## Abstract

Cell size regulation has been extensively studied in symmetrically dividing cells, but the mechanisms underlying the control of size asymmetry in asymmetrically dividing bacteria remain elusive. Here, we examine the control of asymmetric division in *Caulobacter crescentus*, a bacterium that produces daughter cells with distinct fates and morphologies upon division. Through comprehensive analysis of multi-generational growth and shape data, we uncover a tightly regulated cell size partitioning mechanism. We find that errors in division site positioning are promptly corrected early in the division cycle through differential growth. Our analysis reveals a negative feedback between the size of daughter cell compartments and their growth rates, wherein the larger compartment grows slower to achieve a homeostatic size partitioning ratio at division. To explain these observations, we propose a mechanistic model of differential growth, in which equal amounts of growth regulators are partitioned into daughter cell compartments of unequal sizes and maintained over time via size-independent synthesis.

## Introduction

During a single cell cycle, bacteria face the demanding task of replicating their genomes, synthesizing sufficient surface area and macromolecular content for their progeny. To ensure the viability of the next generation, bacterial cells also need to partition DNA and determine the plane of division in such a way that the daughter cells end up with morphologies best adapted to their environment. Various models for bacterial cell size control have been proposed and disputed in recent years (1, 2, 3, 4), including the *sizer* model (5)—where cells attain a critical mass before division—and the widely prevalent *adder* model, wherein cells add a fixed length during each division cycle regardless of their initial size (6, 7, 8, 9, 10). While extensive investigation has been focused on the molecular and biophysical mechanisms governing cell size regulation in symmetrically dividing cells (11, 12, 13, 14, 15, 16), the regulatory strategies used by asymmetrically dividing bacteria to control the size ratio of daughter cells remain poorly understood (17, 18, 19, 20).

One of the most well-studied asymmetrically dividing organisms is *Caulobacter crescentus*, a Gram-negative bacterium, which produces two genetically identical but morphologically distinct daughter cells: a motile, flagellated “swarmer” cell and a sessile but replication-competent “stalked” cell that possesses an adhesive stalk on its end (Fig 1A). *C. crescentus* cells have several checkpoints in place to ensure that asymmetric development is coordinated with the cell cycle progression (22). Although most of the cell cycle checkpoints have been established from prior genetics studies (23), the morphology of the cell also provides valuable information about the cell cycle stage (17, 18, 19, 24, 25). As shown previously (17), *C. crescentus* stalked cells have a distinct invagination near the mid-cell, which is noticeable right at the beginning of the staked cell cycle and becomes more prominent as the cell cycle progresses (Fig 1A). This invagination, characterized by the minimum of the cell width (Fig 1B), partitions the cell into swarmer (*Sw*) and the stalked (*St*) compartments. Furthermore, it was shown that the position of the minimum cell width eventually becomes the location of the cell division plane (17). Therefore, the lengths of the stalked and the swarmer compartments can be precisely tracked throughout the division cycle, from birth

In this study, we combined experimental data analysis and mathematical modeling to investigate the control of asymmetric size partitioning in *C. crescentus* cells. We performed statistical analysis of single-cell growth and shape data of *C. crescentus* cells obtained using a previously developed platform that combined temperature-controlled microfluidic chamber and image analysis pipeline for extracting cell shapes (17, 26). This platform facilitated the morphologies of single stalked cells over multiple generations under balanced growth and low density conditions (26). Using this pipeline, we analyzed the morphological changes in the stalked and the swarmer cell compartments throughout the division cycle in *C. crescentus*, by measuring their relative and absolute sizes, as well as their growth rates in nutrient rich medium (PYE) at different temperature conditions.

We find that correction in division plane positioning begins early in the cell division cycle through differential growth of the stalked and swarmer cell compartments. Analysis of single-cell data suggests that the bigger cell compartment grows slower to maintain a homeostatic size ratio between the daughter cell compartments. To explain these data, we developed a mathematical model for asymmetric size control in which growth regulators are partitioned in fixed amounts into unequal sized cell compartments and maintained via size-independent synthesis. This model successfully explains the maintenance of size asymmetry in daughter cell compartments and provides a mechanism for how the swarmer and the stalked cell compartments regulate growth rates to achieve the correct division ratio. In contrast to the well-established adder model for symmetrically dividing cells in which added length is independent of birth length, our model leads to a new concept of size asymmetry control via differential growth. In this paradigm, the difference in the size of the daughter cell compartments at division is maintained at a constant value independent of their birth size, establishing a new mechanism for robust control of asymmetric size partitioning.

## Results

### Differential growth corrects errors in size partitioning

To study the control of asymmetric size partitioning in *C. crescentus* cells, we first analyzed cell growth dynamics during the division cycle with various size discrepancies between the stalked and swarmer compartments. *C. crescentus* cells exhibit a pronounced invagination near the cell center, which is identifiable at the beginning of the division cycle, even before the onset of constriction (17). The location of the invagination, which ultimately becomes the division plane, divides the cell into two compartments—the stalked compartment, with length *C. crescentus* cells exhibit a tight regulation of the division size ratio. The ratio of the stalked-to-swarmer compartment lengths at division is ≈1.2, with a coefficient of variation (*CV*) ≈ 0.095. Similar observations for high precision in division plane positioning have been reported for symmetrically dividing *Escherichia coli* and *Bacillus subtilis* with *CV* of 0.05 and 0.08, respectively (27). However, the ratio of stalked-to-swarmer compartment lengths *CV* = 0.209 (Fig 1C), which may be attributed in part to the shallowness of the minimum width at early times, making it difficult to precisely determine the location of primary invagination. However, as the cell cycle advances, the invagination becomes more pronounced, facilitating a more accurate determination of *C. crescentus* cells achieve a high precision in

From a previously published dataset of 2,448 individual cell generations at 24°C (17), we extracted a subset where the stalked compartment size is substantially larger than the swarmer compartment size at birth (*L*^{St}/*L*^{Sw} > 1.4, *n* = 403), and conversely a subset (*L*^{St}/*L*^{Sw} < 1.0, *n* = 347) where the swarmer compartment is larger. Surprisingly, at 60% of the division cycle period τ, both the subsets reach the average value *t*_{c} = 0.63*τ*) between lateral cell-wall growth and septal growth in *C. crescentus* (18), which marks the onset of rapid cell-wall constriction. This suggests that the cell corrects deviations from the average value for

To investigate how individual compartments dynamically regulate their size, we computed the instantaneous growth rate κ for the whole cell using the formula, *L*(*t*)/*L*(0)] versus *t*) indicate the instantaneous growth rates for the whole cell and its two compartments. At the beginning of the cell division cycle, the swarmer compartment in the “high” *L*^{St}/*L*^{Sw} subset (*L*^{St}/*L*^{Sw} > 1.4 at birth) has a higher growth rate than the stalked compartment, whereas, the swarmer compartment grows slower than the stalked compartment in the “low” *L*^{St}/*L*^{Sw} subset (*L*^{St}/*L*^{Sw} < 1.0 at birth). Towards the end of the division cycle, both compartments achieve similar growth rates, regardless of the initial size discrepancy (Fig 1E and G). This implies that at the beginning of the division cycle (before *t*_{c}), the stalked and the swarmer compartments grow at different rates to correct the deviations from the average value for *L*^{St}/*L*^{Sw} (Fig 1F). The difference in their growth rates (pre-*t*_{c}) also increases as *t*_{c}, both growth rates reach similar values, with that of the swarmer compartment being slightly higher, regardless of *t*_{c}, leading to a higher whole-cell growth rate, consistent with previous measurements (18). Whereas it is known how *C. crescentus* cells globally control cell size (18), the existing models for cell size control are agnostic about how individual cell compartments regulate their size. The differential growth of the stalked and swarmer compartments implies the existence of a negative feedback between compartment size and their respective growth rates. This begs the question of how each cell compartment regulates its size during the division cycle.

### Asymmetric size control ensures asymmetric size partitioning at division

Cell size control in *C. crescentus* has recently been studied for individual stalked and swarmer cells (7, 12, 18, 25, 26). Analysis of single-cell growth and morphological data revealed that *C. crescentus* stalked cells follow the mixer model for cell size homeostasis that combines an adder and a timer component (18): *L*_{b} is the cell length at birth (i.e., at the beginning of the stalked cell cycle), *L*_{d} is the cell length at division (Fig 2C), and the parameters *a* and Δ depend on the growth conditions. This model, however, does not reveal how individual cell compartments regulate their size to ensure proper size asymmetry between the daughter cells.

At any time, *L*^{St}, is negatively correlated with the length added in the swarmer compartment Δ*L*^{Sw} (Fig 2A). With an adder model for size control (*a* = 1), Δ*L* = Δ*L*^{St} + Δ*L*^{Sw} is a constant. Thus, we would predict a slope −1 for the correlation between Δ*L*^{St} and Δ*L*^{Sw}. However, the data significantly deviate from the adder model predictions (Fig 2A). The division lengths of stalked and swarmer compartments (

Because, *L*_{d} = 1.49 + 1.24 *L*_{b} for the same set of cells (Fig 2C, top left), the regression coefficients for *L*_{b}. This suggests a simple mathematical model for asymmetric size control:

where *δ* = 0.93 μm, Δ = 1.49 μm, *a* = 1.24. The above relations are in excellent quantitative agreement with the least square linear fit to the mean trend in the experimental data

Indeed, experimental data show that there is no correlation between *L*_{b} (Fig 2C, bottom right). Furthermore, the mean of the binned data is in excellent agreement with the predicted value 2*δ* − Δ = 0.37 μm, at all temperatures. Thus, the asymmetric size control models given by Equations (3) and (4) lead to a constant size difference between the stalked and the swarmer compartments at the time of cell division.

### Size-independent partitioning of growth regulators ensures robust asymmetric size partitioning

Experimental data (Fig 1F) suggest a negative feedback between the size of individual compartments and their respective growth rates, such that the larger compartment grows at a slower rate to correct for initial size discrepancies. This differential rate of change in compartment length could arise from differential elongation rates in each compartment, or active movement of the division plane relative to the cell. In the absence of experimental evidence for the latter, we constructed a mathematical model for differential growth. In this model, we assume that the growth rate of the cell is proportional to the concentration of a regulatory molecule (which we call a *growth regulator*), whose abundance at time *t* is given by *ɛ*(*t*). At the beginning of the stalked cell cycle, these growth regulators can be partitioned between the stalked and swarmer compartments in two possible ways: *partitioning by size* and *partitioning by amount* (Fig 3A). In the first model, the growth regulators are partitioned such that their abundance in each compartment is proportional to the size of that compartment, resulting in equal concentration in each compartment. The growth rate, being proportional to the concentration, is the same in each compartment and thus independent of compartment size. Partitioning by size, therefore, cannot account for size-dependent growth such that the bigger compartment grows slower to achieve the correct size partitioning ratio at division (Fig 1F).

By contrast, if the growth regulators are partitioned in fixed amounts between the two compartments, then the regulator concentration is lower in the bigger compartment compared to the smaller compartment (Fig 3A). As a result, we would expect a negative correlation between growth rate and the compartment size, consistent with experimental data.

While our model does not explicitly identify the growth regulators, we can narrow down the potential candidates based on the physical model. To maintain differential concentration, it is necessary that these growth regulators do not significantly diffuse through the cytoplasm since diffusion would tend to equalize concentration. Furthermore, diffusion barrier in *C. crescentus* is set just before cell division (28). Therefore, one possible hypothesis is that the growth regulators are immobile by being bound to the genome, which is partitioned evenly between the stalked and the swarmer cells and DNA replication begins early in the S-phase of the cell cycle (29). Possible candidates for these growth regulators include RNA polymerases, mRNAs and actively translating ribosomes, which are known to regulate bacterial growth rate (30, 31, 32, 33), and display limited mobility owing to their localization at the sites of transcription (34). Experimental observations indicate minimal mobility of active ribosomes in *C. crescentus*, characterized by a low-micrometer-scale diffusion coefficient (34, 35). This limited diffusivity can be attributed to the localization translating ribosomes at mRNA transcription sites, anchored to their corresponding genes (34, 36). Thus, the chromosome can serve as an internal template for amount-based partitioning of growth regulators into daughter cells, much like chromatin-based partitioning of cell size regulators in budding yeast and plant cells (20, 37). Aside from ribosomes, differential recruitment of MreB and PBP2 could also play a role in differential growth of cell compartments.

#### Mathematical model for asymmetric growth control

We first start with a mathematical model for whole-cell growth, where the rate of cell elongation is proportional to the abundance of the regulators *ɛ*,

where *α* is a constant that depends on cell geometric parameters and the speed of peptidoglycan insertion, and *λ* is the number density of the regulatory molecules that coordinate cell growth. If *λ* has a specific spatial profile such that the growth is localized to a fixed region on the cell surface (2, 19, 38, 39, 40, 41), then *L*. This results in a negative feedback between cell length and growth rate *κ* = *αɛ/L*, consistent with data for *M. smegmatis* cells where growth is localized to the cell’s poles (42) (Fig S3A). By contrast, *E. coli* cells exhibit uniform lateral growth along the cell, such that *ɛ* ∝ *L*. As a result, there is negligible correlation between growth rate and cell length (Fig S3B). On the other hand, *C. crescentus* cells grow by inserting peptidoglycan both laterally and at the septal plane (18, 38, 44). *C. crescentus* cell growth is uniform before the beginning of the constriction phase (*t* < *t*_{c}), whereas growth is localized to the septum after *t* > *t*_{c} (18). Our model then predicts no correlation between cell length at birth and the growth rate before *t*_{c}, and a negative correlation between cell length at *t*_{c} and growth rate after *t*_{c}, which is consistent with experimental data (Fig S3C).

We apply the whole-cell growth model to each compartment of the cell, assuming that the rate of change in length of the stalked and the swarmer compartments are given by

where *i* = {St, Sw} and *ɛ*^{i} is the abundance of the growth regulators in compartment *i*. Instantaneous growth rate of each compartment is then given by

If the growth regulators are partitioned in proportion to compartment size then *ɛ*^{i} ∝ *L*^{i}, leading to a constant size-independent growth rate in each compartment, inconsistent with experimental data. By contrast, if the growth regulators are partitioned by amounts independent of size, then *ɛ*^{i} does not dependent on *L*^{i}. It then follows from Equation (8) that *κ*^{i} ∝ 1/*L*^{i}. To quantify the relationship between growth rate and compartment length, we note that the cell compartments grow at equal rates if the stalked-to-swarmer length ratio at birth is equal to its average value (Figs 1F and G and 3A). Therefore,

where *γ** is defined as the average size ratio between stalked and swarmer cell compartments at birth. The numerical value for *γ** is determined by the size control parameters *a*, *b*, *δ* and Δ, as defined in Equations (3) and (4).

When the septal invagination is formed at distance *x* with respect to the average septum location (Fig 3B), such that

The above equations suggest a negative correlation between the growth rates of individual compartments and their respective birth lengths, which fit very well to experimental data (Fig 3C). Differential growth rate of daughter cells upon asymmetric division has been recently reported in *E. coli* (45 *Preprint*), where negative correlation between cell size and growth rate arises from equipartitioning of ribosomes that localize near the cell poles.

#### Size-independent synthesis of growth regulators

Next, we prescribe the dynamics of *ɛ*^{St}(*t*) and *ɛ*^{Sw}(*t*) to predict how the cell dynamically corrects deviations in stalked-to-swarmer size ratio from their homeostatic values. If the growth regulators are synthesized in proportion to cell size such that d*ɛ*^{i}/d*t* (*i* = {St, Sw}) is proportional to *L*^{i}, then such a model would accelerate the growth of the bigger compartment relative to the smaller compartment, unlike what is observed in data. We therefore considered a model of size-independent synthesis such that

where *k* is a constant rate of synthesis. Using Equations (7) and (12) we derive the time-dependence of the stalked and swarmer compartment lengths, given by,

The above equation predicts that the compartments elongate at different rates unless *γ*(*t*) = *L*^{St}(*t*)/*L*^{Sw}(*t*) is equal to the homeostatic value

which predicts a negative correlation between the growth rate ratio and the ratio between the stalked and swarmer compartment lengths, which is in excellent agreement with experimental data (Fig 3D). As *γ*(*t*) approaches *γ** for *t* > *k*^{−1}, both the compartments grow exponentially at equal growth rates.

Combining (Equation (6)) for whole-cell elongation with Equation (12) for size-independent synthesis of growth regulators, we can derive the time evolution of cell length as *C. crescentus* and *E. coli* (46, 47). Our model is thus relevant for other cell types that exhibit super-exponential growth, with appropriate modifications in the patterns of growth and cell size partitioning ratio.

### Differential growth maintains division size asymmetry in cell population

Having developed a quantitative model for asymmetric size partitioning in single cells, we asked if differential growth-mediated size correction is sufficient to achieve tight regulation of cell division ratio at the population level. To this end, we performed stochastic single-cell simulations of growth and division for a population of asynchronous *C. crescentus* cells. Briefly, we simulated a collection of *n* = 10^{4} cells where each cell consists of a stalked and a swarmer compartment that can grow at differential growth rates as given by Equation (8). At the beginning of the division cycle, we chose the position of the pre-cleavage furrow from a Gaussian distribution such that *t*_{c} = 0.6 *τ* (Figs 4B and S2A–D), after which they grew exponentially at constant rates independent of their size (Figs 4C and S4A–D). Interdivision time *τ* is computed using the formula,

Our simulation results were contrasted with two other models for growth control, where both compartments either grew at the same rate or randomly chosen growth rates independent of their initial lengths (Fig 4D). As expected, high discrepancies from experimental data were observed for non-differential growth models (same growth rates or random growth rates), whereas the empirically observed differential growth model quantitatively matched the experimentally observed distribution for cell division ratio quite well. Interestingly, the precision of cleavage positioning for differential growth model (CV = 0.06) was slightly smaller than for experimental data (CV = 0.09), suggesting additional sources of noise for division septum positioning. Taken together, the stochastic simulations based on the differential growth model show that size-dependent regulation of stalked and swarmer compartment growth is sufficient to quantitatively explain asymmetric division control and tight regulation of daughter cell size ratios in *C. crescentus* cells.

## Discussion

In this work, we studied the regulation of asymmetric cell division in *C. crescentus* using quantitative modeling and morphological analysis of single-cell data. Division site selection in *C. crescentus* is controlled by a bipolar gradient of MipZ that inhibits FtsZ polymerization (48). FtsZ assembly is localized near the mid-cell where MipZ has the lowest concentration. However, the mechanisms by which cells regulate precise positioning of the division site remain unclear. Because of concentration fluctuations in the noisy environment of a cell, divisome localization is prone to errors. There must therefore be a robust mechanism to control the precision of asymmetric size partitioning and to correct errors in size partitioning ratio.

Our results show that early in the division cycle, the growth rates of the stalked and the swarmer cell compartments can be very different depending on the size partitioning ratio. If the stalked-to-swarmer length ratio at the beginning of the division cycle is larger than the desired size asymmetry at division, then the stalked compartment grows slower than the swarmer compartment to achieve the correct length ratio at division, and vice versa. This indicates that the insertion of peptidoglycan can be biased to either side of the division plane depending on the septum positioning. The regulation of such bias requires the cell to sense the sizes of both compartments and adjust their elongation rates accordingly.

We propose that, within an individual cell, the growth rate of each daughter cell compartment is coordinated by regulatory molecules, whose concentration decreases as the cell size increases. The concentration of the growth regulators dictates the rates for peptidoglycan production and insertion. Thus, differential growth requires concentration discrepancies for the growth regulator. Our model assuming amount-based partitioning of growth regulators is able to explain such discrepancy. It would be intriguing to test our model using the *tipN* deletion strain in *Caulobacter*, which disrupts the positioning of the division plane while leaving DNA segregation unaffected (49). To comprehensively validate our model, quantitative assessment of model predictions at various time points in mutants with division site positioning defects would be essential.

Our proposed theory is agnostic regarding the molecular identity of the growth regulators. Consequently, the molecular mechanisms underlying the spatial distribution of these regulators, ensuring equal amounts in compartments at the beginning of the division cycle, remain unknown. One possibility is that the growth regulatory molecules are bound to the genomes that are distributed in equal amounts between the two cell compartments. Chromatin-based partitioning of regulatory proteins has been recently proposed as a mechanism to control asymmetric cell division in budding yeast (20) and can serve as a robust mechanism to maintain regulatory molecules in correct proportion by limiting their diffusion. In *C. crescentus*, mRNA molecules are localized to chromosomes, which restricts their mobility (34). Because actively translating ribosomes are associated with mRNAs, they also display limited mobility and are known to set bacterial growth rate (30, 33). Thus, chromatin-based equipartitioning of translation and transcription machineries may underlie the maintenance of differential concentration of growth regulators in the stalked and the swarmer cell compartments. This prediction could be tested by immunofluorescent imaging of ribosomes and mRNA distribution across the cell at different timepoints during the cell cycle. Our proposed model for asymmetric growth control has to be contrasted with a recently proposed model for asymmetric size control in *E. coli* and *B. subtilis* (45 *Preprint*), where ribosomes are segregated from the chromosomes and are found localized to the cell poles (36, 37, 50). Future experiments targeting translation and transcription machineries in different spatial compartments will elucidate the molecular origins of asymmetric cell division and size control in *C. crescentus*.

## Materials and Methods

### Acquisition of experimental data and image analysis

Single-cell data for *C. crescentus* were acquired as described in detail in reference 17, 26. In the main text, we used the same dataset as reference 17, 26, comprising 2,448 individual cell generations at 24°C. Supplementary figures contain the data and analyses of cell shape for other temperatures. Phase-contrast images obtained were analyzed using a Python custom routine (17). This routine was used to determine the minimum cell width, *w*_{min}, by sampling equidistant normal lines to the cell’s midline axis (Fig 1B). The length of the cell’s midline axis, from the swarmer pole to the minimum cell width (*w*_{min}), defines the length of the swarmer cell compartment (*L*^{Sw}), whereas the length from *w*_{min} to the other end of the cell defines the length of the stalked cell (*L*^{St}). Intra-generational and intergenerational dynamics of cell length and growth rate (Figs 1–3) were analyzed using custom codes written in Mathematica.

### Multivariate regression

The multivariate linear least square fits in Equations (1) and (2) were determined using RStudio. Within the dense region of the data cloud in Fig 2, single-generation data were grouped by *L*_{b} and binned accordingly. These bin means were then fitted with a line using Mathematica.

## Data Availability

The data supporting the findings of this study are available from the corresponding author upon request.

## Acknowledgements

We thank Aaron Dinner and Norbert Scherer (University of Chicago) for providing the single-cell growth and shape data for *Caulobacter crescentus* cells and Vahid Shahrezaei (Imperial College London) for sharing the data for *M. smegmatis* cells. The authors also thank Javier López-Garrido and Octavio Reyes-Matte for many useful discussions. S Banerjee acknowledges support from the Royal Society (RGF/EA/181044), the National Institutes of Health (NIH R35 GM143042), and the Shurl and Kay Curci Foundation.

### Author Contributions

TW Ng: formal analysis, investigation, visualization, methodology, and writing—original draft.

N Ojkic: conceptualization, formal analysis, validation, investigation, visualization, methodology, and writing—original draft, review, and editing.

D Serbanescu: formal analysis, validation, and investigation.

S Banerjee: conceptualization, data curation, supervision, funding acquisition, validation, investigation, visualization, methodology, project administration, and writing—original draft, review, and editing.

### Conflict of Interest Statement

The authors declare that they have no conflict of interest.

- Received January 12, 2024.
- Revision received May 9, 2024.
- Accepted May 10, 2024.

- © 2024 Ng et al.

This article is available under a Creative Commons License (Attribution 4.0 International, as described at https://creativecommons.org/licenses/by/4.0/).