The bistable mitotic switch in fission yeast

  1. Béla Novák
  2. John J. Tyson

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Abstract

In favorable conditions, eukaryotic cells proceed irreversibly through the cell division cycle (G1-S-G2-M) in order to produce two daughter cells with the same number and identity of chromosomes of their progenitor. The integrity of this process is maintained by “checkpoints” that hold a cell at particular transition points of the cycle until all requisite events are completed. The crucial functions of these checkpoints seem to depend on irreversible bistability of the underlying checkpoint control systems. Bistability of cell cycle transitions has been confirmed experimentally in frog egg extracts, budding yeast cells and mammalian cells. For fission yeast cells, a recent paper by Patterson et al. (2021) provides experimental evidence for an abrupt transition from G2 phase into mitosis, and we show that these data are consistent with a stochastic model of a bistable switch governing the G2/M checkpoint. Interestingly, our model suggests that their experimental data could also be explained by a reversible/sigmoidal switch, and stochastic simulations confirm this supposition. We propose a simple modification of their experimental protocol that could provide convincing evidence for (or against) bistability of the G2/M transition in fission yeast.

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  1. Version published to 10.1091/mbc.e24-03-0142
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    Reply to the reviewers

    Reply to Reviewers

    We are grateful to the three reviewers for their careful and constructive critiques of our preprint. We will address all of their comments and suggestions, which help to make our paper more precise and understandable. In our replies, we use 'Patterson, eLife (2021)' as shorthand for Patterson, Basu, Rees & Nurse, eLife 2021:10.

    Reviewer #1 (Evidence, reproducibility and clarity (Required)): Novák and Tyson present a model-based analysis of published data that had claimed to demonstrate bistable activation of CDK at the G2/M transition in fission yeast. They point out that the published data does not distinguish between ultra-sensitive (switch-like, but reversible) and bistable (switch-like, but irreversible) activation. They back up their intuition with robust quantitative modeling. They then point out that, with a simple experimental modification, the published experiments could be repeated in a way that would test between the ultra-sensitive and bistable possibilities.

    This is an accurate and concise summary of our paper.

    Therefore, this is a rare paper that makes a specific modeling-based prediction and proposes a straightforward way to test it. As such, it will be of interest to a broad range of workers involved in the fields cell cycle and regulatory modeling.

    We agree that our work will be of interest to a broad range of scientists studying cell cycle regulation and mathematical modeling of bistable control systems.

    Nonetheless, attention to the following points would improve the manuscript. The authors should be more careful about how they describe protein abundance. They often refer to protein level. I believe in every case they mean protein concentration, but this is not explicitly stated; it could be interpreted as number of protein molecules per cell. The authors should either explicitly state that level means concentration or, more simply, use concentration instead of level.

    A valid criticism that has been addressed in the revised version.

    The authors should explain why they include stoichiometric inhibition of CDK by Wee1 in their model. Is it required to make the model work in the wild-type case, or only in the CDK-AF case? My intuition is it should only be required in the AF case, but I would like to know for sure. Also, they should state if there is any experimental data for such regulation.

    Bistability of the Tyr-phosphorylation switch requires 'sufficient' nonlinearity, which may come from the phosphorylation and dephosphorylation reactions that interconvert Cdk1, Wee1 and Cdc25. The easiest way to model these interconversion reactions is to use Hill- or Goldbeter-Koshland functions for the phosphorylation and dephosphorylation of Wee1 and Cdc25, but this approach is not appropriate for Gillespie SSA, which assumes elementary reactions. Both Wee1 and Cdc25 are phosphorylated on multiple sites, which we approximate by double phosphorylation; but this level of nonlinearity is not sufficient to make the switch bistable. In addition, stochiometric inhibition is a well-known source of nonlinearity, and in the Wee1:Cdk1 enzyme:substrate complex, Cdk1 is inhibited because Wee1 binds to Cdk1 near its catalytic site. In our model, stoichiometric inhibition of Cdk1 by Wee1 is required for bistability even in the wild-type case because the regulations of Wee1 and Cdc25 by phosphorylation are not nonlinear enough. There is experimental evidence that stoichiometric inhibition of Cdk1 by Wee1 is significant: mik1D* wee1ts* double mutant cells at the restrictive temperature (Lundgren, Walworth et al. 1991) are less viable than AF-Cdk1 (Gould and Nurse 1989). Furthermore, Patterson (eLife, 2021) found weak 'bistability' when they used AF-Cdk1 to induce mitosis. This puzzling observation suggests a residual feedback mechanism in the absence of Tyr-phosphorylation. Our model accounts for this weak bistability by assuming that free CDK1 can phosphorylate and inactivate the Wee1 'enzyme' in the Wee1:Cdk1 complex, which makes CDK1 and Wee1 mutual antagonists. This reaction is based on formation of a trimer, Cdk1:Wee1:Cdk1, which is possible since CDK1 phosphorylation of Wee1 occurs in its N-terminal region, which lies outside the C-terminal catalytic domain of Wee1 (Tang, Coleman et al. 1993). These ideas have been incorporated into the text in the subsection describing the model (see lines120-125).

    The authors should explicitly state, on line 131, that the fact that "the rate of synthesis of C-CDK molecules is directly proportional to cell volume" results in a size-dependent increase in the concentration of C-CDK.

    The accumulation of C-CDK molecules in fission yeast cells is complicated. In general, we may assume that larger cells have more ribosomes and make all proteins faster than do smaller cells. Absent other regulatory effects, the number of protein molecules is proportional to cell volume, and the concentration is constant. But, in Patterson's experiments, the number of C-CDK molecules is zero at the start of induction and rises steeply thereafter (see lines 147-148), and the rate of increase (#molec/time) is proportional to the size of the growing cell.

    The authors should explain, on line 100, why they are "quite sure the bistable switch is the correct interpretation".

    Line 105-106: "Although we suspect that the mitotic switch is bistable,.."

    On line 166, include the units of volume.

    Done

    On lines 152 and 237, "smaller protein-fusion levels "should be replaced with "lower protein-fusion concentrations".

    Done

    **Referee cross-commenting** *I concur with the other two reviews.

    Reviewer #1 (Significance (Required)): *The paper is significant in that it points out an alternative interpretation for an important result in an important paper. Specifically, it points out that the published data is consistent with activation of CDK at the G2/M transition in fission yeast could be ultra-sensitive (switch-like, but reversible) instead of bistable (switch-like, but irreversible). The distinction is important because it has been claimed, by the authors of the submitted manuscript among others, that bistability is required for robust cell-cycle directionality. *

    We agree with this assessment.

    However, activation of CDK at the G2/M transition in other species has been shown to be bistable and the authors state that they are "quite sure the bistable switch is the correct interpretation". So, the paper is more likely an exercise in rigor than an opportunity to overturn a paradigm.

    We were the first authors to predict that the G2/M switch is bistable (J. Cell Sci., 1993) and among the first to prove it experimentally in frog egg extracts (PNAS, 2004). Our models (Novak and Tyson 1995, Novak, Pataki et al. 2001, Tyson, Csikasz-Nagy et al. 2002, Gerard, Tyson et al. 2015) of fission yeast cell-cycle control rely on bistability of the G2/M transition; so, understandably, we believe that the transition in fission yeast is a bistable switch. But the 'bistable paradigm' has never been directly demonstrated by experimental observations in fission yeast cells. The Patterson paper (eLife, 2021) claims to provide experimental proof, but we demonstrate in our paper that Patterson's experiments are not conclusive evidence of bistability. Furthermore, we suggest that a simple change to Patterson's protocol could provide convincing evidence that the G2/M switch is either monostable or bistable. We are not proposing that the switch is monostable; we would be quite surprised if the experiment, correctly done, were to indicate a reversible switch. Our point is simply that the published experiments are inconclusive. The point we are making is neither a mere 'exercise in rigor' nor a suggestion to 'overturn a paradigm.' Rather it is a precise theoretical analysis of a central question of cell cycle regulation that should be of interest to both experimentalists and mathematical modelers.

    Reviewer #2 (Evidence, reproducibility and clarity (Required)): Summary: The manuscript asks whether the data reported in Patterson et al. (2021) is consistent with a bistable switch controlling the G2/M transition in fission yeast. Patterson et al. (2021) use an engineered system to decouple a non-degradable version of Cyclin-dependent kinase (CDK) from cell growth and concomitantly measure CDK activity (by the nuclear localization of a downstream target, Cut3p). They observe cells with indistinguishable CDK levels but two distinct CDK activities, which they posit shows bistable behavior. In this study, the authors ask if other models can also explain this data. The authors use both deterministic and Gillespie based stochastic simulations to generate relationships between CDK activities and protein levels for various cell sizes. They conclude that the experiments performed in Patterson et al. are insufficient to distinguish between a bistable switch and a reversible ultrasensitive switch. They propose additional experiments involving the use a degradable CDK construct to also measure the inactivation kinetics.

    This is an accurate summary of our paper.

    They propose that a bistable switch will have different forward (OFF->ON) and backward (ON->OFF) switching rates. A reversible ultrasensitive switch will have indistinguishable switching rates.

    Our analysis of Patterson's (2021) experiments is based on the well-known fact that the threshold for turning a bistable switch on is significantly different from the threshold for turning it off (in Patterson's case, the 'threshold' is the level of fusion protein in the cell when CDK is activated), whereas for a reversible, ultrasensitive switch, the two thresholds are nearly indistinguishable. The 'rate' at which the switch is made is a different issue, which we do not address explicitly. In the experiments and in our model, the switching rates are fast, whether the switch is bistable or monostable. The results are interesting and worth publication in a computational biology specific journal, as they might only appeal to a limited audience.

    We think our results should also be brought to the attention of experimentalists studying cell cycle regulation, because Patterson's paper (eLife, 2021) presents a serious misunderstanding of the existence and implications of 'bistability' of the G2/M transition in fission yeast. Whereas Patterson's work is an elegant and creative application of genetics and molecular biology to an important problem, it is not backed up by quantitative mathematical modeling of the experimental results. In that sense, Patterson's work is incomplete, and its shortcomings need to be addressed in a highly respected journal, so that future cell-cycle experimentalists will not make the same-or similar-mistakes.

    Several ideas need to be clarified and additional information needs to be provided about the specific parameters used for the simulations: Major comments: #1 The parameters need to be made more accessible by means of a supplementary table and appropriate references need to be cited.

    Two new supplementary tables (S1 and S2) summarize the dynamic variables and parameter values.

    It is not clear why Michaelis Menten kinetics will not be applicable to this system. Has it been demonstrated that the Km s of the enzymes are much greater than the substrate concentrations for all the reactions? If yes, please cite.

    MM kinetics are not appropriate for such protein interaction networks because one protein may be both an enzyme and a substrate for a second protein (e.g., Wee1 and CDK, or Cdc25 and CDK). So, the condition for validity of MM kinetics (enzyme concen ≪ substrate concen) cannot be satisfied for both reactions. Indeed, enzyme concen ≈ substrate concen is probably true for most reactions in our network. Hence, it is advisable to stick with mass-action rate laws. Furthermore, MM kinetics are a poor choice for 'propensities' in Gillespie SSA calculations, as has been shown by many authors (Agarwal, Adams et al. 2012, Kim, Josic et al. 2014, Kim and Tyson 2020).

    It will not be surprising if the simulation with Michaelis Menten would alter the dynamics shown in this study. A reversible switch with two different enzymes (catalyzing the ON->OFF and OFF->ON transitions) having different kinetics can give asymmetric switching rates. This would directly contradict what has been shown in Figure 7A-D.

    We don't follow the reviewer's logic here. The two transitions, off → on and on → off, are already driven by different molecular processes (dephosphorylation of inactive CDK-P by Cdc25 and phosphorylation of active CDK by Wee1, respectively). Positive feedback of CDK activity on Cdc25 and Wee1 (++ and −−, respectively) causes bistability and asymmetric switching thresholds. Switching rates, which are determined by the kinetic rate constants of the up and down processes, are of secondary importance to the primary question of whether the switch is monostable or bistable.

    #2 Line 427: The authors use a half-time of 6 hours in their model as Patterson et al. used a non-degradable construct. It is not clear why dilution due to cell growth has not been considered. The net degradation rate of a protein is the sum of biochemical degradation rate and growth dilution rate. The growth dilution rate seems significant (140 mins doubling time or 0.3 h-1 dilution rate) relative to assumed degradation rate (0.12 h-1). Please clarify why was the effect of dilution neglected in the model or show by sensitivity analysis this does not change the predicted CDK activation thresholds.

    The reviewer highlights an important effect, but it is not relevant to our calculations. In the deterministic model used to calculate the bifurcation diagrams, both cell volume and the concentration of the non-degradable Cdc13:Cdk1 dimer are kept constant; therefore, there is no dilution effect. The stochastic model deals with changing numbers of molecules per cell; the dilution effect is taken into account by the appearance of cell volume, V(t), at appropriate places in the propensity functions. In other words: in the deterministic model, which is written for concentration changes, the dilution term, −(x/V)(dV/dt), is zero because V=constant; in the stochastic model, written in terms of numbers of molecules, dilution effects are implicit in the propensity functions.

    *#3 Line 402 The authors state that the production rate of the Cdk protein is 'assumed' proportional to the cell volume. The word 'assumed' is incorrect here as a simple conversion of concentration-based differential equation (with constant production rate) to molecular numbers would show that production rate is proportional to the volume. This is not an assumption. *

    Correct; we modified the text (see line 450-462). The role of cell volume in production rate is more relevant to the case of Cdc25, where we assume that its production rate, Δconcentration/Δt, is proportional to V, because the concentration of Cdc25 in the cell increases as the cell grows. We added two references (Keifenheim, Sun et al. 2017, Curran, Dey et al. 2022) to justify this assumption. In the stochastic code, the propensity for synthesis of Cdc25 molecules is proportional to V2.

    #4 Line 423 Please cite the appropriate literature that shows that fission yeast growth during cell division is exponential. If the dynamics are more complicated, involving multiple phases of growth during cell division, please state so.

    We now acknowledge that volume growth in fission yeast, rather than exponential, is bilinear with a brief non-growing phase at mitosis (Mitchison 2003). However, we suggest that our simplifying assumption of exponential growth is appropriate for the purposes of these calculations. See line 473-476: "In our stochastic simulations, we assume that cell volume is increasing exponentially, V(t) = V0eμt. Although fission yeast cells actually grow in a piecewise linear fashion (Mitchison 2003), the simpler exponential growth law (with doubling time @ 140 min) is perfectly adequate for our purposes in this paper.."

    *#5 Line 250 The authors convert the bistable version of the CDK switch to reversible sigmoidal by assuming that Wee1 and Cdc25 phosphorylation is proportional to the CDK level rather than activity, which seems biochemically unrealistic. This invokes an altered circuit architecture where inactive CDK has enough catalytic activity to phosphorylate the two modifying enzymes (Wee1/Cdc25) but not enough to drive mitosis. This might be possible if the Km of CDK for Wee1/Cdc25 is lower relative to other downstream substrates that drive mitosis. The authors can reframe this section of the paper to state this possibility, which might be interesting to experimentalists. *

    The reviewer is correct that the molecular biology underlying our 'reversible sigmoidal' model is biochemically unrealistic. But, in our opinion, this is the simplest way to convert our bistable model into a monostable, ultrasensitive switch while maintaining the basic network structure in Fig. 1. Our purpose is to show that a monostable model-only slightly changed from the bistable model-can account for Patterson's experimental data equally well. If Nurse's group modifies the experimental protocol as we suggest and their new results indicate that the G2/M transition in fission yeast is bistable, then our reversible sigmoidal model, having served its purpose, can be forgotten. If they show that the transition is not bistable, then both experimentalists and theoreticians will have to think about biochemically realistic mechanisms that can account for the new data...and everything else we already know about the G2/M transition in fission yeast.

    #6 It is difficult to phenomenologically understand a bistable switch just based on differences in activation and inactivation thresholds. For example, a reversible ultrasensitive switch also shows a difference in activation and inactivation thresholds (Figure 7D). How much of a difference should be expected of a bistable switch versus reversible switch?

    We show how much of a difference can be expected by contrasting Fig. 7 to Fig. 8. For the largest cells (panel D of both figures), the difference is small and probably undetectable experimentally. For medium-sized cells (panel C), the difference is larger but probably difficult to distinguish experimentally. Only the smallest cells (panel B) provide an opportunity for clearly distinguishing experimentally between monostable and bistable switching.

    *Moreover, as the authors clearly understand (line 275), time-delays in activation and inactivation reactions can inflate these differences. In the future, if the authors can convert the equations to potential energy space as done in Acar et al. 2005 (Nature 435:228) in Figure 3c-d, it will be useful. Also, predicting the distribution of switching rates from the Gillespie simulation might be informative and can be directly compared to experimental measurements in the future (if the Cut3p levels in nucleus and cytosol equilibrates fast enough or other CDK biosensors are developed). *

    The famous paper by Acar et al. (2005) is indeed an elegant experimental and theoretical study of bistability ('cellular memory') in the galactose-signalling network of budding yeast. We have included a comparison of Patterson et al. with Acar et al. in our Conclusions section (lines 353-368):

    "It is instructive, at this point, to compare the work of Patterson et al. (2021) to a study by Acar et al. (Acar, Becskei et al. 2005) of the galactose-signaling network of budding yeast. Combining elegant experiments with sophisticated modeling, Acar et al. provided convincing proof of bistability ('cellular memory') in this nutritional control system. They measured PGAL1-YFP expression (the response) as a function of galactose concentration in the growth medium (the signal), analogous to Patterson's measurements of CDK activity as a function of C-CDK concentration in fission yeast cells. In Acar's experiments, the endogenous GAL80 gene was replaced by PTET-GAL80 in order to maintain Gal80 protein concentration at a constant value determined by doxycycline concentration in the growth medium. The fixed Gal80p concentration in Acar's cells is analogous to cell volume in Patterson's experiments. In Fig.3b of Acar's paper, the team plotted the regions of monostable-off, monostable-on and bistable signaling in dependence on their two control parameters, external galactose concentration and intracellular Gal80p concentration, analogous to our Fig.4. Because Acar's experiments explored both the off → on and on → off transitions, they could show that their observed thresholds (the red circles) correspond closely to both saddle-node bifurcation curves predicted by their model. On the other hand, Patterson's experiments (as analyzed in our Fig.4) probe only the off → on transition."

    The purpose of our paper is to show that Patterson-type experiments can and should be done so as to probe both thresholds, as was done by van Oudenaarden's team. They went further to characterize their bistable switch in terms of 'the concept of energy landscapes'. We think it is premature to pursue this idea in the context of the G2/M transition in fission yeast until there is firm, quantitative data characterizing the nature of the 'presumptive' bistable switch in fission yeast.

    Minor comments: #1 Line 2: Please replace "In most situations" to "In favorable conditions"

    Done.

    **Referee cross-commenting** I agree with Reviewer 1 that this falls more under pointing out an alternative interpretation of a single experiment than challenging widely supported orthodoxy about how the eukaryotic cell cycle leaves mitosis.

    As we said earlier, our 1993 paper in J Cell Sci is the source of this orthodox view, and it is widely supported at present because there is convincing experimental evidence for bistability in frog egg extracts, budding yeast cells and mammalian cells. Patterson's paper is not sound evidence for bistability of the G2/M transition in fission yeast cells. It is important for experimentalists to know why the experiments fail to confirm bistability, and important for someone to do the experiment correctly in order to confirm (or, what would be really interesting, to refute) the expectation of bistability at the G2/M transition in fission yeast cells.

    Reviewer #2 (Significance (Required)): Suitable for specialist comp bio journal eg PLoS Comp Bio

    Reviewer #3 (Evidence, reproducibility and clarity (Required)):

    The paper by Novak and Tyson revisits a recent paper from Nurse group on the bistability of mitotic switch in fission yeast using mathematical modelling. The authors extend their older models of mitotic entry check point and implement both deterministic and stochastic version of new model. They show this model does indeed possess bistability and show that combined with stochastic fluctuations the model can show bimodality for the cyclin-CDK activity at a particular cell size consistent with the recent experimental data. However, the authors also show alternative model that has mono-stable ultrasensitivity can also explain the data and suggest experiments that can prove the existence of hysteresis and therefore bistability.

    Right on.

    While the biological implication of the study is well explained, the authors can improve the presentation of their model and the underlying assumptions. I have the following comments and suggestions for improvement of the paper.

      • The cartoon of the mathematical model is confusing at places, for example the wee1-CDK complex according to the equations either dissociates back to wee1 and CDK or gives rise to pCDK and wee1, the arrow below is confusing as it implies it can also give rise to wee1p, the CDK phosphorylation of wee1 is already included in the diagram. Also, the PP2A is put on the arrow for all reactions but for wee1p2 to wee1p its action shown with a dashed line. Also, I wondered if wee1p and wee1p2 can also bind CDK and sequester or phosphorylate CDK?* We are sorry for the confusion and have improved Fig. 1.
    • The rates and variables in the ODEs are not fully described. Also sometimes unclear what is parameter and what is a variable, I had to look at the code.*

    We now include tables of variables and parameter values, with explanatory notes.

    • The model has quite a few parameters, but these are not at all discussed in the paper. How did the authors come up with these particular set of parameters, has there been some systematic fitting, or tuning by hand to produce a good fit to the data? I could only see the value of the parameters in the code, but perhaps a table with the parameters of the model, what they mean and their value (and perhaps how the values is obtained) is missing.*

    The parameters were tuned by hand to fit Patterson's data, based, of course, on our extensive experience fitting mathematical models to myriad data sets on the cell division cycles of fission yeast, budding yeast, and frog egg extracts. We now provide a table of parameter values.

    • The authors are using the Gillespie algorithm with time varying parameters (as some rates depend on volume and volume is not constant). Algorithm needs to be modified slightly to handle this (see for example Shahrezaei et al Molecular Systems Biology 2008). *

    A valid criticism, but the rate of cell volume increase is very slow compared to the propensities of the biochemical reactions. We write (lines 492-498):

    "In each step of the SSA, the volume of the cell is increasing according to an exponential function, and, consequently, the propensities of the volume-dependent steps are, in principle, changing with time; and this time-dependence could be taken into account explicitly in implementing Gillespie's SSA (Shahrezaei, Ollivier et al. 2008). However, the step-size between SSA updates is less than 1 s compared to the mass-doubling time (140 min) of cell growth. So, it is warranted to neglect the change in V(t) between steps of the SSA, as in our code."

    • The authors correctly point out, ignoring mRNA has resulted in underestimation of noise, however another point is that mRNA life times are short and that also affects the timescale of fluctuations and this may be relevant to the switching rates between the bistable states. *

    A valid point, but to include mRNA's would double the size of the model. Furthermore, we have little or no data about mRNA fluctuations in fission yeast cells, so it would be impossible to estimate the values of all the new parameters introduced into the model. Finally, the switching rates between bistable states (or across the ultrasensitive boundary) are not the primary focus of Patterson's experiments or our theoretical investigations. So, we propose to delay this improvement to the model until the relevant experimental data is available.

    • In the introduction add, "In this study" to "Intrigued by these results, we investigated their experimental observations with a model of bistability in the activation of cyclin-CDK in fission yeast." *

    Done

    Reviewer #3 (Significance (Required)): Overall, this is an interesting study that revisits an old question and some recent experimental data. The use of stochastic modelling in explaining variability and co-existence of cell populations in the context of cell cycle and comparison to experimental data is novel and of interest to the communities of cell cycle researchers, systems biologists and mathematical biologists.

    We agree. Thanks for the endorsement

    References

    Acar, M., A. Becskei and A. van Oudenaarden (2005). "Enhancement of cellular memory by reducing stochastic transitions." Nature 435(7039): 228-232.

    Agarwal, A., R. Adams, G. C. Castellani and H. Z. Shouval (2012). "On the precision of quasi steady state assumptions in stochastic dynamics." J Chem Phys 137(4): 044105.

    Curran, S., G. Dey, P. Rees and P. Nurse (2022). "A quantitative and spatial analysis of cell cycle regulators during the fission yeast cycle." Proc Natl Acad Sci U S A 119(36): e2206172119.

    Gerard, C., J. J. Tyson, D. Coudreuse and B. Novak (2015). "Cell cycle control by a minimal Cdk network." PLoS Comput Biol 11(2): e1004056.

    Gould, K. L. and P. Nurse (1989). "Tyrosine phosphorylation of the fission yeast cdc2+ protein kinase regulates entry into mitosis." Nature 342(6245): 39-45.

    Keifenheim, D., X. M. Sun, E. D'Souza, M. J. Ohira, M. Magner, M. B. Mayhew, S. Marguerat and N. Rhind (2017). "Size-Dependent Expression of the Mitotic Activator Cdc25 Suggests a Mechanism of Size Control in Fission Yeast." Curr Biol 27(10): 1491-1497 e1494.

    Kim, J. K., K. Josic and M. R. Bennett (2014). "The validity of quasi-steady-state approximations in discrete stochastic simulations." Biophys J 107(3): 783-793.

    Kim, J. K. and J. J. Tyson (2020). "Misuse of the Michaelis-Menten rate law for protein interaction networks and its remedy." PLoS Comput Biol 16(10): e1008258.

    Lundgren, K., N. Walworth, R. Booher, M. Dembski, M. Kirschner and D. Beach (1991). "mik1 and wee1 cooperate in the inhibitory tyrosine phosphorylation of cdc2." Cell 64(6): 1111-1122.

    Mitchison, J. M. (2003). "Growth during the cell cycle." Int Rev Cytol 226: 165-258.

    Novak, B., Z. Pataki, A. Ciliberto and J. J. Tyson (2001). "Mathematical model of the cell division cycle of fission yeast." Chaos 11(1): 277-286.

    Novak, B. and J. J. Tyson (1995). "Quantitative Analysis of a Molecular Model of Mitotic Control in Fission Yeast." J Theor Biol 173: 283-305.

    Patterson, J. O., S. Basu, P. Rees and P. Nurse (2021). "CDK control pathways integrate cell size and ploidy information to control cell division." Elife 10.

    Shahrezaei, V., J. F. Ollivier and P. S. Swain (2008). "Colored extrinsic fluctuations and stochastic gene expression." Mol Syst Biol 4: 196.

    Tang, Z., T. R. Coleman and W. G. Dunphy (1993). "Two distinct mechanisms for negative regulation of the Wee1 protein kinase." EMBO J 12(9): 3427-3436.

    Tyson, J. J., A. Csikasz-Nagy and B. Novak (2002). "The dynamics of cell cycle regulation." Bioessays 24(12): 1095-1109.

  3. Review Commons

    Note: This preprint has been reviewed by subject experts for Review Commons. Content has not been altered except for formatting.

    Learn more at Review Commons

    Referee #3

    Evidence, reproducibility and clarity

    The paper by Novak and Tyson revisits a recent paper from Nurse group on the bistability of mitotic switch in fission yeast using mathematical modelling. The authors extend their older models of mitotic entry check point and implement both deterministic and stochastic version of new model. They show this model does indeed possess bistability and show that combined with stochastic fluctuations the model can show bimodality for the cyclin-CDK activity at a particular cell size consistenent with the recent experimental data. However, the authors also show alternative model that has mono-stable ultrasensitivity can also explain the data and suggest experiments that can prove the existence of hysteresis and therefore bistability.

    While the biological implication of the study is well explained, the authors can improve the presentation of their model and the underlying assumptions. I have the following comments and suggestions for improvement of the paper.

    1. The cartoon of the mathematical model is confusing at places, for example the wee1-CDK complex according to the equations either dissociates back to wee1 and CDK or gives rise to pCDK and wee1, the arrow below is confusing as it implies it can also give rise to wee1p, the CDK phosphorylation of wee1 is already included in the diagram. Also, the PP2A is put on the arrow for all reactions but for wee1p2 to wee1p its action shown with a dashed line. Also, I wondered if wee1p and wee1p2 can also bind CDK and sequester or phosphorylate CDK?
    2. The rates and variables in the ODEs are not fully described. Also sometimes unlcear what is parameter and what is a variable, I had to look a the code.
    3. The model has quite a few parameters, but these are not at all discussed in the paper. How did the authors come up with these particular set of parameters, has there been some systematic fitting, or tuning by hand to produce a good fit to the data? I could only see the value of the parameters in the code, but perhaps a table with the parameters of the model, what they mean and their value (and perhaps how the values is obtained) is missing.
    4. The authors are using the Gillespie algorithm with time varying parameters (as some rates depend on volume and volume is not constant). Algorithm needs to be modified slightly to handle this (see for example Shahrezaei et al Molecular Systems Biology 2008).
    5. The authors correctly point out, ignoring mRNA has resulted in underestimation of noise, however another point is that mRNA life times are short and that also affects the timescale of fluctuations and this may be relevant to the switching rates between the bistable states.
    6. In the introduction add, "In this study" to "Intrigued by these results, we investigated their experimental observations with a model of bistability in the activation of cyclin-CDK in fission yeast.

    Significance

    Overall, this is an interesting study that revisits an old question and some recent experimental data. The use of stochastic modelling in explaining variability and co-existence of cell populations in the context of cell cycle and comparison to experimental data is novel and of interest to the communities of cell cycle researchers, systems biologists and mathematical biologists.

  4. Review Commons

    Note: This preprint has been reviewed by subject experts for Review Commons. Content has not been altered except for formatting.

    Learn more at Review Commons

    Referee #2

    Evidence, reproducibility and clarity

    Summary: The manuscript asks whether the data reported in Patterson et al. (2021) is consistent with a bistable switch controlling the G2/M transition in fission yeast. Patterson et al. (2021) use an engineered system to decouple a non-degradable version of Cyclin-dependent kinase (CDK) from cell growth and concomitantly measure CDK activity (by the nuclear localization of a downstream target, Cut3p). They observe cells with indistinguishable CDK levels but two distinct CDK activities, which they posit shows bistable behavior. In this study, the authors ask if other models can also explain this data. The authors use both deterministic and Gillespie based stochastic simulations to generate relationships between CDK activities and protein levels for various cell sizes. They conclude that the experiments performed in Patterson et al. are insufficient to distinguish between a bistable switch and a reversible ultrasensitive switch. They propose additional experiments involving the use a degradable CDK construct to also measure the inactivation kinetics. They propose that a bistable switch will have different forward (OFF->ON) and backward (ON->OFF) switching rates. A reversible ultrasensitive switch will have indistinguishable switching rates.

    The results are interesting and worth publication in a computational biology specific journal, as they might only appeal to a limited audience. Several ideas need to be clarified and additional information needs to be provided about the specific parameters used for the simulations:

    Major comments:

    1. The parameters need to be made more accessible by means of a supplementary table and appropriate references need to be cited. It is not clear why Michaelis Menten kinetics will not be applicable to this system. Has it been demonstrated that the Km s of the enzymes are much greater than the substrate concentrations for all the reactions? If yes, please cite. It will not be surprising if the simulation with Michaelis Menten would alter the dynamics shown in this study. A reversible switch with two different enzymes (catalyzing the ON->OFF and OFF->ON transitions) having different kinetics can give asymmetric switching rates. This would directly contradict what has been shown in Figure 7A-D.
    2. Line 427: The authors use a half-time of 6 hours in their model as Patterson et al. used a non-degradable construct. It is not clear why dilution due to cell growth has not been considered. The net degradation rate of a protein is the sum of biochemical degradation rate and growth dilution rate. The growth dilution rate seems significant (140 mins doubling time or 0.3 h-1 dilution rate) relative to assumed degradation rate (0.12 h-1). Please clarify why was the effect of dilution neglected in the model or show by sensitivity analysis this does not change the predicted CDK activation thresholds.
    3. Line 402 The authors state that the production rate of the Cdk protein is 'assumed' proportional to the cell volume. The word 'assumed' is incorrect here as a simple conversion of concentration-based differential equation (with constant production rate) to molecular numbers would show that production rate is proportional to the volume. This is not an assumption.
    4. Line 423 Please cite the appropriate literature that shows that fission yeast growth during cell division is exponential. If the dynamics are more complicated, involving multiple phases of growth during cell division, please state so.
    5. Line 250 The authors convert the bistable version of the CDK switch to reversible sigmoidal by assuming that Wee1 and Cdc25 phosphorylation is proportional to the CDK level rather than activity, which seems biochemically unrealistic. This invokes an altered circuit architecture where inactive CDK has enough catalytic activity to phosphorylate the two modifying enzymes (Wee1/Cdc25) but not enough to drive mitosis. This might be possible if the Km of CDK for Wee1/Cdc25 is lower relative to other downstream substrates that drive mitosis. The authors can reframe this section of the paper to state this possibility, which might be interesting to experimentalists.
    6. It is difficult to phenomenologically understand a bistable switch just based on differences in activation and inactivation thresholds. For example, a reversible ultrasensitive switch also shows a difference in activation and inactivation thresholds (Figure 7D). How much of a difference should be expected of a bistable switch versus reversible switch? Moreover, as the authors clearly understand (line 275), time-delays in activation and inactivation reactions can inflate these differences. In the future, if the authors can convert the equations to potential energy space as done in Acar et al. 2005 (Nature) in Figure 3c-d, it will be useful. Also, predicting the distribution of switching rates from the Gillespie simulation might be informative and can be directly compared to experimental measurements in the future (if the Cut3p levels in nucleus and cytosol equilibrates fast enough or other CDK biosensors are developed).

    Minor comments:

    1. Line 2: Please replace "In most situations" to "In favorable conditions"

    Referee cross-commenting

    I agree with Reviewer 1 that this falls more under pointing out an alternative interpretation of a single experiment than challenging widely supported orthodoxy about how the eukaryotic cell cycle leaves mitosis.

    Significance

    Suitable for specialist comp bio journal eg PLoS Comp Bio

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    Referee #1

    Evidence, reproducibility and clarity

    Novák and Tyson present a model-based analysis of published data that had claimed to demonstrate bistable activation of CDK at the G2/M transition in fission yeast. They point out that the published data does not distinguish between ultra-sensitive (switch-like, but reversible) and bistable (switch-like, but irreversible) activation. They back up their intuition with robust quantitative modeling. They then point out that, with a simple experimental modification, the published experiments could be repeated in a way that would test between the ultra-sensitive and bistable possibilities. Therefore, this is a rare paper that makes a specific modeling-based prediction and proposes a straightforward way to test it. As such, it will be of interest to a broad range of workers involved in the fields cell cycle and regulatory modeling. Nonetheless, attention to the following points would improve the manuscript.

    The authors should be more careful about how they describe protein abundance. They often refer to protein level. I believe in every case they mean protein concentration, but this is not explicitly stated; it could be interpreted as number of protein molecules per cell. The authors should either explicitly state that level means concentration or, more simply, use concentration instead of level.

    The authors should explain why they include stoichiometric inhibition of CDK byWee1 in their model. Is it required to make the model work in the wild-type case, or only in the CDK-AF case. My intuition is it should only be required in the AF case, but I would like to know for sure. Also, they should state if there is any experimental data for such regulation.

    The authors should explicitly state, on line 131, that the fact that "the rate of synthesis of C-CDK molecules is directly proportional to cell volume" results in a size-dependent increase in the concentration of C-CDK.

    The authors should explain, on line 100, why they are "quite sure the bistable switch is the correct interpretation".

    On line 166, include the units of volume.

    On lines 152 and 237, "smaller protein-fusion levels "should be replaced with "lower protein-fusion concentrations".

    Referee cross-commenting

    I concur with the other two reviews.

    Significance

    The paper is significant in that it points out a alternative interpretation for an important result in an important paper. Specifically, it points out that the published data is consistent with activation of CDK at the G2/M transition in fission yeast could be ultra-sensitive (switch-like, but reversible) instead of bistable (switch-like, but irreversible). The distinction is important because it has been claimed, by the authors of the submitted manuscript among others, that bistability is required for robust cell-cycle directionality. However, activation of CDK at the G2/M transition in other species has been shown to be bistable and the authors state that they are "quite sure the bistable switch is the correct interpretation". So, the paper is more likely an exercise in rigor than an opportunity to overturn a paradigm.

  6. Version published to 10.1101/2024.01.23.576917v1 on bioRxiv