Modelling the Mechanical Properties of Single Suspension-Cultured Tomato Cells

d Background and Aims The relationship between composition and structure of plant primary cell walls, and cell mechanical properties is not fully understood, partly because intrinsic properties of walls such as Young’s modulus cannot be obtained readily. The aim of this work is to show that Young’s modulus of walls of single suspension-cultured tomato cells can be determined by modelling force-deformation data. d Methods The model simulates the compression of a cell between two ﬂat surfaces, with the cell treated as a liquid-ﬁlled sphere with thin compressible walls. The cell wall and membrane were taken to be permeable, but the compression was so fast that water loss could be neglected in the simulations. Force-deformation data were obtained by compressing the cells in micromanipulation experiments. d Key Results Good ﬁts were obtained between the model and low-strain experimental data, using the modulus and initial inﬂation of the cell as adjustable parameters. The mean Young’s modulus for 2-week-old cells was found to be 2·3 6 0·2 GPa at pH 5. This corresponds to an instantaneous bulk modulus of elasticity of approx. 7 MPa, similar to a value found by the pressure probe method. However, Young’s modulus is a better parameter, as it should depend only on the composition and structure of the cell wall, not on bulk cell behaviour. This new method has been used to show that Young’s modulus of cultured tomato cell walls is at its lowest at pH 4·5, the pH optimum for expansin activity. d Conclusions The linear elastic model is very suitable for estimating wall Young’s modulus from micromanipulation experiments on single tomato cells. This is a powerful method for determining cell wall material properties.


INTRODUCTION
The relationship between the composition and structure of plant primary cell walls, and cell mechanical properties is not fully understood. The strength of the cell wall is considered to be mainly due to cellulose micro®brils (Fry, 1995) but the contribution of other wall components and their interaction with cellulose is not clear (Cosgrove, 2000a). Ideally, one would relate the material properties of the cell wall to its composition and structure, preferably using a functional genomics approach, and consider how and why these properties respond to the cell environment. In this respect, the relative simplicity of undifferentiated single cells from suspension cultures makes them a useful tool. Using single cells, one could conceive of using functional genomics to de®ne the roles of different cell wall polymer types in determining cell wall mechanical behaviour. However, no reliable method of measuring the material properties of walls of single cells has yet been reported.
Using mainly uniaxial stretching, mechanical tests on isolated walls have indicated that the cell wall material is non-linear viscoelastic-plastic (e.g. Kawamura et al., 1995). Attempts have been made to simulate turgor and multiaxial tensions in some experiments (Kamiya et al., 1963;Richmond et al., 1980), but it would be preferable to study walls in situ with the actual turgor and original geometry of living cells. The properties of living and dead cells with growing or non-growing walls may be different. A micro-penetration technique has been used for mechanical testing of potato tuber parenchyma tissue (Hiller et al., 1996;Davies et al., 1998), leading to calculated values of cell wall stiffness. Unfortunately these values depended on a dif®cult estimation of cell radius. In any case, this method cannot be used on single cells, although it is a potentially valuable method for studies on tissue.
The pressure probe is an excellent tool for studying plant water relationships and can provide some mechanical property measurements (Tomos, 2000). However, it has only rarely been used on single suspension cells (Tomos and Leigh, 1999;Hukin, 2002). Furthermore, the pressure probe method can only cause small deformations, not deformations leading to cell bursting. Therefore it cannot be used to investigate the full range of cell wall mechanical behaviour. A method of studying particle mechanics that has recently been used on many cell types is micromanipulation, in which a single cell is compressed between¯at, parallel surfaces to provide information on the wholecell mechanical response to an applied compressive load (Zhang et al., 1991;Thomas et al., 2000). Initial work on tomato cells from suspension cultures has been undertaken using such compression experiments (Blewett et al., 2000), and it has been shown that forcedeformation data can be obtained up to cell bursting. Key measurements are bursting force and percentage deformation at bursting, but these are not`intrinsic' properties of the walls, i.e. they have values that depend on the measurement method. Intrinsic wall material parameters such as Young's modulus are required if the mechanical properties of the wall are ever to be related to its composition and structure. It would be useful therefore to be able to extract these properties from force-deformation data, and this can be achieved by mathematical modelling. As pointed out by Smith et al. (1998), this task is facilitated if the form of the cell wall constitutive equation is known a priori. The cell wall material properties can then be determined by matching the force-deformation response of the mathematical model to that obtained in a compression experiment.
Several investigators have modelled the compression mechanics of thin-walled, liquid-®lled spheres. Thin walls in this mechanical context are often referred to as`membranes'. Such a membrane is a thin material element that cannot support bending stresses across its thickness. A cell wall might be considered to be a (mechanical) membrane. Feng and Yang (1973) considered the problem of the deformation and the consequential stresses in an in¯ated, non-linear elastic, gas-®lled spherical membrane compressed between two frictionless rigid plates. Lardner and Pujara (1980) extended this model further by considering the sphere to be ®lled with an incompressible liquid rather than gas. Their model was able to predict accurately the deformation of sea urchin eggs, as previously reported by Yoneda (1973). Liu et al. (1996) improved the computational algorithm, and applied the model to data on microcapsules. None of these studies allowed for water loss from the sphere. Smith et al. (1998) created a ®nite element model in which volume loss was included, and applied this to compression data from yeast cells (Smith et al., 2000). Using a ®nite element method, it was possible in principle to consider any cell wall material constitutive equation, although in practice Smith et al. (2000) only considered the linear elastic case.
There are only two reports of use of the compression experiment to characterize intrinsic material properties of plant primary cell walls. Liu (1995) attempted to estimate Young's modulus from compression experiments on single pericarp tissue tomato cells, but obtained few data. In an extensive study, Blewett (2000) developed the experimental method used here, but found it dif®cult to derive Young's modulus by mathematical modelling. Bounds of 100± 2300 MPa were found for single suspension-cultured tomato cells. In the present work, a linear elastic material model was used to determine Young's modulus of walls of single undifferentiated tomato cells from suspension cultures. The quality of the ®t of this model to experimental data, and the sensitivity of the model to changes in parameter values, were investigated.

Tomato cell suspensions
Cell cultures and isolation of single tomato cells have been described elsewhere (Blewett et al., 2000). The tomato cell line Lycopersicon esculentum vf36 was used as a model system. This is a suspension culture derived from a root radicle callus (provided by Unilever Research Colworth, Bedford, UK). Cells were subcultured weekly in medium consisting of MS basal salts (43 g for 10 L), 30 g sucrose L ±1 , 2 mg 2,4-dichlorophenoxy acetic acid (2,4-D) L ±1 , 1 mg 2,isopentyladenine L ±1 , and 50 mL vitamin solution (100 mg nicotinic acid, 40 mg glycine, 10 mg pyridoxine hydrochloride, 10 mg thiamine hydrochloride, 10 mg folic acid, 1 mg biotin, 20 mL water). The pH was initially 6´0. Cells were grown in a horizontal rotary shaker set at 100 rpm and 25°C with low light. Suspension cultures obtained by this method are nearly in synchronous growth, and so the cells within a sample are of similar age (Blewett et al., 2000).
Single cells were evident in the culture from week 2 and passage through a 72 mm mesh separated them from any aggregates. The single cells were suspended in 0´03 M mannitol at pH 5, the same osmolality and pH as the medium at harvest. They were held at 27 T 1°C during compression testing.

Compression experiment
The basic experimental apparatus and procedure were described by Blewett et al. (2000). Cells were compressed between the¯at end of an optic ®bre (the micromanipulation probe) and a glass surface (see Fig. 1). The probe was mounted on a force transducer, which in turn was mounted on a micromanipulator (MicroInstruments Ltd, Oxford, UK). This allowed the probe to be driven at a chosen speed towards the slide, compressing the cell. The chosen speed here was 23 mm s ±1 . The voltage and the time of deformation could be recorded on a PC-30D data acquisition card (Amplicon Live-line, Brighton, UK) in a personal computer. Figure 2A shows typical raw data. A ®ve-point moving average was used to smooth the data and to determine the exact start of the compression. Using the speed of motion of the probe, and allowing for compliance (Mashmoushy et al., 1998), the probe displacement and the cell deformation could be calculated, and a force-deformation curve generated, as shown in Fig. 2B.

The parameters required and the measurements taken
To remove some of the risks of the model producing nonunique solutions that would ®t the experimental curve using different combinations of parameter values (Smith et al., 1998), some of the parameters were ®xed by using values previously measured on cells grown under the same conditions. These are described below.
Initial thickness of the cell wall. Using freeze-fracture scanning electron microscopy, the mean cell wall thickness of 2-week-old tomato cells from suspension cultures has been found to be 126 T 16 nm (Blewett, 2000). This value was used as the initial wall thickness in the model.
Internal osmotic pressure of the cell and the initial turgor pressure. The mean initial osmotic pressure of the cells under the conditions of test was measured directly by singlecell sampling followed by freezing point depression (Tomos et al., 1994). The mean initial turgor pressure was calculated by subtracting the external osmolality from the internal osmolality. For the 2-week-old tomato cells, the mean initial osmotic pressure difference and therefore the mean initial turgor pressure was found to be 0´363 MPa.

Model de®nition
To derive material properties from force-deformation data, a mechanical model must be developed. The model formulated here is based on that of Lardner and Pujara (1980), considering the plant cell wall and membrane as a liquid-®lled sphere. Unlike Lardner and Pujara (1980), the wall is considered to be permeable. The outer radius of the unin¯ated sphere (equivalent to the cell at incipient plasmolysis) is r 0 ( Fig. 3A; see Table 1 for list of symbols used) and its initial wall thickness is h 0 . The sphere is in¯ated by an internal pressure to an outer radius of r i (Fig. 3B). Then the initial stretch ratio l s is de®ned as l s = r i /r 0 . The in¯ated sphere is then compressed between twō at, parallel surfaces, with the top surface moving at a constant velocity towards the bottom surface. Following Lardner and Pujara (1980), it is also assumed that the cell is axi-symmetrical around its vertical axis before and during compression, and is symmetric across its equatorial plane. The geometry of this problem is shown in Fig. 3C.
The liquid is taken to be incompressible, but the wall is considered permeable and so the internal volume decreases with compression. Equation (1) describes the volume change (Kedem and Katchalky, 1958).
where V is the instantaneous internal volume, t is time, L p is the hydraulic conductivity of the cell wall, A is the area available for¯ow, DP = P external ± P internal is the hydrostatic pressure difference and Dp = p external ± p internal is the osmotic pressure difference. As the probe velocity (v) is constant, the volume change can be related to the distance that the cell has been compressed, rather than to time (Smith et al., 1998): where z is half the distance the cell has been compressed (see Fig. 3C). Equation (2) assumes that solutes do not pass through the cell membrane, that the cytoplasm is a dilute and ideal solution, and that the hydraulic conductivity of the wall, L p , is constant throughout compression. It is further assumed here that the osmotic pressure difference (Dp) is constant during compression, i.e. the water¯ows into or out of the cell cause negligible changes in internal cell solute concentrations. The area available for¯ow is taken to be the exposed area, i.e. the area not in contact with the two¯at surfaces.
To model the cell, the cell wall material properties (its constitutive equation) need to be considered. Here the cell wall is considered to be isotropic and its constitutive equation to be linear elastic and compressible (Poisson's ratio n is not 0´5). The behaviour is also assumed to be timeindependent, i.e. no viscoelasticity. It should be noted that the time-scale of the compression experiments is of the order of 1 s, far too rapid for many physiological responses that may change the cell wall material characteristics, e.g. the creep phenomenon described by Cosgrove (2000b).

The theoretical analysis and numerical simulation
The derivation of the equations governing the deformation of the cell is based on analysis by Feng and Yang (1973) and Lardner and Pujara (1980). The derivation is given in the Appendix.
In the numerical simulations, the governing equations were solved by a Runge±Kutta method, using the Matlab (MathWorks Inc.) ode45 solver. As the chosen constitutive equation for the wall material is time-independent, the compression of the cell was solved as a series of static equilibrium problems following the calculation procedure of Liu et al. (1996). In each step, the model simulates the displacement of the probe corresponding to the time between steps. The probe displacement deforms the cell from its previous shape. During the time taken for the step there may be water loss, depending on the wall permeability. At the end of the step, the volume of the deformed sphere is determined from the calculated cell boundary coordinates r and h (see Fig. 3C). The equations are given in the Appendix. As in the method of Liu et al. (1996), the simulated pressure is then adjusted until the new volume equals the previous volume minus any estimated water loss. This pressure is then the starting pressure for the next step. The simulations lead to predicted force-deformation data, which depend on the values of the model parameters. These are the hydraulic conductivity (L p ), which determines the water¯ow from the cell; Young's modulus (E); Poisson's ratio (n), which describes the compressibility of the cell wall material; and the initial stretch ratio (l s ), which is the diameter of the cell at the start of compression compared with that at incipient plasmolysis.

Dimensionless analysis
To reduce the number of variables and to gain insight, a dimensionless analysis was conducted. The force has a dimensionless form of F/(Eh 0 r 0 l 0 2 ) and the deformation X = z/(r 0 l s ), where z is half the distance the cell has been compressed, i.e. half the displacement of the probe. F I G . 3. Schematic diagram showing: (A) the geometry of the initial sphere; r 0 is the initial radius of the cell before in¯ation; (B) the geometry of the in¯ated sphere before compression; r i is the radius of the cell after in¯ation; and (C) the geometry of a compressed sphere; h is the half-distance between the probe surface and the plate, r is the maximum radius of the deformed wall and membrane. In compression, a typical point like A, at an angle y to the vertical axis of symmetry, moves to point B. The angle G identi®es points that move to the edge of the contact region, which has radius r c .

RESULTS
One aim of developing the model was to run a series of simulations that would show how changes in the parameters alter the shape and magnitude of the force-deformation curves generated by the model. These simulations were carried out by changing the hydraulic conductivity (L p ), Poisson's ratio (n) and initial stretch ratio (l s ). Other parameters were either ®xed a priori (h 0 , Dp i and v) or were to be found by ®tting (E).

Effect of hydraulic conductivity (L p )
Typical simulation results are shown in Fig. 4. It can be seen that increasing the hydraulic conductivity (L p ) produced a lower force for a given displacement. If the cell membrane has a higher hydraulic conductivity, then the resistance to¯ow is lower, and this will lead to a lower increase in pressure during the deformation because of the greater water loss. It has been reported that L p values of different plant cell types range from 2 Q 10 ±8 to 10 ±5 ms ±1 MPa ±1 (Maurel, 1997). For single tomato cells, the hydraulic conductivity is about 10 ±7 ms ±1 MPa ±1 (Maggio et al., 1995;Hukin et al., 2002). The two curves in Fig. 4 for L p = 0 and L p = 10 ±7 ms ±1 MPa ±1 almost coincide, especially at smaller cell deformations (up to about 20 %). So, although the wall is considered permeable, it is reasonable in the case of rapid compression at small deformations to neglect water loss in simulations.

Effect of Poisson's ratio (n)
In previous analyses of compression of a liquid-®lled sphere or a cell (Lardner and Pujara, 1980;Smith et al., 1998), the wall was taken to be incompressible, equivalent to a Poisson's ratio of 0´5. In this study cell wall material is considered to be compressible, with a Poisson's ratio of less than 0´5. However, no data are available concerning the Poisson's ratio of tomato cell walls and there is no reliable method to measure this parameter. Therefore a range of values from 0´3 to 0´5 was used in simulations to investigate the in¯uence of Poisson's ratio on the force-deformation curves. Figure 5 shows typical simulation results. There is no great difference in the force at a given deformation for a change in Poisson's ratio from 0´3 to 0´5, especially at small deformations. This is consistent with the low sensitivity to Poisson's ratio of the calculated elastic modulus of cell walls of potato tuber parenchyma tissue, found by a micropenetration technique (Hiller et al., 1996). To con®rm that the results are insensitive to Poisson's ratio, the simulated deformed pro®les of the compressed cells were compared at a different Poisson's ratio. No difference was observed that could be distinguished easily (e.g. by image analyses of cell pro®les). Thus, it was not possible to obtain a reliable value of Poisson's ratio from ®ts to force-deformation data, but as the choice is not critical, a Poisson's ratio of 0´4 was chosen for all subsequent model ®ts.

Effect of initial stretch ratio (l s )
Among the factors that might in¯uence the forcedeformation curves, some were ®xed a priori. These were the initial thickness of the cell wall (126 nm), the osmotic pressure difference across the cell wall (0´363 MPa), and the probe motion velocity (23 mm s ±1 ). As described above, hydraulic conductivity and Poisson's ratio were not found to be of signi®cance. However, the initial stretch ratio, l s , i.e. the in¯ation of cell above incipient plasmolysis, was potentially important. It is very dif®cult to measure the initial stretch ratio of tomato cells accurately and, in any case, it is not practical to measure this ratio for individual cells before each compression experiment. Therefore a series of initial stretch ratios were tried in simulations. Figure 6 shows the resulting simulated force-deformation curves. The in¯uence of initial stretch ratio on forcedeformation curves was clearly signi®cant. A check on the simulated shape of the cell boundary con®rmed this (see Fig. 7). F I G . 4. Simulation of how hydraulic conductivity changes can alter the dimensionless force-deformation curves generated with the linear elastic model (initial stretch ratio, 1´01; Poisson's ratio, 0´40; initial cell radius, 30 mm; compression speed, 23 mm s ±1 ; Young's modulus, 2´30 GPa; initial wall thickness, 126 nm).
F I G . 5. In¯uence of Poisson's ratio on the force-deformation curves generated with the linear elastic model (initial stretch ratio, 1´01).
Given the signi®cance of l s , and the dif®culty of its measurement, it was decided to treat it as an adjustable parameter to be identi®ed during data ®tting.

Matching force-deformation curves
The remaining unknowns, i.e. initial stretch ratio l s and Young's modulus, E, can be found by matching forcedeformation curves from simulations with data obtained from compression experiments. A series of simulations with different initial stretch ratios were generated. The values chosen were 1´005, 1´01, 1´015, 1´02, 1´03, 1´04 and 1´05. Figure 6 shows data for initial stretch ratios of 1´005, 1´01 and 1´015 only. In each case simulated and experimental force-deformation curves were compared for deformations up to 20 %, with Young's modulus adjusted to give best ®t, using a minimum root mean square error. For each experimental force-deformation curve, this method resulted in a value of Young's modulus for each l s value. Root mean square errors were then used to choose overall best ®t, and therefore to determine the best value of l s and Young's modulus.
An example of such ®tting is shown in Table 2 and Fig. 8. As can be seen from Table 2, the best value of l s and Young's modulus for these data were 1´015 and 2´41 GPa, respectively. The good ®t, which is typical, indicates that the linear elastic model is suitable for determining the Young's modulus of the walls of the tomato cells used in this study. Fitting data from 22 cells, the mean Young's modulus of 2-week-old tomato cell walls at pH 5´0 and 27°C was found to be 2´3 T 0´2 GPa, which is comparable with values reported in the literature, as discussed later. An average initial stretch ratio of 1´01 T 0´002 was obtained, consistent with a value of 1´01 T 0´002 found using image analysis on similar cells (Blewett, 2000). This shows that it is possible to determine the initial stretch ratio of tomato cells through modelling force-deformation data.
Because the micromanipulation method does not allow meaningful hysteresis experiments on rapidly adapting cells, the appropriate deformation range for elastic behaviour of the cell walls could not be found directly. The chosen upper bound of 20 % cell deformation was validated by determining Young's modulus for the typical data of Fig. 2, with upper bounds of 10, 15, 20 and 25 % cell deformation. Table 3 shows the results. The modulus was not signi®cantly different whether 10, 15 or 20 % was used, but appeared to drop at 25 %. This suggested elastic behaviour up to 20 % cell deformation. It should be noted that the correlation coef®cient for all the bounds was very high (>0´95). It was assumed that an upper bound of cell deformation of 20 % could be used in all subsequent analyses, and this was con®rmed by continued good ®ts of experimental and simulation data. 20 % cell deformation represents a wall strain of about 0´07 at the cell equator.

DISCUSSION
Compression of cells by micromanipulation can be used to obtain force-deformation data up to cell bursting. However, bursting force and percentage deformation at bursting are not intrinsic parameters, as they depend on bulk cell behaviour, not only on cell wall composition and structure. In the present study, a linear elastic material model has been used to extract Young's modulus of cell walls of single suspension tomato cells from low-strain force-deformation data. Although the cell wall and membrane were taken to be permeable and the model could therefore account for any volume loss by¯ow from the cell, this was found to be insigni®cant on the time scale of the compression (1 s), assuming a reasonable value for the hydraulic conductivity of single tomato cells (10 ±7 ms ±1 MPa ±1 ). Water loss through the cell wall was therefore neglected in simulations. However, given the range of L p values for plant cells in general (2 Q 10 ±8 to 10 ±5 ms ±1 MPa ±1 ; Maurel, 1997), this assumption might not be true for compression tests on other plant cell types nor indeed on single tomato cells under other conditions. Should the assumption not be valid, F I G . 6. In¯uence of initial stretch ratio on the dimensionless force versus deformation curves generated with the linear elastic model (Poisson's ratio, 0´40).
F I G . 7. In¯uence of initial stretch ratio on the cell wall boundary shapes generated with the linear elastic model.
simulations could still allow for water loss during compression, using a separately determined value of L p . Preliminary studies have shown that L p might be found by compressing cells and holding them, and measuring volume loss directly using image analysis (Blewett, 2000). This would be a powerful adjunct to the present method. The cell wall was taken to be compressible in this study, which meant that Poisson's ratio could not be 0´5, as often assumed in other compression experiment studies (e.g. Lardner and Pujara, 1980;Smith et al., 2000). However, it was shown in the simulations that there was no great difference in the force at a given deformation for changes in Poisson's ratio from 0´3 to 0´5. This is unfortunate, as it does not seem likely that the Poisson's ratio of plant cell walls could ever be estimated from force-deformation data, or even from a study of the cell shape during compression. On the other hand, the insensitivity of the modelling to this ratio means that Young's modulus might be found without this knowledge. The choice of a Poisson's ratio of 0´4 for model ®tting seems reasonable, although it would be equally acceptable from a modelling viewpoint to assume a value of 0´5, i.e. incompressible cell walls.
Compared with Poisson's ratio, the in¯uence of Young's modulus and initial stretch ratio on the force-deformation curves was found to be very signi®cant. Adjusting these parameters allowed very good ®ts between model and lowstrain experimental data. Low strain in this study was taken to mean <20% cell deformation. Typically this represents wall strains of <0´07. The good ®tting up to these strains indicates that the linear elastic model is very suitable for determining primary cell wall Young's moduli from micromanipulation experiments, at least for these undifferentiated tomato cells. It is not known if this observation is applicable to other plant cell types.
The mean Young's modulus of the walls of 2-week-old cultured tomato cells was found to be 2´3 T 0´2 GPa. This value cannot be compared with values for similar cells in the literature, as such data are not available. Besides Blewett (2000), who estimated values for single suspension-cultured tomato cells of 100±2300 MPa, the nearest study was on potato tuber parenchyma tissue cells, using a micropenetration technique (Hiller et al., 1996). A value of 3 GPa was indicated by the analysis, assuming a cell wall thickness of 100 nm. For giant algal cell walls, the modulus has been found to be 407±662 MPa for Chara corallina (Toole et al., 2001) and up to 4 GPa for Nitella opaca (Probine and Preston, 1962). Other than these relatively direct determinations, many investigators have also studied the mechanical properties of plant cell wall components or wall analogues. The modulus of individual cellulose micro®brils has been both measured and calculated (Marho Èfer et al., 1996;Ishikawa et al., 1997) giving a modulus range of 0´7±3´5 GPa. These calculated values are close to those obtained from measurements made on cellulosic networks under uni-axial tensile conditions, e.g. moduli of 2´7 GPa for softwood ®bres at a relative humidity of 50 % (Hamad, 1998) or 1±3 GPa for a cast network of sugar beet cellulose micro®brils (Dufresne et al., 1997). By deforming cell wall analogues based on Acetobacter xylinus cellulose under equi-biaxial tension, moduli ranging from 200 MPa to 500 MPa have been obtained for a celluloseonly system (Chanliaud et al., 2002). Overall, it appears that Young's modulus estimated by the present method gives values that are reasonable compared with prior estimations on related materials.
It should be noted that the value of Young's modulus from model ®tting depends critically on the chosen value of the initial cell wall thickness, as also reported by Hiller et al. The minimum root mean square error was 0.0038. The chosen initial stretch ratio, ls, was therefore 1.015, with a corresponding Young's modulus of 2.41 GPa. F I G . 8. An example of a ®t of the experimental data to the model up to bursting using low-strain parameters (parameters for the model: initial stretch ratio, 1´015; Poisson's ratio, 0´40; initial thickness of the cell wall, 126 nm; L p = 0 ms ±1 MPa ±1 ; Young's modulus for the cell wall, 2´41 GPa).  (1996) for modelling of micro-penetration tests on potato tuber parenchyma tissue cells. Indeed, the modulus is inversely proportional to wall thickness for given forcedeformation data. Freeze fracture scanning electron microscopy was used to measure the wall thickness used in this study (Blewett, 2000), which should give a reliable value. However, it should be noted that not all of the cell wall is necessarily load-bearing. It may therefore be wisest to treat Eh 0 as one parameter combining modulus and wall thickness. This should be invariant for any set of data. Furthermore, it was necessary to use a mean value of the thickness, as measurements of thickness for speci®c cells are not possible during the compression experiment. Some possible effects of variations around the mean are shown in Table 4. Using the pressure probe, measurements have been made of the instantaneous bulk modulus of elasticity, e, of single suspension tomato cells (Tomos and Leigh, 1999;Hukin, 2002). e i is de®ned by: As the name and de®nition imply, e i is not an intrinsic material property of a cell as it depends on bulk parameters such as cell size. Nevertheless, it is possible to use the Young's modulus found in the present work to estimate e i for the cells under test. Assuming linear elastic behaviour during pressure probe measurements, e i was estimated to be about 7 MPa. This value agrees very well with direct pressure probe measurements (5´9±7´1 MPa; Hukin, 2002). This is further validation of the present approach and the value of Young's modulus determined here.
This method for determining Young's modulus of single undifferentiated plant cells has great potential for further studies, speci®cally in investigating how the elasticity of primary plant cell walls depends on their composition and structure, and environmental conditions. Table 5 shows some preliminary data obtained using the method, which show how Young's modulus varies with pH. Although these data need proper interpretation, it is interesting to note that the optimum pH for expansin activity is between 3´5 and 4´5 (McQueen-Mason et al., 1992), while at pH 4´5 Young's modulus is lowest, i.e. the cell wall is least stiff.
It is sometimes suggested that plant cell walls might be viscoelastic (Preston, 1974), although it is dif®cult to separate water¯ows and true viscoelasticity for tissue samples. Such time-dependent behaviour was not considered here. The compression experiment is relatively fast, and creep and enzyme responses act on longer time-scales (Haughton et al., 1968;Cosgrove, 2000b). By using additional techniques such as image analysis of cell shape, and different compression speeds, it may eventually be possible to use the compression experiment to investigate any true time-dependent behaviour, as well as measuring cell hydraulic conductivity directly.

CONCLUSIONS
Compressing cells by micromanipulation to obtain forcedeformation data has been used widely to investigate the mechanical properties of cells, including single plant cells from suspension cultures (Blewett et al., 2000). Normally it can only provide the bursting force and percentage deformation at bursting to characterize the mechanical properties of the cells. The intrinsic material properties of cell walls cannot be obtained directly from these compression experiments. In this study, it was shown that a linear elastic model could be used to determine the Young's modulus of single cultured tomato cell walls from micromanipulation data. It was surprising how well this simple model ®tted the data. Because of the speed of the compression, it appears water loss due to cell permeability was not signi®cant. Although the Poisson's ratio could not be determined, it seemed that Young's modulus did not depend greatly on its value, and a reasonable value of the modulus was found.
Combining the compression experiment with mathematical modelling is a powerful method for investigating plant cell wall mechanics. It should be useful in determining how the elasticity and the strength of plant cell walls depend on their composition and structure, and how these change under different environmental conditions.

Theoretical analysis of cell compression
For a linear elastic material, Cheng (1987) gives an expression for the generalized Hooke's law strain energy function (W). This is important as it can be used to identify the tensions in a cell wall, if this can be assumed to show linear elastic behaviour. In the case of a (mechanical) membrane, a simpli®ed expression for W is given by eqn A1, where bending and shear effects can be neglected. At any given compression, W Eh 0 21 À n 2 l 1 À 1 2 l 2 À 1 2 2nl 1 À 1l 2 À 1 n o A1 where W is the strain energy per unit volume caused by deformation; l 1 and l 2 are the principal stretch ratios, each de®ned (in a given direction) as the ratio of the length of a small section of membrane after deformation, to its original length in the unin¯ated state. Subscripts 1 and 2 refer to the meridian and circumferential direction, respectively (see