Mechanical control of bacterial cell shape: supplemental material

Transcription

1 Mechanical control of bacterial cell shape: supplemental material Hongyuan Jiang 1, Fangwei Si 1, William Margolin 2, Sean X. Sun 1,3 I. Table of Contents II. Introduction...2 III. Geometry of the cell wall...3 IV. Stress fields in the cell wall...4 V. Total free energy...6 VI. Growth dynamics...7 VII. Forces applied by MreB bundle and influence on cell wall stress and total free energy...9 VIII. Growth- induced instability of a cylindrical cell wall along the axial direction and the suppression of the instability by MreB...13 IX. The stability of a cylindrical cell wall along the circumferential direction...16 X. Effect of MreB helix pitch...18 XI. Computational details...19 XII. Experimental procedures Department of Mechanical Engineering and Whitaker Biomedical Engineering Institute, Johns Hopkins University, Baltimore, Baltimore, MD 21218, USA 2 Department of Microbiology and Molecular Genetics, University of Texas Medical school at Houston, Houston, TX 77030, USA 3 Johns Hopkins Physical Science Oncology Center, Johns Hopkins University, Baltimore, Baltimore, MD 21218, USA 1

2 II. Introduction The theoretical model we consider describes the interplay of the cell wall surface with the helical MreB bundle. The wall is also experiencing growth by adding new peptidoglycan (PG) material. This growth is driven by favorable addition of activated PG subunits to the existing PG network. We model the PG network as an isotropic elastic surface with a Young's modulus and a Poisson ratio. The isotropic assumption is motivated by recent electron microscopy images, which showed that the PG network E. coli cells are disordered, with no obvious orientational organization in the PG strands [1], although the exact nature of the cell wall constitutive relation is not crucial to the general conclusions. We also model the MreB bundle as an elastic filament with an effective bending constant (see below). The bundle is helical with a preferred radius,, and pitch. Because the preferred radius of the MreB bundle is smaller than the cell wall radius, the bundle exerts a force on the cell wall. With these assumptions, we are able to explain how MreB controls the bacterial cell shape. There has been previous work that sought molecular level explanation of the bacterial cell shape. Hotjie, Vollmer and others, based on biochemical studies in vitro and the fact that PG synthesis involves the turnover of existing PG material, proposed the 3-for-1 mechanism of cell wall growth [2, 3]. This model assumes that an enzyme complex, possibly involve PBPs and hydrolases, insert 3 PG strands while removing one existing strand. This model is appealing in several ways, but connections to the global shape of the cell are difficult to establish. Modern electron microscopy results [1] motivated Huang et al to develop a static polymer model of the E. coli PG layer [4]. The model was able to show that the cell shape is quite robust to the cell wall damage and appearance of network defects. The model argues that common bacterial cell shapes are results of simple spatial patterning of cell wall defects. However, the model neglects growth and remodeling of the cell wall, which has been shown to be an essential aspect of PG life cycle. Furthermore, how prokaryotic cytoskeletal proteins influence the morphology of bacterial cells has not been discussed. In this paper, we use a mechanochemical model based on continuum theories to describe the cell shape. Even though there is no explicit molecular detail, mechanical properties of the PG layer and the MreB bundle are indirect descriptions of their molecular organization. Moreover, the general principle of mechanical forces influencing growth kinetics is best described by continuum theories, and any detailed molecular model must also satisfy this principle. Therefore, the proposed model is a general framework upon which more sophisticated descriptions can be built. We begin by describing the geometrical shape of curved surfaces and the mechanical energy of the cell 2

3 wall. We then describe growth equations based on the total free energy of the cell wall. We calculate forces exerted by a helical filament adhered to the cell wall and estimate the pressure from MreB. Instabilities predicted by the growth equations are then discussed, along with how MreB can suppress this instability. Finally, experimental conditions where the growth-induced instability is observed are explained. III. Geometry of the cell wall The undeformed mid-plane of a cell wall can be described by a three-dimensional surface, where is a curvilinear coordinate system [5]. The surface tangential vectors at any point are given by (S1) The unit vector normal to the surface is (S2) Therefore the orientation of an arbitrary point is characterized by. This basis is called the covariant basis. Because and are in general not orthogonal, it is convenient to introduce a second set of basis vectors defined so that, where is the Kronecker delta symbol, i.e. for and zero otherwise. In this paper, we use Greek letter to indicate index 1 or 2 and use English letter to indicate index 1, 2 or 3. This second set of vectors is the contravariant basis for the coordinate system. The metric tensor is defined by The Christoffel symbols of the first kind and second kind are defined (S3) (S4) (S5) The curvature tensor is defined by 3

4 (S6) The relation between the Christoffel symbols and curvatures is. The mixed component of curvature tensor is given by. The deformed mid-plane of the cell wall can be described by another threedimensional surface. In the same way, we introduce two sets of basis and. And the metric tensor, the Christoffel symbols and the curvature tensor can be introduced similarly. It is necessary to distinguish the deformed and undeformed geometry if the deformation is large. In this paper, we assume the deformation is small and simply use the deformed geometry to represent the cell shape. IV. Stress fields in the cell wall The thickness of the cell wall is relatively small compared to the characteristic size of the cell so that the thin shell theory can be used. In general, the stress resultant and moment tensor in a shell are not symmetric. The asymmetry is small and ignored in the simplified shell theory. For the membrane theory of the thin shell, the transverse shear stress resultants and the internal moments are neglected since the shell is so thin and the stress resultant is symmetric. Such a system is said to be statically determinate and thus independent of compatibility or constitutive considerations. The mechanical equilibrium equations of a thin shell are [6] (S7) (S8) where is the external force per unit area, is the stress resultant. In this case, we have. For a bacterial cell wall without any internal or external constraints, the only nonzero external loading is the osmotic turgor pressure. If we assume the shape of the cell wall is axisymmetric and use the following cylindrical coordinate (See Fig. S1) (S9) 4

5 and solved and the solution is can be calculated from Eq. (S5) and (S6). Then Eq. (S7) and (S8) can be Figure S1. Coordinate system and parameterization of the bacterial cell wall surface. The shape is a surface of revolution about the axis. and are the local tangential vectors on the cell wall. is the unit vector normal to the surface. Here the prime indicates the derivative with respect to the stress resultant is, i.e., and the mixed component of Here we also give the mixed component of the stress resultant convenient for the energy expression. since it is more For a cylinder with radius, the stress resultants are and since. For a sphere with radius, can be interpreted as the rotational angle so that and. Therefore, the stress resultants are. The 5

6 stress can be derived from stress resultant as, where is the thickness of the thin shell. V. Total free energy We model the total energy of the cell wall as a combination of mechanical strain energy and chemical bond energy (Eq. 1 in the main text), or. The competition between the increased mechanical energy and the released chemical energy will determine the growth of the cell wall. We assume the growth process is quite slow. Therefore the cell wall is always in mechanical equilibrium. This implies that the work done by the turgor pressure is equal to the increase of the strain energy in the cell wall during this quasi-static process. The total energy of the system can be written as (S10) where (S11) is the stretch (the first term) and bending energy (the second term) density of the cell wall. is the thickness of the thin shell and is the curvature change tensor. Here we define an elastic tensor following reference [6] (S12) where and are lame constants. and are Young's modulus and Poisson's ratio, respectively. These lame constants are slightly different from the standard definition due to the specificity of shell. The mid-plane Lagrange strain tensor is defined by. The stress resultant tensor is given by the constitutive relation, where is the thickness of the cell wall. For the membrane theory of the thin shell, the system is statically determinate so that we can solve the stress field directly from Eq. (S7) and (S8). It is not necessary to solve the strain and displacement fields. If the spontaneous curvature of the cell wall is nonzero, the ratio between the bending energy and stretch energy is proportional to [7], where is the radius of the spontaneous curvature. So the bending energy can be neglected if. If the 6

7 spontaneous curvature of the cell wall is zero, the cell wall should be flat in the relaxed state. In this paper, we assume the spontaneous curvature is zero. The density of the strain energy becomes (S13) For small deformations, there is no difference between and. Thus, the density of the strain energy can be simplified to (S14) where is the mean curvature and is the Gaussian curvature. The two constants, and are similar to the bending modulus and the Gaussian rigidity of a lipid membrane. It should be noted that is negative, which is consistent with the common assumption for a membrane bilayer. VI. Growth dynamics We define the driving force for growth,, as the energy decrease when the cell wall grows a unit length. By making the variation with respect to, we have (S15) where the driving force proportional to the driving force, i.e.. The local growth velocity should be (S16) where process. is a constant which is determined by the kinetics of the microscopic growth could vary spatially due to nonuniform distribution of PG synthesis proteins. If during cell wall growth, the shape of the cell is the same, only one or more dimensions of the shape are changing, then mathematically the cell shape can be described by parameters. For example, rod-like bacteria maintain a constant radius, but elongate with time in the axial direction. The shape of such a cell can be characterized by its length and radius if the caps are neglected [8]. In this case, the change in total energy is (S17) 7

8 where is defined as the driving force corresponding to the parameter. The growth velocities of parameter can be described by (S18) where is the phenomenological growth constant corresponding to parameter. The above growth laws are usually suitable for growth at a specific region in the cell. However, if growth occurs everywhere as in E. coli, the growth velocity should be proportional to the surface area or length. Therefore the growth law can be modified to (S19) The underlying mechanism in Eq. (S18) and Eq. (S19) are the same. The only difference between them is how the growth velocity changes with time. For a rod-like bacterial cell, the length of bacteria grows linearly with time if we use Eq. (S18) and exponentially with time if we use Eq. (S19). If we use the same cylindrical coordinate that we used in Eq. (S9), the total energy can be written as (S20) where is the free energy density and the prime represents the derivative with respect to. We introduce a new variable since the energy does not contain explicitly. Therefore, the growth equation can be given as (S21) (S22) where and are two growth constants corresponding to and, respectively. We use Eq. (S21) and (S22) for all the simulations in this paper. 8

9 VII. Forces applied by MreB bundle and influence on cell wall stress and total free energy The above growth laws have been used to study the growth of spherical and cylindrical cells [8]. The model explained why rod-like bacteria can maintain a specific radius, but grows with time in the axial direction. However, the radius was assumed to be uniform along the cell length and the effects of cytoskeletal proteins were neglected. In this paper, we will study the stability of a cylindrical cell wall and focus on how MreB bundles affect cell shape. Figure S2. Coordinate system of a MreB helix bundle. The axis of the helix is in the direction. and are the helix angle and the length of helix, respectively. is arclength of the helix and is the radius of the helix. The MreB helical bundle can be regarded as an elastic spring under force in the radial and/or axial directions. The solution of an elastic helical spring under force and torque at the end of the spring is given by Love [9]. Here we use the same method to solve this problem with different boundary conditions. The centerline of a helix with helical angle and radius (See Fig. S2) is (S23) where is arclength of the helix. The tangent, normal and binormal vectors are 9

10 and they are related by the Frenet-Serret equations [10] as where is the local curvature and is the local twist. The total strain energy of an helix spring [6, 10] is (S24) where and are the curvature and twist of the helical spring in the relaxed state. and are the preferred radius and helical angle of the spring. and are the Young's modulus and shear modulus of the helix. and are area moment of inertia and polar moment of inertia of a circular cross section with diameter. The length of helix in the relaxed state and the deformed state are and, where is the total length of the rod. We consider the helix under the action of a line force along direction and a point force at the end. The response of the helix to these forces will determine the force-displacement relationship of the helix. If the radius and helical angle in the deformed state are and, the forces needed are When the deformation is small we may write and, where. Therefore, we have 10

11 When the helical angle is small, is small and the two forces are approximately decoupled. (S25) (S26) In the cell, the MreB helix is a highly dynamic structure. The time scale of the MreB motion is much smaller than the time scale of cell wall growth. Therefore, the radial force exerted by MreB should be averaged over the cell wall surface and modeled as a pressure field. The average pressure should be, where is the pitch of the helix. Thus, we write the traction applied by MreB along the radial direction as a linear function of displacement, i.e. where as the current radius of the cell wall. is the effective stiffness of MreB in the radial direction and (S27) When MreB is present, the external forces in Eq. (S7) and (S8) should be a combination of turgor pressure and the traction applied by MreB filaments, i.e. Notice that. So is not a unit vector. Therefore,. Then we have, and. In this case, the solution to Eq. (S7) and (S8) is 11

12 (S28) And the mixed component of the stress resultant is (S29) When the cell grows, MreB helix also grows along with the cell. The growth of the MreB bundle is also determined by the competition between chemical energy and mechanical energy. To account for MreB growth, we can add an additional term in the chemical energy. Since the force applied by MreB is modeled as a pressure field, the chemical energy per unit length of MreB is also averaged over cell wall surface. Therefore, the effective chemical energy per unit area is, where is the chemical energy of MreB per unit area. So the growth of MreB can be included as an effective parameter, we only need to use in the simulation. To account for the effects of MreB, the total energy (Eq. (S10)) of the system is modified as (S30) where the strain energy density (Eq. (S14)) is modified to (S31) The last part of the above equation is the mechanical energy of MreB helix. In summary, the cell wall and MreB constitute a reinforced composite material. Eq. (S30) allows us to compute the growth of this composite material, and not just the bare cell wall. 12

13 VIII. Growth-induced instability of a cylindrical cell wall along the axial direction and the suppression of the instability by MreB Assume that there is a cosine wave perturbation along the axial direction of an infinite cylindrical cell wall so that it remains axisymmetric. The radius of bacteria can be written as (S32) where is the initial radius of the cell wall. and are the amplitude and wavenumber of the perturbation. Assume and the position of cell wall is (S33) where indicates the growth of cell wall in axial direction with initial condition. The average free energy in one wave length of the perturbation is (S34) where is the determinant of the metric tensor. If MreB is present, the above equation should be modified to (see Eq. (S30) and (S31)) to include the influence of MreB. The total energy can also be written as. Then the growth equations are (S35) (S36) (S37) where, and are the growth constants for, and, respectively. Eq. (S35) and (S36) represent the growth dynamics of a cylindrical cell wall and Eq. (S37) define the growth behavior of a small perturbation. If we consider leading order terms of, Eq. (S35)-(S37) are expressed as 13

14 (S38) (S39) (S40) where is defined as growth factor of the perturbation., and are defined as (S41) (S42) (S43) Note that does not enter the above expression since the perturbation is small so that the Gaussian bending energy appears in higher order terms of. If and, Eq. (S35) and (S36) are reduced to Eq. (13) and (14) in our previous paper [8]. We find that a cylindrical cell wall itself is unstable. But MreB bundles can suppress the 14

15 instability. The dependence of on for various MreB effective stiffnesses is plotted in Fig. 2 in the main text. To obtain further insights into this instability, we can perform a simple analysis by comparing the energy of a cylinder and a series of spheres. Without considering MreB reinforcements and bending deformations, the strain energy of a growing cylinder can be derived from Eq. (S14) as (S44) where and are the length and radius of the cylinder. The strain energy of a growing sphere with radius is (S45) If we assume the cylinder and the sphere have the same surface area, i.e., we have. The chemical energies of these shapes are identical and therefore drop out. Substituting this into the above energy expression, we have (S46) If, or, then the strain energy of the sphere is lower. can be regarded as the wavelength of the perturbation, i.e.. This estimate implies that there is a wavelength regime where the spherical shape is more favorable than the cylinder. But it cannot tell us whether the cell wall is stable or not. Actually, if the wavelength of the perturbation is very small, the cell wall with small perturbation has higher stretch and bending energy compared to the cylindrical cell wall. Therefore, the cylindrical cell wall is stable although the spherical shape has lower energy. However, if the wavelength of the perturbation is bigger but still satisfying the condition, not only the energy of the spherical shape, but also the energy of the perturbed cylindrical cell wall is lower than that of the straight cylindrical cell wall. So the cylindrical cell wall is unstable in this case. Therefore, the cylindrical cell wall is only unstable for an intermediate wavelength of the perturbation (Fig 2 of the main text). The complete analysis of shape perturbation growth is given in Eq. (S40), where the full wavelength dependence is derived and plotted in Fig. 2 of the main text. 15

16 IX. The stability of a cylindrical cell wall along the circumferential direction In the previous section, we examined the growth instability along the axial direction of an infinitely long cylindrical cell wall. Similarly, instability along the circumferential direction could also occur. Consider a cosine wave perturbation along the circumferential direction of an infinitely long cylindrical cell wall. In this case, the radius is the function of polar angle and can be written as (S47) where n=1, 2, 3, and is the amplitude of the perturbation. is the initial radius of the cell wall. Assume and the position of cell wall is (S48) where has the same meaning as in last section. Figure S3. Stability of a growing cylindrical cell wall in the circumferential direction. The growth factor is plotted as a function of n for several stiffnesses of the MreB bundle. The cell wall is always stable even without MreB. E. coli parameters (see Table S1 in SM) are used in the calculation. 16

17 In this case, we do not need to solve Eq. (S7) since only one component of curvature tensor is nonzero so that can be simply obtained from Eq. (S8). Here, we give the mixed component of the stress resultant as (S49) where the series solution of is given by the force balance along the axial direction, i.e.,, where and are the circumference and area of the cross-section. If MreB is present, has another term as in Eq. (S29). The average free energy in one wavelength of the perturbation along the circumferential direction is. If MreB is present, this energy is modified to (S31)). The total energy can also be written as to include the influence of MreB (see Eq. (S30) and and the growth equations are defined as previously in Eq. (S35)-(S37). If we only consider the leading order terms of, we find the growth equations for, and are the same as Eq. (S38)-(S40) except the growth factor is where the constants are (S50) (S51) 17

18 (S52) The static radius can be solved from Eq. (S38) by setting. Substituting the static radius to Eq. (S50)-(S52) and assume and, we can show that since with Poisson s ratio. Therefore, the growth factor is always non-negative and the cell wall is stable with respect to the perturbation along the circumferential direction even without additional forces from MreB bundles. The growth factor is also plotted as a function of n when, and (see Fig. S3). We see that the circular cross section is stable. In experiments, we do not observe any instability that breaks the circular symmetry of the rod-like cell, which confirms our predictions. X. Effect of MreB helix pitch In the discussion so far, the force exerted by MreB is modeled as a pressure field. Now we show that bacteria can still maintain a cylindrical shape and constant radius even if this assumption is not valid. For simplicity, we regard the MreB helix as an ensemble of rigid tori with identical radii. The distance between the neighboring tori, which is an analog of the pitch of the helix, remains constant during growth. The steady state shape of cell wall is simulated for different values of ring separation (Fig. S4). The cell wall tends to bulge due to the instability we discussed above. For relatively large pitch, the cell wall is ripple-shaped with a period defined by the ring separation. However, the amplitude of the ripple becomes small and the cell wall can be roughly considered as a cylinder if the ring separation is smaller than 0.4 (Fig. S4). In reality, MreB bundles are highly dynamic. Even if the time average of MreB helix cannot be regarded as a continuous structure, the bundle still can re-enforce the cell wall and maintain a smooth cylinder. 18

19 Figure S4. The maximum radius of the bacterial cell wall as a function of the MreB ring separatin, which is an approximation of MreB helix pitch. The preferred radius of MreB,, is taken to be 0.4 (blue line) and 0.5 (red line), respectively. The insets are the steady 3D shapes of a cylindrical cell wall when and the ring separation equals 0.2, 0.4 and 0.8. The approximate pitch of MreB in the cell is 0.3. E. coli parameters (see Table S1) are used in calculation. XI. Computational details We compute the growth of bacterial cells under various conditions by solving the PDEs in Eq. (S21) and (S22). In the simulation, we use a fixed domain ; the actual cell length is given by we use mechanical energy density. Starting with a short cylindrical cell shape, in Eq. (S14) to compute the growth dynamics. The short rod-like cell will grow into a sphere. Then using the spherical shape as a new initial shape, we compute the transformation from a sphere to a rod using the new energy density in Eq. (S31), which includes influence from MreB. The computed shapes are shown in Fig. 5 in the main text. To account for the constriction of the Z-ring, another energy term from the contraction force,, must be added to Eq. (S10). We assume the Z-ring contracts inwards and the energy can be written as, where is the radius at midcell and is a constant force applied by Z-ring. 19

20 Table S1. Summary of the parameters used in the simulation Parameter Description E. coli Young's modulus of cell wall, MPa 50 Poisson s ratio of cell wall 0.3 Thickness of cell wall, nm 7 Chemical energy released per unit area, 0.4 Turgor pressure, MPa 0.2 Growth constant in the radial direction, Growth constant in the axial direction, 2000 Effective stiffness of MreB, 1.5 Preferred radius of MreB, 0.4 The dimension of is different from that of since is dimensionless from its definition. The cell poles are believed to be inert or static in E. coli [11, 12], which indicates that the growth rate constants and at the poles should be zero or at least much smaller than that the rest of the cell. To account for nonuniform growth, we use a spatially varying growth rate constant: (S53) where and are the growth constants at the end ( ) and at the position of the cell, represents 1 or 2, and are two parameters specifying the shape of the function. The effects of and are showed in Fig. S5. In the simulation, we use,,, (see Table S1) and. Notice that the dimension of is different from that of since is dimensionless from its definition. Therefore, the growth constants are zero at the ends and nonzero in the middle of the cell. The above function will converge to a squarewave-like function as. More complicated distribution can be created by the superposition of the above function. We use to this spatially varying growth rate to simulate E. coli growth in Fig. 4D in the main text. 20

21 Figure S5. The spatially varying growth rate constant M is plotted as the function of η. Both M and η are normalized their maximum value. (A) The location of the maximum value of M is determined by η0. (B) The width of the peak is determined by σ. Other parameters used: M e /M max =0, M m /M max =1 and m=2. XII. Experimental procedures E. coli strain WM1283 was used for the observation of morphological changes of filamentous cells induced by A22. WM1283 has a deletion of the FtsZ gene and carries a plasmid expressing FtsZ that is thermosensitive for replication; therefore, FtsZ is depleted from cells upon shift from 30 to 42 after several generations. WM1283 cells were grown in Luria-Bertani (LB) broth at 30 for 12 hours while shaken at 225 rpm. Two hours before imaging, a low density (OD < 0.1) culture was shifted to 42 to deplete FtsZ and the cells become filamentous. Before imaging, the bacterial cells were transferred to a microscope slide containing a thin layer of LB agar supplemented with 10, 20, 40, 60, 80 and 200 of A22 to depolymerize the MreB filaments. Time-lapse images are collected at 4 minute intervals. Supplemental References 1. Gan, L., Chen, S., and Jensen, G. J. (2008). Molecular organization of Gramnegative peptidoglycan. Proc. Natl. Acad. Sci. USA 105, Holtje, J.V. (1998). Growth of the stress-bearing and shape-maintaining murein sacculus of Escherichia coli. Microbiol. Mol. Biol. Rev. 62,

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