In general, there are two classes of mechanisms that cells use to control their size that can be separated in terms of whether cell division or cell growth per se is regulated by cell size. Note that in this work, we will use mass, size, and volume interchangeably. 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 (Cadart et al., 2019; Ginzberg et al., 2018; Miettinen and Björklund, 2016; Nordholt et al., 2020; Tzur et al., 2009). Such slower growing cells are then outcompeted by cells closer to the optimum size even when divisions occur purely by chance (Conlon and Raff, 2003). While size-dependent growth mechanisms exist and do support size homeostasis, such mechanisms rely on inefficient growth in all the cells away from the optimum size (Ginzberg et al., 2018; Miettinen and Björklund, 2016; Nordholt et al., 2020). To avoid such inefficient growth, many types of cells use active size control mechanisms to accelerate progression through the cell cycle in larger cells (Zatulovskiy and Skotheim, 2020). In our simulations, we keep cell growth rates constant over a physiological range of cell sizes. This allows us to focus on the common features of the molecular networks in which increasing cell size drives changes in molecular activities to trigger cell division. We assume that cell volume $V$ grows at a rate $\lambda \left(V\right)\times V$, so that growth is exponential when $\lambda$ is a constant. Volume is divided by 2 at each division after which we follow one of the two daughter cells. The growth rate sets the time scale for the system dynamics as it defines the doubling time $\tau =ln\left(2\right)/\lambda$. Any interdivision time shorter than $\tau$ will see the cell volume shrink at the next generation while any interdivision time larger than $\tau$ will see the cell volume grow. We also use the chemistry square bracket convention such that any protein X’s concentration is denoted by $\left[X\right]$ . Correspondingly, its quantity is denoted by $X$ only and is defined as $X=\left[X\right]\times V$.

We initialize our network evolution simulations with a very simplified model of the cell-cycle (Figure 1A). We model two independent phases of a symmetrically dividing cell, G1 and S/G2/M, separated by a commitment point at the end of G1 and division at the end of S/G2/M (Figure 1A, Chandler-Brown et al., 2017). We encode this cell cycle state information via a binary switch variable we call ‘S/G2/M Switch’ that is 0 in G1 and 1 in S/G2/M. In all simulations, we follow an inhibitor model (Heldt et al., 2018; Schmoller et al., 2015; Zatulovskiy et al., 2020) and assume that the probability of passing the G1/S transition is controlled by the quantity of a transcription regulator $I$. One way the quantity rather than the concentration of a molecule could be sensed is through its titration against a fixed cellular quantity such as the genome, which is part of a general class of titration-based cell size sensing mechanisms (Amodeo et al., 2015; Heldt et al., 2018; Si et al., 2019; Wang et al., 2009). A lower quantity of this inhibitor $I$ means a higher probability of a G1/S transition at the current time step of the simulation (Appendix 1—figure 2). Like all other proteins, the quantity $I$ is produced with a rate proportional to volume, degraded at a constant rate, diluted by cell growth, and equally partitioned between mother and daughter cells at division (see Materials and Methods). We found that due to the volume scaling assumption, $\left[I\right]$’s concentration alone was largely independent of volume and could not trigger a size-dependent G1/S transition, which is why we opted for the quantity of $I$ instead (Appendix 1—figure 3). Upon passing the G1/S transition, we assume cells are committed to division and there is a fixed time delay before they divide thus modeling S/G2/M as a timer (Chandler-Brown et al., 2017; Doncic et al., 2015). We initially fixed the timer duration to be roughly equal to 50% of the doubling time $\tau$ with some uniform noise such that G1 and S/G2/M durations would be the same at equilibrium. Regulation of the quantity $I$ during the cell cycle thus controls the precise timing of the G1/S transition, but it is not always perfect since the transition is probabilistic. This, along with the noise in S/G2/M timer duration, creates natural cell to cell variability in volume that needs to be compensated for by the evolved mechanism. We note that we initialized most of our simulations with one added interaction in which production of $\left[I\right]$ is activated by the S/G2/M Switch variable to reset its concentration to a higher level before the next generation. We initially ran simulations without this specific interaction but found that it systematically appears in the initial stages of evolution simulations. We therefore included it in the initial network to speed up our simulations. We refer the reader to the Appendix 1 for more details. The models used in this study are publicly available (Proulx-Giraldeau and François, 2022).

Typical dynamics of this simple cell-cycle model are represented in Figure 1. These dynamics are similar to models of cell cycles based on relaxation oscillators (Cross, 2003; Tsai et al., 2008). The left and right slow branches correspond to G1 and S/G2/M, respectively, and the fast horizontal branches represent G1/S and division. An intermediate fictitious nullcline is shown as a line that connects the average concentration $\left[I\right]$ at G1/S and at division. Starting with cell birth, the system goes down the left-G1 branch because of degradation, then jumps to the right-S/G2/M branch below the threshold for the G1/S transition, stays there while moving up due to production by the S/G2/M Switch, until it jumps back to the left branch at the end of the timer phase. We note that there is no explicit volume control in this initial model since the only control comes from the quantity of $I$ which does not initially depend in any way on the volume. This initial quantity sensing oscillator does not perform size control and instead results in unstable growth where size deviations are amplified at each generation instead of being corrected (Appendix 1—figure 3) as had been previously described for a size scaling inhibitor dilution model (Barber et al., 2017; Willis et al., 2020). Thus, the network needs to evolve some other interactions and/or parameters to go beyond a simple G1 inhibitor driven by production in S/G2/M to create a viable cell lineage.

To examine how networks controlling cell size could evolve, we ran evolutionary simulations that optimize both the number of divisions $N_{Div}$ and the coefficient of variation of the size distribution at birth $C{V}_{Birth}$ (see Materials and Methods for algorithm details). Figure 2A illustrates the behavior and results of a typical evolutionary run, with axes defined by both fitnesses used. Simulations successfully evolving size control mechanisms typically follow the same pattern. Networks initially cluster in two regions: region [ii] where cells have low $C{V}_{Birth}$ but grow too small and die after a few divisions, and region [i] where cells grow too big and reach our cut-off for fitness 2 (y-axis). Notice that our Pareto evolutionary algorithm maximizes network diversity, so that those two clusters are at first maintained during evolution (rank 1 Pareto networks Warmflash et al., 2012). As the number of epochs increases, networks in cluster [ii] have more and more divisions, but still grow too big, so that those cells are therefore penalized (see details in Appendix 1). At some point in evolution (around epoch 700 for this particular simulation), some weak control mechanism suddenly evolves, preventing cells from becoming too big without imposing a tight control on the average volume (see also Figure 5 for an explicit example of how this is done). Thus, fitness 2 collapses and the number of divisions is optimized simultaneously. Cells later optimize the control to give a lower $C{V}_{Birth}$ . The optimal networks, at the right most end of this line, both maximize $N_{Div}$ and minimize $C{V}_{Birth}$ .

Figure 2

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Evolution simulations are in part reproducible and most often lead to similar network topologies. The evolution trajectory leading to Model A1 is a typical example (Figure 2B, see also variations of this network in Model A2 in Figure 2H and models A3-6 in Appendix 1—figures 8–11). The minimal network common to all those models is very simple. One gene, $R$, is added to the seed network and is both repressed and titrated by $I$ forming the network motif known as a Mixed Feedback Loop (François and Hakim, 2005). Size control can then be understood intuitively as follows.$\left[I\right]$ represses $\left[R\right]$, which is thus only produced in the narrow window of the cycle when $\left[I\right]$ is low, i.e., when the cell is close to the G1/S transition and in early S/G2/M. But, since the quantity $I=\left[I\right]\times V$ is fixed at G1/S by design, the concentration $\left[I\right]$ is inversely proportional to the volume of the cell at the G1/S transition (V_G1/S) as shown in our simulations (Figure 2C). Because of this, the $\left[I\right]$ dependent synthesis rate of $\left[R\right]$ and therefore its subsequent concentration are (linear) functions of the volume of V_G1/S (Figure 2C), allowing for the cell to keep a memory of its volume at G1/S via the $\left[R\right]$ variable (this holds even once $\left[R\right]$ is constantly degraded for the remainder of the cycle). This has two effects. First, during S/G2/M, $I$ synthesis rate is proportional to volume by hypothesis (and thus to V_G1/S), and $I$ is titrated by $\left[R\right]$, also proportional to V_G1/S. Both effects even out so that cells are born with a fixed quantity of inhibitor $I$ that is independent of volume (Figure 2D). Second, after division, production of $I$ is 0 by hypothesis, but $I$ still is titrated in G1 by the remaining $\left[R\right]$ (still proportional to V_G1/S). Because $I$ quantity at the beginning of G1 is size independent, this ensures that daughter cells reach the $I$ quantity threshold of G1/S earlier if they were born larger, thus ensuring size control. To confirm this, we examine the change in quantity of $I$ as a function of time spent in G1 and of V_G1/S, and we see the slope of these two quantities is volume-dependent (Figure 2E). We notice that this evolved size mechanism likely is the simplest possible allowed by our formalism: on the one hand, it entirely captures the volume dependency in a single variable $\left[R\right]$, and on the other hand, it ensures proper scaling of $I$ both at birth and at G1/S for size control with the help of a single titration. Notice that such simple control also explains the sudden evolutionary “jump” of Pareto front around epoch 700 on Figure 2A, which corresponds to when the Mixed Feedback Loop motif first appears.

These evolved size control networks, while relying on quantity sensing, are conceptually similar to the budding yeast network relying on concentration sensing of the cell cycle inhibitor Whi5 since there is a constant quantity of $I$ present right after division (just like Whi5). In budding yeast, the Whi5 protein is passively diluted in G1 to increase the stochastic rate of progression through the G1/S transition. The time spent in G1 depends on the initial concentration of Whi5 at birth, which scales as $\frac{1}{V}$ , to promote a sizer mechanism. Here, the concentration of $\left[I\right]$ at birth scales as $\frac{1}{V}$ , but so does the threshold concentration of $\left[I\right]$ regulating the G1/S transition. This is precisely why an active titration mechanism is required to obtain G1 size control in our setup. Such homeostatic control ensures cell size returns to its steady state distribution following an artificial perturbation as soon as a volume deviation is detected at G1/S (Figure 2F). When we plot the amount of volume added $\Delta V$ for different phases of the cell cycle as a function of the initial volume at the beginning of these phases, we find an approximate adder over the whole cycle that results from weak sizer in G1 followed by a timer in S/G2/M as has been found in budding yeast (Chandler-Brown et al., 2017; Di Talia et al., 2007; Soifer et al., 2016, Figure 2G).

While we chose one simple model to illustrate the control mechanism common to our set of evolved networks (Model A1 shown in Figure 2B), other evolved networks were more elaborate but illustrated a similar principle. For example, Model A2 contains extra interactions for the volume sensing gene $\left[R\right]$, where $\left[R\right]$ is repressed by the S/G2/M Switch (meaning its production is completely shut down in S/G2/M leading to sawtooth-like dynamics). Furthermore, $\left[R\right]$ represses the synthesis of $\left[I\right]$, adding another layer of repression to promote size control beyond the previously described titration by $\left[R\right]$ (Figure 2H). If we perturb cell size to examine the dynamics of the return to steady state and look at the added volume during the cell cycle, we again see overall a weak adder behavior similar to that found in Model A1 (Figure 2I–J). We give additional examples of similarly evolved networks in Appendix 1—figures 8–11 where we can see the sensing and the feedback mechanism being implemented in slightly different ways. Yet, despite these mechanistic differences in feedback regulation the resulting function of the evolved networks were similar as indicated by their $C{V}_{Birth}$ (Appendix 1—table 1).

To study the mechanisms implicated in cell size control, we modify the control at G1/S and introduce the control volume $V_C$ . This control variable is independent from the biochemical network and is maintained fixed allowing us to disconnect the actual cell volume $V$ from the biochemical network and by forcing the G1/S transition to be triggered once $I_C=\left[I\right]\times V_C$ is low enough. We then numerically integrate the differential equations of the model and measure the period $T\left(V_C\right)$ of the simulated cell-cycle for this control volume. Use of the control volume allows us to break the size feedback system and distinguish its input, $V_C$ , from its output, the induced cycle period T (Angeli et al., 2004). We compute $T\left(V_C\right)$ for Models A1 and A2 and compare their responses with the analytical curves of the archetypical timer, adder, and sizer (Figure 3A and C). Those curves intersect at the point where the induced period is exactly equal to the population doubling time ($\tau =ln\left(2\right)/\lambda$), which defines the equilibrium volume achieved by our cell size control network corresponding to ${⟨V}_{G1/S}⟩$. Examination of the size control in different cell cycle phases indicates the contributions of G1 and S/G2/M to the overall system behavior (Figure 3B and D). We note that we later examine statistics of ensembles of evolved models but that Models A1 and A2 are both typical examples of evolved feedback-based models.

Figure 3

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Quantifying precisely how the cell cycle period depends on the control volume at G1/S allows us to see that the $T\left(V_C\right)$ for Model A1 overlaps with the theoretical adder curve over a broad range of volumes. In contrast, Model A2 behaves as a sizer for volumes smaller than the equilibrium volume. However, at higher volumes it behaves more like an adder/timer similar to Model A1. Model A2’s equilibrium is thus ‘tuned’ by evolution to be in a regime corresponding to the minimum of the $\Delta V_{Cycle}$ curve extrapolated from the $T\left(V_C\right)$ as shown in Figure 3D, precisely when the system transits from a sizer at small volumes to an adder-like behavior at higher volumes. Similar behavior was observed experimentally in budding yeast (Chandler-Brown et al., 2017; Delarue et al., 2017). Thus, both Models A1 and A2 approximate an adder near the equilibrium size, but their behavior differs further from equilibrium where for smaller volumes, Model A1 is still an adder while Model A2 is a sizer.

So far, our evolutionary algorithm selects networks that implement adders rather than sizers near the equilibrium size. This is surprising because sizers are in principle better than adders at controlling cell size and reducing the CV of the size distribution at birth, which is one of our fitness functions. That adders are more frequently observed in nature than sizers (Cadart et al., 2018; Eun et al., 2018; Jun et al., 2018; Westfall and Levin, 2017; Willis and Huang, 2017; Zatulovskiy and Skotheim, 2020), is consistent with our evolution simulations, but adds to the mystery as to why this takes place.

To gain insight into the underlying reason for the prevalence of adders, we considered what might be exceptional in the cases where sizers occur. The best studied, and highly accurate sizer, is found in the fission yeast S. pombe (Fantes, 1977; Sveiczer et al., 1996). In contrast to budding yeast and human cells, where the size control takes place largely in G1 phase, fission yeast exerts size control later in the cell cycle at the G2/M transition. That size control took place later in the cell cycle in fission yeast suggested that this structural feature of its cell cycle might be responsible for its stronger size control. To test this, we inverted our seed network so that the G1 phase was a timer, while cell size control could evolve in S/G2/M (see Materials and Methods). We compare these results to the evolutionary simulations starting with the ‘control’ seed network with G1 size control and an S/G2/M timer (Figure 2). We note that S. pombe growth rates have been reported to deviate from exponential (Wood and Nurse, 2015). While slower than exponential growth would aid cell size control, our analysis here is restricted to exponential growth.

To determine how the seed network structure influences the subsequent evolution of cell size control, we performed 120 independent evolutionary simulations for the two network structures initialized with the Model A1 topology and parameters (Figure 4). Sixty simulations were performed using Pareto optimization and another 60 simulations were performed using individual fitness optimization based on the number of cell divisions $N_{Div}$ . For each simulation’s most fit model after 500 epochs, we calculate the CV of the volume distribution at birth, $C{V}_{Birth}$ , and the slope of the linear fit of the volume added in the entire cell cycle as a function of the cell size at birth, Slope $\Delta V_{Cycle}$ (we remind the reader that a slope of –1 corresponds to a sizer, 0 to an adder, and 1 to a timer). We chose to use 500 epochs in our simulations because in our previous experience this was sufficient for networks to evolve to be near the optimum, but not so much that they were forced to extensively explore the effects of neutral mutations near the optimum. Model A1’s initial and Slope before evolution are indicated by the dashed black line. In the control experiment where G1 is a sizer and S/G2/M is a timer, most evolutionary simulations with two fitness functions (Pareto) yield models close to the adder regime with a low $C{V}_{Birth}$ . However, when only the number of divisions ($N_{Div}$) is used as a fitness, the evolutionary simulations are closer to the sizer regime, albeit with a slightly higher $C{V}_{Birth}$ (Figure 4A, Welch’s t-test p<10^–4). When the cell cycle structure is inverted so that G1 is a timer and S/G2/M is a sizer like it is in the fission yeast S. pombe, we found that more sizer-like networks evolve than in the control experiment (Figure 4B, Welch’s t-test on agglomerated data p=0.03). This shows that having a network structure like the fission yeast S. pombe promotes sizers, while having the size control portion of the cell cycle earlier, as in the budding yeast S. cerevisiae, promotes adders. Thus, performing simulations using cell cycle network structures of these two yeasts results in the evolution of size control mechanisms that reflect those that are naturally occurring.

Figure 4

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The general notion that having cell size control in G1 results in adder-like mechanisms, while control later in the cell cycle results in more sizer-like mechanisms fits most observations of human cell lines grown in culture, budding yeast, and fission yeast. However, a recent study examining mouse epidermal stem cells growing and dividing in the skin found both a strong sizer and that this sizer was largely due to the size-dependent regulation of G1 (Mesa et al., 2018; Xie and Skotheim, 2020). This raised the question as to how a network performing size control in G1 could result in a sizer for the entire cell cycle. One important difference between mammalian cells grown in culture and the mouse epidermal stem cells growing in an animal (in vivo) is the change in the relative durations of the G1 and S/G2/M phases of the cell cycle. While the S/G2/M phase of the cell cycle is similar in duration in cultured cells and the epidermal stem cells in vivo at ~12 hr, the G1 phase extends ~fivefold from ~10 hr in culture to ~50 hr in vivo (Cadart et al., 2018; Xie and Skotheim, 2020). This suggests the hypothesis that the overall size control behavior can be dominated by the relatively longer cell cycle phase, as is likely the case for the G1 phase of epidermal stem cells. To test this hypothesis, we performed evolutionary simulations with size control in G1 and a timer in S/G2/M but where we changed the duration of the S/G2/M timer phases of the cell cycle to be significantly shorter or longer than the G1 phase at equilibrium. When a timer in S/G2/M is relatively shorter compared to G1, we generally see more sizer-like behavior can evolve (Figure 4C, Welch’s t-test on agglomerated data p=4 x 10^–3), while when it is relatively longer, we see more adder-like or even timer-like behavior (Figure 4D, Welch’s t-test on agglomerated data p<10^–4).

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 (Appendix 1—figure 5). 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 (Appendix 1—figures 6–7).

Taken together, our simulations show how the structural features of the cell cycle are important for determining what type of size control ultimately evolves. G1 control is more conducive to the evolution of adders, while S/G2/M control is more conducive to sizers. Moreover, size control can be dominated by the cell cycle phase of longest duration and is modulated by the specific selection criteria.

From our series of evolution simulations, we found a somewhat paradoxical inverse correlation between the added volume slope quantifying the degree of size control (sizer vs. adder) and the $C{V}_{Birth}$ (Figure 4B–C). This was surprising because it means that there is a broader distribution of volumes in the sizer regime where control should be more effective in theory. To better understand this inverted correlation between Slope $\Delta V_{Cycle}$ and $C{V}_{Birth}$ , we revisited our evolutionary simulations to examine the evolutionary pathways through which the networks progressed through the simulated epochs.

A typical evolutionary pathway for a Pareto simulation of the S. pombe-like network structure presented in Figure 4B is shown in detail in Figure 5. Here, Model A1 topology is conserved throughout evolution although individual parameter values change. In the early stages of the evolution (epoch 650), we typically see dynamics where small cell size triggers an overshoot to a large cell size, which is then reduced through a series of rapid divisions (Figure 5A). Because of this small-size-triggered overshoot, the system behaves more like a sizer when the average behavior is analyzed (Figure 5B). However, the high degree of variability in the cell cycles also leads to a broad distribution of volumes (Figure 5C). The variability in volume is attenuated in later epochs where there are fewer and smaller volume overshoots triggered by small cell size (Figure 5D–I). Since small cell size no longer triggers a dramatic amount of cell growth, the slope of $\Delta V_{Cycle}$ is increased and the system converges towards a weak adder in which the distribution of volume at birth is more Gaussian and the $C{V}_{Birth}$ is lower (Figure 5H–I). Taken together, these analyses suggest a two-step evolutionary pathway, consistent with the evolutionary dynamics first seen in Figure 2A. First, a strong but imprecise sizer mechanism evolves where, because of noise in the system, small variations in volume lead to a dramatic overcorrection and overshoot of the target volume. The variability in volume produced by this overshoot is then reduced by attenuating the strength of the size control response. Indeed, the overall weaker size control allows the system to respond more mildly to size deviations, thus yielding a lower $C{V}_{Birth}$ overall which we select for. Thus, selecting for a smaller $C{V}_{Birth}$ , i.e. better size control, can end up selecting for adders rather than sizers. This paradoxical result is consistent with the fact that when we select only for the number of cell divisions ($N_{Div}$), one sees that more sizer-like behavior can evolve (Figure 4A–C). The typical behavior before optimization of $C{V}_{Birth}$ is a strong sizer as illustrated in Figure 5A.

Figure 5

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One of the main features of smaller cells is that they have fewer proteins and mRNA. If some aspects of protein synthesis and degradation are subject to Poisson fluctuations, we expect such fluctuations to produce larger concentration fluctuations in smaller cells. For example, let us assume that the balance of synthesis and degradation of a generic protein results into a Poisson distribution with parameter $\frac{\rho V}{\delta }$ , where $\rho$ is the synthesis rate in number of proteins per unit of time for a reference volume of 1, $V$ is the volume, and $\delta$ is the degradation rate. The average concentration of this protein in an exponentially growing cell will be $\frac{\rho }{\delta }$ , which is independent of the cell volume $V$ as expected from the production rate scaling. However, following the Bienaymé formula, the variance in the concentration is $\frac{\rho V}{\delta }\times \frac{1}{V^2}=\frac{\rho }{\delta V}$ (Figure 6A), which decreases with volume. This result makes intuitive sense because bigger cells have to produce more proteins to keep concentrations constant, so that the fluctuations in the relative number of proteins (and thus concentration) are smaller (see Jia et al., 2021 for a complete analytical study of how in general variance scales differently from mean when volume varies). Thus, if the cell could sense the size of concentration fluctuations in some way, it would be able to harness the cell size-dependence of such stochastic fluctuations to regulate cell division and control cell size.

Figure 6

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To test if we can evolve networks that control cell size through sensing Poissonian fluctuations, we first initialized our simulation with a network similar to that shown in Figure 1C, but with an added self-activating gene $A$ that can activate the production of the $I$ inhibitor. We then ran evolutionary simulations using the cell cycle structure of a sizer controlling G1 and timer in S/G2/M, but where the G1/S transition is regulated by the concentration $\left[I\right]$ instead of its quantity. We also used Pareto fitness optimization of $N_{Div}$ and of $C{V}_{Birth}$ . Importantly, we use the stochastic version of our equations with the molecular noise modeled using a Langevin noise term with a variance inversely proportional to the volume as explained above. We then extracted the most fit network and optimized it further using Pareto optimization of $N_{Div}$ and of the $\Delta V_{Cycle}$ slope to push the models towards the sizer regime. The most fit network of those combined evolutionary simulations is presented in Figure 6B and demonstrates that we can indeed evolve fluctuation-based cell size control (Model B).

The mechanism for size control that evolved based on size-dependent fluctuation sensing is remarkably similar to what we observed for models without size-dependent fluctuations (Figure 5A). For large volumes, the cycle has a constant period which corresponds to approximately 85% of the doubling time $\tau$. This ensures that in the high-volume regime, the system shrinks over time. When the volume is small however, fluctuations allow the concentration $\left[A\right]$ to cross the threshold of the highly non-linear transcriptional activation of $\left[I\right]$ by $\left[A\right]$. This results in a massive increase of $\left[I\right]$ that needs to be degraded to progress further into the cell cycle. Thus, the low volume regime occasionally leads to a considerable increase in G1 length and a correspondingly very large cell at division. These very large cells then reliably and deterministically re-enter multiple, rapid cell cycles with short G1 until the cell is small again and the concentration fluctuations again become large enough to trigger the activation of $\left[I\right]$ by $\left[A\right]$. This mechanism thus appears very similar to the early sizer mechanism observed in other quantity sensing simulations shown in Figure 5. However, here the mechanism is based only on size-dependent fluctuations in protein concentration and the overall behavior is closer to an adder (Figure 6E).

The system dynamics that evolved to perform fluctuation-based cell size control produce volume distributions that are long-tailed due to the stochastic occurrence of occasional exceptionally long G1 phases. Interestingly, the probability distribution of cell cycle durations follows a power law (Figure 6F), which is due to the very broad distributions of G1 duration at lower cell volumes. A more controlled analysis specifying the initial conditions showed that cell cycles get increasingly long, and their distributions widen with decreasing control volume $V_C$ (Figure 6G). Such non-Gaussianity is the hallmark of critical behavior, suggesting that the evolution of fluctuation-based cell size control is based on self-organized criticality (SOC). SOC is defined as a system where an order parameter feeds back on a control parameter (Sornette et al., 1995; Vidiella et al., 2021). The canonical example of SOC is the sandpile to which grains of sand are added on top. As the sand accumulates, the slope steepens, and the angle of the pile (control parameter) increases. Eventually, this triggers avalanches (order parameter) that feedback to dramatically reduce the angle of the pile. This ensures that the system dynamically tunes itself at the critical value of the angle of the pile where avalanches can occur.

We conclude that our evolved size control network exhibits SOC based on several observations. Starting from a high volume, multiple divisions at a rate faster than it takes to double the biomass reduce cell volume $V$ just like the addition of grains of sand gradually increases the slope of the pile. Then, for small enough volumes, bursts of $\left[A\right]$ drive an extended G1 that greatly increases cell size, which, like the sandpile avalanches, resets the system’s control parameter (volume of the cell or angle of the sandpile). Interestingly, evolution tuned the system to be near a bifurcation (Figure 6I). If we consider the deterministic regime, in which the fluctuations in $\left[A\right]$ are small, the cycle is unperturbed and oscillates with a period roughly equal to 85% of the doubling time (Figure 6J). In contrast, if we consider the noisy regime, in which the fluctuations in $\left[A\right]$ are large, the cycle disappears, and the system stays locked in a prolonged G1 state with a high value of $\left[I\right]$ which is akin to a bifurcation destroying the cycle (Figure 6K). This bifurcation takes place because the position of the G1 attractor for $\left[I\right]$ becomes larger with increasing $\left[A\right]$ and eventually overcomes the concentration required to induce the stochastic G1/S transition (Figure 6H). Then, the system remains stuck in a state where G1/S cannot be triggered, and cells effectively exit the cycle. As growth occurs, noise dies down and so does the position of the G1 attractor, eventually becoming smaller than the $\left[I\right]$ concentration required to induce the G1/S transition which allows cells to re-enter the cell cycle. Thus, the system is critical from a dynamical systems standpoint and also fits the general observation that SOC systems tune themselves to be right at the point where the order parameter is non zero, but infinitesimal (Sornette et al., 1995). In our case, the bifurcation corresponds exactly to the point where $\left[A\right]$ can sufficiently activate the production of $\left[I\right]$ to prevent the G1/S transition.

To model gene networks, we follow a standard ODE based formalism, where we simulate dynamics of the concentrations of proteins. We use Hill functions for transcriptional interactions, and standard mass action kinetics for protein-protein interactions. We also assume that all proteins are degraded at a constant rate. For most of the simulations presented in the paper, we use deterministic ODEs for simplicity. Importantly, cell-to-cell variability arises from the precise timing of cell cycle progression events. This allows for a natural way to generate noise on cell volume that should then be compensated for by the evolved network. In the last part of the paper, we explicitly include Langevin noise for biochemical reactions that are modeled using a classical tau-leaping formalism (Gillespie, 2007). Thus, each biochemical reaction takes place with a rate that corresponds to the deterministic rate, to which we add one white Gaussian noise with a variance equal to that rate. For example, given a deterministic biochemical rate $k$ and a time interval of size $\Delta t$, we consider a tau-leaping change of ${\displaystyle k\mathrm{\Delta }t+\mathcal{N}\left(0,k\mathrm{\Delta }t\right)}$ where ${\displaystyle \mathcal{N}\left(0,k\mathrm{\Delta }t\right)}$ is a random gaussian variable of mean 0 and variance $k\Delta t$.

Volume influences protein dynamics in three ways. First, protein production rates are generally proportional to cell volume so that proteins reach and maintain a constant concentration that is independent of the cell volume (Chen et al., 2020; Elliott and McLaughlin, 1978; Newman et al., 2006; Swaffer et al., 2021). 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. Thus, the only deterministic influence of volume on concentration dynamics is on the dilution rate, which is proportional to the cell growth rate $\lambda \left(V\right)$ (see details in Appendix 1). At cell division, we also assume that proteins are equally partitioned between the daughter cells, that is, the concentration is the same before and after division. Note that we scale all our variables so that a concentration of one arbitrary unit corresponds roughly to 1000 proteins in a 100fL cell (Milo et al., 2010). Additionally, we scale the time variable so that 1 arbitrary time unit corresponds roughly to 30 min (Di Talia et al., 2007).

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 $\frac{1}{\sqrt{V}}$ contribution to the added Gaussian noise term (see more complete description in the Appendix 1).

We also checked that our results are largely invariant when adding other sources of noise (see Appendix 1—figures 5–7). 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).

To evolve networks regulating cell size, we use the $\phi$-evo software (Henry et al., 2018) with a modified numerical integrator accounting for volume dynamics and volume dependencies as described above (Figure 1B). $\phi$-evo simulates the Darwinian evolution of a population of gene networks. A network is encoded with the help of a bipartite graph connecting biochemical species (typically proteins) and interactions between them (we use a custom-made Python library). Networks are converted into an ensemble of stochastic differential equations using a Python to C interpreter. This code is then compiled and integrated on the fly to compute the behavior of the networks.

Each selection step in the algorithm is referred to as an epoch rather than the more commonly used term generation because we use the term generation to refer to cell divisions in the simulations. At each epoch of the algorithm, each gene network is simulated, and its fitness computed. Based on the fitness function(s) (see below), half of the networks are selected and duplicated, while the other half is discarded to maintain a constant population size. The duplicated networks are then randomly mutated. From the most to least probable, mutations consist in random changes of parameters of the network, random removal of interactions, and random additions of interactions or new proteins. Absolute mutation rates are adjusted as a function of the number of evolutionary epochs so that all networks in a population are mutated on average once per epoch. This implements a numerical equivalent of the biological Drake’s rule that mutation rates adjust with genome size (Lynch, 2007). Practically, this prevents the known phenomenon of code-bloating in evolutionary simulations (Foster, 2001) and also means that the total number of epochs is a good proxy of the number of mutations (in random directions) needed to evolve the best networks. All of this is easily made with our customized Python library encoding networks. For more details on technical aspects and implementations of the $\phi$-evo software, we refer the reader to Henry et al., 2018.

Realistic evolutionary processes select for multiple phenotypes in parallel. While trade-offs between those phenotypes are non-trivial, it has been observed that phenotypes typically define an evolutionarily Pareto front (Shoval et al., 2012; Warmflash et al., 2012). We thus perform network selection using a Pareto mode (Warmflash et al., 2012), in which two distinct fitness functions are computed. During the selection step, networks are first Pareto ranked. For example, consider two networks A and B and two fitness functions f¹ and f². f¹_A refers to the fitness of network A calculated with function f¹. Assuming fitness functions are to be maximized, we say network A Pareto-dominates network B if both f¹_A > f¹_B and f²_A ≥ f²_B. Rank 1 networks are networks which are not dominated by any other networks, Rank 2 networks are networks dominated only by Rank 1 networks, and Rank 3 networks are only dominated by Rank 1 and Rank 2 networks and so on. The algorithm then selects half of the population of Rank 1 networks using a fitness sharing algorithm to maximize population diversity (see details in Warmflash et al., 2012). One advantage of Pareto selection is the increased flexibility of the evolutionary process. Multiple fitness functions can provide different optimization paths in parameter space, which prevents the selection process from getting stuck in a local optimum of a single fitness function. We also perform a few simulations with only one fitness function, in which case networks are simply ranked based on their fitness.

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 $N_{Div}$ . 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 $N_{Div}$ generations, which we call $C{V}_{Birth}$ 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. We further imposed fitness penalties to prevent way too small or too big cells, see Appendix 1. There, we also study alternative fitness functions, such as least-square residual function to minimize volume variation about a target size, and the fitted slope of the amount of volume added at each cycle to be minimized to drive models toward being a sizer.

We also ran evolutionary simulations with different cell cycle structures. For evolutionary simulations where the G1/S transition was controlled by the concentration of $\left[I\right]$ , we simply change the probability to pass the G1/S transition to depend on concentration $\left[I\right]$ instead of its quantity $I$. For evolutionary simulations with a cell cycle structure similar to that found in the fission yeast S. pombe, we invert the cell cycle network structure. In this case, $I$ quantity controls division and the Switch is turned on for a fixed amount of time in G1. In terms of the relaxation oscillator, this means that the left branch is now S/G2/M and the right branch is G1.

Here we briefly describe the $\phi$-evo evolutionary algorithm from Henry et al., 2018 that we used to evolve size control networks. We refer the reader to the original publication’s main text and supplementary material for a more thorough description of the algorithm. A schematic representation of the algorithm’s architecture is shown in Appendix 1—figure 4.

First, an initial seed network is selected by the user as the starting point of the evolution simulation. $\phi$-evo then clones this first individual to create a population of networks. At each epoch, mutations are randomly applied to the networks of the population. Those mutations vary from topological changes to the network, where biochemical species or interactions can be added or removed, to non-topological changes, where the networks’ kinetic parameter values are modified. Following mutations, networks are ranked based on their performance at accomplishing the biological function we select for. This performance is encoded via a user-defined fitness function that is problem specific. We give details about the specific implementation of the fitness functions for cell size control in the following subsection. After ranking the networks, $\phi$-evo proceeds to select the most fit half of the network population. The less fit half is then discarded and replaced by a copy of the most fit half to maintain a constant population size. With this, $\phi$-evo completes the first epoch of the evolutionary process. We use the term epoch here rather than the term ’generation’ which we retain to describe a cell lineage. A predetermined number of epochs of mutation and selection are then performed after which a final population of networks is extracted.

Appendix 1—figure 4

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As mentioned in the main text and in this Appendix, 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. We first included noise in the cell cycle phases, specifically in the timing of the G1/S transition and in the length of the S/G2/M phase. Then in the later parts, we introduced protein production noise modeled as Langevin noise.

In the simulations presented in the main text, we chose not to include noise in the growth rate and in the division ratio as the recorded noise level for in experiments for these measures is lower than that for the timing of the cell cycle and the protein concentration noise (Di Talia et al., 2007; Newman et al., 2006; Zatulovskiy et al., 2020). Nevertheless, those are crucial assumptions that we made that we chose to investigate in more details here.

In subsection S/G2/M noise, we investigate how the level of noise in S/G2/M affects the conclusions drawn in the main text. Then, in subsection Growth rate noise we do the same for noise in the growth rate and in subsection Division ratio noise for the division ratio.

Appendix 1—table 1 shows the values of $C{V}_{Birth}$ of the three models presented in the main text compared to the values reported in the literature for budding yeast (Di Talia et al., 2007), fission yeast (Sveiczer et al., 1996) and mouse epidermal stem cell grown in the animal (Xie and Skotheim, 2020).

S/G2/M noise

Here, we perform similar evolution experiments to those reported in Figure 4 of the main text to examine the effect of modulating the noise in the S/G2/M timer. We thus perform three independent experiment where we set the CV in the timer period to 0%, 5%, and 8% corresponding to no, medium, and high noise respectively. For reference, the CV of the timer period in the control condition where Model A1 was evolved is 3%. Note that we maintain the average duration of the timer to be about half the time it takes to double the cell’s volume. Having specified the S/G2/M timer parameters and starting from the initial seed network of Model A1, we perform evolution and select networks as previously. We compare ensembles of 60 networks for each noise level, half of them evolved under the Pareto optimization of $N_{\text{Div}}$ and $C{V}_{\text{Birth}}$ and the other half under the single objective optimization of $N_{\text{Div}}$. The results are shown in Appendix 1—figure 5.

Increasing the noise, progressively leads to a loss of the sizer signature and increases the $C{V}_{\text{Birth}}$. 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). Thus, an increasing level of noise becomes associated with a worse accuracy in the size control mechanism which leads to loss of the sizer signature and increased $C{V}_{\text{Birth}}$ as can be seen in Appendix 1—figure 5D,E. Other results from the main text remain unchanged.

Appendix 1—figure 5

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Growth rate noise

Here, we perform similar evolution experiments as we did for the noise in S/G2/M but this time by adding noise in the growth rate $\lambda$. Specifically, at each generation of a cell’s lineage, we sample a growth rate from a Gaussian distribution centered around 0.25, the initial value we used in the rest of this project. We perform three evolution experiments with coefficient of variations for the growth rate distributions set to 3%, 5% and 8% corresponding to low, medium and high noise respectively. We perform 30 independent evolution runs with the Pareto optimization framework for each $\lambda$ noise level, each of them starting from the initial seed network of Model A1. The results are shown in Appendix 1—figure 6.

We note that on average, the growth rate will remain centered around the same value, thus not affecting the optimal fitness score networks are able to achieve. However, individual variations at each cycle perturb the ability of the system of accomplishing size control by always modifying the doubling time $\tau =\ln (2)/\lambda$. This leads to progressive loss of the sizer signature and also increases the $C{V}_{\text{Birth}}$. This increased noise in the system sometimes sends the cell volume towards 0 or $\mathrm{\infty }$ as volume is kicked outside of the control mechanism’s working range. Since this behavior is highly penalized in the fitness functions scoring, the evolution finds a way to prevent this from happening via different strategies. Interestingly, at higher noise levels, strong sizers can still evolve but are not the most common phenotype. Instead, evolution seems to favor timers that reliably ensure timely cell division generation after generation. There is however a trade-off and these models exhibit a higher $C{V}_{\text{Birth}}$ due to the lower amount of size control. Adders can also be evolved at all tested noise levels and provide good size control with reliably low $C{V}_{\text{Birth}}$.

Appendix 1—figure 6

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Division ratio noise

Here, we perform similar evolution experiments as we did for the noise in S/G2/M and in $\lambda$, but this time by adding noise in the division fraction $f$. Specifically, at each generation of a cell’s lineage, we sample $f$ from a Gaussian distribution centered around 2, the initial value we used in the rest of this project for symmetrical divisions. We perform three evolution experiments with coefficient of variations for the growth rate distributions set to 2%, 4% and 8% corresponding to low, medium and high noise respectively. We perform 30 independent evolution runs with the Pareto optimization framework for each $f$ noise level, each of them starting from the initial seed network of Model A1. The results are shown in Appendix 1—figure 7.

The results of this experiment are very similar to those for noise in the growth rate $\lambda$ described in the previous subsection. Indeed, changing the division fraction $f$ does not change the doubling time $\tau$ directly like for $\lambda$. Instead, it changes the cycle period around which cells see their volume shrink or grow over successive generations. Indeed, for a division fraction $f$, this equilibrium time between shrinking and growth becomes ${\tau }_f=\ln (f)/\lambda$. Intuitively, if $f$ is bigger than 2, then cells need to spend a little bit more time in their cell cycles for growth to occur in order to compensate for this increased division fraction. This increased noise in $f$ leads to progressive loss of the sizer signature as seen before and also increases the $C{V}_{\text{Birth}}$. The division fraction noise has a big effect on premature cell death. Indeed, at the highest noise level tested where $C{V}_f=8\%$, we saw many runs end with premature cell death to the point where three evolution runs were completely unable to find a mechanism able to prevent this. As a result of this strong pressure to avoid premature cell death, evolution turns once again to timers instead of sizers in the noisier regime. As before, adders seem to be the most reliable size control phenotype exhibiting low $C{V}_{\text{B}irth}$.

Appendix 1—figure 7

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In this section, we give the full set of equations and parameter values of the models from the main text. We remind the reader that we scale all our variables so that a concentration of one arbitrary unit corresponds roughly to 1000 proteins in a 100fL cell (Milo et al., 2010). Additionally, we scale the time variable such that 1 arbitrary time unit corresponds roughly to 30 min (Di Talia et al., 2007).