Getting into shape: how do rod-like bacteria control their geometry?
GGetting into shape: how do rod-like bacteria control their geometry?
Ariel Amir ∗ Department of Physics, Harvard University, Cambridge, MA 02138, USA
Sven van Teeffelen † Groupe Croissance et Morphog´en`ese microbienne, Institut Pasteur, Paris, France (Dated: September 17, 2018)Rod-like bacteria maintain their cylindrical shapes with remarkable precision during growth. How-ever, they are also capable to adapt their shapes to external forces and constraints, for example bygrowing into narrow or curved confinements. Despite being one of the simplest morphologies, we arestill far from a full understanding of how shape is robustly regulated, and how bacteria obtain theirnear-perfect cylindrical shapes with excellent precision. However, recent experimental and theoreti-cal findings suggest that cell-wall geometry and mechanical stress play important roles in regulatingcell shape in rod-like bacteria. We review our current understanding of the cell wall architectureand the growth dynamics, and discuss possible candidates for regulatory cues of shape regulationin the absence or presence of external constraints. Finally, we suggest further future experimentaland theoretical directions, which may help to shed light on this fundamental problem.
INTRODUCTION
Cells of all organisms and kingdoms face a commonchallenge of regulating their own shapes to facilitate via-bility and growth, but also being able to react to externalspatial constraints and mechanical forces that eventuallyrequire adaptive changes in cell-shape or cellular growth,see [1] for an excellent review of the diverse strategiesused by organisms with cell walls. In single-celled bac-teria, cell shape is often very precisely controlled, as il-lustrated in Fig. 1. Bacteria come in a broad range ofshapes and sizes (see Fig. 1 of [2] for a striking graphicalrepresentation), yet despite decades of research, our un-derstanding of how these shapes are controlled and regu-lated at a molecular level is far from complete. Given thelarge difference in length scales between the macroscopiccell shape ( µ m) and the microscopic proteins, enzymes,and molecules responsible for cell shape (nm) – how issuch precise control over shape achieved?In a given growth medium, various rod-shaped bacteriasuch as the canonical Gram-negative Escherichia coli orthe well studied Gram-positive
Bacillus subtilis elongatewhile maintaining a constant diameter. Strikingly, manyrod-like bacteria elongate by expanding their cell enve-lope all along the cell envelope, as compared to growingfrom the tip only. These cells maintain their diametereven if cell division is inhibited and cell length reachesdozens of microns [4].Here, we focus on this example of rod-like growth, anddiscuss our current understanding of cell-shape regula-tion. The discussion will mainly consist of a physicistpoint-of-view, where the molecular machinery of cell-wallinsertion has been “coarse-grained”. We will not dis-cuss in detail the particular action of specific enzymesor the biochemical properties of the peptidoglycan (PG).Rather, we shall focus on the mechanics of the cell walland the sensory cues, which might enable the tight regu-
FIG. 1: Electron microscopy images of
B. subtilis , taken byThierry Meylheuc. In a given growth medium, the differentbacterial cells have a smooth, highly reproducible cylindricalshape, with relatively small fluctuations in length and radius.The image is reproduced from Ref. [3], courtesy of A. Chas-tanet. lation of shape. One important cellular component forshape regulation we will highlight is the bacterial cy-toskeleton.Bacterial cell shape is not only under auto-regulationbut is also subject to external mechanical perturbations(such as geometric confinement or external forces). Cellsare known to adapt their growing shapes to these forces[5, 6]. Learning about the cellular response to externalforces may be important to understand the intra-cellularregulation of shape in unconstrained environments. Thestudy of auto-regulation and external perturbation of cellshape thus requires an interdisciplinary effort of biolo-gists, physicists and materials scientists, as it requiresan understanding of the non-trivial mechanical problemsassociated with thin, elastic media: while in many casesin biology a qualitative understanding of a phenomenonis sufficient to understand the crux-of-the-matter, shape a r X i v : . [ q - b i o . S C ] M a r regulation may involve the sensing of geometric cues andof mechanical stresses and strains, which, in turn, areintegral parts of regulatory feedbacks.There are additional, fascinating questions associatedwith the intersection of mechanics and bacterial growth,that we shall not discuss here, such as the forces exertedby the Z-ring in the bacterial division process [7–10], therole of crescentin in shaping curved cells [11], and thegrowth of curved and helical bacteria [12, 13], to namebut a few. NECESSITY FOR REGULATION
Even for the seemingly straightforward mode of elon-gation of rod-shaped cells, maintaining the rod shape is anon-trivial task – simple“templating” mechanisms whereglycan strands are placed in parallel to the existing oneswould not be stable to the random fluctuations of growth,especially in light of the disorder in the mesh, which weshall elaborate on further down. To gain intuition, con-sider a different regulatory problem – how does a growingleaf stay flat? It turns out that it is a non-trivial task tobe flat. It was shown that a negative feedback regulatorycircuit is required to avoid a bumpy leaf structure, whichis distinctively different from the smooth, flat leaves weare used to and take for granted [14].For the cylindrical growth, previous works started totackle this problem by comparing the robustness of var-ious growth mechanisms [15], and found that uniformlydistributed, helical insertions are quite robust. Yet intheir study no strategy was proven to be robust in thetrue sense, i.e., were we to start from a spherical cell,it is unlikely that the cell would adapt to its rod-shapewhen using any of the proposed strategies, nor would arod-shaped cell maintain its diameter over many roundsof division. Bacterial cells do precisely that – as wasshown for
E. coli : after the cells were significantly dis-torted when grown in a chamber thinner than their di-ameter, they recovered their native shape after severalrounds of division [6]. A recent study by Ursell et al. suggests that cytoskeletal MreB in
E. coli could play animportant part in regulating cell shape. They found thatMreB filaments serve as a local sensor of bacterial enve-lope curvature and thus direct cell-wall insertion to thesesites. The authors show that this mechanism could helpmaintain a cylindrical cell straight [16]. Potentially, thissame curvature-sensing mechanism could also play an im-portant role in maintaining cell diameter. We will comeback to this further down. In the following we summa-rize what is known about cell-wall synthesis, the majorstress-bearing component of the cell envelope, and howit might lead to stable cell shape. We will then discusshow auto-regulation and cell-shape response to externalforces could come about.
FIG. 2: Details of peptidoglycan organization as obtained bycryo-electron tomography; reproduced from Gan et al. [18];courtsey of Grant J. Jensen. Computational reconstructionsof the three-dimensional electron density of a cell-wall sac-culus of the Gram-negative bacterium
C. crescentus reveala circumferential orientation of the cell-wall glycan strands:Shown are two overlapping cell-wall sacculi (A), outlined bygreen or violet dotted lines. The boxed region in (A) is mag-nified in panels (B-D). Panel (D) is an iso-density plot, whichshows long circumferentially oriented structures that are pre-sumably individual glycan strands. The inset in (D) displays asuperposition of the blue-boxed glycan strand and an atomicmodel of a 9-subunit-long glycan strand, for comparison ofscale.
MICROSCOPIC CELL-WALL STRUCTURE ANDMOLECULAR MODE OF CELL-WALL GROWTH
The bacterial cell shape is physically determined bythe PG cell wall, a covalently bonded network of sugarstrands cross-linked by short peptide bridges. The rigidPG meshwork counteracts the high turgor pressure setby the difference in osmotic potentials between the celland its environment [17]. In Gram-negative bacteriacryo-electron tomography images of isolated cell-wall sac-culi suggest that the PG forms a monolayer with gly-can strands running in a near-circumferential directionaround the long axis of the cell [18] (see Fig. 2). Thisobservation is in agreement with atomic force microscopy(AFM) measurements on isolated cell-wall sacculi [19],which have revealed that the elastic constants of the cellwall are anisotropic. This anisotropy is expected becauseof the difference in stiffness between rather rigid circum-ferentially oriented glycan strands and the comparativelyfloppy peptide bonds. Interestingly, there is also a two-fold difference of cell-wall mechanical stresses betweenthe circumferential and axial directions that comes aboutdue to the cylindrical geometry of the cell. Theoreticalmodelling suggests that the large turgor pressure drivesthe cell wall elasticity to the non-linear regime [20, 21].In Gram-positive bacteria the cell wall is much thickerthan in Gram-negatives (e.g., in
B. subtilis the cell wallis approximately 30 nm thick [22, 23]). The thicknessof the cell wall has prevented molecular-resolution imag-ing of the glycan strands. However, recent cryo-electronmicroscopy and surface atomic force microscopy experi-ments have revealed circumferential furrows in the cell-wall surface [22, 24, 25] with a spacing of roughly 50 nm.While this observation is in agreement with the model ofcircumferential glycan strands it also suggests a higher-order three-dimensional structure, which is not under-stood yet. For both Gram-positive and Gram-negativebacteria biological text books often depict the architec-ture of the PG cell wall as a regular lattice. However,the short length of the glycan strands of several nm [26]suggests that the structure is much more disordered.The structure of the newly synthesized cell wall hasbeen revealed in a different, indirect manner: Propercell-wall synthesis depends on the bacterial cytoskeleton,particularly on one or multiple isoforms of the widely con-served actin-homologue MreB [27]. MreB forms filamentsin the cytoplasm that are attached to the cytoplasmicmembrane in both Gram-negative [28] and Gram-positivebacteria [29–31]. The length of the MreB filaments iscurrently under debate, in particular because native-expression-level MreB filaments have not been detectedin whole cells by electron microscopy [32]. Irrespective oftheir exact length, it has been shown by fluorescence mi-croscopy that MreB filaments rotate around the long cellaxis in a processive manner in Gram-negative [33] andGram-positive bacteria [30, 31, 34, 35]. This rotationdepends on PG synthesis and proceeds at a speed com-patible with processive insertion of single glycan strandsinto the PG meshwork [33], as already suggested by Bur-mann and Park in the 1980s [36]. It is thus plausiblethat MreB filaments are physically linked to the enzymesresponsible for cell-wall insertion. In fact, some of thecell-wall-synthesis enzymes have been seen to move in asimilar manner as MreB filaments in the Gram-positive
B. subtilis [34, 35], supporting the hypothesis of physicalinteraction. In Gram-negative
E. coli , at least one im-portant synthesising enzyme, the transpeptidase PBP2,moves rapidly and diffusively, showing no processivity onthe sub-second time scale [37], thus suggesting a moretransient interaction of the cell-wall synthesis proteins.Ref. [38] finds that the timescales, at which disruptingMreB affects cell wall elasticity are similar to the growthtime, in consistence with this interpretation. Further-more, filaments have recently been reported to move witha filament-length dependent speed [30]. The speed-lengthrelationship observed is compatible with a simple modelof synthesis complexes effectively consituting motor pro-teins that randomly attach to MreB filaments and exerta force in either of the two circumferential directions.Accordingly, the speed as a function of length displaysa maximum at finite filament length of a few hunderednanometers [30].Interestingly, the trajectories of cytoskeletal filamentsobserved in
E. coli are slightly helical on average [33],suggesting an average helical organization of the cellwall as a whole. This helicity of the cell wall has since been supported by experiments of combined mi-croscopy and optical trapping [39]: Wang et al. attachedfluorescent beads to the envelope of elongated
E. coli or B. subtilis cells using optical tweezers. They thentracked the bead position before and after osmoticallyup-shocking the cells in a flow cell. First, they findthat the cells shrink much more along the long axis thanalong the radial direction – in accordance with the afore-mentioned anisotropy of elastic constants. Furthermore,they also find that the beads follow helical trajectoriesduring the shape transition, which suggests a slight he-lical anisotropy and, thus, a helical orientation of thePG meshwork – in agreement with the helical trajecto-ries of MreB motion. With combined fluorescence mi-croscopy on MreB filaments and computational elastic-network simulations (coarse-grained molecular-dynamicssimulations), the authors argue that the helicity mightbe caused by the orientation of MreB filaments belowthe cylindrical surface of the cell wall. The orientation ofthe MreB filaments with respect to the cell envelope, inturn, could be caused by the filament-intrinsic curvatureand twist in combination with a curved surface of thecylindrical cell envelope[40, 41].Linking cell-wall synthesis to MreB filaments is veryinteresting from a physics perspective: Multiple indepen-dent studies have suggested that MreB filaments assumeon average macroscopic lengths of few hundred nanome-ters [29–31, 42]. The mechanical stiffness of these fila-ments [4] could facilitate the macroscopic organization ofthe cell-wall synthesis machinery and might thus providea key ingredient for a robust cell-shape feedback mecha-nism (see discussion below). Computational simulationsby Furchtgott et al. have already shown that the stiffnessof MreB could provide the cell with a mechanism to avoidan unfavorable positive feedback of macroscopic cell-wallbulges, i.e., local departures from the intended perfectlycylindrical geometry [15]. Their argument goes as fol-lows: If cell-wall insertion was only dependent on theavailability of PG substrate, i.e., if new PG was insertedwith equal probability at any potential site of insertion,local cell-wall bulges would grow, as they contain a highernumber of potential insertion sites. Conversely, sites withlower cell-wall density would be depleted of cell-wall ma-terial, while the surrounding meshwork would expand.The cytoskeleton could render insertion independent ofthe local density of PG, simply by bridging small devia-tions from the cylindrical envelope due to polymer stiff-ness. Related ideas regarding the role of MreB in stabi-lizing cylindrical growth are provided in Refs. [10, 43],which also illustrate theoretically and experimentally amechanical instability which can occur in the absence ofMreB. However, while this mechanism could prevent lo-cal deviations from a flat cylinder surface regulation ofcell shape requires a mechanism that measures large-scaledeformations of the cell envelope – either directly in formof cell-envelope curvature (suggested by Ursell et al. [16])or indirectly, e.g., via a modified mechanical stress in thecell wall (see discussion below).
UNDERSTANDING CELL-SHAPE REGULATIONBY CELL-SHAPE PERTURBATIONS
Looking at sub-cellular components such as the PG cellwall and the MreB cytoskeleton have fundamentally im-proved our understanding of the organizing principles ofthe cell wall in the steady state of rod-like growth. A dif-ferent approach to understanding cell-shape regulation isto perturb cylindrical cell shape and observe how the cellreacts to the perturbation – both during and after theperturbation [5, 6, 44, 45]. Such an approach is particu-larly appealing from a physics perspective, as the cell wallis a partially ordered elastic sheet that undergoes elas-tic and plastic deformations upon mechanical forces andduring growth, respectively [44]. We note that by “plas-tic” we mean irreversible due to a change of the covalentpeptide and glycan bonds. This change comes aboutdue to the cleaving of existing bonds through enzymes(as opposed to ripping) and possibly through the inser-tion of new PG material. Thanks to the non-uniform,possibly adaptive growth process the residual stresses inthe cell wall can be much smaller than in plastic de-formations happening in non-living materials (e.g: met-als). This allows for the controlled test of molecular andphysical models of cell-wall insertion and cell-wall elasticproperties. Besides helping us to understand cell-shapeauto-regulation during normal growth, cell-shape defor-mation experiments also allow us to study how the cellreacts to mechanical and geometric constraints, such asconfining spaces (see Fig. 3). Ultimately, the molecu-lar mechanisms underlying cellular response to pertur-bations and the mechanisms underlying cell-shape auto-regulation might be the same; however there might alsobe components of adaptation and auto-regulation, re-spectively, that compete against each other, a possibilitythat we shall later elaborate on.In a first such experiment, Takeuchi et al. found thatfilamentous
E. coli cells grown in small, cylindrical con-fining chambers maintain their shapes after release fromthe chamber (see Fig. 3). Thus,
E. coli is able to adaptits shape instead of growing as a rod-like cylindrical cellwhen grown in confined environments. In another exper-iment M¨annik et al. observed that
E. coli cells can growand even divide in shallow confining slits. In both casescells revert their shapes after sufficient additional growthoutside the confining geometry [6].For another beautiful example of a biophysical ap-proach, consider the work of Ref. [45]: in this workthe naturally curved gram-negative bacterium
Caulobac-ter crescentus is manipulated genetically to be straight.The dynamics of the straightening process, when startingfrom a curved cell, was measured in time using optical mi-
FIG. 3: Microscopy image of
E. coli , reproduced from Ref.[5], courtsey of G. M. Whitesides. (a) The cell was grown ina narrow circular channel. (b) The cell length grows expo-nentially in time with the physiological growth rate, showingthat the cell is at least locally close to its normal growthconditions, yet the cell adapts to the shape of the channel.(c) When taken out of the channel the cell maintained a de-formed shape, illustrating that the cell is able to adapt itsshape to the confinement during growth, without building uplarge one-sided stress that would relax by straightening afterrelease from the chamber. croscopy. Careful analysis of the mechanics involved ledthe authors to rule out several models for the observedstraightening, and to conclude that processive, circum-ferential insertions of glycan strands into the cell wall atrandom locations explain their measurements – consis-tent with the more recent and more direct evidence ofcircumferential insertion as described in Sec. . The dif-ferential geometry and mechanics used in this work is farfrom the standard toolbox of a biologist, yet the conclu-sions reached are intuitive and understandable – as wellas highly relevant – to any biologist interested in bacte-rial morphology.In a recent study, a large bending torque was appliedto growing filamentous
E. coli and
B. subtilis cells [44],using a viscous drag, in order to study the elastic andplastic deformations of the cell wall during growth. Theauthors concluded that mechanical stresses are involvedin the regulation of shape in
E. coli : the cell grew morecell wall on the side of the flow, where a tensile stressstretched the cell wall, and grew less on the oppositeside where the external stress was compressive (see Fig.4). Thus, the cell reacts to an external force by adapt-ing its shape. The observed plastic shape deformationsduring growth were consistently interpreted in terms ofthe dislocation-mediated growth theory [46, 47]. In thisformalism the circumferential insertions are interpretedin terms of edge dislocations in the PG mesh, buildingon concepts developed in the context of the physics ofdefects in metals. This mechanism of plastic deforma-tions might also be responsible to the circular cell shapesobserved in cells grown in confinement (Fig. 3), and isreminiscent of the role of “smart autolysins” proposed byKoch [48].Interestingly, when the external stress (due to the flow)is switched off, the cell straightens. While this seemsin accord with the previously described experiment on
C. crescentus straightening, there is a crucial differencebetween the two: in the previous experiment the curva-ture of the cell centerline was measured to decay expo-nentially during the straightening process, however, at arate lower than the rate of exponential cell elongation dueto growth. Thus, the filamentous cells never reached astraight configuration. According to the authors’ modelof random processive insertion the decay of the centerlinecurvature is a monotonically increasing function of thelength of newly inserted glycan strands (the amount ofprocessivity). Yet, even for glycan strands much longerthan the cell circumference (infinitely processive inser-tions) the decay rate would saturate at a finite value(which happens to be the growth rate). The shape ofthe bacteria (were it not to divide) would be self-similar[49], i.e., a curved cell, which does not divide would main-tain a curved shape. On the contrary, filamentous
E. coli cells in the flow-cell experiment straighten more than bythe maximum straightening rate in the random-insertionprocess. This result suggests the presence of an addi-tional straightening mechanism. Further work is neededin order to establish whether this is an “active” mech-anism, through which
E. coli attempts to “correct” forcell shape deformation, or a result of the coupling to theresidual mechanical stresses in the cell wall, which persisteven after the external force is switched off. The emerg-ing picture of cell bending and straightening in
E. coli isillustrated in Fig. 4, showing the way a cell responds tocurvature and external forces.
FIG. 4: Experiments suggest that cell wall growth dependson the situation: typically, cells would grow in a manner thatwould straighten them, as shown in (a). However, when underexternal mechanical stress (are a result of a confining environ-ment [5] or an applied bending torque [44]), the cells wouldplastically deform to adapt to the new environment.
POSSIBLE FEEDBACK MECHANISMS
All experiments and computational simulations de-scribed above leave us with two questions: How – mech-anistically – does the cell adapt its shape to the influenceof external forces, and secondly, how does it restore andauto-regulate cylindrical shape during normal growth?The plastic deformation of the bacterial cell shape dur-ing long-term application of a torque suggests that thebacterium preferentially inserts new PG material on theside of the cell facing the flow (where it experiences ahigher stress in its cell envelope). Alternatively, the cellcould grow a less dense PG meshwork on the flow-facingside. In either case the local cell-wall synthesis machinerymust react to the mechanical stresses applied.How do cells maintain their cylindrical shape or re-acquire it after perturbation? Recent work by Ursell etal. [16] suggests that cells are able to sense the localcell-envelope curvature through cytoskeletal MreB fila-ments. As discussed above, membrane-associated MreBfilaments are stiff and could thus favor their own local-ization at sites of particular cell-envelope curvature, thuseffectively constituting a curvature sensor. Ursell et al. find that MreB localizes at positions of cell-wall inden-tations, i.e., at negative Gaussian curvature of the en-velope. Monitoring the local expansion of the cell wall(using a cell-wall stain as fiducial marker) and imagingboth cell shape and the localization of MreB at the sametime, Ursell et al. propose that MreB filaments are phys-ically linked to sites of PG insertion. Thus, MreB mayguide the PG insertion machinery to sites of preferredcurvature. Furthermore, they found that these regions ofthe cell envelope flipped curvature sign after PG inser-tion. The curvature-based insertion scheme could thusprovide a way to help maintain cell shape during rod-likegrowth by providing an inherent feedback mechanism be-tween the PG-insertion machinery, which determines cellshape, and the cell shape, which, in turn, determines thelocation of the PG-insertion machinery. Indeed, a coarse-grained computational simulation suggests that couplingprocessive cell-wall insertion to cell-envelope curvaturehelps keeping a cell straight [16].However, the curvature-based growth mechanism alonecannot account for the aforementioned bending exper-iments [44]: if only curvature sensing is present, uponbeing elastically deformed to the right, the cell would at-tempt to add more material on its right side, since it hasa negative curvature. This implies that when the exter-nal force which led to the bending is switched off, thecell should be deformed to the left since more materialwas inserted on the right hand side. However, the experi-ments show that the cell is deformed to the right. There-fore the sign of the differential growth expected from acurvature-based mechanism is opposite of what is exper-imentally observed. Further work is needed in order toestablish the connection between these two observations,and the relative importance of mechanical stress and ge-ometric curvature.
FUTURE PROSPECTS
The previous examples of cell-shape experiments andmodeling illustrate the effectiveness of combining theo-retical modelling and novel experimental techniques toimprove our understanding of cell wall regulation andthe dynamics of growth. There is lots of room to furtherexplore both of these avenues.On the theoretical side, the attempts to study the ro-bustness of growth have been primarily numerical. Com-putational simulations from KC Huang’s lab have demon-strated how global helical cell-wall structure and localcell-wall integrity can emerge from mesoscopic cytoskele-tal filaments [4]. However, the computational resources,which are at our disposal at the moment do not allowfor the modelling of the full number of interacting unitsin the peptidoglycan mesh, and can only provide intu-ition as to the true robustness of a particular model. Al-ternative more “coarse-grained” approaches have beenrecently introduced [46, 47, 50], in which the relativelysmall number of active growth sites correspond to mov-ing dislocations in the peptidoglycan mesh, yet in theseprevious studies a perfectly cylindrical geometry was as-sumed. In general, the theory describing thin inter-faces such as the bacterial cell wall, shallow shell theory,involves highly non-linear partial differential equations,making analytic progress challenging. One possible di-rection would be to adapt the existing equations, whichare commonly used by engineers to study thin shells (theDonnell-Mushtari-Vlasov equation, which generalize theF¨oppl-von K´arm´an equations) to incorporate growth , andtest the stability of the equations to perturbations usinglinear stability analysis. A second theoretical tool, whichwas recently introduced is the use of a metric to describecurved surfaces [51]. The non-uniform growth can becast in terms of its effect on the “target metric”, and fora thin interface the shape is determined by the Gaussiancurvature of that metric. This tool has proven usefulin calculating the metric necessary to achieve a desiredshape, which can then be prescribed onto a thin polymersheet, leading to remarkable control of three-dimensionalobjects [52, 53]. The deformations of bacterial cells inthe microfluidic experiment described above [44] can, infact, also be described using the effect on the Gaussiancurvature of a metric [54], and this could be a power-ful theoretical tool to handle the problem of cylindricalstability.On the experimental side, it seems that further re-search is needed in order to establish the relative role ofboth curvature-related and stress-related regulation. Re-peating both microfluidic experiments described in Sec. while following the dynamics of MreB would providemore information regarding the differential growth. Withnew possibilities to track the metric of the cell wall di-rectly in live cells [16] we can now quantitatively under-stand where new material is being deposited and corre-late it with the stress distribution on the cell wall. Similarapproaches in the very different context of tissue morpho-genesis have proven useful; remarkably, also in this case,mechanical stress have been shown to play an importantregulatory role [55]. Making even larger perturbations isanother experimental route, which may lead to new in-sights: both in gram-positive and gram-negative bacteriathe production of cell wall can be damaged such that ina low osmolarity medium the cell is still viable in spite oflack of cell wall, leading to spherical cells [56]. Recently,recoveries from such spherical cells into the native rod-shaped forms have been observed in
B. subtilis [57]. Howdoes a sphere grow to be a cylinder? The observed pathof recovery shows a distinct morphology, which providesimportant constraints for theoretical models – not onlydo they have to predict a robust cylindrical growth, butthe form of recovery must also agree with these experi-mental findings.A complete theory of bacterial cell shape should alsoaccount for the magnitudes of both radius and length; theregulation of these two is, however, of very different na-ture: References [58–60] suggest a robust mechanism ofmaintaining cell length in bacteria, consistent with theexperimentally observed correlations and distributions,invoking a simple biophysical mechanism that does notcouple to mechanics or curvature. This mechanism is ob-viously decoupled from that of radius maintenance, asis proven by the possibility of having extremely long fil-amentuous cells, which nevertheless maintain their con-stant radius [44]. Various approaches have been used toexplain the origin of the micron-scale diameter of
E. coli and
B. subtilis , including an energy minimization scheme[61] and the natural curvature of MreB filaments [16].In contrast to
E. coli and
B. subtilis various bacte-ria such as Mycobacteria, Streptomyces, are tip-growers[62]. How is rod-shape maintained for tip-growers? It isplausible that a different mechanism will be necessary inthis scenario. In this case drawing an analogy with thegrowth of plants and fungi could be helpful, since theyorganisms are also tip-growers [63]. Extensive work hasbeen done on modelling tip-growth and the role of me-chanics [64, 65], and it is intriguing to see whether theseconcepts could apply for bacteria as well.We are still far from unravelling the fundamental “en-gineering” challenges that biology has to overcome inshaping single cells as well as multi-cellular tissues. Yetthe rapid development of new theoretical, computationaland experimental techniques in these fields, combinedwith the recent fruitful collaborations between biologists,physicists and engineers, suggest a promising and excit-ing future.AA was supported by the Harvard Society of Fellowsand the Milton Fund. SvT was supported by a Hu-man Frontier Science Program Postdoctoral Fellowship.The authors acknowledge useful discussions and feedbackregarding the manuscript from E. Efrati, O. Amster-Choder, Y. Eun, K. C. Huang, D. R. Nelson, J. Pauloseand T. Ursell. ∗ [email protected] † [email protected][1] J. Dumais, Journal of Experimental Botany (2013). DOI10.1093/jxb/ert268[2] K.D. Young, Microbiology and Molecular Biology Re-views (3), 660 (2006)[3] A. Chastanet, R. Carballido-Lopez, Front Biosci (ScholEd) , 1582 (2012)[4] S. Wang, H. Arellano-Santoyo, P.A. Combs, J.W. Shae-vitz, Proceedings of the National Academy of Sciences , 91829185 (2010)[5] S. Takeuchi, W.R. DiLuzio, D.B. Weibel, G.M. White-sides, Nano Letters (9), 1819 (2005)[6] J. Mnnik, R. Driessen, P. Galajda, J.E. Keymer,C. Dekker, Proceedings of the National Academy of Sci-ences (35), 14861 (2009)[7] A.J. Egan, W. Vollmer, Annals of the New YorkAcademy of Sciences (1), 8 (2013)[8] Y. Li, J. Hsin, L. Zhao, Y. Cheng, W. Shang, K.C.Huang, H.W. Wang, S. Ye, Science (6144), 392 (2013)[9] O. Piro, G. Carmon, M. Feingold, I. Fishov, Environ-mental Microbiology (2013)[10] S.X. Sun, H. Jiang, Microbiology and Molecular BiologyReviews (4), 543 (2011)[11] M.T. Cabeen, G. Charbon, W. Vollmer, P. Born, N. Aus-mees, D.B. Weibel, C. Jacobs-Wagner, The EMBO Jour-nal (9), 1208 (2009)[12] L.K. Sycuro, Z. Pincus, K.D. Gutierrez, J. Biboy, C.A.Stern, W. Vollmer, N.R. Salama, Cell (5), 822 (2010)[13] A. Typas, M. Banzhaf, C.A. Gross, W. Vollmer, NatureReviews Microbiology (2), 123 (2011)[14] E. Sharon, private communications .[15] L. Furchtgott, N.S. Wingreen, K.C. Huang, MolecularMicrobiology (2), 340 (2011)[16] T.S. Ursell, J. Nguyen, R.D. Monds, A. Colavin,G. Billings, N. Ouzounov, Z. Gitai, J.W. Shaevitz, K.C.Huang, Proceedings of the National Academy of Sciencesp. 201317174 (2014)[17] Y. Deng, M. Sun, J.W. Shaevitz, Physical Review Letters (15), 158101 (2011)[18] L. Gan, S. Chen, G.J. Jensen, Proceedings of the Na-tional Academy of Sciences (48), 18953 (2008)[19] X. Yao, M. Jericho, D. Pink, T. Beveridge, J Bacteriol. (22), 6865 (1999)[20] A. Boulbitch, B. Quinn, D. Pink, Phys. Rev. Lett. ,5246 (2000)[21] Y. Deng, M. Sun, J.W. Shaevitz, Phys. Rev. Lett. ,158101 (2011)[22] M. Beeby, J.C. Gumbart, B. Roux, G.J. Jensen, Molec-ular Microbiology (4), 664 (2013)[23] G. Misra, E.R. Rojas, A. Gopinathan, K.C. Huang, Bio- physical Journal (11), 2342 (2013)[24] G. Andre, S. Kulakauskas, M.P. Chapot-Chartier,B. Navet, M. Deghorain, E. Bernard, P. Hols, Y.F.Dufrˆene, Nature communications , 27 (2010)[25] E.J. Hayhurst, L. Kailas, J.K. Hobbs, S.J. Foster, Pro-ceedings of the National Academy of Sciences (38),14603 (2008)[26] H. Harz, K. Burgdorf, J.V. Hltje, Analytical Biochem-istry (1), 120 (1990)[27] J.W. Shaevitz, Z. Gitai, Cold Spring Harbor perspectivesin biology (9) (2010)[28] T. Kruse, J. Møller-Jensen, A. Løbner-Olesen, K. Gerdes,EMBO Journal (19), 5283 (2003)[29] L.J. Jones, R. Carballido-Lopez, J. Errington, Cell (6), 913 (2001)[30] P.v. Olshausen, H.J. Defeu Soufo, K. Wicker, R. Heintz-mann, P.L. Graumann, A. Rohrbach, Biophysical Jour-nal (5), 1171 (2013)[31] C. Reimold, H.J. Defeu Soufo, F. Dempwolff, P.L. Grau-mann, Molecular biology of the cell (2013)[32] M.T. Swulius, G.J. Jensen, Journal of bacteriology (23), 6382 (2012)[33] S. van Teeffelen, S. Wang, L. Furchtgott, K.C. Huang,N.S. Wingreen, J.W. Shaevitz, Z. Gitai, Proceedings ofthe National Academy of Sciences , 15822 (2011)[34] E.C. Garner, R. Bernard, W. Wang, X. Zhuang, D.Z.Rudner, T. Mitchison, Science (6039), 222 (2011)[35] J. Dom´ınguez-Escobar, A. Chastanet, A.H. Crevenna,V. Fromion, R. Wedlich-Sldner, R. Carballido-Lpez, Sci-ence (6039), 225 (2011)[36] L.G. Burman, J.T. Park, Proceedings of the NationalAcademy of Sciences (6), 1844 (1984)[37] T.K. Lee, C. Tropini, J. Hsin, S.M. Desmarais, T.S.Ursell, E. Gong, Z. Gitai, R.D. Monds, K.C. Huang,Proceedings of the National Academy of Sciences p.201313826 (2014)[38] H.H. Tuson, G.K. Auer, L.D. Renner, M. Hasebe,C. Tropini, M. Salick, W.C. Crone, A. Gopinathan, K.C.Huang, D.B. Weibel, Molecular Microbiology (5), 874(2012)[39] S. Wang, L. Furchtgott, K.C. Huang, J.W. Shaevitz, Pro-ceedings of the National Academy of Sciences (10),E595 (2012)[40] S. Wang, N.S. Wingreen, Biophysical Journal (3),541 (2013)[41] S.S. Andrews, A.P. Arkin, Biophysical journal (6),1872 (2007)[42] T. Kruse, J. Bork-Jensen, K. Gerdes, Molecular Micro-biology (1), 78 (2004)[43] H. Jiang, F. Si, W. Margolin, S.X. Sun, Biophysical jour-nal (2), 327 (2011)[44] A. Amir, F. Babaeipour, D. R. Nelson and S. Jun,arXiv:1305.5843 (2013), to appear in PNAS.[45] O. Sliusarenko, M.T. Cabeen, C.W. Wolgemuth,C. Jacobs-Wagner, T. Emonet, Proceedings of the Na-tional Academy of Sciences , 10086 (2010)[46] D.R. Nelson, Annual Review of Biophysics (1), 371(2012)[47] A. Amir, D.R. Nelson, Proceedings of the NationalAcademy of Sciences (25), 9833 (2012)[48] A.L. Koch, Bacterial growth and form (Springer, Berlin,2001)[49] R. Mukhopadhyay, N.S. Wingreen, Phys. Rev. E (6),062901 (2009) [50] A. Amir, J. Paulose, D.R. Nelson, Phys. Rev. E ,042314 (2013)[51] E.S. E. Efrati, R. Kupferman, Soft Matter , 8187 (2013)[52] Y. Klein, E. Efrati, E. Sharon, Science (5815), 1116(2007)[53] J. Kim, J.A. Hanna, M. Byun, C.D. Santangelo, R.C.Hayward, Science (6073), 1201 (2012)[54] J. Paulose and A. Amir, to be published.[55] C. Guillot, T. Lecuit, Science (6137), 1185 (2013)[56] P.A. de Boer, R.E. Crossley, L.I. Rothfield, Proceedingsof the National Academy of Sciences (3), 1129 (1990)[57] E. Garner, private communications .[58] M. Osella, E. Nugent, M.C. Lagomarsino, Proceedings ofthe National Academy of Sciences (9), 3431 (2014)[59] L. Robert, M. Hoffmann, N. Krell, S. Aymerich,J. Robert, M. Doumic, BMC biology (1), 17 (2014) [60] Cell size regulation in microorganisms, A. Amir, arXiv:1312.6562 (2014).[61] H. Jiang, S.X. Sun, Phys. Rev. Lett. , 028101 (2010)[62] K. Fl¨ardh, D.M. Richards, A.M. Hempel, M. Howard,M.J. Buttner, Current opinion in microbiology (6), 737(2012)[63] A. Geitmann, M. Cresti, I.B. Heath, Cell biology of plantand fungal tip growth , vol. 328 (IOS Press, 2001)[64] M. Keijzer, A. Emons, B. Mulder, in
Root Hairs , PlantCell Monographs , vol. 12, ed. by A. Emons, T. Ketelaar(Springer Berlin Heidelberg, 2009), pp. 103–122[65] A. Goriely, M. Robertson-Tessi, M. Tabor, R. Vandiver,in