Abstract
Membrane lysis, or rupture, is a cell death pathway in bacteria frequently caused by cell wall-targeting antibiotics. Although several studies have clarified 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, where, strikingly, the formation of an initial bulge (“bulging”) after cell wall digestion occurs on a characteristic timescale as fast as 100 ms and the growth of the bulge (“swelling”) occurs on a slower characteristic timescale of 10-100 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 experimentally observed bulge shapes are consistent with model predictions. We then show that swelling can involve both the continued flow of water into the cytoplasm and 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. Our results contrast biological membrane physics and the physics of thin shells, reveal principles of how all bacteria likely function in their native states, and may have implications for cellular morphogenesis and antibiotic discovery across different species of bacteria.
Antibiotic resistance is one of the largest threats to global health, food security, and development today.1 Its increasing prevalence2 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 mechan-otransduction in eukaryotes3 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 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 E. coli, the two-dimensional cell wall is sandwiched between the inner and outer membranes (IM and OM), while in Gram-positive species the cellular envelope comprises an inner membrane 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, including penicillin and β-lactams, bind to transpeptidases to inhibit cross-linking. Inhibition of peptide bond formation, combined with mislocalized wall degradation by PG hydrolases, has been thought to result in large defects in the cell wall which precede bulging of the IM and OM and eventual cell lysis.6,7,18,19
Results
Dynamics of bacterial cell lysis
Inspired by previous work,16 we degraded wild-type E. coli cell walls with cephalexin, a β-lactam antibiotic, and observed typical cells to undergo the morphological transitions shown in Fig. 1A-C and Supplementary Video 1. Bulging—defined here as the development of a protrusion under an approximately constant volume, which is accompanied by a noticeable shrinking of the cell length—was observed to occur on a timescale as fast as 100 ms. Swelling, defined here as the growth of the protrusion, which may involve more variation in cytoplasmic volume, was observed to occur on a timescale of 10-100 s (Fig. 1D).16
Physical modeling can elucidate the mechanisms which drive membrane bulging and swelling, and, as suggested above, understanding this is important because mechanical failure of the cell wall is a poorly understood, yet typical and exploitable, cell death pathway.16,18,19,20,21 Furthermore, the role of the cellular envelope here may entail different physics than that of eukaryotic blebbing.22,23,24,25,26,27 In a recent modeling study,28 a critical 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 the aforementioned β-lactam experiments, where it can be shown that the bending energies are negligible compared to the stretching energies and that shortening of the cell contributes to bulge growth (Fig. 1C-D). 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 experimental observations.
Cell envelope mechanics
We model the cell wall, IM, and OM as elastic shells in contact. Although fluid membranes cannot support in-plane shears,29 we do not consider shear strains and stresses in this work and the strain energy is effectively that of an elastic shell. Importantly, we also suppose that, on timescales longer than that of the elastic response, the membrane geometries can vary due to membrane fluidity while conserving their reference surface areas. This contrasts 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 where the superscripts w, i, and o denote wall, IM, and OM quantities, respectively, Estretch and Ebend 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. Here only water molecules are assumed to be outside the cell and S = −k(ns ln xs + nw ln xw), where k is Boltzmann’s constant, xs and xw are the number fractions of solute and water molecules, respectively, and ns and nw are the numbers of solute and water molecules, respectively. We assume an ideal, dilute solution in this work and note that the presence of the entropic term implies that, when the chemical potentials of water are equal both inside and outside the cell, the mechanical stresses in the cellular envelope are proportional to p = kTC, where C is the number density of solutes inside the cell and p is defined as the turgor pressure (Supplementary Information, SI). Assuming characteristic parameter values, the bending energies are negligible compared to the stretching energies, as is typically the case for thin shells.30,31,32 We therefore discard the bending energies in the expressions below and verify in the SI that they do not change our results. We assume linear, isotropic constitutive relations for the IM and OM and an orthotropic constitutive relation for the cell wall, building on evidence for a larger elastic modulus in the circumferential direction than the axial direction.33,34 Estretch can then be expressed as , where dA is an area element and
Here (Y, ν) are the two-dimensional Young’s modulus and Poisson’s ratio of the membranes, are the orthotropic analogues for the cell wall, and denote in-plane stresses in the axial and circumferential directions, respectively, of the α component of the cellular envelope. We relate Yi and Yo to the area-stretch moduli Ka of lipid bilayer membranes by . Values of Ka have been estimated to be in the range of Ka ≈ 0.03-0.25 N/m for E. coli spheroplasts depending on external osmolarity and size35 and Ka ≈ 0.2-0.4 N/m for red blood cells (RBCs) and giant unilamellar vesicles,36,37 and these values are expected to be similar for bacterial membranes.35
As the membrane stresses may vary due to the in-plane rearrangement of phospholipids,38,33 we assume that they can be determined in a healthy cell by minimizing over the possible reference states. The existence of excess membrane area does not significantly change our results and involves calculations similar to that considered below. For a range of parameter values believed to be relevant to the E. coli cellular envelope, and are similar in magnitude to Yi and Yo. Because is quadratic in the in-plane stresses, the minimization of in Eq. (1) predicts that both the cell membranes and the cell wall are load-bearing. In particular, over the cylindrical bulk of a spherocylindrical cell for which with α ∊ {i, o, w} and for characteristic values of ,35,33,39,40 ,33,34 and Poisson’s ratios all set to 0.2,41,42 we find that is minimal when , , , and , where r is the cell radius. This result 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.
When is minimized over the cellular dimensions, 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 Lp,43 defined so that the instantaneous volumetric flow rate through a membrane is dV/dt = LpAtotp, where p is the turgor pressure and Atot is the total membrane surface area.43,44 Estimates of Lp vary depending on membrane structure: studies involving osmotically shocked bacteria,45 liposomes with aquaporin-1, and RBCs have found Lp ≈ 10−12 m3/N · s, while studies for liposomes and other bilayers without water channels have indicated Lp ≈ 10−13 m3/N · s.43,46 Below, we find that the larger value of Lp is consistent with a volume increase on the order of 1-10% of the initial cell volume during bulging and that water flow can also contribute to swelling.
Model of bulging
We now show that, over a timescale of 100 ms, removal of a piece of cell wall can result in bulging. As the amount of excess membrane surface area is finite,35 we assume that the surface areas of the reference states of the IM and OM remain unchanged over the timescale of bulging. We therefore consider a quasi-equilibrium state in which the membrane reference surface areas limit bulging, for which the envelope stresses correspond to those caused by turgor pressure loading. Furthermore, since osmoregulation is believed to occur on a timescale of ~1 min for osmotic shocks applied over less than 1 s,44,47,48 we assume the number of solute molecules to remain constant after cell wall degradation. 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 over the cell geometry and the cellular dimensions, assuming that the membrane reference surface areas are fixed.
Suppose that an area A of the cell wall is removed. For simplicity, we assume A to be a circle of radius rd. If the IM and OM were at equilibrium without bulging, then ; we denote the stresses satisfying force balance over A with the subscript A and quantities of the unbulged state with the subscript u. As discussed below, the solutes may be diluted due to water flow, so that pu = pVcell/Vu, where Vcell = 2πrL is the volume of a healthy cell, L is its length, and Vu is the volume of the unbulged state. Assuming the linear strain-displacement relations and for the cell wall, where (uxx, uyy) denote in-plane strains and (L0, r0) the reference length and radius, respectively, the dimensions (Lu, ru) of the cylindrical bulk in the unbulged state, which may differ from the dimensions (L, r) of a healthy cell, completely determine the stresses in the cell wall. The condition of force balance then constrains the stresses in the IM and OM over the remainder of the cellular envelope. Thus, the free energy of the unbulged state is where is the area removed, Acell,u = 2πruLu − A is the remaining surface area, ignoring the end-caps, S(Vu) is the entropy of mixing corresponding to , with α ∊ {i, o, w}, and the in-plane stresses are compactly denoted by σ = (σxx, σyy). (Lu, ru) are related to (L, r) by the condition of fixed reference membrane areas, which for simplicity is considered only for the IM in this work and requires , where the scripted symbols denote corresponding reference quantities and , and Acell = 2πrL is the surface area of a healthy cell. Due to water flow, the volume enclosed by the IM may increase, and we show below that the amount of volume increase is consistent with the timescale of bulging. We now wish to find cell envelope geometries for which the free energy is lowered.
As an ansatz which will later be supported by comparison with experiments, we suppose the formation of an ellipsoidal bulge with radii (a, a, b) and some circular cross-section coinciding with A, as described by the parametric angle θ = sin−1(rd/a) (Fig. 2A). We neglect the bending energy of the neck, as detailed in the SI (see also Fig. S1). As the reference membrane surface areas are conserved, bulge formation requires area to be appropriated from the cylindrical bulk. Below, we examine the change in stretching energy of the cylindrical bulk under the assumption that a reference membrane area is removed and relegated to the bulge, and then include the change in stretching energy due to the formation of an ellipsoidal bulge over A with reference area (whose calculation for an ellipsoidal shell is discussed in the SI).
To determine the change in stretching energy of the cylindrical bulk once a reference area is removed, we first examine how the bulk contracts. As above, the dimensions (Lb, rb) of the cylindrical bulk after bulging completely determine the stresses in the cell wall, and the condition of force balance constrains the stresses in the IM and OM. The stretching energy of the cylindrical bulk after bulging can be expressed as where the in-plane stresses of the IM and OM satisfy with α ∊ {i, o, w}, pb = pVcell/Vb, where Vb is the volume of the bulged cell, and Acell,b = 2πrbLb − A is the surface area of the bulk. Here the subscript b denotes quantities of the bulged state. For any , the minimizers of subject to the constraint determine the dimensions of the bulk, where . Given that the cylindrical bulk shrinks, and that the reference area by which it shrinks is relegated to the ellipsoidal bulge, the free energy of the bulged state can then be expressed as where the independent variables satisfy the reference area constraint. Here the aspect ratio ε = b/a, S(Vb) is the entropy of mixing corresponding to a bulged volume is the volume of the bulge, and the stretching energy of the bulge is where, for α ∊ {i, o}, and
are the total in-plane stresses of the bulge as functions of parametric coordinates (x, y) = (θ′, φ). θ′ and φ denote the parametric angles along and around the axis of symmetry, respectively, and dAbulge = a sin θ′[a2 + (b2 – a2) sin2 θ′]1/2d φ dθ′ is the surface area element of the bulge (SI).
Bulging is energetically favorable due to the relaxation of the entropic and stretching energies when . For characteristic parameter values relevant to E. coli, is calculated numerically as detailed in the SI and plotted in Fig. 2B. The individual contributions to of both the changes in entropic energy and stretching energies of the bulk and the bulge are examined in the SI and Fig. S2. Although diverges as θ → 0—corresponding to the formation of a small bulge with diverging radii of curvature—there are alternate paths to the energetic minima for which no energetic barriers are present and bulging occurs spontaneously. One such path, in which a partial spherical bulge of fixed radius protrudes from the defect area before filling the defect area completely, is shown in Fig. 2C and analyzed in the SI. Importantly, for a wide range of rd, the configuration in which no bulging occurs is unstable, and the predicted bulge geometries appear consistent with experimental measurements (Fig. 3). For larger values of rd, bulges corresponding to smaller defects are, intriguingly, predicted to subtend larger angles, while the predictions for small rd can be supported by analytical calculations (SI). Moreover, our model predicts a volume increase on the order of 1-10% for characteristic values of rd, with larger volume changes corresponding to larger rd, and is consistent with a relaxation process in which the membranes slide against the wall due to the different strain rates of envelope components and the cell wall shrinks in the axial direction due to the membranes bearing larger loads (SI).
We note three further implications of our analysis. First, bulging can be energetically favorable for a defect radius as small as rd = 10 nm (SI and Fig. S2). Second, over a large range of rd, the bulge volume V* corresponding to minimizers of depends weakly on L (Fig. S3). The insensitivity to L suggests a physical explanation for the observation that cell length appears uncorrelated with bulge volume,16 and our model further predicts that bulge size does not significantly change over a broad range of r. Third, balancing the energy dissipation with the viscous drag on the bulge results in a timescale much smaller than 100 ms (SI), suggesting that the relaxation time may be limited by water flow: assuming the parameter values in Materials and Methods, the larger value of Lp ≈ 10−12 m3/N · s implies a volumetric flow rate of approximately 20% of the initial cell volume per second and is consistent with a volume increase on the order of 1-10% of the initial cell volume during bulging. We anticipate further experiments, for instance ones which modulate membrane permeability during β-lactam killing, to elucidate the origins of these fast dynamics and test other predictions of our model.
Model of swelling
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 flow49 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 is small (Fig. 4A). In contrast, image analysis reveals that bulges grow at a rate consistent with theory when the defect radius, rd, also increases (Fig. 4B), implicating defect growth as the limiting step of bulge growth before lysis. Since the mean bulge radii at lysis are a ≈ 0.9 µm and b ≈ 0.7 µm, assuming the parameter values summarized in Materials and Methods and that turgor pressure has not changed due to osmotic stress responses44,45,47,48,50,51,52 suggests an upper bound for the yield areal strain of the E. coli IM and OM as approximately 20%, which lies within the empirical range of RBCs under impulsive stretching53 and exceeds that under quasi-static loading.54
Discussion
To summarize, we have used a continuum, elastic description of the cellular envelope to model membrane bulging and found that both continued water flow into the cytoplasm and defect enlargement can contribute to 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 broad 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. A growth mechanism which regulates pore size could help cells avoid lysis, in addition to regulating wall thickness and straight, rod-like morphology.30
Beyond bacterial morphogenesis, the combination of theory and experiment in our work has revealed a novel description of biological membrane physics and underscored the importance of mechanical stresses in cells. By being free to change their reference geometries, biological membranes differ from 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. This paves the way for investigating and manipulating similar, rich interactions of fluid membranes with elastic surfaces38 and the material nature of living cells.
Materials and Methods
Model parameters
Unless otherwise specified, in this work we assume , , , p = 0.5 atm, T = 300 K, r = 0.5 µm, L = 10 µm, and rd= 0.6 µm. The curves in Fig. 2B are drawn with Lb determined by the condition of fixed reference membrane area, rb = 0.5 µm, and ε = 1, approximately the values at which the minimum of over θ, rb, and ε is achieved.
Bacterial strains and growth
The wild-type strain used in this study is E. coli MG1655, and we verified that the morphological dynamics are statistically indistinguishable in two other wild-type strains, JOE309 and BW25113. Cells were grown in liquid LB (LB: 10g/L tryptone, 5g/L yeast extract, 10g/L NaCl) supplemented with no antibiotics. LB media containing 1.5% Difco agar (w/v) was used to grow individual colonies. Cells were taken from an overnight culture, diluted 100 to 1000-fold, and grown in LB at 37°C in a roller drum agitating at 60 rpm to an absorbance of approximately 0.3 to 0.6 (λ = 600 nm). Cells were then concentrated by centrifugation at 3000 rpm for 5 min and resuspended. We added 1 µL of the bacterial culture to No. 1.5 coverslips (24×60 mm) and placed 1 mm thick LB agarose (1.5%) pads containing 50 µg/mL of cephelaxin, a β-lactam antibiotic, on top for imaging. Cells were imaged immediately afterwards.
Microscopy
We used a Nikon Ti inverted microscope (Nikon, Tokyo, Japan) equipped with a 6.5 µm-pixel Hamamatsu CMOS camera (Hamamatsu, Hamamatsu City, Japan) and a Nikon 100x NA 1.45 objective (Nikon, Tokyo, Japan) for imaging. All cells were imaged at 37°C on a heated stage. The time between each frame during timelapse measurements ranged from 10 ms to 2 s, and the duration of timelapses varied from 10 min to 3 h. Images were recorded using NIS-Elements software (Nikon, Tokyo, Japan).
Image analysis
Image sequences were compiled from previous work16 and from ten replicate experiments described above, which resulted in raw data for over 500 cells. These sequences were annotated manually in ImageJ (National Institutes of Health, Bethesda, MD) to obtain cell dimensions, bulge radii, defect radii, and subtended bulge angles. Bulged cells were fit to cylinders with protruding ellipsoids with radii (a, a, b), as described in the main text, to determine cell volumes and membrane areas. For Fig. 1D, a subset of 30 cells were chosen among cells which bulged on a timescale ~100 ms. Here and below, all cells considered bulged in the imaging plane. For Fig. 3, a subset of 134 cells were chosen among cells which bulged on a timescale ~1 s. For Fig. 4, a subset of 112 cells were chosen among cells which bulged on a timescale ~1 s and for which the cellular dimensions could be determined, and relevant statistics were measured or computed at two or three time points until ~10 seconds after bulging. This choice of timescale was made to mitigate the potential influence of cellular stress responses such as transport of solutes out of the cytoplasm,44 which could confound volumetric measurements. We discarded data points lying beyond the ranges plotted in Figs. 3 and 4, which corresponded to outliers, and applied a trailing moving average filter of 10 to 20 points to generate the moving average curves.
Additional Information
Correspondence and requests for materials should be addressed to A.A.
Author Contributions
F.W. and A.A. conceived the project, performed modeling, analyzed data, and wrote the paper. F.W. performed experiments.
Competing Financial Interests
The authors declare no competing financial interests.
Supplementary Videos
Supplementary Video 1: Lysis dynamics of E. coli cells. Supplementary Video 1 shows a population of wild-type E. coli cells bulging, swelling, and lysing under antibiotic treatment. The time between frames is 30 seconds, the timelapse covers a period of approximately 1 hour, and the field of view is 100 µm × 80 µm.
Supplementary Figures
Acknowledgements
F.W. was supported by the National Science Foundation Graduate Research Fellowship under grant no. DGE1144152. A.A. was supported by the Materials Research and Engineering Center at Harvard, the Kavli Institute for Bionano Science and Technology at Harvard, the Alfred P. Sloan Foundation, and the Volkswagen Foundation. We thank John W. Hutchinson for numerous extended discussions, John W. Hutchinson, L. Mahadevan, Shmuel M. Rubinstein, Haim Diamant, Roy Kishony, Michael Moshe, Ugur Çetiner, and Zhizhong Yao for helpful feedback, Ethan C. Garner, Sean Wilson, and Georgia Squyres for microscopy assistance, Thomas G. Bernhardt and Sue Sim for the BW25113 strain, and Po-Yi Ho, Jie Lin, and Michael Moshe for comments on the manuscript.