Evolution of cell size control is canalized towards adders or sizers by cell cycle structure and selective pressures

  1. Felix Proulx-Giraldeau
  2. Jan M Skotheim
  3. Paul François

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    Evaluation Summary:

    This paper develops evolutionary simulations to identify the type of molecular networks that can give rise to size control. We now know a lot about the functional consequences and underlying molecular biology of different cell size control strategies, but comparatively less about which factors select for particular mechanisms. The authors address this point in an evolutionary framework. They show that the evolution of a specific cell size control mechanism is dependent on the cell cycle structure. The paper will interest researchers in development, evolution, and physics of biological systems.

    (This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 and Reviewer #2 agreed to share their name with the authors.)

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Abstract

Cell size is controlled to be within a specific range to support physiological function. To control their size, cells use diverse mechanisms ranging from ‘sizers’, in which differences in cell size are compensated for in a single cell division cycle, to ‘adders’, in which a constant amount of cell growth occurs in each cell cycle. This diversity raises the question why a particular cell would implement one rather than another mechanism? To address this question, we performed a series of simulations evolving cell size control networks. The size control mechanism that evolved was influenced by both cell cycle structure and specific selection pressures. Moreover, evolved networks recapitulated known size control properties of naturally occurring networks. If the mechanism is based on a G1 size control and an S/G2/M timer, as found for budding yeast and some human cells, adders likely evolve. But, if the G1 phase is significantly longer than the S/G2/M phase, as is often the case in mammalian cells in vivo, sizers become more likely. Sizers also evolve when the cell cycle structure is inverted so that G1 is a timer, while S/G2/M performs size control, as is the case for the fission yeast S. pombe . For some size control networks, cell size consistently decreases in each cycle until a burst of cell cycle inhibitor drives an extended G1 phase much like the cell division cycle of the green algae Chlamydomonas . That these size control networks evolved such self-organized criticality shows how the evolution of complex systems can drive the emergence of critical processes.

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  1. Version published to 10.7554/elife.79919 on eLife
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    Author Response

    Reviewer #1: “(Public Review):

    The main result of the paper is a statistical dependence between the evolved size control strategy and the structure of the cell cycle, in that size control that manifests early (later) in the cell cycle tends to give adder- (weakly sizer-) like strategies. Notably, even when the final evolved network shows weak adder or weak sizer-like behaviour, they find strong sizer-like control in the evolutionary transient. Finally, they constrain the evolutionary algorithm to sense cell size only through stochastic fluctuations of protein concentrations and uncover a strategy that exhibits hallmarks of self-organised criticality.

    The questions studied by the authors are both interesting and timely, and their results are intriguing and well documented. On the whole, the conclusions are convincingly argued, and the authors do an excellent job of extracting qualitative features from their evolved networks. However, the manuscript is a little difficult to read, with the figures being crowded and difficult to parse. In addition, while there is a lot of detail in some places (as in the description of one particular feedback control strategy), other results are less fleshed out (such as statistical summaries of the different simulations). The manuscript would benefit from a sharper presentation of the results.’

    We have done our best to tighten the writing and better focus on the main results of the paper. We have done this in response to the specific criticisms of the reviewers, however, most of the comments indicated that our manuscript was rather dense and so important points had been lost. Therefore, in the revision, we have mostly focused on increasing the clarity rather than condensing our prose further.

    A particularly interesting question addressed in the paper is why adders are more commonly found when sizers are believed to be better at controlling cell size. Here, the authors' simulations give two answers: first, that sizers tend to appear when cell size control is exerted later in the cycle (as in S. pombe). Second, that even when adders eventually evolve, the evolutionary transient passes through a strong sizer strategy. As the adder-vs-sizer question is repeatedly raised, it would strengthen the paper to have a longer and sharper discussion on (a) why early cell size control favours adders, and (b) why sizers appear as transients when fluctuations in cell size are large?’

    We now clarify these key points and extend our discussion. The question as to why sizers appear as transients when fluctuations in cell size are large is more complex. We see repeatedly that sloppy sizers evolve first. But, these sizers are not necessarily that good at giving a low CV. Then, as the system continues to evolve, adders appear that are better at reducing CV than the noisy sizers. This emphasizes that the contribution to reducing the CV comes from two parts, first the slope contribution defining the relationship between the amount of growth in the cell cycle and the cell size at birth, and second, the amount of noise in this process, i.e., how variable the result will be for two cells born the same size. The system proceeds from a noisy sizer to a less noisy adder while reducing the CV as selected for. Thus, we speculate that in the later stages of evolution, where the system has already significantly reduced cell size variability, the ability to more accurately perform size control with less noise reduces the selection pressure on the slope so that adders tend to emerge. To address the comment, we have extended our discussion as to why early cell size control favors adders. We have broken the penultimate paragraph in the discussion into two parts where we now write:

    “Our evolution simulations gave insight into factors that bias evolution towards sizer or adder type control mechanisms (Fig. 4). First, it is worth noting that our evolution simulations were not deterministic. There was no one-to-one correspondence between a given evolutionary pressure and any one specific cell size control mechanism. Rather, our claims represent an average behavior observed over the course of many simulations. It is first worth noting that size control, as measured by the CV at a particular point in the cell cycle, has contribution both from the slope of the correlation between cell size and the amount of cell growth and from the amount of noise characterizing the differences between cells that are initially the same size (Di Talia et al., 2007). It is therefore possible that a low noise adder can produce a lower CV than a higher noise sizer. This is reflected in the evolutionary paths of some of our simulations, which traverse from a noisy sizer to a less noisy adder (Fig. 5). However, we anticipate even noisy sizers will be better than adders at controlling cell size in response to large deviations away from the steady state distribution. This is because sizers will always return the cell size to be within the steady state distribution within a cell cycle.

    In the selection of a size controlling G1 network followed by a timer in S/G2/M, we observed a prevalence of adders that is consistent with the prevalence of adders reported in the literature. While fewer in number, sizers have also been observed. That the most accurate sizers have been observed in the fission yeast S. pombe (Fantes, 1977; Sveiczer et al., 1996; Wood & Nurse, 2015), and that this organism performs cell size control at G2/M rather than at G1/S led us to explore the effect of cell cycle structure on the evolution of cell size control. We found that controlling cell size later in the cycle in S/G2/M biases evolution away from adders and towards sizers. In retrospect, this result can be rationalized since any size deviations incurred earlier during the timer period can be compensated for by the end of the cycle with the sizer. However, when the order is inverted, any size deviations escaping a G1 control mechanism would only be amplified by exponential volume growth during the S/G2/M timer period. A second recent case exhibiting sizer control was found in mouse epidermal stem cells, which exhibit a greatly elongated G1 phase and a relatively short S/G2/M phase (Mesa et al., 2018; Xie & Skotheim, 2020). We found that if we increased the relative duration of G1 in our simulations by shortening the S/G2/M timer, we also see a bias towards sizer control. In essence, by extending G1 to a larger and larger fraction of the cell cycle the control system is gradually approaching a size control taking place at the end of the cell cycle, i.e., an S/G2/M size control. Taken together, these simulations suggest the principle that having size-dependent transitions later in the cell cycle selects for sizers, while having such transitions earlier selects for adders.”

    The final part of the paper, which describes a strategy based on sensing size through concentration fluctuations, is very interesting but brief, which is understandable given the quantity of results presented earlier in the paper. Nonetheless, it provides an excellent example of the power of the authors' approach.

    Overall, the results in this paper are a compelling addition to the recent interest in cell size control.’

    We thank the reviewer for their careful reading of our manuscript and their support.

    Reviewer #2 (Public Review):

    The use of evolutionary models to understand the emergence of cell size control is novel and interesting. One strength of the approach is that simulations do not impose any mechanistic model for cell size control, rather the feedback motif for size control emerges from optimisation of chosen fitness functions. This allows the authors to come up with various size control motifs for given evolutionary pressures and model rules. Interestingly, the authors find that there is no one-to-one correspondence between specific size control mechanisms and evolutionary pressures, rather size control mechanisms are dependent on cell cycle structures. The authors also evolve a size control model based on the sensing of protein concentration fluctuations. This model exhibits interesting features such as self-organized criticality and the existence of very large cells that achieve size homeostasis by undergoing rapid cell divisions. The authors' model, however, comes with many arbitrary choices and assumptions that need further justifications and theoretical results should be compared with experimental data to establish the applicability of the model.

    We thank the reviewer for their careful reading of our manuscript and have worked to address its previous shortcomings as described below.

    Major Comments:

    1. Fitness function choices: Two fitness functions are used for the majority of this paper, number of cell divisions and CV_birth. What motivates the choice of these fitness functions and how do they relate to single-cell fitness?

    We added some text describing the choice of fitness function in the Supplement in the S3A - Fitness subsection. Using the number of cell divisions as a fitness makes sense since the higher the number of divisions in a given window of time, the bigger the population, which corresponds to the classical Darwinian fitness. Adding CV as an extra fitness specifically pushes the system towards better size control, which is the problem we aim to study, and also helps the optimization process. This is an effective way to include in our simulated evolution all observed detrimental effects observed when cell size is not controlled well. In the methods section we write:

    “We impose two evolutionary selection pressures in the form of two fitness functions. The first fitness function is simply the number of cell divisions during a long period, which we call NDiv . This is consistent with the classical definition of fitness as optimizing the number of offspring and is to be maximized by the algorithm. The second fitness function is the coefficient of variation of the volume distribution at birth for those NDiv generations, which we call CVBirth and is to be minimized by the algorithm. This penalizes broad distributions of volume at birth, which are detrimental to cell size homeostasis, which is what we aim to examine here.”

    Since the selection for tight size distribution is enforced via minimization of CV_birth, the model is unlikely to explain the timer control that is observed in some parts of the cell cycle. The authors discuss how a single fitness function results in all-or-nothing selection in the evolutionary algorithm, however, a third simultaneous fitness function is not considered. Are the results of this paper robust with respect to the addition of other selection pressure (for instance, optimization of growth rate)? This is a crucial question that is not addressed in the text.

    While we could always add more fitness functions, we have to start somewhere. The two fitness functions we use make most sense for the problem we are interested in, and allows us to obtain some clear results from the examination of an already complex starting point. Adding more than two fitness functions greatly increases the complexity of the problem. In fact, we are not aware of any work in the field of computational evolution using more than two fitness functions. One reason is that simulated evolution under control of two fitness functions is already not well understood in general (as we discussed previously in Francois & Siggia, Physical Biology 2008; Henry et al Plos Comp Bio 2018). We hope our simulations will inspire other work in this direction.

    1. Cell-cycle structure not considered to be changeable in evolution: Based on the presented details of the evolutionary algorithm, the network topology parameters are varied but not the temporal structure of the cell cycle, i.e. timer in G1/S and sizer S/G2/M or sizer in G1/S and timer in S/G2/M, etc. How do you justify evolution in one part of the cell cycle but not in the other? Do your results hold when the temporal structure is permitted to evolve?

    We are very interested in how the network structure affects the results. To address this point, we did invert size-dependence of the cell cycle phases as suggested by the reviewer i.e., we considered a fission yeast-like network with a timer in G1 and a sizer in S/G2/M (see Fig. 4,5, and S10). The possibilities of performing different types of evolution experiments is almost endless. We therefore restricted our examination to cases inspired by naturally occurring networks in well studied model organisms such as budding and fission yeasts. While it is in principle possible that size control could take place in multiple cell cycle phases, we do not yet know of a naturally occurring example and so chose not to explore this possibility at the present time. Nevertheless, the reviewer is raising a very interesting question as to why evolution selecting for cell size control tends to pick one or another cell cycle phase, but possibly not both, in a particular organism. We do not know the answer to this question at present and refrain from attempting to address it since our manuscript is already quite dense. Future work can explore this interesting direction.

    1. Noise sources: The authors consider noise protein quantity or concentration while neglecting noise in growth rate or division. Can the assumption that growth noise is negligible compared to protein production noise be supported by experimental data? This is a crucial assumption that is not supported by a discussion of physical values or citations. In addition, it is assumed later in the supplement (S132-133) that there is no division noise without presenting justification for why that noise is negligible on the scale of protein production noise.

    As for many other points raised by the referee, there is a necessary balance to achieve between biochemical realism and simplifying assumptions to theoretically study such problems. Of course we fully agree with the reviewer that there are multiple sources of noise in the system. In this study, we chose a hierarchical way of introducing noise in the system, starting with the biggest contributing factor and incrementally adding sources of noise if needed. We chose to first focus on noise in the cell cycle phases themselves whose CV can be as high as 50% (cf Fig. 1 in Di Talia et al 2007 Nature). For this reason, we first introduced noise in the precise timing of the G1/S transition as well as in the timing of the S/G2/M phase duration. Next, we introduced protein production noise because it is larger than the noise associated with cell division and cell growth rate in several cases where it has been measured. For example, the CV of cell growth rate in a diploid budding yeast is ~14% (Di Talia et al 2007 Nature; cf Table S12). The noise in partitioning at cell division is easier to measure in symmetrically dividing cells. For human cells grown in culture, division noise is ~10% (cf Fig. 3G in Zatulovskiy et al 2020 Science). In contrast, noise in protein concentrations is typically higher. This can be seen in the examination of molecular noise across all GFP labeled proteins in budding yeast (Newman et al, Nature 2006, PMID: 16699522). The CV in concentration of regulatory proteins in similarly sized cells is ~20-30% which is larger than noise in division by partitioning or noise in cell growth rate. We therefore next focused our analysis on the effects of protein production noise.

    In revising our manuscript, we now also consider noise in cell growth rate and noise in partitioning of mass at division as suggested by the reviewer. This results in slightly lower control, and more noise in alignment with our intuition. However, broadly speaking, our results are unchanged (see new supporting figures Fig. S6-S7 shown below). We now describe the logic of our series of simulations of increasing complexity in the methods section, which has two new paragraphs that reads as follows: “In this study, we chose a hierarchical way of introducing noise in the system, starting with the biggest contributing factor and incrementally adding additional sources of noise in subsequent analyses. All simulations presented include noise (stochastic control of G1/S transition and timing of S/G2/M, see below) in the cell cycle phases, whose CV has been found to be as high as 50% (Di Talia et al., 2007). Then, we introduced protein production noise via Langevin noise because the CV of regulatory protein concentrations is typically 20-30% (Newman et al., 2006). Importantly, the cell volume also contributes to stochastic effects, which are larger in smaller cells with fewer molecules. Thus, for stochastic simulations, we include a multiplicative 1/√V contribution to the added Gaussian noise term (see more complete description in the Supplement).

    We also checked that our results are largely invariant when adding other sources of noise (see Figs. S5-S7). In these simulations, we also included noise in cell growth rate (CV ~15%; e.g. (Di Talia et al., 2007), and in mass partitioning at cytokinesis (CV ~10%; e.g. (Zatulovskiy et al., 2020).”

    1. Types of biochemical interactions considered: It is assumed that inhibitor protein production rate scales with cell volume. Is this assumption supported by data? The assumption is contrary to the production rate of the inhibitor protein Whi5 in budding yeast, which does not scale with cell volume.

    In general, most proteins are at relatively constant concentration as cells grow. This means that their production rate (measured in number of proteins per time) has to scale in proportion to cell volume. As noted by the reviewer, Whi5 in the budding yeast is an exception to the general rule where the production rate does not scale with cell volume. This Is why Whi5 is diluted by growth, leading to a sizer in G1. However, allowing the network to generate size control with a diluted inhibitor starting point is basically too simple because it would start with a size sensor and does not need to evolve any feedback mechanism. Here, we are focused on exploring how cell size control can be done by a network with multiple feedbacks rather than just the concentration of a single protein. We made those points more explicit in the text, which now included the following sentences in the methods section: “We note that we are not allowing the cell to employ proteins such as Whi5 in budding yeast whose production is independent of cell size so that its concentration is a direct readout of cell size (Schmoller et al 2015; Swaffer et al 2021). We chose to do this because we want to explore how cell size control can be done by a network with multiple feedbacks rather than just the concentration of a single protein with a special dedicated synthesis mechanism.”

    1. Comparisons to data: Currently no attempt has been made to compare the model predictions quantitatively with experimental data that are easily available. For instance, how does the CV of cell birth size predicted by the model compare with cell size distribution in budding yeast or in the fission yeast? The same goes for the scaling of added volume with initial cell volume in different phases of the cell cycle. Furthermore, the noise parameters should also be calibrated to reproduce the cell size variability seen in experiments.

    To facilitate the comparison of our evolution simulations with model organisms we have included Table S1 in the supporting material, where we show the published results for budding yeast, fission yeast, and mammalian cells grown in culture and mouse epidermal stem cells growing in the animal. In fact, it turns out that distribution and CV that we obtained in our simulations are relatively similar in some cases to what is observed experimentally, but can also be much lower and exhibit a tighter control when optimized. However, the comparison is not perfectly fair since the model organisms were grown in laboratory conditions rather than their natural environment for which they are likely more optimized.

    Reviewer #3 (Public Review):

    In this paper, Proulx-Giraldeau et al. develop evolutionary simulations to study how size control can emerge. In the first part of the paper, the authors initiate cell cycle simulations with a simple network that does not allow cell size sensing and ask what molecular networks can lead to size control after evolution. Results show that a wide range of network types allows size control, some of which are comparable to experimentally identified networks such as the dilution inhibitor model in budding yeast. In the second part of the paper, the authors use their framework to ask how the structure of the cell cycle, including the duration of G1 vs. S/G2/M and the form of size control in each of these phases (i.e. 'sizer' or 'adder'), affects the overall size control. While this is a very important question and the authors bring comprehensive and interesting answers, it is less clear that framing the findings in the context of evolution is meaningful. Indeed, the solutions for how the combination of strength of size control, noise levels, and respective duration of the phases can be found analytically/with simulations that are not 'evolving' the cell cycle structure. Additionally, the finding that a sizer in G1 can lead to an overall adder if it is followed by a timer in S/G2/M is only true if a significant amount of noise is added during the timer phase. At present, this finding is discussed as a result of 'evolution' which is confusing and the dependency of this conclusion on the level of noise during S/G2 does not appear very clearly.

    With more cautiously formulated conclusions and a better discussion of already established theoretical and experimental work, this paper will become more accessible to experimentalists and will be a very valuable contribution to the field of cell size control.

    We thank the reviewer for their careful reading of the manuscript and their thoughtful comments.

    Major suggestions:

    1. Fig 4-5. While the use of the evolution simulation seems interesting to identify which underlying network(s) can result in size control, the use of the same framework to compare the result of sizer+timer vs. timer+sizer is less easy to interpret. Previous analytical/simulation approaches have explored how noise & duration of the timer phase can alter the 'sizer' or 'adder' signature (see doi.org/10.1016/j.celrep.2020.107992, doi.org/10.3389/fcell.2017.00092, for example) and what evolutionary simulations add to this question is unclear.’

    We thank the reviewer for pointing out this highly relevant work, which we now cite where appropriate at various places in the manuscript. We agree that several of our results could have been derived from non-evolutionary analysis as performed in this work (such as the conclusion that a sizer followed by a timer can yield an adder). However, many of our other results cannot. For example, we are interested in how a network based on constant concentrations of proteins can measure cell size. Our evolution simulations yield highly non-trivial networks which we then proceed to analyze. We now clarify the distinction between our approach using evolution simulations to the more traditional analytical approach in the discussion. We added the following text: “We note that these generic results of how sizers and adders can govern cell size homeostasis can be derived from more traditional analytical methods (Barber et al., 2017; Willis et al., 2020). However, our evolution simulations are particularly useful because the molecular networks that evolved give non-trivial insights into how the observed size homeostasis dynamics can be regulated.”

    – What is the authors' interpretation of why the optimization of Pareto vs. number of divisions yield different size control results (Fig. 4A)? Is it possible that these different fitness parameters allow for the evolution of different levels of noise/duration of the timer phase?

    This relates to what we discuss in section “A two-step evolutionary pathway for cell size control”. We think the effect is intuitive : if there is no selection on CV, there is no reason for the system to evolve good noise control in general. Then in the absence of secondary effects such as size dependent growth rates, etc…, networks such as the one presented in Fig 5 A are essentially optimum for the number of divisions, and are pure sizers. This is not related to the timer phase as far as we can see. We added a few words at the end of that section to make this more explicit.

    – In the conclusion: 'G1 control is more conducive to the evolution of adders, while G2 control is more conducive to sizers', do the authors really believe that this is an evolutionary acquired trait, or are their observations instead the natural consequence of having a noise-adding phase (timer + multiplicative noise) after a phase with size control?

    We believe what the reviewer says, ie, adder is a consequence of noise-adding phase after the size control. We do not think this is necessarily an evolutionary acquired trait. As discussed above, and now in our discussion, this result could have been found using traditional analytical approaches. That the result is similar in a computational evolution simulation is interesting because the flexibility of the PhiEvo algorithm might have allowed for different phenomenological results to emerge. That they did not do so further strengthens the intuition built up from the analytical approach.

    – A perfect sizer in G1, followed by a timer (with exponential growth) in S/G2/M would simply give an overall 'noisy sizer' (i.e. the slope of final volume vs. initial volume would still be 0 but with some variability around the slope). Only beyond a certain level of noise added in S/G2/M, would the sizer signature be lost. Would it be possible for the authors to perform simulations with different levels of noise (on the timer in S/G2) to help understand this conclusion better? This conclusion could be one of the most valuable to experimentalists studying different organisms.

    This is an excellent suggestion by the reviewer and we have performed these evolution experiments examining the effect of modulating the noise in the S/G2/M timer. We consider a CV in the timer of 0, 5, and 8% corresponding to no, medium, and high noise respectively. The average duration of the timer is half the time it takes to double the cell’s volume. Having specified the S/G2/M timer parameters, we then evolved and selected networks as previously, and compared ensembles of 60 networks for each noise level. The results are in line with our and the reviewer’s intuition. Increasing the noise, progressively leads to a loss of the sizer signature and increases the CV of cell size at birth. These results are described in a new paragraph in the results section modulating cell cycle structural constraints selects for sizers and adders, which reads as: “We next considered the effect of changing the amount of noise in the timer phase of the cell cycle. To do this, we examined the evolution of networks performing size control in G1 and where the S/G2/M phase with an increasing amount of noise. Increasing the noise in the timer progressively reduced the amount of size control done by the network (Fig. S5). This is likely because the fixed duration of S/G2/M allows the system to accurately reset protein concentrations for the subsequent cell cycle to promote accurate G1 control (Willis et al., 2020). We also examined the effects of adding noise to the cellular growth rate and to volume partitioning at division and found similar results (Fig. S6-S7).”

    The results are shown in the new supporting figure 5.

    1. Some aspects of the mathematical formalism were unclear:
    • Working with the hypothesis that growth is exponential and at a constant rate is reasonable. However, the description of the scenario where growth modulation contributes to size homeostasis is incorrect. E.g. the statement 'cells further from the optimum size grow slower' is not accurate. If size control occurs via growth regulation, what is expected is a negative correlation between size and growth rate (big cells grow slow, small cells grow fast).

    To clarify this point, we have modified the sentence to read as: “In the first class, it is crucial that the growth rate per unit mass of a cell depends on cell size so that cells that are significantly larger than the optimum cell size grow slower.”

    – The quantity I is produced with a rate proportional to volume, degraded at a constant rate, diluted by cell growth': why is I diluted? Concentration should be constant if I increases at the same rate as volume. 'the quantity of I does not initially depend in any way on the volume'. Does the quantity of I not increase with volume (since concentration is constant)?

    The equation for the amount of I does not have a dilution term, but the equation for the concentration of I does. This is easy to see if you consider stopping synthesis of I but continuing cell growth. In the case where I is stable, the concentration of I would decrease in proportion to the growth rate of the cell, which is the dilution term. In the case of constant synthesis of I, the concentration is indeed constant at equilibrium and reflects a balance between protein synthesis and dilution and degradation (e.g., see Eq. S4).

    Fig. 3, The rescaling of the variables to tau and Veq was difficult to understand. Fig. 3A: If T_S/G2/M is at ~0.5 of the doubling time tau, how relevant is it to look at the behaviour of T_(Vc) for values of T_(Vc)/tau above 0.5 (and beyond 1)? Fig 3B: for which value of T(Vc) is the prediction made?

    Time is rescaled to the amount of time it takes to double the biomass. Volume was rescaled to the average volume at the G1/S transition for a population of cells at the size distribution's steady state. We realize now that this nomenclature is unclear, and have replaced Veq with <VG1/S>, which we believe is more clear.

    Because of the timer constraint, T_(Vc)/tau has to be at least 0.5, which corresponds to a G1 phase with 0 duration. But, in principle, T_(Vc)/tau could have any value larger than 0.5. The range of T_(Vc)/tau is set by the size control mechanism after we specify the range of Vc that we wish to examine. To clarify this, we now denote what parts of the plot correspond to cells increasing or decreasing in size.

    The prediction is the solid line and is made for a bit more than the range of cell sizes that we see in the steady state simulation. We think there is confusion about our nomenclature for a single point indicated on each line as ‘Added Veq’. This point represents the average amount of volume added at steady state. To clarify this we now label this as <∆V>.

    1. Discussion:

    – Including a discussion of previous theoretical work that explored the consequences of varying the relative duration of the timer and sizer phases would be valuable.’

    As discussed above, we have now cited the previous theoretical work in the introduction, results, and discussion. We thank the reviewer for pointing out this omission.

    – A reason commonly evoked to explain why cells might show sizer vs. adder behaviour is the role of the growth mode: S. pombe is a sizer but is thought to grow linearly, E. coli behaves like a sizer when it grows slower than usual (see Walden et al. 2015). It would be helpful to mention this when discussing S. pombe and remind the reader that the findings of this paper are limited to exponential growth mode.

    As suggested, we clarify that our analysis is restricted to exponential growth rates and that S. pombe growth rates have been reported to deviate from exponential.

    – The paper seems to be focusing on the noise of the size control mechanism (i.e. probability of transitioning through G1/S based on levels if I) but does not address the question of other sources of noise (i.e. asymmetry at division). What do the authors think about the role of such sources of noise as selective pressure on size control mechanisms evolution?

    This point was also raised by referee 2. There is a necessary balance to achieve between biochemical realism and simplifying assumptions to theoretically study such problems. Of course we fully agree with the reviewer that there are multiple sources of noise in the system. In this study, we chose a hierarchical way of introducing noise in the system that starts with the biggest contributing factor and incrementally adding sources of noise if needed.

    In revising our manuscript, we now also consider noise in cell growth rate and noise in partitioning of mass at division as suggested by the reviewer. This results in slightly lower control, and more noise in alignment with our intuition. However, broadly speaking, our results are unchanged (see new supporting figures Figs. S6-S7). We now describe the logic of our series of simulations of increasing complexity in the methods section, which has a new paragraph that reads as follows: “In this study, we chose a hierarchical way of introducing noise in the system, starting with the biggest contributing factor and incrementally adding additional sources of noise in subsequent analyses. All simulations presented include noise (stochastic control of G1/S transition and timing of S/G2/M, see below) in the cell cycle phases, whose CV has been found to be as high as 50% (Di Talia et al., 2007). Then, we introduced protein production noise via Langevin noise because the CV of regulatory protein concentrations is typically 20-30% (Newman et al., 2006). Importantly, the cell volume also contributes to stochastic effects, which are larger in smaller cells with fewer molecules. Thus, for stochastic simulations, we include a multiplicative 1/√V contribution to the added Gaussian noise term (see more complete description in the Supplement).

    We also checked that our results are largely invariant when adding other sources of noise (see Figs. S5-S7). In these simulations, we also included noise in cell growth rate (CV ~15%; e.g. (Di Talia et al., 2007), and in mass partitioning at cytokinesis (CV ~10%; e.g. (Zatulovskiy et al., 2020).”

  3. eLife

    Evaluation Summary:

    This paper develops evolutionary simulations to identify the type of molecular networks that can give rise to size control. We now know a lot about the functional consequences and underlying molecular biology of different cell size control strategies, but comparatively less about which factors select for particular mechanisms. The authors address this point in an evolutionary framework. They show that the evolution of a specific cell size control mechanism is dependent on the cell cycle structure. The paper will interest researchers in development, evolution, and physics of biological systems.

    (This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #1 and Reviewer #2 agreed to share their name with the authors.)

  4. eLife

    Reviewer #1 (Public Review):

    The main result of the paper is a statistical dependence between the evolved size control strategy and the structure of the cell cycle, in that size control that manifests early (later) in the cell cycle tends to give adder- (weakly sizer-) like strategies. Notably, even when the final evolved network shows weak adder or weak sizer-like behaviour, they find strong sizer-like control in the evolutionary transient. Finally, they constrain the evolutionary algorithm to sense cell size only through stochastic fluctuations of protein concentrations and uncover a strategy that exhibits hallmarks of self-organised criticality.

    The questions studied by the authors are both interesting and timely, and their results are intriguing and well documented. On the whole, the conclusions are convincingly argued, and the authors do an excellent job of extracting qualitative features from their evolved networks. However, the manuscript is a little difficult to read, with the figures being crowded and difficult to parse. In addition, while there is a lot of detail in some places (as in the description of one particular feedback control strategy), other results are less fleshed out (such as statistical summaries of the different simulations). The manuscript would benefit from a sharper presentation of the results.

    A particularly interesting question addressed in the paper is why adders are more commonly found when sizers are believed to be better at controlling cell size. Here, the authors' simulations give two answers: first, that sizers tend to appear when cell size control is exerted later in the cycle (as in S. pombe). Second, that even when adders eventually evolve, the evolutionary transient passes through a strong sizer strategy. As the adder-vs-sizer question is repeatedly raised, it would strengthen the paper to have a longer and sharper discussion on (a) why early cell size control favours adders, and (b) why sizers appear as transients when fluctuations in cell size are large?

    The final part of the paper, which describes a strategy based on sensing size through concentration fluctuations, is very interesting but brief, which is understandable given the quantity of results presented earlier in the paper. Nonetheless, it provides an excellent example of the power of the authors' approach.

    Overall, the results in this paper are a compelling addition to the recent interest in cell size control.

  5. eLife

    Reviewer #2 (Public Review):

    The use of evolutionary models to understand the emergence of cell size control is novel and interesting. One strength of the approach is that simulations do not impose any mechanistic model for cell size control, rather the feedback motif for size control emerges from optimisation of chosen fitness functions. This allows the authors to come up with various size control motifs for given evolutionary pressures and model rules. Interestingly, the authors find that there is no one-to-one correspondence between specific size control mechanisms and evolutionary pressures, rather size control mechanisms are dependent on cell cycle structures. The authors also evolve a size control model based on the sensing of protein concentration fluctuations. This model exhibits interesting features such as self-organized criticality and the existence of very large cells that achieve size homeostasis by undergoing rapid cell divisions. The authors' model, however, comes with many arbitrary choices and assumptions that need further justifications and theoretical results should be compared with experimental data to establish the applicability of the model.

    Major Comments:

    1. Fitness function choices: Two fitness functions are used for the majority of this paper, number of cell divisions and CV_birth. What motivates the choice of these fitness functions and how do they relate to single-cell fitness? Since the selection for tight size distribution is enforced via minimization of CV_birth, the model is unlikely to explain the timer control that is observed in some parts of the cell cycle. The authors discuss how a single fitness function results in all-or-nothing selection in the evolutionary algorithm, however, a third simultaneous fitness function is not considered. Are the results of this paper robust with respect to the addition of other selection pressure (for instance, optimization of growth rate)? This is a crucial question that is not addressed in the text.

    2. Cell-cycle structure not considered to be changeable in evolution: Based on the presented details of the evolutionary algorithm, the network topology parameters are varied but not the temporal structure of the cell cycle, i.e. timer in G1/S and sizer S/G2/M or sizer in G1/S and timer in S/G2/M, etc. How do you justify evolution in one part of the cell cycle but not in the other? Do your results hold when the temporal structure is permitted to evolve?

    3. Noise sources: The authors consider noise protein quantity or concentration while neglecting noise in growth rate or division. Can the assumption that growth noise is negligible compared to protein production noise be supported by experimental data? This is a crucial assumption that is not supported by a discussion of physical values or citations. In addition, it is assumed later in the supplement (S132-133) that there is no division noise without presenting justification for why that noise is negligible on the scale of protein production noise.

    4. Types of biochemical interactions considered: It is assumed that inhibitor protein production rate scales with cell volume. Is this assumption supported by data? The assumption is contrary to the production rate of the inhibitor protein Whi5 in budding yeast, which does not scale with cell volume.

    5. Comparisons to data: Currently no attempt has been made to compare the model predictions quantitatively with experimental data that are easily available. For instance, how does the CV of cell birth size predicted by the model compare with cell size distribution in budding yeast or in the fission yeast? The same goes for the scaling of added volume with initial cell volume in different phases of the cell cycle. Furthermore, the noise parameters should also be calibrated to reproduce the cell size variability seen in experiments.

  6. eLife

    Reviewer #3 (Public Review):

    In this paper, Proux-Giraldeaux et al. develop evolutionary simulations to study how size control can emerge. In the first part of the paper, the authors initiate cell cycle simulations with a simple network that does not allow cell size sensing and ask what molecular networks can lead to size control after evolution. Results show that a wide range of network types allows size control, some of which are comparable to experimentally identified networks such as the dilution inhibitor model in budding yeast. In the second part of the paper, the authors use their framework to ask how the structure of the cell cycle, including the duration of G1 vs. S/G2/M and the form of size control in each of these phases (i.e. 'sizer' or 'adder'), affects the overall size control. While this is a very important question and the authors bring comprehensive and interesting answers, it is less clear that framing the findings in the context of evolution is meaningful. Indeed, the solutions for how the combination of strength of size control, noise levels, and respective duration of the phases can be found analytically/with simulations that are not 'evolving' the cell cycle structure. Additionally, the finding that a sizer in G1 can lead to an overall adder if it is followed by a timer in S/G2/M is only true if a significant amount of noise is added during the timer phase. At present, this finding is discussed as a result of 'evolution' which is confusing and the dependency of this conclusion on the level of noise during S/G2 does not appear very clearly.

    With more cautiously formulated conclusions and a better discussion of already established theoretical and experimental work, this paper will become more accessible to experimentalists and will be a very valuable contribution to the field of cell size control.

    Major suggestions:

    1. Fig 4-5. While the use of the evolution simulation seems interesting to identify which underlying network(s) can result in size control, the use of the same framework to compare the result of sizer+timer vs. timer+sizer is less easy to interpret. Previous analytical/simulation approaches have explored how noise & duration of the timer phase can alter the 'sizer' or 'adder' signature (see doi.org/10.1016/j.celrep.2020.107992, doi.org/10.3389/fcell.2017.00092, for example) and what evolutionary simulations add to this question is unclear.
      - What is the authors' interpretation of why the optimization of Pareto vs. number of divisions yield different size control results (Fig. 4A)? Is it possible that these different fitness parameters allow for the evolution of different levels of noise/duration of the timer phase?
      - In the conclusion: 'G1 control is more conducive to the evolution of adders, while G2 control is more conducive to sizers', do the authors really believe that this is an evolutionary acquired trait, or are their observations instead the natural consequence of having a noise-adding phase (timer + multiplicative noise) after a phase with size control?
      - A perfect sizer in G1, followed by a timer (with exponential growth) in S/G2/M would simply give an overall 'noisy sizer' (i.e. the slope of final volume vs. initial volume would still be 0 but with some variability around the slope). Only beyond a certain level of noise added in S/G2/M, would the sizer signature be lost. Would it be possible for the authors to perform simulations with different levels of noise (on the timer in S/G2) to help understand this conclusion better? This conclusion could be one of the most valuable to experimentalists studying different organisms.

    2. Some aspects of the mathematical formalism were unclear:
      - Working with the hypothesis that growth is exponential and at a constant rate is reasonable. However, the description of the scenario where growth modulation contributes to size homeostasis is incorrect. E.g. the statement 'cells further from the optimum size grow slower' is not accurate. If size control occurs via growth regulation, what is expected is a negative correlation between size and growth rate (big cells grow slow, small cells grow fast).
      - 'the quantity I is produced with a rate proportional to volume, degraded at a constant rate, diluted by cell growth': why is I diluted? Concentration should be constant if I increases at the same rate as volume. 'the quantity of I does not initially depend in any way on the volume'. Does the quantity of I not increase with volume (since concentration is constant)?

    3. Fig. 2, The rescaling of the variables to tau and Veq was difficult to understand. Fig. 2A: If T_S/G2/M is at ~0.5 of the doubling time tau, how relevant is it to look at the behaviour of T_(Vc) for values of T_(Vc)/tau above 0.5 (and beyond 1)? Fig 2B: for which value of T(Vc) is the prediction made?

    4. Discussion:
      - Including a discussion of previous theoretical work that explored the consequences of varying the relative duration of the timer and sizer phases would be valuable.
      - A reason commonly evoked to explain why cells might show sizer vs. adder behaviour is the role of the growth mode: S. pombe is a sizer but is thought to grow linearly, E. coli behaves like a sizer when it grows slower than usual (see Walden et al. 2015). It would be helpful to mention this when discussing S. pombe and remind the reader that the findings of this paper are limited to exponential growth mode.
      - The paper seems to be focusing on the noise of the size control mechanism (i.e. probability of transitioning through G1/S based on levels if I) but does not address the question of other sources of noise (i.e. asymmetry at division). What do the authors think about the role of such sources of noise as selective pressure on size control mechanisms evolution?

  7. Version published to 10.1101/2022.04.12.488093v3 on bioRxiv
  8. Version published to 10.1101/2022.04.12.488093v2 on bioRxiv
  9. Version published to 10.1101/2022.04.12.488093v1 on bioRxiv