Singlecell growth inference of Corynebacterium glutamicum reveals asymptotically linear growth
Abstract
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 singlecell growth dynamics. Using this approach, we find that C. glutamicum exhibits asymptotically linear singlecell growth. To explain this growth mode, we model elongation as being ratelimited 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 singlecell 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 (LaneClaypon, 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 singlecell 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 (TaheriAraghi et al., 2015; Mir et al., 2011; IyerBiswas et al., 2014; Yu et al., 2017; Godin et al., 2010).
Singlecell 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 ratelimiting for growth. Cells with different ratelimiting 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 superexponential 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 rodshaped bacterium grows its cell wall exclusively at the cell poles, allowing, in principle, for deviations from exponential singlecell growth (Figure 1). The dominant growth mode depends on the ratelimiting step for growth, which is presently unknown for this bacterium. Nonexponential 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 productionorganism for aminoacids 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 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 rodshaped firmicutes and γproteobacteria, where cellwall 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 divisionsite 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 singlecell 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 singlecell exponential growth law. Instead, these Corynebacteria exhibit asymptotically linear growth. We develop a mechanistic model, termed the ratelimiting apical growth (RAG) model, showing that these anomalous elongation dynamics are consistent with the polar cell wall synthesis being the ratelimiting step for growth. Finally, we demonstrate a connection between mode of growth and the impact of singlecell 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 singlecell growth in C. glutamicum.
Results
Measuring elongation trajectories using microfluidic experiments
To measure the development of single C. glutamicum cells over time, we established a workflow combining singlecell epifluorescence microscopy with semiautomatic image processing. Cells were grown in a microfluidic device. We used wild type cells and cells expressing the scaffold protein DivIVA 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 Vsnapping 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 Vsnapping 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 brightfield and fluorescentimages over 3min intervals, which were subsequently processed semiautomatically with a workflow developed in FIJI and R (Schindelin et al., 2012; R Development Core Team, 2003). For each individual cell per timeframe, the data set contains the cell's length, area and estimated volume, the DivIVAmCherry 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 singlecell 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 rodshaped 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).
Populationaverage test suggests nonexponential growth for C. glutamicum
A standard way of characterizing singlecell bacterial growth, is to determine the average relation between birth length ${l}_{\text{b}}$ and division length ${l}_{\text{d}}$ (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 TaheriAraghi, 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 singlecell growth. To this end, we applied a previously developed analysis on bacterial elongation data (Logsdon et al., 2017), by plotting $\mathrm{ln}\left(\frac{{l}_{\text{d}}}{{l}_{\text{b}}}\right)$ versus the growth time (Figure 3B). For an exponential grower, with the same exponential growth rate α for all cells, the averages of $\mathrm{l}\mathrm{n}\left(\frac{{l}_{d}}{{l}_{b}}\right)$ per growth time bin are expected to lie along a straight line with slope α 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 nonexponential elongation behavior. However, the significance and implications of these deviations for singlecell 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 nonexponential 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 singlecell trajectories, but simultaneously does not average out potentially relevant growth features such as timedependence and birth length variability.
Growthinference 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 noisereducing properties of multiplecell conditional averaging. The key idea is to obtain an average dependence of the cellular length $L\left(t,{l}_{\text{b}}\right)$ on the time $t$ since birth and birth length ${l}_{\text{b}}$, by first obtaining the average dependence of $L\left(t,{l}_{\text{b}}\right)$ on ${l}_{\text{b}}$ for each discrete value of $t$ individually. This yields an average elongation curve for each birth length ${l}_{\text{b}}$, without the need to perform inference on noisy $L\left(t\right)$ 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 ${l}_{\text{b}}.\text{}$ 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 $l}_{\text{b}$. 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). Higherorder fitting functions result in a negligible improvement of the goodnessoffit, 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 linear growth $L\left(t,\text{}{l}_{b}\right)=\alpha t+{l}_{b}$, whereas for exponential growth $L\left(t,\text{}{l}_{b}\right)={l}_{b}\mathrm{exp}(\alpha t)$ (Appendix 2—figure 3). From the family of relations, we compute a series of points $\left\{L({t}_{0},\text{}{l}_{\mathrm{b}}),\text{}L({t}_{1},\text{}{l}_{\mathrm{b}}),\text{}L({t}_{2},\text{}{l}_{\mathrm{b}})\right\}$ yielding the average growth trajectory of a cell starting out at length $l}_{\text{b}$ (Figure 4C). Note, we must remove a bias in the $l}_{\text{b}$ 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 latetime 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, fastergrowing cells divide earlier than slowergrowing 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). Subpopulation 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 slowerdividing 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 2σ 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 singlecell 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 DivIVAmCherry (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 DivIVAmCherry fusion protein, indicating that the DivIVAmCherry 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 $L\left(t{t}_{d},\text{}{l}_{d}\right)$ as a function of the time until division $t{t}_{d}$ and division length $l}_{d$. The construction is analogous to that of $L\left(t,\text{}{l}_{\mathrm{b}}\right)$ (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 $\leftt{t}_{d}\right$ exceeds the shortest singlecell 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 DivIVAlabeled 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 FtsZdependent divisome propagates the invagination of the septum under the consumption of cell wall precursors (e.g. LipidII), we hypothesized that the appearance of the additional sink for cellwall building blocks could lead to coincidental levelingoff of the elongation rates (Scheffers and Tol, 2015). To test this hypothesis, we used the moment of a sharp increase in the average DivIVAmCherry 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 cellwall building blocks.
Polar cell wall formation is the ratelimiting step for growth, leading to a linear growth regime
To provide insight into the anomalous singlecell growth behavior, we model singlecell elongation as being ratelimited by the apical cell wall formation mechanism. To formulate this ratelimiting 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 LipidII, and the integration of new material into the PGmesh is mediated by transglycosylases (TGs) located at the cell pole. At the TG sites, LipidII is translocated across the plasma membrane by the LipidII flippase MurJ (Sham et al., 2014; Kuk et al., 2017; Butler et al., 2013). After PG building blocks provided by LipidII 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 LipidII and TG sites by MichaelisMenten 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 $L\left(t\right)$ the cell length at time $t$, $C\left(t\right)$ the concentration of LipidII, $K}_{m$ the Michaelis constant for this reaction, and $\alpha$ is a proportionality constant.
To gain insight into the cellcycledependence of $N(t)$ and $C(t)$, we made use of the cyan fluorescent Dalanine 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 midcell in nondividing 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 μm 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 singlecell 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 $C(t)$ and $N(t)$ within our model in Equation (1), we argue for a scenario where either $C(t)$ is constant or $C(t)\gg {K}_{m}$. 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) $C(t)\gg {K}_{m}$ or $C(t)$ is constant across cell lengths, and the number of transglycosylases at the tip ${N}_{\mathrm{t}\mathrm{i}\mathrm{p}}(t)$ is constant, or (2) ${N}_{\mathrm{t}\mathrm{i}\mathrm{p}}(t)$ and $C(t)$ anticorrelate in such a way to produce constant insertion.
However, we consider constant ${N}_{\text{tip}}\left(t\right)$ as biologically the most plausible scenario. This is supported by noting that the concentration of LipidII is the same directly before and after division, such that $C\left(t\right)$, and by implication ${N}_{\text{tip}}\left(t\right)$, is similar for the shortest and the longest cell lengths (Appendix 2—figure 9). In our subsequent analysis, we will therefore assume that either $C\left(t\right)$ is constant, or $C\left(t\right)\gg {K}_{m}$. This implies that $\frac{dL\left(t\right)}{dt}$ in Equation (1) is directly proportional to $N\left(t\right)$.
To derive an expression for $N\left(t\right)$, we first note that the old and new cell pole in the cell need to be treated differently. We assume the number of polar TGsites to saturate within one cellular lifecycle, such that the new pole initiates with $N\left(t\right)$ 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 $N\left(t\right)$
Here, $N}^{max$ is the maximum number of sites at the cell poles, and $\beta$ 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 curvaturesensing 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 $N}_{\mathrm{D}}(t)={N}_{\mathrm{D}}(0){e}^{\gamma t$. This results in a modified kinetic description for $N\left(t\right)$ (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, $\gamma$, encoding the selfrecruitment 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 ratelimiting step for cellular growth. To test this assumption, we repeated our experiment with a rodA knockout (Sieger et al., 2013). The SEDSprotein RodA is a monofunctional 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 shakingflask (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 ratelimiting step of growth. Specifically, we expect the knockout of rodA to mainly affect the efficiency of LipidII integration into the murein sacculus. Within our RAG model, this translates to a lowering of the cell wall production per transglycosylase site $\alpha$. This would imply elongation rate curves of similar shape for the ΔrodA mutant, only scaled down by a factor $\alpha}^{WT}/{\alpha}^{\mathrm{\Delta}rodA$. 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 $N(0)$ and $N}^{\mathrm{m}\mathrm{a}\mathrm{x}$. Specifically, from Equation (2), it can be shown that the difference between $N\left(t\right)$ and $N}^{\mathrm{m}\mathrm{a}\mathrm{x}$ is halved every $\mathrm{l}\mathrm{n}(2)\beta$ minutes, which amounts to ~8 min given fitted value of $\beta $ (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 wildtype and the ΔrodA mutant (~65% on average), but significantly higher for DivIVAlabeled 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 DivIVAmCherry fusion more transglycosylase sites are present in absolute numbers.
Birth length distribution of linear growers is more robust to singlecell growth variability
After obtaining average singlecell 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 cellsize distribution for a purely exponential grower (Marantan and Amir, 2016). For an asymptotically linear grower, however, we would expect singlecell 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 ${l}_{d}={l}_{b}\mathrm{exp}(\alpha ({t}_{t}+\mathrm{\Delta}t))+\mathrm{\Delta}l$, with α the exponential elongation rate, $t}_{t$ the target growth time, $\mathrm{\Delta}t$ a timeadditive noise term and $\mathrm{\Delta}l$ a sizeadditive 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 singlecell growth times, but due to the asymptotically linear growth mode, the populationlevel 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 ratelimiting step for growth. The RAG model is further validated by experiments with a ΔrodA 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 ratelimiting 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 binningrelated 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 NETOlike 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 singlecell growth mode for cellsize 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 growthrelated noise levels for the cell. Indeed, if singlecell 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 rodshaped 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, rodshaped 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 surfacetomass ratio was reported to remain constant (Oldewurtel et al., 2019). It is unknown how singlecell 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 ΔrodA 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 celltype 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 singlecell bacterial growth, and suggests the presence of novel growth regulatory mechanisms.
Materials and methods
Culture and livecell timelapse imaging
Request a detailed protocolExponentially growing cells of C. glutamicum WT, C. glutamicum divIVA::divIVAmCherry and C. glutamicum divIVA::divIVAmCherry ΔrodA 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 fourcolor InSightSSI module and an environmental chamber heated to 30°C. Images were taken in a threeminute interval for 10 hr with a 100×/1.4 oil PSF UPlan SApo objective and a DSredspecific filter set (32% transmission, 0.025 s exposure).
Staining of newly inserted peptidoglycan and visualization in demographs
Request a detailed protocolFor 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 µl PBS, together with 0.25 µl of 5 mM HADA dissolved in DMSO. The cells were incubated at 30°C in the dark for 5 min, followed by a twotime washing step with 1 ml PBS and finally resuspended in 100 µl PBS. To obtain still phasecontrast and fluorescent micrographs, 2 µl of the cell suspension were immobilized on an agarose pad. For microscopy, an Axio Imager (Zeiss) equipped with EC PlanNeofluar 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 singlecell 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
Request a detailed protocolFor image analysis, a custommade algorithm was developed using the opensource programs FIJI and R (Schindelin et al., 2012; R Development Core Team, 2003). During the workflow unique identifiers to singlecell 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/MorpholyzerGenerationTracker (copy archived at swh:1:rev:d01d362ea53b9be6027f29fb85668a0ed418398a, Morpholyzer, 2021).
Appendix 1
Singlecell growth mode for apical cell wall formation as a ratelimiting step for growth
To study growth limited by polar cell wall formation, we start by considering the MichaelisMenten equation describing this formation process (Main Text Equation (1)):
with $L\left(t\right)$ the cell length at time $t$, $C\left(t\right)$ the concentration of cell wall building blocks in the cytosol, $N\left(t\right)$ the number of transglycosylases at the cell pole, ${K}_{m}$ the Michaelis constant for this reaction, and α a proportionality constant.
In Main Text Figure 1, we consider two scenarios. (1) Abundant availability of cell wall building blocks, that is $C\left(t\right)\gg {K}_{m}$, and (2) scarcity of cell wall building blocks, that is, $C\left(t\right)<{K}_{m}$.
A1.1 Building block insertion as a ratelimiting step for growth
In scenario (1), Equation (A1) reduces to $\frac{dL\left(t\right)}{dt}=\alpha N\left(t\right)$. In the regime of a constant number of transglycosylases at the pole, this implies that $\frac{dL\left(t\right)}{dt}$ is constant, resulting in linear growth.
A1.2 Building block availability as a ratelimiting 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 $n\left(t\right)$, 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 $\frac{dA\left(t\right)}{dt}\propto \frac{dV\left(t\right)}{dt}$. To connect Equation (A2) to Equation (A1), we note that $C\left(t\right)=\frac{n\left(t\right)}{V\left(t\right)}$. Restricting ourselves to the regime $C\left(t\right)\ll {K}_{m}$, we can rewrite Equation (A1) to
where we made use of $\frac{dL\left(t\right)}{dt}\propto \frac{dV\left(t\right)}{dt}.$ 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 timeevolution of $V\left(t\right).$ These equations have no simple analytic solution; however, we can numerically explore the dependence of $V\left(t\right)$ on the differential equation parameters. To do this, we first absorb $c$ into $n\left(t\right)$, leaving us with two free parameters and two boundary conditions. The boundary conditions we set by imposing $V\left(0\right)=1$ and $V\left(1\right)=2$. 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 singlecell growth mode is dependent on the physiology of building block creation and depletion in the cell.
We can further constrain the solution space by demanding that the concentration of building blocks $C\left(t\right)=\frac{n\left(t\right)}{V\left(t\right)}$ is the same at birth and division. In this scenario, the observed variation in elongation curves is smaller (solid lines Appendix 1—figure 1B), however the corresponding elongation rates (dashed lines Appendix 1—figure 1B) still show marked qualitative differences between parameter choices.
Appendix 2
Supplementary figures
Appendix 3
Measurement noise estimate
To obtain an estimate for the measurement noise from our timeseries growth data, we make use of length measurements at subsequent time intervals. For short enough time intervals, the variance of the length differences between intervals can be used as a measure of the measurement noise. However, since we expect cellular growth to also significantly contribute to this variance within the 3 min measurement interval, we have to separate out the two contributions.
To separate out the two contributions to the variance in subsequent length measurements, we write this variance as
with ${l}_{m}\left(t\right)$ the measured length at time $t$, $l\left(t\right)$ the actual length at time $t$, and ${\sigma}_{\text{n}}$ the standard deviation of the measurement noise. This expression can be derived by noting that for a single elongation trajectory, we have
with $\xi $ the measurement noise. A solution for ${\sigma}_{\text{n}}$ can be found if the functional form of $\text{Var}\left(l\left(t\right),l\left(t+\mathrm{\Delta}t\right)\right)$ is known, by obtaining values for multiple $\mathrm{\Delta}t$ and treating ${\sigma}_{\text{n}}$ as a fitting parameter. To obtain this functional form, we make use of the observed linear growth regime after ~20 min (Main Text Figure 5). We observe that the elongation rate is approximately constant in this regime for cells of all birth lengths, and now assume that this is also true for cells individually within this regime. The contrary would imply that nonconstant singlecell elongation rates precisely cancel out across time and birth lengths to produce linear growth, which seems biologically implausible.
For linearly growing single cells, the standard deviation of $l\left(t+\mathrm{\Delta}t\right)l\left(t\right)$ is proportional to $\mathrm{\Delta}t$, implying that the term $\text{Var}\left(l\left(t\right),l\left(t+\mathrm{\Delta}t\right)\right)$ is of the form
with $c$ an unknown parameter. To simultaneously obtain $c$ and ${\sigma}_{\text{n}}$, we fit Equation (A7) under substitution of Equation (A9) to the DivIVAlabeled cell data over the regime between the onset of linear growth (18 min, black dashed line Appendix 3—figure 1) and the first division event (36 min, gray dashed line Appendix 3—figure 1). From this fit, we obtain the estimates ${\sigma}_{n}=$ 0.060 ± 0.018 μm and $c=4.5x{10}^{5}\pm 0.47{\mathrm{x}\mathrm{m}}^{2}\text{}{\mathrm{m}\mathrm{i}\mathrm{n}}^{2}$, where the error margins are determined via bootstrapping. This value of ${\sigma}_{n}$ is used in the correction procedure for assigned birth lengths described in Appendix 4.
Appendix 4
Bias correction procedure for assigned birth lengths
Before calculating average elongation rate curves, a statistical bias arising in the assignment of birth lengths to each curve needs to be corrected for. This bias is not specific to the inference method introduced in this paper, but arises in any procedure involving the assignment of lengths to a cells within a population, if there is noise in the measurement of individual cell lengths.
Due to measurement noise, cells will be assigned to birth lengths that systematically differ from their actual birth lengths. Specifically, given that the birth lengths in the population follow a symmetric, unimodal distribution, cells with a measured birth length larger than the population mean will on average be assigned a birth length that is larger than their actual length. Conversely, cells with a birth length smaller than the population mean will on average be assigned a birth length that is smaller.
The magnitude of the systematic deviation in the assignment of birth lengths is calculated as follows. Given that the cellular birth lengths follow a Gaussian distribution ${P}_{l}({l}_{b})$ with mean ${\mu}_{l}$ and standard deviation ${\sigma}_{l}$, and the measurement noise follows a Gaussian distribution ${P}_{n}(\mathrm{\Delta}l)$ with mean 0 and standard deviation ${\sigma}_{n}$, the distribution of measured lengths will again be a Gaussian, with mean ${\mu}_{m}={\mu}_{l}$ and standard deviation ${\sigma}_{m}=\sqrt{{\sigma}_{l}^{2}+{\sigma}_{n}^{2}}$.
For a given measured birth length ${l}_{m}$, we now consider the probability distribution of corresponding actual birth lengths ${P}_{l}\left({l}_{b}{l}_{m}\right)$. This distribution is given by
The product of two Gaussian distributions is again Gaussian, with a mean equal to
Equation (A5) thus provides the transformation needed to remove the systematic bias in the assignment of birth lengths, and to determine the most likely birth length ${l}_{b}$ to a cell with a measured birth length ${l}_{m}$. For an estimation of the experimental measurement noise, see Appendix 3.
For the length increase since birth, there is no systematic bias once the bias in birth length has been removed. We can see this as follows. For each singlecell elongation trajectory, the measured length ${l}_{m}\left(t\right)$ at time $t$ is given by
with $\xi $ the measurement noise and $\mathrm{\Delta}{l}_{t}$ the length increase since birth at time $t$. As the measurement noise $\xi $ has a zero mean, there is no systematic bias in length increases after birth, provided that we have an unbiased estimate for the birth length ${l}_{b}$.
To test the derived correction procedure for assigned birth lengths, we performed a simulation of a population of growing cells, with the length measurement subject to noise. The measurement noise was sampled from a Gaussian, with the same standard deviation as estimated for experiment (Appendix 3). The singlecell growth mode was chosen as an input parameter. We analyzed two choices for input growth mode: linear (Appendix 4—figure 1A,C, dashed lines) and exponential (Appendix 4—figure 1B,D, dashed lines), with elongation rates comparable in magnitude to measured elongation rates.
For each singlecell growth mode, we applied our elongation rate inference procedure to simulated cell lengths subject to measurement noise. Without correcting for a bias in assigned birth lengths, we find a systematic deviation between inferred elongation rates and input elongation rates in both cases (Appendix 4—figure 1A,B). With the implementation of the correction for assigned birth lengths, the input elongation rates are, however, accurately recovered (Appendix 4—figure 1C,D).
Minor deviations from the input elongation rates can still be seen for exponentially growing cells (Appendix 4—figure 1D), arising from applying a Gaussian smoothing to elongation curves that are locally nonlinear due to limited time resolution. However, this effect is small compared to the uncertainty on the inferred elongation rates.
Appendix 5
Smoothing of elongation curves
We obtain elongation rate curves (Main Text Figure 5 and Figure 6C) by taking a numerical derivative of smoothed growth trajectories. For the smoothing, a Gaussian smoothing procedure was used. In this procedure, a moving average is applied twice over groups of three subsequent time stamps of average elongation curves. As a check of the validity of the smoothing procedure, we also compare elongation rates before and after smoothing (Appendix 5—figure 1).
Appendix 6
Calculating mean elongation curves as a function of time until division
The construction of the average elongation curves $L\left(t{t}_{d},{l}_{\text{d}}\right)$ as a function of the time until division $t{t}_{d}$ and division length ${l}_{d}$ is as follows. We relate the length at time $t{t}_{d}$ to the division length ${l}_{\text{d}}$ for all cells, and use linear fits to obtain a family of curves ${L}_{t{t}_{d}}\left({l}_{\text{d}}\right)$ for each $t{t}_{d}.$ From this family of relations ${L}_{t{t}_{d}}\left({l}_{\text{d}}\right)$, we can subsequently compute $L\left(t{t}_{d},{l}_{\text{d}}\right)$ for any choice of ${l}_{\text{d}}$. The resulting mean elongation rate curves are shown in Appendix 6—figure 1.
Appendix 7
Testing the elongation rate inference procedure
To test our elongation rate inference procedure, we generated a simulated data set with elongation rates as inferred by our inference procedure for DivIVAlabeled cells (Main Text Figure 5). The distribution of birth lengths and division lengths of the simulated cells are taken to match the experimentally observed distributions. On each simulated data point, a measurement noise as determined in Appendix 3 is applied. On the simulated data set subject to noise, we apply the assigned birth length correction procedure as described in Appendix 4, and subsequently apply our elongation rate inference procedure. We find that the input elongation rates are accurately recovered (Appendix 7—figure 1), demonstrating the internal consistency of our inference approach.
Appendix 8
Prediction of average elongation rate as a function of cell length
To calculate the predicted average elongation rates shown in Main Text Figure 6C, we make use of our timeseries data for wildtype cells, and the inferred mean elongation rates shown in Main Text Figure 5B.
We start by calculating the timeaveraged elongation rate ${\overline{l\text{'}}}_{i}$ for each cell $i$ in the wildtype data set, where the prime denotes a time derivative, by dividing the length added between birth and division by the total growth time. We then assume that the elongation rate for a cell at a time $t$ is approximately given by a rescaling of the populationaveraged elongation rates $L\text{'}\left(t,{l}_{b}\right)$ by the timeaveraged elongation rate of the cell ${\overline{l\text{'}}}_{i}$. Specifically, we calculate the estimated elongation rate at time $t$ by
with ${n}_{\text{i}}$ the number of time intervals in the growth trajectory of cell $i$, and ${t}_{\text{div}}^{i}$ its division time. For times $t$ later than the first population division event ${T}_{\text{div}}$, we obtain a value for $L\text{'}\left(t,{l}_{b}\right)$ by extrapolating the linear growth regime, setting $L\text{'}\left(t,{l}_{b}\right)$ = <$L\text{'}\left(t,{l}_{b}\right){}_{20\text{min}t{T}_{\text{div}}}$.
From the ensemble $\left\{l{\text{'}}_{i}\left(t\right)\right\}$ of estimated elongation rates of all cells at each time since birth, we calculate the average elongation rate as a function of cell length by taking a moving average over the corresponding measured $\left\{{l}_{i}\left(t\right)\right\}$. The standard error on the mean is calculated from the standard deviation and the number of cells of each moving average bin.
Appendix 9
RAG model fitting procedure
The model fits shown in Main Text Figure 6E–G are obtained via the ParametricNDSolve function in Mathematica. The obtained parameter values are shown in Appendix 9—tables 1 and 2.
Appendix 10
Population simulation method
The goal of the population growth simulations is to obtain the distribution of cellular birth lengths assuming two different growth modes: asymptotically linear and exponential elongation. Both simulations extract all necessary growth parameters and distributions from the experimental data. For the asymptotically linear growth mode, the simulation serves as a check whether the assumed growth mode indeed recovers the correct cellular length distribution. For the exponential growth scenario, the simulation reveals the cellular length distribution an exponential grower would have if it had inherent noise levels similar to C. glutamicum allowing for a fair comparison. Both simulations start with a single cell and continue for 20 generations, after which the birth lengths of the last generation are binned and plotted. Repeated simulations with different lengths of the starting cell do not show discernable differences.
Exponential growers
For the exponential growers, cells are assumed to elongate according to
The exponential growth rate $\alpha$ is chosen as the slope of the linear fit of $\mathrm{l}\mathrm{n}\left(\frac{{l}_{d}}{{l}_{b}}\right)$ versus $t}_{d$ that intersects the origin, as shown in Main Text Figure 3B. A sizeadditive noise term is indicated by $\zeta (t)$, which will be specified below at the time of division. For a cell with a given birth length $l}_{b$, the target final length $l}_{t$ is determined via a linear fit of $l}_{b$ versus $l}_{d$, as shown in Main Text Figure 3A. The target growth time $t}_{t$ is then given by ${t}_{t}=\frac{1}{\alpha}\mathrm{l}\mathrm{n}\left(\frac{{l}_{t}}{{l}_{b}}\right)$. A time additive noise term $\mathrm{\Delta}t$ is added to $t}_{t$ according to experimentally observed growth time variations (Appendix 10—figure 1D). Additionally, a sizeadditive noise term $\mathrm{\Delta}l$ encodes the division length variation due to $\zeta (t)$, which is also directly obtained from experiment (Appendix 10—figure 1E).
The full expression for the division length $l}_{d$ is then given by
At division, the characteristic Vsnap of C. glutamicum occurs, separating the two daughter cells. During this Vsnap, the length of the daughter cells rapidly increases: the average measured birth length is 0.57 times the average measured division length (2.3 µm and 4.0 µm respectively), instead of the expected ratio of 0.5. To account for this Vsnap effect, we calculate the distribution of added lengths during the Vsnap. We find that the average added length depends on the division length: longer cells add less length during the Vsnap than shorter cells (Appendix 10—figure 1B). To take this length dependence into account, we subdivide the data set into three division length bins, and obtain a distribution of added lengths during the Vsnap for each bin. When a simulated cell divides, an added length during Vsnap is randomly drawn from the distribution corresponding to its division length.
After division, the length asymmetry of the two daughter cells is chosen by drawing a random value from the experimentally observed division asymmetry distribution (Appendix 10—figure 1C) corresponding to the obtained division length. This distribution is found to be narrower for the shortest birth lengths (Appendix 10—figure 1C), thus two distributions are used.
Asymptotically linear growers
For asymptotically linear growers, cells are assumed to elongate according to
which is obtained by inserting Main Text Equation 3 into Main Text Equation 1, integrating and grouping constant terms into $\lambda$ and $\gamma$. An additive noise term $\eta (t)$ is added to this to account for singlecell variability around the inferred average growth trajectory. We assume the cells to have the 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 $\gamma$ can be approximated as being constant in time, simplifying the growth mode to linear growth (Main Text Figure 5A). A timeadditive noise term $\mathrm{\Delta}t$ will then act as sizeadditive noise and can thus be absorbed into one additive noise term $\mathrm{\Delta}l$, 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 Vsnap effect are incorporated in the same way as for the exponential grower simulation.
Data availability
All data generated during this study are included in the manuscript and supporting files.
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Decision letter

Aleksandra M WalczakSenior and Reviewing Editor; École Normale Supérieure, France
Our editorial process produces two outputs: (i) public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.
Acceptance summary:
The paper presents a new and interesting method for inferring the growth dynamics of bacteria from noisy singlecell data and applies this to C. glutamicum. The method which is original, is promising for other systems.
Decision letter after peer review:
Thank you for submitting your article "Singlecell growth inference of Corynebacterium glutamicum reveals asymptotically linear growth" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Aleksandra Walczak as the Senior Editor. The reviewers have opted to remain anonymous.
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
The work is nicely executed and it should be published in eLife. Generally the method is potentially more notable than the result, so strengthening the method's validation is a good idea. There are two main remaining issues (please also see the reviewer comments below and address all of them):
1. The issue of whether seeing nonexponential growth is a notable result in itself as advertised in the paper, especially in light of recent work by Bruggeman's group (Biphasic CellSize and GrowthRate Homeostasis by Single Bacillus subtilis Cells, Current Biology 2020) that shows nonexponential growth for the wellstudied model bacterium B. subtilis. This example of nonexponential growth is not discussed at length in the paper under review.
2. Regarding the analysis method, it would be useful to spell out the central novelty and the method should be validated more appropriately. For example, in Bruggeman's paper cited above they used a different analysis method (growth rate vs. age plot) that seems to work well, and the two methods were not compared or discussed.
Reviewer #1 (Recommendations for the authors):
This is an interesting study, with a strong method, but also nice results for the specific system. Here are a few points where the authors could improve the manuscript:
Main issues:
– Change the text to remove the repeated references to the consensus of exponential growth, this is likely less dominant than claimed here. E.g. abstract, on p. 2 bottom, p.4 line 80 ("universality") etc.
– Add some analysis to test whether linear growth continues into the phase where the first cells have started to divide. I imagine that removing the earliest dividing cells before the elongation rate determination should give the opportunity to analyze the remaining ones for a longer time. This could be done successively, and while the growth rate would be reduced as more and more cells are removed, one should see that growth rate remains constant up to later times.
Reviewer #2 (Recommendations for the authors):
Some comments:
1. The trend of elongation rate at birth seems to be different for different mutants. Figure 5A shows that in C. glutamicum divIVA:divIVAmCherry the elongation rate is higher for cells with longer size at birth. Can the authors comment on this and how this is to be interpreted within their model?
2. Similarly, could the authors explain why the asymptotic linear elongation rate is smaller in C. glutamicum divIVA:divIVAmCherry such cells as compared to WT using the RAG model?
3. In Figure 7, the authors seem to compare the birth length distribution obtained from theory and experiment. However, the fitness advantage or disadvantage of a longertailed distribution is unclear. In my opinion, the section can be removed from the main text for a better reading of the paper.
[Editors' note: What follows are minor comments after acceptance.]
Reviewer #2 (Recommendations for the authors):
The authors have done a good job of addressing the comments and I find the paper adequate for publication in eLife. I would like to raise a potential issue with the agreement of the model and data implicated by Author response image 1 of the response letter: In the biorxiv paper referenced in the updated version (by Kar et al.), a plot of the form of Author response image 1 is shown to be linear (y=x) both for exponential and linear growth. It therefore seems that for the model of growth proposed in the work (exponential crossing to linear) one would also obtain a simple y=x dependence. However Author response image 1 shows significant deviations from such dependence, suggesting that the proposed model might be inconsistent with the data?
https://doi.org/10.7554/eLife.70106.sa1Author response
Essential revisions:
The work is nicely executed and it should be published in eLife. Generally the method is potentially more notable than the result, so strengthening the method's validation is a good idea. There are two main remaining issues (please also see the reviewer comments below and address all of them):
1. The issue of whether seeing nonexponential growth is a notable result in itself as advertised in the paper, especially in light of recent work by Bruggeman's group (Biphasic CellSize and GrowthRate Homeostasis by Single Bacillus subtilis Cells, Current Biology 2020) that shows nonexponential growth for the wellstudied model bacterium B. subtilis. This example of nonexponential growth is not discussed at length in the paper under review.
We thank the reviewers for pointing out these recent developments. Whereas cellcycle averaged measures of growth have revealed exponential single cell growth for a range of bacterial species over the last decade, detailed inspection of growth rate and elongation rate have revealed deviations from exponential growth over the last year. Using this method, such deviations have been found recently in (Nordholt et al., Current Biology 2020), but also in a preprint that appeared on the BioRxiv about a week ago in (To bin or not to bin: analyzing singlecell growth data, Kar et al., BioRxiv, 2021). In light of these results, we have modified our introduction and our discussion of exponential growth. We made the following changes to the Main Text:
Page 2, lines 2324:
“So far, single bacterial cells have been found to grow predominantly exponentially, which implies the need for tight regulation to maintain cell size homeostasis.”
Page 2, lines 4245:
“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(TaheriAraghi et al. 2015; Mir et al. 2011; IyerBiswas et al. 2014; Yu et al. 2017; Godin et al. 2010).”
Page 3, lines 5055:
“Recently, detailed analysis of the mean growth rate throughout the cell cycle revealed deviations from pure exponential growth. For B. subtilis (Nordholt, van Heerden, and Bruggeman 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 superexponential 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.”
Page 27, lines 558560:
“The inferred asymptotically linear growth of C. glutamicum deviates from the predominantly found exponential singlecell bacterial growth, and suggests the presence of novel growth regulatory mechanisms.”
2. Regarding the analysis method, it would be useful to spell out the central novelty and the method should be validated more appropriately. For example, in Bruggeman's paper cited above they used a different analysis method (growth rate vs. age plot) that seems to work well, and the two methods were not compared or discussed.
We have now extended our Discussion section to include a comparison of our method to that of (Nordholt et al., Current Biology 2020), and of (Kar et al., BioRxiv, 2021). The paragraph in which we discuss this has been revised to (page 25, lines 501511):
“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, van Heerden, and Bruggeman 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 binningrelated 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.”
In (Kar et al., bioRxiv, 2021), it was further shown that the test used for exponential growth shown in Figure 3B is unreliable in the presence of noise in the exponential growth rate. We made the following change to discuss this point (page 11, lines 190192):
“Furthermore, it was recently shown that exponentially growing cells can appear nonexponential with this test in the presence of noise in the exponential growth rate (Kar et al. 2021).”
This shortcoming of this exponential growth test does not impact subsequent results, since this test is only used as a motivation to study elongation rates in more detail. In (Kar et al., bioRxiv, 2021), an alternative test is proposed which bins over the $\mathrm{ln}(\frac{{l}_{d}}{{l}_{b}})$ axis rather than the time axis, which is reported to not be subject to this bias in the presence of noise. Performing this test on our data still shows a deviation from exponential growth (Author response image 1).
Reviewer #1 (Recommendations for the authors):
This is an interesting study, with a strong method, but also nice results for the specific system. Here are a few points where the authors could improve the manuscript:
We appreciate these supportive comments and constructive suggestions.
Main issues:
– Change the text to remove the repeated references to the consensus of exponential growth, this is likely less dominant than claimed here. E.g. abstract, on p. 2 bottom, p.4 line 80 ("universality") etc.
As mentioned in our response to Essential Revision 1, we modified our discussion of exponential growth to include these developments in our revised manuscript.
– Add some analysis to test whether linear growth continues into the phase where the first cells have started to divide. I imagine that removing the earliest dividing cells before the elongation rate determination should give the opportunity to analyze the remaining ones for a longer time. This could be done successively, and while the growth rate would be reduced as more and more cells are removed, one should see that growth rate remains constant up to later times.
In the supplement we had included one analysis, in which we computed conditional elongation curves, conditioned on cells that grew for at least 51 minutes. Following the reviewer’s suggestion, we now expanded this analysis to show mean elongation curves for a range of choices for the cutoff time. The results are shown in the new Appendix 2—figure 4. We find the resulting curves to be consistent with linear growth. However, the error on the inferred mean elongation rates increases with higher cutoff times due to fewer cells being included in the analysis.
To provide stronger evidence for the continuation of linear growth until division, we adapted our inference procedure to describe mean elongation curves as a function of the time until division and division length. Using this procedure, we indeed find that linear growth continues until division. We explain this procedure as follows in the Main Text, page 15, lines 289296:
“To further test if the linear growth mode persists until division, we adapt our inference procedure to obtain average elongation curves $L\left(t{t}_{d},\text{}{l}_{d}\right)$ as a function of the time until division $t{t}_{d}$ and division length $l}_{d$. The construction is analogous to that of $L(t,{l}_{b})$ (Appendix 6). Calculating the corresponding elongation rate curves, we find 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 $t{t}_{d}$ exceeds the shortest singlecell total growth time. Hence, for our analysis we only consider elongation rate curves until this point.”
The results are shown in the new inset of Figure 5 and in the new SI Appendix 6.
Reviewer #2 (Recommendations for the authors):
Some comments:
1. The trend of elongation rate at birth seems to be different for different mutants. Figure 5A shows that in C. glutamicum divIVA:divIVAmCherry the elongation rate is higher for cells with longer size at birth. Can the authors comment on this and how this is to be interpreted within their model?
The inferred mean elongation rates indeed suggest a change in the dependence of elongation rate at birth on birth length. To see how this could be interpreted within our model, we first note that at the end of the cell cycle, cells with a larger birth length grow faster on average. As the old poles are inherited by the daughter cells, we would thus expect cells with a larger birth length to have a faster growing old pole. To still have similar elongation rates across birth lengths, this would imply that the new pole is on average less saturated at birth for longer cells, and more saturated at birth for shorter cells.
It is, however, unclear if such differences in birth elongation rates between our strains are indeed present. The error margins on the elongation rates at birth are significant: A spread of mean elongation rates over birth lengths similar to that of the DivIVA labelled cells would still be within the error margins for both the wild type and the ΔrodΔ strains. Because of this, we decided not to discuss this point in our manuscript.
2. Similarly, could the authors explain why the asymptotic linear elongation rate is smaller in C. glutamicum divIVA:divIVAmCherry such cells as compared to WT using the RAG model?
Within the RAG model, there could be several possible explanations for the lowering of the elongation rates in the DivIVA labelled cells compared to wild type:
– An overall decrease of the amount of transglycosylase sites N(t) at the cell tip
– A lowering of the insertion efficiency of the transglycosylases α
– A lowering of the local concentration of lipidII C(t) at the tip.
Without further experiments to elucidate how the DivIVA labelling affects the apical growth mechanism, it is not possible to narrow this down further. Therefore, we decided not to discuss this labelling in the context of the RAG model in the present manuscript. This result is consistent, however, with earlier findings of an overall decrease of the average singlecell elongation rate for DivIVA labelled cells. To emphasize this, we added the following to the Main Text on page 15, lines 285288:
“This likely reflects a disturbance in the interaction between RodA or bifunctional PBPs and the DivIVAmCherry fusion protein, indicating that the DivIVAmCherry fusion is not fully functional. This is consistent with findings we reported earlier (Donovan et al., PLOS One 2013)”
3. In Figure 7, the authors seem to compare the birth length distribution obtained from theory and experiment. However, the fitness advantage or disadvantage of a longertailed distribution is unclear. In my opinion, the section can be removed from the main text for a better reading of the paper.
While it is difficult to determine with certainty which fitness advantages this provides for C. glutamicum, we find it an interesting observation how the growth mode affects this distribution (if all other parameters remain the same). The main point we intended to make with this figure, is that linear growth can act as a cellsize regulation mechanism.
To emphasize this point, we made the following addition in the Main Text on page 24, lines 478481:
“C. glutamicum has a high degree of variation of division symmetry (Appendix 9—figure 1C) and singlecell growth times, but due to the asymptotically linear growth mode, the populationlevel variations in cell size are still relatively small. This indicates that linear growth can act as a regulator for cell size.”
[Editors' note: What follows are minor comments after acceptance.]
Reviewer #2 (Recommendations for the authors):
The authors have done a good job of addressing the comments and I find the paper adequate for publication in eLife. I would like to raise a potential issue with the agreement of the model and data implicated by Author response image 1 of the response letter: In the biorxiv paper referenced in the updated version (by Kar et al.), a plot of the form of Author response image 1 is shown to be linear (y=x) both for exponential and linear growth. It therefore seems that for the model of growth proposed in the work (exponential crossing to linear) one would also obtain a simple y=x dependence. However Author response image 1 shows significant deviations from such dependence, suggesting that the proposed model might be inconsistent with the data?
Indeed in Kar et al, it is shown that a plot of the form of Author response image 1 obeys y=x both for exponential growth and linear growth within an adder model. This, however, does not imply that a model that switches between these growth modes during growth also obeys the same y=x trend. To illustrate this, we performed simulations of cells growing purely exponentially, purely linearly, and switching from exponential to linear growth at a fixed moment in the cell cycle (Author response image 2). We find that purely exponentially growing cells and purely linearly growing cells obey the y=x trend, in agreement with Kar et al. By contrast, simulated cells performing mixed growth deviate from this trend, indicating that mixed models, as we show to be the case for Corynebacterium, in general do not obey the y=x trend.
Article and author information
Author details
Funding
LudwigMaximiliansUniversität München (Graduate Student Stipend)
 Joris JB Messelink
Deutsche Forschungsgemeinschaft (TRR 174 project P06)
 Joris JB Messelink
 Chase P Broedersz
Deutsche Forschungsgemeinschaft (TRR 174 project P05)
 Fabian Meyer
 Marc Bramkamp
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
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.
Senior and Reviewing Editor
 Aleksandra M Walczak, École Normale Supérieure, France
Publication history
 Preprint posted: May 26, 2020 (view preprint)
 Received: May 6, 2021
 Accepted: October 1, 2021
 Accepted Manuscript published: October 4, 2021 (version 1)
 Accepted Manuscript updated: October 6, 2021 (version 2)
 Accepted Manuscript updated: November 3, 2021 (version 3)
 Version of Record published: November 16, 2021 (version 4)
Copyright
© 2021, Messelink et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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