Bornholdt, Boolean network model predicts cell cycle sequence of fission yeast. Technical monograph Google Scholar. Klipp, B. Nordlander, R. Gennemark, S. Hohmann, Integrative model of the response of yeast to osmotic shock. Li, T. Long, Y. Lu, Q. Ouyang, C. Tang, The yeast cell-cycle network is robustly designed. Okabe, M. Sasai, Stable stochastic dynamics in yeast cell cycle. Radmaneshfar, M. Thiel, Recovery from stress: a cell cycle perspective. Radmaneshfar, D.
Kaloriti, M. Gustin, N. R Gow, A. P Brown, C.
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Grebogi, M. Romano, M. Reiser, K. Ly-Sha, A. Amon, The stress-activated mitogen-activated protein kinase signaling cascade promotes exit from mitosis. Cell 17 7 , — CrossRef Google Scholar. Robert, Discrete iterations: a metric study. Saez-Rodriguez, L. Simeoni, J. Lindquist, R. Hemenway, U. Environmental changes are transmitted by molecular signaling networks, which coordinate their actions with the cell cycle.
Our models condense a vast amount of experimental evidence on the interaction of the cell cycle network components with the osmotic stress pathway. Importantly, it is only by considering the entire cell cycle that we are able to make a series of novel predictions which emerge from the coupling between the molecular components of different cell cycle phases.
The model-based predictions are supported by experiments in S.
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Modelling the molecular mechanisms of aging
This component, in turn, is responsible for initiating mitosis and cytokinesis. In order to simulate this model on a timescale appropriate for the S. The first detailed model of the budding yeast cell cycle considered is that of Chen et al [ 26 ] more specifically, the moderately simplified version of this model considered in [ 35 ] , referred to here simply as the Chen model.
These aspects make the original Chen model substantially different from the other models considered here. However, multiple simplifications of the Chen model were derived by Battogtokh et al [ 35 ] for the purpose of bifurcation analysis—the most complex variation is used here in order to represent the Chen model. This model includes 9 variables and 63 parameters. The Barik model of the cell cycle was based upon the previous models of the Tyson group, with several modifications [ 30 ].
The model consists of mass-action kinetics, and as a result represents many more molecular species e. This model includes 61 variables and 70 parameters. The sensitivity of an observable quantity, Q , to relative changes in a parameter, k , is defined by: 2. It should be noted that this property is only defined for successions of daughter cells. This is because mother cells increase in mass at birth in each successive generation both in reality [ 24 ], and in the above models.
These sensitivities can be calculated by simulation, as described in S1 Text. The quantity Q in Eq 2 can be any of several observable quantities, such as the relative phases of the peaks of different cyclins, the magnitude of the peak level of cyclin inhibitors, or the timing of cell cycle events such as kinetochore attachment. Such a general approach has been taken in the analysis of circadian rhythms [ 32 , 42 ].
However, in the case of S. This connection to macroscopic, experimentally measured features motivates the specification of absolute rather than relative sensitivities of Q as in [ 42 ]. Thus, the sensitivity coefficients have intuitive interpretations e. Dynamic sensitivity analysis of an oscillating system can be performed in a variety of ways, and can provide a variety of different types of information.
The objective is to understand how the cell cycle models respond to step changes in parameters applied at different times during the cell cycle. This response is characterised by changes in the duration of the cell cycle, and the size of the cell at budding and division, over several generations. Sensitivity analysis has previously been applied to understand cell cycle dynamics, for example in identifying times at which these dynamics are unstable [ 43 ], and demonstrating common sensitivities of the dynamics of molecular components [ 44 ].
The present analysis complements these by linking the dynamics of cell cycle regulation to macroscopic phenotypes across multiple generations. In general, the subsequent cycles following a perturbation may differ from both the initial cycle and the final cycle. For a cell cycle characteristic, Q , in the i th subsequent cycle following the application of the step change at time t defined as the time since cell division , the dynamic sensitivity of Q i to perturbations in parameter k is given by: 3 The response dynamics to a step change in parameters are illustrated for the case of V div and T div responding to a step change in k s , bS in the Barik model in S1 Fig.
This shows how the transient response of the cell cycle to a step change in parameters at a particular time can be simulated across multiple subsequent generations, until the behaviour stabilises. This also provides a simple way of computing the dynamic sensitivities i. It is also possible to represent the same information in the form of a sensitivity to perturbations of infinitesimal duration at time t , as given by: 4.
This gives an idea of how the function changes over time during the cell cycle. The dynamic sensitivities can be related to the sensitivities under constant conditions calculated in the previous section: 5. This represents the lasting changes in timing of cell cycle events following perturbations applied at different times [ 45 ]. For two cells under the same conditions, the cell cycle phase difference is equal to the difference in their timing of division.
Observation of CP Violation in B± → DK± Decays
For two cells subjected to a change in parameters at different times, this phase difference is evaluated after the system has had sufficient time for transient changes to disappear. As above Eq 4 , the instantaneous effect of a parameter change at time t is characterised by , given by: 7. Sensitivity analysis provides a straightforward way of understanding how combinations of parameter perturbations change cell cycle behaviour.
In particular, we can approximate changes in behaviour in the linear regime by the linear combination of changes elicited by each perturbation, following [ 42 ]. For example, in the case of changes in V dau in generation i following a perturbation in parameter k at time t , we have: 8. Thus, for changes in multiple parameters k 1 , k 2 , …, k n , we have: 9. An assessment of the accuracy of this approximation to changes in model behaviour away from the basal parameter set is shown for 8 parameters in the Barik model in S2 Fig.
While the approximations are generally good, the highly non-linear nature of the model dynamics means that the range of parameter values for which this approximation is accurate is limited in some cases. However, even in these cases the qualitative changes in behaviour are matched across a wide range of parameter values.
This demonstrates the utility of sensitivity analysis for understanding changes in model behaviour in a wide regime of parameter space. We begin by evaluating the steady-state parameter sensitivities of the models, focussing on the macroscopic observable quantities such as the cell cycle duration T div and cell volume at division V div. First, we note that, for a particular growth rate, the macroscopic cell cycle observables can be calculated in terms of only T div and V div. For example, for V dau and T G 1 : As a result, the sensitivity of the cell cycle to changes in parameters can be understood in terms of changes in T div and V div alone or, equivalently T G 1 , V dau.
This makes it natural to visualise the distribution of sensitivities in 2-dimensional scatter plots for each model, with each parameter shown as a point with position or, similarly,. This as shown in Fig 2A. This allows comparison across models of the properties of particular parameters, and identification of general trends across many parameters and models.
B Proportions of parameters for which the sensitivities of V dau and T G 1 have the same sign light bars and opposite sign dark bars. C Negative correlation between average volume and unbudded fraction for cells grown at constant rate under different limiting nutrients i. Data from [ 12 ]. D Negative correlation between V dau and T G 1 for cells grown at constant rate under different nutrient and genetic perturbations.
Data from [ 46 ]. Some parameters of particular interest are those representing the regulation of cyclin synthesis and degradation. Cln3 activity has been hypothesised to increase with cell size, and to therefore communicate cell size information to the cell cycle [ 47 , 48 ]. Parameters representing the synthesis of Cln3 are present in both the Chen and Barik models, and an analogous parameter can be identified in the Pfeuty model see S1 Text for details. As can be seen in Fig 2A , increasing the rate of synthesis of Cln3 acts to reduce the cell size in all three models, consistent with its role in cell size sensing.
While changes in V div are consistent across models, T div is sensitive to changes in this parameter only in the Chen model. Other species of interest are the mitotic cyclins. Mitotic cyclins increase through the G2-M transition, and are rapidly degraded by the APC upon exit from mitosis [ 49 ]. Parameters representing the synthesis of mitotic cyclins and the synthesis of APC subunits Cdc20 and Cdh1 are present in the Chen and Barik models analogous components are not present in the Pfeuty model; see S1 Text for details.
As can be seen in Fig 2 , in both models these parameters act primarily to change T div in opposing directions, with increased mitotic cyclin levels leading to a longer cell cycle period. While this is consistent across models, it should be noted that the changes in V div predicted by the models are not.
In all three models, increasing the growth rate reduces the duration of the cell cycle, and increases the size of the daughter cell S3 Fig , in agreement with experimental observations [ 10 , 12 , 50 , 51 ]. This qualitative agreement has previously been noted for other cell cycle models [ 28 ]. In summary, it is clear that the models broadly agree on some, but not all, qualitative features of regulation by particular parameters.
Beyond specific parameters, it is also interesting to look at patterns observed across all parameters. It is clear from Fig 2A that in all three models most parameters act to modulate V dau and T G 1 in opposite directions with a few clear exceptions in the case of the Chen model. This is quantified in Fig 2B. As a result, most combinations of parameter perturbations are expected to either increase V dau and decrease T G 1 , or vice versa.
This suggests that, for cells growing at the same rate under different conditions i. A dataset that is useful for evaluating this model prediction was generated by Brauer et al. In their experiments, cells were grown in chemostats at 6 different growth rates 0. Average cell volume denoted , proportional to V dau and the fraction of unbudded cells denoted F G 1 , proportional to T G 1 see S1 Text for derivation were measured.
Analysis of these data reveals a negative correlation between and F G 1 at all 6 growth rates, as shown in Fig 2C. Similarly, a recent study by Soma et al. Finally, recently a high-throughput screen of cell cycle behaviour by Soifer et al. Considering only those mutants which were classified as having wild-type growth rates, this correlation was again observed S4 Fig. The consistency of the qualitative behaviour of all three models with these experimental data suggests that they share essential dynamics that correctly describe cell cycle progression.
While the steady-state sensitivity analysis allows the characterisation of cell cycle models under constant conditions, it is also interesting to ask how the cell cycle responds to dynamic changes in parameters. Dynamic sensitivity analysis allows us to understand the complex dynamic behaviour which the cell cycle is capable of producing on its own. This provides a foundation for understanding how signalling networks with their own complex dynamics interface with the cell cycle. As detailed above, dynamic sensitivity can be characterised by the change in cell cycle characteristics down generations to a sustained step change in a parameter, starting at a particular time t.
Two characteristics are apparent in this example, and are seen frequently in many parameters across all models: the dependence of the response on the timing of the perturbation, and the non-monotonic dynamics of this response. This sensitivity can also be visualised as a continuous function of the time of perturbation, as shown in Fig 3D. A and for all parameters in the Barik model, with the parameter representing mitotic cyclin synthesis marked in green. C As in B , for a step-change in parameters applied at mins.
D and down generations as functions of time. Points from B,C are marked by blue and red circles, respectively. Vertical dashed lines represent the time of the G1-S transition. As before, it is also instructive to consider the biological significance of this particular example. First, the qualitative characteristics of the response change depending on the time at which the perturbation is applied.
This can be understood by the role played by mitotic cyclins: their level must first increase to initiate mitosis, but must then decrease to allow the cell cycle to restart. Increasing mitotic cyclin synthesis at a time when cyclin levels need to decrease might be expected to temporarily delay cell cycle progression, as demonstrated by this sensitivity analysis.
While this sensitivity analysis is qualitatively consistent with known biology, we note that an assessment of how mitotic cyclins drive the cell cycle in S. In summary, dynamic sensitivity analysis provides a useful tool for understanding the range of behaviours which the cell cycle is capable of producing. In all three models considered here, nontrivial dynamic behaviours were identified, including nonmonotonic changes in cell size down generations.
It has been observed qualitatively in many studies that the duration of the G1 phase of the cell cycle is especially sensitive to changes in conditions. This manifests itself in a change in the fraction of unbudded cells in populations [ 10 , 50 ]. As a result, there has naturally been significant interest in understanding how signals determine progression through this transition.
In this section, we investigate how the duration of the G1 phase changes under parameter perturbations of the models. From this, we identify the relationship: Where f denotes the fraction of cell mass taken by the daughter cell upon division see S1 Text for derivation. This demonstrates how parameter changes which alter the duration of the pre- and post-budded phases of the cell are fundamentally coupled to one another in the model.
This relationship is depicted for all three models in Fig 4A. The strict proportional relationship is clear in all cases. B Examples of monotonic changes in T G 1 down generations. C Examples of nonmonotonic changes in T G 1 down generations. D Comparison of fractions of parameters exhibiting monotonic and nonmonotonic changes in T G 1 for all three models see S1 Text for details of calculation. Therefore, at steady-state, the duration of a particular phase of the cell cycle may be altered by perturbations that act during other phases.
As discussed above, it is also commonly observed that moving cells into a stress condition can result in a transient accumulation of cells in G1 before the cell population eventually returns to its original state. At the single-cell level, this corresponds to a transient increase in T G 1. One interpretation of this behaviour is that the cells take time adapt to the stress, during which cell cycle dynamics are perturbed, before the cells eventually return to their original state and their original cell cycle behaviour.
In the context of the analysis presented here, this would be analogous to changing model parameters for some time while the cells are experiencing stress before returning them to their original values after the cells have adapted to the stress. However, we previously noted that a step-change in parameters can result in complex cell cycle dynamics, including transient changes away from the eventual behaviour. This was observed in the examples of and given previously Fig 3 , and is also true of changes in T G 1. Since growth rate is held constant in these simulations, this behaviour is not the result of temporary changes in growth rate that might also be expected to accompany some changes in conditions.
This suggests that transient responses of the cell cycle to changes in conditions must be interpreted with some caution. There are cases in which transient signalling appears to give rise to transient changes in cell cycle dynamics e. However, the models suggest that transient signalling or changes in growth rate are not required for this behaviour to be observed. In conclusion, these results demonstrate two causes for caution in the interpretation of changes in cell cycle dynamics in different conditions. First, that in cases where cells are grown under constant conditions, it is difficult to identify the cause for a change in cell cycle timing.
This is because the duration of one cell cycle phase might change significantly as a result of regulation occurring during a different phase. Second, that transient changes in the duration of the G1 phase are a generic property of these models, and do not imply that signalling to the cell cycle is itself transient. The core yeast cell cycle oscillator interacts with other cellular oscillators, including the yeast metabolic cycle YMC [ 22 ], and is postulated to entrain slave oscillators such as oscillations in Cdc14 activity [ 59 ] and a transcriptional oscillator [ 60 ].
In addition, it is possible to partially mode-lock the cell cycle to an external periodic signal [ 24 ]. In other organisms, additional oscillator interactions have been identified, for example gating of cell cycle transitions by circadian clocks [ 61 — 63 ]. In this context, it is interesting to ask how dynamic perturbations alter the timing of cell cycle events. This has been investigated previously in the context of cell cycle responses to periodic forcing signals [ 64 , 65 ]. Here, we are able to link control of cell cycle timing to the modulation of macroscopic cell cycle variables.
This can also be calculated according to: This enables us to predict the mode-locking behaviour of the cell cycle to periodic forcing. The phase shift between two cells can be related to differences in the mass fraction donated to the daughter cell down generations. In particular, consider a perturbation which causes a temporary change in the fractions of mother cell volume donated to the daughter cell.
Then the phase shift is given by: In practice this limit converges rapidly within a few generations. This establishes a link between how a parameter changes the mass of daughter cells, and how it changes the phase of the cell cycle. This correspondence is demonstrated in Fig 5A and 5B. We note that this is independent of any details of the models considered here, and applies to any asymmetrically dividing cell growing exponentially at a constant rate.
A Phase responses for three exemplar parameters in the Barik model. B Sensitivity of the fraction of cell volume donated to the daughter cell for the three parameters shown in A. The high similarity of the functions in A and B follows from the correspondence between phase shifts and daughter cell size fraction Eq C,D Examples of predominantly biphasic C and monophasic D phase response curves for parameters in the Barik model.
E Distribution of biphasic extent of parameters for all three models, evaluated according to Eq F Distribution of peak phase sensitivities for all three models. One notable qualitative feature of some of these phase response curves is that they are biphasic i. This property can be quantified for a parameter k by the metric B k : This gives values of B k ranging between 0 and 1.
B k is 0 for a completely monophasic pattern of sensitivity, as is strictly positive or negative, so. The distribution of B k across the parameters of all models are shown in Fig 5E.
From this, it is clear that many parameters in all models display this property. This is a property shared with other biological oscillators, for example circadian and neuronal oscillators [ 34 , 66 ]. Another observation that can be made is that the phase shifts are most pronounced when perturbations are applied later in the cell cycle from T G 1 onwards.
The distributions of the times of peak sensitivity of the parameters of all models are shown in Fig 5F. In all models there are two main groups of parameters—those peaking around T G 1 and those peaking around T div —with very few parameters displaying peak sensitivity before T G 1. This is somewhat counter-intuitive given the noted sensitivity of T G 1 to parameter changes see above. The robustness of the cell cycle model behaviour to perturbations during G1 has been observed previously in the case of the Chen model [ 43 ]. In summary, these results show that the cell cycle models consistently predict a preponderance of biphasic phase response curves, and further illustrate the qualitative differences in sensitivity observed before and after the G1-S transition.
The analysis presented above provides a framework for understanding the effects of perturbations on the dynamics of cell cycle progression. In order to demonstrate how the analysis presented can be applied to understanding signalling to the cell cycle, it is useful to consider a specific example. Here, we investigate how glucose-sensing signalling pathways might affect cell cycle progression.
Glucose sensing is particularly important in this context, as the extra- and intracellular glucose levels are key determinants of nutrient availability. As such, several pathways have been identified through which glucose affects cell cycle components, both through direct sensing [ 13 , 14 , 16 , 67 , 68 ], and indirect effects via metabolism and growth rate [ 15 , 16 ]. Here, we consider the effects of direct signalling pathways, and note that their effects can be separated from indirect, growth-rate-mediated effects in conditions where growth rate does not change in response to glucose levels.
An example of this was recently demonstrated in experiments by Soma et. We consider three particular forms of cell cycle regulation by glucose Fig 6A. The first mechanism of cell cycle regulation by glucose involves the control of translation of Cln3—a cyclin responsible for inducing G1-S transition. The regulation of Cln3 translation is mediated in part through the direct regulation of the translation initiation factor eIF4E [ 69 ], and can also be controlled through the relief of competition for translation initiation factors e.
The rate of translation of Cln3 is represented in the Barik model by the parameter k s , n 3. The second mechanism we consider is the repression of Cln2 expression by glucose [ 71 ].
The rate of ClbS transcription is represented by the parameter k s , mbS. Finally, it is known that signalling through the TOR kinase complex is capable of modulating the activity of the PP2A phosphatase complex [ 72 , 73 ]. Upon phosphorylation by the TOR1C complex, this phosphatase dephosphorylates a wide range of targets, including Net1 [ 74 ]. Net1, in turn, is responsible for sequestering the cell-cycle phosphatase Cdc14, which is required for progression through mitosis.
The dephosphorylation of Net1 in the Barik model is represented by the constitutive activity of a generic phosphatase, Ht1. The model parameters representing this activity are k d , t 1 and k d , nt , regulating free Net1 and Net1 in the RENT complex, respectively. A natural assumption is that regulation of this pair of parameters is coupled, and therefore that they are modulated proportionally to one another. A A schematic of glucose regulation. Glucose is known to act on the cell cycle and many other processes through a diverse range of signalling pathways.
We evaluate glucose modulation of the Barik model through regulation of Cln2 transcription k s , mbS , Cln3 translation k s , n 3 , and Net1 dephosphorylation k d , t 1 , k d , nt. The attainable range of behaviours is represented by the shaded region. This is consistent with experimental observations dark blue line; see text for details. C The dynamic sensitivities of V dau to the three coregulating pathways are shown. The consequences of different balances of these three parameter perturbations are shown, all of which have the same eventual change in behaviour.
The inclusion of strong regulation in mitosis through Net1 allows dynamic response to changes in glucose levels late in the cell cycle. The above summary of some regulatory mechanisms is by no means complete, partly as a result of some regulatory components not being present in this model e. However, since it includes components involved in regulating different cell cycle phases, it provides a useful starting point for understanding the range of behaviours that might be achieved by glucose regulation of the cell cycle. These constraints imply a certain attainable range of responses in V dau and T G 1 , meaning that only particular changes in V dau and T G 1 are possible in response to increases in G.
The shaded region represents the space spanned by linear, positive sums of these vectors, which is the attainable range of responses. Here, the regulatory mechanisms that we consider are limited to speeding up the cell cycle with increasing glucose levels i. Additionally, while this form of regulation can freely decrease the cell size without having a significant impact on the cell cycle period, there must be a decrease in period in order to effect an increase in cell size. The consistency of the attainable region with experimental observations can be assessed by evaluating measured V dau and T G 1 values under different glucose concentrations and constant growth rate, as reported in [ 46 ].
The linear correlation between these values at three glucose concentrations 0. Note that this makes no assumption about the explicit relationship between glucose levels and the magnitude of parameter perturbations. The corresponding sensitivity to changes in glucose is then given by: As shown in Fig 6B , this lies within the attainable region, confirming that this simple combination of regulations is consistent with the observed changes in behaviour.
An interesting aspect of the attainable region is that it is bounded by the opposing effects of stimulation of Cln3 translation and inhibition of ClbS transcription by glucose. This means that regulation of Net1 dephosphorylation does not broaden the range of behaviours that can be brought about through the pathway under constant conditions. These different combinations of parameter perturbations will, by construction, have identical cell cycle behaviour under constant conditions, but may have distinct behaviours under dynamic changes in conditions.
In order to evaluate the potential for diverse dynamics in this system, we fix the change in behaviour achieved by parameter perturbations according to experimental observations Eq 19 , and consider three cases: no, weak, and strong up-regulation of Net1 dephosphorylation with increasing glucose levels. The resultant changes in the dynamic sensitivity shown in Fig 6D are the result of differences in the timing of sensitivity of the cell cycle to the different parameters see Fig 6C for the individual sensitivity profiles.
Regulation of Cln2 transcription and Cln3 translation alone is only capable of modulating cell cycle progression around the G1-S transition, while regulation of Net1 dephosphorylation modulates progression through mitosis. Therefore, regulation of Net1 in the model allows for a faster response to changes in glucose levels by extending the time window of responsiveness to glucose levels.
It has been noted previously that glucose levels act predominantly to modulate duration of the G1 phase of the cell cycle [ 76 ], as discussed above in the more general case. An important conclusion arising from the work presented here is that this form of regulation does not exclude active regulation of processes occurring during mitosis or other phases of the cell cycle. Indeed, as long as counteracting pathways can be modulated in tandem, regulation of processes occurring in mitosis may be a useful strategy for dynamic adjustment of cell cycle characteristics after a change in conditions.
In the particular example of strong Net1 regulation shown in Fig 6D , this is seen to lead to a more rapid modulation of V dau than would be possible if only Cln2 and Cln3 were regulated. As discussed above, observations of cell populations under constant conditions e. Furthermore, the effects of such perturbations may only be observable in experiments in which response dynamics are observed. Cell cycle sensitivity during mitosis has been observed experimentally in response to sudden nutrient starvation or application of rapamycin [ 77 , 78 ], suggesting that investigation of nutrient signalling under constant conditions can indeed mask important regulation.
Cell cycle progression is a highly regulated process. This is a result of the importance of the processes it coordinates, and of the fine-tuned response required in changing conditions.