of Sic1 by Cdc28, considering the total concentration of Cdc28, which is bound to cyclins: where V kpc1 is described by Eq.(102) of Section 4. V kpc1 is a function of the total Cdc28 concentration (represented by cyclin concentration in our model) and the total concentration of Sic1.
(iv) During the M phase, active APC complexes degrade the B-type cyclins (Clb5, Clb2). These degradations cause the release of Sic1 from the C5 and C2 complexes in addition to spontaneous disassociation of Sic1 from these two complexes. We modelled the freeing of Sic1 from C2 and C5 complex by the following equations: capture the dynamics of APC dependent release of Sic1 from C5 and C2, and k dib5 and k dib2 are the disassociation rates of Sic1 from C5 and C2, respectively. We assumed that the same mechanisms freeing Sic1 apply to the PTrim (Sic1-Cdc28P-Clb2) complex. The following equation describes our model for the dissociation of Sic1 from PTrim and also the release of Sic1 from PTrim when Clb2 is degraded by APC dependent mechanisms: where (v) Protein phosphatase Cdc14 is required for exit from mitosis. Cdc14 dephosphorylates Sic1. We modelled this dephosphorylation by mass-action kinetics. Regulation of Cdc14 is described by Eq.(62) of where k ppc1 is the dephosphorylation rate of Sic1P by Cdc14.
(vi) Hog1PP directly phosphorylates Sic1 [20]. This phosphorylation reduces the binding of Sic1 to the SCF complex and thus hinders efficient Sic1 degradation [20]. Phosphorylation of Sic1 by Hog1PP occurs at its Thr173 site, which is different from the Cdc28-dependent phosphorylation site. Therefore, we distinguish between the Sic1 which is phosphorylated by Cdc28 (Sic1P) and the one which is phosphorylated by Hog1PP (Sic1h). Sic1h is more stable compared with Sic1P [20]. We used the following Hill function to model the phosphorylation of Sic1 by Hog1PP: where k ash1 corresponds to maximal level of Sic1h, and kkk ash1 represents the concentration of Hog1PP needed to phosphorylate Sic1. We assumed that Cdc14 is the phosphatase that reverses the phosphorylation of Sic1 by Hog1PP. This is a reasonable choice, since Cdc14 has many substrates in a cell and has been reported to dephosphorylate many Cdc28-Clb substrates [24]. We used mass-action kinetics to model this dephosphorylation: where k h1ppc1 is the dephosphorylation rate of Sic1h by Cdc14. By similar arguments we modelled the regulation of Sic1P and Sic1h. We also assumed that simultaneous phosphorylation of Sic1 by Cdc28 and Hog1PP is possible, the product of which is denoted by Sic1hP.

Sensitivity analysis of the estimated parameters
We studied the influence of the estimated parameters on the delay duration upon application of 1 M NaCl osmostress at different time points throughout the cell cycle. We focussed on the set of new parameters which were added to the model to link the osmotic stress response to the cell cycle, since the rest of the parameters was available from the literature [4,20,40,41,43,46,[50][51][52][53][54].
To investigate the sensitivity of the parameters we took the following approach: (i) we chose a time point in each of the phases for application of 1 M NaCl (t = 20 minutes for the G1 phase, t = 50 minutes for the S phase, t = 75 minutes for the DNA re-replication window and t = 90 minutes for the M phase). (ii) At each time point we generated 100 sets of randomly chosen parameters uniformly distributed in the interval from 0.1 to 10 times the value in Table S2 (this is done for the parameters involved in the interaction between Hog1PP and the cell cycle components active during each of the investigated phases, highlighted in Table S2. For example, for the G1 phase we randomly vary the 22 parameters describing the interaction of Hog1PP with Sic1, Cln2 and Cln3.) (iii) Finally, we calculated the change in delay duration with respect to the estimated set of parameters, namely ∆ τ = τ o − τ r , where τ o is the delay duration caused by 1 M NaCl for the estimated set of parameters (Table S2) and τ r is the delay duration caused by 1 M NaCl with the random set of parameters. The results for each of phases are shown in Figure S8. The y-axis shows ∆τ |τo| and the x-axis represents the sets of randomly chosen parameters (100 sets for each phase). If the cell cycle duration obtained for one of the randomly chosen set of parameters was longer than 300 min, we stopped the simulation and represented it by a red point. The results for each of the phases show that the obtained delay duration is very robust against variations in the parameters (see Figures S8A (G1 phase), S8B (S phase), S8C (DNA re-replication window) and S8D (M phase)). This is especially the case for the G1 and S phase. In the case of the DNA re-replication window, there are a few random sets which yield a non-dividing cell, but the general results show that the predicted delay is very robust. The most sensitive parameters seem to be the ones related to the M phase. Note that with our set of estimated parameters, our model predicts an acceleration of the cell cycle, instead of an arrest, if stress is applied during the M phase. 37% of the chosen random sets show an arrest of the cell cycle instead of an acceleration, indicating the sensitivity of the chosen parameters in the M phase. In all those sets, the parameter governing the downregulation of CLB2 by Hog1PP happen to be very small compared with the estimated value, showing that the prediction of accelerated exit from mitosis is very sensitive to the value of this parameter.          [55] kk sb5 = 0.0008 [50] kkk sb5 = 0.005 [50] kkk db2 = 0.4 [50] k d3c1 = 1 [50] k d2f 6 = 1 [50] kkksswi = 0.08 [40] kk db5 = 0.01 [50] kkk db5 = 0.16 [40] kk sb2 = 0.001 [50] kkk sb2 = 0.04 [50] kk db2 = 0.003 [50] k db2p = 0.15 [50] kksc1 = 0.012 [40] kkksc1 = 0.12 [40] k d1c1 = 0.01 [40] k d2c1 = 1 [40] kppc1 = 4 [40] kk sf 6 = 0.024 [40] kkk sf 6 = 0.12 [40] kkkk sf 6 = 0.004 [40] k d1f 6 = 0.01 [40] k d3f 6 = 1 [40] k ppf 6 = 4 [40] k asb5 = 50 [46] k dib5 = 0.06 [46] k asf 5 = 0.01 [46] k asb2 = 50 [46] k dib2 = 0.05 [46] k asf 2 = 15 [46] k dif 2 = 0.5 [46] kksswi = 0.005 [46] k dif 5 = 0.01 [40] kks20 = 0.006 [40] k dcdh = 0.01 [40] k d14 = 0.  [40] k dirent = 1 [40] k direntp = 2 [40] ksppx = 0.1 [40] kk dppx = 0.17 [40] kkk dppx = 2 [40] kkk s1pds = 0.03 [40] kkk s2pds = 0.055 [40] kk d1pds = 0.01 [40] kkk d2pds = 0.2 [40] kkk d3pds = 0.04 [40] k diesp = 0.5 [40]   This phosphorylation occurs via Cdc28-Clb2 and the Hsl1-Hsl7 complex. If the phosphorylation is caused by Cdc28-Clb2, the product is denoted by Swe1P in the diagram, whereas if the phosphorylation is mediated by the Hsl1-Hsl7 complex, it is denoted by Swe1M. The substrate Swe1MP is highly unstable. We assume that the total concentration of Swe1 inhibits the activity of Cdc28-Clb2; see text for further details. The formation of the complex Hsl1-Hsl7 coincides with the formation of the bud. This timing is shown by the variable denoted by BUD in the figure. BUD denotes a mathematical function which describes the observation of bud formation and depends on Cln2, Cln3 and Clb5. The degradation of Hsl1-Hsl7 is APC dependent, which has not been shown in this figure, but it has been considered in the model. The influence of Sic1 accumulation on the G1 and S phase delay duration has been experimentally investigated, which confirms our simulation results. A) A wild-type cell is subjected to 0.4 M NaCl. B) A sic1∆ cell is subjected to 0.4 M NaCl. Deletion of Sic1 makes the G1 phase of the cell shorter in untreated conditions. It also makes the G1 phase cell less adaptable to the osmo-condition, whereas the delay duration for the S phase sic1∆ cell is approximately equal to that of wild-type. C) The Cdc6 levels increase when Clb5 activity is reduced by Hog1PP. This effect is more pronounced for higher stress doses. For illustration purposes the focus is on reactivation of Cdc6 upon presence of Hog1PP. After Hog1PP returns to its basal level, Clb5 starts increasing again. The downregulation, following by an upregulation of Clb5 can lead to DNA re-replication.  Table S2 for; A) G1 phase: 1 M NaCl is applied at t = 20 minutes, B) S phase: 1 M NaCl is applied at t = 50 minutes, C) DNA replication window: 1 M NaCl is applied at t = 75 minutes, D) M phase: 1 M NaCl is applied at t = 90 minutes. See section 3 for further details of analysis. The y-axis shows ∆τ |τo| and the x-axis represents the sets of randomly chosen parameters (100 sets for each phase). The red dots represent the cells which are not dividing with the given set of parameters. They are set to -1.5 for visualisation purposes. Cdc28-Clb2 Cdc28-Clb5 Figure S 9. Mathematical definition of the cell cycle phases. The G1 phase starts immediately after cell division, and finishes when the level of Cdc28-Clb5 crosses the level of Sic1, which indicates initiation of DNA replication and the start of the S phase. The S phase finishes when the level of Swe1 is less than the level of Cdc28-Clb2. Note that we do not distinguish between G2 and M phase, as the G2 phase is very short for S. cerevisiae.