View Article Online / Journal Homepage / Table of Contents for this issue Soft Matter Dynamic Article LinksC<

Cite this: Soft Matter, 2012, 8, 7446 www.rsc.org/softmatter PAPER Growth of curved and helical bacterial cells

Hongyuan Jiang and Sean X. Sun*

Received 26th February 2012, Accepted 17th May 2012 DOI: 10.1039/c2sm25452b

A combination of cell wall growth and cytoskeletal protein action gives rise to the observed bacterial cell shape. Aside from the common rod-like and spherical shapes, bacterial cells can also adopt curved or helical geometries. To understand how curvature in bacteria is developed or maintained, we examine how Caulobacter crescentus obtains its crescent-like shape. Caulobacter cells with or without the cytoskeletal bundle crescentin, an intermediate filament-like protein, exhibit two distinct growth modes, curvature maintenance that preserves the radius of curvature and curvature relaxation that straightens the cell (Fig. 1). Using a proposed mechanochemical model, we show that bending and twisting of the crescentin bundle can influence the stress distribution in the cell wall, and lead to the growth of curved cells. In contrast, after crescentin bundle is disrupted, originally curved cells will slowly relax towards a straight rod over time. The model is able to quantitatively capture experimentally observed curvature dynamics. Furthermore, we show that the shape anisotropy of the cross-section of a curved cell is never greater than 4%, even in the presence of crescentin.

1. Introduction forces applied by external constraints generate curved cells. Strikingly, the growth modes of the cell with or without cres- Bacterial cell walls are built through a complex biochemical centin are different17,18 as shown in Fig. 1. Here, based on process under high internal turgor pressure; therefore mechanical a mechanochemical model of cell wall growth, we provide forces and biochemistry are both important in determining the quantitative explanations of how crescentin controls cell overall cell shape. Mechanisms of growth and shape control have 1–4 curvature.

Downloaded by Johns Hopkins University on 08 January 2013 been extensively discussed in the literature. Recently, a number

Published on 15 June 2012 http://pubs.rsc.org | doi:10.1039/C2SM25452B of cytoplasmic cytoskeletal proteins have been implicated in bacterial cell shape maintenance.4–7 For example, MreB, a prokaryotic actin homologue, is the primary cytoskeletal protein in rod-like bacteria for maintaining the cylindrical shape.4–9 Disassembly of MreB filaments in bacteria leads to a morphological transformation from a rod to a sphere.8–13 Crescentin is an intermediate filament-like protein found in Caulobacter crescentus. Crescentin forms a bundle on the concave side of the crescent-shaped C. crescentus (Fig. 1). Experiments14,15 have shown if the crescentin bundle is detached from the cell wall, it collapses into a helix and appears to move freely in the cell. After crescentin is disrupted, initially curved cells gradually lose their characteristic curved shapes and even- tually become straight rods.14,15 Remarkably, artificially producing crescentin in Escherichia coli leads to helical cells.15 Not only crescentin but also physical forces applied by external Fig. 1 Growth of bacterial cell walls (green surface) is influenced by constraints can change the cell shape. Experiments showed that cytoskeletal bundles such as crescentin (red bundle). Curved bacterial straight rod-like E. coli cells grown in circular micro-chambers cells can have two different growth modes: (I) detachment of crescentin in C. crescentus cells,14,15 or the release of curved E. coli cells from circular become curved or even helical due to the confinement of micro- micro-chambers,15,16 leads to curvature relaxation (cell straightening). chambers.15,16 Moreover, these cells slowly return to their native 15,16 The curvature radius of the cell centerline, R, increases with time, but the straight rod-like shape after escaping from these chambers. span angle q increases slowly or remains constant. (II) When crescentin is This indicates that both the crescentin bundle and the mechanical attached to the cell wall, a curved rod-like shape develops if cells are short, but a spiral shape is obtained if cells grow longer.14,15 In this case, Department of Mechanical Engineering and Whitaker Biomedical the curvature radius of the cell centerline R and helical pitch p remain Engineering Institute, Baltimore, MD 21218, USA. E-mail: [email protected] unchanged, but the span angle q increases with time.

7446 | Soft Matter, 2012, 8, 7446–7451 This journal is ª The Royal Society of Chemistry 2012 View Article Online

2. Mechanical stresses in the cell wall Pr 22 ¼ ; T 2 (5) Due to a high internal concentration of osmolytes, bacterial cells 2ðR þ r cos fÞ maintain a high turgor pressure. This implies that the growing cell wall is under tension, and the growth dynamics is strongly T12 ¼ T21 ¼ 0. (6) influenced by the mechanical stress.19 Here, we can compute the stress field in the cell wall for curved and helical cell shapes using The mixed components of the stress resultant tensor can b ¼ rb b ¼ br a mathematical model. Since the cell wall is a stretchable network be given by Ta garT and T a T gra. In this of polysaccharides, it can be modeled as a thin elastic shell. The Prð2R þ r cos fÞ Pr case, T 1 ¼ T 1 ¼ , T 2 ¼ T 2 ¼ and other undeformed mid-plane of a cell wall can be described by a three 1 1 2ðR þ r cos fÞ 2 2 2 r dimensional surface (x1,x2), where (x1,x2) is a curvilinear coor- components are zero. We give the mixed component of the stress dinate system.20 The tangential vectors of the surface at any point b a resultant tensor Ta and T b since they are more convenient to m ¼ r are given by a v /vxa, where a ranges from 1 to 2. The unit use for the energy expression. As R goes to infinity, the two m ¼ m m m vector normal to the surface can be defined as 3 1 2/| 1 resultant stresses reduce to Pr and Pr/2, which are resultant m 2|. Therefore, the orientation of an arbitrary point is char- stresses of a cylindrical cell wall. acterized by the co-variant basis (m1,m2,m3). Notice that for certain geometry, like a helical cell wall, m and m are not 1 2 2.2. Stress field in curved rod-like cells with an elliptic cross- orthogonal. So it is convenient to introduce a second set of basis section vectors, i.e., the contravariant basis (m1,m2,m3), defined so that i i i i m $mj ¼ dj. Here dj is the Kronecker delta symbol, i.e. dj ¼ 1 for i In the previous section, we assumed that the cross-section of ¼ j and zero otherwise. The metric tensor is defined by gij ¼ mi$mj a curved rod-like cell is a circle. However, the shape of the cell and gij ¼ mi$mj. In this paper, we use Greek letters to indicate cross-section may deviate from a circle if other mechanical index 1 or 2 and use English letters to indicate index 1, 2 or 3. elements such as the crescentin bundle are attached to the cell The deformed mid-plane of the cell wall can be described by wall. To simplify the problem, we can assume that the cross- another three dimensional surface r(x1,x2). In the same way, we section is an ellipse and examine any possible shape anisotropy. 1 2 3 introduce two sets of basis (m1,m2,m3) and (m ,m ,m ). And the In this case, the shape of a curved rod-like cell with elliptic cross- metric tensor can be introduced similarly. section can be described as The mechanical equilibrium equations of a thin shell are r ¼ [(R + acos f)cos q,(R + acos f)sin q, bsin f] (7) vT ab þ ab g þ ag b þ b ¼ ; T Gag T Gag p 0 (1) vxa where a and b are the major and minor semi-axes of the cross- section. R, q and f share the same definition as in the last section. ab 3 T kab ¼ p , (2) The solution to eqn (1) and (2) is Pað2R þ a cos fÞ Downloaded by Johns Hopkins University on 08 January 2013 i ab 11 where p ¼ p mi is the external force per unit area, T ¼ T ma 5 T ¼ pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (8) ð þ Þ ð 2 þ 2Þð 2 2Þ Published on 15 June 2012 http://pubs.rsc.org | doi:10.1039/C2SM25452B ab 2b R a cos f a b a b cos 2f mb is the stress resultant tensor and T is symmetric. v2r v2r  à ¼ m $ i ¼ mi$ 2 2 2 Giab i and Gab are the Christoffel P ab ða b Þð2R cos f þ a cos 2fÞ vxavxb vxavxb 22 ¼ pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; vm T 2 (9) symbols of the first kind and second kind. k ¼ m $ 3 is the 2bðR þ a cos fÞ ða2 þ b2Þða2 b2Þcos 2f ab a vx curvature tensor. b T 12 ¼ T 21 ¼ 0. (10) 2.1. Stress field in curved rod-like cells with a circular cross- section It is easy to verify that the above solution can be reduced to eqn (4)–(6) when a ¼ b ¼ r. For a bacterial cell wall without any internal or external constraints, the only non-zero external loading is the turgor pressure p3 ¼ P. If we neglect the cell poles and assume that the 2.3. Stress field in a helical cell wall shape of the cross-section is a circle, the shape of a curved rod- For a helical cell, we can describe the cell shape using like cell can be described as one part of a torus:  Lr sin q sin f r ¼ ðR þ r cos fÞcos q þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðR þ r cos fÞsin q r ¼ 2 þ 2 [(R + rcos f)cos q,(R + rcos f)sin q, rsin f] (3) R L Lr cos q sin f Rr sin f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Lq þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : (11) where R is the curvature radius of the centerline, r is the radius of R2 þ L2 R2 þ L2 the cross-section, q is the span angle of the torus and f is the rotational angle along the circumferential direction of the torus Here R is the radius of the centerline, r is the radius of the cross- (see Fig. 1). The solution to eqn (1) and (2) is section, q is the span angle of the helix, f is the rotational angle along the circumferential direction of the helix, and p ¼ 2pL is ð þ Þ the pitch of the helix (see Fig. 1). In this case, the two basis 11 P 2R r cos f T ¼ ; (4) vectors m1 and m2 are not orthogonal and the resultant shear ð þ fÞ 2r R r cos stress is not zero anymore. It is not possible to compute the stress

This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 7446–7451 | 7447 View Article Online

resultant tensor by solving eqn (1) and (2) directly. However, we of transpeptidases, transglycosylases and hydrolases. Chemically can assume L/R 1 and examine leading order expansions of active PG monomers are inserted into the cell wall, making new the stress field as a function of the small parameter: bonds and releasing stored chemical energy. At the same time, initially stress-free PG subunits are stretched by the cell turgor 11 ¼ 11 11 11 2 11 3 . T Ta + Tb L + Tc L + Td L + (12) pressure, P. Therefore, the growth of cell wall is determined by the competition between the increased strain energy and the 22 ¼ 22 22 22 2 22 3 . 8,9,25 T Ta + Tb L + Tc L + Td L + (13) released chemical energy. The total free energy without considering crescentin can be written as T12 ¼ T12 + T12L + T12L2 + T12L3 + . (14) a b c d ð​ G ¼ ð f 3ÞdA; (18) By substituting the above expansions to the equilibrium eqn (1) 1 and (2) and requiring that the coefficients of each power of L sum where 3 is the chemical bond energy per unit area during the to zero, we obtain a series of recurrence ODEs. The series solu- growth process. f is the strain energy density of the cell wall, tion of eqn (1) and (2) is h h3 which can be given by f ¼ Dabrmg g þ DabrmDk Dk . Pð2R þ r cos fÞ PðR cos f r sin2fÞ 2 ab rm 24 ab rm T 11 ¼ þ L2 D 2rðR þ r cos fÞ 2RðR þ r cos fÞ3 Here, gab and kab are the mid-plane Lagrange strain tensor and h 3 2 2 2 2 2 the curvature change tensor. is the thickness of the cell wall. Pðr 2rR þ Rðr þ R Þcos f þ rR cos fÞ 28 4 Dabrm ¼ lgabgrm + m(gargbm + gamgbr) is the elastic tensor. Here, l 4 L 2R4ðR þ r cos fÞ ¼ 2 ¼ À Á nE/(1 n ) and m E/2(1 + n) are two constants, where E and n þ O L5 (15) are Young’s modulus and Poisson ratio of the cell wall. The stress resultant tensor is related to the mid-plane Lagrange strain 2 2 2 ab abrm Pr Prð3r 2R þ r cos 2fÞ tensor by the constitutive relation T ¼ hD grm. We can use T 22 ¼ þ L2 2ðR þ r cos fÞ2 4R2ðR þ r cos fÞ4 the results from the previous section to explicitly compute the h i strain tensor and the strain energy density. 2 2 2 4 2 2 2 2 Pr 2ðR r Þ R þRrðr R Þcos f r R cos 2f If the deformation in the cell wall is small, and the deformed þ 4 5 L r 2R5ðR þ rcos fÞ shape is close to the reference shape (x1,x2), the strain energy À Á density can be explicitly computed: þ O L5 (16) 2ðl þ mÞT a T b lðT a Þ2 k f ¼ b a a þ 1 ð2HÞ2þk K (19) 8mðl þ mÞh 2 2 Pr2 cos f Pr2 cos fð2r2 5R2 þ rR cos fÞ T 12 ¼ L þ L3 The first term of eqn (19) is the stretch energy and the other terms ð þ Þ3 4ð þ Þ4 2R R r cos f 4R R r cos f a À Á are the bending energy. T b is the mixed component of the stress 5 þ O L resultant tensor. H and K are the mean curvature and Gaussian

Downloaded by Johns Hopkins University on 08 January 2013 (17) curvature, respectively. Here, we assume that the spontaneous 3 Published on 15 June 2012 http://pubs.rsc.org | doi:10.1039/C2SM25452B curvature of the cell wall is zero. Two constants k1 ¼ h (l +2m)/ The above solution can be reduced to eqn (8)–(10) when L ¼ 0. 3 12 and k2 ¼h m/6 are similar to the bending modulus and the Gaussian rigidity of a lipid membrane. For a thin shell, the 3. Free energy of the cell wall bending energy is generally small compared to the stretch energy and is therefore negligible. Note that the strain energy now only The bacterial cell wall is a network of peptidoglycan (PG) depends on the reference shape r(x1,x2). Therefore the free energy strands. PG strands are constantly inserted into or deleted from of the cell wall is simplified. Also, it should be noted that the 21 the network by enzymes. Computational modeling showed that energy of the cell membrane is neglected here since it is signifi- removal of peptide bonds in a static PG network can lead to cantly softer than the cell wall. curved cells. Although changes in peptidoglycan (PG) cross- If we further consider the energy contribution of pressure and 22 linking seem to promote Helicobacter pylori’s helical shape, volume changes, the total free energy without considering cres- there are no discernible variations in crosslinking, thickness or centin can be modified to read 15 composition in curved C. crescentus cell wall. Therefore, ð ð although the microstructure of the cell wall may influence the cell G1 ¼ ð f 3ÞdA PdV; (20) shape, the helical shape of C. cresentus is mainly created and ð maintained by differential cell wall growth induced by cres- where PdV is the energy associated with internal osmotic centin.15 Previous models have considered the elastic interaction ð of the crescentin bundle with the cell wall.21,23,24 In these models, pressure. We found that the addition of PdV only changes the cell wall is treated as a static elastic structure. However, in the simulation results quantitatively for a helical cell. The similar living cells, the wall is constantly growing and reorganizing; thus conclusion can be drawn for spherical and cylindrical bacterial a non-equilibrium theory must be the starting point of analysis. cell walls.8,25 For example, for a spherical cell, the free energy Such a dynamic growth model that includes both mechanics ¼ p 2 4 2 is G1 4 [P R /8h(l +. m) 3R ], which yields a steady and some elements of biochemistry for the bacterial cell wall has pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð þ Þ been proposed for rod-like cells.8,9,25 The model considers an radius Rs 4h3 l m P if we do not include the contribu- elastic PG network undergoing subunit turnover from the action tion of pressure and volume changes.25 In contrast, if the term

7448 | Soft Matter, 2012, 8, 7446–7451 This journal is ª The Royal Society of Chemistry 2012 View Article Online ð parameters: the radius of the cross-section r, the curvature radius PdV is considered, the free energy is modified to G ¼ 4p 1 of the centerline R and the span angle of the torus q. Therefore, [P2R4/8h(l + m) 3R2 PR3/3]. In this case, the steady radius still the free energy can be written as G ¼ G(r, R, q) from eqn (20) and h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii . (21). For the helical shape, one additional parameter, the helical exists and it is Rs ¼ 1 þ 1 þ 43=ðl þ mÞh ðl þ mÞh P. pitch p, is required (Fig. 1). In this case, the free energy can be Crescentin is an intermediate filament-like protein that ¼ 14,26 written as G G(r, R, q, p). The poles of E. coli cells are rigid and assembles into stiff and stable filamentous structures in vitro, 30 inert. Therefore, we assume that they are also rigid and inert in and adheres to the concave side of the cell wall in vivo14,15 (Fig. 1). C. crescentus cells and neglect the cell poles in the growth By replacing the wild-type crescentin with non-functional CreS- calculation. The change of the total energy can be written as GFP,14 or inhibiting peptidoglycan cross-linking with b-lactam P dG ¼ F da , where a are shape parameters of the cell, i.e., r, antibiotic mecillinam in Dbla cells,15 the crescentin bundle can be i i i i R, q or p; F ¼vG/va is defined as the driving force detached from the cell wall. The detached crescentin bundle i i corresponding to the parameter ai. The growth velocity can be adopts a left-handed helical shape with a preferred helical pitch, 8 14,15 defined as p0 ¼ 1.4–1.6 mm. The preferred radius of the crescentin helix a R0 is not measured in experiments, but the radius of the helix is 1 d i ¼ MiFi: (22) close to the radius of C. crescentus after it is detached from the ai dt cell wall.14,15 Therefore, the preferred radius is assumed to be where Mi is a phenomenological growth constant corresponding similar to the radius of C. crescentus cells. The width of cres- to parameter ai. At the molecular scale, cell wall growth is centin helix is estimated to be 100–300 nm. The mechanical regulated by the spatio-temporal activity of cell wall synthesis properties of crescentin are unknown. However, crescentin is an and lytic enzymes. At the continuum level, the spatio-temporal intermediate filament-like protein. Therefore, we assume that activity of the enzymes is characterized as the growth constants. Young’s modulus of the crescentin filament is similar to that of In our calculation, the growth constants are assumed to be ¼ 27 intermediate filament, i.e., Ec 300–900 MPa. independent of time and space. In general, however, we can use To understand how crescentin influences cell wall growth, we spatio-temporally varying growth constants to model the spatio- model the crescentin bundle as a helical elastic rod adhered to the temporal regulation of cell wall growth. Indeed, recently we used inner side of the elastic cell wall (Fig. 1). So the shape of the a spatially varying growth constant to examine shape instabilities ¼ p r ¼ crescentin is given by eqn (11) when f , i.e., [(R r)cos in growing E. coli cells.8 q,(R r)sin q, Lq]. Assuming that the bundle can slide freely on the cell wall, bending and twisting are more important than bundle stretch. The strain energy of the helical bundle28,29 is 5. Results ð   L E I m J We first consider curved rod-like cells without the influence of G ¼ c ðk k Þ2þ c ðs s Þ2 ds (21) 2 2 0 2 0 crescentin. Similar to rod-like cells, curved rod-like cells can 0 8,25 R p =ð2pÞ maintain a constant radius of cross-section (Fig. 2a). In this ¼ 0 s ¼ 0  where k0 2 and 0 2 are the case, the resultant stresses along the circumferential direction Downloaded by Johns Hopkins University on 08 January 2013 R2 þ p2 ð2pÞ R2 þ p2 ð2pÞ 0 0 0 0 and axial direction are Pr(2R + rcos f)/[2(R + rcos f)] and Pr/2, Published on 15 June 2012 http://pubs.rsc.org | doi:10.1039/C2SM25452B curvature and twist of the spring in the relaxed state. R and p 0 0 respectively (see Section 2.1). f is the rotational angle along the are the preferred radius and helical pitch of the circumferential direction (see Fig. 1). On the concave side of the R r p=ð2pÞ spring. k ¼ . and s ¼ . cell (f ¼ p), the resultant stress along the circumferential ðR rÞ2 þ p2 ð2pÞ2 ðR rÞ2 þ p2 ð2pÞ2 are the curvature and the twist in the deformed state, respec- tively. R is the curvature radius of the cell centerline and r is the radius of the cross-section (Fig. 1). R r and p are the radius and

helical pitch of the spring in deformed state. Ec and mc are Young’s modulus and shear modulus of the helix. I ¼ pd4/64 and J ¼ pd4/32 are the area and polar moments of inertia of a circular cross-section with diameter d. The total energy of the cell wall– crescentin system is then a sum of the individual contributions,

i.e., G ¼ G1 + G2. Therefore, the cell wall and crescentin constitute a composite material. In this paper, we study the growth of this composite material, not just the cell wall itself.

Fig. 2 Time evolution of (a) the radius of the cross-section r, (b) the 4. Growth laws curvature radius of the centerline R and (c) the span angle of the torus q in

a curved rod-like cell after crescentin bundle is disrupted (EcI ¼ 0). In the Having obtained an explicit expression for the cell wall free calculation, the crescentin is removed at time t ¼ 0. The same variables energy as a function of the cell reference shape, we now can are plotted in (d), (e) and (f) for curvature maintenance in normal cells 2 compute the evolution and growth of the reference shape driven (EcI ¼ 100 nN mm ). The parameters used are P ¼ 0.3 MPa, E ¼ 30 MPa, 2 1 1 by this free energy. To simply characterize the shape and growth n ¼ 0.3, h ¼ 3 nm, 3 ¼ 0.05 J m , R0 ¼ 0.5 mm, MrMR ¼ 90 mJ s , Mq of a curved bacteria cell (Fig. 1), we need three dynamic ¼ 0 (red), 9 106 (green) and 2.7 107 (blue) J1 s1.

This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 7446–7451 | 7449 View Article Online

direction is Pr(2R r)/[2(R r)], which is larger than the resultant stress Pr(2R + r)/[2(R + r)] on the convex side (f ¼ 0). Since the cell wall growth is stress-dependent (eqn (1)), the effect of the stress-dependent growth is that the curvature radius R

always increases with time (Fig. 2b) no matter what Mq is used (Fig. 2c). Therefore, cells tend to straighten and the curvature relaxes during growth, which is consistent with the experimental observations.15,16 Our model predicts that the curvature declines with time exponentially (Fig. 3a) which can be compared with the experimental measurement for C. crescentus cells with disrupted 15,18 crescentin bundle. The value of the growth constant MR is fitted to experimental data from ref. 15 and 18 (see Fig. 3a). From Fig. 4 The shape anisotropy of the cross-section of a curved cell. When the experimental data, we know how the curvature of cell the crescentin bundle is present, the cell cross-section is more elliptical in centerline 1/R evolves with time. In the simulation, the growth shape. The shape anisotropy is defined as the ratio of the two semi-axes of dynamics of R is mainly determined by the value of MR and we the ellipse a and b. As the crescentin bundle stiffness increases, the cross- found an appropriate MR to reproduce the curvature relaxation section becomes more anisotropic, but the ratio a/b is never smaller curve found in the experiment (see Fig. 3a). The growth than 0.96. dynamics of r and q is unavailable in the experiment. Therefore, M and M are unknown, and their values are estimated in this r q It is also interesting to ask whether the crescentin bundle has calculation. an influence on cross-sectional shape of the cell. One might When the crescentin bundle is attached to the cell wall, expect that the cell cross-section is not circular if crescentin exerts curvature still tends to relax in a growing cell. However, the significant forces. In our model, we can investigate an elliptical tendency is counterbalanced by the bending and twist of the cross-section (see Section 2.2) and ask the equilibrium values of crescentin bundle. Therefore, there is a steady curvature radius elliptical axes, a and b. We predict that the circular shape is quite for the cell centerline R (Fig. 2e). Thus, in C. crescentus, the cell robust, and the shape anisotropy, 1 a/b, is never greater than wall and the crescentin bundle should be regarded as a composite 4% (Fig. 4). Interestingly, the long axis of the elliptical cross- material. The steady curvature radius is determined by the section is predicted to be perpendicular to the plane of curvature mechanical properties of this composite. The stiffness of cres- of the cell, although in this calculation, MreB is not considered centin bundle E I is proportional to the 4th power of the cross- c and may further stabilize the circular cross-sectional shape.8 section diameter of bundle since I ¼ pd4/64. Therefore, thicker Including MreB in the model would only quantitatively change crescentin bundles lead to smaller steady radius R as shown in the results. Fig. 3b. We predict that the steady radius can be altered by The model can be extended to helical curved cells where the changing the number of filaments in the crescentin bundle, which 15 helical pitch of the cell is described by p. The growth of this

Downloaded by Johns Hopkins University on 08 January 2013 can be achieved by changing the expression level of crescentin. helical cell can be characterized by R, q, r and p (Fig. 1). In this Strains lacking crescentin are straight rods (R / N), while Published on 15 June 2012 http://pubs.rsc.org | doi:10.1039/C2SM25452B 15 case, material basis vectors are not orthogonal and the resultant overproducing of crescentin results in smaller curvature radius. shear stress is not zero. We can only obtain a series solution if we This observation is consistent with our prediction. Furthermore, assume p/(2p) R (see eqn (15)–(17) in Section 2.3). Fig. 5b and the crescentin bundle does not have to be the same length as the c show that the curvature radius R and the helical pitch p increase cell. In fact, if the cell becomes very long, the crescentin bundle with time if crescentin bundle is disrupted in C. crescentus. might grow slower than the cell wall, and is shorter than the cell. In this case, the cell is straight at the two ends, but remains helical in the middle of the cell (Fig. 3D and E in the ref. 31). All these findings are in accord with our predictions.

Fig. 5 Time evolution of (a) the radius of the cross-section r, (b) the curvature radius of the centerline R, (c) the helical pitch and (d) the span

Fig. 3 (a) The curvature of cell centerline, 1/R, decreases exponentially angle of the helix q in a helical cell after crescentin bundle is disrupted (EcI with time for C. crescentus cells with disrupted crescentin bundle. The ¼ 0). In the calculation, the crescentin is removed at time t ¼ 0. The same simulation result is compared with experimental data from ref. 15 and 18. variables are plotted in (e), (f), (g) and (h) for curvature maintenance in 2 (b) The static curvature radius R as the function of crescentin bundle normal cells (EcI ¼ 100 nN mm ). In the simulation, p0 ¼ 1.6 mm and other stiffness. The same parameters are the same as in Fig. 2 except Mq ¼ 0. parameters are the same here as in Fig. 2.

7450 | Soft Matter, 2012, 8, 7446–7451 This journal is ª The Royal Society of Chemistry 2012 View Article Online

Therefore, similar to Fig. 2, the cell always straightens no matter 5 J. Moller-Jensen and J. Lowe, Curr. Opin. Cell Biol., 2005, 17, 75–81. 6 Y. L. Shih and L. Rothfield, Microbiol. Mol. Biol. Rev., 2006, 70, 729– what Mq is used. In contrast, if crescentin is intact (Fig. 5e–h), R 754. and p approach fixed values during cell growth so that the cell 7 M. T. Cabeen and C. Jacobs-Wagner, J. Cell Biol., 2007, 179, 381– can maintain a spiral shape. In both cases, cells always develop 387. a constant cross-sectional radius r (Fig. 5a and e). 8 H. Y. Jiang, F. W. Si, W. Margolin and S. X. Sun, Biophys. J., 2011, 101, 327–335. 9 H. Y. Jiang and S. X. Sun, Microbiol. Mol. Biol. Rev., 2011, 75, 543– 6. Concluding remarks 565. 10 T. Kruse, J. Møller-Jensen, A. Løbner-Olesen and K. Gerdes, EMBO We have described a mechanochemical explanation of curved J., 2003, 22, 5283–5292. bacterial cells with a stiff PG cell wall. The model shows that C. 11 N. Iwai, K. Nagai and M. Wachi, Biosci., Biotechnol., Biochem., 2002, crescentus cells can develop curvature in the presence of the 66, 2658–2662. 12 R. M. Figge, A. V. Divakaruni and J. W. Gober, Mol. Microbiol., crescentin bundle. When the bundle is disrupted, cell curvature 2004, 51, 1321–1332. relaxes exponentially with time. Two distinct growth modes 13 Z. Gitai, N. Dye and L. Shapiro, Proc. Natl. Acad. Sci. U. S. A., 2004, observed in experiments are fully explained by a single model. 101, 8643–8648. 14 N. Ausmees, J. R. Kuhn and C. Jacobs-Wagner, Cell, 2003, 115, 705– The model provides an explicit relationship between mechanical 713. stress and growth of the PG network. 15 M. T. Cabeen, G. Charbon, W. Vollmer, P. Born, N. Ausmees, The continuum model presented here, however, does not have D. B. Weibel and C. Jacobs-Wagner, EMBO J., 2009, 28, 1208– molecular details, which requires further in vivo elucidation. For 1219. 16 S. Takeuchi, W. R. DiLuzio, D. B. Weibel and G. M. Whitesides, example, it was found recently that PBP enzymes add new cell Nano Lett., 2005, 5, 1819–1823. 32–34 wall material in a processive MreB-dependent manner. Also, 17 R. Mukhopadhyay and N. S. Wingreen, Phys. Rev. E: Stat., the cell wall probably grows by creating PG defects using Nonlinear, Soft Matter Phys., 2009, 80, 062901. hydrolases and filling up the defects using PBPs. The concen- 18 O. Sliusarenko, M. T. Cabeen, C. W. Wolgemuth, C. Jacobs-Wagner and T. Emonet, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 10086– tration of molecular defects may also depend on the mechanical 10091. stress. This implies that the cell wall mechanical properties are 19 A. Typas, M. Banzhaf, C. A. Gross and W. Vollmer, Nat. Rev. not constant, and should depend on the dynamics of molecular Microbiol., 2012, 10, 123–136. 20 D. R. Nelson, T. Piran and S. Weinberg, Statistical Mechanics of scale defects. These molecular-level details can be incorporated Membranes and Surfaces. World Scientific, Singapore, 2004. into the present continuum framework by considering 21 K. C. Huang, R. Mukhopadhyay, B. Wen, Z. Gitai and a dynamics constitutive relationship for the cell wall, and a more N. S. Wingreen, Proc. Natl. Acad. Sci. U. S. A., 2008, 105, 19282– complex model for the growth constants M in eqn (22). 19287. i 22 L. K. Sycuro, Z. Pincus, K. D. Gutierrez, J. Biboy, C. A. Stern, However, these molecular details still must obey the energy W. Vollmer and N. R. Salama, Cell, 2010, 141, 822–833. balance criterion discussed in this paper. Indeed, the growth law 23 J. S. Kim and S. X. Sun, Biophys. J., 2009, 96, L47–L49. of eqn (22) is a statement on the kinetics of cell wall addition and 24 C. W. Wolgemuth, Y. F. Inclan, J. Quan, S. Mukherjee, G. Oster and M. A. Koehl, Phys. Biol., 2005, 2, 189–199. how this kinetics is influenced by the global stress in the cell wall. 25 H. Y. Jiang and S. X. Sun, Phys. Rev. Lett., 2010, 105, 028101. Refinement of the present model combined with new experi- 26 O. Esue, L. Rupprecht, S. X. Sun and D. Wirtz, PLoS One, 2010, 5, Downloaded by Johns Hopkins University on 08 January 2013 mental data will provide further connections between the e8855.

Published on 15 June 2012 http://pubs.rsc.org | doi:10.1039/C2SM25452B  molecular level and the cell level to explain the bacterial cell 27 C. Guzman, S. Jeney, L. Kreplak, S. Kasas, A. J. Kulik, U. Aebi and L. Forro, J. Mol. Biol., 2006, 360, 623–630. shape. 28 A. F. Bower, Applied Mechanics of Solids. CRC Press, 2009. 29 S. S. Antman, Nonlinear Problems of Elasticity. Springer, 2005. References 30 J. J. Thwaites and N. H. Mendelson, Adv. Microb. Physiol., 1991, 32, 173–222. 1 J. V. Holtje, Microbiol. Mol. Biol. Rev., 1998, 62, 181–203. 31 G. Charbon, M. T. Cabeen and C. Jacobs-Wagner, Genes Dev., 2009, 2 K. D. Young, Microbiol. Mol. Biol. Rev., 2006, 70, 660–703. 23, 1131–1144. 3 W. Vollmer and U. Bertsche, Biochim. Biophys. Acta, Biomembr., 32 J. Dominguez-Escobar, et al., Science, 2011, 333, 225–228. 2008, 1778, 1714–1734. 33 E. C. Garner, et al., Science, 2011, 333, 222–225. 4 M. T. Cabeen and C. Jacobs-Wagner, Nat. Rev. Microbiol., 2005, 3, 34 S. van Teeffelen, et al., Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 601–610. 15822–15827.

This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 7446–7451 | 7451 Addition and correction

View Article Online

Note from RSC Publishing

This article was originally published with incorrect page numbers. This is the corrected, final version.

The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.

Downloaded by Johns Hopkins University on 08 January 2013 Published on 15 June 2012 http://pubs.rsc.org | doi:10.1039/C2SM25452B