At the single-cell level, noise arises from multiple sources, such as inherent stochasticity of biomolecular processes, random partitioning of resources at division, and fluctuations in cellular growth rates. How these diverse noise mechanisms combine to drive variations in cell size within an isoclonal population is not well understood. Here, we investigate the contributions of different noise sources in well-known paradigms of cell-size control, such as adder (division occurs after adding a fixed size from birth), sizer (division occurs after reaching a size threshold), and timer (division occurs after a fixed time from birth). Analysis reveals that variation in cell size is most sensitive to errors in partitioning of volume among daughter cells, and not surprisingly, this process is well regulated among microbes. Moreover, depending on the dominant noise mechanism, different size-control strategies (or a combination of them) provide efficient buffering of size variations. We further explore mixer models of size control, where a timer phase precedes/follows an adder, as has been proposed in Caulobacter crescentus. Although mixing a timer and an adder can sometimes attenuate size variations, it invariably leads to higher-order moments growing unboundedly over time. This results in a power-law distribution for the cell size, with an exponent that depends inversely on the noise in the timer phase. Consistent with theory, we find evidence of power-law statistics in the tail of C. crescentus cell-size distribution, although there is a discrepancy between the observed power-law exponent and that predicted from the noise parameters. The discrepancy, however, is removed after data reveal that the size added by individual newborns in the adder phase itself exhibits power-law statistics. Taken together, this study provides key insights into the role of noise mechanisms in size homeostasis, and suggests an inextricable link between timer-based models of size control and heavy-tailed cell-size distributions.

Our data analysis shows thatfollows the power-law distribution (see Fig. S1 ) with exponent Supporting Material , Section S6). It can be noted thatis much smaller than m and the 95% confidence interval for its estimate,, is overlapping with that of. Although we are unable to definitively conclude that the power-law distribution fit of, with, is significantly better than the exponential distribution fit, in either case, the exponent is higher thanand does not play a role in determining the exponent of size distribution ( Supporting Material , Section S6). In summary, although C. crescentus newborn sizes do indeed follow a power-law distribution, as predicted by the mixer model, their exponent is likely explained by the power-law statistics of the size added by single cells in the adder phase.

Recall that the mixer model assumes all moments ofandto be finite, and relaxing this assumption could explain the discrepancy betweenand m. For example, consider the size fold change, whereis the growth rate, andis the duration of the timer phase in the nth cell cycle. Ifis drawn independently from a Gaussian distribution, then size fold change is lognormally distributed with finite moments. Sinceis positively valued, perhaps a better approximation would be to drawfrom a Gamma distribution, in which case the size fold change follows a power-law distribution. Thus, in principle,andcould themselves follow power-law distributions with exponentsand. The exponent of the size distribution would then be essentially determined by the minimum of all the exponents m,, and. To intuitively see this, consider the size added,, having infinite variance; then, as per Eq. 24 , the steady-state size variance will also be infinite. Thus,, irrespective of the value of m.

Since the experiment only tracks the stalked daughter cells that are slightly larger than the mobile daughter cells, the definition ofin Eq. 5 needs to be modified to account for the biased selection. It can be seen that in this case,represents the partitioning errors in these cells ( Supporting Material , Section S1). The mean and noise incan be inferred from experimental data. In particular, using the data from (), we infer the partitioning parameters asand. Other relevant model parameters can also be estimated from these measurements. For example, based on measurements of, we estimate a higher noise in the timer phase,, compared to the partitioning errors. We refer the reader to Tables S1 in Supporting Material , Section S5, for precise parameter values with confidence intervals. Based on these parameters, the mixer model () predicts an exponent of, where the 95% confidence interval is obtained based on uncertainties in the estimates of underlying parameters (see Supporting Material , Section S5 for details). Intriguingly, consistent with the theoretical predictions, data reveal that the C. crescentus newborn cell size also follows a power-law distribution ( Fig. 4 ). However, the power-law exponent estimated from data using a maximum-likelihood method () is found to be Supporting Material , Section S6), and considerably smaller thanas predicted by the mixer model structure.

Consistent with model predictions, the C. crescentus newborn cell size follows a power-law distribution. The probability density function (pdf) is estimated by first allocating newborn sizes into different bins and then finding the number of samples in each bin divided by the product of the total number of samples and bin width. The error bars show the 95% confidence interval for bin height, with the medians as circles. The higher cutoff threshold,472 cells), after which the pdf follows a power-law distribution, is obtained by minimizing the statistical distance (KS statistic) between distribution of the power-law fit and empirical distribution. As this value is usually an overestimate, we assume aerror in the cutoff, with the lower estimate being1290 cells). The obtained power-law exponent is, with the higher cutoff yieldingand the lower cutoff giving(the ± symbol denotes the 95% confidence interval obtained using bootstrapping). These exponents are estimated using a maximum likelihood estimate method (more details in Supporting Material , Section S6). The raw data are binned starting atusing bin widths of, with the last binwide. To see this figure in color, go online.

C. crescentus is a gram-negative bacterium where individual cells can exist in two forms—a mobile cell capable of swimming, and a “stalked” cell that has a tubular stalk enabling it to adhere to surfaces. Only the stalked cells are capable of cell division, and they divide asymmetrically to produce a stalked and a mobile cell, with the former being slightly larger in size. Size monitoring of individual stalked cells across multiple generations reveals a size homeostasis strategy based on a mixer model, where the timer is followed by a pure adder. More specifically, newborns grow exponentially for a size-independent time,, producing a-size fold change in the timer phase, with). Although there are considerable cell-to-cell differences in, they are independent of the newborn size,). This is consistent with our model assumption that in each cell cycle,is drawn independently from a fixed distribution. The end of the timer phase marks the start of cell-wall constriction, and division occurs after adding a fixed size from the timer phase (). Similar to, the size added by individual cells is also observed to be independent of. Since in C. crescentus, growth rates exhibit considerable correlations across generations (), independence of size added andimply thatin (). This results in the following mixer model describing the stochastic dynamics of newborn sizewhere, andare iid random variables describing stochasticity in the timer phase, size added during the adder phase, and the partitioning process, respectively. Althoughhas finite moments by construction, all moments ofare assumed to be bounded. We explore whether the C. crescentus newborn size follows a power-law distribution with an exponent similar to that predicted by the mixer model.

To investigate the effect of partitioning errors, we modelvia a Beta distribution with mean,, and coefficient of variation squared,. Although analytical approximations for m are tractable in certain limits ( Supporting Material , Section S4), we numerically study m as a function ofand Fig. 3 c). Our results show that whereas the power-law exponent, m, decreases with increasing levels of either noise source, it is much more sensitive to partitioning errors ( Fig. 3 d). As observed in the previous section, the power-law exponent is independent of, and the order in which size control is executed. In summary, the power-law statistics arises solely due to the multiplicative effect of partitioning errors and timer-phase noise on newborn cell size. Moreover, the probability of cells having large sizes is more sensitive to the partitioning errors compared to timer-phase noise. Next, we test these predictions with single-cell size measurements in C. crescentus that follows a mixer model for size homeostasis ().

It turns out that the above result on infinite size variance ( Eq. 26 ) can be generalized, in the sense that, for any non-zero noise levels, some higher-order moment ofgrows unboundedly over time. This occurs despite the fact that all moments of iid random variablesare assumed to be finite. For example, consider the ideal case of zero partitioning errors, i.e.,with probability 1, and low timer-phase noise, then(see Supporting Material , Section S4). Consistent with the finding of (), divergence of higher-order moments in Eq. 29 generates a tail of the distribution for the newborn cell size that follows a power law. In particular, the steady-state distribution ofsatisfiesfor values of x higher than a cut-off (). We refer to m as the power-law exponent arising from the mixer model structure, and it decreases with a longer and noisier timer phase (increasingand, as shown in Fig. 3 a. Furthermore, the probability of cells having large sizes is higher with a lower power-law exponent ( Fig. 3 b). For example, the probability of having a cell-size an order of magnitude higher than the cutoff is sevenfold less withcompared to the case with. For a sizer, the power-law exponent is, which implies that the probability of having an extremely large cell is zero. The exponent becomes finite and decreases as size control shifts toward the adder.

The power-law exponent of the newborn cell-size distribution decreases with increasing size fold change during the timer phase, noise in the timer phase, and partitioning errors. (a) Color plot of the exponent m using Eq. 29 as a function of the mean size fold change in the timer phase,, and the timer-phase noise,, when partitioning errors are absentwith probability 1). A higher proportion of size increase occurring during the timer phase results in lower values of m, and hence, lower-order moments of cell size grow unboundedly over time. (b) The probability distribution of the normalized cell size following the power-law tail is shown for different exponents. The normalization is done with respect to the value of size cutoff of the tail. For a lower exponent, the probability of finding cells with large cell sizes is higher. (c) Color plot of m as a function of the timer-phase noiseand partitioning errorsforand. Values of m are obtained numerically using Eq. S4.10. The increase in the magnitude of noise sources causes reduction in the power-law exponent. (d) The power-law exponent is obtained by varying one of the noise sources and keeping the other constant with a magnitude 0.05. The exponent reduces rapidly due to an increase in partitioning error compared to increases in timer-phase noise, showing that the exponent is more sensitive to partitioning errors. To see this figure in color, go online.

Ultimately, all sensitivities (Eq. S3.12) diverge at the same value of, and this divergence corresponds to. This phenomenon of size variance growing unbounded over time is a unique feature of mixer models that is absent in the generalized adder with a <1. The exact condition for the blow-up can be traced back to the denominator of Eq. 24 becoming negative:Since Eq. 26 requires the presence of at least one noise mechanismand/orat sufficiently high levels. For example, in a timer-adder mixer model, with aincrease in size in the timer phase Eq. 26 holds whenNote that the extent of partitioning errors required to destabilize the size variance is significantly smaller than the level of timer-phase noise. Forany amount of partitioning errors or timer-phase noise will cause the blow-up, which is characteristic of a pure timer. Finally, it is important to point out that the condition for variance divergence ( Eq. 26 ) depends on the timer-phase characteristics and partitioning errors, but is invariant of additive noise mechanism, and whether the timer follows or precedes the generalized adder.

Assuming, the variation in newborn sizecan be linearly decomposed into individual noise contributions using Eq. S3.12. To study the sensitivity ofto different noise sources, the linear coefficients are plotted as a function offor two different levels of the timer-phase noise, Fig. 2 ). Our analysis shows that the sensitivity ofto partitioning errors increases dramatically with the inclusion of a timer phase, and the sensitivity blows up for finite values of. In contrast, sensitivity toandfirst decreases and then increases, exhibiting a U-shape profile, with the dip in the profile being more pronounced for low levels of timer-phase noise ( Fig. 2 ). An important implication of this is that the inclusion of a precise timer phase in the generalized adder can reduce size variations when partitioning errors are negligible.

The contributions of different noise mechanisms to cell-size variations in mixer models diverge with an increase in fold change in the timer phase. (Left) Plots of the individual noise contributions using Eq. S3.12 as a function of(average increase in size in the timer phase) for two levels of(timer-phase noise) when a timer precedes an adder. Contributions are normalized by their value at. The growth-rate fluctuations are correlated with. Although the noise contribution of partitioning errors increases unboundedly with, contributions from cell-division noise and growth-rate fluctuations first decrease and then increase. All noise contributions blow up at the same value of, which decreases with increasing. (Right) The plot of cell-size variation for the mixer in Eq. 24 as a function offor both timer preceding an adder and vice versa. The cell-size variation in the mixer is normalized by that at. We consider equal contributions from each source of noise. The growth-rate fluctuations are correlated with. The cell-size variation in the mixer increases with fold change and finally diverges at high values of the fold change. Furthermore, a mixer model with adder preceding timer gives higher variation than a mixer model with timer preceding adder. For comparison with the generalized adder, the noise levels for sizerand adderare shown. To see this figure in color, go online.

It is worth pointing out that the motivation behind separating the cell-size control into size-based (a) and time-based controlis to study the mixer model (such as that observed in C. crescentus) (). Although there is some equivalence between the mixer model and the linear map considering a andas random variables, the latter is unable to distinguish the biphasic control ( Supporting Material , Section S3). For example, an overall control strategyis possible withor. Next, we quantify and investigate cell-size variations in mixer models.

Consider a newborn cell that grows in the timer phase fromto, whereis an iid random variable drawn from an arbitrary positive-valued distribution. Based on this formulation,is the average fold change in cell size in the timer phase, and we assume(size increase is less than twofold). Furthermore, the noise in thte timer phase is quantified by, the coefficient of variation squared of. A generalized adder controlling size from the end of the timer phase to cell division, yields the mixer modelAlthough Eq. 20 assumes that the timer precedes the generalized adder, the opposite scenario of the timer following the generalized adder results inNote that both models converge to Eq. 10 in the limit of. The average newborn sizes, corresponding to Eqs. 20 and 21 , are given byrespectively. Recall that in Eq. 14 , the interpretation ofin the generalized adder changes with a. In the mixer model, we could write a similar expression asThus, for a given mean cell size,must decrease with. Notice that the above relation reproduces the traditional timer forandasfor these parameter values. Furthermore,also becomes zero in the limitand. This limit represents an alternate interpretation of a timer through the mixer-model framework, wherein there is a mixture of an adder phase withwith the fold change in the timer phase,

In many organisms like Caulobacter crescentus, size control is exerted in only a part of the cell cycle (). More specifically, the cell cycle can be divided into two phases. There is a timer phase, where the cell grows exponentially for a size-independent duration of time. In essence, this phase represents time required to complete a certain cell-cycle step (such as genome replication), irrespective of size. The timer is followed by a generalized-adder phase that regulates size as per Eq. 10 . We refer to such biphasic size controls as mixer models, and we investigate how the duration and noise in the timer phase alters the noise contributions in Eq. 17

In summary, simple stochastic models based on recurrence equations provide critical insights into contributions of alternative noise mechanisms to size fluctuations and what control strategies provide the best buffering of fluctuations.

Stochastic variations in cell size are generally well regulated such thatvalues are ≪1. For example, size variation across a clonal population of E. coli cells has been reported to be Eq. 15 allows decomposition ofinto components representing contributions of individual noise mechanisms,where the linear coefficients can be interpreted as the sensitivity ofto different noise sources. For example,is the sensitivity ofto partitioning errors, and lower sensitivities correspond to more effective control of size variations. For, the sensitivities in Eq. 17 always satisfyimplying that size variations are most sensitive to partitioning errors, consistent with the findings in Fig. 1 . Note that the sensitivity oftoandis identical for, with the former increasing as growth-rate fluctuations become more correlated across cell cycles. With increasing a, the sensitivity ofto partitioning errors increases and finally diverges as; however, its sensitivity to noise mechanisms represented by(i.e., cell-division noise and growth-rate fluctuations) decreases to zero ( Fig. 1 ). This effect is exemplified by noting that the sensitivities in the decompositionassume the valuesfor the sizerfor the adder, andfor the timer. Therefore, the adder has higher (lower) sensitivity to partitioning errors (cell-division noise and growth-rate fluctuations) compared to the sizer. Moreover, the adder provides more effective noise buffering against cell-division noise and growth-rate fluctuations, whereas the sizer is a better strategy against partitioning errors. Intuitively, in sizer-based control Eq. 10 , the newborn size is only affected by noise in the previous cell cycle. In contrast, size is affected by noise in all previous cell cycles in the adder. This dependence on history in the adder is a double-edged sword: it allows effective averaging of additive noise mechanisms likeacross multiple cell cycles, but this comes at the cost of amplifying multiplicative noise mechanisms like. Given this trade-off, when both partitioning and cell-division noise are present at comparable levels,values are minimized by a combination of adder and sizer, i.e., at intermediate values of a ( Fig. 1 ).

Sincealways appears in Eq. 15 with, for the remainder of the manuscript we redefine the extent of growth-rate fluctuations as

A similar analysis for the second-order moment yields the following noise level ( Supporting Material , Section S1):Note that in light of Eq. 13 , defined in Eq. 8 , can now be interpreted as the log sensitivity of the mean newborn cell size to the average growth rate,. To gauge the effect of different noise mechanisms, we plotas a function of individual noise magnitudes assuming other noise sources are absent. Results show that forincreases most rapidly with Fig. 1 ). Thus, effective regulation of partitioning errors is a key ingredient for buffering stochastic variations in size. Perhaps not surprisingly, Eq. 15 further shows thatincreases as 1) fluctuations in the growth rate become correlated across generations (i.e., increasing), 2)becomes more sensitive to(i.e., increasing), and 3) the size added,, becomes correlated withat the level of individual cells (i.e., increasing c).

Stochastic variations in newborn cell size are most sensitive to partitioning errors. (Left) The extent of the newborn size variations in Eq. 15 as a function of the noise in partitioning, cell division, and growth rate for the addercase. For each line, the rest of the noise sources are set to zero. For the plot ofagainst, we consider highand moderatecorrelations between two consecutive cell cycles. Other parameters used are. (Right) The noise contributions in Eq. 17 are plotted as functions of a. Each contribution is normalized by its value for, and the noise contribution from growth-rate fluctuations is plotted for. The contribution from the partitioning errors increases with a, whereas the contributions from the growth-rate fluctuations and cell-division noise decrease with a. The combined contribution from cell-division noise and partitioning errors foris minimized at an optimal value of a (shown in the inset). To see this figure in color, go online.

C V V 2 = ( C V β 2 + 1 ) ( 2 − a ) 2 4 − a 2 − a 2 C V β 2 ( ( c 2 S α ¯ 2 C V α 2 + 1 ) ( 1 + C V Δ 2 ) + 2 a ( 1 2 − a + c 2 S α ¯ 2 C V α 2 ρ α 2 − a ρ α ) ) − 1 .

Let the steady-state mean and noise (as quantified by the coefficient of variation squared) levels of the newborn cell size be denoted byrespectively. Using the fact thatis drawn independently ofand, it is straightforward to show from Eq. 10 thatwhich, usingand, reduces toThus, the mean newborn size increases with growth rate as per the function F. Another point to note is thatcan be written aswhich implies that for a fixed mean newborn cell size across various size control strategies,is determined by the parameter a. This is consistent with the earlier observation that the interpretation of, and therefore, changes with a:is zero for the timer strategy, and in the sizer strategy, its value is twice that in the adderstrategy.

Among these three noise sources impacting cell size, cell-division noise and growth-rate fluctuations are additive noise mechanisms implemented via, whereasis a multiplicative noise mechanism. Our goal is to quantify the steady-state moments of V, and investigate how they are impacted by the magnitude of noise sources(cell-division noise),(partitioning errors), and(growth-rate fluctuations). To guide the reader, a description of the various symbols used in this article are provided in Table 1

To enhance the repertoire of size-control strategies, the discrete-time model () is expanded towhere, andis the cell size just before division. A simpler form of Eq. 10 , which does not have the noise properties captured byand, has been referred to as the noisy linear map () or the imperfect adder (). A key feature of this model is that it allows deviations from the adder strategy of size control. Although the sizer and the timer are represented byand, respectively, intermediate values of a allow combinatorial size control (). Note that the biological interpretation ofchanges with a—it is the size added (size threshold) for division to occur as per the adder (sizer) control strategy that is realized when, whereasfor a timer strategy. To summarize, the overall model describing the stochastic dynamics of newborn cell size is given by Eqs. 6 8 , and 10

A natural question that arises at this point is how to connectto the size added by a single newborn cell before division. In many growth conditions, the size added by single E. coli newborns shows little or no dependence on), in which case growth-rate fluctuations can be ignored. However, single-cell correlations between the size added andhave been reported for some growth mediums (). To capture this effect, we take a phenomenological approach and modifyin Eq. 4 towhere. Here,corresponds to the scenario where the size added and the growth rate are connected via their population averagesbut are independent at the single-cell level. In contrast,represents strong coupling between them at both the population and single-cell levels. In the limit of small fluctuations inwhereis the log sensitivity of the function F to the growth rate, and is given bywhen F takes a linear (exponential) form (). Note that if fluctuations inare uncorrelated across cell cycles, thenis completely independent of the newborn cell size, V. However, weak dependencies betweenand Vmay arise forand

Having introduced two different mechanisms for driving stochastic variations in cell size (cell-division noise and partitioning errors), we consider the third and final noise mechanism, fluctuations in the growth rate. Such fluctuations can potentially arise from noisy expression of metabolic enzymes, and they may also exhibit memory over successive generations (). A simple model that incorporates the memory over multiple generations is a discrete-time autoregressive process. Denoting the growth rate of a newborn cell at the start of the nth cell cycle as, its time evolution can be written aswheredenotes the correlation of growth rates across cell cycles,are iid zero-mean Gaussian random variables with variance, andis the mean growth rate. A key quantity of interest is the coefficient of variation of, which measures the magnitude of fluctuations in the growth rate.

To see howandare related to the errors in partitioning, consider the scenario whereis obtained by picking either of the daughter cells with equal probabilitywhereis a random variable that represents errors in partitioning, i.e., the daughter cells areand. Note thatirrespective of symmetricor asymmetricpartitioning. Furthermore, it could also be seen that for symmetric division,, the noise inis exactly the same as the noise in partitioning,. The relationship between the mean and the noise ofandis slightly more involved if one of the daughter cells is picked preferentially, and we refer the reader to the discussion in the Supporting Material , Section S1. For the rest of the article, we will assume that the cell divides symmetrically, i.e.,and, unless stated otherwise.

Recall that in Eq. 1 , the volume of a cell is partitioned equally between two daughter cells. However, the partitioning process is often not perfect and is subject to stochastic effects. To account for the stochasticity in partitioning, Eq. 1 can be modified towhereis an iid random variable with meanand coefficient of variation squared. It is important to point out that the volume right before the division event is given by, and this volume is partitioned into two cells. However, from an experimental standpoint, the volumeis found by selecting one of the two daughter cells. Thus, the variableeffectively accounts for both the errors in partitioning and the likelihood of picking a specific cell.

These findings motivate the following form for the added size:whereis an independent and identically distributed (iid) random variable that follows a size-independent distribution. The mean and the coefficient of variation squared ofare denoted byrespectively, where the angled brackets denote the expected value. Throughout the manuscript, the subscript n is dropped when describing the statistical moments of an iid random variable like. Randomness inessentially encompasses noise inherent in the processes of the cell cycle and timing of cell division. In light of this, we refer toas the extent of cell-division noise. The notation F in Eq. 2 represents a non-decreasing function of the growth rate, and empirical data suggest that it can have a linear,), or an exponential,, form (). Next, we incorporate another noise source that critically impacts size fluctuations—errors incurred in the partitioning of the mother-cell volume between two daughters.

To adopt a form for, we consider two key experimental insights from E. coli: 1) For a given, the histograms of the added size corresponding to different newborn sizes collapse on top of each other (). This implies that not only the mean, but the entire distribution ofis invariant of V. 2) Although the mean size addeddepends on, the distribution of size added normalized by its meanbecomes invariant of it (). Thus, varying growth conditions essentially rescales the distribution of size added by its corresponding mean.

Consider tracking an individual cell undergoing cycles of exponential growth and division. Let Vdenote the size of the (newborn) cell at the start of the nth cell cycle. Between successive division events, size increases exponentially with rate. Then, as per the adder strategy,where the random variabledenotes the size added to Vjust before the mother cell divides. The division event results in two equally sized daughters, and either of them could be picked as the newborn cell size,

A key question of interest is whether stochastic variation in cell size is more sensitive to some noise sources than others. Another related issue is to examine how this sensitivity to noise mechanisms changes across size-control strategies. We investigate these questions in the context of the recently uncovered “adder strategy” for size homeostasis. As per this strategy, division is triggered after newborn cells add (on average) a constant size to their size at birth (). Assuming exponential growth in cell size over time, the adder strategy implies that larger newborns divide earlier (i.e., the constant size is accumulated in shorter time) than smaller newborns. The generality of this strategy can be underscored by the fact that it has been reported in many microbial species, such as, Escherichia coli (), Bacillus subtilis (), Pseudomonas aeruginosa (), Synechocystis sp. (), and Desulfovibrio vulgaris Hildenborough (). We begin by describing the stochastic formulation of the adder model that encompasses different noise sources, consistent with findings of recent single-cell studies (). Later on, this model is expanded to “the generalized adder,” that encapsulates the adder, the sizer (division occurs upon reaching a size threshold), and the timer (division occurs after a fixed time from birth) paradigms of cell-cycle control ().

Unicellular organisms employ diverse control strategies to maintain size homeostasis, i.e., to ensure that they do not become abnormally large (or small) (). It is well known that cells within an isoclonal population, which presumably follow identical size-control strategies, can exhibit significant cell-to-cell variation in size (). Here, we systematically explore how such stochastic variation in cell size is impacted by various underlying noise sources, such as 1) noise in partitioning of volume among daughter cells during mitosis and cytokinesis (), 2) random fluctuations in the cell growth rate that potentially have memory over multiple generations (), and 3) stochasticity in the biomolecular processes associated with the cell cycle that generates randomness in the timing of cell-division ().

Results and Discussion

a = 1 corresponds to a pure adder (division occurs after adding size Δ n , α n from birth), and a = 0 , a sizer (division occurs when size reaches Δ n , α n ) . To our knowledge, a novel assumption in our model is the specific form for Δ n , α n that is motivated by experimental findings ( 15 Taheri-Araghi S.

Bradde S.

Jun S.

et al. Cell-size control and homeostasis in bacteria. Δ n , α n was assumed to be a product of an iid random variable, Δ n , with finite moments that is drawn independently of size, and a (linear or exponential) function of the cellular growth rate ( Diverse cell types employ different size-control strategies to maintain an optimal cell size. How effective are these strategies in regulating stochastic variation in cell size that arises from various physiologically relevant noise sources? We addressed this question in the context of the generalized adder, a recently uncovered size-homeostasis mechanism in microbes, where timing of cell division is regulated such that successive newborn cell sizes are related via Eq. 10 . In this framework,corresponds to a pure adder (division occurs after adding sizefrom birth), and, a sizer (division occurs when size reaches. To our knowledge, a novel assumption in our model is the specific form forthat is motivated by experimental findings (). In particular,was assumed to be a product of an iid random variable,, with finite moments that is drawn independently of size, and a (linear or exponential) function of the cellular growth rate ( Eq. 8 ).

( C V V 2 ) with C V β 2 (errors in the partitioning process), C V α ˆ 2 (growth-rate fluctuations), and C V Δ 2 (noise in Δ n , or cell-division noise). In the limit of low noise, C V V 2 can be decomposed as a linear function of the different noise sources ( C V V 2 to the corresponding noise sources. This formula reveals that for 0 < a < 2 , the size variation is most sensitive to partitioning errors; not surprisingly, this process is highly regulated in many microbes. Cell division in bacteria is mediated by the septal ring, and spatial precision in ring formation essentially dictates the partitioning error. In E. coli, the positioning of the septal ring at the cell midpoint is actively regulated using the Min protein system ( 13 Marr A.G.

Harvey R.J.

Trentini W.C. Growth and division of Escherichia coli. 14 Männik J.

Wu F.

Dekker C.

et al. Robustness and accuracy of cell division in Escherichia coli in diverse cell shapes. C V V 2 to C V β 2 increases with a, sensitivities to other noise sources (growth-rate fluctuations and cell-division noise) decrease with a ( ( a = 1 ) is effective in minimizing C V V 2 from the additive noise source, Δ n , α n , but it is susceptible to multiplicative noise arising through the partitioning process. In comparison, the sizer ( a = 0 ) performs better in buffering size variations from partitioning errors ( C V V 2 is minimized at an intermediate value of a when multiple noise sources are present ( Our main result for the generalized adder connects the stochastic variation in cell sizewith(errors in the partitioning process),(growth-rate fluctuations), and(noise in, or cell-division noise). In the limit of low noise,can be decomposed as a linear function of the different noise sources ( Eq. 17 ), with the coefficients representing the sensitivities ofto the corresponding noise sources. This formula reveals that for, the size variation is most sensitive to partitioning errors; not surprisingly, this process is highly regulated in many microbes. Cell division in bacteria is mediated by the septal ring, and spatial precision in ring formation essentially dictates the partitioning error. In E. coli, the positioning of the septal ring at the cell midpoint is actively regulated using the Min protein system (), and mutations in the Min proteins can significantly amplify cell-to-cell size variations due to large partitioning errors ( Eq. 25 ). Intriguingly, although the sensitivity oftoincreases with a, sensitivities to other noise sources (growth-rate fluctuations and cell-division noise) decrease with a ( Fig. 1 ). This implies that the adderis effective in minimizingfrom the additive noise source,, but it is susceptible to multiplicative noise arising through the partitioning process. In comparison, the sizerperforms better in buffering size variations from partitioning errors ( Fig. 1 ). Finally, given that sensitivities in Eq. 17 have different functional dependencies on a, it is easy to generate scenarios whereis minimized at an intermediate value of a when multiple noise sources are present ( Fig. 1 ).

C V β 2 , which typically corresponds to asymmetric partitioning ( ( C V β 2 ≈ 0 ) , where cell-division noise or growth-rate fluctuations may be the dominant noise mechanisms, the adder provides better suppression of stochastic size deviations. C. crescentus and budding yeast cells would then be expected to follow the sizer rule in the generalized adder phase owing to the asymmetric partitioning (high C V β 2 ) . However, this is not the case, as seen for C. crescentus in ( 36 Banerjee S.

Lo K.

Scherer N.F.

et al. Crossover in the dynamics of cell wall growth controls bacterial division times. 28 Soifer I.

Robert L.

Amir A. Single-cell analysis of growth in budding yeast and bacteria reveals a common size regulation strategy. C V β 2 ( Overall, this analysis suggests that organisms with large, which typically corresponds to asymmetric partitioning ( Supporting Material , Section S1), should employ sizer-based control to regulate variations around an optimal size. On the other hand, in case of organisms with symmetric and highly regulated partitioning, where cell-division noise or growth-rate fluctuations may be the dominant noise mechanisms, the adder provides better suppression of stochastic size deviations. C. crescentus and budding yeast cells would then be expected to follow the sizer rule in the generalized adder phase owing to the asymmetric partitioning (high. However, this is not the case, as seen for C. crescentus in () and budding yeast in (). One possibility in which this apparent discrepancy could be resolved is that there may be an evolutionary bias to reduce the cell-to-cell variation solely in one of the daughter cells capable of cell division. This would then lower the magnitude of Supporting Material , Section S1) and the adder phase may buffer cell-size variation as seen in Fig. 1

41 Vargas-Garcia, C. A., M. Soltani, and A. Singh. 2016. Conditions for cell size homeostasis: a stochastic hybrid systems approach. arXiv, arXiv:1606.00535. https://arxiv.org/abs/1606.00535. m − 1 and higher grow unboundedly over time, where m is given by 38 Marantan A.

Amir A. Stochastic modeling of cell growth with symmetric or asymmetric division. 42 Kessler, D. A., and S. Burov. 2017. Effective potential for cellular size control. Published online January 6, 2017. http://arxiv.org/abs/1701.01725. C V V 2 ( 1 + f n = exp ( α n t n ) and growth-rate. The growth-rate α n can be manipulated by using temperature ( 40 Iyer-Biswas S.

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et al. Scaling laws governing stochastic growth and division of single bacterial cells. 43 Harris L.K.

Theriot J.A. Relative rates of surface and volume synthesis set bacterial cell size. 36 Banerjee S.

Lo K.

Scherer N.F.

et al. Crossover in the dynamics of cell wall growth controls bacterial division times. We further expanded the size computations to mixer models of size control, where a timer phase precedes or follows a generalized adder. It is well known that a simple timer mechanism for controlling cell-division does not provide size homeostasis, in the sense that the variance in cell size grows unboundedly over time (). Our results show that mixing a timer with a generalized adder results in similar instabilities—all moments of the newborn cell size of orderand higher grow unboundedly over time, where m is given by Eq. 29 . This leads to the cell size following a power-law distribution with exponent m (), which decreases with increasing a (size control shifts toward the adder), and for increasing stochasticity in the partitioning process and timer phase ( Fig. 3 c). Interestingly, when the partitioning errors are negligible, adding a timer phase can reduce Fig. 2 ). This leads to a counterintuitive result that for some optimal timer-phase fold change, mixer models may have lower stochastic variations in cell size compared to the generalized adder, even though higher-order moments diverge in the former but may not in the latter. Moreover, we also find that the generalized adder followed by a timer is more noisy than the reverse scenario (see Eq. 25 ), but both systems exhibit the same power-law exponent m. The feature of reduced variation in cell size at the optimal fold change in the timer phase can be potentially verified via experiments by leveraging the relationship between fold changeand growth-rate. The growth-ratecan be manipulated by using temperature () or using ribosome inhibitors (), and the change in variation in newborn cell size can be measured using a combination of microfluidics and phase-contrast microscopy, as in ().

40 Iyer-Biswas S.

Wright C.S.

Scherer N.F.

et al. Scaling laws governing stochastic growth and division of single bacterial cells. m V ∈ [ 11.7 , 19.6 ] ( 36 Banerjee S.

Lo K.

Scherer N.F.

et al. Crossover in the dynamics of cell wall growth controls bacterial division times. 40 Iyer-Biswas S.

Wright C.S.

Scherer N.F.

et al. Scaling laws governing stochastic growth and division of single bacterial cells. f n (size fold change in the timer phase) or Δ n (size added during the adder phase) itself following power-law distributions, i.e., their moments diverge beyond a certain order. It turns out that in C. crescentus, the power-law exponents of f n and Δ n are much smaller than predicted by the mixer-model structure, and hence dominate in terms of determining the power-law exponent of the size distribution ( 44 Stumpf M.P.H.

Porter M.A. Mathematics. Critical truths about power laws. 39 Clauset A.

Shalizi C.R.

Newman M.E.J. Power-law distributions in empirical data. The prediction of power-law distribution in mixer models was tested with single-cell data on C. crescentus, where size regulation is mediated through a timer followed by an adder (). Consistent with theory, we find strong evidence of the newborn cell size following a power-law distribution with an exponent of Fig. 4 ). Surprisingly, independent measurements of noise sources and cell-cycle parameters from () predicted an exponent fourfold higher ( Supporting Material , Section S5). We provide a simple mechanistic explanation for this discrepancy that lies in statistical fluctuations in(size fold change in the timer phase) or(size added during the adder phase) itself following power-law distributions, i.e., their moments diverge beyond a certain order. It turns out that in C. crescentus, the power-law exponents ofandare much smaller than predicted by the mixer-model structure, and hence dominate in terms of determining the power-law exponent of the size distribution ( Fig. S1 ). A caveat of the power-law fits used for the above analysis is that, as a rule of thumb, the candidate data should span at least two orders of magnitude, which is not the case for our data set (). Nonetheless, we are able to provide both statistical support based on () and mechanistic rationale for the existence of a power law in the single-cell data.