Contrasting mechanisms of growth in two model rod-shaped bacteria

How cells control their shape and size is a long-standing question in cell biology. Many rod-shaped bacteria elongate their sidewalls by the action of cell wall synthesizing machineries that are associated to actin-like MreB cortical patches. However, little is known about how elongation is regulated to enable varied growth rates and sizes. Here we use total internal reflection fluorescence microscopy and single-particle tracking to visualize MreB isoforms, as a proxy for cell wall synthesis, in Bacillus subtilis and Escherichia coli cells growing in different media and during nutrient upshift. We find that these two model organisms appear to use orthogonal strategies to adapt to growth regime variations: B. subtilis regulates MreB patch speed, while E. coli may mainly regulate the production capacity of MreB-associated cell wall machineries. We present numerical models that link MreB-mediated sidewall synthesis and cell elongation, and argue that the distinct regulatory mechanism employed might reflect the different cell wall integrity constraints in Gram-positive and Gram-negative bacteria.

We now introduce ( ) as the length of the cell cylinder where PG synthesis is active. Since the poles have a hemispherical shape then ( ) and ( ) are related by this simple relation: Assuming that is small enough, we can then deduce from expression (*) the following differential equation: where ( )/ denotes the time derivative of the length of the cell cylinder at the time . We found that for both B. subtilis and E. coli, remains constant for a given growth rate, therefore: Expression (**) can then be rewritten as = × × × ( ). We then deduce that for every in [0, ] we have ( ) = 0 exp( × × × ) where 0 is the length of the cell cylinder at birth. By definition, for =  we have ( ) = 2 0 and then × × × = ln (2) which can be rewritten as × × = where is the growth rate. Consequently, for a given growth rate we have: 

= ln 2 × ×
where all the constants depend possibly on the growth rate  Using the two methods described above, we can estimate the width of the PG band inserted per patch, in nm), for each MreB fusion and growth medium (See Supplementary Table   S9).

Supplementary Note 2. Mechanistic models of cell wall synthesis in bacteria
Here, we propose a mechanistic model that describes PG synthesis in Gram-positive bacteria.
The complexity of the multilayered cell wall of Gram-positive bacteria has, to our knowledge, prevented attempts in building mathematical or even mechanistic models so far. The mechanistic model that we propose is consistent with published observations and would allow to maintain a sustainable, thick PG meshwork in Gram-positive bacteria. For Gramnegative bacteria, we have retained the prevailing model in the field (as justified below).
Assumptions of PG structure and insertion shared by both B. subtilis and E. coli.
The detailed ultrastructure of the sacculus and the precise biochemical reaction(s) coupled to MreB-associated PGEMs are not yet established 2,3 . Here, we will assume that parallel circumferential glycan strands cross-linked by peptide bridges run perpendicular to the long axis of the cell 4 , which is the prevailing model of PG organization ( Supplementary Fig. 15a).
We will also assume that E. coli has a monolayered PG (even though its thickness is compatible with 1 or 2 layers) and that B. subtilis has several concentric layers of PG 2 .
Finally, and based on the early work of Pooley, Mobley, Koch, Doyle and others, we will presuppose an "inside-to-outside" mode of growth for the PG of B. subtilis, where new layers of PG are added to the innermost face of the sacculus, pushing outwards the previous layers, which will be eventually degraded, accounting for the observed PG turnover [5][6][7][8][9] .
Potential mechanism for growth of the monolayered PG meshwork of E. coli.
In E. coli, several possibilities have been envisioned along the years to explain how new glycan strands are inserted in the PG layer: 1-the "cut and insertion" strategy from Burman and Park 10,11 , where cleavage of the peptide bonds between 2 glycan strands are coordinated with insertion of one or two new strands ( Supplementary Fig. 15b, left panel); 2-the "3-for-1" mechanism 11 , which hypothesizes that three new glycan strands are co-inserted and replace one pre-existing strand, which is degraded ( Fig. 6b and Supplementary Fig. 15b, right panel). Insertion of a single strand is not favored because its stem peptides could not face properly the stem peptides of the two adjacent strands, introducing stress on the cross-links.
This was recently confirmed in the comprehensive study by Nguyen and co-workers in which a mathematical model describing PG synthesis in E. coli was generated 12 . Simultaneous insertion of 2 glycan strands should theoretically suppress this issue, as it does the "3-for-1" model (where there is a net insertion of 2 strands). This prediction was confirmed, and both modes of growth were shown to be compatible with robust cell wall increase in their model 12 .
However, all "cut and insertion" strategies failed to explain the high rate of PG recycling per generation in E. coli, while the "3-for-1" model imposes that 50 % of the glycan strands are released during one cell cycle. We will therefore consider the "3-for-1" model as the most compelling in E. coli.
Potential mechanism for growth of the multilayered PG meshwork of B. subtilis.
The "3-under-2" model. In B. subtilis, insertion strategies such as the "3-for-1" mechanism do not apply because of the assumptions that the PG meshwork is multilayered and that growth occurs in a push-up, inside-to-outside way. Insertion of glycan chains in a pre-existing layer would expand it laterally and quickly "dilute" potential upper layers, irrevocably leading to a monolayered PG and excluding an inside-to-outside mode of growth. Our two original assumptions impose a mechanism where at least one complete new layer of PG is generated per cell cycle, using the innermost pre-existing PG layer as scaffold. If a single PG layer is added per cell cycle and the cell has to double its length, then the simplest model is that newly inserted PG layers will have twice as many glycan strands than their template layer. In this scenario, every other glycan strand of the new layer will be connected to the upper (older) layer, in a "3-under-2" geometry ( Fig. 6b and Supplementary Fig. 15c).
Expansion through hydrolysis of lateral cross-links. Addition of new PG layers will not allow expansion without a somewhat synchronized cleavage of peptide cross-links on the upper layers. Such cleavage will transfer the tension created by the osmotic pressure onto the newer, innermost layer(s) (containing twice as much strands and cross-links), thereby allowing local lateral expansion ( Supplementary Fig. 15c). In our "3-under-2" model, we make the hypothesis that the stress-bearing layer is the innermost PG layer ( Supplementary   Fig. 15c, left panel), but it is also conceivable that an upper layer (or several layers) bears the stress ( Supplementary Fig. 15c, right panel). In this situation, maximum expansion of the cell would be constrained by the uppermost stress-bearing layer, and the lower stress-bearing layer(s) could act as a safety net in case of potential injuries in the meshwork. They would also constitute a significant amount of material: if for example the 3 rd layer bears the stress by being at its maximal extension, the 2 nd layer would represent twice the surface of the cell and the 1 st layer would compact 4 times the material needed to entirely cover the cell. In addition to obvious crowding issues (can a cell compact cross-linked glycan strands at 1.25 nm or even 2.5 nm distance?), there would also be an energetic cost for producing so many "unused" glycan strands. We will therefore consider the most simple and thrifty case where only the innermost layer is stress-bearing ( Supplementary Fig. 15c, left panel).
Mechanistic consequences of the "3-under-2" model on PGEMs and PG assembly.
The "3-for-1" model of PG insertion in Gram-negative bacteria is mechanistically simple.
Three new glycan strands replace an old one. Whether transglycosylation occurs before transpeptidation or simultaneously would not change the mechanism 12 . However the "3under-2" model for Gram-positive bacteria raises mechanistic questions regarding how the innermost new layer is connected to the scaffold, and this has important implications for the structure of the cell wall. Because three strands constitute the minimal geometrical unit connected to the upper layer, it is tempting to imagine that, like in the "3-for-1" model, there is co-synthesis or at least attachment of three glycan strands together (i.e. as a triplet) to the meshwork. A direct consequence of the 3-under-2 geometry is that addition of one triplet would enlarge the cell by one crosslink length only ( Supplementary Fig. 15c, see also Fig.   6b), and that a cell should replace every single crosslink in order to double its length. This differs from the 3-for-1 mechanism, where only half the crosslinks need to be replaced per generation. Another consequence of this mechanism is that bundles of glycan strands could be co-inserted. While it is hard to imagine insertion/replacement of more than one triplet of glycan strands in the "3-for-1" model due to the local crowding, attachment to the upper layer could allow larger bundles of glycan strands to be inserted simultaneously in the "3-under-2" model.

Consequences of the mechanistic and mathematic models on the PGEMs
Our numerical data allowed to calculate the net (apparent) width of PG inserted per turn of MreB patch, [, which derives from the net elongation of the sidewall (without any assumption on the mechanism driving this elongation). Based on the mechanistic models of PG insertion described above, it is then possible to speculate on the theoretical number of glycan strands inserted per MreB patch. In E. coli, if the "3-for-1" model applies, one old glycan strand is replaced by three new glycan strands. While the net increase per insertion event would be two peptide crosslinks, the actual synthesis would be 1.5 x larger (three crosslinks). In this scenario, the number of glycan strands inserted should be 1.5 x bigger than the number deduced from the apparent width of the PG band inserted. A similar reasoning applies for B. subtilis. If we make the hypothesis that synthesis follows a "3-under-2 model", cells are producing per cell cycle twice the surface present at the beginning of the cycle. In other words, locally the actual PG produced must be twice the quantity deduced from the apparent extension.
Based on these models, the amount of PG inserted per MreB patch falls into a narrow window of 6 to 10 glycan strands for both organisms (See Supplementary Table 10

Construction of a mutant strain (RCL78) inactivated for mbl
We noticed that the published B. subtilis strain 4261, a widely used mbl mutant 13

Construction of strains with mreBH fused to a SPA tag
Since anti-MreBH antibodies are not available, we created an in situ translational fusion between the 3" end of mreBH and a SPA sequence (sequential peptide affinity) 15 for detection by Western Blotting using anti-flag antibodies. A 426bp fragment corresponding to the 3" end of mreBH (devoid of its stop codon) was PCR-amplified using Phusion polymerase and primers AC938/933, and cloned into the plasmid pMUTINspa 16 , using the HindIII and NcoI restriction sites. The resulting plasmid pAC616 was transformed into B. subtilis 168 and Campbell-like insertions into the mreBH locus were selected by resistance to erythromycin.
The resulting strain ABS1324, was grown in the presence of IPTG to insure expression of the gene downstream of mreBH. Chromosomal DNA of strain ABS1324 was subsequently transformed into strains NC103, 2521, 3725 and RCL78 (see Supplementary Table 5).

Luciferase assays
Luciferase experiments were carried out as previously described 17  Multilabel Reader while luminescence and OD 600 were recorded at 5 min intervals.

Kymograph visualization of mobile and non-mobile patches
Kymograph analysis of MreB patches was performed as described previously 18

Automatic classification of patch movement based on mean-squared displacement (MSD) analysis
Automatic classification of patch movement as directed motion, random diffusion or constrained diffusion requires MSD analysis. Random walks and directed movements (in 2D) were simulated in MATLAB using parameters identical to those used in our experimental conditions (lag time=1s, trace duration = 15s). First, directed trajectories were generated ( = 1 000, speed = 50 nm s -1 ) (typical traces are shown in Supplementary Fig. 8a).
These and appeared flat as compared to averaged MSD curves obtained with directed motion and Brownian diffusion. We concluded that constrained patches could be easily separated from mobile patches considering the maximal value of the MSD curve ( < 0.05 µm 2 s -1 ).
In summary, tracked patches were classified without any supervision based on the value of the MSD curve and its fitting ( Supplementary Fig. 8i). Patches with MSD maximum < 0.05 µm 2 s -1 were classified as constrained. Otherwise, the behavior of patch movement was classified as directed motion if 2 ≥ 0.8 and 2 > 2 , as random diffusion if 2 ≥ 0.8 and 2 > 2 and unclassified if 2 < 0.8 and 2 < 0.8.

Analysis of MreB trajectories using cumulative distribution functions (CDFs)
Cumulative distribution functions (CDFs) of displacements is an alternative standard approach to obtain apparent diffusion coefficients by fitting them to an exponential function corresponding to 2D Brownian motion 19,20 . Using trajectories obtained with u-track and classified as diffusive by MSD analysis, frame-to-frame displacements ( ) were calculated and pooled together for each movie. The CDFs of the displacement magnitudes was fitted using an analytical function that takes into account a two-state model: where ( , ) is the cumulative probability of a displacement of magnitude given the observation period =1s, diffusion coefficients are 1 and 2 , and the relative fraction between the two states is 1 . Finally, the apparent diffusion coefficients measured for each cell were averaged. All calculations were performed on MATLAB (Mathworks, R2014b), using a nonlinear least-square algorithm.

Determination of the minimum width and the average distance between patches
In our analysis based on u-Track, comet detection algorithm used for patch identifications could not provide measurements of positioning errors and patch widths, mostly because this method relies on intensity thresholding and watershed. To circumvent this, patches detected using u-track were individually fitted to measure their lateral dimensions and to estimate the accuracy of centroid localization using a 2D anisotropic Gaussian: To determine the spacing between patches, the distance between the centroids of closest neighboring patches was measured ( > 1 000 per condition).

Measurement of cell dimensions (for Supplementary Tables 1 and 2)
Phase contrast images were acquired on exponentially growing cells (OD 600 ~ 0.3). Intensity profiles were obtained from lines manually drawn across the long and short axis of cells.
Drastic intensity changes appeared where lines crossed the cell contour. The median value of the maximum and minimum intensity was used to produce two intersecting points. The distance between these two points yields cell length and width ( Supplementary Fig. 12). All image processing was performed using MATLAB (Mathworks, R2014b).

Image acquisition and processing for microfluidics experiments
In order to follow B. subtilis and E. coli cells during media switch, we used a microfluidics flow chamber (CellASIC, EMD Millipore) and B04A microfluidics plates. B. subtilis cells carrying GFP-Mbl and E. coli cells carrying MreB-msfGFP SW were first grown in S medium in shaking plates into early exponential phase (OD 600 ~ 0.1), introduced into microfluidics plates filled with S medium, and then allowed to adapt for 60 minutes before LB was injected to replace S medium. Cell loading was achieved under pressure (5 psi) for 6 seconds, and constant media flow was maintained at 0.5 psi, which corresponds to roughly a flow rate of 10 µl h -1 . All experiments were carried out at 37 °C. Bright field (snapshot) and TIRFM movies (100 ms exposure every second for 1 minute) were acquired 5 minutes before, right after (T=0") and every 5 minutes after the nutrient upshift. All TIRFM/SPT analyses were performed as described for other acquisitions. To calculate cell parameters (width, length, area and growth rate), we first obtained binary images excluding background noises, by applying background subtraction (ball radius = 50 pixels = 3.2 µm) and intensity threshold (based on Otsu's method) (Supplementary Fig. 13a). Next, cells were individualized using watershed segmentation while objects others than bacterial cells, such as square pillars in CellASIC bacterial plates, were removed manually. Cell dimensions (length and width) were measured by an ellipse fitting. Cell area was calculated as the number of pixels within the cell contour multiplied by pixel area (Supplementary Fig. 13a). Single-cell growth rates were then TIRF imaging acquisition parameters and SPT analysis were as for cells under steady-state growth. Image processing was done using Fiji 21 .

Bootstrap analysis
The bootstrap allows to create many new sets of data from the original dataset by sampling (with replacement) and leads to an empirical distribution of possible mean values, allowing to estimate a confidence interval 22,23 . Bootstrap samples were generated as follows: (i) all tracks confidence intervals were calculated (Supplementary Table 3). All calculations were performed on MATLAB (Mathworks, R2014b).

Statistical analysis
All statistical tests were performed using GraphPad Prism 6.05 and a non-parametric statistical test (two-tailed Mann-Whitney test with an alpha level of 5%).       The density of diffused patches is not significantly (n.s.) different between rich and poor media for all three fluorescent fusions. However, the density of constrained patches is highly increased in poor medium relative to rich medium in E. coli. Note that the density of constrained patches also significantly (but to a much lesser extent) increases in poor medium in B. subtilis. This weak variation is likely to result from a better assignment of 'unclassi ed' patches due to slower motion at a lower growth rate. In red are plotted averages and standard deviations. Data are a compilation of at least 2 independent experiments. Plotted averages and standard deviations are shown in red. Distributions in LB and S media were compared using the Mann-Whitney non parametric statistical test (****, p<0.0001; ***, 0.0001<p<0.001; **, 0.001<p<0.01; *, 0.01<p<0.05; n.s., p>0.05).  Zoomed-in are views from inside the cells, but cytoplasmic membranes have been partially removed for clarity. In the Gram-positive bacterium, an activated complex (blue dot) inserts new PG strands below the stress-bearing PG layer. In the Gram-negative bacterium, a deformation of the membrane drives the localization of the active PGEM that inserts new PG bands between existing strands in a "3for-1" mechanism (see Supplementary Fig. 15). PG, Peptidoglycan; d, discontinuous PG layers; s, stress-bearing PG layer; n, new uncompleted innermost PG layer; o. mb, outer membrane. Strategies of glycan strand insertion in a 2D (x, y) peptidoglycan layer. In a "cut-and-insertion" strategy (left), peptide bridges are cleaved, allowing new strands to be inserted. Odd numbers of strands (e.g. single strand insertion) impose a stress on the cross-links due to un-alignment (dotted lines) of the stem peptides, while even numbers (e.g. double strand insertion) do not. In the "3-for-1" insertion, a triplet of glycan strands is replacing an existing strand of the meshwork. The net increase in strand number being even, no stress is introduced. Each insertion involves the recycling of old material representing after one cell cycle (one doubling) 50% of the initial peptidoglycan mass, and requires producing 1.5-fold this initial mass. Red lines represent peptide bridges, brown tubes for GluNac-MurNac sugar chains. (c) "3-under-2" glycan strand insertion modes in a multilayered peptidoglycan meshwork in 2D side view (x, z). Two possibilities are explored in which stress either is borne by the innermost (newest) layer only (left), or by the penultimate (n -1 ) layer (right). Note that the number of glycan chains inserted per fully extended crosslink is two-fold bigger when stress is borne on layer n -1 . Arbitrarily, only five layers of PG are presented (numbered n 0 to n -4 from the innermost to the outermost). It is assumed that bundles of 3 to 9 glycan strands are inserted. Peptide bridges are shown in gray, in a relaxed conformation (broken lines), in full extension (straight lines) or hydrolyzed (dashed lines). Circles indicate cross-sections of glycan chains newly (blue) or previously inserted (orange).

Supplementary Table 3. Influence of the minimal number of steps used for tracking and MSD analysis
Step # 2.0 (*) Estimated peptidoglycan unit length = 4.5 nm (estimated width of a stretched sugar strand plus a peptide cross-link).