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Single-cell growth inference of Corynebacterium glutamicum reveals asymptotically linear growth

Regulation of growth and cell size is crucial for the optimization of bacterial cellular function. So far, single bacterial cells have been found to grow predominantly exponentially, which implies the need for tight regulation to maintain cell size homeostasis. Here, we characterize the growth behavior of the apically growing bacterium Corynebacterium glutamicum using a novel broadly applicable inference method for single-cell growth dynamics. Using this approach, we find that C. glutamicum exhibits asymptotically linear single-cell growth. To explain this growth mode, we model elongation as being rate-limited by the apical growth mechanism. Our model accurately reproduces the inferred cell growth dynamics and is validated with elongation measurements on a transglycosylase deficient ΔrodA mutant. Finally, with simulations we show that the distribution of cell lengths is narrower for linear than exponential growth, suggesting that this asymptotically linear growth mode can act as a substitute for tight division length and division symmetry regulation.


Introduction
Regulated single-cell growth is crucial for the survival of a bacterial population. At the population level, fundamental laws of growth were discussed as early as the beginning of the 20th century, and distinct population growth phases were identified and attributed to bacterial growth (Lane-Claypon, 1909;Buchanan, 1918;Monod, 1949). At the time, however, growth behavior at the singlecell level remained elusive. This changed only over the last decade, as evolving technologies enabled detailed measurements of single-cell growth dynamics. Extensive work was done on common model organisms, including Escherichia coli, Bacillus subtilis, and Caulobacter crescentus, revealing that averaged over the cell cycle, single cells grow exponentially for these species (Taheri-Araghi et al., 2015;Mir et al., 2011;Iyer-Biswas et al., 2014;Yu et al., 2017;Godin et al., 2010).
Single-cell exponential growth is expected if cellular volume production is proportional to the protein content (Amir, 2014), as shown to be the case for E. coli (Belliveau et al., 2020). Importantly, however, such a proportionality will only be present if cellular volume production is rate-limiting for growth. Cells with different rate-limiting steps could display distinct growth behavior.
Recently, detailed analysis of the mean growth rate throughout the cell cycle revealed deviations from pure exponential growth. For B. subtilis (Nordholt et al., 2020), a biphasic growth mode was observed, where a phase of approximately constant elongation rate is followed by a phase of increasing elongation rate. For E. coli, a new method provides evidence for super-exponential in the later stages of the cell cycle (Kar et al., 2021).
A promising candidate for uncovering strong deviations from exponential growth is the Grampositive Corynebacterium glutamicum. This rod-shaped bacterium grows its cell wall exclusively at the cell poles, allowing, in principle, for deviations from exponential single-cell growth ( Figure 1). The dominant growth mode depends on the rate-limiting step for growth, which is presently unknown for this bacterium. Non-exponential growth modes may have important implications for growth regulatory mechanisms: while exponential growth requires checkpoints and regulatory systems to maintain a constant size distribution (Mir et al., 2011), such tight regulation might not be needed for other growth modes.
Corynebacterium glutamicum is broadly used as a production-organism for amino-acids and vitamins and also serves as model organism for the taxonomically related human pathogens Corynebacterium diphteriae and Mycobacterium tuberculosis (Hermann, 2003;Antoine et al., 1988;Schubert et al., 2017). A common feature of Corynebacteria and Mycobacteria is the existence of a complex cell envelope. The cell wall of these bacteria is a polymer assembly composed of a classical bacterial peptidoglycan (PG) sacculus that is covalently bound to an arabinogalactan (AG) layer (Alderwick et al., 2015). Mycolic acids are fused to the arabinose and form an outer membrane like Figure 1. Growth mode analysis for four possible rate-limiting steps for cellular volumegrowth in the apically growing C. glutamicum. Here, V is the cellular volume, A is the cell wall area, and C t ð Þ is the concentration of membrane building blocks in the cytoplasm. A constant cell width is assumed throughout, implying dA dt / dV dt . A fixed production capacity per unit volume is assumed for the rate-limiting steps 'cell mass production' and 'formation of cell wall building blocks'. For the rate-limiting step 'assembly of cell wall', a constant insertion area at the cell poles is assumed. For an analysis of the single-cell growth mode if cell wall building block formation is the rate-limiting step for growth, see Appendix 1. Cell mass production, specifically ribosome synthesis, has previously been indicated as the rate-limiting step for growth in E. coli (Belliveau et al., 2020;Scott et al., 2010;Amir, 2014). Linear growth is observed if the rate-limiting step for volume growth is the cell wall assembly (shown here in a simplified representation). The protein DivIVA serves as a scaffold at the curved membrane of the cell pole for the recruitment of the Lipid-II flippase MurJ and several mono-and bifunctional trans-peptidases (TP) and -gylcosylases (TG). In the process of elongation, peptidoglycan (PG) precursors are integrated into the existing PG sacculus, which serves as a scaffold of the synthesis of the arabinogalactan-layer (AG) and the mycolic-acid bilayer (MM).
bilayer, rendering the cell surface highly hydrophobic (Puech et al., 2001). The mycolic acid membrane (MM) is an efficient barrier that protects the cells from many conventional antibiotics.
C. glutamicum's growth and division behavior is vastly different to that of classical model species. In contrast to rod-shaped firmicutes and g-proteobacteria, where cell-wall synthesis is dependent on the laterally acting MreB, members of the Corynebacterianeae lack a mreB homologue and elongate apically. This apical elongation is mediated by the protein DivIVA, which accumulates at the cell poles and serves as a scaffold for the organization of the elongasome complex (Letek et al., 2008;Hett and Rubin, 2008;Sieger et al., 2013; Figures 1 and 2A,B). Furthermore, a tightly regulated division-site selection mechanism is absent in this species. Without harboring any known functional homologues of the Min-and nucleoid occlusion (Noc) system, division typically results in unequally sized daughter cells (Donovan et al., 2013;Donovan and Bramkamp, 2014). Lastly, the spread in growth times between birth and division is much wider than in other model organisms, suggesting a weaker regulation of this growth feature (Donovan et al., 2013). These atypical growth properties suggest that this bacterium is an interesting candidate to search for novel growth modes. To reveal the underlying growth regulation mechanisms, it is necessary to study the elongation dynamics of C. glutamicum.
Here, we measure the single-cell elongations within a proliferating population of C. glutamicum cells, and develop an analysis procedure to infer their growth behavior. We find that C. glutamicum deviates from the generally assumed single-cell exponential growth law. Instead, these Corynebacteria exhibit asymptotically linear growth. We develop a mechanistic model, termed the rate-limiting apical growth (RAG) model, showing that these anomalous elongation dynamics are consistent with the polar cell wall synthesis being the rate-limiting step for growth. Finally, we demonstrate a connection between mode of growth and the impact of single-cell variability on the cell size distribution of the population. For an asymptotically linear grower, these variations have a much smaller impact on this distribution than they would for an exponential grower, which may suggest an evolutionary explanation for the lack of tight regulation of single-cell growth in C. glutamicum.

Measuring elongation trajectories using microfluidic experiments
To measure the development of single C. glutamicum cells over time, we established a workflow combining single-cell epifluorescence microscopy with semi-automatic image processing. Cells were grown in a microfluidic device. We used wild type cells and cells expressing the scaffold protein Div-IVA as a translational fusion to mCherry. DivIVA is used as a marker for cell cycle progression, since it localizes to the cell poles and to the newly formed division septum in C. glutamicum (Letek et al., 2008;Donovan et al., 2013).
For the choice of microfluidic device, we deviate from the commonly used Mother Machine (Long et al., 2013), which grows bacteria in thin channels roughly equaling the cell width. The Mother Machine is not ideally suited for C. glutamicum growth, as the characteristic V-snapping at division could lead to shear forces and stress during cell separation, affecting growth (Zhou et al., 2019). Indeed, in some cases, the mother machine has been shown to affect growth properties even in cells not exhibiting V-snapping at division, due to mechanical stresses inducing cell deformation (Yang et al., 2018). Therefore, we instead used microfluidic chambers that allow the growing colony to expand without spatial limitations into two dimensions for several generations ( Figure 2C,D, Materials and methods). Within the highly controlled environment of the microfluidic device, a steady medium feed and a constant temperature of 30˚C was maintained. We extracted bright-fieldand fluorescent-images over 3-min intervals, which were subsequently processed semi-automatically with a workflow developed in FIJI and R (Schindelin et al., 2012;R Development Core Team, 2003). For each individual cell per time-frame, the data set contains the cell's length, area and estimated volume, the DivIVA-mCherry intensity profile, and information about generational lineage ( Figure 2E-G). We used these data sets to further investigate the growth behavior of our bacterium. Thus, using this procedure, we obtained data sets containing detailed statistics on single-cell growth of C. glutamicum.
For subsequent analysis, the measured cell lengths were used, because of their low noise levels as compared to other measures (Appendix 2- figure 1B). Importantly, the increases in cell length are proportional to the increases in cell area (Appendix 2- figure 1A), suggesting that cellular length increase is also proportional to the volume increase. This proportionality is expected since the rod-shaped C. glutamicum cells insert new cell wall material exclusively at the poles, while maintaining a roughly constant cell width over the cell cycle (Schubert et al., 2017;Daniel and Errington, 2003).

Population-average test suggests non-exponential growth for C. glutamicum
A standard way of characterizing single-cell bacterial growth, is to determine the average relation between birth length l b and division length l d (Amir, 2014). For C. glutamicum, we find an approximately linear relationship between these birth and division lengths, with a slope of 0.91±0.16 (2xSEM, Figure 3A). This indicates that on a population level, C. glutamicum behaves close to the adder model, in which cells on average grow by adding a fixed length before dividing (Jun and Taheri-Araghi, 2015;Amir, 2014).
To investigate the growth dynamics from birth to division, we first tested if our cells conform to the generally observed exponential mode of single-cell growth. To this end, we applied a previously developed analysis on bacterial elongation data (Logsdon et al., 2017), by plotting ln ld lb versus the growth time ( Figure 3B). For an exponential grower, with the same exponential growth rate a for all cells, the averages of ln ld lb per growth time bin are expected to lie along a straight line with slope a intersecting the origin. By contrast, there appears to be a systematic deviation from this trend, with cells with shorter growth times lying above this line and cells with longer growth times lying below it, suggesting non-exponential elongation behavior. However, the significance and implications of these deviations for single-cell growth behavior are not clear from this analysis. There are several quantities that could be highly variable between cells that are averaged out in this representation, such as possible variations in exponential growth rate as a function of birth length, or variations in growth mode over time. Furthermore, it was recently shown that exponentially growing cells can appear non-exponential with this test in the presence of noise in the exponential growth rate (Kar et al., 2021). Thus, a more detailed analysis of the growth trajectories is needed to rule out exponential growth, and to quantitatively characterize the growth dynamics. The variability of key growth parameters is not easily extracted from individual growth trajectories due to the inherent stochasticity of the elongation dynamics and measurement noise ( Figure 3C). In fact, it has been estimated that to distinguish between exponential and linear growth for an individual trajectory, the trajectory needs to be determined with an error of~6% (Cooper, 1998). Distinguishing subtler growth features may require an even higher degree of accuracy, which is presently experimentally unavailable (Appendix 3). Therefore, an analysis method is needed that is less noisesensitive than an inspection of the single-cell trajectories, but simultaneously does not average out potentially relevant growth features such as time-dependence and birth length variability.

Growth-inference method yields average elongation rate curves
To obtain quantitative elongation rate curves as a function of time and birth length, despite the high degree of individual variation, we developed a data analysis procedure that exploits the noise-reducing properties of multiple-cell conditional averaging. The key idea is to obtain an average dependence of the cellular length L t; l b ð Þ on the time t since birth and birth length l b , by first obtaining the average dependence of L t; l b ð Þ on l b for each discrete value of t individually. This yields an average elongation curve for each birth length l b , without the need to perform inference on noisy L t ð Þ singlecell curves.
The analysis procedure is as follows. First, for all cells in our data set, we determine the time since birth t, the cellular length L at time t, and the birth length l b : Subsequently, we relate the length at time t to the birth length, yielding a series of scatter plots for each measurement time ( Figure 4A). Importantly, these scatterplots suggest a simple apparently linear relationship between L and l b . For each such plot, we thus make a linear fit through the data, yielding a family of curves for each time since birth t ( Figure 4B). Higher-order fitting functions result in a negligible improvement of the goodness-of-fit, while increasing the mean error on inferred elongation rates (Appendix 2- figure  3). Note that for both purely linear and purely exponential growth, would depend linearly on: for lin- . From the family of relations, we compute a series of points Lðt 0 ; l b Þ; Lðt 1 ; l b Þ; Lðt 2 ; l b Þ f g yielding the average growth trajectory of a cell starting out at length l b ( Figure 4C). Note, we must remove a bias in the l b associated with each average trajectory, arising from measurement noise in the cell lengths at birth (Appendix 4). In summary, this procedure allows us to obtain an unbiased interference of the average elongation trajectories as a function of the cell's birth length.

Elongation rate inference reveals asymptotically linear growth mode
Our inference approach yields the functional dependence of the average added length on growth time and birth length. We find that the average length steadily increases initially, but levels off and shows pronounced fluctuations for larger growth times ( Figure 4C). This late-time behavior (dashed lines in Figure 4C) is caused by decreasing cell numbers due to division events ( Figure 4D), which also introduces a bias in the averaging procedure. After the first division event, the average inferred growth would be conditioned on the cells that have not divided yet. For a given birth length, faster- growing cells divide earlier than slower-growing cells (Appendix 2-figure 2) causing this conditional average to underestimate cellular elongation rates for the whole population after the first division. Because our aim is to infer elongation curves that characterize the whole population, ranging from slow to fast growers, for further analysis only the part of each trajectory before the first division event is used ( Figure 4D). Sub-population elongation curves can also be obtained that extend past the first division event, but only if the entire analysis for these curves is performed only on these slower-dividing cells (Appendix 2- figure 4).
We obtain elongation rate curves by taking a numerical derivative of smoothed growth trajectories (Appendix 5). To determine the associated error margins of the elongation rates, we use a custom bootstrapping algorithm (Efron, 1979). The resulting 2s bounds are shown as semitransparent bands. Despite the high noise level of individual elongation trajectories, the inferred average elongation rates have an error margin of around 8%. Thus, our approach robustly infers average elongation trajectories from single-cell growth data. Elongation rates of cells with larger birth length are consistently higher than the elongation rates of cells with smaller birth length. Strikingly, the elongation rate curves initially increase, but then gradually level off toward a linear growth mode ( Figure 5). We note a slight difference in the cell elongation rates between the strain expressing DivIVA-mCherry ( Figure 5A) and wild type cells ( Figure 5B). Importantly, this difference does not qualitatively change the mode of growth, but does show that a translational fusion to DivIVA tends to lower elongation rates. This likely reflects a disturbance in the interaction between RodA or bifunctional PBPs and the DivIVA-mCherry fusion protein, indicating that the DivIVA-mCherry fusion is not fully functional. This is consistent with findings we reported earlier (Donovan et al., 2013).
To further test if the linear growth mode persists until division, we adapt our inference procedure to obtain average elongation curves L t À t d ; l d ð Þ as a function of the time until division t À t d and division length l d . The construction is analogous to that of L t; l b ð Þ (Appendix 6). Calculating the corresponding elongation rate curves, we find that that linear growth indeed extends until the division time across division lengths (inset Figure 5B,SI, Appendix 6-figure 1). Note that with this construction, elongation rates become biased once t À t d j j exceeds the shortest single-cell total growth time. Hence, for our analysis we only consider elongation rate curves until this point.
To test the performance of our proposed inference method, we simulated a population of growing cells with a presumed growth mode from which we sample cells lengths as in our experiments, including measurement noise (Appendix 3). We ran simulations for cells performing linear growth, exponential growth, and the growth mode inferred here for DivIVA-labeled cells ( Figure 5A). We find that our inference method is able to recover the input growth mode with high precision in all cases (Appendix 4, Appendix 7), demonstrating the accuracy and internal consistency of our inference method.
Onset of the linear growth regime does not consistently coincide with septum formation A central feature of the obtained elongation rate curves is a transition from an accelerating to a linear growth mode after approximately 20-25 min ( Figure 5). One possibility is that this levelling off is connected with the onset of division septum formation. Given that the FtsZ-dependent divisome propagates the invagination of the septum under the consumption of cell wall precursors (e.g. Lipid-II), we hypothesized that the appearance of the additional sink for cell-wall building blocks could lead to coincidental leveling-off of the elongation rates (Scheffers and Tol, 2015). To test this hypothesis, we used the moment of a sharp increase in the average DivIVA-mCherry signal at the cell center as a proxy for the moment of onset of septum formation (Appendix 2-figure 7): the inward growing septum introduces a negative curvature of the plasma membrane, leading to the accumulation of DivIVA (Lenarcic et al., 2009;Strahl and Hamoen, 2012). We observe that the onset of septum formation does not consistently coincide with the moment at which the elongation rate levels off ( Figure 5A): for smaller cells, the onset of septum formation occurs much later. Therefore, it seems implausible that the observed linear growth regime is due the septum acting as a sink for cell-wall building blocks.
Polar cell wall formation is the rate-limiting step for growth, leading to a linear growth regime To provide insight into the anomalous single-cell growth behavior, we model single-cell elongation as being rate-limited by the apical cell wall formation mechanism. To formulate this rate-limiting apical growth (RAG) model, we first consider the biochemical pathway that leads to cell wall formation in C. glutamicum, as illustrated in Figure 1. The key process for cell wall formation in C. glutamicum is polar peptidoglycan (PG) synthesis. PG intermediates are provided by the substrate Lipid-II, and the integration of new material into the PG-mesh is mediated by transglycosylases (TGs) located at the cell pole. At the TG sites, Lipid-II is translocated across the plasma membrane by the Lipid-II flippase MurJ (Sham et al., 2014;Kuk et al., 2017;Butler et al., 2013). After PG building blocks provided by Lipid-II are incorporated into the existing cell wall by transglycolylation, transpeptidases (TP) conduct the crosslinking of peptide subunits, which contributes to the rigidity of the cell wall (Scheffers and Pinho, 2005;Valbuena et al., 2007;Schleifer and Kandler, 1972). During growth, the area of the PG sacculus, and thus the number of TG sites, is extended by RodA and bifunctional penicillin binding proteins (PBPs), recruited by DivIVA (Letek et al., 2008;Sieger et al., 2013).
To model this growth mechanism, we assume that the rate of new cell wall formation is proportional to the number of TG sites. We describe the interaction between Lipid-II and TG sites by Michaelis-Menten kinetics ( Figure 6A). Specifically, if the cell length added per unit time is proportional to the cell wall area added per unit time, we find with L t ð Þ the cell length at time t, C t ð Þ the concentration of Lipid-II, K m the Michaelis constant for this reaction, and a is a proportionality constant.
To gain insight into the cell-cycle-dependence of NðtÞ and CðtÞ, we made use of the cyan fluorescent D-alanine analogue HADA (see Materials and methods) to stain newly inserted peptidoglycan. Exponentially growing C. glutamicum cells were labeled with HADA for 5 min before imaging. The HADA stain will mainly appear at sites of nascent PG synthesis. As expected, HADA staining resulted in a bright cyan fluorescent signal at the cell poles and at the site of septation. Still images were obtained with fluorescence microscopy and subjected to image analysis (Figures 2A and  6B, Materials and methods).
We first verify that the HADA intensity profile at the cell poles can be used as a measure for the peptidoglycan insertion rate. To do this, we assume that the HADA intensity profile has two relevant contributions: fluorescent probe present in the cell plasma, and fluorescent probe attached to newly inserted peptidoglycan. We use the minimum of the HADA intensity profile, consistently located around mid-cell in non-dividing cells, as an estimate of the contribution from the cell plasma in each cell, and subtract this from the entire cellular profile to obtain the corrected HADA profile (Appendix 2-figure 8). We then define the polar regions where we use the corrected HADA intensity to measure newly inserted peptidoglycan as the portions of the cell within 0.78 mm of the cell tips. Our results are, however, not strongly dependent on this polar region definition (Appendix 2- figure  10). Subsequently, we compute a moving average of the corrected polar HADA intensity as a function of cell length ( Figure 6C). These polar HADA intensities are approximately proportional to the inferred average single-cell elongation rates (Appendix 8), as shown in the inset of Figure 6C. Thus, the polar HADA intensities can be used as a measure for the cellular elongation rate. Assuming a proportional relationship between elongation rate and peptidoglycan insertion rate, this implies the polar HADA intensities are also approximately proportional to the peptidoglycan insertion rate. Deviations of~10% from proportionality within the error margins observed over the range of tip intensities do not affect subsequent conclusions from the HADA intensity data.
Analyzing the HADA intensity profile for smaller segments within the polar region, we find that the increase in intensity is unevenly distributed ( Figure 6D). Close to the cell tip, the HADA intensity remains approximately constant across cell lengths, whereas a linear increase over cell lengths is seen further from the tip. Considering the implications of these measured intensities for CðtÞ and NðtÞ within our model in Equation (1), we argue for a scenario where either CðtÞ is constant or CðtÞ ) K m . Our reasoning is as follows. From Equation (1), we see that the approximately constant intensity at the cell tip can be produced in two ways: (1) CðtÞ ) K m or CðtÞ is constant across cell  Figure 5A. Solid lines: best fit of elongation model from Equation (2), which assumes constant transglycosylase recruitment. Dashed lines: best fit of elongation model from Equation (3), which assumes an exponential increase of transglycosylase recruitment. lengths, and the number of transglycosylases at the tip N tip ðtÞ is constant, or (2) N tip ðtÞ and CðtÞ anticorrelate in such a way to produce constant insertion.
However, we consider constant N tip t ð Þ as biologically the most plausible scenario. This is supported by noting that the concentration of Lipid-II is the same directly before and after division, such that C t ð Þ, and by implication N tip t ð Þ, is similar for the shortest and the longest cell lengths (Appendix 2- figure 9). In our subsequent analysis, we will therefore assume that either C t ð Þ is constant, or C t ð Þ ) K m . This implies that dL t ð Þ dt in Equation (1) is directly proportional to N t ð Þ. To derive an expression for N t ð Þ, we first note that the old and new cell pole in the cell need to be treated differently. We assume the number of polar TG-sites to saturate within one cellular lifecycle, such that the new pole initiates with N t ð Þ below saturation, while the old pole -inherited from the mother cell -is saturated. Letting the number of TG sites increase proportional to the number of available sites, we arrive at the following kinetic description for N t ð Þ Here, N max is the maximum number of sites at the cell poles, and b is a rate constant. This result, together with Equation (1), defines our RAG model. The predicted elongation rates provide a good fit to the experiment for all studied genotypes ( Figure 6E-G), although the data appear to exhibit a stronger inflection.
Instead of assuming a constant recruitment of TG enzymes, we can construct a more refined model that takes TG recruitment dynamics into account. There is evidence that transglycosylase RodA and PBPs are recruited to the cell pole via the curvature-sensing protein DivIVA (Letek et al., 2008;Sieger et al., 2013). As shown in Lenarcic et al., 2009, DivIVA also recruits itself, leading to the exponential growth of a nucleating DivIVA cluster. Therefore, we let the recruitment rate of TG enzymes be proportional to the number of DivIVA proteins N D ðtÞ ¼ N D ð0Þe gt . This results in a modified kinetic description for N t ð Þ (Equation (2)): This refined model can capture more detailed features of the measured elongation rate curves ( Figure 6E-G), including the stronger inflection, with an additional free parameter, g, encoding the self-recruitment rate of DivIVA.
The central assumption of our RAG model is that the growth of the cell poles, mediated via accumulation of TG enzymes, is the rate-limiting step for cellular growth. To test this assumption, we repeated our experiment with a rodA knockout (Sieger et al., 2013). The SEDS-protein RodA is a mono-functional TG (Meeske et al., 2016;Emami et al., 2017;Sjodt et al., 2018), whose deletion results in a phenotype with a decreased population growth rate in the shaking-flask (Sieger et al., 2013). The cells' viability is nonetheless backed up by the presence of bifunctional class A PBPs capable of catalyzing transglycoslyation and transpeptidation reactions. We expect this knockout to lower the efficiency of polar cell wall formation, thus slowing down the rate-limiting step of growth. Specifically, we expect the knockout of rodA to mainly affect the efficiency of Lipid-II integration into the murein sacculus. Within our RAG model, this translates to a lowering of the cell wall production per transglycosylase site a. This would imply elongation rate curves of similar shape for the DrodA mutant, only scaled down by a factor a WT =a DrodA . Indeed, we observe such a scaling down of the elongation rate curves ( Figure 5C), lending further credence to our model for C. glutamicum growth.
A striking feature observed across growth conditions and birth lengths, is the onset of a linear growth regime after approximately 20 min ( Figure 5A-C). The robustness of this timing can be understood from the RAG model: the regime of linear growth is reached via an exponential decay of the number of available TG sites until saturation is reached. This exponential decay makes the moment of onset of the linear growth regime relatively insensitive to variations in Nð0Þ and N max . Specifically, from Equation (2), it can be shown that the difference between N t ð Þ and N max is halved every lnð2Þb minutes, which amounts to~8 min given fitted value of b (Appendix 9-table 1).
Finally, our RAG model makes a prediction for the degree of transglycosylase saturation of the cell poles at birth, relative to the saturation in the linear growth regime. We find that this saturation is comparable between wild-type and the DrodA mutant (~65% on average), but significantly higher for DivIVA-labeled cells (~80% on average) (Appendix 9-tables 1 and 2). Note that the percentage of the saturation levels are relative values and do not suggest that in the DivIVA-mCherry fusion more transglycosylase sites are present in absolute numbers.

Birth length distribution of linear growers is more robust to single-cell growth variability
After obtaining average single-cell growth trajectories, we next asked how this growth behavior at the single cell level affects the growth of the colony. It was shown that asymmetric division and noise in individual growth times results in a dramatic widening of the cell-size distribution for a purely exponential grower (Marantan and Amir, 2016). For an asymptotically linear grower, however, we would expect single-cell variations to have a much weaker impact.
To quantify the difference between asymptotically linear growth and hypothetical exponential growth for C. glutamicum, we performed population growth simulations for both cases. For the asymptotically linear growth, we assumed the elongation rate curves obtained from our model. For exponential growth, we assumed the final cell size to be given by l d ¼ l b expðaðt t þ DtÞÞ þ Dl, with a the exponential elongation rate, t t the target growth time, Dt a time-additive noise term and Dl a size-additive noise term. All growth parameters necessary for the simulation were obtained directly from the experimental data (Appendix 10). From this simulation, the distribution of initial cell lengths was determined for each scenario.
The resulting distribution of birth lengths for the asymptotically linear growth case closely matches the experimentally determined distribution (Figure 7). By contrast, the distribution for exponential growth is much wider, and exhibits a broad tail for longer cell lengths. This suggests a strong connection between growth mode and the effect of individual growth variations on population statistics. C. glutamicum has a high degree of variation of division symmetry (Appendix 10-figure 1C) and single-cell growth times, but due to the asymptotically linear growth mode, the population-level variations in cell size are still relatively small. This indicates that linear growth can act as a regulator for cell size.

Discussion
By developing a novel growth trajectory inference and analysis method, we showed that C. glutamicum exhibits asymptotically linear growth, rather than the exponential growth predominantly found in bacteria. The obtained elongation rate curves are shown to be consistent with a model of apical cell wall formation being the rate-limiting step for growth. The RAG model is further validated by experiments with a DrodA mutant, in which the elongation rate curves look functionally similar, but with a downward shift compared to wild type ( Figure 5B,C), as expected based on our model. For C. glutamicum, apical cell wall formation is a plausible candidate for the rate-limiting step of growth, because synthesis of the highly complex cell wall and lipids for the mycolic acid membrane is cost intensive and a major sink for energy and carbon in Corynebacteria and Mycobacteria (Brennan, 2003).
An analysis of elongation rates as a function of time and birth length has previously been done in B. subtilis by binning cells based on birth length (Nordholt et al., 2020). Applying this method to our data set yields elongation rates averaged over cells within a binning interval (Appendix 2-figure 5). Averaging our inferred elongation rates over the same bins, we find the two methods to yield consistent results. The binning method, however, involves a tradeoff: a smaller bin width results in a larger error on the inferred elongation rates, whereas a larger bin width averages out all variation within a larger birth length interval. Our method does not suffer from this binning-related tradeoff, and it provides detailed elongation rate curves at any given birth length. In other recent work (Kar et al., 2021), average growth rate curves were calculated as a function of cell phase. Our method provides additional detail by extracting the dependence of elongation rate on birth length as well as time since birth. Our proposed growth model shares some similar features to recent experimental observations on polar growth in Mycobacteria (Hannebelle et al., 2020). Polar growth was shown to follow 'new end take off' (NETO) dynamics (Hannebelle et al., 2020), in which the new cell pole makes a sudden transition from slow to fast growth, leading to a bilinear polar growth mode. In our proposed growth model for C. glutamicum however, the new pole gradually increases its average elongation rate before saturating to a constant maximum. The deviation of C. glutamicum from NETO dynamics can also be seen by comparing each of the pole intensities in the HADA staining experiment, which does not show any signatures of NETO-like growth (Appendix 2- figure 11). It remains unclear which molecular mechanisms produce the differences in growth between such closely related species. However, the mode of growth described here for C. glutamicum might well be an adaption to enable higher growth rates.
To investigate the implications of our inferred single-cell growth mode for cell-size homeostasis throughout a population of cells, we performed simulations of cellular growth and division over many generations. We found that our asymptotically linear growth model accurately reproduces the experimental distribution of cell birth lengths. By contrast, a model of exponential growth predicts a much broader distribution with a long tail for larger birth lengths. This indicates a possible connection between mode of growth and permissible growth-related noise levels for the cell. Indeed, if single-cell growth variability is reduced by a factor 3, the distributions corresponding to both growth modes show a similarly narrow width (Appendix 10- figure 2). However, an asymptotically linear grower is able to maintain a narrow distribution of cell sizes even for higher noise levels, whereas for an exponential grower this distribution widens dramatically (Figure 7).
The enhanced robustness of the length distribution of linear growers is interesting from an evolutionary point of view. Most rod-shaped bacteria use sophisticated systems, such as the Min system, to ensure cytokinesis precisely at midcell (Bramkamp et al., 2009;Lutkenhaus, 2007). Bacteria encoding a Min system grow by lateral cell wall insertion. In contrast, rod-shaped bacteria in the Actinobacteria phylum such as Mycobacterium or Corynebacterium species, grow apically and do not contain a Min system, nor any other known division site selection system (Donovan and Bramkamp, 2014). C. glutamicum rather couples division site selection to nucleoid positioning after chromosome segregation via the ParAB partitioning system (Donovan et al., 2013), and has a broader distribution of division symmetries. We speculate that due to C. glutamicum's distinct growth mechanism, a more precise division site selection mechanism is not necessary to maintain a narrow cell size distribution.
The elongation rates reported in this work reflect the increase in cellular volume over time. However, the increase in cell mass is not necessarily proportional to cellular volume. In exponentially growing E. coli, the cellular density was recently reported to systematically vary during the cell cycle, while the surface-to-mass ratio was reported to remain constant (Oldewurtel et al., 2019). It is unknown how single-cell mass increases in C. glutamicum, but it would follow exponential growth if mass production is proportional to protein content. This raises the question how linear volume growth and exponential mass growth are coordinated. The presence of a regulatory mechanism for cell mass production that couples to cell volume is implied by the elongation rate curves obtained for the DrodA mutant. As the elongation rate is lower in this mutant, average mass production needs to be lowered compared to the WT in order to prevent the cellular density from increasing indefinitely.
Our growth trajectory inference method is not cell-type specific, and can be used to obtain detailed growth dynamics in a wide range of organisms. The inferred asymptotically linear growth of C. glutamicum deviates from the predominantly found exponential single-cell bacterial growth, and suggests the presence of novel growth regulatory mechanisms. Culture and live-cell time-lapse imaging Exponentially growing cells of C. glutamicum WT, C. glutamicum divIVA::divIVA-mCherry and C. glutamicum divIVA::divIVA-mCherry DrodA respectively, grown in BHI-medium (Oxoid) at 30˚C and 200 rpm shaking, were diluted to an OD 600 of 0.01. According to the manufacturer's manual cells were loaded into a CellASIC-microfluidic plate type B04A (Merck Milipore) and mounted on a Delta Vision Elite microscope (GE Healthcare, Applied Precision) with a standard four-color InSightSSI module and an environmental chamber heated to 30˚C. Images were taken in a three-minute interval for 10 hr with a 100Â/1.4 oil PSF U-Plan S-Apo objective and a DS-red-specific filter set (32% transmission, 0.025 s exposure).

Staining of newly inserted peptidoglycan and visualization in demographs
For the staining of nascent PG, 1 ml of exponentially growing C. glutamicum ATCC 13032 cells, cultivated in BHI-medium (Oxoid) at 30˚C and 200 rpm, were harvested, washed with PBS and resuspended in 25 ml PBS, together with 0.25 ml of 5 mM HADA dissolved in DMSO. The cells were incubated at 30˚C in the dark for 5 min, followed by a two-time washing step with 1 ml PBS and finally resuspended in 100 ml PBS. To obtain still-phase-contrast and fluorescent micrographs, 2 ml of the cell suspension were immobilized on an agarose pad. For microscopy, an Axio Imager (Zeiss) equipped with EC Plan-Neofluar 100x/1.3 Oil Ph3 objective and a Axiocam camera (Zeiss) was used together with the appropriate filter sets (ex: 405 nm; em: 450 nm). For single-cell analysis and the visualization in demographs, custom algorithms, developed in FIJI and R (Schindelin et al., 2012;R Development Core Team, 2003), were used. The code is available upon request.

Image analysis
For image analysis, a custom-made algorithm was developed using the open-source programs FIJI and R (Schindelin et al., 2012;R Development Core Team, 2003). During the workflow unique identifiers to single-cell cycles are assigned. The cell outlines are determined manually. Individual cells per timeframe are extracted then from the raw image and further processed automatically. The parameters length, area and relative septum position are extracted and stored together with the genealogic information and the timepoint within the respective cell cycle. The combination of image analysis and cell cycle dependent data structuring yields a list that serves as a base for further analysis. The documented code is available at: https://github.com/Morpholyzer/MorpholyzerGeneration-Tracker (copy archived at swh:1:rev: d01d362ea53b9be6027f29fb85668a0ed418398a, Morpholyzer, 2021).

Acknowledgements
This work was further funded by grants from the Deutsche Forschungsgemeinschaft (project P05in TRR174, granted to MB and project P06 in TRR174, granted to CB). JM is supported by a DFG fellowship within the Graduate School of Quantitative Biosciences Munich (QBM). We thank our colleagues from CB and MB groups for discussions, feedback and comments on the manuscripts. The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication. Single-cell growth mode for apical cell wall formation as a rate-limiting step for growth

Author contributions
To study growth limited by polar cell wall formation, we start by considering the Michaelis-Menten equation describing this formation process (Main Text Equation (1)): with L t ð Þ the cell length at time t, C t ð Þ the concentration of cell wall building blocks in the cytosol, N t ð Þ the number of transglycosylases at the cell pole, K m the Michaelis constant for this reaction, and a a proportionality constant.
In Main Text Figure 1, we consider two scenarios.
(1) Abundant availability of cell wall building blocks, that is C t ð Þ ) K m , and (2) scarcity of cell wall building blocks, that is, C t ð Þ< K m .

A1.1 Building block insertion as a rate-limiting step for growth
In scenario (1)

A1.2 Building block availability as a rate-limiting step for growth
In scenario (2), the dynamics of building block creation, usage, and dilution need to be considered to determine the cellular elongation rate behavior. For the number of building blocks in the cytosol as a function of time n t ð Þ, we can write the following differential equation: Here, a encodes building block production rate per unit volume, and b encodes building block usage by the cell wall formation mechanism, making use of dA t Here, c encodes the proportionality between volume increase and the concentration of building blocks.
Combining Equation (A2) with Equation (A3), we obtain a set of coupled nonlinear differential equations governing the time-evolution of V t ð Þ: These equations have no simple analytic solution; however, we can numerically explore the dependence of V t ð Þ on the differential equation parameters. To do this, we first absorb c into n t ð Þ, leaving us with two free parameters and two boundary conditions. The boundary conditions we set by imposing V 0 ð Þ ¼ 1 and V 1 ð Þ ¼ 2. In Appendix 1-figure 1A, we see that depending on the choice for a and b we can have either sublinear, approximately linear, or superlinear growth. This demonstrates that the single-cell growth mode is dependent on the physiology of building block creation and depletion in the cell. elongation rate curves. The values shown in column 4 and 6 are an average over the four birth lengths of each condition. same target final length l t as in the exponentially growing scenario, determined via a linear fit of l b versus l d . For cells close to observed division times, the term proportional to g can be approximated as being constant in time, simplifying the growth mode to linear growth (Main Text Figure 5A). A time-additive noise term Dt will then act as size-additive noise and can thus be absorbed into one additive noise term Dl, obtained from the experimental distribution of final sizes l d around the target final sizes (Appendix 10- figure 1F). The expression for the division length is thus given by The division asymmetry and V-snap effect are incorporated in the same way as for the exponential grower simulation.
Appendix 10-figure 1. Input used for simulations of exponential and asymptotically linear growth. For both simulations, a linear fit of the division length versus birth length is used to define a target length (A). The length added during the V-snap at division is randomly drawn from the distribution corresponding to the division length of the simulated cell (B). The experimental data is divided into three subpopulations according to division length (red, green, and orange distributions), as the average length added during V-snap decreases with division length (dashed lines). The asymmetry of the daughter cells is randomly drawn from the distribution corresponding to the combined length of the simulated daughters (C). As the asymmetry is lower for the smallest daughter cells, the experimental data is divided into two subpopulations (red and green distributions). For the simulation of exponential growth, two noise sources are needed as input. The time-additive noise is randomly drawn from the distribution of deviations from target growth times (D). This distribution is obtained from the deviations of single-cell growth times from the average of their birth length bin. All growth variability not captured by growth time variations is calculated for four narrow birth length bins (blue, orange, green, and red points) (E). From the distribution of deviations of added lengths from a linear fit for each initial size bin, a size-additive noise term is randomly drawn. For the linear growth simulation, only a single additive noise term is required, which is randomly drawn from the distribution of deviations of cells lengths at division from the target division length (F).