Mechanical Control of Bacterial Cell Shape

Abstract

In bacteria, cytoskeletal filament bundles such as MreB control the cell morphology and determine whether the cell takes on a spherical or a rod-like shape. Here we use a theoretical model to describe the interplay of cell wall growth, mechanics, and cytoskeletal filaments in shaping the bacterial cell. We predict that growing cells without MreB exhibit an instability that favors rounded cells. MreB can mechanically reinforce the cell wall and prevent the onset of instability. We propose that the overall bacterial shape is determined by a dynamic turnover of cell wall material that is controlled by mechanical stresses in the wall. The model affirms that morphological transformations with and without MreB are reversible, and quantitatively describes the growth of irregular shapes and cells undergoing division. The theory also suggests a unique coupling between mechanics and chemistry that can control organismal shapes in general.

Introduction

The apparent shape of a bacterium is determined by the geometry of its growing cell wall (1–4). Recently, a number of prokaryotic cytoskeletal proteins, such as FtsZ, MreB, and crescentin, have been shown to be important for shaping the bacterial cell (4–7). These proteins regulate visible morphological changes that require cell wall growth and remodeling. The growth process, which occurs slowly over many minutes, involves the insertion and removal of cell wall building blocks, and appears to be sensitive to mechanical forces. For example, FtsZ seems to exert a contractile force, facilitating cell division (8). The division furrow is generated over tens of minutes. If A22, a small molecule that depolymerizes MreB bundles in the cell (9), is added to the growth medium, Escherichia coli can transform from a rod-like shape to a spherical shape (10–12). The cell shape is not altered until long after the disappearance of MreB, indicating that the shape change results from cell wall remodeling rather than direct mechanical deformations. A22 causes similar morphological transformations in Caulobacter crescentus (13,14), where crescent-shaped cells transform into round lemon-shaped cells. Of interest, the rod-like shape is recovered if MreB bundles are restored (15). In similarity to MreB, if crescentin in Caulobacter crescentus is deleted, the cells lose their characteristic curved shapes and become straight rods (16,17). The underlying molecular mechanism for these morphological changes mediated by cytoskeletal proteins is still unclear.

In this work, we use a theoretical model to describe the interplay of cell wall growth, mechanics, and cytoskeletal filaments in shaping the bacterial cell. Based on known mechanisms of cell wall assembly and the influence of forces on the assembly, we postulate that MreB bundles exert additional forces on the cell wall. The model predicts that a growing rod-like cell by itself is unstable, but this instability can be suppressed by bundles of MreB. MreB can mechanically reinforce the cell wall, and the composite structure composed of MreB and cell wall can resist the onset of instability. We performed experiments to verify these predictions, and the results agreed quite well with the predictions. Simulations demonstrate that our model explains a range of MreB functions, revealing that 1), depletion of MreB leads to a reversible transformation from a short rod to a sphere; 2), overexpression of MreB results in the filamentation of bacterial cells; 3), depolymerization of MreB helix around the septum seems to be a prerequisite for cell division; and 4), nonuniform growth and disassembly of MreB can lead to bulges in filamentous cells. Taken together, these findings suggest a unique coupling between mechanics and chemistry that can control organismal shapes in general.

Materials and Methods

Competition between mechanical and chemical energy in the bacterial cell wall

To develop a general understanding of bacterial cell shape, it is necessary to combine molecular-level biochemistry of cell wall assembly with mechanical influences from turgor pressure and cytoskeletal filaments. For Gram-negative bacteria, such as E. coli, the cell wall is a living and growing network of peptidoglycan (PG) strands cross-linked by short peptide bonds (1–4). PG subunits are synthesized at the cytoplasmic side of the inner membrane and translocated to the periplasm. The exported subunits store chemical energy derived from their synthesis in the cytoplasm. This stored energy is released during the addition of new subunits to the existing network (18). Enzymes such as transpeptidases, transglycosylases, and hydrolases constantly modify the network, adding new glycan strands and connections, and deleting existing strands (1,3). The details of these enzymatic reactions are complex, but at the simplest level, the insertion and deletion of PG subunits can be viewed as a reversible assembly reaction (Fig. 1). These reactions are driven by chemical free energy released during PG subunit addition, and do not require additional input of energy from ATP or GTP hydrolysis. When forces, such as those from cellular turgor pressure, are applied to the reactant and product, the chemical equilibrium of assembly will be affected. To develop a quantitative understanding, let us examine the process by which the wall area increases slightly, or $A\to A+dA$ (Fig. 1). This reaction is energetically favorable from the insertion of new PG subunits. The favorable chemical energy change during this growth process can be modeled as ${\varepsilon }_{0}dA$, where ${\varepsilon }_0$ is the released chemical energy per unit area from the activated PG subunits in the undeformed configuration (${\varepsilon }_0$ can be estimated from the chemical activation energy of the PG subunits and the bond energies in the PG network).

Figure 1.

Bacterial cell wall is a growing network of PG strands. PG subunits are inserted at random points along the cell by enzymes. The network is also under mechanical stress from turgor pressure and cytoskeletal influences. The cartoon shows the reversible assembly reaction whereby the cell wall area increases from A to $A+dA$. The total energy change of this reaction is modeled by Eq. 1. Cytoskeleton filaments such as MreB can exert forces on the cell wall and modify the equilibrium of the assembly process. We model the MreB bundle as a helical rod with a preferred radius R₀ and pitch p. The current cell radius R > R₀. The helical bundle is dynamic and thus exerts additional pressure on the cell wall.

At the same time, previously stress-free PG subunits are stretched and inserted into the existing PG network. Therefore, the mechanical energy of PG subunits increases. We can estimate the total change in energy as

$$dG=G\left(A+dA\right)-G\left(A\right)=dU-{\varepsilon }_{0}dA,$$ (1)

where dU is the change in the strain energy of the network under constant pressure. Experiments indicate that the PG layer of E. coli is a disordered network of polysaccharides linked by peptide bonds (19). Thus, for simplicity, we use an isotropic model of an elastic thin shell to describe the mechanical strain energy, U, in Eq. 1. One can study an anisotropic cell wall model by modifying the elastic tensor (see Supporting Material), but this complication will not be considered here. Here, the strain energy of the cell wall includes both the stretch energy and bending energy. We assume that the growth process is relatively slow. Therefore, the cell wall is always in mechanical equilibrium. This implies that the work done by the turgor pressure is equal to the increase of the strain energy in the cell wall during this quasi-static process. Mathematical details are given in the Supporting Material. Previous studies used models that treat the PG layer as a static elastic body, and measured the mechanical properties of isolated PG sacculus (20,21). Here, we introduce dynamic PG assembly and disassembly by adding ${\varepsilon }_{0}dA$ to the total energy, which leads to new predictions, as discussed below. Note that the insertion of new PG subunits may also change the stress state of the old network. Therefore, dU depends on the shape, size, and growth direction of the cell wall. This indicates that there could be a size and shape where the increased strain energy exactly balances the decreased chemical energy (dG = 0) when the cell grows in a particular direction. When this configuration is reached, assembly and disassembly reactions will exactly balance and the cell will stop growing in that direction. For example, it is possible to find a static radius for a cylindrical cell, where dG = 0 and further variation in radius is unfavorable. However, at this radius, growth in the axial direction is favorable and dG < 0. This explains why rod-like bacteria can maintain a specific radius but grow linearly or exponentially with time in the axial direction (22).

Growth equations

To describe the cell wall growth dynamics, one can use kinetic equations with forward and backward reaction rates $k_{on}$ and $k_{off}$ to model the local PG assembly. The equilibrium constant for the reaction is related to the net free-energy change, dG in Eq. 1, which is in turn affected by the local elastic energy density, dU. For arbitrary three-dimensional surfaces, however, it is more convenient to describe growth using a driving force. This method has been widely used by investigators in the field of material science to study the morphology evolution of microscopic surfaces of solid materials (23), such as the interface migration of a solid particle in contact with its vapor. In such systems, the motion of the interface that decreases the free energy is favorable. Therefore, a driving force can be defined as the free-energy decrease associated with adding a unit volume of atoms to the particle. Furthermore, the velocity of the interface motion is proportional to the driving force. Such a nonequilibrium thermodynamic process is analogous to growth of the cell wall. Therefore, we define the driving force for growth, F, as the free-energy decrease when the wall grows a unit length, i.e.,

$$\mathbf{F}\left(\overline{\mathbf{r}}\right)=-\delta G/\delta \overline{\mathbf{r}},$$ (2)

where $\overline{\mathbf{r}}$ is a material point on the undeformed surface and $\delta G/\delta \overline{\mathbf{r}}$ is the functional derivative of G. It is clear that F = 0 corresponds to the above-mentioned dG = 0. For example, after a cylindrical cell reaches its static radius, the driving force along the circumferential direction is zero, but the driving force along the axial direction is nonzero (22). Therefore, the cell can keep growing along the axial direction and maintain a constant radius, although the growth is not an equilibrium process. The rate of wall growth should be proportional to the driving force of adding more materials to the cell wall (22,23). Therefore, we define the growth velocity as

$$\partial \overline{\mathbf{r}}/\partial t=M\mathbf{F}\left(\overline{\mathbf{r}}\right),$$ (3)

where M is a positive phenomenological constant that is determined by the kinetics of the growth mechanism, and could depend on the spatial location of growth. For actual computations, a cylindrical coordinate system is used, and Eq. 3 simplifies to Eqs. S21 and S22 in the Supporting Material.

It is perhaps interesting to ask whether the morphological evolution of the cell is governed by energetic considerations at all. To answer this, it is important to note that this model does not describe the energy of the whole cell. Rather, it describes the strain energy of the cell wall alone, under the condition of constant turgor pressure and material turnover. This shape change of the cell wall must conform to physical constrains. If PG synthesis enzymes do not consume energy, and PG bonds are made as well as broken, then the rates of assembly and disassembly must be governed by the competition of mechanical strain energy with chemical energy. Another way to describe this is to say that the rates of assembly and disassembly, $k_{on}$ and $k_{off}$, depend on the total energy in the PG network. With this idea, the net material flux is $J=k_{on}-k_{off}$, which also depends on the total energy change. The growth law derived from the flux approach is equivalent to the driving force approach derived above.

During cell wall growth, if the shape of the cell remains the same, only one or more dimensions of the shape are changing. Mathematically, then, the cell shape can be described by parameters a_i. For example, rod-like bacteria maintain a constant radius but elongate in the axial direction. The shape of such a cell can be characterized by its length and radius if the caps are neglected (22). In this case, the change in total energy is $dG={\sum }_i-F_{i}d{a}_i$, where F_i is the driving force corresponding to the parameter a_i, i.e.,

$$F_i=-\partial G/\partial a_i.$$ (4)

The growth velocities of cell dimensions can be described by

$$d{a}_i/dt=M_i{F}_i,$$ (5)

where M_i is the positive phenomenological growth constant corresponding to a_i.

Results

A cylindrical cell wall itself is unstable and the instability is suppressed by MreB

Using the growth equations, it is interesting to determine whether common cell shapes are stable during growth. For a nearly cylindrical cell wall, let us examine how small variations in the cell shape along the axial direction would change over time. We write the cylinder radius as $R\left(\eta ,t\right)=R_s\left(t\right)+B\left(t\right)\cos \left(q\eta \right)$, where R_s is the current cell radius, and B and q are the amplitude and wave number of a cosine-wave perturbation, respectively. We assume $B<<R_s$. The position of the cell wall is $\mathbf{r}=\left(R\cos \varphi ,R\sin \varphi ,L\left(t\right)\eta \right)$, where ϕ is the rotational angle of an axisymmetric cell (see Fig. S1 and Section S8), and L(t) measures the growth of the cell in the axial direction with the initial condition L(0) = 1. In this case, the average free energy in one wavelength is $G=q/2\pi {\int }_0^{2\pi /q}{\int }_0^{2\pi }\left(f_0-{\varepsilon }_0\right)\sqrt{g}d\varphi d\eta$, where f₀ is the strain energy density of the cell wall. The expression for f₀ and the stress field are described in the Supporting Material. In this case, the free energy can also be written as $G=G\left(R_s,L,B\right)$. The growth equations for the cell radius, cell length, and perturbation amplitude are as follows:

$$\frac{d{R}_s}{dt}=-M_R\frac{\partial G}{\partial R_s},$$ (6)

$$\frac{dL}{dt}=-M_L\frac{\partial G}{\partial L},$$ (7)

$$\frac{dB}{dt}=-M_B\frac{\partial G}{\partial B},$$ (8)

where M_R, M_L, and M_B are the growth constants for R_s, L, and B, respectively. If we only consider the leading order term of B, Eqs. 6 and 7 are only functions of R_s and L. Eq. 8 then becomes

$$\frac{dB}{dt}=-\alpha B,$$ (9)

where $\alpha =M_B\left(C_1+C_2{q}^2+C_3{q}^4\right)$ is the growth factor of the perturbation amplitude. The expressions for α, C₁, C₂, and C₃, are given explicitly in the Supporting Material. The perturbation will increase with time if α < 0, which implies that the cylindrical shape is unstable with respect to growth. Otherwise, the shape is stable and $B\to 0$.

The blue line in Fig. 2 shows the theoretical predictions for the growth factor, α, for a bare cell wall, which is negative for wavelengths $2\pi /q\approx 2.8-10.4\mu \mathrm{m}$. This indicates that the cylindrical cell wall with no influences other than turgor pressure is unstable. Small variations in the radius will grow with time. Indeed, simple back-of-the-envelope estimates indicate that the cylinder shape is unfavorable when compared with rounded shapes (see Supporting Material). Wild-type E. coli, however, does not exhibit this instability and maintains a constant rod radius. What other elements in the cell could be suppressing this instability?

Figure 2.

Instability in a growing cylindrical cell wall along the axial direction and suppression of the instability by the MreB bundle. The growth factor α in Eq. 9 is plotted as a function of spatial frequency of shape perturbations for different effective stiffnesses of the MreB bundle. The cell wall is stable when α > 0; otherwise, it is unstable. Without MreB ($k_m=0$), the cell wall is unstable for perturbations with wavelength $2\pi /q\approx 2.8-10.4\mu \mathrm{m}$. E. coli parameters (see Table S1) are used in the calculation. Insets show pictures taken from cell culture of WM1283 with and without the treatment of A22. The scale bar represents 10 μm.

Here we propose that the MreB cytoskeletal bundles exert additional forces on the cell wall and suppress the growth-induced instability. This force could arise from an ATP-driven curvature change of the MreB bundle, which has been observed for purified filaments (24). In this work, we model the MreB bundle as an elastic helical filament adhered to the cell wall. The mechanical forces exerted by the helical filament in the radial and helical axis directions are described in the Supporting Material. In particular, we show that the force in the radial direction of the cell is $F_1\sim E_{m}Il\delta R/R_0^4$, where E_m is the Young's modulus of MreB, $I=\pi d^4/64$ is the area moment of inertia, d is the diameter of the bundle, l is the contour length of the helical bundle, and R₀ and δR are the preferred radius and the radius change of the MreB bundle, respectively.

In the cell, the MreB helix is a highly dynamic structure that moves and turns over with time (25). The timescale of the MreB motion is much smaller than the timescale of cell wall growth. Therefore, the radial force exerted by MreB should be averaged over the cell wall surface and modeled as a pressure field. If the helical angle of the MrB helix is small (see Fig. S2), the average pressure is approximately $F_1/\left(lp\right)$, where p is the pitch of the helix. Thus, the traction applied by MreB along the radial direction can be written as a linear function of radius change, i.e.,

$$P_m=F_1/\left(lp\right)=k_m\left(R-R_0\right)$$ (10)

where $k_m=E_{m}I/\left(p{R}_0^4\right)$ is the effective stiffness of MreB in the radial direction and R is the current radius of the cell wall. This extra pressure modifies the stress field in the cell wall and changes the elastic energy, U (see Supporting Material). Thus, the cytoskeleton bundle can influence the chemical equilibrium of cell wall assembly by exerting forces on the cell wall.

In similarity to the cell wall, the competition between chemical energy and mechanical energy also affect the growth of MreB. To include MreB growth, we can add an additional term in the chemical energy. Because the force applied by MreB is modeled as a pressure field, the chemical energy per unit length of MreB is also averaged over the cell wall surface. Therefore, the effective chemical energy per unit area is $\varepsilon ={\varepsilon }_0+{\varepsilon }_m$, where ${\varepsilon }_m$ is the chemical energy of MreB per unit area. The effective parameter ɛ models both the wall and MreB chemical energy. The mechanical energy of the MreB bundle depends on the cell shape and also contributes to the total energy of the system (see Supporting Material).

Because MreB is an actin homolog, the persistence length of MreB protofilaments is likely similar to that of F-actin (26), i.e., $E_{m}I/\left(k\mathrm{B}T\right)\sim 15\mu \mathrm{m}$. The diameter of MreB protofilament is 3.9 nm (27). Therefore, the Young's modulus of MreB, E_m, is ∼5 GPa. In vitro, MreB assembles into filamentous bundles that can spontaneously form ring-like structures (24,27) with relatively uniform diameters of 100∼200 nm. In vivo, MreB forms a helical bundle that lies underneath the PG surface and has the same radius as the cell (10,13,28,29). By combining the above data, we estimate the preferred curvature radius of the MreB bundle, R₀, to be 0.1∼0.5 μm. From fluorescence images, the width of the MreB bundle is between 100 and 300 nm (10,13,28,29). Therefore, the effective stiffness of MreB, $k_m$ is ∼${10}^3-{10}^7\mathrm{P}\mathrm{a}/\mu \mathrm{m}$. Note that some proteins are colocalized with MreB in bacteria. For example, RodZ, Mbl, and MreBH also form helical structures (4–7,30–32). These proteins also can apply forces and increase the bundle stiffness beyond previous estimates (26). Experiments suggest that the MreB bundle contributes one half of the stiffness of the whole cell (33). In our calculations, we use $k_m=1.5\mathrm{M}/\mu \mathrm{m}$ for E. coli.

The model predicts that the growth instability is suppressed by the MreB helix and the cell shape is stable with respect to perturbations (red curve in Fig. 2). The critical stiffness, below which the cell wall is unstable, is also shown in Fig. 3. With the appropriate bundle stiffness, $k_m$, and preferred radius, R₀, physiological cell radius is achieved (Fig. 3). The force applied by MreB is in the opposite direction of the force applied by the turgor pressure. Therefore, the turgor pressure can be counterbalanced by MreB so that the cell wall can grow larger. Thus, the bacterial cell wall can be regarded as an elastic sheet with MreB reinforcements, much like structures such as fiber-reinforced composites. The static radius of the cell R_s would approach R₀ when $k_m\to \mathrm{\infty }$. In this case, the static radius is mainly determined by MreB because MreB helix is much more rigid than the cell wall. In the opposite limit with no MreB, R_s is only determined by the properties of the cell wall and the turgor pressure.

Figure 3.

Static radius of the cell as a function of the effective stiffness of the MreB helix, $k_m$. The preferred curvature radius of MreB, R₀, is 0.4 μm. A range of ɛ-values between 0.05 and 0.4 $J\cdot m^{-2}$ is plotted, giving the shaded region. The red line represents the critical stiffness, below which the cell wall is unstable. E. coli parameters (see TableS1) are used in the calculation. We see that a range of parameters can give the natural E. coli radius of 0.5 μm.

So far, we have examined instability along the axial direction of an infinite cylindrical cell wall by applying a shape perturbation $R\left(\eta ,t\right)=R_s+B\left(t\right)\cos \left(q\eta \right)$. Similarly, instability along the circumferential direction is also possible; the cell can potentially develop a noncircular cross section. We consider a cosine wave perturbation along the circumferential direction of an infinitely long cylindrical cell wall. In this case, the radius is a function of the angle ϕ along the circumference and can be written as $R\left(\varphi ,t\right)=R_s+B\left(t\right)\cos \left(n\varphi \right)$, where n = 1, 2, 3…. Again, the total energy can be written as $G=G\left(R_s,L,B\right)$. If we consider the leading order terms of B, we find that the growth equations for R_s, L, and B are the same as what we obtained for the axial case, except that the growth factor is $\alpha =M_B\left(C_4+C_5{n}^2+C_6{n}^4\right)$, where constants C₄, C₅, and C₆ are given in the Supporting Material. Given that the initial cell radius is the static radius determined by $d{R}_s/dt=0$, we find that the growth factor α is always positive and the cell wall is stable with respect to the perturbation along the circumferential direction, even when MreB bundles are not present (see Section S9 and Fig. S3). Therefore, our model predicts that growth instabilities only occur along the axial direction for a cylindrically shaped cell.

Experimental verification

To test our model predictions and experimentally verify the growth-induced instability, we culture E. coli cells with depleted FtsZ, which form long filamentous rods (experimental procedures are given in the Supporting Material). Without A22, the filamentous cells have a uniform cell radius (Fig. 2, insets). However, upon addition of A22, which partially disassembles MreB, wave-like bulges are formed after a couple of hours of growth (Fig. 2, insets). A22 affects the cell morphology in a dose-dependent manner (14), such that the MreB helix is only partially disassembled at low A22 concentrations. Therefore, the calculated fastest growth wavelength should depend on the stiffness of MreB helix, which in turn depends on the number of MreB filaments. We find that the fastest growth wavelength decreases as the effective stiffness of MreB helix decreases (Fig. 4). In this case, the purple curve and green curve in Fig. 2 approximately describe the experimental situation.

Figure 4.

Results from the A22 titration experiment and comparison with theory. The blue axes (left) and colored curves denote the fastest growth wavelength versus the effective stiffness of the MreB helix, $k_m$. The closed circles represent the critical stiffness, beyond which the cell wall is stable. E. coli parameters (see Table S1) are used in the calculation. The black axes (right) and black line show the growing bulge size as a function of the A22 concentration; 27, 31, 25, 34, and 31 bulges were measured for A22 concentrations of 10, 20, 40, 60, and 80$\mu g/ml$, respectively. Although the MreB helix stiffness should decrease as the A22 concentration increases, the explicit relationship is unknown; therefore, we use the A22 concentration as the x axis. The standard error is used to calculate the error bar. The bulge size is represented by the length of the bulge along the cell axis, and the size is measured 1.5 h after the addition of A22.

To further quantify these predictions, we repeat the growth experiment for six different A22 concentrations: 10, 20, 40, 60, 80, and 200 $\mu g/ml$. We find that the average bulge size decreases as A22 concentration increases (Fig. 4). This result is consistent with our model prediction that the fastest growth wavelength will decrease as the effective stiffness of MreB helix decreases. Here, the bulge size is roughly the experimental counterpart of the fastest growth wavelength. As A22 concentration increases, we expect the MreB stiffness, k_m, to decrease. We also find no bulges when A22 concentration is ≥200 $\mu g/ml$, which agrees with the earlier observation that high A22 concentrations do not change cell morphology (14). For very high A22 concentrations, cell wall growth and remodeling presumably are stopped completely, and therefore no shape changes are observed. Finally, we also do not observe any instability that breaks the circular cross-sectional symmetry of the cell in our experiment. Taken together, these results suggest that our model is at least consistent with experimental observations.

Simulations

Using growth equations (Eqs. S21 and S22), we can also simulate bulge formation in the middle of long filamentous E. coli cells after A22 is added. In the inset of Fig. 2, the overall radius of the cell increases but the poles appear to be unchanged. The cell poles are believed to be inert or static in E. coli (34,35), which indicates that the growth constants M₁ and M₂ defined in Eqs. S21 and S22 at the poles should be zero, or at least much smaller than the other part of the cell wall. In the simulation, we use a spatially varying growth constant (see Supporting Material) and decrease the effective stiffness of MreB to zero. Depending on the width of the actively growing region, one bulge or multiple bulges can appear in the middle of the cell (Fig. 5 D). The observed cell shapes in our experiment are in accord with the simulations.

Figure 5.

Simulations based on our theoretical model show how MreB bundles can affect cell shape. In all figures, MreB helices are shown in red. (A) Left: Depletion of MreB leads to the transformation from a short rod to a sphere. Right: The transformation is reversible, and the simulation shows a recovery from a sphere to a rod after A22 is removed. These simulations are consistent with experimental observations (10–15,33). (B) Cell division after depolymerization of MreB around the septum. The MreB helix and two MreB rings are represented by the red helix and blue rings, respectively. (C) Cells cannot divide and become filamentous if MreB bundles are not disassembled between the blue rings. These results can be compared with images in de Pedro et al. (35). (D) Nonuniform growth and the disassembly of MreB lead to one or more bulges in the middle of a filamentous E. coli cell. The number of the bulges depends on the width of the actively growing region that is defined by spatially varying growth rate constant M (see Supporting Material). E. coli parameters (Table S1) are used. The images are from cultured WM1283 cells treated with A22 (Supporting Material).

The model predicts that the fastest growth wavelength will be larger than the natural length of E. coli. Therefore, normal E. coli cells tend to transform into spheres instead of multiple bulges upon MreB disassembly. Thus, the proposed model is also able to explain the observed reversible transformation from short rods to spheres in wild-type E. coli (10–13,15,36). We use a spherical cell as the initial starting shape. Once the influence of MreB is added in the simulation, the spherical cell gradually transforms to a rod-like cell with a constant radius (Fig. 5 A). This calculated radius is consistent with the static radius predicted in Fig. 3. With the calculated rod-like shape used as the initial condition, a short rod-like cell transforms back into a spherical cell when MreB is deleted in the simulation (Fig. 5 A). It should be noted that although almost all rod-shaped bacteria contain MreB or MreB homologs, some (e.g., Corynebacteria) can maintain a rod-like shape without MreB (37). Such bacteria grow by inserting new material at the cell poles while the lateral cell wall remains inert. The new cell wall becomes inert soon after the growth process is completed. This again suggests that morphological changes require the reversible remodeling of cell wall as depicted in Fig. 1.

During cell division, the cell shape deviates substantially from the rod-like shape. Experiments have shown that the MreB helix is dynamically remodeled during cell division in a FtsZ-dependent manner (38). At the beginning of cell division, a single ring of MreB forms at midcell, and the other parts of the MreB helix, which extends much of the length of the cell cylinder, become thinner. The MreB ring soon splits into two separate rings that flank the FtsZ ring. At the same time, MreB helices between the two rings begin to disappear. The two rings gradually move apart during cell division and finally disappear after division is completed. These observations suggest that the MreB helix around the septum must disassemble during cell division. In our model, we disassemble MreB by setting $k_m=0$ between the two rings, and apply a constant FtsZ-ring contractile force. Fig. 5 B shows the shape evolution during cell division. The septum region elongates in the axial direction such that two MreB rings are pushed away from the midcell, which is consistent with experimental observations (38). The computed cell shape is similar for Z-ring forces ranging from several piconewtons to several hundred piconewtons. Thus, Z-ring force is only needed to initiate cell division by changing the local stress in the cell wall, and to guide the growth in the radial direction. After initiating contraction, the cell wall can divide spontaneously without a contractile force from the Z-ring. This is consistent with the theoretical prediction that a small Z-ring force (<10 pN) is sufficient to accomplish cell division (8). Experiments in reconstituted systems showed that Z-rings in lipid tubules change the membrane curvature only slightly (39). This is consistent with our prediction that a weak contractile force from the Z-ring is sufficient for division. In contrast, we find that cells cannot divide if MreB bundles are not disassembled between the two rings. This might explain why overexpression of MreB inhibits cell division and leads to the formation of multinucleated filamentous cells (10,40) (see Fig. 5 C).

Discussion and Conclusions

The major concept we propose in this work is that mechanical forces play an important role in regulating the biochemistry of cell wall growth. For PG networks in bacteria, because the cell wall is continuously built and broken, the net flux of PG subunits should be controlled by mechanical tension. Using a simple isotropic elastic model, we demonstrate that a growing cylindrical cell wall itself is unstable. The observed shapes in filamentous cells treated with A22 are consistent with the predicted instability. This result suggests that MreB probably exerts further inward pressure on the cell wall. Treating MreB as a helical bundle with a preferred curvature, we show that the instability can be suppressed by MreB. The computed cell shapes from the model agree well with the observed shapes of cells in experiments. These results suggest that the bacterial cell wall acts as a composite material, i.e., the elastic PG network is probably mechanically reinforced by the MreB bundle. Recent experiments (33) suggest that MreB contributes significantly to the cell wall rigidity, which is in accord with our conclusions. Previous studies examined similar growth-induced instability in other systems; however, those studies focused mainly on spherical cells, such as observed in unicellular algal growth (41), dendritic branching in neuronal growth (42), differential growth in elastic shells (43), and membrane-cytoskeleton systems (44). There are also connections between the growth-induced instability reported here and the pearling instability observed in cylindrical membrane vesicles (45,46), although the physical origins of these instabilities are all different.

A significant point of this study is that the shape of the cell wall is determined by the dynamics of PG assembly and disassembly. A number of recent works showed that a variety of bacterial cell shapes could result from spatial patterning of PG defects, but the dynamics of PG assembly was not studied explicitly (47,48). Given the known PG turnover process, dynamical models such as the one outlined here should be considered. Similar mechanochemical modeling ideas have been used extensively in studies involving active gel theory (49–51) and models of growing tissues (52,53). In such studies, the cells and tissues are viscoelastic, but the flux and turnover of the biological material are explicitly considered, and similarly to the work presented here, living matter is described at the continuum level and the coupling between forces and molecular chemistry has important macroscopic influences. The mechanochemical principle likely has general implications for all scales in biology.

In this study, we have assumed that the cell wall is an isotropic elastic material. In reality, however, the bacterial cell wall could be anisotropic, especially in bacteria such as Bacillus subtilis. In such cells, the elastic tensor $D^{\alpha \beta \rho \upsilon }$ (defined in the Supporting Material) must be modified to account for this anisotropy, and additional elastic constants are required. Nevertheless, the basic modeling framework remains the same. The inclusion of cell wall anisotropy also does not change the main conclusion of this study. The bacterial cytoskeleton may also play additional roles in coordinating cell wall growth. For example, MreB can be used as a scaffold for the cell wall synthesis machinery and affect the insertion and turnover of PG strands. Within our model, this can be included by considering additional complexity in the growth parameter M in Eq. 3 and the possibility of a dynamically changing constitutive relation for the cell wall. Therefore, more experiments are needed to test the model. It should also be noted that when mechanical tension is too high and the cell size extends past a critical value, mechanical failure can occur. Our model currently does not describe this regime.

The model suggests that other cytoskeletal bundles, such as crescentin, have a role similar to that of MreB. Indeed, if crescentin is modeled as a helical filament attached to one side of the filamentous C. crecentus, a helical cell will develop. Crescentin also acts in concert with MreB to generate the crescent cell shape in Caulobacter. Our model is able to explain the helical shapes of growing cells under confinement, and the subsequent relaxation of the helical shape when cells are released from confinement (54). The model can also be applied to investigate shape changes in B. subtilis cells, which have a different set of parameters.

Other proteins, such as PBP2 (55), RodA (56,57), and RodZ (30–32), can also affect the morphology of bacteria. These proteins colocalize with MreB to form a helical complex and are indispensable for the proper assembly of the MreB helix. Depletion of these proteins can also disrupt the MreB helix and lead to altered cell shapes. The roles of these proteins are currently unknown, but some possibilities can be raised. For example, RodA might be an anchor for MreB and connect it to the inner membrane and the cell wall. PBPs are critical for forming specific PG network geometries, and mutations in PBPs would result in altered PG networks. Therefore, biochemical measurements of the structure of the PG network during normal and mutant growth should be performed. Our theoretical model provides a framework in which biochemistry, protein assemblies, and mechanics can be combined to elucidate the molecular mechanism of cell shape determination.