Wind And Gravity To Anchoring Themselves Biology

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On a mechanical level, plants are hierarchical structures made of materials with subtle properties that can be changed by the plant. In understanding the mechanics of

plants, quantitative descriptions or models have great value, because a model enables the ideas underlying it to be tested by experiment. Any model is an idealization of the real system, so its predictive abilities are constrained by the realism with which the structure and materials are abstracted. It may be adequate in the case of modelling the bending of a tree limb under its own weight to con- sider the limb as a tapering beam made of continuum solid material, perhaps with location-dependent properties. But this approach is less tenable when the mechanics of a leaf or shoot are under scrutiny because there may be only a few cells across the smallest dimension. At this scale, the continuum approach becomes unacceptable because the response to external loading or imposed deflection of the structure depends on the interplay between the walls and fluid contents of a few cells, so modelling of the cellular nature of the structure is called for. At the next higher magnification, the subtleties of the behaviour of cell walls make them a modelling challenge, in which the objective may be to quantify how the structural polymers give rise to wall mechanical properties. Linking across these hier- archies of structure to summarize some functionality derived from a lower level is a further modelling challenge. This paper reviews the modelling of the mechanics of plant materials from the level of cell wall to tissue, and illustrates how modelling can contribute to the under- standing of these materials. The mechanical properties of the material of plants originates in the cell wall. Unless the wall or the intercellular bond fails, the wall and its interaction with the cell contents determine the mechan- ical behaviour, so this review focuses to a large extent on modelling of wall properties. This is supplemented by less extensive sections on modelling the mechanics of cells

and tissue.


Wall structure and composition differ markedly between species and between tissues, specialized as they

are, from the thin, hydrated walls of parenchyma cells (e.g. the flesh of a potato tuber) through thickened-wall, hydrated, collenchyma cells (e.g. celery stem fibre) to cells that have developed a massively thick wall of closely packed cellulose fibres; fibre cells of wood are bound and waterproofed with lignin, whereas sclerenchyma fibres found in flax and cotton have little lignin. The mechanical properties of these cell types are very different, but all arise from their composition and architecture. For a detailed explanation of wall biochemistry and physiology, and a useful summary of wall mechanical behaviour, the reader is referred to Brett & Waldron (1996).

The primary wall comprises a fibrous network of cellu- lose microfibrils and hemicellulose that is coextensive with a matrix of pectic polymers. The plasma membrane within the wall gives the structure low permeability to water so that the fluid take-up, driven by osmosis, can develop the wall stress needed to support stems and leaves against gravity. The high tensile strength of the cellulose fibrils enables the wall to withstand this stress. Structure and hence properties of the primary wall are influenced by the cell's bioprocesses, under genetic control. Structure and composition of primary cell walls are reviewed by Car- pita & Gibeaut (1993) and McCann & Roberts (1994). Conceptual models have been hypothesized of primary wall structure at the molecular scale, e.g. by Carpita & Gibeaut (1993), but no conceptual model has yet, to my knowledge, been validated or used as the basis of a math- ematical model.

Regarding the relation between growth and turgor, it is

now known that growth is not initiated by an increase in

turgor. The essence of cell growth is extension of the wall

under the control of the cell (McQueen-Mason 1995).

Relaxation of stress in the wall, achieved by rearrangement

of wall polymers and mediated by enzyme and acidity in

a complex manner, reduces turgor. This initiates the

uptake of water and hence the expansion of the cell.

Introgression of new polymers into the wall follows.

Although wall behaviour is well described from experi-

ments on excised, de-natured sections of, for example

hypocotyls, a sound theoretical basis for modelling the

mechanics of the wall has yet to be established. This is

arguably the next major challenge for plant cell modelling.

A good summary of recent experimental work on the regu-

lation of physical and biochemical changes to cell walls in

growing plants is given by Schopfer (2000).

In those cells that form a secondary wall, this occurs by

thickening of the original primary wall by the addition of

closely packed cellulose fibres. The fibre structure deter-

mines properties; for walls with a helicoidal structure, in

which the fibre orientation alters between successive lay-

ers, the properties in the plane of the wall are isotropic,

whereas with a helical fibre structure, in which orientation

is consistent from layer to layer, the properties are highly

anisotropic, being dependent on helix angle. In some

tissues, for example wood and sclerenchyma, the pectin

matrix is replaced by lignin, which bonds the fibres more

strongly than the pectin it replaces. Lignification elimin-

ates water from the wall to result in a structure that is

hydrophobic and rigidified. Removal of water eliminates

the viscosity of the wall and so results in a material that

is able to resist compression, bending and shear; forces

off-axis to the cellulose fibril direction are transmitted to

the fibres through shear properties of the matrix. In other tissues, for example flax stem fibre, from which linen is manufactured, and cotton, the wall becomes thickened, but little lignin deposited. For more details of the bio- chemistry of secondary wall formation, the reader is referred to Brett & Waldron (1996).

In mechanical terms, a tissue is simply a conglomerate of a similar type of cells adhered together. The attachment between cells ranges from strong enough such that failure occurs by wall rupture, e.g. in potato tuber, to sufficiently weak that cells separate without rupturing, e.g. a 'mealy' textured apple. Hence tissue properties are very depen- dent on cell attachment, and a tissue model must, implicitly or explicitly, include this factor.

Readers interested in pursuing the subject of the mech- anics up to the scale of organs and plants are directed to Niklas (1992).


The mechanical behaviour of materials may be divided into two categories, namely fundamental physical proper- ties of a material, such as elastic modulus, and properties that depend on the particular test piece of material, such as strength. All the material in a simple test piece is involved in producing the extension in response to an applied force, whereas the strength is determined by the stresses in the vicinity of flaws in the particular test sam- ple. For test samples it is not usually possible to know the number or size of flaws, and as a consequence strength can be predicted only if a statistical description of flaws is available. For these reasons, most modelling of cellular tissue in plants has focused on predictions related to the force-deflection behaviour or, put more generally, the state of stress and strain in the modelled entity. This is despite the fact that some applications for a good under- standing of the mechanics of fresh plant tissue relate to aspects where tissue failure is crucial, e.g. cracking and bruising of fruit and vegetable tissue. However, because strength is a function of both stress and flaw dimension, the prediction of stress is certainly of value in strength calculations. Readers interested in fracture are directed to Jeronimidis (1980, 1991) and Vincent (1990).

When modelling cellular plant material, an important

question is, at what level of hierarchy is a material con-

sidered as a continuum rather than a structure? For a

model of a whole single cell or of tissue, a continuum

material description of cell wall may be adequate, whereas

at a higher scale, it may be satisfactory to treat tissue as a

continuum material. Within the wall, the complex struc-

ture of polysaccharides can be modelled with theories

derived for fibre-matrix composites or for entangled poly-

mers. At this level there is the possibility of modelling

some of the biological activity of the wall, for example, the

loosening of the wall by enzymes to allow extension at a

turgor that the wall would otherwise contain. A model

may allow several levels of structural hierarchy to be con-

nected; a description of cell wall behaviour based on its

polymeric nature may be summarized into a constitutive

relation for a continuum material appropriate for a math-

ematical model.

All mechanics of primary plant cells is essentially a problem of interplay between turgor and wall properties. One approach is to describe the mechanical properties of cell walls by using theory developed for continuum, iso- tropic materials. In a perfectly linear elastic material where engineering strain is proportional to true stress, the only parameter that determines the stiffness of the material is the modulus of elasticity, E. The Poisson's ratio of the material, , becomes involved if the sample is not free to expand or contract. Conventional elastic theory is restric- ted to strains typically less than a few per cent, but theory of membranes allows large deformations in thin samples to be modelled. The form of primary plant cell walls, which are thin relative to their area, suggests that the theory of membrane mechanics is an appropriate tool for modelling cell walls, and various analyses of biological materials have been conducted using membrane theory. The usual method is to assume, or determine, a strain energy function, explained in the following paragraph, and then to obtain the force-stretch relations by partial differ- entiation. A set of differential equations is thus obtained, the solution of which yields the membrane shape and stresses due to the deformation. Although the term 'linear' in describing a constitutive relation refers to the linearity of the stress-strain equation for the material, it may avoid confusion to note that classical theory of elastic-plastic deformations derives a linear stress-strain equation from a quadratic strain energy equation (Sokolnikoff 1956). When dealing with large deformations of membranes, it is convenient to describe deformation in terms of 'stretch ratios' than strains because a stretch ratio depends only on the state of strain and not on the choice of reference axis. The stretch ratio approach simplifies the analysis and allows three strain invariants, I, to be defined. The strain energy function for a Mooney-Rivlin type of material is defined in terms of the strain invariants and material con- stants that need to be determined by experiment.

There are two main approaches to model formation. The classical theory of Newtonian mechanics results in force and momentum balances. Identified forces perform work on the system and result in motion and deformation. In models of plant growth the driving force is presumed to arise from cell turgor, an increase in which also serves

to balance external compressive forces in studies of defor- mation. This approach has two drawbacks. First, it pre- sumes that all forces acting on the system are identified. Second, form of the constitutive relations usually has to be assumed. This limits the model's ability to test hypotheses about mechanical and biochemical processes occurring in plant tissue. The second main approach to modelling cell growth or response to deformation consists of combining an energy balance statement with a constitutive strain relation. This approach, which arises from theories of ana- lytical mechanics, uses the change of thermodynamic potential energy as the process performing the work. Although these theories have no advantages in simple mechanical systems where the forces are easily identified, they are to be preferred where the forces driving motion in a complex system are not completely known. Compared with the force balance method, the energy balance approach can be applied more generally, in that the ther- modynamic relations of both mechanical and biological

processes are included in the analysis, and the form of the constitutive relation need not be defined.

Once the governing equations and boundary conditions that describe a particular problem have been formulated, numerical approaches to solution of the equations have advantages over analytical approaches in that problems can be easily defined and complex geometries and irregu- lar material properties can be used.

Because of the practical difficulties of characterizing cell wall properties by manipulation of the wall in any plant material other than, for example, the large cells of the alga Nitella, the cell wall properties have been inferred by some authors from measurements on single cells or samples of tissue. Calculation of wall mechanics requires a model, formulated in terms of the constitutive relation of the cell wall material, of the deformation of the test sample. Where the sample is an isolated sphere compressed between two parallel plates, or in an osmotically manipulated environ- ment, the model will express the balance between tension in a pressurized spherical membrane and pressure within the cell. Although this approach at first sight seems straightforward, it has limitations. First, the form of the constitutive relation may not be known, so the data need to be good enough to determine the form as well as the (several) parameters. Some parameters are required that cannot be easily determined for the test cell, for example wall thickness and initial radial stretch of the inflated cell, so values may have to be assumed. The determination of the material characteristics and properties of test samples is clearly vital for a mathematical model to be verified, and readers are referred to Smith et al. (1998) who review a range of biophysical approaches. Further consideration of the characterization of materials is beyond the scope of this review.


First, models are considered in which the wall is treated as a continuum material, then as a material with properties that depend on its polymeric nature. For those interested in the current state of knowledge on cell wall architecture, Cosgrove (2000) reviewed current models (conceptual rather than mathematical) of the cell wall for their ability to account for the mechanism of cell wall enlargement.

The continuum approach was taken by Hettiaratchi & O'Callaghan (1974) who developed a model that describes cell extension, in which the walls of the cells were modelled as thin shells subjected to an internal inflationary pressure. The cell wall was represented by a rubber-like material with a linear elastic stress-strain characteristic, the molecular structure of which resembled that of the wall, given the extent to which wall structure was known at the time. The authors identified that finding a suitable expression for the strain energy function was the major difficulty with this approach. Having no evidence to support more than the simplest formulation of the strain energy function, they used a linear elastic material as did, for example, Pitt & Davis (1984). For tomato cell wall, Lardner & Pujara (1980) chose a constitutive model of the Mooney-Rivlin type, commonly used to describe rubber-like materials that are incompressible and can undergo large elastic deformations (Mooney 1940; Riv- lin 1948).

Davies et al. (1998) developed a model of deflection of a membrane by a probe acting normally to the membrane at its centre, and solved it analytically. They worked with a Mooney-Rivlin constitutive model with two material constants, though for verification only one was estimated with experimental data, the other being zero because a linear elastic material was assumed. Having verified the technique on a rubber membrane, which was also tested in a uniaxial manner to check the calculated properties, the authors calculated the single parameter of the consti- tutive model for the walls of potato tuber parenchyma cells.

Cell walls would be expected to behave differently from rubber because they contain relatively inextensible microfibrils. To account for the presence of microfibrils, Hettiaratchi & O'Callaghan (1978) and Wu et al. (1988) developed models of fibre-reinforced rubbers by introduc- ing stiffening factors. However, the assumptions that the microfibrils were inflexible and that they did not slip dur- ing cell expansion did not allow for realistic volume changes to occur. The modelled extension was only appropriate to an artificially induced increase in cell vol- ume as a result of manipulating its osmotic environment. Recognizing that cellulose fibrils are the major compo- nent of the cell wall with an identifiable structure, authors have attempted to explain the characteristics of cell wall as a two-component material of fibre and matrix. Wu et al. (1985) developed the work of Hettiaratchi & O'Callaghan (1978) to describe the pressure-volume relation for pressurized spherical and cylindrical cells. Their work was based upon the stress-strain relation for a polymeric material established by Wu & Sharpe (1979). They assumed two phases of cell expansion, the first occurring without the need for stressing the microfibrils and the second as a result of microfibril extension, the transition being at the point of incipient plasmolysis. Chaplain (1993) extended and simplified the theory of Wu et al. (1985, 1988) by characterizing the elastic properties of the ideal, isotropic cell wall in terms of a general strain energy function, so as to be able to describe better the nonlinear relation between pressure and volume in cell expansion. The advantage of this general function is that as the cell expands, the wall thins and the microfibrils introduce shear interactions. Chaplain (1993) also distinguished between the two components of the cell wall and defined a two-term strain energy function, one term each for the matrix and the microfibril phases, thus producing a model of the wall as a fibre composite material. The two most important variables in cell expansion were shown to be microfibril extensibility and matrix shear modulus. He also noted that the action of enzymes known to mediate cell wall extensibility (for one family of enzymes see McQueen-Mason (1995)) could be incorporated into the model by assuming shear modulus to be some function of enzyme concentration. Wall viscosity would appear to be

a more appropriate characteristic to choose.

That plant cell walls are viscoelastic has been demon-

strated experimentally by several authors (e.g. Preston

1974; Sellen 1980; Nolte & Schopfer 1998; Kohler &

Spatz 2002), but modelling of cell or tissue mechanics in

which the walls have a viscous component added to their

to their solid properties has not yet been addressed. This

is partly because the mathematical formulation and sol-

ution are challenging, and partly because the viscous term is difficult to determine with useful precision from experi- mental data, though Kohler & Spatz (2002) have made progress in this area. A method of calculating the viscous term from the structure and properties of the polymers in the walls is needed, but wall structure is not yet well enough defined to allow theories developed for structured and entangled polymers to be fully applied. However, Veytsman & Cosgrove (1998) have modelled plant cell wall extension by using concepts of thermodynamics of polymer mixtures. They formulated a simple model for a cylindrical plant cell where the free energy of the cell wall was the sum of the contribution of the free energy of the cellulose microfibrils and that of the hydrogen bonds, by which hemicellulose is attached to the surface of the cellu- lose. Their analysis accounted for aspects of polymer structure such as the number of rotatable link lengths. It was shown that macroscopic properties of cell walls are explicable in terms of the microscopic properties of inter- penetrating networks of cellulose and hemicellulose. Such work makes incorporation of the action of wall-loosening enzymes in a model a more realistic possibility.

Wall loosening induced by topical application of enzymes has been observed to initiate production of leaf primordia in the tomato (Fleming et al. 1997) but the phenomenon has not yet been studied mathematically. Chaplain & Sleeman (1990) examined how the form of the strain energy function can allow bifurcation to occur, which may be sufficient to initiate a new growing tip in a unicellular marine alga.

Smith et al. (1998) progressed beyond an elastic description of walls in their model. They used a constitut- ive model that is linear elastic with an elastic limit, at which a transition to plastic behaviour takes place, and with a finite hydraulic conductivity. They examined how far it is possible to determine uniquely the form of the constitutive relation from experimental data on com- pression of isolated spherical yeast cells, and conclude that high-quality data on parameters of the cells being used, together with a comprehensive model including hydraulic conductivity of the wall, are needed to be able to calculate both the form and the constants of the constitutive relation. It would therefore be a significant advance if the form of a constitutive relation could be calculated from wall structure and polymer composition.

For modelling based on events at molecular scale, it is

not feasible to specify the forces acting on the system so

an energy-based approach is appropriate. McCoy (1989)

presented a model based on energy balances in which a

change in thermodynamic potential energy was the driver

for cell wall extension. The model allows for water uptake

and biosynthesis as well as mechanical deformation, and

does not assume a form for the strain energy function.

This work brings together the purely mechanical aspects

of cell mechanics with important biological aspects, and

points the way towards models that integrate the two.

Hepworth & Bruce (2000) avoided assuming the form

of the constitutive relation a priori, but only by working

within a deformation time-scale of 15 s to avoid any effects

of cell wall viscosity and hydraulic conductivity. They

ascribed the cell wall tensile properties to the fibre compo-

nent of the wall and deduced the stress-strain curve of

this fibre component by fitting to experimental data on

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