Mechanical Behaviour of Bacterial Cell Walls

Mechanical Behaviour of Bacterial Cell Walls JOHN J . THWAITES" and NEIL H . MENDELSONb "Department of Engineering. University of Cambridge. Cambridge CB2 I PZ. UK. and bDepartment of Molecular and Cellular Biology. University of Arizona. Tucson. AZ 85721. USA

1. Introduction . . . . . . . . . . A . The cell surface . . . . . . . . B . The mechanical role of the cell wall . . . . I1 . Thestructureofcell-wallmaterial . . . . . A . Peptidoglycan . . . . . . . . B . Peptides and cross-linking . . . . . . C . Other polymers . . . . . . . . 111. Thephysicalstateofthecellwall . . . . . . A . Chargedcell-wallpolymers . . . . . B . The dynamic structure of the cell wall . . . C. Order in cell-wall material . . . . . . D . Thesignificanceofmacrofibres . . . . . IV . Mechanical properties . . . . . . . . A . Early observations . . . . . . . B. Direct measurements by means of bacterial thread C . Measuredproperties . . . . . . . D . Environmental effects . . . . . . . E . Visco-elasticity . . . . . . . . F. Molecular arrangement in the cell wall . . . V . Cell-wall models . . . . . . . . . A . Theaimsofmechanicalmodelling . . . . B . Geometrical models . . . . . . . C . Models involving surface tension-like stress . . D . A model involving anisotropic cell-wall material . E. A cell-wall growth model . . . . . . VI . Conclusions . . . . . . . . . . References . . . . . . . . . . .

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I. Introduction A. THE CELL SURFACE

The external boundaries of living cells provide the interface between two very different environments, one highly regulated by homeostatic biochemical mechanisms within cells, the other highly variable to extremes in the chemical and physical nature of the environment. The functions of the boundary are complex: it plays a structural role, it accommodates the selective movement of materials through itself, it undergoes changes made necessary by growth within or in response to forces from without and it transfers information about the environment into the cell. The materials that serve as cell boundaries consist of: (i) lipids organized into bilayer membranes, (ii) sugar polymers built into exoskeletal girdles (cell walls) and (iii) proteins (Rogers et al., 1980). The properties of these materials dictate the kinds of function each is suited to perform. We will not deal with membranes here. Sugar polymers, such as cellulose and peptidoglycan that comprise the cell-wall materials of plant and bacterial cells, respectively, are made of covalently linked, sometimes multilayered, sheets of material that form porous networks of considerable strength. They derive their integrity from the chemical nature of the linkages that bind the monomers together into polymers, or that link polymers to one another, as well as from hydrogen-bonding, electrostatic and hydrophobic interactions. Thus, the strength of cell-wall materials, a most significant feature, originates in the strength of the bonds between the components and the structural organization of the components in the wall (Marquis, 1988). The properties of proteins also have bearing on bacterial cell-wall mechanics. The folded three-dimensional shape of a protein polymer involves helical and sheet regions held in position by a combination of cross-bridges, charge interactions and hydrodynamic factors (Cantor and Schimmel, 1980). The final configuration is highly inflexible but not totally unable to change shape in response to diverse stimuli. Many of the mechanical properties of peptidoglycan described in this article reflect the behaviour of the peptide portion of the polymer. It is a considerable challenge, therefore, to understand the behaviour of bacterial cell walls as a material. The effort is warranted in view of the significant role cell walls play in cell growth and survival.

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B THE MECHANICAL ROLE OF THE CELL WALL

I . Mechanical Requirements

When dealing with bacterial cellular processes that involve the cell surface it is often difficult to separate the contributions of the cell membrane and the

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cell wall. When it comes to the mechanical role of surface components, however, distinctions become much clearer owing to the physical nature of the materials involved. For example, stiffness in the cell surface, which is required to maintain shapes other than spherical, can only be a property of the wall. This is easily shown by removing the wall from a rod-shaped cell under osmotically stabilized conditions. The resulting wall-less form is always spherical ,indicating that surface-tension forces predominate and the cell membrane assumes a minimum surface area (Gilpin and Nagy, 1976). Similarly, if cell walls from rod-shaped cells are purified under conditions where they remain intact, the collapsed empty structures clearly resemble the shape of the original cylindrical cells (Beveridge, 1981). Wall stiffness must therefore maintain shape in living cells even under conditions where cells grow, which raises the question of whether stiffness and shape might influence the growth process itself. Bacterial cells of all shapes have a kind of structural polarity that becomes evident during growth and division when their structures gradually change in an ordered way from that of the initial shape to elongated versions of the same form. As new cell wall is assembled a continuity of shape is preserved. The cell wall is stiff enough to maintain that shape and, at the same time, ductile enough to permit the expansion made necessary by synthesis of cellular materials and consequent growth of the cell within. Growth polarity appears to be an essential aspect of an ordered cell cycle, possibly because the cell surface plays a role in segregating DNA to progeny cells. The ability to maintain shape while expanding in one dimension and at the same time retain flexibility suggests that the structural organization of the cell-wall polymers cannot be highly crystalline. Instead the wall acts mechanically more like an elastic gel. Its flexibility governs its response to transient deformations and its elasticity is shown by its ability to recover from such temporary shape changes. Unlike an inert gel, however, a cell wall can also undergo changes linked to wall metabolism or growth that persist for generations because the force imposed by growth continues and the material itself becomes altered biochemically. The ability to undergo adaptive change is the hallmark of many biological materials, including bacterial cell walls. Perhaps the most obvious requirement in the cell wall is strength. It must provide protection for the cell membrane from outside forces. It must also, like the wall of a pressure vessel, be strong enough to withstand the turgor pressure within the cell. Turgor is essentially a hydrostatic pressure, maintained osmotically, which forces the membrane outwards against the inside of the cell wall (Csonka, 1989). It is almost certain that there are no internal cytoplasmic structures that can bear any of the osmotic force, and the membrane itself is relatively weak. Finally, the wall must withstand the electrostatic repulsion between its parts. For it is highly charged and, under

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normal growth conditions, the charges are not entirely neutralized. There are some complicating factors, however. It is not absolutely clear how the cell membrane engages the wall and thus how the turgor pressure is transferred to the wall polymers. Also, when the external osmolality is varied experimentally to change turgor pressure, cells respond by homeostatic mechanisms that keep the turgor pressure within a limited range, suggesting an important role for turgor in cell physiology. These difficulties notwithstanding, turgor-pressure measurements reveal that bacterial cellwall material is very strong. And, as might be expected, the more material that is in a wall, the greater is the turgor-pressure load it can bear. 2. Other Functions The cell wall has other functions for which its mechanical properties are important and which are of significance in cell physiology. Three of these are: (i) permeability through the wall, (ii) anchorage of structures in the cell wall or their protrusion through it and (iii) the possible role cell walls might play in sensing and/or transferring information into cells. Everything that passes into or out of bacterial cells has got to traverse the wall as well as the cell membrane (or membranes). Normally, walls are thought to act as sieves that cannot pass materials larger than a molecular weight of 1200, or that have a radiusgreater than 1.1nm (Scherrer and Gerhardt, 1971). There are, however, exceptions and its appears likely that the charge structure of the wall and related features of its physical state, such as its degree of contraction, almost certainly play a role in controlling what can and cannot pass through it. Flagella and pili are protein-containing structures that penetrate the cell wall yet do not jeopardize its structural integrity. They somehow effectively plug the opening through which they penetrate, thus preventing turgor pressure from extruding the membrane through it. The means by which they d o so is not clear. Transfer of information into bacteria through the cell wall is a subject of new interest and great potential significance in that it provides a means by which the physical nature of the environment can be linked directly to regulation of cell processes. The precise role of the wall is not yet understood, however, as can be seen by considering some recent examples. A number of systems have been described in which bacteria regulate expression of their genes in response to mechanical rather than chemical stimuli. The cell surface is involved in all of these, so the cell wall may play an important role in transfer of information into cells. Stress in, or deformation of, the wall structure may be part of the process. When the marine bacterium Vibrio parahuemolyticus swims in solutions of the same viscosity as its normal ocean habitat, it uses a single polar flagellum for propulsion. If the

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viscosity of the solution is increased or the cells made to grow on a surface rather than in solution, they respond by turning on genes otherwise not being expressed, at least one of which leads to production of lateral flagella. Other phenotypic changes also arise in addition to the new means of propulsion (Belas et al., 1986). Evidence that the large polar flagellum may act as a dynamometer comes from studies of mutants with defects in its structure or function in which lateral-flagella genes were always expressed regardless of medium viscosity (McCarter el al., 1988). It appears, therefore, that the normal functioning of the polar flagellum is needed to keep lateral-flagella genes turned off. The signal leading to gene regulation could be the surface drag engendered by moving through the solution, the torque applied to the cell body as a result of polar flagellar rotation, or the power required by the polar flagellum to drive the cell through the solution. In another system, gene expression in Escherichia coli is affected by changes in osmotic pressure. The cell wall obviously participates, but whether as a sensor, transducer or just scaffolding for them is not known (Csonka, 1989). Both turning on and turning off of genes in response to osmotic changes have been found. The results suggest that the state of turgidity in a bacterial cell provides information in some mechanical way for gene regulation. Changes in turgidity set into play a cascade of biochemical reactions leading eventually to changes in gene expression. A mechanical component is needed to initiate the cascade. A third bacterial system in which physical stimuli have been found to initiate changes in gene expression involves a marine organism that inhabits deep-sea environments (Bartlett et al., 1989). These cells respond to barometric pressure. The amount of gene expression observed is positively correlated with the barometric pressure under which the cells are grown. These three examples, and also the newly discovered “touch-sensitive genes” in plants (Braam and Davis, 1990), show that forces acting on the cell surface can evoke responses at the level of gene expression. The mechanical properties of the cell wall must, therefore, be involved in transfer of information. To understand how cell walls participate in the processing of such information requires knowledge of their material properties as well as their structure and composition. 11. The Structure of Cell-Wall Material A. PEF’TIDOGLYCAN

A considerable amount is known about the structure, composition and synthesis of walls from both Gram-positive and Gram-negative bacteria. As

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one would expect on evolutionary grounds, diverse materials and organizations are utilized by different species (Murray et al., 1965), but the common aspects of them all are not difficult to identify. The key material is a heteropolymer, murein or peptidoglycan, consisting of a polysaccharide backbone to which short peptides are covalently linked. This is the strengthbearing polymer of virtually all eubacteria. Its molecular composition consists of a glycan backbone made of disaccharide (p-( 1+4)-linked alternating residues of N-acetylglucosamine and N-acetylmuramic acid). The glycans are cross-linked by covalent connections between short peptides that emanate from some o r all N-acetylmuramic acid residues along the backbone. The meshwork or sacculus so formed is the material which governs the mechanical nature of the wall. The organization of peptidoglycan in the bacterial wall was revealed using the electron microscope. In Gram-negative cells, only a thin 2-3 nm layer is present lying between the cell membrane and an outer membrane (Murray et al., 1965). Gram-positive cells, in contrast, lack the outer membrane but have much more peptidoglycan, on average a layer about 30 nm thick (Shockman and Barrett, 1983). Calculations based upon the amount of peptidoglycan in each cell, ideas about the molecular structure of peptidoglycan, its density and the surface area that must be covered, led initially to the idea that the thin layer in Gram-negative cells was barely sufficient to form a monomolecular sheet (Braun et al., 1973). Re-examination of the molecular models later suggested that perhaps three layers could be formed (Schwarz and Glauner, 1988). Recent findings about turnover and reutilization of peptidoglycan, and also details of its cross-linking, support the idea that at least two layers are present in Gram-negative cells (Goodell and Schwarz, 1985). New microscopical techniques have also contributed to a revision of ideas about the way in which the Gram-negative cell wall may be organized. The full volume between inner and outer membranes might be filled with a gel-like peptidoglycan material (Hobot et al., 1984). Whether in the form of a sheet or a gel, the glycan backbones must lie predominantly in the plane of the surface, for they are too long to stick straight out radially (Rogers etal., 1980). The orientation of the glycans on the surface, however, has not yet been definitively determined. Partially fragmented and digested purified walls, when examined by electron microscopy, reveal a meshwork pattern suggesting an alignment primarily perpendicular to the cylinder axis (Verwer et al., 1980). But the overall organization in a circumferential pattern appears not to be highly regular. There is considerable evidence against any crystalline order of the polymers within bacterial walls (Labischinski et al., 1983). Other polymers, however, are known to lack crystallinity yet still, on average, be highly ordered. It is reasonable to suppose that peptidoglycan may fall into this class of material. The walls of Gram-positive bacteria (such as Bacillus subtilis) maintain

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their thickness by a turnover process that requires constant synthesis and shedding (Pooley, 1976 a,b). The structure of the wall is therefore in a constant dynamic state. The organization of glycans within the wall is, again, not certain. The length of chains, however, precludes their sticking straight out in a radial fashion unless folded, A more likely arrangement is as in Gram-negative cells; the glycans lie parallel to the cell surface. Chemical analysis of glycan lengths in B. subtih (which, if anything, is likely to underestimate, owing to cleavage by autolytic enzymes) indicates a range of 30-600 nm (30-590 disaccharide repeats) (Shockman and Barrett, 1983). Even the longest strands cannot, therefore, stretch the entire length (about 3-4 pm) or circumference (about 2.5 pm) of a cell. A linked network is needed, therefore, to cover the entire surface. However, there is anything but agreement as to how the glycans are arranged in the linked cell-wall network. Everything from a totally random net, resembling a non-woven material (Koch 1989), to a highly precise crystalline organization (Burman and Park, 1984), has been proposed. Recent molecular models depict the glycan backbone as a right-handed helix with a repeat distance of four disaccharide units per turn. The twisted nature of the backbone enables the peptides to point in any direction. And, since cross-linking of one glycan to another can only arise when the two peptides that join are within a prescribed distance from each other, the backbone geometry limits the kind of regular three-dimensional structure that can be formed. Even with this limitation, however, there remain at least seven different possible arrangements of the glycans relative to one another, known as topotypes. Each topotype has its own characteristic level of possible cross-linking, its own number of layers of glycans and its own predicted extensibility(Labischinski et al., 1985). The latter is based on the idea that the backbone glycans are rigid and unable to be stretched or compressed along their length very much before rupturing bonds, whereas the peptides are thought to lie in a ring-like arrangement flat on the backbone with the ability to assume a straightened configuration, thereby extending the distance between neighbouring glycans by a factor of about three. Other degrees of freedom which might contribute to extension have not been considered in these topotypes. B. PEPTIDES AND CROSS-LINKING

Although the chemical composition of the glycan portion of peptidoglycan is essentially the same for all organisms from which the polymer has been isolated," the composition of the peptide moiety, the types of linkage that Occasional variationsinvolve acetylation or phosphorylationof muramic acid residuesor loss of N-acetyl groups in the endospore cortex conversion of a muramic acid residue to a lactam, and in some mycobacteriaand corynebacteriareplacement of an N-acetyl group on a muramic acid residue by an N-glycol group (Heymer et ul., 1985).

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join peptides, and the presence or absence of interpeptide bridge structures, differ greatly among various bacterial species (Ghuysen, 1968). Regardless of precisely which residues are present in the stem peptides, they always appear to consist of alternating L- and D-amino acids with an L form linked to the glycan. For example, in B. subtilis, the stem pentapeptide consists of residues of L-alanine, D-glutamic acid, rneso-diaminopimelic acid and Dalanyl-malanine. It is not certain what function this pattern serves, but it may confer resistance to protease hydrolysis which would otherwise lead to solubilization of the wall network (Schleifer and Kandler, 1983). At least three variations on linkage structure between peptides have been found. A common pattern involves linkage between the third-position amino-acid residue from one stem peptide to the fourth-position amino-acid residue from another, as for example the linkage of a diaminopimelic acid residue from stem A to a D-alanine residue in stem B in B. subtilis and E. coli. A second linkage pattern involves bond formation between the secondposition amino-acid residue from one stem peptide and the fourth-position amino-acid residue from another. The third variety, recently found in E. coli, involves a direct connection of third-position amino-acid residues to one another (Schwarz and Glauner, 1988). These differences and others involving the joining of stem peptides through interpeptide bridge material (Schleifer and Stackebrandt, 1983) are very likely to influence the mechanical behaviour of peptidoglycan and thus the material properties of the walls from different species. Two very important structural parameters in polymers are chain length and degree of cross-linking; these parameters for peptidoglycan are significant in terms of cell-wall biomechanics. Predictions of the crosslinking index for peptidoglycans have been made from models which take account of the restraints imposed by the geometry of the glycan backbones. Values obtained for different topotypes range from 24% in a type that could only form monolayers to 72% in a type that could form a multilayer network similar to the polymer in the wall of a typical Gram-positive bacterium. Measurements of the cross-linkingindex give values ranging from 20 to 93% in different bacterial species. Even the lowest values for peptidoglycan cross-linking are high in comparison with other polymers. Peptidoglycan chain length, however, is much shorter. A relationship between chain length and degree of cross-linkinghas recently been noted from studies of polymers in E. coli mutants. It appears that, as chain length decreases, the degree of cross-linking goes up (Schwarz and Glauner, 1988). It has also been suggested that such variations may exist in different parts of the same cell, such as in rod-shaped cells, the cylinder and the poles. If so, one might expect the corresponding regions to behave quite differently mechanically. There are other reasons for believing that the poles and cylindrical portions

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of rod-shaped cells may differ mechanically. The walls in these two regions are assembled in a different manner in Gram-positive cells; they have vastly different rates of turnover, and they also bear the turgor pressure in different ways. C. OTHER POLYMERS

The proportion of cell wall that is peptidoglycan can range from 40% (by weight) to as high as 95% (Marquis, 1988). Most of the remainder of the wall in Gram-positive cells is a negatively charged accessory polymer, namely teichoic acid or teichuronic acid (Rogers et al., 1980). The presence of these anionic polymers has a profound effect on cell-wall structure. If, for example, in B. subtilis, the glycerol teichoic acid polymer normally covalently attached to C-6 of the N-acetylmuramic acid residue in the peptidoglycan is lost by mutation, the remaining wall becomes highly disorganized, thickened, and extremely roughened on its outer surface. Cell shape changes under these conditions from rod to sphere (Cole et al., 1970). Similar phenotypes have been observed in other situations where cell walls become deficient in negatively charged polymers (Rogers, 1979). The mechanical properties of walls must also be influenced by the electrostatic nature of these polymers. Accessory polymers become inserted into the wall attached to new peptidoglycan, and are shed into the medium with it during turnover (Mauck and Glaser, 1972; Boylan et al., 1972). Their spatial arrangement within the wall is not certain. Those on the surface project radially from it (Birdsell et al., 1975), while those in the peptidoglycan network probably lie parallel to the surface (Archibald et al., 1973). If so, the chains within the wall would be in a position to interact with one another as well as with charged groups on the peptidoglycan peptide. Very little is known about how other polymers that are found in, or associated with, cell walls might contribute to their mechanical behaviour. In walls of B. subtilis, a dozen or so proteins ranging in mass from 14to over 200 kDa have been found (Studer and Karamata, 1988; Doyle et al., 1977). Neither their function nor their location is well enough understood to permit an assessment of their role, if any, in wall mechanics. The same can be said for the cell membrane, which, in Gram-negative cells, is known to penetrate in regions through the peptidoglycan (Bayer’s adhesions; Bayer, 1979). Wall-associated polysaccharides, such as exocellular capsules and slimes, could be significant in terms of mechanics, but virtually nothing is yet known about their mechanical properties (Isaac, 1985; Costerton et al., 1981).

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111. The Physical State of the Cell Wall A. CHARGED CELL-WALL POLYMERS

Structural studies have not yet provided a clear picture of the physical state of the cell wall, but other experimental approaches have given an outline of the situation. Key factors are: (i) the charged nature of wall polymers, (ii) the dynamic nature of wall structure, and (iii) the degree of order in it. The wall appears to be a somewhat open porous network, fully hydrated, in which counter-ions maintain neutrality (Marquis, 1988). In vivo, the network is under pressure from within the cell. It is stretched, but it can expand or contract and, as described later, generally behaves as a viscoelastic material. The electronegative nature of the bacterial wall is due to dissociation of charged groups, which can be identified by titration. Using the electrophoretic mobility of whole cells and of purified walls as functions of pH value, in order to measure ionization, carboxyl groups in E. coli and both carboxyl and phosphate groups in B. subtilis were shown to be the source of electronegativity (Neihof and Echols, 1968). Carboxyl groups in peptidoglycans are located on residues of D-glumatic acid, meso-diaminopimelic acid and terminal D-alanine residues of the peptide. Phosphate groups are present along the backbone of teichoic acids. At neutral pH values, cell surfaces are highly negatively charged because of these groups. Electrostatic repulsion between negative forces in the wall represents a significant factor in terms of wall mechanics. If the charges were unneutralized, the repulsion they generate would far exceed the strength of the wall, essentially blowing it apart. Two sources of cations are available to neutralize negative charges, namely amino groups in the matrix of peptidoglycan, and counter-ions from the environment. In walls of some Gram-positive bacteria there are so many anionic groups to be neutralized that the capacity of the walls to bind cations is on a par with that of commercial ion-exchange resins (Marquis et al., 1976). Walls from other species are not so electronegative, however. Factors such as the degree of cross-linking (each cross-link eliminates two charged groups), amidation of glutamic acid residues, which converts the carboxyl to an uncharged amide, or alanyl substitutions on teichoic acids, which neutralize neighbouring phosphate groups, all serve to decrease negative charge. Even so, in all cases, peptidoglycan appears to behave like a flexible polyelectrolyte gel that swells or shrinks in response to electrostatic factors and, thus, the density and porosity of the cell wall must vary accordingly (Marquis, 1968). The compactness of the cell wall varies from organism to organism. The range of measured dextran-impermeable volumes is between 2 and 13 ml

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(g dry weight)-'. These values can be used in conjunction with the known density of hydrated walls to determine the fraction of wall volume occupied by polymer and by water. The wet density of walls has been measured and found to be very low, less than 1.1g ml-' even in highly compact walls (Ou and Marquis, 1972). Thus, the cell wall appears to consist of about 74% water by weight when the walls are contracted and about 93% water when expanded (Marquis, 1988). If these values are an accurate estimate of the state in vivo, there does not appear to be any chance for wall polymers to be organized in a crystalline fashion. Instead, a flexible, open, sponge-like material is indicated. One would not expect hydrogen bonding or hydrophobic interactions to be significant under these circumstances since the molecular chains within the network are not close enough to one another. The effects of temperature and urea on cell-wall compactness do, however, show that weak interactions are present though not extensive (Ou and Marquis, 1972). The bacterial cell wall is quite different, therefore, from those of plants and fungi that consist of cellulose and chitin. The latter both form mechanically rigid structures containing crystallites rather than flexible gels. The peptide side chains in the bacterial wall, short as they may be, are long enough to allow formation of a very open flexible network. B . THE DYNAMIC STRUCTURE OF THE CELL WALL

The cell wall is in a dynamic state. The most obvious manifestation of this is that the cell grows. In rod-shaped cells, for example, the length increases rapidly while the diameter remains remarkably constant (Trueba and Woldringh, 1980). At any given time, the wall material must be stretched. Evidence that the wall is stretched in living cells comes from measurements of cell shrinkage caused by eliminating turgor pressure (Koch, 1984; Koch et al., 1987). The stretched surface is apparently between 20 and 46% greater in area than it is without turgor. The stretching effect of turgor pressure may play a major role in expansion of the wall during growth, possibly even in regulation of growth itself (Koch, 1983). The magnitude of turgor pressure in bacterial cells has been estimated from: (i) measurement of the total weight of internal solutes in crude cell extracts (from which water activity could be calculated), (ii) using vapour-pressure diffusion equilibrium methods, (iii) quantitative determination of the osmolarity that induces plasmolysis, and (iv) using collapse of intracellular gas vacuoles as a function of applied hydrostatic pressure (for a survey see Csonka, 1989). All of the methods used have been criticized for one reason or another, so the values reported are probably not sufficiently accurate to warrant critical predictions of how strong the cell wall must be in order to bear the turgor pressure. Estimated values of turgor pressure in Gram-positive cells under normal

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growth conditions are in the range of 1.5-2.0 MPa. The values for Gramnegative cells are lower (about 0.1-0.5 MPa). The upper value of about 20 atm is remarkably high to be contained by an open-structure wall that is mainly water! Many of the other aspects of the dynamic nature of the wall have been explored in Gram-positive cells. The basic pattern involves a constant loss and replacement of wall polymers during growth. In B. subtilis, newly inserted peptidoglycan starts to be shed approximately one cell-generation time after it has been inserted into the wall. The rate of turnover (shedding) is proportional to growth rate, the major factor apparently being the time it takes for polymer in the cylindrical wall to be pushed up to the surface by new material inserted below it (Pooley, 1976a). During steady-state growth of wild-type cells of B. subtilis 168, about 50% of the cylindrical wall material is shed in each generation. For reasons not yet clear, very little wall material is turned over in the cell poles; 15-20 generations are required to turn over pole material compared to between three and four for cylindrical wall (Doyle and Koch, 1987; Doyle et al., 1988). Pole and cylindrical wall appears to be controlled differently and, perhaps, the enzymes involved are less effective on polar wall. Prevalent as it is, cell-wall turnover is not required at the rate usually found in wild-type cells. Bacillus subtilis mutants, such as the FJ7 strain that carry a lyt2 defect which results in a decrease in autolytic enzyme activity to between 5 and 10% of that of the wild-type, grow perfectly well, but turn over their walls at less than 20% of the wild-type rate (Mobley et al., 1984). The only obvious phenotypic defects noted in such strains are an inability to separate cells following septation, which leads to formation of chains of cells and a lack of flagella, presumably because there is insufficient autolysin to open the wall in order to permit their penetration (Fein, 1979). There is now evidence that peptidogylcan in walls of Gram-negative bacteria undergoes constant turnover. About 50% of the peptides in mature peptidoglycan in E. coli is released during each cell generation (Goodell and Schwarz, 1985). The outer membrane, however, blocks it so that only 8% of the peptidoglycan breakdown products are shed into the environment. The remainder is trapped in the periplasm and is re-utilized. Surprisingly, some of it, consisting of tetra- and tripeptides, appears to be recycled directly into new peptidoglycan instead of being processed first to individual amino acids. The relationship of synthesis to turnover in walls of Gram-negative bacteria must be very precisely regulated in view of the very small amount of material present and the seemingly indispensable role it plays in maintenance of cell structure and in cell growth. Breakdown of peptidoglycan either in the periplasm of Gram-negative cells or at the surface of the wall on Gram-positive bacteria requires cleavage

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of covalent bonds. Stresses in the wall due to hydrostatic pressure and expansion due to growth may contribute, but the primary mechanism of peptidoglycan breakdown involves cleavage by specific enzymes. Such enzymes are potentially lethal to cells; their activities must be well regulated. Four means to do so have been proposed. One involves inhibition by lipoteichoic acids, a second relies upon the association of autolysin and an activator protein into a functional complex, the third is based on the idea that an energized cell membrane inhibits nearby autolysins, and, lastly, extracellular proteases may inactivate or activate autolysins at the cell surface (Shockman and Barrett, 1983). In B. subtilis, two autolysins have been identified. One cleaves the glycan backbone, the other severs the connection of stem peptides to the backbone. Both these activities are decreased in lyt mutants. Strains totally devoid of them have never been found, however, which suggests that some peptidoglycan remodelling must be required during growth. In E. coli, both kinds of peptidoglycan hydrolases have also been found. Both activities are needed to generate the tetra- and tripeptide breakdown products already described. There is much similarity, therefore, between the dynamic behaviour of cell-wall polymers in Gram-positive and Gram-negative bacteria. C. ORDER IN CELL-WALL MATERIAL

One of the most compelling indications that there must be some order in the arrangement of peptidoglycan in the cell wall is the fact that bacterial cells are usually not spherical. An isotropic wall would respond to turgor pressure during growth by expanding to a sphere just as protoplasts do. The tendency to produce a spherical shape must, therefore, be resisted by an anisotropic material in order that shapes such as quasi-spherical, cylindrical, helical or differentiated, as in the stalked cells of Caulobacter spp., can be produced and maintained. From a mechanical perspective, there is no alternative short of eliminating turgor pressure, which is apparently the case in the square bacterium Halobacterium sp. (Walsby, 1980), but not in others. Another is the observation that rod-shaped cells twist as they elongate. This has been inferred from the behaviour of long multicellular filaments and of twisted multifilament structures (macrofibres, see Section 1II.D) which form as the result of filament plying (Fein, 1980; Mendelson, 1982a). In these, the growth process is itself anisotropic in a controlled way. The obvious conclusion is that cell walls which maintain their structural integrity are also anisotropic, either because of the mode of insertion of new polymer or perhaps because of the twisted nature of the peptidoglycan molecule itself. Measurement of twist in multicellular structures indicates that, for B. subtilis, the helix angle of the cell-wall twist can be as large as about 8" right

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hand and 6" left hand (Mendelson et af.,1984) and, depending on a number of a factors, anywhere between. All of the factors appear to operate by controlling some aspect of the insertion of new polymer into the cell wall (Mendelson and Thwaites, 1988). The result is that a given cell in a given environment grows with a particular geometry, one of many that it is capable of producing. From these observations we infer that the bacterial cell wall must be able to assume a range of structural states, each perfectly stable and able to pass from parent to progeny so long as the environment remains constant. Studies of twist in macrofibres provide evidence that peptidoglycan metabolism, the integrity of the glycan backbone and electrostatic interactions in the cell wall of individual cells are involved in the establishment and maintenance of cell-wall twist (Mendelson and Thwaites, 1988; Favre et al., 1986; Mendelson et al., 1985; see also Section 1II.D). None of these, however, requires that the molecular architecture of the wall be crystalline or otherwise highly regular. Amorphous polymers are well known to possess order and to have anisotropic properties (Ferry, 1980). D. THE SIGNIFICANCE OF MACROFIBRES

Macrofibres are self-assembling twisted structures formed in various cultures of the same cell separation-suppressed mutants of B. subtilis as those used to make bacterial thread (Mendelson, 1978, 1982a). During growth, the ends of individual cellular filaments consisting of long chains of cells have for many years been observed to rotate in opposite directions as the filaments elongate. This phenomenon has been observed in Mendelson's laboratory since 1977 in time-lapse films. As the filaments twist, they also bend, so producing a writhing motion which persists until a filament touches itself, whereupon it plies into a two-fold twisted structure (Fein, 1980). The two-fold structure behaves in exactly the same way as a single filament so that four-fold, and eventually manifold, twisted structures are formed. Although the twist has not yet been measured for a single filament, it can be calculated from measurements of surface-helix angle and diameter in micrographs for macrofibres of all orders from about four-fold upwards. For a given environment, no matter what the order, the twist (turns per unit length) is the same. Twist can also be obtained dynamically, i.e. by measuring the relative rotation rate of the ends of a macrofibre fragment and relating this to the elongation rate. Such measurements made from stopaction microcinematograph films agree with measurements from still images (Mendelson et al., 1984). Since the behaviour of individual filaments is qualitatively the same as that of the multifilament structures of all orders, the obvious conclusion is that the filament twist has the same value as macrofibre twist.

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The simplest explanation for formation and development of macrofibres is that, when a filament (or a macrofibre) touches itself, its twisting-withelongation is blocked (Mendelson, 1976). This requires an axial torque of opposite handedness to the twist, which is provided in some way by the contact. A long filament under torque but under no tension (indeed, since it is growing through a fluid, under slight thrust) is not in a stable state, even less so when in a loop. The mechanical torsion is relieved by plying in just the same way as textile yarns ply together. The ply twist is in the opposite direction to the torque, i.e. of the same handedness as the blocked twisting of the filament. Thus, for a given environment, the twist of macrofibres of all orders is the same. Other possible explanations have been advanced involving much less plausible assumptions. Koch (1988), for example, finding difficulty with the idea that cells at the centre of a macrofibre can metabolize properly, suggests that the outer filaments, growing faster, might be restrained by the inner and so buckle in a helical fashion. This would not explain the plying of single filaments or of macrofibres of low order, where there are no central cells. Nor can it explain the range of twist, both left- and right-hand, that can be engendered in macrofibres by changes in environment (see, for example, Mendelson et al., 1985; Mendelson and Thwaites, 1988a). In addition, buckling due only to thrust never takes a helical or twisting form, whereas torsion is relieved in this way. Centrally placed cells in large macrofibres probably do metabolize at a slower rate but, since each filament migrates due to the twist in a macrofibre, from inside to outside, all filaments behave in essentially the same way. Although the origin of the twisting of cellular filaments is reasonably well understood (see Section V), filament bending during growth is less so. It may in part be due to random differential variations in growth rate in different segments of the cell wall round the circumference. However, observations of filament behaviour following an imposed disturbance suggest that it may be a dynamic material response. A major cause must be the thrust developed by filament growth through the culture medium. Although the viscosity is approximately that of water, the ends of a long filament are moving apart at a speed of the order of one cell diameter per second. The thrust developed is not sufficient to cause buckling in the classical sense, but it does not take much thrust to cause writhing, as can be observed in the motion of the hosepipes that are used for cleaning (by their motion) the bottoms of swimming pools. The thrust in this case is due to the water efflux at the pipe tip. However, when the viscosity of the culture medium is artificially increased lOW-fold, macrofibres develop by classical buckling of an almost straight structure without contact of any kind. It can be shown that the thrust due to growth is all that is required for such a phenomenon (Mendelson and Thwaites, 1990).

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Macrofibres can grow with a wide range of twist. The corresponding cellwall helix angle can lie anywhere in a spectrum extending from about 8”right hand to about 6” left hand (Mendelson and Thwaites, 1988). The cell surface must, therefore, assume a corresponding range of structural states. Since cells growing with twist appear to be completely normal in growth and heritability, their walls must be perfectly functional. Once a twist state has been established, it passes from parent to progeny cells as long as environmental conditions remain the same. This suggests that the cell surface is assembled with a particular structural or mechanical anisotropy that governs the characteristic pattern of twisting with growth. Evidence that peptidoglycan is involved in both the establishment and maintenance of twist comes from: (i) the fact that twist establishment is blocked by low concentrations of penicillin G (Zaritsky and Mendelson, 1984), (ii) the discovery that twist establishment is influenced by the concentration of Dalanine, or its antagonist, D-CyClOSerine, in the growth medium (Mendelson, 1988) and (iii) the finding that lysozyme digestion of live cells causes rapid changes in twist before liberation of sphaeroplasts (Favre et al., 1986). Peptidoglycan cross-linking may, therefore, play a role in twist establishment. The integrity of the glycan backbone is necessary for twist maintenance. Wall charge must also play a significant role, for changes in pH value and ions such as bromide also induce rapid twist changes (Mendelson et al., 1985). The significance of macrofibre experiments in relation to the mechanical behaviour of cell walls is that they magnify certain cell-wall behavioural phenomena such as twisting-with-elongation.Not only this, like the bacterial thread system, they offer a means of quantitatively describing this behaviour, for example, by being able to measure twist. They are also very sensitive indicators of change in the cell wall (see, for example, Favre et al., 1985,1986). So they provide information about cell shape and wall assembly that relates to single cells, and which is not obtainable by other means. The information obtained does not just relate to mutants. There is no reason to suppose that cell walls of B. subtilis strain FJ7, for example, are significantly different from those of the standard 168 strain of B. subtilis which will, if grown in very low-density cultures, form filaments (Zaritsky and MacNab, 1981). Mutants are used because their filaments are much more stable. Macrofibre formation has been observed in at least six other species, namely B. cereus (Roberts, 1938), B. licheniformis (Robson and Baddiley, 1977),an unnamed “new caldoactive filamentous bacterium’’ (Hudson et al., 1984), Mastigocladus laminosus (Hernandez-Muniz and Stevens, 1988), Clostridium autobutylicum (M. Young, private communication, 1986) and B. sphaericus (Y. Hoti, private communication, 1987). There is every reason to believe that the cell walls of these organisms can be successfully investigated by means of macrofibre experiments.

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IV. Mechanical Properties A. EARLY OBSERVATIONS

Until very recently little was known about the mechanical properties of bacterial cell walls. Manipulation of cells (Wamoscher, 1930) had shown walls to be extensible, flexible and, in some sense, elastic. By mounting on films which were then stretched they were shown to be highly ductile (Knaysi et al., 1950). The first measurements of deformation in relation to its cause showed that walls could shrink by up to 50% in volume in salt solutions depending on the bacterial species, due to electrostatic forces (Marquis, 1968; Ou and Marquis, 1970). Later, attempts were made to relate deformation to turgor (Koch, 1984; Koch et al., 1987). Contractions in length in the range 12-17% were inferred for E. coli when turgor was suppressed, but there were complicating factors in these experiments which make the figures unreliable as estimates of the extent of deformation due to stress in walls alone. All of these observed deformations were much less than the 100% that could be imposed (slowly) on the plasma membrane (Corner and Marquis, 1969) and the theoretical figure of about 200%, based on models of peptide configuration in peptidoglycan (Labischinskiet al., 1983). The only knowledge of cell-wall strength was that it was large enough to bear turgor pressure (Mitchell and Moyle, 1956) but, since the magnitude of this was a matter for some dispute and accurate measures of the cell-wall cross-section were not initially available, there was little point in calculating a value for cell-wall stress. Because of the small size of bacteria the only methods of approach to mechanical properties in normal cultures were indirect, often with tenuous lines of inference. Direct measurements became possible with bacterial thread, which is a fibrillar fibre made from cultures of a cell separation-suppressed mutant that can be investigated by standard fibre-testing techniques. The existence of filament-forming mutants has so far restricted measurements to B. subtilis but, in principle, the technique could be used on any filament-forming micro-organism. B. DIRECT MEASUREMENTS BY MEANS OF BACTERIAL THREAD

The mutant used is FJ7 of the 168 strain of B. subtilis (Mendelson and Karamata, 1982; Mendelson et al., 1984). Under the right growth conditions (Mendelson and Thwaites, 1989b) the bacteria grow in long cellular filaments, containing on average about 300 cells, in a structure like a textilefibre web. From this a “thread” can be drawn into which the individual filaments are drawn radially and compressed together into a multifilament fibre with a circular cross-section. A standard thread is produced by

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FIG. 1. Scanning electron micrographs of bacterial threads. (i) A standard thread, i.e. with culture medium removed by washing in water. (ii) An unwashed thread, i.e. as drawn from the culture. (iii) A lysozyme-treatedthread, i.e. washed a second time in lysozyme (15 minutes at 50 pg 1-' concentraton).Bars are equal to 5 pm for (i) and (ii) and 10 pm for (iii). Micrographs (i) and (ii) were orignially published in Mendelson and Thwaites (1989b).

resuspension in a large volume of de-ionized water in order to wash off residual culture medium. Although the filaments separate upon immersion, they retain enough cohesion for a thread to be redrawn in which they are closely packed and highly aligned parallel to its axis (Fig. 1). They also adhere strongly to one another so that, when a thread is extended, it behaves integrally in the same way that naturally occurring fibrillar fibres, such as wool, do. Threads produced in this way can be up to 0.5 m in length and 100 pm in diameter, i.e. they contain about SO00 filaments and more than lo9 cells. These are similar dimensions to those that can be obtained without washing (Thwaites and Mendelson, 1989). Tensile tests are made on threads using equipment of a standard kind placed in an atmospheric enclosure in which the humidity is maintained constant. (Thwaites and Mendelson, 1989) Specimens which are stored in

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the laboratory atmosphere are equilibrated in the enclosure before testing. It has been established that such a procedure produces the same results as when specimens are drawn and maintained at the relative humidity of the test. A standard test consists of extending a specimen to break, at a constant speed of about half the elongation rate of cells in vivo, while recording the tension as a function of the extension. Using an estimate of the wall crosssectional area (Thwaites and Mendelson, 1989), the tension is converted to wall stress and extension is expressed as a proportion of the initial length by use of the term strain. The reasons why the properties being measured are those of the cell wall and are not assemblage-related are given by the authors (Thwaites and Mendelson, 1989). The main ones are as follows. There are no signs of interfilament slippage in any stresdstrain curve. Almost complete recovery can be otained from extensions that are substantial fractions of the breaking extension, even at high relative humidities. In fact, the observed stress/strain behaviour is just like that of other polymeric materials. The cytoplasm can be shown to exert negligible effect and the individual filaments break, not by the pulling apart of septa, but in the cylinder wall. C. MEASURED PROPERTIES

Much information about material behaviour can be obtained from observation of the way in which stress depends on strain. Typical stressktrain curves for a cell wall are shown in Fig. 2 (J. J. Thwaites and U. C. Surana, unpublished observations). At lower relative humidities (less than about 50%) the curves are approximately straight lines with much steeper slopes, and breaks occur at slightly more than 1%strain. This behaviour is typical of any glassy polymer but, at high levels of humidity, the shape of the stress/ strain curve, the ductile nature of the wall and its low strength are all characteristic of a polymer above its glass transition, i.e. when it has become rubbery. Representative parameters used to describe behaviour are the extensibility and the strength (i.e. the strain and the stress at break, the stress being calculated from the area of the cross-section at break) and also the initial (Young’s) modulus, calculated from the slope of the stresdstrain curve at zero strain. Experimental results for these are shown in Figs 3 and 4, in the relative humidity range 2Q-95%. Cell walls are strong (about 300 MPa) but brittle at low relative humidities. They are also stiff (modulus about 15 GPa). These properties are maintained up to about 50% relative humidity but, as humidity is further increased, they become ductile (extensibility rises to the range 60430%) and strength and modulus fall dramatically, to values extrapolated to 100% humidity of 13 and 30 MPa, respectively.

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FIG. 3 A plot of the tensile strength of cell walls in threads of Bacillussubtilis FJ7 as a function of relative humidity. The response of a standard (i.e. washed in water) wall is shown by A and that of an unwashed wall upshifted by 18% relative humidity by 0.The tensile strength is the breaking load divided by the cell wall cross-sectional area at break point.

These properties are, in part, similar to those of other biopolymers, but differ in significant ways. When dry, they are, except for extensibility, comparable with those of cellulose (Meredith, 1959). When wet, cell-wall strength is about the same as that of chitin (Thor and Henderson, 1940), and the modulus is about ten times that of elastin (Gotte et al., 1968). However, although the polymer backbones of chitin, and to a lesser extent of cellulose, are similar to that of peptidoglycan, both are crystalline to some degree and are much higher polymers. Cellulose, for example, has about 100 times as many residues in each chain. Its properties change with humidity, but by much smaller factors than do those of a bacterial cell wall. On the other hand, although the variation in modulus of elastin with humidity is similar to

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that of a cell wall, elastin is an insoluble protein rubber. Because peptidoglycan and its accessory polymers have many ionizable and other hydrophilic groups, there is ample opportunity for water to associate with the cell wall, competing for the hydrogen-bond sites which provide links between peptides when it is dry. This could explain why a stiff polymer network becomes increasinglymore flexible as humidity increases. However, it is possible that increasing ordering of water molecules, as occurs in elastin and other proteins, may also contribute (Scheraga, 1980).

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If the extrapolated values for strength can be taken as appropriate for fully hydrated walls, the cells in vivo should be able to withstand a turgor pressure of about 2.6 MPa, which is comparable with values deduced for other Grampositive bacteria (Mitchell and Moyle, 1956; Marquis and Carstensen, 1973). That is, provided the wall is stronger in the circumferential (hoop) direction, the hoop stress in a cylinder wall due to internal pressure is twice the longitudinal stress. This does not seem unreasonable. The wall is clearly stiffer in this direction or else the diameter could not be maintained so constant during growth. It should be noted that the conclusion about turgor pressure does not depend on the fact that the wall cross-sectional area is an estimate; the same factor (four times the wall thickness divided by the cell diameter) is involved in calculating wall stress from turgor pressure as it is from thread tension. There are, however, other caveats. (i) The stress is, because of inside-to-outside wall growth, unlikely to be uniformly distributed through the wall so that, for a maximum stress of 13 MPa on the outside, the average wall stress and, therefore, the internal pressure would be less. (ii) The material is visco-elastic, and its properties depend upon the speed of deformation (see Section 1V.E); it may therefore be stronger in vivo. (iii) It is by no means certain that extrapolation to 100% relative humidity represents the fully hydrated state. (iv) Turgor pressure may not be the only source of stress in cell walls. Electrostatic forces have been shown to exert a strong influence on cell-wall deformation (Marquis, 1968). Their influence during growth is unknown. Some of these difficultiescan be addressed by the use of mathematical models of the cell wall (see Section V). Meantime, it is worth noting that Gram-negative bacteria have both thinner walls and lower turgor pressure than do Gram-positive bacteria. Taking the latter to be 0.3 MPa (Stock et al., 1977), and assuming that the periplasm has no osmotic function, the stress in the cell wall of E. coli should be about the same as that in B. subtilis. D. ENVIRONMENTAL EFFECTS

The chief effect is, of course, that due to water; the variation in properties with the degree of hydration has already been described. It is to be expected that the ionic environment might also have a profound effect. At first, this was thought to be the reason for differences in cell-wall properties measured using standard threads and unwashed threads, to which residual culture medium adheres (Fig. 1). The first property measurements were made on such threads (Thwaites and Mendelson, 1985, 1989; Mendelson and Thwaites, 1989b) and shows a similar variation with humidity to those derived from standard threads. With fill data sets available for both types, it is clear that all of the properties of unwashed preparations are the same as

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the standard properties at about 18% higher relative humidity. Figure 2 shows that this relates not just to the representative parameters but to the whole stresshtrain curve. Although the ions in the residual culture medium are a possible cause, the closeness of the agreement suggests two more likely causes. Firstly, changes in mechanical properties may be related to water content rather than water activity as given by relative humidity. This would require two quite different sorption isotherms, which seems unlikely, but measurements of this kind have not been made. Secondly, and more likely, since the dried culture medium is very hygroscopic, the cell walls, which are surrounded by it within the thread, experience an atmosphere which is more humid than the bulk atmosphere. Neither of these hypotheses explains the fact that properties obtained from the differently treated threads, when extrapolated to 100%relative humidity, differ by substantial factors (ten for the moduli). If 100% humidity were the equivalent of full hydration, this would not be so. It may not be equivalent for measurement of mechanical properties. It certainly cannot be for diffusion, and it is only by diffusion that culture medium is removed in order to make standard threads. A further possibility which does not have this difficulty, but which otherwise is somewhat unlikely because the strain is fyt-, is that there is a small amount of autolysin present in the residual growth medium and that the water activity at high relative humidity is sufficient for some enzyme activity. This might explain why properties for the differently treated threads are not the same at 100% relative humidity. However, another mechanism would be needed to explain the differences in the middle range of relative humidity. The effects of ions on mechanical properties have been demonstrated by immersing standard threads a second time, firstly in water of different pH values. There is an increase in cell-wall ductility in the pH range from 3.3 to 9.0, but no significant changes in either strength or initial modulus (J. J. Thwaites, U. C. Surano and A. M. Jones, unpublished observations). The increase in ductility, however, leads to a significant decrease in average modulus as the pH value is increased. The effects of inorganic ions have been shown by a second immersion in solutions of known concentration and the drawing out of salt-washed threads. Their behaviour, like that of standard threads, varies with the humidity of the atmosphere. Apart from the response at relatively low humidity levels, the general effect is to make walls more ductile, less strong and less stiff (Fig. 5 ) . However, the effects cannot be described by a simple shift in relative humidity, which makes it unlikely that the residual culture-medium effect already discussed is due to ions. Ions probably influence mechanical properties through changes in polymer conformation due to ion binding. It is known to diminish the electrostatic forces between charged groups (Marquis, 1968) and it is

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Salt concentration (MI

FIG. 5. Initial (Young's) modulus of cell walls in threads of Bacillus subtilis FJ7 as a function of salt concentration at three values of relative humidity (mean k SD). Treatment was with sodium chloride, shown by V ,or ammonium sulphate, shown by V. Controls are shown by A . Values on the graph indicate percentage relative humidity.

thought that, for concentrations greater than about 0.2 M, polysaccharides behave like uncharged polymers. The effects of salt on mechanical properties increase in magnitude as salt Concentration increases, with some indication, certainly, in the strength and extensibility changes, of saturation at about 0.5 M. An attempt has been made to change peptidoglycan structure in a more obvious way by treatment with lysozyme both in the culture prior to drawing

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threads and by washing standard thread in lysozyme solutions. There is no difference in any cell-wall property, including relaxation behaviour (see below), between treated and standard threads. This is so, even though the lysozyme concentration was only just below the value at which it became impossible to draw threads. The threads are certainly changed in appearance (Fig. 1).There are distinct signs of preferential attack near the cross walls in the cellular filaments and even of cell separation. It is clear that adhesion of filaments to each other is very strong and is, even with some cell separation, enough to maintain threads as integral fibres during testing. Although the properties of cell walls vary with relative humidity, with salt concentration and because of the presence of residual culture medium, there is one constant feature. Their strength is related to their extensibility always in the same way; Fig. 6 shows this. Allowing for the spread, the points lie on the same “failure locus”, whatever their treatment. This indicates

FIG. 6. Tensile strength of walls in threads of Bacillus subtilis FJ7 as a function of extensibility (breaking strain). The response of a standard wall is shown by A , unwashed by 0, salt-treated by V and lysozyme treated by A.

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that, despite the amount of molecular flexibilty that must exist in the wall (see Section IV.F), the average amount of disentanglement required in order that a given number of bonds be broken is the same whatever the molecular conformation might be. E. VISCO-ELASTICITY

The properties of cell walls depend upon the speed of deformation; at higher extension rates they appear to be stiffer and more brittle, and vice versa. This is a consequence of their visco-elastic nature, which is common to most other polymers. Visco-elastic parameters can be measured in several ways, the simplest being by observing either (a) creep, i.e. continuing extension at a constant stress, towards, after very long times, an asymptotic final extension (Thwaites and Mendelson, 1985) or (b) stress relaxation, i.e. reduction in stress at a constant extension towards an asymptotic relaxed stress. (Thwaites and Mendelson, 1989). The decay in stress with time explains why cell walls appear less stiff when extended more slowly. A typical stress-relaxation curve for a cell wall resembles, but only superficially, an exponential decay. However, there is no single time constant; it is as if the time constant were getting longer with time! An initial value of this characteristic time can be estimated provided a good estimate of the fully relaxed stress can be obtained. For cell walls an estimate has been obtained after 30 minutes. The initial time constant lies in the range 21-84 seconds, depending on humidity (J. J. Thwaites and U. C. Surana, unpublished observations). The relaxed modulus, derived from the relaxed stress and the imposed (constant) strain, also varies with humidity, with a minimum value of about 0.2 times the initial modulus, at about 75% relative humidity, which corresponds to the minimum characteristic time. The full range of relaxed modulus as a function of initial modulus is shown in Fig. 7. The corresponding data for cell walls in threads with residual culture medium also lie essentially on the same locus, i.e. within the scatter. If the stress-relaxation curves are plotted against the logarithm of time, it becomes clear that 30 minutes is not long enough to achieve a true relaxed modulus. (Thwaites and Mendelson, 1989). The data suggest a very wide distribution of relaxation time constants, a strictly phenomenological concept (Ferry, 1980, Chapter 3). For many polymers, there is a temperaturetime equivalence which allows a master relaxation curve to be drawn for the whole range of times (and temperatures). For some, e.g. nylon, there is a humidity-time equivalence. Using this idea and a plausible shift factor, data for cell walls can tentatively be plotted on a master curve extending over 22 decades of time (Thwaites and Mendelson, 1990). The form of the curve suggests that a fully relaxed modulus might only exist in the state of full

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hydration. The visco-elasticbehaviour of some polymers can be described in terms of very sharp transitions as temperature, or humidity, varies. The socalled glass-transition temperature is often used (Ferry, 1980, Chapter l l). Although the behaviour of cell wall changes from glass-like when dry to rubber-like when wet, the transition is not a sharp one and it is unlikely that a glass-transition temperature could be posited. This is a common feature of other highly cross-linked (amorphous) polymers and, although peptidoglycan in B. subtilis is less highly cross-linked than that of many other bacteria, it is highly cross-linked by the standards of man-made polymers.

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F. MOLECULAR ARRANGEMENT IN THE CELL WALL

Despite the lack of measurements other than in the axial direction, there is much evidence to indicate that the mechanical properties of the cell wall are anisotropic. The way in which rod-shaped bacteria maintain a constant diameter during growth suggests that the cell wall is much stiffer in the hoop direction. The twisting-with-elongation growth pattern of B. subtilis, established by observations of macrofibres (Mendelson et al., 1984), suggests that the anisotropy is helical. This implies some order in the structural polymer peptidoglycan, that is the backbones are on average in a preferred direction. But this does not indicate a regular structure, as has been proposed for some bacteria (Burman and Park, 1984). Order is a feature of all amorphous polymers. The mechanical behaviour of cell walls is in all respects just like that of amorphous polymers and suggests no other arrangement of peptidoglycan than an entanglement network. Where peptidoglycans differ from more well-known polymers is in being polyelectrolytes, i.e. having a lot of interaction between charged groups. Marquis (1968) has shown this, and it is also most likely that this is why mechanical properties are so much influenced by ions. The conformational changes required in order to effect this influence require a lot of molecular flexibility which would not be possible with a regular molecular arrangement. V. Cell-Wall Models A. THE AIMS OF MECHANICAL MODELLING

The general purpose behind wall modelling is to investigate states of wall deformation and stress with a view to explaining how cell shape is determined and maintained in terms of surface assembly and mechanics. A subsidiary purpose is to understand how molecular structure leads to the mechanical properties required for such behaviour. A long-term goal is to try to determine whether, and if so how, forces generated during growth govern other vital cell processes. A cell-wall model is, or should be, the quantitative embodiment of some hypothesis. Its usefulness lies in predicting phenomena that can be measured, so that the validity of the hypothesis can be tested and ideas modified or discarded. It should also suggest which experiments are likely to prove most fruitful in terms of further understanding. The models described in Sections V.D and E may, because they are couched in the language of applied mechanics, strike microbiologists as unusual, but they result in

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several predictions that can be checked experimentally as well as explaining already observed phenomena. They also open up the possibility of investigating the dynamics of disturbed growth and of the control processes at work. Some of the questions that should be addressed first are as follows: How does a rod-shaped bacterium combine \ongitudina\ ductility

with such high circumferential stiffness? Why do the cells of B. subtilis and, by implication, several other rodshaped bacteria (see Section 1II.D) twist as they grow? Why is the rod shape so stable? Indeed, how do different organisms maintain their characteristic shape not only during growth but also during division? Is growth a semi-autonomous process in which the cell wall simply responds to the stresses within it or is it necessary for it to be under close control by some other process? If the latter, what kind of control signals are required? What varying parameters might be sensed by cells in order to exercise control?

In the past, some attention has been given to questions (a), (b), (e) and even (d). The models described later bear on questions (a), (b), (c) and (e). B. GEOMETRICAL MODELS

Models which take no account of stress due to turgor, and possibly other body forces, should not be decried; understanding geometry is a prerequisite for good mechanical understanding. For example a rod-growth model in which new material is added to the cylindrical surface in a helical pattern explains in an elementary way how cell-twisting with elongation might come about (Mendelson, 1976). It also shows that, if the rate of growth varies circumferentially round the rod, the result is not merely that the rod becomes curved, as it would if material were added parallel to the cylinder axis, but that it takes up a helical shape (Fig. 8), as is observed not only in the filaments af macrofibres but also in individual filaments (Tilby, 1977). The argument is that, if a certain treatment produces differential growth, the helical wall build would automatically establish a helical shape. A variant of this model shows how cells could relate temporal order to structure (Mendelson, 1982b). If the material is added in strings side-by-side so that the helical pitch must change in order to accommodate them, the rate of twisting in the rod must decrease eventually to zero. This change in speed

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FIG.8. Elongation and bending of a cylinder due to the addition of new material, in

parallel alignment (above) or helically (below). Bending results from non-uniform growth rate round the circumference. Helical addition results in relative rotation of the ends and (with non-uniform growth rate) helical shape. From Mendelson and Thwaites (1989a).

or even the string geometry could be sensed by the cell and used to trigger other processes. The general idea behind such models is that, although the rules may be established genetically and can be changed by genetic manipulation, the actual process could thereafter be autonomous. We shall see that this is also a possibility for more sophisticated models. Geometrical models of this and similar kinds (see, for example, Burman and Park, 1984) suffer not only from lack of consideration of the forces involved but also in that they predict a very regular molecular structure in the wall. This is, as remarked in Section 111, not detectable by ultrastructural investigation. Furthermore, all of the evidence about mechanical properties (Section IV) indicates the wall to be an amorphous polymer, with some order, but not this kind of regularity.

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C. MODELS INVOLVING SURFACE TENSION-LIKE STRESS

Although the existence of turgor pressure had been established for many years (Mitchell and Moyle, 1956), the first serious attempt to relate it to bacterial shape was made only in 1981 by Koch and his colleagues. In their model, pressure is linked to surface curvature and a surface tension-like stress and, as for a soap bubble, the mechanics of the situation is described by what they call an energy-conservation equation. It is really a statement about mechanical work, of a kind often used in applied mechanics. During growth the situation is a changing one and, therefore, for each incremental change, the work done by the turgor pressure in expanding the cell volume (p dV) is equal to the work done against the surface tension in extending the surface area (T dA). Unfortunately, the problem attempted is an explanation of the shape of a coccal pole, forming from a splitting septum. It is a very difficult one of its kind, the geometry of which almost certainly implies continuous variation in stress over the whole surface, whereas surface tension would be uniform. In order to make headway, artificial constraints are introduced, such as that the material becomes “rigid” immediately it is externalized from the septum. Not surprisingly, the fitting of the predicted geometry to observed shape is not good. Nor is the rationalization of the differences very convincing. Nonetheless the basic idea is a good one. As subsequently stated by Koch (1983), it is that growth and division are driven, not in some mysterious way by the molecular architecture of wall insertion, but by the stress in the cell wall due to turgor. This is of course another version of the “autonomous process following rules” idea already referred to. Development of the idea has, unfortunately, despite the number of papers devoted to it, not been thorough, largely because the state of stress in walls has not been properly analysed. The fact that the stress level normally varies over the cell surface has been dealt with by introducing a variable surface tension-like stress (variable T) related to localized differences in the polymerization process (Koch, 1983). Clearly, all manner of soap bubble-like shapes can (and have been) produced by this convenient device. The organism must of course know what shape it needs to be and act accordingly! No doubt this is just possible but it defies one of the basic tenets of the so-called “surfacestress theory”, and relies on the (forsworn) mysterious wall-insertion process. Furthermore, it is obvious, and was well known when the ramifications of the basic idea were first worked on (Koch 1983), that the two stress components in the bacterial cell wall are not, in general, equal. For example, in the cylindrical part of a rod-shaped bacterium subject to turgor pressure, the hoop stress is twice the longitudinal stress, wheras surface tension is the

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same in all directions. Apart from this, it means that the simple work equation, basic to the “surface-stress theory” is not applicable; more complicated and (to some at least) well-known relations between stress and strain must be used. This has not been done. Instead, artificial constraints have usually been devised in order to surmount difficulties. For example, it is alleged that cylindrical extension in the Gram-positive rod is stable provided only that turgor pressure and surface-tension stress are constant and that the poles are rigid (Koch, 1983). This is in fact not true; a mass balance is also required and the rigidity of poles leads, in response to the slightest disturbance, to “barrelling” of the cylinder; but the artificiality of the requirement of pole rigidity is astounding. For, in other attempted explanations of shape, the septum must split and stretch in order to take up the required pole shape (Burdett and Koch, 1984). This shape, which is of course not a hemisphere, nonetheless maximizes the enclosed volume (for a given surface area) under the constraint that there is no cirumferential expansion where the pole joins the cylinder (Koch and Burdett, 1986); because of the rigidity of the cylinder, presumably. But one cannot argue that cylinder rigidity maintains pole shape and at the same time that pole rigidity is responsible for cylinder stability. This is not to say that attempts to develop the basic idea have been worthless. Consideration of stress levels has given a unique explanation of why there is much less turnover in the poles of Gram-positive rods than in the cylinder wall. The argument is based on the notion that bonds subjected to greater stress are more readily cleaved. The wall, in changing from a flat septum to a curved pole, is subject to stress which varies little through its thickness whereas, because of upwelling during growth, the stress on the outside of the cylinder is much greater than the average stress and, therefore, much greater than the pole stress. The “split and stretch” idea is also attractive, but it applies to somewhat intractable problems. Nascent poles are, in general, not the same shape as completed poles. It is often difficult to solve the equations that result from analysis of deformation from known geometry, let alone ones in which the geometry is itself an unknown. Recently, attention has turned to possible cell-wall material properties, not by the experimental method of Section IV, but again in terms of the relation of stress to the breaking of bonds, (Koch, 1988) and to the cracking of cell surfaces. The fact that the exterior surface of the walls of Grampositive bacteria is extremely rough presents the modeller with a problem, but there is little point in attempting to analyse a cracked cell surface if the corresponding whole surface cannot be properly dealt with. Our inference, on the basis of much experimental work on the twisting of macrofibres and individual filaments (see, for example, Mendelson 1976, 1982a; Mendelson

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et al., 1984), that B. subtilis cells twist as they elongate appears finally to have been agreed. It is natural to try to explain this by analysis of the stresses in the cell wall. We do so below. Koch (1989) makes the attempt by arguing that the high levels of stress on the outside of the cylinder wall must lead to cracks. There is no experimental evidence for this but, since the stress must be non-uniform, it could be so. He then proposes, without analysis, that a subtle combination of hoop stress and high longitudinal strain somehow produces helical cracks. This proposition has no foundation in mechanics whatsoever. The helical cracks are further supposed to propagate and, in so doing, in an unexplained manner, to produce rotation of one end of the cell relative to the other. This is a poor hypothesis; it is apparently aimed at explaining one already known phenomenon. It leads to no prediction concerning, for example, the amount of twist, nor even its direction. It cannot be applied to explain the wide spectrum of twist, both left and right hand, that has been observed. This would still be true even if there were any convincing stress analysis relating to the alleged cracks. In fact, the major shortcomings of attempts to date to deal with cell-wall stress are that the problems are not properly posed mechanically, and that there has been no proper stress analysis. Some of the ideas are good and their limitations can be explored by a relatively simple analysis of stress. The same approach would soon lead to the discarding of others. This, we believe, is “what the microbiologist can learn from the textile engineer” (Koch, 1988) or, indeed, any mechanical engineer. It is what we attempt to do in the following sections. We choose as a first example one of the less difficult geometrical problems. D. A MODEL INVOLVING ANISOTROPIC CELL-WALL MATERIAL

1. Analysis of Stress

The simplest geometrical shape, apart from a sphere, in which stress can be analysed is a circular cylinder. Since rod-shaped bacteria are essentially cylinders with polar caps this is a good starting point. It is well known that the hoop stress, o h , due to internal pressure in the wall of a closed cylinder is twice the longitudinal stress q,i.e. the stress on the faces of axial planes in the wall material is twice that on the faces of radial planes (Fig. 9). This follows only from consideration of the equilibrium of forces. It does not mean that the hoopwise deformation is twice as great as the lengthwise deformation, even for an isotropic elastic material. This is because the elastic strain, i.e. extension divided by length, in each direction depends not only on the stress component in that direction but also on a fraction (Poisson’s ratio) of the component in the perpendiculardirection. Nonetheless,

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less, for an isotropic material, the two deformations are of the same order. This is clearly not the case for rod-shaped bacteria in which the cylinder extends by growth while the diameter remains remarkably constant. In order to explain this, indeed to investigate theoretically Koch’s proposition that growth is a semi-autonomous process in which the cell wall simply extends in response to stress, the material must be highly anisotropic, being much stiffer in the hoop direction than in the longitudinal direction. It is likely that the stiffer direction represents the average orientation of the peptidoglycan backbone.

FI i . 9. Diagram illustrating longitudinal stress 0, and hoop stress ohin the wall of circular cylinder under internal pressure p.

I

This is not all, however. Twisting with growth has been observed in filaments of B. subtilis and other species, and measurements of twisting rate have been deduced from experiments on macrofibres (Mendelson et al., 1984; see also Section 1II.D). This too may be an essentially mechanical response of the cell wall to stress. When a helically reinforced pipe is subjected to internal pressure, it twists, so that it is sensible to consider a model in which the stiff direction in the wall material is inclined slightly to the cylinder hoop direction. This has little effect on hoopwise stiffness, but does allow us to investigate whether the proposition about twisting with growth is theoreticaly possible. It is certainly a more plausible proposition than either of those already described. Finally, the state of stress in the wall need not be due solely to turgor pressure. The wall is, in general, quite highly charged (Marquis, 1968). Much of the charge lies on teichoic acids and, in B. subtilis, teichoic aciddeficient mutants are frequently misshapen (Cole etal., 1970).This suggests that electrostatic forces are significant mechanically. Even if only a tiny fraction of wall charge remains unneutralized, the resulting electrostatic repulsive force could engender stresses of the same order as those due to turgor (see below). The effect would not be the same as that of an internal

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pressure nor would it necessarily be the same at all points in the cylinder wall, even if the net charge were to be uniformly distributed, which, for B. subtilis and most likely for other bacteria, may not be so (Sonnenfeld et al., 1985). The effect would be to add substantially more to the longitudinal component of wall stress than to the hoop component, so that the combined effect of turgor and electrostatic repulsion could produce a longitudinal stress greater than the hoop stress (J. J. Thwaites, unpublished observation). Thus, an appropriate “thin-shell’’ model of the cylinder wall is one in which the material has helical anisotropy with the major axis (the stiffer direction) inclined at a small angle a to the hoop direction, and where the cylinder is subjected to internal pressure, p l , together with a longitudinal tension, which it is convenient to think of as a pressure,p2, acting only on the closed ends of the cylinder. The hoop and longitudinal stress components are then given, respectively, by and where h is the ratio of wall thickness to cylinder radius. In order to obtain the resulting components of deformation, the stress components on planes perpendicular to the material axes must be found, and the resulting deformations transformed back in terms of cylinder axes (Thwaites, 1977). For a small angle, a, this is a simple matter. The wall-material properties are both non-linear and time-dependent (see Section IV), but the present state of knowledge would not justify the effort required in order fully to model these features. Instead, the material is here assumed to be either a linear elastic solid, i.e. one obeying a generalized Hooke’s law, or a linear viscous gel of high viscosity. The property values chosen are averages of those measured or, where not measured, plausible estimates. True cell-wall behaviour should lie somewhere between those predicted by these two models. For a linear elastic material, the strains in the hoop and longitudinal directions, &h and differ negligibly from what they would be if the material axes co-incided with the cylinder axes. They are given by and where E is the elastic modulus in the less stiff principal direction for the material, i.e. approximately along the axial direction, e is the ratio of E to the elastic modulus along the stiff material axis, and v is Poisson’s ratio.

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There is also a shear strain which results in a relative rotation of the cylinder ends through an angle 8 given by (J. J. Thwaites, unpublished calculation)

where a and 1 are the cylinder radius and length, respectively, and G is the shear modulus for the material. There are similar relations for the viscousgel material, giving strain rates and rotation rate in terms of stress and corresponding viscous-material parameters. The measurements of the mechanical properties of walls of B. subtilis described in Section IV show that the average modulus, E, must be less than the initial modulus. For model purposes it is taken to be 20 MPa; p 1 is taken to be 1 MPa. This is less than generally accepted for turgor pressure but, since the longitudinal stress must not exceed the observed breaking stress, it is a reasonable value. Thus, for a thicknesshadius ratio of 0.1, q,is 10 MPa. No measurements of transverse modulus or Poisson's ratio have been made but, for other highly anisotropic polymers, v is small, of the order of 0.1. Also, there is good reason to believe that, like many other polymers, cell walls are approximately incompressible (see Section V.E), in which case the material properties are not independent and e = 2v (Thwaites, 1977). The transverse modulus is therefore taken to be 100 MPa. Using these values and where x = q h h = 0.5(l+p2/pl). For equal pressures the stress components are equal, &h = 0.05 and = 0.45;forpz = 1.4p1, i.e. o1= 1.20h, the strains are 0.04 and 0.55, respectively. The longitudinal strains are, not surprisingly,of the order of those measured (Section IV). The hoop strains of a few per cent indicate why the cell diameter can remain approximately constant. Corresponding results giving strain rates for a viscous material are obtained by taking a viscosity (instead of E) of 20 MPa h. This is a reasonable value representing the stress-relaxation behaviour as measured. The longitudinal strain rate El is then 0.55 h-', which implies a length doubling time of about 75 minutes, i.e. of the same order as B. subtilis growing at 20°C. Strictly, these figures relate only to a cylinder consisting of a given amount of material and take no account of addition of new wall or of

MECHANICAL BEHAVIOUR OF BACTERIAL CELL WALLS

21 1

turnover. Nonetheless, they are substantially correct when the model is modified in order to take account of these effects. In the above analysis, the choice of pressure ratios (p2/p1)of 1 and 1.4, respectively, was somewhat arbitrary, but the size of p2 implied is not unreasonable. The extra longitudinal pressure for a cylinder with uniform surface-charge density, q , depends on the cylinder lengthhadius ratio; it is of the order of q2/2c0 where c0 is the dielectric permittivity. This is 1.4 MPa when q = 0.45 C m -2, i.e. for walls of B. subtilis about 11 mEq 1-’. Thus, taking Marquis’ value for charge (derived from Doyle et al., 1980), only about 5% of wall charge needs to be unneutralized to achieve comparability with turgor. This does not appear to be unreasonable, but the question of charge neutralization in relation to cell-wall stress needs further research. The inclusion of electrostatic repulsion provides an explanation of a hitherto puzzling finding, namely that, when turgor pressure is removed, cells (strictly cellular filaments) shrink in length much less than one would expect (Koch, 1984; Koch et al., 1987). Using the property values already referred to and taking p2 to be 1.4 MPa, removal of turgor (pl = 0) lowers the value for cl from 0.55 to 0.30, i.e. a shrinkage (based on the original length) of 16.1%. If p2 is taken to be 1 MPa, the value would be 17.2%. These values are close to the shrinkage of 17% observed. It is true that the observations were made on filaments of E. coli but, as remarked in Section IV.B, it is not unreasonable to suppose that the stress in its wall is of comparable magnitude to that in the wall of B. subtilis. A shrinkage of the same order should be observable if p 2 were to be removed, for example by changing the pH value. Shrinkage of roughly 10% was observed during the experiments determining the effects of neutral salts (at 50 mM) on mechanical properties (Section 1V.B.3), but no proper measurements were made. Small changes in turgor or electrostatic repulsion could not be detected in this way because their effect would be difficult to measure with any confidence. Changes in twist, however, are much more readily observable. 2. Cell-Wall Twist The models predict (equation (3)) that, even in the absence of external forces, the cell should twist under the influence of p1 andp2. The unknown factor is the shear modulus, G, which has yet to be measured for B. subtilis. For very open networks such as the peptidoglycan network in cell walls (Marquis, 1968; Ou and Marquis, 1970), it must be much less than Young’s moduli. This is true of materials in sheet form and also of many textile structures (including non-wovens) which are good analogues of open-

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network sheets. For the purposes of calculation, G is taken to be 2 MPa. A value of 0.1E for G is somewhat arbitrary, but many of the following conclusions do not depend on this particular value. If the shear modulus were larger, relatively small changes in the ratiop2/pl would lead to the same conclusions. Using this and values already used, the twist is given by

aell = ( 3 . 9 ~- 4.7)~ which is zero forp, = 1.4pl. Moreover, for changes in p 2 of only +lo%,ae-1 takes values of f d4,i.e. the twist changes sign without the need for a to do so. It seems distinctly possible that the positive and negative (right- and lefthand) twists in cells, inferred from observations of macrofibres, are a direct result of helical anisotropy in the cell wall, and that the handedness is controlled in part by quite small changes in the net unneutralized charge, but not necessarily by changes in the handedness of the material anisotropy. Rapid changes in twist, i.e. non-growth-related changes, can be induced by lysozyme attack, by changing the pH value and by neutral salts (Mendelson et al., 1985). Lysozyme achieves its effect by cleaving certain glycosidic bonds of the peptidoglycan backbone, in model terms by decreasing the transverse modulus, i.e. increasing e. Corresponding changes in 8 follow from d9

_ de

2ohla

-

Ea

so that, provided a is positive, i.e. the helix of material anisotropy is right handed, twisting towards the right hand is predicted whatever the existing twist. This is as observed (Favre et al., 1986). Rapid changes in twist ought also to result from removal of turgor pressure. They were not reported by Koch (1984), but this may be only because it is difficult to observe such twisting in relatively short single filaments. However, changes in turgor should produce observable twisting in macrofibres: e.g. for the property values used above, a 20% decrease inpl should give an increase in aW1 of about 0.53~.Attempts to observe such changes by using sucrose to change medium osmolarity have been inconclusive. A decrease in electrostatic repulsion due to charge neutralization, by changing the pH value for example, should produce positive twisting of the same order as that produced by reducing turgor. Unfortunately, lowering the pH value not only decreases the unneutralized charge but, since the cell wall becomes less ductile, it increases the average modulus. This, by equation (3), has a contrary effect so that it is not possible to make a good prediction. However macrofibres, and by implication cell walls, do twist rapidly when the pH value is decreased (Mendelson et al., 1985). They also twist when treated with certain ions. The theoretical situation is similar in

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this case. Mechanical properties are changed (Section IV.D), perhaps by a different mechanism involving water ordering, and, presumably, charge neutralization takes place also. Until more is known about the relative strength of these contrary effects, there is no point in speculating about the predictive value of the model in these situations. The model does, however, show clearly that it is not necessary to change the helix handedness of material anisotropy in order to achieve a reversal in cell-wall twist. Such a mechanism, though quite possible during growth and, therefore, slow, would be an unlikely one for the rapid changes observed. In the same way, the continuous twisting observed during growth need not be related to different degrees (or even helix handedness) of anisotropy. The twist could be controlled by factors which affect the material properties and the unneutralized charge only. Strictly, continuous twisting can only be described by a model which takes account of new wall insertion, but this effect can be thought of as follows. Each new cylinder element (of length l) extends by l q and one end moves circumferentially, in the cylinder surface, relative to the other by a distance ue. The helix angle of continuous twisting is therefore uO/lq. For a viscous material model, even without new wall insertion and certainly with it, this helix angle is u6/ltl,where the dots denote rates of change. The helix angle y~of twisting is, therefore, given by the same expression, derived from equations ( 2 ) and (3): _y~ -- (x-l)g+2(e-vx) - 2 (4) a (x-v) where g and e are corresponding ratios of either moduli or viscosities. For the parameter values already used, w changes from approximately - d 2 to + d 2 for a +lo% change inpz about the value 1 . 4 ~Since ~ . the twist angles inferred from macrofibre measurements on B. subtilis mutant FJ7 are in the range -5" to +6" (Mendelson et uf., 1984), the helix angle of anisotropy would not need to be more than about 10" if this were the cause of twist changes. But the variety of influences on macrofibre twist, and by implication cell-wall twist, suggests that changes in material properties, and even in a, must also be involved. (Mendelson and Thwaites, 1988). A final quantitative prediction concerns the forces needed to suppress twisting during growth and bears on the argument (see Section 1II.D) that sufficient torque is engendered by contact between filaments for them to ply and eventually form macrofibres. For a twist 0/1, the torque required is approximately 21tGu~he.l.With values already used and a equal to 10" ,this torque could be produced by a tangential force at the cell surface of about 0.8 X lo-' N, i.e. a little under 1 pg weight. Whether or not this is a reasonable force in bacterial terms must await suitable experiments on inhibition of twist.

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In summary, a “thin-shell’’ model of the cylinder wall of rod-shaped bacteria, on the assumption of helical anisotropy and using known material properties or plausible estimates, can be used to predict quantitatively, or explain, changes in length and diameter and also rapid twisting due to the following: (i) changes in turgor pressure; (ii) changes in effective wall charge engendered by variation in p H value or by ions; (iii) changes in material properties, due, for example, to enzyme attack. Several of these phenomena have yet to be measured. In addition, it can be adapted to predict the rate of twisting-with-growth and the way in which it depends on (i) to (iii). In particular, it explains why it could be possible to have changes in twist handedness without needing a change in the material anisotropy. Strictly, this adaption needs justifying by what follows. E. A CELL-WALL GROWTH MODEL

Although a “thin-shell” model can be used to describe rapid changes in response to various stimuli and might be adapted, with much greater geometrical sophistication, in order to describe aspects of growth in which material is inserted in given bands, it cannot describe fully what might be happening in walls in Gram-positive cells. It has been well established for many years that growth is accompanied by upwelling of wall material from the plasma membrane and by turnover of material at or near the outer surface (Archibald, 1976; Archibald and Coapes, 1976; Pooley, 1976 a,b). This involves increasing deformation during upwelling, with consequent changes in the state of stress. One deals with this by means of a so-called “thick-shell’’ model, which is not to imply that the wall thicknesskell radius ratio is any greater, but simply that variations through the wall are taken into account. The basis for such a model is the observation that wall material in B. subtilis is inserted uniformly over the inner surface of the cylindrical portion (though not also at the poles) and that it wells up uniformly too (Pooley, 1976a; Merad et al., 1989), so that material at a given radius in the wall is all of the same age. While this may not be precisely true, there is sufficient averaging over the cylinder surface for it to be a good basis. The model is made tractable by the further assumption that the material is incompressible, i.e. that it does not change in volume due to application of stress. As remarked earlier, this is true of many polymers, and it is argued that it must be approximately so for cell walls, for they contain a large proportion of water, the bulk modulus of which (about 2 GPa) is so very much greater than the measured tensile modulus (and by implication the transverse modulus). Volumetric strains due to the levels of stress inferred (and those permissible in measurements such as those described in Section IV) would be minute.

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FIG. 10. Diagram of the cross-section of a cylindrical cell wall showing mass flow. Material initially inserted at the membrane (radius a ) is, after time t , at radius r. Insertion rate: h per unit volume; shedding rate (radius b): p per unit volume.

So we consider the material that was initially inserted at the membrane (radius a in Fig. 10) to be, at a time t later, at radius r. The volume occupied by a length 1 of material inserted during that time (shaded in Fig. 10) is increasing at a rate R d(? - a ')f/dt. This must equal the rate of insertion of new material which is taken to be h per unit wall volume. The resulting differential equation is 2rt - 2ad

+ (3 - a2)q = (b2 - a2)h

(5)

where q is the longitudinal strain rate ill and dots denote rates of change. Note that q is the same for all values of r, but is not necessarily constant in time. There is a similar equation for the whole thicknessof the wall, which shows that the proportional rate of increase of the wall cross-sectional area, x(b'-a'), is equal to h-p-q, where pis the rate of wall turnover per unit volume. There is an obvious balance when the rate of insertion minus the rate of loss due to turnover is equal to the rate of volume increase due to extension. Equation ( 5 ) allows expression of the hoop strain rate, f / r ,as a function of r. The hoop strain, In (du), can be calculated directly; it increases from zero at the membrane to approximately h (the thicknesdradius ratio) at the outer surface. The longitudinal strain is also zero at the membrane, but increases to In (l+q/p) at the outer surface. Since, particularly in lyt- mutants, p must

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be much smaller than q, this implies much larger strain than can be borne by the cell wall in experiments (Section IV). The basis for stress analysisfor this model, the details of which will appear elsewhere (J. J. Thwaites, unpublished observation), is that there are equations of the same form as equations (2) and (3) relating strains and twist (or their rates of change) to stress at a general radius r. Average values of longitudinal and hoop stress, which are identical with those used in the thinwall model, are found by integration over the whole wall cross-section, i.e. with respect to varying r. The variation in stress with radius is then given by equations analogous to equations (2). It is substantial for the elastic-material model. At the membrane, all stress components are compressive and equal to the pressure pl. This state is called “hydrostatic”. Both the hoop and longitudinal stresses become tensile and increase rapidly with radius, reaching values at the outer surface which are several times the average values. This is illustrated, for the case p2 = 1.4pl, in Fig. 11. The viscous

to”

Rod 0

I

Axis

Rod b

FIG. 11. The state of stress at the membrane (radius a ) and the outer surface of the wall (radius b ) , according to wall-growth models for (i) elastic material and (ii) viscous-gel material. The figures are the ratios of the stresses to the average hoop stress o h .

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model produces a quite different stress distribution with almost no variation in longitudinal stress with radius, and hoop stress slowly decreasing from inside to outside (Fig. 11). A point to note is that wall material, although it starts without strain, is not stress-free at any time during its journey through the wall. Given the measured properties for walls of B. subtilis, the predicted level of stress at the outer surface, for an elastic material, is greater than can be borne in our experiments. Some of the material would be fractured, so that average values of moduli would effectively be lower than those used in the calculations. It is not at all clear, despite assertions to the contrary, that this kind of fracture does occur (Koch, 1989). If it does, the mode of fracture would certainly not be as Koch suggests. In any case, the material is visco-elastic; the levels of stress must lie between those predicted for elastic and viscous materials. A visco-elastic model is the subject of current research. Other conclusions to be drawn from analysis of wall-growth models are, firstly, that equations (2) holds provided the stresses are the average stresses given by equations (1) and the strains are average strains (or strain rates). For the viscous-material model, this gives a direct expression for the longitudinal strain rate, i.e. the rate of growth, in terms of average stresses, which is the same as predicted by thin-shell modelling, i.e. q = (0, - v o h ) / E rwhere , E' is the viscosity. The average longitudinal strain for the elastic material is, however, because of the relation between radius and time, a more complicated expression which yields quite a different answer for the overall longitudinal strain rate. It is interesting to observe that a purely elastic-material model, because of continuous material insertion and turnover, can predict a strain rate, but a full visco-elastic treatment is required in order to obtain a better expression for q. Secondly, provided the helix angle of anisotropy does not vary much from the inside to the outside of the wall, the helix angle of twisting-with-growth is given by equation (4)with, since the material has been taken to be incompressible, e = 2v, so that conclusions relating to twisting can all be obtained by essentially thin-shell arguments. There is little experimental evidence of any kind concerning material anisotropy in the wall, so further analysis does not seem to be appropriate. It should be stated, however, that polymeric materials, when subject to the levels of strain envisaged in the model, usually become more highly oriented. For example, a high degree of anisotropy is achieved in fibres by drawing. A further conclusion follows from the inclusion in the upwelling equation (5) of a term for variation in overall cylinder radius with time and derivation of an expression for the rate of change of wall cross-sectional area. This is that the wall thicknesdradius ratio, h, tends with time, after a disturbance, towards an equilibrium value 2v(20h-o,)lEr(h+p). This, of course, alters

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with changes in the ratio p 2 / p I ,but the changes need not be large. Taking p = 0.2h (Pooley, 1976a), and giving h its equilibrium value, a +lo% change in p 2 of about 1 . 4 produces ~ ~ approximately f13% change in h about the value 0.1. The ratio h tends towards its equilibrium value with a time constant l/(h+p). If the system is in balance in terms of material insertion, this is of the order of the length-doubling time which is long in terms of maintaining control over the wall. Nonetheless, h tends towards its equilibrium value automatically without the need for a control signal. Consequently, the cylinder radius is under control provided the wall area is constant, i.e. provided h - p = q. Maintaining this material balance is a matter requiring a control signal. One might speculate that the cell senses stresses in its wall, or strain rates, and controls the rate of insertion of wall accordingly. To summarize, a wall-growth model in which the stress analysis given in Section V.D is applied to the wall of Gram-positive bacteria, taking account of upwelling and turnover, first of all confirms the twisting-with-growth conclusions obtained with a thin-shell model. It also can be used to predict the dynamics of change during growth, in length, diameter, wall thickness and twist, due to the following: (i) changes in turgor; (ii) changes in effective charge (variation in p H value; ions); (iii) changes in material properties (e.g. enzyme attack); (iv) changes in material insertion and turnover rates. Again, many of these phenomena have yet to be measured. In addition, it predicts parameters that cannot be measured, but are nonetheless useful conceptually, such as stress distribution through the wall. Possible future developments include: (a) better modelling of the wall material, e.g. including compressibility and variations in properties through the wall; and (b) quantitative investigation of control mechanisms for maintaining diameter and wall thickness. The more difficult problem of non-cylindrical shape, in particular its establishment during division, should also be attacked. VI. Conclusions

In the preceding sections, we have attempted to show the value of investigating the mechanical behaviour of bacterial cell walls and also how it can be done both experimentally and theoretically. We now suggest the directions that future research should take. Concerning material properties, the anisotropy of walls needs to be examined. Shear behaviour will be measured in torsional experiments on bacterial threads, but the more difficult measurement of transverse modulus should also be attempted. A fuller investigation of visco-elasticity in walls is needed, including the effects of temperature. The effects of variations in wall polymers should be

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examined by using treatments that modify them in known ways structurally and possibly by using walls of different polymeric composition as a result of physiological or genetic manipulation. Fully hydrated material needs to be tested; this will require more sensitive instrumentation, for example to experiment on the mechanical properties of macrofibres, possibly in vivo. Single-cell measurements will not be possible in the near future. Mechanical modelling suggests experiments involving dynamic changes in the observable geometry of live cells, such as wall thickness, cell length and diameter, and twisting rate in some cases. How these change in response to stimuli that affect the rate of insertion of new wall material, for example, or that change the mechanical properties in known ways, should be measured. These experiments include the possibility of observing the effects of upwelling of new material, following an abrupt change, of different properties, as with reversal of twist handedness in macrofibres following a rapid change in environment. Other experiments involve growth under constraint. The aim should be to determine whether an imposed deformation could be set, i.e. that it would persist when the constraint was removed. The implication would be that the growth pattern had changed owing to the imposition of stress and that the cell-wall material had acquired a memory of form even though this had been externally imposed. Theoretical modelling iself, after improvements suggested in the last section, should be aimed at the determination of shape. The difficulties of doing so should, however, not be underestimated. The problems of working with an unknown geometry, unknown that is in the sense that it is what is to be determined, are mathematically difficult, but not intractable. Models involving possible feedback links, using mechanical signals such as wall stress to control, via plausible biochemical means, such parameters as the insertion rate of new material and even, to some extent, its properties (e.g. by variations in cross-linking) should also be investigated. Models of a somewhat different kind should aim at the molecular organization of the cell-wall polymers and the way in which the wall is assembled. Finally, it should be emphasized that all of the tools of genetics, biochemistry and structural biology can be applied in support of mechanical investigations, by providing controlled variations against which changes in mechanical behaviour can be evaluated. Once cell-wall mechanical behaviour is understood, its role in the regulation of cellular processes can be examined. Micro-organisms successfully integrate such processes and mechanics. We should try to do likewise.

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