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Edited by: Mikael Bjorklund, University of Dundee, United Kingdom

Reviewed by: Naama Brenner, Technion - Israel Institute of Technology, Israel; Hanna Salman, University of Pittsburgh, United States

*Correspondence: Andrew W. Murray

Ariel Amir

This article was submitted to Cell Growth and Division, a section of the journal Frontiers in Cell and Developmental Biology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Organisms across all domains of life regulate the size of their cells. However, the means by which this is done is poorly understood. We study two abstracted “molecular” models for size regulation: inhibitor dilution and initiator accumulation. We apply the models to two settings: bacteria like

Symmetrically dividing budding cells are unable to regulate their size effectively using either an inhibitor dilution or initiator accumulation strategy. Simulations demonstrate increases in mean and standard deviation of cell sizes up to 100-fold higher than an asymmetrically dividing control for both inhibitor dilution and initiator accumulation models.

Based on the correlation between volume at birth and division, both inhibitor dilution and initiator accumulation models can yield robust adder behavior in asymmetrically dividing, budding cells. This is consistent with observed adder behavior in budding yeast, and as such we cannot exclude either model from consideration as a viable size regulation strategy in this organism.

It is unlikely that bacteria that display adder behavior use an inhibitor dilution strategy to regulate their cell size, since implementing such a strategy in cells that grow fully before setting their plane of division does not produce adder correlations that are robust to noise.

An initiator accumulation model in bacteria is consistent with the experimentally observed adder behavior, provided cells grow in the regime where noise in their timing of DNA replication initiation is much greater than noise in the time from initiation of DNA replication to cell division.

The paper is structured as follows:

Section 2.1 provides necessary background on the cell cycle in both bacteria and budding yeast, and details assumptions made throughout the text about the growth morphologies of these organisms. In section 2.2 we address the necessary background on size regulation in both budding yeast and bacteria. Section 2.3 discusses the approach of the paper. Finally, section 2.4 provides mathematical definitions of the two models of size regulation studied.

Section 3 addresses cells that grow by budding, with an application to budding yeast. The growth models used for this cell type are outlined in section 3.1. We study this growth morphology for cells that divide asymmetrically in section 3.2 and for symmetrically dividing, budding cells in section 3.3. Within these subsections we apply the inhibitor dilution (sections 3.2.1 and 3.3.1) and initiator accumulation (sections 3.2.2 and 3.3.2) models to the relevant cell types.

Section 4 addresses non-budding cells, with an application to certain bacteria including

Table

Model definitions reference table.

Inhibitor dilution | Noisy synthesis rate | Section 2.4.1 |

Noisy integrator | Section 2.4.1 | |

Initiator accumulation | – | Section 2.4.2 |

Growth models | Noisy asymmetry | Section 3.1 |

Noisy timing | Section 3.1 | |

Budding morphology | – | Section 3.1 |

Non-budding morphology | – | Section 4.1 |

Organisms across all domains of life regulate their cell size, coupling growth and division to constrain the range of cell sizes produced. Despite this ubiquity, understanding how size control is implemented on a molecular level has remained an active area of research for several decades (Pritchard et al.,

In this work we apply these distinct models of size regulation to organisms that adopt two distinct modes of growth: cells that produce offspring by budding, such as the budding yeast, and non-budding cells such as the bacteria

Illustration of the cell cycles and growth morphologies.

Figure

Here we present the necessary background on size regulation in budding yeast and bacteria. For a broader discussion of these topics and size control in other organisms, we direct readers to the following review articles: (Chien et al.,

Statistical correlations on single cell data now allow us to explore the connections between phenomenological models of size regulation and molecular mechanisms for cell cycle progression (Campos et al.,

In bacteria the initiator accumulation model has recently been shown to allow simultaneous regulation of both cell size and the number of origins of replication, provided that DNA replication is initiated upon accumulation of a critical abundance of initiator protein per origin of replication (Amir,

This work builds on existing phenomenological models of size regulation using two distinct “molecular” mechanisms, focusing on the effect of these size regulation strategies on the observed correlations between cell volume at birth and division. As noted earlier, the adder phenomenon has been observed within all domains of life. This observation is remarkable, given the great evolutionary distance separating organisms that have adopted this size regulation strategy. As such, for an adder size regulation strategy to be biologically relevant we expect that it should be robust to the introduction of biological noise, and we use the classification of whether this robust adder behavior is observed in order to characterize the models we consider. A consistent theme therefore will be the evaluation of whether these adder correlations are robust to coarse grained noise in the cell cycle. We will perform this analysis for a selection of different cell growth morphologies. Assuming a given growth morphology, we will evaluate robustness by studying adder correlations within a biologically relevant region of parameter space that was selected based on experimental observations. In this region we tested deviations from adder behavior based on the slope of a linear regression between _{b} and _{d} (volume at birth and at division). As described previously, a slope greater than 1.0 implies poorer size control relative to the adder model, while a slope less than 1.0 brings us closer to the strongest form of size control: a cell size threshold (Amir, _{b} vs. _{d} slope of roughly 1.0 ± 0.1 across a selection of different growth media, producing a variety of different physiological states (Soifer et al., _{b} vs. _{d} slopes in the sampled portions of parameter space to determine domains in which deviations from adder behavior were consistent with this experimentally observed variation. We defined the adder behavior to be robust provided that these domains were not limited to fine-tuned ranges of model parameters, i.e., they spanned broad ranges of parameter space rather than discrete pockets. This is consistent with previous studies in this area, which have demonstrated robustness by showing limiting behavior in certain regions of phase space, such as the observation that an initiator accumulation model in bacteria yields adder behavior provided that noise in initiation of DNA replication is much greater than noise in the duration of the _{Δ} or σ_{K}), or relating to the molecular mechanisms regulating initiation of DNA replication (i.e., σ_{i} or σ_{s}) we do not have experimental data that could provide an accurate range of noise strengths. As such, we used an estimate based on measurements of other cell cycle noises, taking

The inhibitor dilution model assumes that passage through Start or initiation of DNA replication occurs upon the dilution of an inhibitor molecule _{1}. This inhibitor's expression pattern is cyclical, being synthesized exclusively in the period following the initiation of DNA replication (Pritchard et al.,

Here _{i} is the volume at initiation of DNA replication, _{d} is the inhibitor abundance at division, _{b} is the inhibitor abundance at birth, and _{1} has the effect of setting the scale of average cell size in combination with _{s}. Note that at this stage we have not made any assumptions about the distribution of

In this model a cell initiates DNA replication upon accumulation of a sufficient absolute quantity of some initiator protein _{2} of initiator protein. Here _{2} is a scaling factor with units of concentration that sets the scale of the size distribution in a similar manner to _{1}. As in the inhibitor dilution model, we assume that at cell division the initiator protein is distributed to both progeny according to their relative volumetric fractions. This process is defined by Equation (2), where _{d} is the initiator abundance at division, _{b} is the initiator abundance at birth, and

Here the first line comes from the definition of initiator synthesis for a given cell volume increment, and the assumption that initiator is degraded entirely at initiation of DNA replication. The behavior of the model depends on this assumption for the particular forms of noise studied here. However, the implications of not degrading initiators following initiation have not been thoroughly investigated at this point. Note that the total new cell growth between initiation of DNA replication and cell division is Δ_{d} − _{i}. The second equality comes from setting the abundance of initiator at the subsequent Start event (i.e., the sum of initiator abundance at birth = _{b} and new initiator produced through growth = _{2} (_{i} − _{b})) equal to _{c}. This model is a simplified case of that previously proposed for fast-growing bacteria, where we now restrict the maximum number of DNA replication forks and initiation events per cell cycle to one (Ho and Amir,

In budding yeast it has been shown that after passage through Start, virtually all cell growth occurs in the bud (called the daughter cell in the subsequent generation) while the main cell body remains at a roughly constant volume until the subsequent G1 period (Soifer et al.,

Note the use of superscripts _{i} between initiation of DNA replication and division, with

Both of these are consistent with the observation that the division asymmetry ratio ^{λt} − 1. We note that setting cell growth noise to zero in either of these cell growth scenarios (i.e., σ_{t} = σ_{λ} = 0 or σ_{x} = 0, respectively) allows us to construct a mapping from a noisy synthesis rate model directly to a noisy integrator model. We may do so by defining σ_{Δ} ≡ σ_{K}

Here we focus on asymmetric division, where in the limit of small noise terms, the slope of a linear regression between volume at birth and at division becomes exactly 1 for the inhibitor dilution and initiator accumulation models discussed above. We now ask whether these models yield robust domains of adder behavior within the biologically relevant regimes for 〈

Of the inhibitor dilution models considered for this work, the one which predicted the greatest domain for adder behavior assumed a noisy integrator in inhibitor synthesis and noisy asymmetry in cell growth (see Table _{λ} and G2 timing σ_{t} into one noise term in the division asymmetry σ_{x}, and from the decoupling of inhibitor synthesis from noise in cell growth. A numerical comparison with other variants of an inhibitor dilution model is provided in Figure _{1} = 1, the parameter _{x}/〈

Figure _{x}/〈_{b} vs. _{d} slope shows little to no change with increasing noise in passage through Start _{x}/〈_{r}/〈_{b} vs. _{d} slope on both σ_{x}/〈_{r}/〈_{x}/〈_{r}/〈

An inhibitor dilution model can yield robust adder behavior in asymmetrically dividing, budding daughter cells for biologically relevant parameter values and is consistent with experimental observations of adder behavior. Simulations were performed assuming noisy integrator synthesis and noisy asymmetry growth (section 3.1). _{d} values, averaged over data points binned with respect to _{b}. Noise values are: _{x}/〈_{x}/〈_{b} vs. _{d} for daughter cells only. Variation is with respect to _{x}/〈

The simulations presented in Figure _{b} vs. _{d} linear regression slope intractable. Further, approximate calculations of linear regression slopes which ignored this additional condition deviated from the simulated behavior sufficiently that they did not warrant discussion. Despite this, we can readily understand the cause of this subpopulation being born ready to pass through Start immediately. The generation of cells with a low inhibitor concentration at birth is caused by the growth of mother cells over successive generations. In the limit of small noise, mother volume at Start will increase over successive generations following the recursion relation

Note the re-adoption of _{b} evolution have fixed points

As previously noted, we have a choice of how to introduce noise into the growth morphology for yeast cells obeying the accumulation model. Here we assume a noisy asymmetry growth model, and explore the effect of varying 〈_{x}/〈_{i}/〈_{c}〉 in an initiator accumulation model. Again, under the assumption that _{2} = 1, 〈_{c}〉 sets the scale for cell size distributions, but _{b} vs. _{d} correlations will be independent of 〈_{c}〉 provided noise strengths are given relative to it.

Figure _{x}/〈_{i}/〈_{c}〉 and that σ_{i}/〈_{c}〉 satisfies σ_{i}/〈_{c}〉 ≤ 0.3. This is qualitatively similar to the observation in bacteria that adder behavior is observed provided σ_{i}/〈_{c}〉 ≫ λσ_{t}, though in bacteria we adopt a noisy timing growth model (see Ho and Amir, _{i}/〈_{c}〉 over the full range of parameter values tested. For 〈_{x}/〈_{x}/〈_{x}/〈_{b} vs. _{d} slope for daughter cells to be suppressed below 1 by noise in _{x}/〈_{x}/〈_{r}/〈_{i}/〈_{c}〉 and a higher range of tolerable values for σ_{r}/〈

An initiator accumulation model can yield robust adder behavior for asymmetrically dividing, budding cells for biologically relevant parameter values, and is consistent with experimental observations of adder behavior. Heat maps are of linear regression slopes from fitting _{b} vs. _{d} for daughter or mother subpopulations in an initiator accumulation model of size regulation.

Combining the results of this section with those of section 3.2.1, we conclude that despite the observation of qualitative differences in behavior, we are unable based on adder-like cell cycle correlation behavior alone to distinguish between an inhibitor dilution and initiator accumulation model within budding yeast cells. However, we also note here that the initiator accumulation model has an additional advantage of inherently producing a longer G1 phase in daughter cells than mother cells, based on their difference in cell size. This is consistent with experimental observations that mother cells have significantly shorter G1 times (Di Talia et al.,

This is calculated neglecting noise in _{b} is the volume at birth. Since for any mother-daughter pair we have that

As in the case of the dilution model, we found it necessary to impose the additional requirement that cell volume should only monotonically increase. This was necessary due to a subpopulation of parent cells producing sufficient initiator before division that their progeny were born with an initiator abundance _{b} > _{c}, and would therefore otherwise decrease in volume prior to Start. We understand the generation of this subpopulation of cells as follows. In the limit of small noise, the volume at Start of mother cells followed through multiple generations follows

Note that as in Equation (6) we adopt

Here we study symmetric division in budding cells within the previously described noisy asymmetry growth model, setting 〈_{db} ≡ log(2)/〈λ〉.

For the inhibitor dilution model we assumed noisy integrator synthesis in addition to noisy asymmetry growth. Noise is inserted in passage through Start σ_{s}, in the amount of inhibitor produced σ_{Δ}, and in the division ratio σ_{x}. We made similar observations to those outlined below when considering noisy integrator synthesis and a noisy timing model (data not shown).

Figure _{b} vs. _{d} slopes of just below 2 across the full range of noise values explored. From this observation it is clear that the majority of the population is being born such that they should go through Start immediately. This may be extracted from the numerics, but it is illustrative to understand how this inherently arises in a symmetric budding growth setting, independent of whether noise in the budded growth is taken to follow a noisy timing or noisy asymmetry rule. We do so by considering the average growth in G1 for a mother-daughter pair. Using

The inhibitor dilution model yields poor size regulation for symmetrically dividing, budding cells. _{b} vs. _{d} for simulated populations of symmetrically dividing, budding cells. ^{5} doubling times. _{b}) vs. time (in _{db}) shows the increase in σ(_{b}) of approximately two orders of magnitude relative to the asymmetrically dividing control, with the mean traces for the 20 replicate simulated populations shown for both symmetric (〈_{b}〉 vs. time (in _{db}). _{b}) vs. time (in _{db}). _{b})) vs. time (in _{db}) shows that the increase in standard deviation is below that expected from a pure geometric random walk in volume, with the mean for 20 repeats shown in bold. Noise was introduced in _{x}/〈

The first equality comes from noting that _{d} = (1 + 〈_{i} = (1 + 〈_{b} = 2_{b}. This would predict a _{b} vs. _{d} slope of 2.

Pure timer based models of symmetric cell division coupled with exponential volume growth on the single cell level will lead to geometric random walks in volume, with a standard deviation that grows arbitrarily large over time (Amir, _{b} (volume at birth). This appears to saturate at a maximal value approximately two orders of magnitude higher than that of the asymmetric control over the extent of these simulations (simulations were run for 2.5 × 10^{5} volume doubling times). Figure _{b}〉, while Figure _{b})/〈_{b}〉 over twofold greater than the asymmetric control. Figure _{b})) with time is inconsistent with a pure geometric random walk in volume. Within the first 1,000 doubling times in which linear increases in log(σ(log(_{b}))) were observed on a log-log scale, a linear regression between log(σ(log(_{b}))) and log(time) yielded a slope of 0.16 for the full population. A geometric random walk in cell size would yield a slope of 0.5. The saturation of σ(_{b}) at a finite value is also inconsistent with the interpretation of a geometric random walk. These deviations from the expectation of a pure timer model of symmetric cell division may be explained by the observation that for non-zero noise values the _{b} vs. _{d} slope is slightly below the value of 2 that would be expected for a pure timer model. We believe this to indicate the presence of some very weak size control. This may constrain the cell size distributions from growing arbitrarily broad and explain the above deviations from geometric random walk behavior. Despite this, the observation of such broad and unconstrained distributions of cell volumes in symmetrically dividing, budding cells illustrates a clear problem associated with symmetric cell division in a budding growth morphology.

We now study the initiator accumulation model for symmetrically dividing, budding cells. As in section 3.3.1, we see in Figure _{b} vs. _{d} slope. Imposing this requirement leads to _{b} vs. _{d} slopes ≤ 2.0, that are reduced below 2.0 by the introduction of finite noise in both _{b} vs. _{d} slope. Motivated by this result we again tested the effect of serial growth and dilution on symmetrically dividing budding cells, this time following an initiator accumulation growth policy. Results are presented in Figures

In this section we consider the case of cells which do not grow by budding, the growth morphology most relevant to bacteria such as _{d} are given to each of the two progeny, respectively (labeled _{1} and _{2} for convenience), so that

We further assume that volume at division is related to volume at initiation of DNA replication by the noisy timing model described in Equation (4), and that the cell cycle period _{db} (Cooper and Helmstetter,

In bacteria, the dilution model does not give robust adder correlations for the variety of models considered. For simplicity, we focus on the case of perfectly symmetric division in slow growing bacteria, in which the

The inhibitor dilution model is not robust for the symmetrically dividing bacterial mode of growth. Heat maps of linear regression slopes from fitting _{b} vs. _{d} for symmetrically dividing bacterial cells. The models simulated are variants of an inhibitor dilution model in which the amount of inhibitor synthesized is _{K} = 0). Black outlines provide a guide to the eye for regions in which adder-like behavior is observed (slope = 1.0 ± 0.1). Adder behavior is seen to be sensitive to noise strength. This indicates that the dilution model is unlikely to be implemented as a means of size regulation in symmetrically dividing bacteria which display adder behavior.

The accumulation model can give robust adder behavior in bacteria. Since rapidly growing bacteria maintain multiple ongoing rounds of DNA replication, we consider an accumulation model in which a constant volume per origin of replication is added between replication initiation events. This model can allow for an extra round of replication initiation late in the cell cycle, through stochastically accumulating a threshold number of initiators before division. The simultaneous regulation of DNA replication and cell division allows the model to robustly recover from these stochastic events (Ho and Amir,

Here, we derive an analytical expression for the slope _{b}, _{d}) between sizes at birth and at division, under the simplifications that cells undergo perfectly symmetric division and that cells do not undergo extra rounds of replication initiation. The slope can be written as a normalized covariance,

The size at birth can be written in terms of the size at the previous DNA replication initiation, _{b} = _{i} exp (λ (〈_{t}))/2, where λ is the noiseless growth rate, 〈_{t} is a Gaussian random variable with standard deviation σ_{t}. We can then write _{b}, _{d}) in terms of _{i} as

where _{t})〉. Since ξ_{t} is a Gaussian random variable, _{i} by writing _{i} is a Gaussian random variable with standard deviation σ_{i}. We have used the simplifications that cells undergo perfect symmetric division and that cells do not undergo extra rounds of replication initiation. Substituting into the expression for the slope, we find after simplification

where _{i}/〈_{c}〉. To lowest order in small variables σ_{i}/〈_{c}〉 and λσ_{t}, the expression becomes _{i}/〈_{c}〉 ≫ λσ_{t}, the slope approaches one, as confirmed by simulations (Figure _{db}). However, Equation (13) and its derivation both hold for the fast-growth case (_{db}) as well. The approximate Equation (13) deviates from numerical results only when the fraction of cells undergoing extra initiations becomes significant at σ_{i}/〈_{c}〉 ⪆ 0.3 (Figure _{i}/〈_{c}〉 ≈ 0.1 and σ_{t}/〈_{db}〉 ≈ 0.1 (Wallden et al.,

The initiator accumulation model is robust for the symmetrically dividing bacterial mode of growth, provided that σ_{i}/〈_{c}〉 ≫ σ_{t}/_{db}. Heat maps of linear regression slopes from fitting _{b} vs. _{d} for symmetrically dividing bacterial cells. The model simulated is the initiator accumulation model. Black outlines provide a guide to the eye for regions in which adder-like behavior is observed (slope = 1.0 ± 0.1). In the limit of σ_{i}/〈_{c}〉 ≫ σ_{t}/_{db} the observed _{b} vs. _{d} slopes approach 1, consistent with experimental observations.

We have presented results on a selection of size regulation mechanisms applied to different growth morphologies, relevant to budding cells (in particular budding yeast) and non-budding cells such as the bacteria

Size regulation summary for budding yeast mode of growth.

Symmetric | Slope ≈ 2 | Slope ≈ 2 |

Asymmetric | Robust adder | Robust adder |

Size regulation summary for bacterial (non-budding) mode of growth.

Symmetric | Not robust | Robust adder |

In asymmetrically dividing, budding cells we observed that both inhibitor dilution and initiator accumulation models can give rise to robust adder behavior within specific noise regimes, and both models are consistent with the observed adder behavior in budding yeast. However, both models failed to regulate size effectively in a budding growth morphology when assuming symmetric division, predicting very weak size regulation with _{b} vs. _{d} linear regression slopes of just less than 2. This failure contrasts with the relative efficacy of these mechanisms in regulating size for a symmetrically dividing bacterial growth morphology, and illustrates the problems that an organism which divides by budding would encounter if it divided symmetrically. We hypothesize that the ineffective size regulation we have predicted for symmetric division in budding cells represents a selective pressure that contributed to the evolution of asymmetric division in budding yeast.

We found that the inhibitor dilution model in budding cells can produce robust adder behavior, provided that noise in cell division asymmetry satisfies σ_{x}/〈_{r}/〈_{x}/〈_{r}/〈

Throughout this work we have assumed that the distribution of inhibitor or initiator between mother and daughter cells at division is done according to their relative volumetric fraction, consistent with these factors being present at the same cell-body-averaged concentration in each cell. This assumption stands in apparent opposition to observations in haploids that the concentration of Whi5 in the bud nucleus is higher than in the main cell nucleus at cell division (Liu et al.,

We found that the initiator accumulation model in budding cells can produce robust adder behavior, provided that noise in cell division asymmetry satisfies σ_{x}/〈_{i}/〈_{c}〉 and that σ_{i}/〈_{c}〉 ≤ 0.3. Our predictions for this model are consistent with the robust adder behavior observed in budding yeast, diploid daughter cells, given the aforementioned uncertainty in the measurements of σ_{r}/〈

Molecular candidates aside, our results on the robustness of cell cycle correlations for daughter cells do not allow discrimination between an inhibitor dilution or initiator accumulation model as the relevant candidate for budding yeast. However, as noted in section 3.2.2, Equation 7, we observed that the initiator accumulation model has the additional advantage of inherently producing a longer G1 phase in daughter cells than mother cells, based on their difference in cell size. In contrast, the inhibitor dilution model predicts identical G1 timing for both cell types, given the assumption maintained throughout this text that the inhibitor is distributed between mother and daughter at cell division in a manner proportional to their relative volume fractions. Note that this is consistent with the inhibitor being present at the same whole-cell-average concentration in both cell types.

In bacteria, we observed that achieving adder behavior in a symmetrically dividing inhibitor dilution model requires fine-tuning of noise in the

Throughout this paper we have made the implicit assumption that size regulation can be described by the abstraction of an organism's cell cycle regulatory network to a small circuit with only a few key components. Another possibility is that size regulation is a systems level phenomena, arising from the interaction of many components in a way that cannot be mapped onto the simple circuits we have outlined. This idea is discussed in work by Robert (

More broadly we observed that the consistency of inhibitor dilution or initiator accumulation models with experimental observations can depend on the assumptions surrounding the structure of those models. Determining whether these models are valid will require further experimentation on the specific molecular candidates for size regulation in a given organism, as is suggested for both budding yeast and bacteria above. We also noted that both the mode of growth and the division asymmetry can lead to significant changes in the robustness of inhibitor dilution models, as is evident when contrasting the asymmetrically dividing, budding yeast case of Figure

FB performed the calculations and designed the simulations relevant to the budding yeast growth morphology. PH performed the calculations and designed the simulations relevant to the bacterial growth morphology. All authors wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

FB and AA wish to thank Ilya Soifer for providing experimental data, as well as for helpful discussions and feedback. FB wishes to thank Bryan Weinstein for helpful discussions regarding implementation of the simulations. FB wishes to thank the staff of MCB graphics for assistance with the cell illustrations in Figure

The Supplementary Material for this article can be found online at: