 Copyright © 2001 American Society of Plant Physiologists
Abstract
We have studied the elasticity and load bearing ability of plant tissue at the cellular level, using onion (Allium cepa) epidermal cells. The Young's modulus and Poisson's ratio of the cells were obtained by loading a tensile force on onion epidermal peels of different turgor pressures, and measuring the elongation and the lateral contraction of the peels. The Young's moduli and the Poisson's ratios ranged from 3.5 to 8.0 MPa and 0.18 to 0.30, respectively. To determine the effects of cell elasticity and turgor pressure on the cell's ability to bear load, we loaded a small glass ball onto a cell and measured the projected contact area between the ball and the cell. Unlike previous studies, we considered the cell as a whole entity, and utilized the Boussinesq's solution to derive the relevant equations that related the elastic parameters and cell deformation. For cells with a turgor pressure ≥ 0.34 MPa, the predicted contact area agreed well with the measured area. The equations could also predict cell turgor pressure with a deviation from the measured value of 0.01 MPa. This study gives strong support to ball tonometry, a new method of measuring cell turgor pressure.
Although the loadbearing properties of woody stems are due in large part to the mechanical properties of the woody component alone, the abilities of herbaceous stems to support external loads is not due to any single structural element. Instead, it is due to the combined effect of cell turgor pressure, cell wall elasticity, and stiffness. A number of different approaches have been used to study the hydraulic and elastic properties of plant cells. Steudle and Zimmermann used an oilfilled cell pressure probe to determine the hydraulic conductivity and volumetric elastic modulus of giant algal cells (Steudle and Zimmermann, 1974; Zimmermann and Steudle, 1974a, 1974b). The cell pressure probe technique was also used to study the volumetric elastic modulus and volumepressure curve ofMesembryanthemum crystallinum epidermal bladder cells (Steudle et al., 1977). Ferrier and Dainty (1977) applied an external force to a piece of onion (Allium cepa) epidermis using a displacement transducer and balancing arm assembly. Vinters et al. (1977) used a special apparatus to conduct combined measurements of transcellular osmosis and relative length changes of Chara corallina cell. Ferrier and Dainty (1978) applied an external force onto multilayered tissues of red beet and artichoke. Green et al. (1979) also applied an external force on the storage organs of red beet. Steudle et al. (1982) combined the external force and the pressure probe techniques to measure the volumetric and transverse elastic extensibilities of C. corallina.
The application of external forces in the studies above, especially with the simultaneous application of the pressure probe technique, has provided much insight into the elastic and hydraulic properties of plant cells. By considering C. corallina internodes as thinwalled cylindrical cells, Steudle et al. (1982) assumed that the shearing strains in the wall and changes in wall thickness were negligible. Thus, they could reduce the number of elastic coefficients necessary to describe the elastic behavior of wall material. The relative changes in the cell's dimensions (length, radius, and volume) were then related to the changes in the principal circumferential and longitudinal tensions in the cell wall, using the corresponding Young's moduli and Poisson ratios. As a consequence, the volumetric elastic modulus can be expressed as a function of cell dimensions, cell wall thickness, the Young's moduli, and the Poisson ratios. This 1982 study further demonstrated that the deformation of the cell wall resulting from a concentrated external force has to be taken into account. Cell deformation dominates the transverse extensibility at low turgor pressures, and can be important at high turgor pressure. This study gives us a much better understanding of cell elasticity and the effects of cell turgor pressure.
Wu et al. (1985) presented an interesting viewpoint regarding the traditional approach of describing plant cell elasticity. They argued, largely from an engineering perspective, that the traditional approach–the bulk modulus or any related concept–was “vaguely defined and incorrectly applied.” After reviewing how the concept of bulk modulus was commonly derived in the earlier literature, they pointed out that “the derivation shown above, while mathematically tractable, is physically meaningless.” They noted that these concepts can only be applied to a system in which material is conserved, and that cytoplasm was not conserved during changes in volume. Considering the polymer makeup of cell walls, they advocated the use of the principles of polymer elasticity as an alternative method of analysis. A similar consideration was also posed by Hettiaratchi and O'Callaghan (1978).
Nilsson et al. (1958) presented another type of approach to plant cell's rigidity, which relied heavily on theoretical physics. They started from a model in which the cells were spherical, filled with liquid, bounded by elastic cell walls, and surrounded by an airfilled intercellular space. The calculation of Young's modulus stemmed from consideration of the energy and work, and the mathematics was done in a curved coordinate system bounded on the cell wall. They further discussed more complicated cell types (e.g. isodiametric tetrakaidecahedron and rhombic dodecahedron). They also performed some experiments (Falk et al., 1958) and observed good agreement between the theoretical and experimental results, although the factors of the higher order terms in the expression for energy density could not be determined in their measurements.
In light of these previous studies, this paper purports a direct application of elasticity theory and its mathematical treatments to the examination of stress and deformation at the cellular level, with both experimental and theoretical aspects. Our work here does not aim to undermine the volumetric modulus, the most widely accepted and pragmatic method of examining plant cell elasticity. Rather, we present another approach to examine whole cell elasticity.
In engineering terms, elasticity for a uniaxially stressed body is described by the twin concepts of Young's modulus (E), and Poisson's ratio (v), which are defined as follows:
The ability to resist an external load is clearly related to the turgor pressure and the elastic properties of the cell wall. Two extreme situations are: (a) A cell with an inelastic rigid cell wall can withstand a load even when the turgor pressure is low, and (b) a highly elastic cell can withstand a load only when its turgor pressure is high. The reality of a living cell lies between these two. Nevertheless, this paper attempts to approach the problem by considering the cell as a whole entity, and thus the resulting elastic parameters accommodate the combined effects of turgor pressure and wall elasticity.
To quantify cell deformation, we loaded a small rigid sphere onto the cell surface with a known force and measured the degree of indentation. If we suppose a spherical cell, what will the contact area between the ball and the cell be at a given force? A cell having a high loading resistance tends to be able to keep its spherical shape, and therefore the contact area between the ball and the cell will be small. A cell having a low loading resistance conversely will be indented by the ball to a greater degree, resulting in a larger contact area.
The present study consists of two experiments: the loading of a glass ball onto an epidermal onion cell, and the stretching of onion epidermal peels. The former experiment studies the load bearing ability of a cell, whereas the latter obtains the Young's moduli and Poisson's ratios. We were aware that cell turgor pressure plays a key role in the cell's ability to bear load, but it was unclear how the elasticity of the cell contributes to this. Because load bearing is reflected in cell deformation, and because Boussinesq's solution can concern deformation and elasticity, we found that Boussinesq's solution was a powerful tool to yield insight into cell elasticity and load bearing ability. On the other hand, the elastic coefficients in the Boussinesq's solution could not be obtained from the ball loading experiment. Therefore, we must rely on the stretching experiment to get the Young's moduli and Poisson's ratios.
In brief, the purpose of this paper is to present experimental and theoretical studies to show that: (a) By treating a cell as a whole entity, and by applying the Boussinesq's mathematical solution to general elasticity problems, we have a way to study cell elasticity and load bearing ability; and (b) the theoretical results obtained from the above considerations quantitatively coincide with experimental data, except for cells that were not sufficiently turgid.
RESULTS
Results of Cell Turgor Pressure Measurements
The turgor pressure of fully turgid cells was 0.54 ± 0.02 MPa (n = 6). This result agreed with previous pressure probe measurements (Lintilhac et al., 2000). With the application of four different mannitol solutions, we then had cells of five different turgor pressures, ranging from 0.54 to 0.14 MPa. These turgor pressure decrements had been verified in the previous study in which we also bathed onion epidermal peels in different mannitol solutions and measured the turgor pressures using a cell pressure probe.
Results of the Contact Area Measurements
Sample images of contact area measurements were shown in Figure1. The results of these contact area measurements were presented in Table I(indicated by ●). These contact areas were presented as their radii assuming that they were perfect circles.
Results of the Elastic Parameter Measurements
The Young's modulus measurements convincingly demonstrated the elastic properties of the cells, both in the longitudinal and transverse directions. Figure 2A shows typical loading and unloading paths of longitudinal strips. We chose to exclude the first loading path because it differed significantly from all subsequent loading and unloading cycles. This deviation may be due to wrinkles in the tissue and/or the presence of bonding substance on the tissue surface. The r ^{2} of data points in the remaining loading and unloading cycles consistently ranged from 0.90 to 0.97. Longitudinal strips remained elastic when the force was ≤18 gf. Transverse strips were more stretchy (i.e. larger strain per unit stress) than longitudinal strips (Fig. 2B), because the alignment of the cylindrical cells caused the outer wall surfaces to be highly convoluted in the transverse direction. Yet, the transverse strips also showed convincing elastic behavior when the force was ≤13 gf. Ther ^{2} of transverse strips, excluding the first deformation path, consistently ranged from 0.90 to 0.95.
The slope in Figure 2 represents the elongation of the strip per 1gf change in load. These slopes, excluding the first deformation path, were used in Equation 1 to calculate the Young's modulus. For longitudinal strips, the slopes were obtained by incrementally loading and unloading with forces up to 15 gf, whereas for transverse strips, the forces were kept below 5 gf to ensure that the test strips remained in their elastic range. Table II shows the average slopes which were grouped according to the different types of strips.
Table III shows the calculated Young's modulus, Poisson's ratio, and k values. In the calculation of the Young's modulus, the tensile force was applied on the cross sectional area of 3 mm × 120 μm (i.e. the width × the thickness of the epidermal strip), and the original length of the strip was 18 mm. The slopes used in our calculation are the averages of longitudinal and transverse strips taken together. In the calculation of the Poisson's ratio, the lateral contraction similarly was measured under a tensile force of 15 gf for longitudinal strips, and 5 gf for transverse strips. The denominator of Equation 2 therefore should be the relative elongation under the corresponding tensile force. Thek value was then calculated using Equation 4 (see “Materials and Methods”).
We ended each run by stretching the tissue into the plastic range and then to failure. Longitudinal strips reached plasticity at a tensile force of about 18 gf and broke at about 40 gf. Transverse strips reached plasticity at about 13 gf and broke at about 20 gf.
The Equations
To simplify the mathematics we treat the outer surface of the target cell as a spherical surface. The contact patch then becomes a circle and the relevant equations have symmetry about the vertical axis. In addition, the equations derived in this paper are only valid when the indentation is not too deep. This requirement was satisfied in our experiments except for cells that were not sufficiently turgid, such as cells of tissues bathed in 0.3 and 0.4 MPa mannitol.
The contact area between the cell and the ball depends on the load applied and the rigidity of the cell. However, the rigidity of the cell depends on the cell turgor pressure and the cell wall properties (Steudle et al., 1977). In the case of a thin and soft cell wall, as in most living cells, we can neglect the supporting effect of the cell wall and consider turgor pressure alone. Therefore, we have an estimation for cell turgor pressure:
However, if we are considering whole cell elasticity, which is a function of both the cell wall properties and turgor pressure, then cell elastic parameters should be taken into account regardless of the thickness of the wall.
From elasticity theory, we derived Equation 13 for the radius of contact area (a), where k is an elastic coefficient determined by the Young's modulus E and Poisson's ratio; and Q is a dimensional coefficient determined by the ball radius R
_{B} and cell radiusR
_{C}. They are defined as follows:
Equation 13 is a key equation in that it predicts the radius of the contact area. The k was obtained by direct measurements of the Young's modulus and Poisson's ratio of the tissue. Notice that the term P, cell turgor pressure, does not appear in Equations 13 and4, yet it has been reflected in the elastic properties of the cell (see Table III). In other words, cell turgor pressure was a hidden variable in Equation 4. Therefore, Equation 13 physically describes the effects of cell elasticity and turgor pressure on cell deformation. The accuracy of Equation 13 would be verified by determining whether the calculated radius of the contact patch coincided with those obtained by direct measurement.
A Summary for the Derivation of Equation 13
A problem of essential importance in elasticity is the behavior of an elastic body when a concentrated force acts on a small area of its surface. Based on its solution, one can, by integration, calculate the effect of any distribution of surface forces. Boussinesq (1885)considered a simplified situation in which the body occupied a semispace (z < 0) and the force (F) acted along thez axis (Fig. 3). The Boussinesq's solution can provide the resulting stress distribution and the displacement of any point in the body. In the present study, we are only interested in the z component of the displacement at the surface (s_{z}
) because this is the one that relates to the indentation of the cell by the ball. This can be obtained by considering the kcomponent of the general displacement solution, and letting z → 0. The result was:
The force (F) is supposed to be a concentrated one acting on the surface at ρ = 0. However, in our case F is the load applied by the ball. Thus, F should be replaced by P(x)ds, which is distributed over a circle centered at ρ = 0. The distribution function P(x) has been given in Equation 6, and the total effect can be calculated by integrating over the whole contact area. The integration gives:
After the ball indents the cell, the z component of the displacement of N can be expressed as a reduction of
̅M̅N̅
̅ by a small quantity ε:
Comparing Equations 8 and 9 yields:
The Predicted Radius of the Contact Area
The predicted radii of contact areas, using Equation 13, were indicated by ♣ in Table I. The Q value for calculation was 0.023 μm^{−1} because R_{B} = 150 μm and R_{C} = 60 μm. (We supposed the cell radius is 60 μm because in a real cell the length was much greater than the width and therefore any possible boundary effect to the loading would mainly come from the cell width, which was approximately 120 μm).
DISCUSSION
The images of the contact patch were elliptical rather than circular (Fig. 1). This is because onion cells are cylindrical, and not spherical. We found no evidence that the anisotropic properties of the cell wall contributed to the elliptical shape of the contact patch.
Figure 2 shows hysteresis during the stretching experiments. This implied that the epidermal strip was not a conserved system, in that water was lost while the strip was under tension. Another possible reason for the hysteresis was that the wall material did not truly remain in its elastic range during stretching.
In the present study, none of the cell elastic parameters was obtained from the ball loading experiments; instead, they were obtained from the stretching experiments. This is because it is impossible to measure cell elastic properties from the loading experiments because turgor pressure rather than wall elasticity dominates the resistance to deformation. Equation 13 served to interpret the results of the ball loading experiments (i.e. the a) using the results of the stretching experiments (i.e. the k).
Equation 13 is selfsufficient in that, given a cell with a certain turgor pressure, the R _{C} and the k are then determined; given the size of the object loading the cell, theR _{B} is then determined. Therefore, Equation 13will be sufficient to calculate the radius of the contact area under a loading force (F). Tracing back the above argument, it is clear that the deformation of a cell is determined by the condition of the cell itself and the external load.
Table I shows that when the cells were fairly turgid (e.g. cells in water, or in dilute mannitol solutions; P_{π} = 0.1 and 0.2 MPa), the contact patch radii calculated by Equation 13 agreed well with the measured radii, indicating that the mathematical approach of this study is correct. However, for flaccid cells in more concentrated mannitol solutions (P_{π} = 0.3 and 0.4 MPa), there was a considerable discrepancy between predicted and measured radii.
The elastic parameter (k) is a measure of whole cell elasticity, including contributions from both the cell wall and the cell turgor pressure. The effect of cell turgor pressure onk can be seen from Table III where strips of different cell turgor pressures had different k values.
The Young's moduli in Table III range from 3.5 to 8.0 MPa, which are one or two orders of magnitude smaller than those of earlier studies (external force) are. In fact, this discrepancy does reflect the stark difference between the previous studies and our own: We treated the cell as a whole entity rather than a thinwalled shell. The mathematical basis of our study, the Boussinesq's solution, concerns a semiinfinite elastic body, not a shell. In accordance, we calculated the Young's modulus by considering that the tensile force in the stretching experiments acted on the cross section of the epidermal strip (see “Results”), and not on the cross section of the cell wall material. Of course, this has led to a lower value of the Young's modulus.
How can Boussinesq's solution, a theory of an elastic body, be applied to interpret the results of a ball loading experiment, a surface event? What is the basis for it? Mathematically, we considered only thek component of the solution and let z → 0 (note: k is the unit vector of z axis; see Fig.3 and below). We substituted the force in the Boussinesq's solution with a pressure distribution function representing the loading of the ball (see Eq. 6). The mathematical treatment of letting z → 0 brought about the assumption that the derived equations were only valid when the indentation was not too deep. Surely this assumption is somewhat vague because how deep is not too deep?
As mentioned earlier, the predicted contact radii for turgid onion cells (turgor pressure ≥ 0.35 MPa) agreed closely with our experimental results, whereas the predicted radii for less turgid cells (turgor pressure < 0.35 MPa) did not agree well. If we take 20 μm as a representative contact area radius for turgid cells (see Table I), then the depth of indentation is 1 μm. In a similar manner, if we take 25 μm as a representative contact area radius for less turgid cells, then the depth of indentation is 2 μm. It is interesting that compared with the 120μm cell thickness, both indentations are relatively small. However, the 1μm indentation represented the practical validity of Equation 13, whereas the 2μm indentation exceeded it.
Despite the fact that the predicted radii and pressures agree with the corresponding experimental results (Tables I andIV), neither the present paper nor Boussinesq's solution enlightens us as to the anisotropic elastic properties of cell walls (the Boussinesq's solution itself is a mathematical relation that determines the state of stress in an isotropic elastic half space). To thoroughly describe the anisotropic elastic properties of cell walls, two or three elastic moduli (or Poison's ratios) must be measured for axisymmetric and orthotropic materials, respectively. It is unfortunate that the present study is not able to measure those elastic coefficients at the level of an individual cell. The stretching experiment (Fig.2) revealed only the elastic coefficients of epidermal tissue. The Young's moduli and Poison's ratios of longitudinal strips were much different from those of transverse strips, mostly because the cells were cylindrical and the orientations of the cells were different in the longitudinal and transverse strips. Therefore, we took the averages of the longitudinal and transverse Young's moduli and Poison's ratios, respectively, to best depict the elastic properties of the presumed “isotropic cell body.”
Cells with higher turgor pressures can sustain greater loads. In fact, the ball tonometry method (Lintilhac et al., 2000) has in some way answered the question of how the stiffness of a living cell depends on its turgor pressure. The method estimated that the contact area between the ball and the cell simply equals to the ratio of the load to the turgor pressure, which in turn provides us a way to measure turgor pressure.
There is a consistent pattern in Table I: The measured radius tends to be slightly less than the predicted one. This discrepancy may be due to a number of factors. Surface irregularities on the target cell may give rise to point loads, which reduce the total area of the contact patch. Another possible cause is a systematic error introduced by a combination of optical effects around the perimeter of the contact patch and the contrast enhancement algorithm used by the image analysis software, which may effectively erode the true contact area. In fact, we have already realized this potential optical problem and efforts have been made to improve the image resolution.
Introducing Equation 13 to somehow correct the radius may provide a method of measuring turgor pressure that has both benefits and drawbacks. On the one hand, the ratio of F toπa
^{2} inferred from Equation13:
One may point out that Equation 14 undermines the principle of the ball tonometry method because the principle states that the ratioF/A must be constant. Also, tonometry does not consider the effects of ball size and cell elastic properties, whereas the ratio revealed by Equation 14 obviously depends on these two factors. Let us now estimate to what extent Equation 14 does devalue the ball tonometry method.
Equation 14 came from Equation 13. It is obvious that Equation 13 gives rise to da/dF being proportional toF ^{−2/3}. Larger loads could reduce this systematic deviation, yet loads that are too great will generate deep indentations that will exceed our assumption. Can we then in practice apply a light enough load and have da/dF attain our desired accuracy? In a similar manner, Equation 13 leads toda/dQ being proportional toQ ^{−4/3}. Problems with Q are more complicated because they involve the size of the cell targeted. Can we then in practice find an appropriate ball size so that the ball can target the cell and also have da/dQ satisfy our desired accuracy?
The answer to these questions is yes. As an example, suppose we use three different sizes of balls and apply three different loads on a fully turgid onion cell. Table V shows the resulting turgor pressures calculated by using Equation 14. The average of these pressures is 0.57 ± 0.04 MPa. Although not perfectly consistent, random errors could easily yield a worse result. It is in this sense that if the size of the ball and the amount of load are within appropriate ranges, the ball tonometry method can generate accurate and consistent results.
The pressure distribution P(x) in Equation 6 plays an important role in using Boussinesq's solution for our purpose. P
_{0}is a mathematic constant that needs to be determined by real conditions. P
_{0} physically represents a hypothetical concentrated pressure applied at the very top point of the cell. The total force generated by thispressure distribution is given by Equation 11, which yields:
Finally, the stiffness of a cell may be described by the term
CONCLUSIONS
The common method of measuring Young's modulus and Poisson's ratio (the stretching experiments), and the application of a relevant theory of elasticity (the Boussinesq's solution) can provide a valid approach to the study of cell elasticity. The present study shows how this approach solved the problem of loadbearing ability of a cell (the ball loading experiment).
A less turgid cell has a smaller Young's modulus and Poisson's ratio values than a turgid cell.
The effect of external load on cell deformation can be described by Equation 13. The predicted turgor pressures deduced from this equation also quantitatively coincided with the measured pressures. However, these equations are only valid for fairly turgid cells.
As shown in Table V, so long as the ball size and the load are within appropriate ranges, the ball tonometry can measure cell turgor pressure with acceptable accuracy and deviation.
MATERIALS AND METHODS
Plant Material
Living, singlecellthick adaxial epidermal peels from Spanish onion (Allium cepa) leaf bases were cut into 18mm × 3mm rectangles so that the long axis of the cut was either parallel or perpendicular to the long axes of the cells (note: for convenience we will call them longitudinal or transverse strips, respectively). Prior to all cell elasticity and loading measurements, these epidermal peels were bathed either in pure water for about 12 min to allow them to reach full turgor pressure, or in a mannitol solution for about 12 min to reduce the cell turgor pressure. The osmotic pressures of the mannitol solutions were 0.1, 0.2, 0.3, and 0.4 MPa. Thus, we were able to prepare tissues of five different cell turgor pressure levels, fully turgid, and 0.1, 0.2, 0.3, and 0.4MPa decrements.
Cell Turgor Pressure Measurements
The turgor pressures of fully turgid cells were measured using the cell pressure probe method (Steudle, 1993). Based upon this pressure value, the cell turgor pressures of tissues bathed in one of the four mannitol solutions could then be reduced by 0.1, 0.2, 0.3, or 0.4 MPa. Because the pressure probe method is able to provide accurate pressure measurements in both positive and negative ranges (Wei et al., 1999), we were able to have an accurate starting point and precise decrements from full turgor.
Loading of the Cell and Projected Contact Area Measurement
The static loading method used in this study was the recently developed ball tonometry (Lintilhac et al., 2000). As shown in Figure5, a small glass ball (D = 300 μm) attached to a loading arm was positioned on a cell. The contact area between the ball and the cell was imaged by spinning disc confocal microscopy (Technical Instruments K2SBIO, San Francisco) and measured by video image analysis (Optimas, Media Cybernetics L.P., Bothell, WA). The force (F) exerted on the cell was adjusted by the addition of the appropriate weights to the loading arm. TheF was 47, 59, and 69 mgf in our experiments. (Note: Because this study involved loading of a weight on a cell, for convenience we used the gravitational unit, mgf, as the unit of force; 1 mgf = 9.81 × 10^{−6} N.)
Young's Modulus and Poisson's Ratio Measurements
Tensile loading of prepared onion epidermal peels was done using a commercial version of a mechanical testing frame previously developed in this laboratory (Vitrodyne V200, Liveco Inc., Burlington, VT). This instrument incorporates a microprocessor based feedback control system capable of operating either in straincontrolled or loadcontrolled mode.
Each end of a cut onion strip was glued (ethyl cyanoacrylate) to a load shim on the testing frame (Fig. 6). Care was taken to ensure that the strip length was 18 mm and that the strip was not yet under tension. The tissue was then placed in water (or a mannitol solution) for about 12 min, and then removed from the solution for tensile loading. The strain measurements were conducted by increasing the tensile stress to a certain value, and then reducing it to the initial value in 1gf increments. We ensured that the process remained within the elastic limits by examining whether the tissue returned to its original length. The entire loading and unloading cycle lasted about 20 min. To keep the tissue moist, the system was enclosed in a glass culture vessel lined with paper towels wetted with the same solution that bathed the tissue.
Data for Poisson's ratio estimation was obtained from two simultaneous measurements. Under a given tensile force, we recorded the elongation of the tissue, given by direct readout of the device, and measured the lateral contraction of the tissue with an optical micrometer in a stereomicroscope.
Because of the uniformity, the unicellular thickness of the tissue, and the lack of intercellular spaces, the elastic parameters determined for the multicellular epidermal peels do in fact represent the parameters of the individual component cells. In other words, the relative increase in length of the epidermal strip was equal to the relative increase in individual cell length. In a similar manner, the relative decrease in the strip's width reflected the relative decrease in cell width. It is worth noting, however, that the above consideration could not take into account the elastic behavior of the middle lamellae.
Footnotes

↵* Corresponding author; email cwei{at}zoo.uvm.edu; fax 802–656–0440.
 Received November 6, 2000.
 Revision received January 17, 2001.
 Accepted March 15, 2001.