Mechanics and Dynamics of Bacterial Cell Lysis

Abstract

Membrane lysis, or rupture, is a cell death pathway in bacteria frequently caused by cell wall-targeting antibiotics. Although previous studies have clarified the biochemical mechanisms of antibiotic action, a physical understanding of the processes leading to lysis remains lacking. Here, we analyze the dynamics of membrane bulging and lysis in Escherichia coli, in which the formation of an initial, partially subtended spherical bulge (“bulging”) after cell wall digestion occurs on a characteristic timescale of 1 s and the growth of the bulge (“swelling”) occurs on a slower characteristic timescale of 100 s. We show that bulging can be energetically favorable due to the relaxation of the entropic and stretching energies of the inner membrane, cell wall, and outer membrane and that the experimentally observed timescales are consistent with model predictions. We then show that swelling is mediated by the enlargement of wall defects, after which cell lysis is consistent with both the inner and outer membranes exceeding characteristic estimates of the yield areal strains of biological membranes. These results contrast biological membrane physics and the physics of thin, rigid shells. They also have implications for cellular morphogenesis and antibiotic discovery across different species of bacteria.

Introduction

Antibiotic resistance is one of the largest threats to global health, food security, and development today (1). Its increasing prevalence (2) begs the question of whether physical principles, which may be more universal than particular chemical pathways, could inform work on novel therapeutics as has been done for mechanotransduction in eukaryotes (3) and tissue growth and fluidity (4, 5). To elucidate such principles, a physical understanding of the cell death pathway caused by many antibiotics, which may complement the knowledge of related biochemical mechanisms (6, 7, 8, 9, 10, 11, 12), is needed.

In many bacteria, cell shape is conferred by the cell wall, which resists the internal turgor pressure and is composed of two- or three-dimensional layers of peptidoglycan (PG) (13, 14, 15). In Gram-negative bacteria such as Escherichia coli, the two-dimensional cell wall is sandwiched between the inner membrane (IM) and outer membrane (OM), whereas in Gram-positive species, the cellular envelope comprises an IM enclosed by a three-dimensional cell wall. PG consists of rigid glycan strands cross-linked by peptide bonds and is maintained through the combined, synchronized activity of enzymes, including transglycosylases and transpeptidases (13, 15, 16, 17). Many antibiotics, such as β-lactams, bind to transpeptidases to inhibit cross-linking. Inhibition of peptide bond formation and cell wall synthesis results in large defects in the cell wall, which precede bulging of the IM and OM and eventual cell lysis (6, 7, 18, 19, 20).

In this work, we show how the response of the bacterial cell envelope to large, micron-scale defects in the cell wall can be modeled physically. By dissecting the dynamics of lysis, we reveal salient features—the emergence of different timescales and the formation of partially subtended, spherical bulges—that require explanation. We then show that a theoretical model comprising turgor pressure and cell envelope stretching is consistent with these features. By clarifying aspects of membrane physics, entropy, and water flow, the model illustrates how lysis arises as a generic, mechanical response and how different cell envelope components interact during bulging. We anticipate these results to be useful for revealing a better understanding of antibiotic action, probing lysis in other experimental contexts, and modeling related systems involving biological membranes and elastic shells, as discussed further in the Conclusions.

Materials and Methods

Model parameters

We briefly discuss the choice of parameter values in this work here, with further details provided below and in Table 1. We model the cell wall as a rigid, orthotropic cylindrical shell with elastic moduli $Y_x^w=0.1\text{N}/\text{m}$ (axial direction, x) and $Y_y^w=0.2\text{N}/\text{m}$ (circumferential direction, y) (21, 22, 23, 24). The Poisson’s ratios are ${\nu }_{xy}^w=0.2$ and ${\nu }_{yx}^w=0.4$ (23), the reference cell wall radius is $r_0^w=0.5\mu \text{m}$, the reference cell wall length is $L_0^w=10\mu \text{m}$, and the reference cell wall area, neglecting the cellular poles, is ${\mathcal{A}}^w=2\pi r_0^w{L}_0^w$. The area stretch moduli of both the IM and OM are set to $K_a^i=K_a^o=0.1\text{N}/\text{m}$ (25, 26, 27); here and below, we use superscripts to denote IM (i), OM (o), or cell wall (w) quantities. The membrane bending rigidity is $k_b^i=k_b^o=20kT$, where k denotes Boltzmann’s constant and T = 300 K is the temperature (27). The number of solute molecules inside the cell is taken to be n_s = 9.5 × 10⁷, corresponding approximately to a turgor pressure of p = 0.5 atm for the cellular dimensions considered in this work (21). To contextualize the choice of parameter values above, Table 1 provides estimated ranges of all parameter values found in the literature.

Table 1.

Variables Used, or Calculated, in This Work for E. coli and Their Estimated Numerical Ranges

Quantity	Estimate	Source
Axial cell wall elastic modulus (3D), $E_x^w$	20–30 MPa	(21, 22, 23).
Circumferential cell wall elastic modulus (3D), $E_y^w$	50–75 MPa	(21, 24).
Cell wall thickness, hw	3–4 nm	(23, 70).
Axial cell wall elastic modulus (2D), $Y_x^w$	0.06–0.12 N/m	Y = Eh
Circumferential cell wall elastic modulus (2D), $Y_y^w$	0.15–0.30 N/m	Y = Eh
Cell wall Poisson’s ratio, ${\nu }_{xy}^w$	0.2	(23).
Cell wall Poisson’s ratio, ${\nu }_{yx}^w$	0.25–1	${\nu }_{yx}^w=Y_y^w{\nu }_{xy}^w/Y_x^w$
Cell membrane area stretch modulus, Ka	0.03–0.24 N/m	(25).
Turgor pressure, p	0.3–2 atm	(21, 71, 72).
Number of solute molecules, ns	(5.7 − 38) × 107 molecules	$p\approx kT{n}_s/\left[\pi {\left(r_0^w\right)}^2{L}_0^w\right]$
Reference cell wall radius, $r_0^w$	0.5 μm	–
Reference cell wall length, $L_0^w$	10 μm	–
Reference cell wall surface area, $\mathcal{A}$w	31.4 μm2	${\mathcal{A}}^w=2\pi r_0^w{L}_0^w$
Reference membrane surface area ratio, γ = $\mathcal{A}$i/$\mathcal{A}$w	1.0–1.2	this work
Membrane bending modulus, kb	10–20 kT	(27).
Temperature, T	300 K	–

2D, two-dimensional; 3D, three-dimensional.

Bacterial strains and microscopy

The wild-type strain used in this study is E. coli MG1655, and we verified that the morphological dynamics is statistically indistinguishable in two other wild-type strains, JOE309 and BW25113. The Supporting Materials and Methods contains further details regarding bacterial growth, microscopy, and image analysis.

Results and Discussion

Dynamics of bacterial cell lysis

Inspired by previous work (16), we degraded wild-type E. coli cell walls with cephalexin, a β-lactam antibiotic, at a concentration of 50 μg/mL and observed typical cells to undergo the morphological transitions shown in Fig. 1, A–C and Video S1. Bulging—defined here as the development of an initial protrusion, which may be accompanied by a noticeable shrinking of the cell length—was observed to occur as fast as 100 ms (16) but on a typical timescale of 1 s. Swelling, defined here as the growth of the protrusion, was observed to occur on a typical timescale of 100 s (Fig. 1 D).

Figure 1.

Experimental observation of membrane bulging, swelling, and lysis. (A) Shown is a phase-contrast image of a population of E. coli cells immediately after antibiotic treatment (see also Video S1). (B) A phase-contrast image of the same population ∼1 h after antibiotic treatment shows that membrane bulging and swelling are common to most cells. (C) Shown is a phase-contrast time lapse of a single E. coli cell during antibiotic killing, with the corresponding stages of lysis denoted. (D) Histograms of the time for bulging and the time between bulging and lysis illustrate the separation of timescales involved. The population mean, standard deviation (SD), and cell number (N) are indicated.

In a recent modeling study (28), a critical cell wall pore size for bulging was found by studying the trade-off between the bending energy cost of bulging and the pressure-volume energy gained. This trade-off appears to be irrelevant for determining bulge size in our experiments, in which it can be shown that the bending energies are negligible compared to the stretching energies, as discussed below. As we shall see, membrane remodeling and the relaxation of the entropic and stretching energies of the cell envelope can predict bulging and are consistent with the separation of timescales shown in Fig. 1 D.

Cell envelope mechanics

We model the cell wall, IM, and OM as linear-elastic shells. Importantly, we suppose that, on timescales comparable to that of bulging, the membrane geometries can vary because of membrane fluidity while conserving their reference surface areas. This contrasts the IM and OM with the rigid cell wall, whose reference configuration is assumed to be a cylinder. The free energy of the cell wall, IM, OM, and the volume enclosed by the IM is as follows:

$$\mathcal{F}=E_{\text{stretch}}^w+E_{\text{stretch}}^i+E_{\text{stretch}}^o+E_{\text{bend}}^w+E_{\text{bend}}^i+E_{\text{bend}}^o-TS,$$ (1)

where the superscripts ^w, ⁱ, and ^o denote wall, IM, and OM quantities, respectively; E_stretch and E_bend are the stretching and bending energies, respectively, of an elastic shell; T is the temperature; and S is the entropy of mixing water and solutes corresponding to the turgor pressure. Here, only water molecules are assumed to be outside the cell, the solute molecules are assumed to be enclosed by the IM, and S = −k(n_s ln x_s + n_w ln x_w), where k is Boltzmann’s constant, x_s and x_w are the number fractions of solute and water molecules inside the IM, respectively, and n_s and n_w are the numbers of solute and water molecules, respectively. Whereas we will assume n_s to be fixed (as discussed below), n_w depends on the volume, Vⁱ, enclosed by the IM as n_w = Vⁱ/m_w, where m_w is the volume occupied per water molecule. We also assume the solution to be ideal and dilute: while we assume n_s = 9.5 × 10⁷ molecules for a typical cell of the dimensions considered here, n_w ≈ 2.6 × 10¹¹ molecules and n_s/n_w ∼ 10⁻⁴.

Because the cell wall, IM, and OM are thin, it is convenient to simplify the stretching energies by integrating over the thicknesses and working with stress resultants. In particular, for the cell wall, the planar Young’s moduli can be expressed in units of force per length. Building on evidence for a larger elastic modulus in the circumferential direction than the axial direction (21, 24), we assume an orthotropic constitutive relation for the cell wall, so that ${\sigma }_{xx}^w=Y_x^w\left(u_{xx}^w+{\nu }_{yx}^w{u}_{yy}^w\right)/\left(1-{\nu }_{xy}^w{\nu }_{yx}^w\right)$ and ${\sigma }_{yy}^w=Y_y^w\left(u_{yy}^w+{\nu }_{xy}^w{u}_{xx}^w\right)/\left(1-{\nu }_{xy}^w{\nu }_{yx}^w\right)$. Here and below, $\left(Y_x^w,Y_y^w,{\nu }_{xy}^w,{\nu }_{yx}^w\right)$ are the two-dimensional Young’s moduli and Poisson’s ratios of the cell wall. $\left({\sigma }_{xx}^{\alpha },{\sigma }_{yy}^{\alpha }\right)$ denote in-plane stresses (or strains, u) in the axial and circumferential directions, respectively, of the α component of the cellular envelope (α∈{i, o, w}); in general, we will also use x and y to denote orthogonal directions for geometries that are not cylindrical. $E_{\text{stretch}}^w$ can then be expressed as

$$E_{\text{stretch}}^w=\frac{1}{2}\int \frac{{\left({\sigma }_{xx}^w\right)}^2}{Y_x^w}+\frac{{\left({\sigma }_{yy}^w\right)}^2}{Y_y^w}-\left(\frac{{\nu }_{xy}^w}{Y_x^w}+\frac{{\nu }_{yx}^w}{Y_y^w}\right){\sigma }_{xx}^w{\sigma }_{yy}^{w}d{A}^w,$$ (2)

where dA is an area element in the deformed state. Unlike the rigid cell wall, the IM and OM are fluid and possess different stretching energies. Consistent with the fact that fluid membranes cannot support in-plane shears (29), we take the membrane shear moduli to be zero, so that the membrane stretching energies comprise of areal penalties alone,

$$E_{\text{stretch}}^{\alpha }=\frac{K_a^{\alpha }}{2}\int {\left(u_{xx}^{\alpha }+u_{yy}^{\alpha }\right)}^{2}d{A}^{\alpha }=\frac{K_a^{\alpha }}{2}\int {\left(\frac{\mathit{\Delta }{A}^{\alpha }}{A^{\alpha }}\right)}^{2}d{A}^{\alpha },$$ (3)

where α ranges over {i, o}, ΔA/A is the fractional change in membrane area, and the equality holds because here and below, we assume a linear theory in which higher-order terms in the strains are neglected. For a vanishing shear modulus, $K_a^{\alpha }$ is equivalent to the first Lamé coefficient in two-dimensional elasticity (30, 31). Values of K_a have been estimated to be in the range of K_a ≈ 0.03–0.24 N/m for E. coli spheroplasts depending on the external osmolarity and size (25) and K_a ≈ 0.2–0.4 N/m for red blood cells (RBCs) and giant unilamellar vesicles (27, 32), and these values are expected to be similar for the IM and OM (25). As shown below, these values imply that the IM and OM can be as load bearing as the cell wall. Finally, for characteristic parameter values, the bending energies of Eq. 1 are negligible compared to the stretching energies, which is usually the case for thin shells (33, 34, 35): whereas the bending energies scale as the third power of thickness, the stretching energies are linear in thickness. We therefore discard the bending energies in the expressions below and verify in the Supporting Materials and Methods that they do not change our results.

Homogeneity of membrane stresses

Before modeling the mechanics of lysis further, it is convenient to note a few properties of membrane stresses. As the reference membrane dimensions are allowed to vary because of fluidity, the strains u_xx and u_yy may vary. The stretching energy of Eq. 3 depends only on the trace of the membrane strain tensor, ΔA/A = u_xx + u_yy. Hence, manifesting the fluid nature of the membranes, mathematically minimizing Eq. 1 over the reference membrane dimensions shows that u_xx = u_yy at equilibrium. Let us write g(x,y) = u_xx = u_yy to denote the strains as a function of general surface coordinates, (x,y), and note that, in a linear theory, the area element of the deformed geometry is related to that of the reference geometry by dA_reference = (1 − 2g(x,y))dA_deformed. Consider now the free energy expression of Eq. 1 and suppose that the cell envelope is in a state of equilibrium in which the reference membrane area constraint applies. As $E_{\text{stretch}}^w$ and −TS do not depend on the reference membrane dimensions, the minimization of $\mathcal{F}$ in Eq. 1 is equivalent to the following:

$$\underset{g\left(x,y\right)}{\text{min}}\int g{\left(x,y\right)}^{2}dxdy,$$ (4)

subject to the constraint of a fixed membrane reference area,

$$\mathcal{A}=\int \left(1-2g\left(x,y\right)\right)dxdy,$$ (5)

where $\mathcal{A}$ is a constant membrane reference area. Minimizing the functional of Eq. 4 under the constraint of Eq. 5 shows that g(x,y) is constant. Thus, regardless of the deformed geometry, the membrane stresses are not only isotropic (36, 37) but also spatially homogeneous at equilibrium. As we will assume the IM and OM to have identical material properties and reference areas, the same argument applies for both the IM and OM and shows that the stresses in these two layers are everywhere identical. Intriguingly, the membrane stresses in other contexts, such as the junctions of epithelial cells and eukaryotic cell blebs, have also been suggested to be spatially uniform (38, 39, 40).

The homogeneity of the membrane stresses places constraints on the bulged geometries considered below. In particular, at equilibrium and without the cell wall, the stresses σ_xx = σ_yy = K_a(u_xx + u_yy) in a membrane are anticipated to satisfy Laplace’s law,

$$\frac{{\sigma }_{xx}}{{\kappa }_x}+\frac{{\sigma }_{yy}}{{\kappa }_y}=p,$$ (6)

where x and y are two principal directions, κ_x and κ_y are the two principal radii of curvature, and p is the turgor pressure. As the stresses are both isotropic and spatially homogeneous, we find that the equilibrium shapes of the membranes possess constant mean curvature. We will use this fact to constrain the shapes of the bulges we consider below as spherical caps.

The healthy state

To model the mechanics of lysis, the type of calculation we undertake below with Eq. 1 will be as follows. We ignore the cellular poles for simplicity and assume the reference surface area of the IM (and OM), $\mathcal{A}$ⁱ ($\mathcal{A}$^o), and the reference radius $r_0^w$ and length $L_0^w$ of the cell wall to be given. As the cell wall is rigid, we suppose that the membranes assume the shape of the cell wall, so that their deformed geometries are cylinders with radii and lengths (rⁱ,r^o) and (Lⁱ,L^o), respectively. This assumption will be supported by the numerical calculations below, which show that the membranes are in contact with the cell wall. Given the material properties of the cell envelope, $\mathcal{A}$ⁱ, $\mathcal{A}$^o, $r_0^w$, and $L_0^w$, we minimize $\mathcal{F}$ over 1) the deformed cylindrical cell wall dimensions, r^w and L^w, 2) the deformed cylindrical membrane dimensions, (rⁱ,r^o) and (Lⁱ,L^o), and 3) the reference membrane cylindrical dimensions, $\left(r_0^i,r_0^o\right)$ and $\left(L_0^i,L_0^o\right)$, which satisfy the constraints $2\pi r_0^i{L}_0^i\le {\mathcal{A}}^i$ and $2\pi r_0^o{L}_0^o\le {\mathcal{A}}^o$. The inequalities in these constraints allow for membrane invaginations and less membrane surface area to be stretched than is available. Because of steric exclusion, we further require rⁱ ≤ r^w ≤ r^o and Lⁱ ≤ L^w ≤ L^o. The 10 foregoing variables and the linear strain-displacement relations u_xx = (L − L₀)/L₀, u_yy = (r − r₀)/r₀ then entirely determine F. To summarize,

$${\mathcal{F}}_e=\underset{r^w,L^w,r^i,L^i,r^o,L^o,r_0^i,L_0^i,r_0^o,L_0^o}{\text{min}}\mathcal{F}\text{subject}\text{to}2\pi r_0^i{L}_0^i\le {\mathcal{A}}^i,2\pi r_0^o{L}_0^o\le {\mathcal{A}}^o,r^i\le r^w\le r^o,L^i\le L^w\le L^o$$ (7)

describes the equilibrium conformation of a healthy, intact cell.

To simplify the analysis further, we assume the IM and OM to share the same reference area for the remainder of this work, so that $\mathcal{A}$ⁱ = $\mathcal{A}$^o and all equations involving the OM are identical to their counterparts for the IM. We now solve for the equilibrium state both analytically and numerically, the former by determining the stresses using Laplace’s law and the reference area constraint and the latter by undertaking the minimization in Eq. 7 explicitly. For the former, we start by assuming r = r^w = rⁱ = r^o and L = L^w = Lⁱ = L^o, so that the envelope layers are in contact, and suppose the membrane reference area to be limiting, so that $2\pi r_0^i{L}_0^i={\mathcal{A}}^i$. As the membrane stresses are isotropic and homogeneous at equilibrium, we set $\sigma ={\sigma }_{xx}^i={\sigma }_{yy}^i={\sigma }_{xx}^o={\sigma }_{yy}^o$. It then follows from Laplace’s law that

$$\frac{pr}{2}={\sigma }_{xx}^w+2\sigma ,pr={\sigma }_{yy}^w+2\sigma ,$$ (8)

where $p=kT{n}_s/\left(\pi {\left(r_0^w\right)}^2{L}_0^w\left(1+u_{xx}^w+2u_{yy}^w\right)\right)$, and henceforth, all equalities will be accurate to the first order in the strains. Substituting the linear strain-displacement relations for the cell wall and solving for L and r, we find two simple expressions,

$$L=L_0^w-\frac{kT{n}_s\left(2Y_x^w{\nu }_{yx}^w-Y_y^w\right)}{2\pi r_0^w{Y}_x^w{Y}_y^w}+2\sigma L_0^w\left(\frac{Y_x^w{\nu }_{yx}^w-Y_y^w}{Y_x^w{Y}_y^w}\right),r=r_0^w+\frac{kT{n}_s\left(2Y_x^w-{\nu }_{xy}^w{Y}_y^w\right)}{2\pi L_0^w{Y}_x^w{Y}_y^w}+2\sigma r_0^w\left(\frac{Y_y^w{\nu }_{xy}^w-Y_x^w}{Y_x^w{Y}_y^w}\right).$$ (9)

It remains to determine σ using the reference area constraint. As u_xx = u_yy = σ/(2K_a) at equilibrium, the reference area constraint can be expressed as follows:

$${\mathcal{A}}^i=2\pi rL\left(1-\frac{\sigma }{K_a}\right).$$ (10)

Substituting Eq. 9 into Eq. 10 yields a single equation in σ, for which the solution is as follows:

$$\sigma =\frac{K_a(-\left({\mathcal{A}}^i-2\pi r_0^w{L}_0^w\right)Y_x^w{Y}_y^w+kT{n}_s\left(2Y_x^w\left(1-{\nu }_{yx}^w\right)+Y_y^w\left(1-{\nu }_{xy}^w\right)\right)}{2\pi r_0^w{L}_0^w\left(2K_a\left(Y_x\left(1-{\nu }_{yx}^w\right)+Y_y\left(1-{\nu }_{xy}^w\right)\right)+Y_x^w{Y}_y^w\right)};$$ (11)

in turn, this solution determines all quantities of the equilibrium state. Building on evidence of finite excess membrane area in E. coli (25, 41), we plot the predicted value of σ in Eq. 11 against $\mathcal{A}$ⁱ in Fig. 2 A for the parameter values summarized in Materials and Methods and note that, when γ = $\mathcal{A}$ⁱ/$\mathcal{A}$^w ≈ 1, the membrane stresses are nonzero and decreasing in γ. For the parameter values considered in this work, this occurs until γ ≈ 1.15, at which point the equality in Eq. 10 is no longer valid—only the cell wall deforms—and the membrane stresses become zero. That the membrane stresses can be nonzero for smaller γ contrasts with the idea that the cell wall is the only load-bearing structure of the cellular envelope and is consistent with experimental observations suggesting that the IM and OM can also be load bearing, as manifested by the known fact that bulging precedes lysis (16). As the IM and OM are fluid, load bearing by the IM and OM does not contradict the fact that E. coli cells become spherical without their cell walls (42, 43, 44).

Figure 2.

Stresses in the cellular envelope. (A) Shown are the stresses for the cell wall (w) and inner and outer membranes (i and o) as functions of the reference membrane area ratio, γ, with both the linear theory predictions (Eqs. 8 and 11) and independent numerical results plotted. The inner and outer membranes are assumed to share identical material properties and exhibit identical stresses. (B and C) Shown is the bar plot representation of two points in (A), as well as the stresses across the entire cell envelope (tot). The dashed lines indicate the linear theory predictions (Eqs. 8 and 11), whereas the bars indicate numerical results. To see this figure in color, go online.

To verify the foregoing calculations, we numerically minimized Eq. 7 for the parameter values summarized in Materials and Methods and different values of $\mathcal{A}$ⁱ starting from γ = 1.0. We found that $\mathcal{F}$ is minimal when $2\pi r_0^i{L}_0^i,2\pi r_0^o{L}_0^o={\mathcal{A}}^i$ and r^w = rⁱ = r^o ≈ 0.53 μm, L^w = Lⁱ = L^o ≈ 9.9 μm, $r_0^i=r_0^o\approx 0.52\mu \text{m}$, and $L_0^i=L_0^o\approx 9.7\mu \text{m}$. In this case, all envelope components are in contact and the stresses are ${\sigma }_{xx}^w\approx 1.5\text{mN}/\text{m}$, ${\sigma }_{yy}^w\approx 12.6\text{mN}/\text{m}$, and ${\sigma }_{xx}^i={\sigma }_{xx}^o={\sigma }_{yy}^i={\sigma }_{yy}^o=K_a\left(u_{xx}^i+u_{yy}^i\right)=K_a\left(u_{xx}^o+u_{yy}^o\right)\approx 4.8\text{mN}/\text{m}$, in good agreement with the linear theory (Fig. 2 B; Table S1). We then repeated the foregoing calculations across a range of larger reference surface areas $\mathcal{A}$ⁱ. We found similar results in all cases, with the membrane stresses being generally dependent on $\mathcal{A}$ⁱ and decreasing in agreement with the linear theory (Fig. 2 C; Table S1). Below, we will also show that our prediction of bulging holds over a range of $\mathcal{A}$ⁱ.

The bulged state

We now show that the removal of a piece of cell wall can result in bulging. Assuming that the membrane reference surface areas remain unchanged over the timescale of bulging, we consider a quasiequilibrium state in which they limit bulging. In contrast to the membrane areas, we do not assume cell volume to be limiting; as discussed below, characteristic timescales of water flow are fast in comparison to bulging. Since osmoregulation is believed to occur on a timescale of ∼1 min for osmotic shocks applied over less than 1 s (45, 46, 47), we also assume the number of solute molecules to remain constant. The free energy may be lowered by water flow and bulging if the IM and OM may assume arbitrary geometries. Hence, we wish to minimize $\mathcal{F}$ over the cell geometry and the cellular dimensions, assuming that the membrane reference surface areas are fixed.

As mentioned above, when $\mathcal{F}$ is minimized, net flow of water into the cytoplasm may be required. The bulk flow of water from the external milieu to the cytoplasm is thought to be characterized by the hydraulic conductivity, L_p (48), defined so that the instantaneous volumetric flow rate through a membrane is dV/dt = L_pA_totΔP, where A_tot is the total (strained) membrane surface area and ΔP is the pressure difference across the membrane, hereafter taken to be the turgor pressure p (45, 48). Estimates of L_p vary depending on membrane structure; studies involving osmotically shocked bacteria (49), liposomes with aquaporin-1, and RBCs have found L_p ≈ 10⁻¹² m³/N⋅s, whereas studies for liposomes and other bilayers without water channels have indicated L_p ≈ 10⁻¹³ m³/N⋅s (48, 50). The larger value of L_p predicts a volume increase of ∼20% of the initial cell volume per second. As our model will predict smaller or similar volume increases, we will assume that water flow is not limiting in the analysis below.

We now suppose that an area A of the cell wall is removed. For simplicity, we assume A to be a circle of radius r_d (Fig. 3 A). As discussed in Homogeneity of membrane stresses, we consider bulge geometries with a constant mean curvature. Koiso proved that the only constant mean curvature surfaces with a circular boundary, which are only contained on one side of the boundary, are spherical caps (51). Consistent with the shapes observed in experiments (Figs. 1, A–C and 3 A), we therefore describe the bulged state by a two-parameter family of geometries in which a spherical bulge, B, of radius R extrudes from A, with the degree of extrusion described by the subtended angle θ (Fig. 3 A). At equilibrium, we require that the bulge fills the defect, so that r_d = R sin θ.

Figure 3.

Stresses and energetics of the bulged conformation. (A) Shown is a schematic of the bulged conformation (left), in which a circular cell wall defect of radius r_d is introduced to the strained state and a spherical bulge forms over the defect (right). (B) Shown are linear theory predictions for the subtended angle (θ), cell radius (r), and cell length (L) as functions of the membrane reference area ratio (γ) and defect radius (r_d) found by solving the bulging equation, Eq. 17. For large defect radii corresponding to large cell wall stresses, the linear theory becomes inaccurate, and the predictions for the cell length deviate. (C and D) Shown are stresses for the cell wall (w), inner and outer membranes (i and o), and across the entire cell envelope (tot) for γ = 1.0 and 1.2. The inner and outer membranes are assumed to share identical material properties and exhibit identical stresses. The dashed lines indicate the linear theory predictions found by solving Eq. 17, whereas the bars indicate numerical results. To see this figure in color, go online.

To simplify the analysis and reduce the number of free variables below, we assume the cell wall and the membranes to remain in contact in the cylindrical bulk, so that their strained dimensions are described by the two parameters r = r^w = rⁱ = r^o and L = L^w = Lⁱ = L^o. Relaxing this assumption does not change our results; in particular, repeating the minimization of Eq. 12 below, but allowing (r^w,rⁱ,r^o) and (L^w,Lⁱ,L^o) to vary independently while satisfying the corresponding steric exclusion constraints, will result in the same minimizers. As the membrane stresses are everywhere equal at equilibrium, the membranes must be in contact in the bulge, so that R and θ do not differ for the IM or OM. The free energy of the bulged state, as denoted by the subscript _b, is then

$${\mathcal{F}}_b\left(r_d\right)=\underset{\begin{array}{c}r,L,R,r_0^i,L_0^i,r_0^o,L_0^o,\\ R_0^i,R_0^o\end{array}}{\text{min}}\left[E_{\text{stretch}}^i+E_{\text{stretch}}^o\right]\left(A_B\right)+\left[E_{\text{stretch}}^i+E_{\text{stretch}}^o+E_{\text{stretch}}^w\right]\left(A_{\text{cell}}\right)-TS\left(V_b^i\right),$$ (12)

where the dependence of the stretching energies on the different geometries are indicated by the areas of the geometries, A_B = 2πR²(1 − cos θ) is the strained bulge area, $A=\pi r_d^2$ is the strained area removed, A_cell = 2πrL − A is the remaining surface area, ignoring the cellular poles, $S\left(V_b^i\right)$ is the entropy of mixing corresponding to an IM volume $V_b^i=\pi r^{2}L+V^{\text{∗}}$, and $V^{\text{∗}}=\pi R^3\left(2-3\text{cos}\theta +{\text{cos}}^3\theta \right)/3$ is the bulge volume. In the cylindrical bulk, the strained membrane and cell wall dimensions are related to the reference dimensions, r₀ and L₀, in the usual manner by u_xx = (L − L₀)/L₀ and u_yy = (r − r₀)/r₀. In the bulge, the strained membrane dimensions are related to the reference membrane dimensions as u_xx,B = u_yy,B = (R − R₀)/R. Accurate to the first order in the strains, the constraint on the reference membrane area for the IM can be expressed as follows:

$$2\pi r_0^i{L}_0^i-A\left(1-u_{xx}^i\right)\left(1-u_{yy}^i\right)+2\pi {\left(R_0^i\right)}^2\left(1-\text{cos}\theta \right)\le {\mathcal{A}}^i,$$ (13)

and analogously for the OM. When θ = 0, the cell exhibits no bulging in response to the defect over A.

We proceed to solve for the equilibrium state corresponding to Eq. 12 and the associated reference membrane area constraint both analytically and numerically. The homogeneity of membrane stresses and Laplace’s law require the following:

$${\sigma }_{xx,B}^i={\sigma }_{yy,B}^i={\sigma }_{xx,B}^o={\sigma }_{yy,B}^o={\sigma }_{xx}^i={\sigma }_{xx}^o={\sigma }_{yy}^i={\sigma }_{yy}^o=\frac{pR}{4},$$ (14)

$${\sigma }_{xx}^w+{\sigma }_{xx}^i+{\sigma }_{xx}^o=\frac{pr}{2},{\sigma }_{yy}^w+{\sigma }_{yy}^i+{\sigma }_{yy}^o=pr,p=\frac{kT{n}_s}{V_b^i},$$ (15)

where $V_b^i=\pi {\left(r_0^w\right)}^2{L}_0^w\left(1+u_{xx}^w+2u_{yy}^w\right)+V^{\text{∗}}$, and, as before, all equalities will be accurate to the first order in the strains. Substituting the strain-displacement relations for the cell wall and solving Eqs. 14 and 15, we find the following:

$$L=L_0^w\left(1+\frac{kT{n}_s\left(\left(R-2r_0^w\right)Y_x^w{\nu }_{yx}^w+\left(r_0^w-R\right)Y_y^w\right)}{2Y_x^w{Y}_y^w\left(\pi {\left(r_0^w\right)}^2{L}_0^w+V^{\text{∗}}\right)}\right),r=r_0^w\left(1+\frac{kT{n}_s\left(Y_x^w\left(2r_0^w-R\right)+\left(R-r_0^w\right)Y_y^w{\nu }_{xy}^w\right)}{2Y_x^w{Y}_y^w\left(\pi {\left(r_0^w\right)}^2{L}_0^w+V^{\text{∗}}\right)}\right).$$ (16)

Assuming the reference area constraint to be an equality, we note that it uniquely determines the equilibrium state and can be rewritten as $\left(2\pi rL-\pi r_d^2\right)\left(1-2u\right)+\left(2\pi R^2\left(1-\text{cos}\theta \right)\right)\left(1-2u\right)={\mathcal{A}}^i$, where, at equilibrium, u = pR/(8K_a). As the bulge fills A, R sin θ = r_d. Combining this with Eqs. 15 and 16, we re-express the reference area constraint as a single, transcendental equation involving the variable θ only:

$${\mathcal{A}}^i=2\pi r_0^w{L}_0^w-\pi r_d^2\left(1-\frac{2}{1+\cos \theta }\right)+\frac{3kT{n}_s}{4K_a{Y}_x^w{Y}_y^w}\times \frac{\mathit{\Phi }\left(\theta \right)}{3{\left(r_0^w\right)}^2{L}_0^w{\sin }^3\theta +r_d^3\left(2+\cos \theta \right){\left(\cos \theta -1\right)}^2},$$ (17)

where $\mathit{\Phi }\left(\theta \right)=2r_0^w{L}_0^w\left(2K_a{r}_0^w{\sin }^3\theta \left(2Y_x^w\left(1-{\nu }_{yx}^w\right)+Y_y^w\left(1-{\nu }_{xy}^w\right)\right)-r_d{\sin }^2\theta \left(Y_x^w{Y}_y^w+2K_a(Y_x^w\left(1-{\nu }_{yx}^w\right)+Y_y^w\left(1-{\nu }_{xy}^w\right)\right)\right)-r_d^3{Y}_x^w{Y}_y^w{\tan }^2\left(\theta /2\right){\sin }^2\theta$. Eq. 17, the bulging equation, is the main result of this work; its solution for θ determines the equilibrium state and all associated variables. Numerical solutions of the bulging equation for different values of r_d and $\mathcal{A}$ⁱ are shown in Fig. 3 B. We find that θ increases with $\mathcal{A}$ⁱ and that the cell length, and not the radius, predominantly shrinks during bulging. For a typical value of r_d = 0.5 μm and γ = 1.0, these results predict the formation of a hemispherical bulge with θ ≈ 1.6 and R ≈ 0.5 μm, which will be compared with full numerical calculations below.

To gain further intuition for the solutions of the bulging equation, we considered two simple cases. First, asymptotically expanding the bulging equation around θ = π, we find the following:

$$\theta \approx \pi -2r_d\sqrt{\frac{\pi }{{\mathcal{A}}^i}},$$ (18)

so that, in this limit, large membrane reference areas give rise to full and large bulges, irrespective of the material properties of the cell envelope. Indeed, for full bulges, Eq. 18 shows that the subtended angle depends only on the ratio of defect to reference areas, $\text{A}$/$\mathcal{A}$ⁱ. Second, we considered a simple case in which $Y_x^w=Y_y^w=Y$ and neglected Poisson’s effect. Accurate to the first order in r_d/R, the solution of the bulging equation reduces to the following:

$$\theta \approx \frac{r_d}{R},R\approx \frac{6kT{n}_s{K}_a{r}_0^w-4\left(\gamma -1\right)K_{a}Y\pi {\left(r_0^w\right)}^2{L}_0^w}{kT{n}_s\left(4K_a+Y\right)}.$$ (19)

Thus, our model predicts that, for small cell wall defects relative to the bulge radius, cells with large membrane reference areas produce full, but small, bulges, whereas cells with small membrane reference areas produce shallow, but large, ones. These results are consistent with Fig. 3 B and the intuition that cells with excess membrane area may form large bulges by “throwing away” the excess area. Nevertheless, a comparison to experimental data will suggest the excess membrane area in typical cells to lie in a limited range (see Model of swelling below).

To support the analytical calculations above, we numerically computed the minimum of Eq. 12 over the 9 independent variables subject to the reference area constraint. We found that, for a range of $\mathcal{A}$ⁱ, the numerical minimizers of Eq. 12 are generally well described by the linear theory (Fig. 3, B–D; Table S1). When r_d = 0.5 μm and γ = 1.0, for instance, we find that θ ≈ 1.5 and R ≈ 0.5 μm, in excellent agreement with the linear theory. These results also predict that the cell length, in contrast to the cell width, shrinks significantly during bulging (Fig. 3 B). Consistent with our assumption that water flow is not limiting during bulging, when r_d = 0.5 μm and γ = 1.0, the fractional volume increase relative to the healthy state is ΔV < 1%, whereas for γ = 1.2, ΔV increases to ΔV ≈ 25%. As mentioned above, typical values of L_p for bacteria predict volume increases on the order of ΔV ≈ 20% per second. Thus, we conclude that bulging is energetically favorable, and the observed timescale of bulging is consistent with water flow into the membrane.

Implications for dynamics

The foregoing analyses show that, for a range of membrane reference areas, bulging corresponds to an equilibrium state: the only stable configuration of the cellular envelope is one in which bulging occurs. In general, our model predicts that partially subtended, spherical bulges form upon the introduction of cell wall defects (Fig. 3 B) and clarify the resulting stresses (Fig. 3, C and D). By elucidating the factors determining bulge size, our results reveal the importance of membrane stretching and contrast with Daly et al.’s study examining critical defect sizes for bulge nucleation (28), in which the authors studied the trade-off between the bending energy cost of bulging and the pressure-volume energy gained.

Our model also assumes that the membranes may slide against the wall because of the differing strain rates of envelope components. For instance, $u_{xx}^w$ typically decreases after bulging, whereas $u_{xx}^i$ and $u_{xx}^o$ remain approximately unchanged (Figs. 2, B and C and 3, C and D). Although molecules such as Braun’s lipoprotein anchor the OM to the cell wall, the estimated number of such OM-wall anchors (∼10⁶) are few in comparison to the estimated numbers (∼10⁷) of phospholipids (52, 53). Hence, free phospholipids could modulate the reference states and allow for membrane reorganization. The assumption of sliding is therefore consistent with physical coupling of the OM to the cell wall.

Importantly, a central prediction of our model is that bulging arises as a relaxation process. Hence, the timescale of bulging is determined by the equilibration of $\mathcal{F}$_b. Balancing the energy dissipation with the viscous drag on the bulge results in a timescale much smaller than 100 ms (Supporting Materials and Methods), suggesting that the relaxation time may specifically be limited by membrane reorganization (54). We anticipate further experiments (for instance, ones that modulate membrane fluidity during β-lactam killing) to better clarify the processes limiting relaxation.

Model of swelling

Having shown that bulging arises as a relaxation process leading to a metastable state, we now demonstrate that swelling—the increase of bulge volume over a much longer timescale of minutes—is consistent with the growth of cell wall defects. As we anticipate that the energetic trade-offs considered above remain relevant on the slower timescale in which cell wall defects grow, the model of bulging also predicts bulge size during swelling, as shown below. The significant difference between the timescales of bulging and swelling (Fig. 1 D) can then be explained by a separation of timescales due to 1) energetic relaxation and 2) defect growth.

During swelling, the amount of water uptake is determined by the same balance of the entropic and stretching energies of the cellular envelope as above; if lysis did not occur, then net flow into the cytoplasm would occur until the membranes are sufficiently stretched. In fact, the small synthesis rate of membrane material relative to water flow (41) suggests that water flow is not limiting and that the membranes are always stretched. To support this notion, we analyzed the swelling of E. coli cells of different lengths over ∼10 s and found that the population-averaged volumetric flow rate does not depend on the membrane surface area (Fig. 4 A), as would be the case if membrane synthesis was fast and water flow became limiting. In contrast, image analysis reveals that bulges grow at a rate consistent with Eq. 12 when the defect radius, r_d, also increases, supporting the notion that the reference membrane areas remain limiting (Fig. 4 B). This result therefore suggests defect growth to be the limiting step of bulge growth before lysis.

Figure 4.

Statistics of swelling cells. (A) Shown is a plot of the volumetric flow rate dV/dt against the total membrane surface area A_tot for 112 swelling cells of different lengths and one or two data points per cell. Bulged cells were fit to cylinders with protruding spheres; see the Supporting Materials and Methods for details on the image analysis methodology. The scatter indicates cell-to-cell variability. As the slope of a linear fit to the data is (2 ± 5) × 10⁻⁹ m³⋅Pa/N⋅s, dV/dt does not increase with A_tot as dV/dt = L_pA_totp, and we conclude that the cellular volume increase during swelling is not governed by water flow. (B) Shown is a plot of the moving average of $V^{\text{∗}}/V_b^i$, the fractional bulge volume, against the defect radius r_d for the same cells in (A), with 1) the linear theory predictions for different γ, as found by solving Eq. 17, and 2) the prediction corresponding to adding a hemispherical bulge of radius r_d irrespective of the model overlaid. To see this figure in color, go online.

We next wondered whether the observed range of bulge volumes suggested a typical value for the excess membrane area; indeed, we found that the empirically observed fractional bulge volumes in Fig. 4 B suggest any pre-existing excess membrane area in healthy cells to be limited. In the case in which γ = 1.5, for instance, the formation of a full bulge of radius R ≈ 1 μm, corresponding to a fractional bulge volume of $V^{\text{∗}}/V_b^i\approx 0.4$, would be energetically favorable even for limitingly small defect radii as the cell initially “throws away” the excess membrane area. Nevertheless, as shown in Fig. 4 B, such large bulge volumes are not observed for defect radii less than ∼0.8 μm. Instead, the data are consistent with excess membrane area in the range of γ = 1.0 to γ = 1.2, and the value γ = 1.05 provides the best fit (Fig. 4 B). Intriguingly, the solution of Eq. 17 predicts a sharp increase in bulge volume when γ = 1.05 and r_d ≈ 0.3 μm, which arises from the transition between small and large subtended angles (Fig. 3 B). For larger values of γ such as γ = 1.2, the subtended angles are large across a broader range of r_d (Fig. 3 B), and hence, the predicted dependence of bulge volume on the defect radius becomes much shallower.

As our model also predicts that the mechanical stresses in the bulge increase because of increasing bulge size, swelling may occur until the cell lyses. Since the mean bulge radius at lysis is R ≈ 0.8 μm, assuming the same parameter values as in Materials and Methods and that the number of solutes has not changed due to osmotic stress responses (45, 46, 47, 49, 55, 56, 57) suggests the yield areal strain of the E. coli IM and OM to be approximately 20%. This estimate is consistent with the empirical range of RBCs and lipid vesicles that are stretched on the timescale of 0.1–100 s (58, 59, 60) and exceeds that of RBCs under quasistatic loading (61). The final step of lysis is, therefore, consistent with material failure of the IM and OM under turgor pressure loading.

Conclusions

To summarize, we have used a continuum, elastic description of the cellular envelope to model membrane bulging and found evidence that defect enlargement underlies swelling. Our results underscore the different roles of each envelope component in resisting mechanical stresses and indicate that bulging can arise as a relaxation process mediated by membrane fluidity and water flow once a wall defect exists. These findings have implications on cellular physiology and morphogenesis. Because bulging and swelling result in eventual lysis and are mediated by cell wall defects, the existence of large pores in bacterial cell walls can be deadly.

In many rod-shaped bacteria, including E. coli, the cell wall is locally and dynamically remodeled by protein complexes that rotate around the cell, but how these protein complexes maintain a cell-spanning rod shape is unknown (62, 63, 64, 65, 66). Our work shows that cell wall remodeling processes must regulate pore size and suggests constraints on how PG synthases can hydrolyze pre-existing PG. Although cell wall remodeling at the scale of the micron-sized defects considered here occurs on a slower timescale than that of bulging (22), a growth mechanism that regulates pore size could preemptively help cells avoid lysis, in addition to regulating wall thickness and straight, rod-like morphology (33). In general, our work illustrates that analyzing mechanical instabilities and failure modes in cells can constrain how physiological growth pathways function. Conversely, exploiting the physical consequences of large cell wall defects may lead to novel approaches for developing new antibiotics.

Beyond bacterial morphogenesis, the combination of theory and experiment in our work has underscored characteristics of biological membrane physics and the importance of mechanical stresses in cells. By being free to change their reference geometries, fluid membranes differ from rigid, elastic shells, and we have shown that this difference has physiological implications on cell envelope mechanics and how mechanical stresses are distributed between membrane-solid layers. Our study therefore paves the way for investigating similar interactions of fluid membranes with elastic surfaces (67) and understanding the material nature of living cells across different contexts.

Author Contributions

F.W. and A.A. conceived the project, performed modeling, and wrote the article. F.W. performed experiments and analyzed data.

Editor: Amy Palmer.

Supporting Citations

References (68, 69) appear in the Supporting Material.