Stress-mediated growth determines Escherichia coli division site morphogenesis

Significance

The bacterial wall is composed of peptidoglycan (PG) and distinguishes bacteria from eukaryotes. Hence, PG synthesis and remodeling are of special interest for the development of antibiotics. Bacterial division represents the most challenging PG remodeling process in the bacterial cell, and its detailed mechanics is not well understood. We propose a simple mechanochemical model that quantitatively reproduces the morphogenesis of the Escherichia coli division site. Its predictions depend on two adjustable parameters associated with the PG recrosslinking dynamics. Changing these parameters also recovers the morphology of divisome mutants, without further adjustments to the model. Moreover, the modifications reflect the expected changes in a rational manner. This indicates that the model captures key aspects of PG remodeling during bacterial division.

Abstract

In order to proliferate, bacteria must remodel their cell wall at the division site. The division process is driven by the enzymatic activity of peptidoglycan synthases and hydrolases around the constricting Z-ring. We introduce a morphoelastic model that correctly reproduces the shape of the division site during the constriction and septation phases of Escherichia coli. In the model, mechanical stress directs the transformation of the bacterial wall. The two constants associated with growth and remodeling respectively are its only adjustable parameters. Different morphologies, corresponding either to mutant or wild type cells, are recovered as a function of the remodeling parameter. In addition, a plausible range for the cell stiffness and turgor pressure was determined by comparing numerical simulations with bacterial cell plasmolysis data.

Bacteria are single-cell organisms that multiply by binary fission. Unlike most eukaryotic cells, bacteria are enveloped by multiple layers. The hardest of these layers, protecting the cell from osmotic lysis, is the bacterial wall formed of peptidoglycan (PG), a peptide-crosslinked glycan polymer. In contrast to cell membranes, the bacterial wall cannot be divided by purely mechanical forces generated by protein assemblies. Instead, bacterial division requires the enzymatic activity of specific PG synthases and hydrolases, as parts of the so-called divisome. Hence, inhibition of the divisome is a major target of antibiotics (1). In recent years our understanding of bacterial growth and division has been enhanced greatly (2–6): bacterial genetics studies have identified the determining factors for cell wall synthesis, homeostasis, and division. While the detailed mechanisms of how the different proteins in the divisome mediate cell division remain not fully understood, cryoelectron tomography (cryo-ET) was able to furnish the ultrastructure of the division site of Escherichia coli as a function of PG synthesis and hydrolysis levels, thereby providing a basis for quantitative analysis of its division mechanism (7).

As of yet, there has not been any successful model of the division mechanism that faithfully reproduces division site ultrastructure and morphogenesis. Moreover, for such a model to be useful for interpreting and predicting the behavior of the cell, it should map genetic modifications to observed morphology via a transparent mechanism. We are not aware of any published attempts to meet the latter challenge. The model should also be physical and parsimonious, which excludes purely phenomenological approaches. To wit, older models often resulted in reasonable morphologies of the bacterial surface, but the constriction dynamics evolved independently of any cell process; moreover, PG was described as thin shell and so, structural details, such as septal wedge formation, could not be modeled (8). Recent coarse-grained particle-based models (9, 10) have introduced a stylized mechanistic understanding, but also lack realistic material properties and are unable to describe the formation and structure of bacterial septa.

E. coli growth and division are responsive to mechanical stimuli, suggesting mechanochemical coupling of the growth apparatus (11), a mechanism well established in tissue growth (12, 13). Models of stress-mediated growth are often formulated within the mathematical framework of morphoelasticity (14, 15), which combines large-strain elasticity with the possibility of a permanent change of volume and shape. Developing a morphoelastic model of bacterial division of E. coli enhances our understanding of the mechanobiology of the bacterial cell envelope: it yields a microscopic and quantitative description of the forces acting at each stage of cell division and the way they determine the ultrastructure of the division site.

Results and Discussion

Mechanical Properties of PG and Turgor Pressure.

There is neither a consensus on the mechanical properties of the bacterial wall, nor on the magnitude of the turgor pressure. The cell wall consists of PG, made of up to $\sim$200 nm long glycan strands (16), $\sim$35 nm on average (17), crosslinked by short peptides (Fig. 1A). In E. coli, PG is mostly single layered with a thickness between $\sim$2.5 and $\sim$6 nm (16–20). The glycan strands are disordered and partially overlap (16). Alignment of glycan strands along the circumferential direction of the bacterium has been widely assumed (9, 21). The anisotropy data by Turner et al. (16) confirm a preferential orientation, despite a large variance (Fig. 1B). Accordingly, the mechanical response of PG is partially anisotropic, depending only on the preferential strand orientation (20, 21). In other words, the material is transversally isotropic, with two Young’s moduli $E_g$, $E_p$, and one Poisson ratio ${\nu }_{\mathit{gp}}$; see Fig. 1C.

Fig. 1.

PG structure and elastic properties. (A) Schematic of PG crosslinked structure. (B) Prevalent fiber orientation. From ref. 16 under CC 4.0 (https://creativecommons.org/licenses/by/4.0/). (C) Three elastic moduli, associated deformations of a transversely isotropic bulk material. (D) Mean strain ${\epsilon }_m$ observed in numerical simulations shown in a log–log plot of the turgor pressure $P_{\mathit{tur}}$ and the cell’s stiffness $E_h$. A $10\%$ interval around the experimental value (${\epsilon }_m\approx 0.1$) is indicated by blue lines. Linear relation traced by dashed black line. Experimentally measured values of either $E_h$ or $P_{\mathit{tur}}$ marked by lines, simultaneously measured values by crosses. (E) Strain difference ${\epsilon }_d$ observed in numerical simulations, shown in a log–log plot of axial and circumferential Young’s modulus $E_p$ and $E_g$. A $100\%$ interval around the experimental value (${\epsilon }_d\approx 0.01$) is marked by blue lines. Correlation depicted by dashed black line. Experimentally measured values marked by crosses.

The disordered long-range structure of PG complicates calculations of its elastic parameters in atomistic simulations (21). Direct experiments on living cells are difficult to interpret due to several active and passive regulatory mechanisms (22). The estimates of the elastic moduli of PG range widely from $5\times {10}^{-2}$ to $2\times {10}^2$ MPa; even atomic force microscopy (AFM) measurements vary by two orders of magnitude (23, 24). Examining sacculi of dead cells increases control over the experimental conditions, but differences in sample preparation strongly affect results, for example by sensitivity to hydration (20).

To determine the magnitude of the turgor pressure, several techniques have been used: collapse of gas vesicles (in other Gram-negative bacteria) (25, 26), AFM indentation (27, 28), and estimation of the total chemical content of the cytoplasm (29). The predicted pressure values vary by more than an order of magnitude, from $0.1$ to $3$ bar. AFM indentation cannot separate the internal pressure from the wall’s elastic contributions and hence provides only an upper bound on the turgor pressure. A lower bound is provided by a study on mutated E. coli (30), where the bacteria have a weakened wall, which may have led them to reduce their internal pressure.

In order to obtain a consistent estimate of PG elasticity and turgor pressure, we took data from an independent plasmolysis experiment (31), that provided longitudinal strain ${\epsilon }_l$ and the radial strain ${\epsilon }_w$ relative to the unpressurized (plasmolyzed) state (Materials and Methods). Then, for varying elastic moduli and turgor pressures, we computed the deformation of the cell wall from a stress-free reference configuration and compared its strain with the experimental values; see Fig. 1F). The mean strain ${\epsilon }_m:=\frac{1}{2}({\epsilon }_l+{\epsilon }_w)$ measures the magnitude of the deformation and is determined by the harmonic mean $E_h=2E_g{E}_p{(E_g+E_p)}^{-1}$ of Young’s moduli that defines the material’s stiffness. The strain difference ${\epsilon }_d:={\epsilon }_l-{\epsilon }_w$ measures the anisotropy of the deformation and is related to the ratio $E_g{E}_p^{-1}$ of Young’s moduli, which describes material anisotropy.

In Fig. 1D, we show computed values of ${\epsilon }_m$ in a log–log plot of the stiffness $E_h$ against the turgor pressure $P_{\mathit{tur}}$, and in Fig. 1E, we show numerically computed values of ${\epsilon }_d$ in a log–log plot of the Young’s moduli $E_h$ and $E_p$. Values reported in the literature are included in the plots: an early review by Beveridge (32) estimates the turgor pressure in the range 3 to 5 atm, referring, e.g., to nephelometric experiments on bacteria with gas vesicles (25). Later atomic force microscopy (AFM) measurements by Yao et al. (28) suggest values 0.1 to 0.2 atm (for the related Gram-negative bacterium Pseudomonas aeruginosa). Elastic moduli were measured via AFM by Yao et al. (20) on sacculi of dead cells and by Eaton et al. (33) on live bacteria. MD simulations of a PG matrix segments were performed by Gumbart et al. (21). An AFM experiment determining separately turgor pressure and elastic moduli was performed by Deng et al. (30).

The comparison in Fig. 1D reveals that the experimentally measured value of ${\epsilon }_m$ determines the ratio of the turgor pressure $P_{\mathit{tur}}$ and the material stiffness $E_h$. For example, a too high $P_{\mathit{tur}}$ at a given $E_h$ will lead to excessive strain values incompatible with experiment. In the same manner, ${\epsilon }_d$ determines the ratio of $E_g$ and $E_p$. The Poisson ratio was measured to be ${\nu }_{\mathit{gp}}=0.48$ in ref. 20, which is up to our knowledge the only available experiment. This value of ${\nu }_{\mathit{gp}}$ is also comparable to MD calculations (21).

Since our model departs from a stress-free state (i.e., unpressurized sacculus to which the turgor pressure is then applied), we take elastic moduli from ref. 20, the only AFM experiment where all three elastic moduli where measured on unpressurized sacculi. We hence have $E_g\approx 42$ MPa, $E_p\approx 23$ MPa, and ${\nu }_{\mathit{gp}}=0.48$. It is noteworthy the ratio $E_g/E_p\approx 1.8$ yields strain difference ${\epsilon }_d$ that is compatible with the plasmolysis experiment (c.f., Fig. 1E), ensuring consistency. The compatible turgor pressure determined by Fig. 1D is then $P_{\mathit{tur}}=0.6$ atm $=6\times {10}^{-2}$ MPa, a realistic value lying within the range reported in literature. Although there have been reports of a stress-stiffening elastic behavior of PG (30, 34), the elastic moduli of pressurized sacculus under $\sim$0.3 atm were measured to be $E_g\approx 49$ MPa and $E_p\approx 23$ MPa, values almost identical to those at zero turgor pressure, confirming our estimate.

Cell Division by Stress-Driven PG Growth and Remodeling.

The divisome directs PG synthases and hydrolases, which build and remodel the PG matrix at the division site. Its key components are 1) the Z-ring (formed by the tubulin-like FtsZ protein), 2) the PG synthesis complexes (FtsW and FtsI synthase (35) regulated by the FtsQLB and FtsN complexes), and 3) the PG hydrolyzing amidases (activated by the FtsEX–EnvC complex) (36). PG precursors are synthesized in the cytoplasm and then flipped across the inner membrane (IM) to face the periplasm where they are subsequently incorporated into the PG matrix by the PG synthase FtsWI. A portion of already existing PG is modified by amidases that remove the peptide crosslinks from the glycan chains.

Z-ring force.

The Z-ring is composed of tubulin-like FtsZ filaments [app. 100 to 200 nm in length (37)] which recruit the PG synthesizing/cleaving apparatus to constrict the cell. Each FtsZ monomer is about 4 nm long. Upon GTP hydrolysis, it bends and contributes 20 to 30 pN to the constriction force, as estimated from MD simulation (10). The number of monomers is believed to lie between $5,000$ and $7,000$ in the whole E. coli cell from which 30 to 40% is incorporated into the Z-ring and the rest is cytoplasmatic (38). The $\sim$2,100 FtsZ molecules in the Z-ring could form a total protofilament $8,400$ nm long, which would encircle a 1-$\mu$m-diameter cell two and a half times. Hence, we expect the Z-ring constriction force line density in units to a few tens of $\text{pN}·{\text{nm}}^{-1}$. The Z-ring width stays approximately at $84\pm 2$ nm during the whole division (37).

Stress-mediated growth.

Building a useful mechanochemical model requires removing unnecessary degrees of freedom. We use a morphoelastic model of PG growth and remodeling within continuum mechanics with constant mechanical properties, including an effective circumferential orientation of the fibers. The phenomenology of remodeling and growth emerges from enzyme kinetics, protein activity, and diffusion happening at the molecular scale (SI Appendix).

Since the elastic relaxation time of the cell wall is $\sim$1 s (39) and the division time is of the order of $\sim$10³ s in the laboratory under favorable conditions (7), neither inertial nor viscous effects are important and a quasistatic approximation is appropriate.

PG growth and insertion.

The growth of PG via precursor insertion leads to local volume increase, local volume decrease is associated with PG recycling and hydrolysis. The global rate of net septal PG volume increase $\mathrm{\Omega }$ was measured to be $\approx$${\text{10}}^3{\text{nm}}^3·{\text{s}}^{-1}$ (7). As the cell wall thickness is on the order of nm, the area increase is ca. $\sim$${\text{10}}^3{\text{nm}}^2·{\text{s}}^{-1}$. As this is much slower than the diffusion coefficient of periplasmic proteins $\sim 3·{10}^6{\text{nm}}^2·{\mathrm{s}}^{-1}$ (40), the precursors diffuse in the porous environment to an equilibrium distribution; consistent with the quasistatic approximation.

The porosity of the PG medium arises as it is de-crosslinked by hydrolases. Hydrostatic pressure can then pull some of the strands apart, enlarging the gap between molecules. This in turn eases the entry of PG precursors into the area. Growth is then accomplished by PG synthases, prominently FtsWI, integrating the new material into the matrix (Fig. 2D). The growth rate $\gamma$ depends on the pressure $P$ (the hydrostatic pressure $p$ which has been rescaled to account for the elastic deformation) through the resistance modulus $\kappa$, which determines the sensitivity of the growth rate $\gamma$ to the pressure difference $\mathrm{\Delta }P$; c.f., Fig. 2B. For further details, see the SI Appendix. Since the volume rate $\mathrm{\Omega }$ is a fixed parameter of the model, the mean value of the growth rate $\gamma$ is fixed via the Lagrange multiplier $\omega$. The location of the divisome (here called active zone) is expressed in the model by the localizing function $\zeta$ (similar to a Gaussian distribution), which depends on the distance $a$ from the middle of the bacterium. We associate the resistance $\kappa$ with the presence and concentration of glycosyltransferase and transpeptidase enzymes.

Fig. 2.

Stress-mediated remodeling and growth during the division of a wild type E. coli and its comparison with experiment (7). (A) Layout of the sacculus with depicted parameters and pressures. (B) Influence of the pressure profile on the resulting growth rate. (C) Panels from top to bottom: First: Detail of the active zone with a distribution of the shear stress (Left) and the normal radial stress (Right) during the constriction phase (Top) and the septation phase (Bottom). Second: Detail of the active zone with a distribution of the pressure (Left) and the growth rate (Right) during the constriction phase (Top) and the septation phase (Bottom). Growth rate is a dimensionless quantity. Third: Detail of the active zone with a distribution of the relative density (Left) and energy density (Right) during the constriction phase (Top) and the septation phase (Bottom). Relative density, as a ratio of the current over the one at relaxed, unstressed state, is dimensionless. (D) Schematic of the effective remodeling. (E) Schematic of the effective growth. (F) Radii of the IM in green and the outer membrane (OM) in purple at the division site. Several stages of the cell division shown below the curves. Transition between the constriction and septation phase is set to the moment when IM narrows two times faster than OM. (G) Curves from the plot in (F) overlaid with experimental kymographs, measured via fluorescently labeled membranes at the division site (7), with depicted radial (xy) and temporal (t) directions. (H) Detailed view of the division site predicted by the model at selected times; OM in purple, PG cut highlighted in blue. (I) PG cut in blue as predicted by the model overlaid with cryoelectron tomography data (7).

PG remodeling.

The solid PG structure is locally disrupted by hydrolysis of some of the peptide crosslinks. The shear stress carried by them is then redistributed to the rest of the matrix as the denuded PG strands slide over each other until a new equilibrium position is reached. The PG chains are finally recrosslinked by transpeptidases, reestablishing a stiff structure. The more enzymes are active, the more denuding occurs and the more the structure will comply with the shear stress, which in turn leads to a faster isochoric remodeling. The PG effectively behaves as a melted polymer composed of reptating chains (41), the remodeling rate $3\times 3$ matrix $\mathrm{\Gamma }$ is driven by the “pressure-free” stress inside the material, the deviatoric part of the Cauchy stress tensor, via the solidity modulus $\eta$. As was the case with the pressure, the stress field has to be transformed in order to account for the elastic deformation; we denote the resulting quantity ${\mathbb{M}}^d$. The letter M represents the Mandel stress tensor, the superscript d indicates the deviatoric part; see SI Appendix. The remodeling is localized within the active zone by function $\zeta$; see Fig. 2D.

Phenomenology of the division process.

At the beginning of the division process, the inward pulling Z-ring force induces two significant stresses: shear stress deflecting the sacculus to the center of the cell, resulting in a bending deformation, and normal stress in the radial direction expanding its thickness; see Fig. 2C. The growth and remodeling reduce the pressure and shear stress within PG by directing material modifications to follow it. Consequently, the cell wall gradually bends inward and thickens from inside, slowly forming a small wedge of new PG. As the division site changes its shape, the shear stress decreases while the normal stress in the radial direction increases. The thicker the PG at the division site gets, the more the normal stress concentrates in the middle of the wedge, as $\zeta$ limits the Z-ring and the active zone onto the forming septum. Hence, instead of bending the whole structure as in the beginning, the Z-ring force rather strains the wedge, which directs the elongation via the PG synthases into the radial direction. This positive feedback loop leads in the end to a rapid closure of the septum.

The growth rate distribution (Fig. 2C) is in the initial stage homogeneous across the thickness, highest in the middle of the cell and decreases smoothly with the arc-length distance from the center of the Z-ring; this reflects the collocation of the growth and remodeling apparatus with FtsZ. Later, approximately when the cell wall triples in thickness, the domain of the maximum growth rate moves from the center in the longitudinal direction. Also the growth rate is higher near the IM and lower at the outer PG boundary; the highest mass increase thus occurs at the neck of the septum.

Roughly speaking, the PG in the active zone behaves as an elastic, almost solid glue which, as being pulled inward, forms a small drop; however, as new PG is being incorporated, no necking occurs, the drop does not separate from the cell wall and, instead, forms a small wedge which later becomes the septum. The septum hence grows rather at the neck from where it is being pulled down by the Z-ring. In particular, the septum does not grow from its tip.

While this finding is in contrast with what is widely assumed in the field, we note that the width of the active zone ($2\delta =90$ nm) agrees with the Z-ring’s width (8, 9) which is necessarily wrapped around parts of the septum. Our model therefore reflects the dimensions and position uncertainty of the growth apparatus.

Quantitative model of wild type cell division.

The predictions of the model and their comparison to cryo-ET and fluorescence microscopy data (7) are shown in Fig. 2F–I; see also Movie S1. An initially slow constriction is followed by a short and accelerating septation (Fig. 2F). This agrees with experimental kymographs obtained by fluorescently labeling inner and outer bacterial membranes (Fig. 2G). Also the final architecture of the division site and the width and thickness of the septum are faithfully recovered, as can be seen by overlay with electron micrographs (Fig. 2 H and I); the full original micrographs are included in SI Appendix; see Materials and Methods for further details. The hydrostatic pressure field $p$ is depicted in the Middle panel of Fig. 2C, relative density (the ratio of the current density over the referential at the relaxed, unpressurized state) together with the energy density are shown in the Bottom panel of Fig. 2C.

Modeling of mutant strains.

The $\mathrm{\Delta }$envC-mutant is defective for one of two amidase activators (42). Reduced amidase activity corresponds in our model to higher solidity $\eta$, because PG remodeling is suppressed; the results are shown in Fig. 3A–D; see also Movie S2. The septum also closes at the late stage of the division, both in the model and experiment (Fig. 3 A and B). However, the constriction deformation is severely impaired, so that the septation is almost vertical, as a natural consequence of reduced PG remodeling that disables the bending-like behavior. The resulting large contact surfaces of the flat septa experimentally lead to a failure of separation of the bacteria and to the formation of filament-like colonies (7). The model also predicts properly the two main differences in the mutant’s division: the wider septum with a proper shape (Fig. 3 C and D) and the longer division time, on average, ca. twice the time of the wild type, also reported earlier (7).

Fig. 3.

Stress mediated remodeling and growth during the division of the $\mathrm{\Delta }$envC-(Top) and ftsN-$\mathrm{\Delta }$SPOR-(Bottom) mutants of E. coli and their comparison with experiment. (A) Radius of the IM in green and the OM in purple at the division site of the $\mathrm{\Delta }$envC-mutant. Several stages of the cell division shown along the curves. (B) Curves from the plot in (A) overlaid with experimentally measured positions of the membranes at the division site of the $\mathrm{\Delta }$envC-mutant (7), with depicted radial (xy) and temporal (t) directions. (C) Detailed view of the division site of the $\mathrm{\Delta }$envC-mutant predicted by the model at a selected time; OM in purple, PG cut highlighted in blue. (D) PG cut in blue as predicted by the model overlaid with cryoelectron tomography images of the $\mathrm{\Delta }$envC-mutant. (E) Radius of the IM in green and the OM in purple at the division site of the ftsN-$\mathrm{\Delta }$SPOR-mutant. Several stages of the cell division shown along the curves. (F) Curves from the plot in (E) overlaid with experimentally measured positions of the membranes at the division site of the ftsN-$\mathrm{\Delta }$SPOR-mutant (7), with depicted radial (xy) and temporal (t) directions. (G) Detailed view of the division site of the ftsN-$\mathrm{\Delta }$SPOR-mutant predicted by the model at a selected time; OM in purple, PG cut highlighted in blue. (H) PG cut in blue as predicted by the model overlaid with cryoelectron tomography images of the ftsN-$\mathrm{\Delta }$SPOR-mutant.

The ftsN-$\mathrm{\Delta }$SPOR-mutant lacks the SPOR domain that recognizes the denuded PG and thereby activates the FtsQLB complex. As this suggests reduced incorporation of PG precursors, we modeled this mutant with reduced volume rate $\mathrm{\Omega }$ (Fig. 3E–I); see also Movie S3. The OM of the ftsN-$\mathrm{\Delta }$SPOR-mutant constricts for most of the division at a similar rate as the IM and the much narrower septum is formed quickly at the latest stage of the process, which is also observed in experiments (7); see Fig. 3E, F, H, and I. However, mere reduction of the volume rate $\mathrm{\Omega }$ is not sufficient as the ftsN-$\mathrm{\Delta }$SPOR-mutant divides slower (7) The three parameters $\eta$, $\kappa$, and $\mathrm{\Omega }$ have to be scaled properly to reproduce the same behavior on a longer time scale; see the next section.

The original micrographs of both mutants are included in SI Appendix; see Materials and Methods for further details on image processing. Note that Cryo-EM images are necessarily of different individual cells, asymmetric due to the experimental method. Considering these limitations, our model agrees with the data.

Sensitivity of the Model to Parameter Variations.

The model’s predictions are stable with respect to the geometric parameters. Varying the length or width of the bacterium in the order of a few tenths of its value leads only to a minor quantitative change of the PG matrix’ elasticity, remodeling, and growth. When reducing the cell wall’s thickness, the elastic moduli have to be increased to maintain the bacterial shape during division, because the elastic response is governed by a product of these.

In order to initiate the constriction, the Z-ring force must overcome the turgor pressure (8, 9). The balance of these forces is described by the constriction number

$$\mathrm{Co}:=\frac{{\mathrm{f}}_{\mathrm{Z}}}{2\delta {\mathrm{P}}_{\mathrm{tur}}}.$$

For $\mathrm{Co}<1$ the cell in the middle expands in radius and its wall bulges, $\mathrm{Co}>1$ causes faster constriction and alters the shape near the septum. The division is driven by three time scales: the first is the characteristic production time, given by the ratio of the active zone volume ${\mathcal{V}}_{\mathrm{a}}$ and the volume rate $\mathrm{\Omega }$

$${\tau }_{\mathrm{p}}:=\frac{{\mathcal{V}}_{\mathrm{a}}}{\mathrm{\Omega }}\approx \frac{2\pi {\mathrm{r}}_{\mathrm{out}}2\delta \mathrm{h}}{\mathrm{\Omega }},$$

where $\mathrm{h}:={\mathrm{r}}_{\mathrm{out}}-{\mathrm{r}}_{\mathrm{in}}$ is the current thickness of the active zone and ${\mathrm{r}}_{\mathrm{out}}$, ${\mathrm{r}}_{\mathrm{in}}$ is the current outer and inner radius respectively, the given approximation is valid in the constriction phase. Every ${\tau }_{\mathrm{p}}$ the PG grows by the volume ${\mathcal{V}}_{\mathrm{a}}$. The second timescale is the characteristic remodeling time, determined by the driving force ${\mathrm{f}}_{\mathrm{Z}}$ and the solidity modulus $\eta$, and the characteristic length $2\delta$, the width of the active zone,

$${\tau }_{\mathrm{r}}:=\frac{2\delta \eta }{{\mathrm{f}}_{\mathrm{Z}}}=\frac{\eta }{{\mathrm{f}}_{\mathrm{Z}}}\frac{\mathrm{\Omega }}{{\mathcal{S}}_{\mathrm{a}}}{\tau }_{\mathrm{p}}=\frac{\eta {\dot{\mathcal{A}}}_{\mathrm{a}}}{{\mathrm{f}}_{\mathrm{Z}}}{\tau }_{\mathrm{p}},$$

where ${\mathcal{S}}_{\mathrm{a}}$ is the area of the active zone’s front and

$${\dot{\mathcal{A}}}_{\mathrm{a}}:=\frac{\mathrm{\Omega }}{{\mathcal{S}}_{\mathrm{a}}}$$

is the arc length rate, i.e., the speed by which the growing PG elongates along the IM. The remodeling time scale expresses how fast the material reshapes under loading. Finally, the growth time is given by the ratio of the “driving” pressure ${\mathrm{P}}_{\mathrm{tur}}$ and the resistance modulus $\kappa$

$${\tau }_{\mathrm{g}}:=\frac{\kappa }{{\mathrm{P}}_{\mathrm{tur}}}=\mathrm{Co}\frac{2\delta }{{\mathrm{f}}_{\mathrm{Z}}}\kappa =\mathrm{Co}\frac{\kappa }{\eta }{\tau }_{\mathrm{r}}.$$

It describes how fast the newly grown PG distribution diverges due to pressure differences within the thin PG cell wall. The three time scales of the division can be related by two dimensionless numbers. The first is an analogue of the Weissenberg number, used in the rheology of viscoelastic fluids (43)

$$\mathrm{Wi}:=\frac{{\tau }_{\mathrm{r}}}{{\tau }_{\mathrm{p}}}=\frac{\eta }{{\mathrm{f}}_{\mathrm{Z}}}\frac{\mathrm{\Omega }}{{\mathcal{S}}_{\mathrm{a}}}=\frac{\eta {\dot{\mathcal{A}}}_{\mathrm{a}}}{{\mathrm{f}}_{\mathrm{Z}}}.$$

It is defined as the ratio of the remodeling and production time scale and equals the ratio of “viscous” friction forces to elastic forces: a virtual remodeling force line density $\eta {\dot{\mathcal{A}}}_{\mathrm{a}}$ (which has the same dimension as a viscous friction force) and the elastic force line density ${\mathrm{f}}_{\mathrm{Z}}$. The second dimensionless number is specific to our model and relates the remodeling and growth process by a simple ratio

$$\mathrm{Su}:=\frac{\eta }{\kappa }=\mathrm{Co}\frac{{\tau }_{\mathrm{r}}}{{\tau }_{\mathrm{g}}};$$

we call $\mathrm{Su}$ the susceptibility number as it is a ratio of the pressure susceptibility ${\kappa }^{-1}$ of the PG deposition and the PG fluidity ${\eta }^{-1}$. As $\mathrm{Wi}$ and $\mathrm{Su}$ fix only the ratio of the three time scales, the total division time can be always set to match experimental data.

In order to investigate the role of $\mathrm{Wi}$ and $\mathrm{Su}$ we report several observables of the division process: the final septum width (relative to the original width of the cell), the mean thickness of the newly grown septal PG (relative to the initial PG thickness), and the ratio of the final septum and the length the newly grown PG measured along the IM (to differentiate constriction from septal division).

All measures are plotted against $\mathrm{Wi}$ and $\mathrm{Su}$ in Fig. 4. The determining parameter is the Weissenberg number, the susceptibility number has only little influence. The variations show the same pattern, for example, thick and wide septa go hand in hand with short constriction phase Fig. 4A–C. All mutants hence lie “on a line,” between the extrema given in Fig. 4 D and K, boundaries between the phenotypes are indicated in Fig. 4A–C. The capability of reproducing several observed phenotypes with one effective parameter that, moreover, has a clear interpretation strongly suggests our model captures the essence of the process.

Fig. 4.

Qualitative analysis of the model’s sensitivity to the Weissenberg ($\mathrm{Wi}$) and susceptibility ($\mathrm{Su}$) number. (A–C) Contour plots showing selected measures and their dependence on $\mathrm{Wi}$ and $\mathrm{Su}$. Points corresponding to the wild type and two mutants marked by red symbols; the three division modes corresponding to them are separated by red isolines. (D–K) Final morphologies of the division site for selected values of $\mathrm{Wi}$ and $\mathrm{Su}$.

Model limitations and outlook.

We have assumed the material keeps its properties constant during the division, meaning the density, stiffness, and anisotropy of PG remain the same. Solidity $\eta$ and resistance $\kappa$ do not vary with the distance from the cytoplasm. The mechanical contribution of the membranes was neglected. Ordinary lipid bilayers can carry only a few % of the load. It has recently been suggested, that the OM is strongly stiffened by cation–sugar interactions of bacterial glycolipids (44). The potential influence of the OM on the division process (45) is beyond the scope of this study. As the model accounts only for continuous deformation, the final cytokinesis phase is out of scope of our current work. We could perform some infinitesimal “surgery” to separate bacteria at the end of the cytokinesis stage and to close off septum formation; however, due to its short duration, there is also very little experimental data on the event of cytokinesis, making validation difficult.

Our model faithfully reproduces the constriction and septation phase of E. coli cell division in a quantitative manner and with a minimal number of arbitrarily adjustable parameters. Moreover, all our model parameters have a clear biophysical interpretation and their variation reproduces phenotypes of mutated bacteria, which are rather limited. Hence, our model captures the essential mechanobiology involved in the constriction and septation phases of cell division. The growth equation is fully coupled with mechanics and coarse-grained, effective enzyme kinetics, it is not prescribed by a separate, artificial evolutionary equation and allows for a variable thickness of the PG. In contrast to other approaches, our model relies on simple assumptions, namely the dependence of chemical potential on stress and a quasi-equilibrium state due to slow growth. Its conceptual simplicity has allowed us to avoid difficult kinetic attribution questions and extensive parameterization. Our results also highlight the commonality and differences to known models of solid and fluid mechanics. A viscoelastic or elastoplastic material will reduce stress and minimize free energy by accommodating external stress. Growth processes can transfer this type of behavior to hard materials by collocation of active remodeling sites and material incorporation. At the same time, growth leads to a different morphology unknown from inert materials, so that, e.g., necking and scission are suppressed in our model. We are optimistic that the insights gained in this way will transfer to, e.g., cocci and perhaps even plant cell division. It remains to be seen whether our model will be able to also treat cytokinesis and the reconstruction of the bacterial poles.

Materials and Methods

Cell Wall Geometry.

According to ref. 31, we set the length and width of an unpressurized sacculus to $L=3$ $\mathrm{\mu }$m and $W:=2R_{\mathit{out}}=1$ $\mathrm{\mu }$m respectively; see Fig. 2A. From the range of experimentally observed values, we chose the thickness of an unpressurized sacculus $\mathrm{H}:={\mathrm{R}}_{\mathrm{out}}-{\mathrm{R}}_{\mathrm{in}}=6\mathrm{nm}$ since it makes our simulations compatible with other studies (20, 30). We exploit the known (approximate) cylindrical symmetry of the rod-like shape of E. coli (which is maintained in the division process) in our model.

Cell Wall Elasticity and Turgor Pressure.

From the plasmolysis experiment (31), to which the Fig. 1 refers, we took values of longitudinal and radial strain measured on E. coli in lysogeny broth (LB), Fig. 1C therein. Since both the values of the strains ${\epsilon }_l$ a ${\epsilon }_w$ lie around $0.1$, the material response is close to a linear regime. As explained in the main text we take $E_g\approx 42$ MPa, $E_p\approx 23$ MPa, ${\nu }_{\mathit{gp}}=0.48$, and $P_{\mathit{tur}}=0.6$ atm $=6·{10}^{-2}$ MPa; see SI Appendix for the precise form of the anisotropic elastic energy.

Z-Ring Force.

The Z-ring pressure $P_Z$ pulls the interior of the PG wall within the active zone, being highest in the middle and vanishing at the distance $\delta$; see Fig. 2A. The Z-ring force line density $f_Z$ points inward along the circumference and is given as a line integral of the Z-ring pressure $P_Z$ along the axial cross-section of the IM within the $\delta$ distance (Fig. 2A), i.e., ${\int }_{-\delta }^{\delta }{P}_Z\mathrm{d}l$. In other words, the Z-ring force applies up to an arc-length of $\delta$ along the inner boundary of the bacterial wall. Modeling the Z-ring as a mechanical force (8, 9), locating it via the arc-length of the shell boundary (8) are natural steps. Based on the discussion about the Z-ring force we set $f_Z=6\text{pN}·{\text{nm}}^{-1}$. and according to literature we take $\delta =45$ nm (37).

Growth and Remodeling.

The global net rate of septal PG (volume) increase estimated by fluorescence measurements is of the order ${10}^3{\text{nm}}^3·{\mathrm{s}}^{-1}$ (7). Since numerical experiments show that $\kappa$ has almost no influence on the division process we set $\kappa =\frac{2}{3}\eta$. The wild type bacterium was modeled with $\eta =66\text{MPa}·\mathrm{s}$, $\kappa =44\text{MPa}·\mathrm{s}$, and $\mathrm{\Omega }={10}^3{\text{nm}}^3·{\mathrm{s}}^{-1}$. The parameters for the $\mathrm{\Delta }$envC-mutant were $\eta =93\mathrm{MPa}·\mathrm{s}$, $\kappa =62\mathrm{MPa}·\mathrm{s}$, and $\mathrm{\Omega }={10}^3{\text{nm}}^3·{\mathrm{s}}^{-1}$. (The adjustment of $\kappa$ was made solely to conserve $\mathrm{Su}$ and thereby facilitate comparison.) The ftsN-$\mathrm{\Delta }$SPOR-mutant was computed with $\eta =66\mathrm{MPa}·\mathrm{s}$, $\kappa =44\mathrm{MPa}·\mathrm{s}$, and $\mathrm{\Omega }=900{\text{nm}}^3·{\mathrm{s}}^{-1}$.

Numerical Modeling.

The morphoelastic model was solved using the finite element method. The solver was implemented using FEniCS (2019.1.0) (46) and uses dynamic remeshing via Gmsh (47). See SI Appendix for full details about the implementation and a summary of modeling assumptions.

Image Processing.

Images were processed as in ref. 7, the bacterial width was estimated using a square frame of the full images shown in SI Appendix, Figs. S3–S7). The model data were then scaled to the width of the image. Note, that we have tested the stability of the model with respect to small variations in the bacterial diameter.

Cryo-EM Specimen Preparation.

Bacterial strains were grown overnight in LB media, back diluted 1:1,000, and incubated shaking at ${37}^{°}$C, $250$ rpm to ${\text{OD}}_{600}=0.3$. Cells were harvested by centrifugation ($2$ min, $5,000\times$g, RT) and resuspended in LB media to a final ${\text{OD}}_{600}=0.6$. This cell suspension ($3\mathrm{\mu }$L) was applied to Cflat-2/1 200 mesh copper or gold grids (Electron Microscopy Sciences) that were glow discharged for 30 s at 15 mA. Grids were plunge-frozen in liquid ethane (48) with a FEI Vitrobot Mark IV (Thermo Fisher Scientific) at RT, 100$\%$ humidity with a waiting time of 13 s, one-side blotting time of 13 s and blotting force of 10. Customized parafilm sheets were used for one-side blotting. All subsequent grid handling and transfers were performed in liquid nitrogen. Grids were clipped onto cryo-FIB autogrids (Thermo Fisher Scientific).

Cryo-FIB Milling.

Grids were loaded in an Aquilos 2 Cryo-FIB (Thermo Fisher Scientific). Specimen was sputter coated inside the cryo-FIB chamber with inorganic platinum, and an integrated gas injection system (GIS) was used to deposit an organometallic platinum layer to protect the specimen surface and avoid uneven thinning of cells. Cryo-FIB milling was performed at a nominal tilt angle of ${14}^{°}$ to ${18}^{°}$ which translates into a milling angle of $7^{°}$ to ${11}^{°}$ (49). Cryo-FIB milling was performed in several steps of decreasing ion beam currents ranging from 0.5 nA to 10 pA and decreasing thickness to obtain 150 to 250 nm lamellae.

Cryo-TEM Data Acquisition.

All imaging was done on a FEI Titan Krios (Thermo Fisher Scientific) transmission electron microscope operated at 300 KeV equipped with a Gatan BioQuantum K3 energy filter (20 eV zero-loss filtering) and a Gatan K3 direct electron detector. Prior to data acquisition, a full K3 gain reference was acquired, and ZLP and BioQuantum energy filter were finely tuned. Micrographs were collected with a nominal defocus of $-$3.5 $\mathrm{\mu }$m. Data collection was performed in the nanoprobe mode using the SerialEM (50) or Thermo Scientific Tomography 6 software.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Animation showing the model results for the division of the E. Coli wild type. Bacterial outline (burgundy) and PG cross-section (blue) are shown superimposed. An initially slow constriction is followed by a short and accelerating septation.

Movie S2.

Animation showing the model output for the division of the △envC mutant. The constriction deformation is severely impaired, so that a wide septum, visible in the PG cross-section (blue), is formed. This is a natural consequence of reduced PG remodeling that disables the bending-like behavior.

Movie S3.

Animation showing the model output for the division of the ftsN-△SPOR mutant. Due to the reduced incorporation of PG precursors the outer and inner membrane constrict for most of the division at a similar rate and the much narrower septum is formed quickly at the latest stage of the process.

Data, Materials, and Software Availability

Software and images will be deposited in Zenodo. Previously published data were used for this work (7).