Despite extensive characterization of the biochemical properties and genetic determinants of the bacterial cell wall, fundamental physical questions about how bacteria maintain their shape and size remain unanswered. To investigate cell-shape maintenance in rod-shaped Gram-negative bacteria, we developed a biophysical model for simulating growth dynamics, allowing for different mechanisms of insertion of new cell-wall material. Using this model, we examined the three main factors dictating the insertion of new glycan strands: (i) geometry, via the spatial patterning of new material, (ii) biochemistry, via the likelihood of strand termination, which dictates average inserted strand length, and (iii) mechanics, via forces exerted by the insertion machinery on new strands. Through a quantitative, multi-dimensional exploration of these three factors, we found that maintenance of cell shape depends on uncoupling the spatial pattern of insertion from the local cell-wall density, while determination of cell width depends on a combination of the average length and the mechanical stretching of new strands during insertion. Moreover, our model can also be used to reveal the failure of qualitative models such as tension-dependent insertion () to maintain cell shape.
Rod-shaped bacteria such as E. coli
, S. typhimurium
, and P. aeruginosa
robustly maintain a cylindrical shape during exponential growth (). Thus, while each species has its own well-defined radius, the precise maintenance of cell shape is likely general across species. Although we have focused primarily on experiments involving E. coli
, our modeling is intended to test general principles of cell growth, rather than the consequences of particular parameters. Moreover, many parameters such as the Young’s modulus may in fact be similar even across evolutionarily distant organisms. For example, despite significant differences between the peptidoglycan architectures of E. coli
and the Gram-positive Bacillus subtilis
, atomic-force microscopy measurements indicate that both species have a Young’s modulus of ~25 MPa (Yao et al., 1999
, Thwaites & Surana, 1991
). In any event, we hope that our computational framework will motivate further experimental measurements of the biophysical properties and organization of peptidoglycan across a range of species.
Intriguingly, E. coli
can revert to a straight morphology even after large morphological perturbations. E. coli
spheroplasts slowly revert back to a rod shape after lysozyme is removed as the cell wall regrows over multiple generations (Schwarz & Leutgeb, 1971
). Filamentous E. coli
cells grown in circular agarose microchambers have a curved morphology determined by the chamber geometry, but grow into an increasingly straight conformation when released from the chamber (Takeuchi et al., 2005
). It has been hypothesized that in order for this growth-dependent straightening to occur, insertion of new peptidoglycan cannot depend only on local self-similar growth, instead there must be a global structure coordinating the insertion of new material (Sliusarenko et al., 2010
, Mukhopadhyay & Wingreen, 2009
). Our simulations also conclude that robust rod-shaped cell-wall growth can be achieved by inserting new material with constant longitudinal and azimuthal density, uncoupled from local peptidoglycan density. We also demonstrated that self-similar growth initiated from random locations along the cell wall results in cell-wall density fluctuations that grow due to positive feedback. Starting from a straight cell, these model cell walls exhibited noticeable departures from a straight morphology after less than one doubling. At later stages of growth, cells were severely bent and bulged outwards with a variation in local width of ~50%. Our results indicate that cell-wall synthesis must be regulated in order to maintain shape, and that spatial control alone is sufficient.
Using our model, we also explored the consequences of peptidoglycan insertion in a fixed helical pattern or along a pattern of short helical segments, both of which are much sparser patterns than uniform insertion in which only a tiny percentage (<1%) of the peptide crosslinks are viable initiation sites at any one time (). We found that rod shape was preserved with similar fidelity to completely uniform insertion. Newly inserted material was shifted progressively farther away from insertion helix by subsequent insertion events, resulting in a flow of new material that homogenized the distribution of insertion. These results suggest that insertion patterns such as patches and short filaments may also be effective at maintaining a rod shape as long as the resulting pattern of inserted material becomes uniform after sufficient growth. Indeed, rod shape was also maintained when the fixed helical pattern was replaced by four dynamic helical segments covering the length of the cell whose angular orientations were independently and randomly selected every time the cell length increased by 33% (Fig. S6
Most rod-shaped bacteria contain cytoskeletal proteins thought to form global structures within the cell. The actin homolog MreB forms filaments (Wang et al., 2010
) that organize into helical patterns when overexpressed (Shih et al., 2005
). MreB also coordinates the localization of key components of the peptidoglycan synthesis machinery such as the transpeptidase PBP2 (Den Blaauwen et al., 2003
), and the flexural rigidity of E. coli
cells drops by 50% under A22-treatment that depolymerizes MreB (Wang et al., 2010
), suggesting that the MreB structure and the cell wall make comparable contributions to cell stiffness. The stiffness of the MreB structure and its localization on the cytoplasmic face of the inner membrane may enable the spatial organization of peptidoglycan insertion to be independent of the existing cell-wall density. E. coli
and C. crescentus
cells treated with A22 initially maintain a rod shape but eventually grow into lemon-shaped morphologies (Supp. Movie 1
) (Varma et al., 2007
, Gitai et al., 2005
). Moreover, E. coli
cells treated with both cephalexin and A22 continue to elongate but have variable width and eventually lyse due to bulging of the cytoplasmic membrane through large pores in the cell wall (, Supp. Movies 2
). Similarly, in our simulations cell walls growing via random insertion initially remain straight, and only lose rod shape later in growth as cell-wall disorganization increases. Taken together, these results suggest a specific role for the MreB cytoskeleton in rod-shape maintenance, namely use of the structural rigidity of MreB to decouple insertion of new strands from local peptidoglycan density.
In addition to maintaining a specific shape, many rod-shaped Gram-negative bacteria faithfully maintain a specific width dependent on species and growth conditions (). By individually varying the biochemical and mechanical characteristics of newly inserted strands, we have elucidated an interplay between strand termination and insertional stretching that would allow cells to modulate their width. In particular, changes in cell width depend inversely on two factors: (i
) the average strand length, which scales inversely with the termination probability during strand synthesis, and (ii
) the stretching force exerted by the insertion complex on new strands as they are crosslinked to the existing network. Differences in the distribution of glycan strands have been experimentally measured across morphological variants of cells of a single species and across cells of different species. E. coli
cells grown in LB in stationary phase are more spherical and have an average glycan strand length of 17.8 disaccharides, whereas E. coli
grown in PB at 42°C have an average strand length of 37.9 (Glauner et al., 1988
). Round E. coli
minicells, formed by polar divisions, have shorter strands and altered concentrations of penicillin binding proteins (Obermann & Holtje, 1994
). Moreover, peptidoglycan glycosyltransferases from different bacterial species produce different glycan chain lengths in vitro
(Wang et al., 2008
). Thus, our modeling suggests that E. coli
cells can alter their width by varying the processivity of strand synthesis, a prediction that could be tested via measurement of the strand length distribution before and after cells are shifted from a low-nutrient medium (e.g., M9-glucose, in which cell width is ~0.6 µm) to a rich medium (e.g., LB, in which width is ~1 µm).
Insertional stretching has been hypothesized as a width-maintenance mechanism in a previous theoretical study that used a continuum model of cell-wall mechanics to argue that the undeformed radius of the newly added material must be less than the current cell radius in order to accommodate pressure-mediated expansion (Lan et al., 2007
). Our studies reveal that roughly 10% stretching is required to maintain a fixed width, and show that increased stretching is correlated with an increase in the fraction of crosslinked glycans and an increase in the rigidity of the cell wall near sites of recent insertion, both of which could serve as experimental signatures of insertional stretching. This degree of insertional stretching would require ~5–10 pN of force per glycan subunit, which could be provided by the chemical energy released by crosslinking of the new strand to the old network (Jiang & Sun, 2010
) or by the insertion machinery during synthesis (Holtje, 1998
). Insertion complexes may also perform a dual mechanical function by exerting forces on newly inserted strands and restraining the expansion of the existing material during hydrolysis to maximize crosslinking potential. Based on our computational result that the cell wall reaches a steady-state level of organization after sufficient elongation using a given growth model (Fig. S3
) and our experimental observation that the width of filamentous E. coli
is identical to wild-type cells, we predict that the glycan strand length distribution in the cylindrical portion of the cell wall is largely unaffected by cephalexin treatment. Taken together, our results suggest that differences in strand length distributions and crosslinking density contribute to differences in cell width. Our computational framework can be used to investigate the rules governing shape determination and the strategies a cell might employ to grow its cell wall in a robust manner.
Our modeling of cell-wall growth reinforce the principle that systematic spatial variation of insertion probability could lead to variant rod shapes such as curved or helical cells. Moreover, our simulations of helical growth predict that old cell-wall material will move apart in a specific pattern that could be observed by pulse-chase labeling of peptidoglycan. Both our computational () and experimental results () predict that in the absence of a global MreB-mediated template for insertion, the loss of rod-shape maintenance is coupled to the formation of large pores that could be measurable via cryo-electron tomography (Gan et al., 2008
) or AFM (Hayhurst et al., 2008
). In the future, incorporating external and cytoskeletal forces into our models will enable the study of peptidoglycan organization for cells growing in confined geometries (Takeuchi et al., 2005
), curved species such as C. crescentus
(Cabeen et al., 2009
), and morphological transitions such as cell division (Osawa et al., 2008
Because the biochemical properties of peptidoglycan are highly conserved across prokaryotes, our results have elucidated mechanisms that may have broad applicability across even highly divergent species. Indeed, homologs to the pbp2 and mreB genes are found in a large fraction of rod-shaped bacteria, and the small-molecule inhibitor A22 has similar impact on cell shape in all species with MreB. Thus, simulations provide an effective tool for assigning morphological consequences to each of the molecular properties of the peptidoglycan synthesis machinery, and for testing the effectiveness of different mechanisms of cell growth. In the future, expanding this study to include bacterial species of a variety of shapes and sizes could reveal general mechanisms for manipulating and controlling bacterial growth and division.