Scaling of stochastic growth and division dynamics: A comparative study of individual rod-shaped cells in the Mother Machine and SChemostat platforms

INTRODUCTION

Time-lapse microscopy is an essential technique for the study of dynamic and stochastic biological processes in individual bacterial cells. One of the key challenges in conducting such experiments lies in the creation of an environment that is suitable for the microculture of bacteria (including replenishment of nutrients) while also permitting high-resolution microscopy (including compatibility with common imaging modalities and fluorescent probes; Moffitt et al., 2012). Agarose pads traditionally served this purpose (Fiebig et al., 2006; Ducret et al., 2009; Young et al., 2011), but in the last two decades have been largely superseded by microfluidic systems (Potvin-Trottier et al., 2018). A plethora of microfluidic designs have been utilized to generate a chemostat-like environment with continuous supply of fresh media and removal of excess cells (Balaban et al., 2004; Mather et al., 2010; Robert et al., 2010; Wang et al., 2010; Moffitt et al., 2012; Boulineau et al., 2013; Long et al., 2013, 2014; Teng et al., 2013; Ullman et al., 2013; Iyer-Biswas et al., 2014; Lambert and Kussell, 2014; Grünberger et al., 2015; Hashimoto et al., 2016; Lin and Kussell, 2016; Yu et al., 2017). Of these, the design that has proved the most popular involves polydimethylsiloxane (PDMS) “dead-end” growth channels oriented perpendicular to a main channel through which growth media is passed (Wang et al., 2010). It has earned the moniker “Mother Machine”, as a single “mother” cell at the end of each growth channel remains trapped throughout the experiment while all offspring eventually move into the main channel and exit the device (Allard et al., 2022). Originally used to study Escherichia coli, variations in the channel dimensions were quickly explored to accommodate the similarly rod-shaped Bacillus subtilis (Norman et al., 2013).

Over a decade later, the Mother Machine remains the design of choice for most single-cell bacterial physiology studies. It has been used to study phenomena as diverse as single-cell aging (Wang et al., 2010; Łapińska et al., 2019) and cell size homeostasis (Taheri-Araghi et al., 2015; Si et al., 2019) in steady-state conditions; gene expression (Kaiser et al., 2018), physiological growth (Basan et al., 2020), starvation adaptation (Bakshi, Leoncini, et al., 2021) in time-varying environments; and antibiotic susceptibility (Bamford et al., 2017), accumulation (Cama et al., 2020), and persistence (Kaplan et al., 2021). The ability to observe large numbers of individual cells in a controlled growth environment has enabled fundamental studies of phenotypic heterogeneity as a function of cell strain (Tanouchi et al., 2015), growth condition (Rochman et al., 2016), cell lineage (Bergmiller et al., 2017), and cell age (Yang et al., 2023), as well as between species (E. coli and B. subtilis; Sauls et al., 2019).

Despite its numerous advantages, the Mother Machine has design constraints whose effects on qualitative and quantitative cell physiology merit closer examination. The loading procedure involves centrifugation of bacteria to force them into the dead-end channels, which may introduce stresses (Tanaka et al., 2023). Most studies have used a dedicated fluorescent reporter to identify and segment cells (Kaiser et al., 2018), but overexpression of a fluorescent marker can introduce a growth burden (Garay-Novillo et al., 2019) and prolonged exposure to requisite bright illumination can lead to phototoxicity (Icha et al., 2017). Phase contrast microscopy provides a suitable alternative (Robert et al., 2018); although used less frequently due to challenges in detecting cells within the close walls of the Mother Machine, a number of tools have been developed for automated analysis of such data (Hardo and Bakshi 2021). Nonetheless, an extensive comparison of these tools noted systematic discrepancies in values obtained by different methods, necessitating care in interpretation of absolute quantities (Thiermann et al., 2024). In addition, unintended gradients in nutrient concentrations may well be unavoidable across each channel, and mechanical confinement in narrow growth channels may disrupt growth and morphology (Yang et al., 2018). That being said, growth of E. coli in suspension and in the Mother Machine have been found to be equivalent, on average (Yang et al., 2018). However, to fully characterize the effects of confinement on interpretation of quantitative Mother Machine studies, it is necessary to benchmark against an alternate single-cell technology that does not suffer from the same limitations.

Such a principled analysis has yet to be conducted, likely due to the lack of a suitable control that would permit the study of noninteracting, statistically identical, unconfined individual cells over multiple generations under constant environmental conditions. Unlike the aforementioned approaches, the SChemostat (Iyer-Biswas et al., 2014) design obviates the need for undesired confinement of cells to create a stable population for long-term imaging, instead relying on controllable surface adhesion to permit observations of a defined population of cells exposed to an identical flow environment within a relatively spacious microfluidic channel. Cell density may be tuned according to experimental considerations but is in general chosen to eliminate neighbor–neighbor contacts and maintain isolated microenvironments. This additionally assures the ability to acquire high-precision, high-resolution phase contrast images, which has enabled the study of cell cycle-dependent changes in cell morphology in the crescent-shaped Caulobacter crescentus (Wright, Bannerjee, et al., 2015). The technology has proved particularly fruitful for studying stochastic phenomena including scaling laws governing stochastic growth and division of individual cells (Iyer-Biswas et al., 2014; Jafarpour et al., 2018), stochastic intergenerational cell size homeostasis (Joshi, Wright, Ziegler, et al., 2023; Joshi et al., 2023a,c) and physiological adaption of single cells to precise time-dependent changes in growth conditions (Joshi, Ziegler, Roy, et al., 2023; Joshi et al., 2023b). Because the SChemostat design eschews the need for overexpression of fluorescent reporters, cell centrifugation, surfactant treatment, or confinement that could result in mechanical stresses or heterogeneous microenvironments, it represents an ideal comparator to the Mother Machine for the measurement of baseline physiological parameters. We therefore developed experimental protocols to collect data from both the Mother Machine and SChemostat, under as similar conditions as possible, and quantified differences in single-cell growth and division dynamics between these approaches. Furthermore, we use these advances to address how the insights gleaned on cell size homeostasis of individual bacterial cells (Joshi, Wright, Ziegler, et al., 2023; Joshi et al., 2023a,c) are affected based on experimental and device modalities.

RESULTS

Experimental Design

Wild-type C. crescentus cells exhibit a characteristic crescent shape, making them unsuitable for growth in the restricted environment of the Mother Machine, which is intended for rod-shaped cells. Deletion of crescentin, the intermediate filament-like protein responsible for imparting cell curvature, results in rod-shaped cells whose average growth rate is indiscernible from that of wildtype cells (Ausmees et al., 2003). Therefore, we generated a version of the C. crescentus strain used in SChemostat experiments (Iyer-Biswas et al., 2014) that lacks functional crescentin (∆creS), resulting in controllably adherent, rod-shaped cells suitable for growth in Mother Machine channels. In contrast to published protocols (Wang et al., 2010; Norman et al., 2013), ultracentrifugation did not result in a significant improvement in cell loading efficiency; we therefore excised this step from the protocol in the interest of minimizing insults to cells before the experiment. We did however find it necessary to use the surfactant Pluronic F108, which reduces nonspecific surface interactions between PDMS channel walls and bacterial cells (Owens et al., 1987; Liu et al., 2002; Lee et al., 2019). Pretreatment of the device and supplementation of the growth media with Pluronic F108 (Lawson et al., 2017; Luro et al., 2020; Bakshi, Leoncini, et al., 2021) significantly reduced undesired adhesion to the PDMS, permitting observation of single cells over multiple generations without clogging of the growth channels.

Although many Mother Machine experiments are conducted using magnifications of 40 × to 60 × to maximize the amount of data acquired, this may result in reduced precision in the determination of cell sizes, which are required to accurately assess single cell growth rates. Therefore, we used the same imaging conditions standard for the SChemostat technology, consisting of a high-NA 100 × objective (plus a 2.5 × expander) to acquire high-resolution single-cell images. As C. crescentus cells divide more slowly than E. coli (∼70 min vs. 20 min, respectively), images acquired at a frame rate of 1 min provide more-than-sufficient data points to precisely determine growth rates. To enable side-by-side comparison of data from the SChemostat and the Mother Machine, we modified the standard SChemostat image analysis routine to extract cross-sectional cell sizes from high-resolution, phase contrast images of cells growing in the Mother Machine. In conclusion, we established a protocol to successfully grow C. crescentus cells in the Mother Machine, and a routine to measure growth and division from phase contrast images (Figure 1).

Standardized growth and preparation of cells is necessary to compare experimental conditions between the SChemostat and Mother Machine platforms (see Figure 2a). A key difference in the established protocol involves Pluronic F108 pretreatment of the Mother Machine before, and media supplementation during, experiments. However, the effects of Pluronic F108 on C. crescentus growth are unknown. To control for Pluronic F108, we observed ∆creS cells growing in the SChemostat in complex media (SC*), cells growing in the SChemostat in complex media supplemented with Pluronic F108 (), and cells growing in the Mother Machine in complex media supplemented with Pluronic F108 (). Finally, to minimize potentially confounding effects due to nutrient depletion or mechanical pressure resulting from neighboring cells within the Mother Machine, here we use data obtained from growth channels occupied by a single C. crescentus cell (Figure 2, b–d).

The size growth of a single C. crescentus cell under balanced growth conditions over one generation is exponential, , where and are the initial and final cell areas, respectively, k is the growth rate, τ is the interdivision time and the subscript indicates the generation number (Iyer-Biswas et al., 2014). We also calculate the division ratio, which gives the fraction of the total area of the mother stalked cell immediately before division inherited by the daughter stalked cell immediately after division. We find that the asymmetric division of C. crescentus cells ensures that irrespective of the initial alignment of the cell trapped at the end of the Mother Machine growth channel, starting from the second generation onwards, the eponymous mother cell is always stalked, with the stalk oriented toward the closed end of the channel. Because imaging and data acquisition start a sufficient time after device loading, all data presented here are thus collected from a defined population of stalked cells within both the Mother Machine and SChemostat approaches.

Glossary

For convenience in interpreting the results presented here, we include the following glossary to distinguish between the three distinct device and experimental configurations considered here:

• SC*: ∆creS cells growing in the SChemostat in complex media

• : ∆creS cells growing in the SChemostat in complex media supplemented with surfactant (Pluronic F108)

• : ∆creS cells growing in the Mother Machine in complex media supplemented with surfactant (Pluronic F108)

The interdivision time distribution has larger mean and variability for cells under mechanical confinement in the Mother Machine setup

Interestingly, while the population-averaged single-cell growth rates of and cells are reduced by ∼12% compared with SC* cells, the and single-cell growth rate distributions are remarkably similar. We deduce that the addition of the surfactant (Pluronic F108), a necessary step for the Mother Machine protocol, results in the observed differences in growth rate distributions (Figure 4a).

Intriguingly, for cells in the SChemostat, the reduction in growth rate due to Pluronic F108 is compensated by an ∼11% increase in interdivision time (Figure 4b), such that the distributions of their product, kτ, for SC* and are very similar (Figure 4c). In contrast, cells exhibited a greater increase in interdivision time than expected due to the surfactant (Pluronic F108) alone (∼22% increase compared with SC* and ∼10% compared with ), indicating that additional factors slow down the time to cell division in the Mother Machine (Figure 4b). This additional delay in the division process ensures that the kτ distribution of cells is distinctly shifted toward a larger mean value and has a greater spread (variance), indicating increased variability compared with both SC* and cells (Figure 4c). The coefficients of variation (CVs) of these quantities are reported in Supplemental Table 2.

To further investigate whether this additional increase in interdivision time within the Mother Machine could result from reduced nutrient diffusion within the growth channels, we compared the distributions of relevant growth and division variables for cells growing in different channel lengths in the Mother Machine. We observed no significant differences between these distributions across channel lengths, implying that mechanical confinement, and not nutrient diffusion as a function of channel length, causes the additional delay in cell division in our Mother Machine setup (with one cell per channel).

Variation in kτ across device and experimental configurations is compensated by complementary variation in the division ratio

Despite differences in the distributions of k, τ, and r (Figure 4, a–d), the distributions of re_k^τ (where r is the division ratio) are remarkably similar for all three variations of the experimental setup (Figure 4, e and f). Furthermore, the coefficient of variation (CV) of re_k^τ is less than the CV of either k or τ, (see Supplemental Table 2), indicating that fluctuations in r and kτ, are anticorrelated. The value of re_k^τ corresponding to a given generation is equal to the ratio of the next generation’s initial size (size at birth) to the current generation’s initial size, because

where the subscripts denote generation numbers and a denotes the initial size. The distribution of re_k^τ must be approximately balanced around one to maintain cell size homeostasis (i.e., to prevent runaway cell sizes). Otherwise, if these values are consistently greater (smaller) than one across generations, initial sizes will increase (decrease) indefinitely, breaking intergenerational cell size homeostasis. Accordingly, we observe that the population-wide distributions of re_k^τ are centered around 1. A similar finding has been reported for E. coli cells in a study of individual lineages (Stawsky et al., 2022).

Emergent simplicities governing stochastic intergenerational homeostasis of mean-rescaled cell sizes are universal across device and experimental configurations

The straightforward requirement for homeostasis that distributions of re_k^τ must be centered around 1 cannot explain why the entire re_k^τ distributions are observed to match across growth conditions (Figure 4f). Instead, we find that this observation is a consequence of a remarkable emergent simplicity: cells in all three variations of the experimental setup follow the same precision kinematics of stochastic intergenerational cell size homeostasis.

Given a generation’s initial cell size, the next generation’s initial size can be obtained by multiplying by the current generation’s value of re_k^τ (see [1)]). Thus, if the intergenerational scaling factor, re_k^τ, follows the same intergenerational dynamics across different devices and conditions (consistent with Figure 4f), we can expect the intergenerational dynamics of initial sizes to also be identical. This clearly does not hold for SC*, , and cells (Figure 5). Instead, when the steady state initial size distributions are rescaled by their respective mean values, we find the emergent simplicity that the resulting distributions are identical for SC*, , and cells (Figure 5). We bridge this discrepancy through the following proposition. We know that re_k^τ in a given generation depends on that generation’s initial size (Joshi, Wright, Ziegler, et al., 2023), otherwise it would result in “timer”-like behavior, which breaks cell size homeostasis (Amir, 2014; Lin and Amir, 2017; Joshi et al., 2023a). Based on the observations that SC*, , and cells have the same re_k^τ distributions but different steady state initial size distributions and same mean rescaled steady state initial size distributions, we propose that re_k^τ must depend on the rescaled initial size instead, rescaled by a scaling factor that is proportional to the population-averaged initial size for the particular growth condition. This allows for the absolute cell sizes to vary across growth conditions (through the difference of a constant scaling factor) while following identical laws determining cell sizes over successive generations. Indeed, despite cells in SC*, , and conditions having different distributions of initial and final sizes (Figure 5, a and c), when the steady-state size distributions are rescaled by their respective mean values, we find that the resulting distributions are identical for SC*, , and cells (Figure 5, b and d)! This remarkable empirical observation implies that once appropriately rescaled, cell sizes in all three conditions have the same homeostatic distributions.

We track the intergenerational evolution of initial cell sizes rescaled by the population mean of the corresponding experiment denoting the rescaled cell size by s. We now turn to the emergent simplicities in the stochastic intergenerational dynamics of cell sizes. Traditionally, deterministic models corresponding to the idealizations of either sizer (constant final size), adder (constant difference between final and initial sizes), or timer (constant ratio of final to initial sizes) have been used to characterize cell size homeostasis in a range of unicellular organisms, notably the adder model for E. coli and B. subtilis (Amir, 2014; Deforet et al., 2015; Jun and Taheri-Araghi, 2015; Tahier-Araghi et al., 2015; Sauls et al., 2016). However, the behaviors of the mythical average cell prove inadequate in capturing basic phenomenologies of stochastic intergenerational cell size homeostasis, including the inter generational scaling law we previously reported in Joshi, Wright, Ziegler, et al., 2023; Joshi et al., 2023a,c (see Figure 3).

C. crescentus in particular deviates substantially from the adder–sizer–timer framework (Jun and Taheri-Araghi, 2015; Joshi, Wright, Ziegler, et al., 2023) but the fully stochastic behavior is precisely captured by the theoretical framework we have presented in Joshi et al., 2023a (Figure 3, a–d). This theory accurately predicts the distribution of initial sizes after n generations for a cell having initial size a₀ in the starting (zeroth) generation (Joshi, Wright, Ziegler, et al., 2023). Accordingly, this distribution is given by:

where is the growth condition-dependent conditional distribution of possible initial sizes after n generations, given that the initial size in the zeroth generation is . is the mean-rescaled distribution of the next generation’s initial size given the current generation’s initial size. That it is independent of the current generation’s initial size is the relevant emergent simplicity (Joshi, Wright, Ziegler, et al., 2023; Joshi et al., 2023a,c). is the conditional average of the next generation’s initial size given the current generation’s initial size; it is observed to be a linear function of the current generation’s initial size (Joshi, Wright, Ziegler, et al., 2023; Joshi et al., 2023c). Rewriting (2) in terms of the mean rescaled initial size, with being the population mean of , the conditional distribution after generations becomes:

where, and thus are different for different growth conditions. Remarkably then, once we mean-rescale by the population means, we find that all growth conditions have the same values of and (Figure 3, e–g), and hence the same . We establish this by comparing the experimentally obtained conditional distribution of mean-rescaled initial sizes after generations, given the mean-rescaled initial size in the zeroth generation, for SC*, , and cells, and using the theoretical predictions for for the SC* cells given by (3). (We note that and are experimentally measured; there are no fine-tuning fitting parameters). The observed and calculated distributions match compellingly for different values of final generation, , and initial rescaled size, (Figure 6). In particular, they converge to the same homeostatic distribution after sufficient generations have elapsed.

Thus, we conclude that despite apparent differences in the distributions of interdivision times, growth rates, division ratios and cell sizes, once cell sizes are appropriately rescaled by the corresponding population averages, they undergo identical stochastic intergenerational dynamics for cells growing in all three modalities: SC*, , and ! The significant implication is that the emergent simplicities previously encountered in the context of intergenerational homeostasis of cell sizes (Joshi, Wright, Ziegler, et al., 2023; Joshi et al., 2023a,c) are conserved across these devices and conditions.

DISCUSSION

The results presented here provide insights into the nature of the robustness of the bacterial growth and division machinery. Despite significant changes in individual growth and division parameters, the determinant for cell size homeostasis, , remains invariant. Furthermore, the fact that the distributions of across devices and conditions themselves align so closely implies that the emergent simplicities governing stochastic intergenerational homeostasis of mean-rescaled cell sizes are universal across device and experimental configurations! With knowledge of this remarkable new emergent simplicity, principled quantitative comparisons of stochastic growth and division dynamics measured in different microfluidic devices are made possible. Follow-up studies in different growth media or at different temperatures, pH, or other growth conditions could serve to further broaden the scope of the empirical findings reported here.

Mechanical confinement is known to affect growth; channels that are too narrow lead to anomalous morphology and larger-than-expected volumes in E. coli (Männik et al., 2012). Even in standard size channels, it has been observed that E. coli adapt to the confined environment, approximately preserving cell volume but adjusting aspect ratio by becoming narrower and longer than cells grown in suspension (Yang et al., 2018). Mechanical forces opposing cell growth (such as friction due to channel walls) can decrease elongation rate in E. coli (Tuson et al., 2012) and may constitute the main factor limiting cell growth in long narrow channels, even though growth rates between standard Mother Machine conditions and bulk culture are, on average, equivalent (Yang et al., 2018). We attribute reduced growth rate of C. crescentus in the Mother Machine to the presence of Pluronic F108; however, this does not account for deviations in the interdivision time distribution. These results demonstrate the need to fully characterize the effects of environmental constraints on observed phenomena, especially when moving from population-level aggregate to stochastic single-cell measurements and attempting to draw inferences from individual-level variations. They also serve to underscore that the mythical “average cell” does not realistically capture the complex behaviors and stochastic dynamics of the individual cell, as is increasingly appreciated in different biological contexts (Iyer-Biswas, 2009; Iyer-Biswas et al., 2009; Hu et al., 2009; Iyer-Biswas and Zilman, 2016; Jafarpour et al., 2017; Sanders, Joshi, et al., 2023; York, Joshi, Wright, et al., 2023; Joshi, York, et al., 2024). Direct genetic perturbations to division machinery such as FtsZ (Si et al., 2019) and Min (Vashistha et al., 2023) have been demonstrated to alter division timing independent of growth rate, but future studies examining the dynamics of specific components of the division machinery are needed to characterize the relationship between mechanical confinement and disruptions in the timing of cell division.

Scaling laws are ubiquitous in nature, and have been reported in the context of fluctuations in growth and division dynamics of different bacterial species. These scaling laws offer insights into the mechanisms underlying growth and division. Previously, intragenerational scaling laws have been reported for many microorganisms (Salman et al., 2012). In particular, the mean-rescaled cell interdivision time distribution (from different growth conditions; Iyer-Biswas et al., 2014), the mean-rescaled cell size distributions (from different times since the last division event; Iyer-Biswas et al., 2014), and the mean-rescaled cell age distributions (from different growth conditions; Jafarpour et al., 2018; Joshi et al., 2023b) undergo scaling collapses for C. crescentus cells in balanced growth conditions. Other known examples of intragenerational scaling laws include the mean-rescaled distributions of initial size, final size, growth rate, and division time across growth conditions for E. coli cells growing in the Mother Machine (Taheri-Araghi et al., 2015). In contrast, here we focus on intergenerational scaling laws, generalizing further the ongoing line of work reported in Joshi, Wright, Ziegler, et al., 2023; Joshi et al., 2023a, c by drawing comparisons across different experimental setups and identifying invariant behaviors. In particular, we find that the mean-rescaled intragenerational growth rate and division time distributions are not invariant across devices but, remarkably, the initial and final cell size distributions still are. This intercondition scaling law, when paired with the invariance of across devices, results in the general validity across experimental realizations of the intergenerational scaling law governing cell size homeostasis (Figure 3).

MATERIALS AND METHODS

Experimental Setup

SChemostat and Mother Machine microfluidic devices were fabricated from PDMS using standard soft lithography techniques. Mother Machine devices were incubated with 0.1 g/ml Pluronic F108 (Sigma-Aldrich, 542342) for 24 h before each experiment. C. crescentus cells were grown to log phase at 30°C in peptone yeast extract (PYE) liquid cultures. For SC* and experiments, cells were exposed to 75 µM vanillic acid (Sigma Aldrich, H36001) for 75 min immediately before loading into the device. For all experiments, cells were loaded into the device and incubated for 60 min either in PYE (SC*) or in PYE supplemented with 1% of 0.1 g/ml Pluronic F108 ( and ). The same growth media was then perfused through the device at 10 µl/min and phase contrast images were captured every 60 s, as described previously (Joshi et al., 2023c).

Image Analysis

SChemostat data were analyzed as previously described (Joshi, Wright, Ziegler, et al., 2023). To adapt this routine to phase contrast images obtained in the Mother Machine, we generated a background image for each movie from a series of registered images sampled over time, then subtracted this background from each image to remove features corresponding to the device. This procedure provided images suitable for pixel-based segmentation as previously described (Joshi et al., 2023c).

FOOTNOTES

SC*	∆creS cells growing in the SChemostat in complex media
	∆creS cells growing in the SChemostat in complex media supplemented with surfactant (Pluronic F108)
	∆creS cells growing in the Mother Machine in complex media supplemented with surfactant (Pluronic F108)

ACKNOWLEDGMENTS

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