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First published online September 24, 2004; 10.1104/pp.104.043752 Plant Physiology 136:3266-3275 (2004) © 2004 American Society of Plant Biologists
The Mechanics of Surface Expansion Anisotropy in Medicago truncatula Root Hairs1Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138 (J.D.); and Department of Biological Sciences, Stanford University, Stanford, California 94305 (S.R.L., S.L.S.)
Wall expansion in tip-growing cells shows variations according to position and direction. In Medicago truncatula root hairs, wall expansion exhibits a strong meridional gradient with a maximum near the pole of the cell. Root hair cells also show a striking expansion anisotropy, i.e. over most of the dome surface the rate of circumferential wall expansion exceeds the rate of meridional expansion. Concomitant measurements of expansion rates and wall stresses reveal that the extensibility of the cell wall must vary abruptly along the meridian of the cell to maintain the gradient of wall expansion. To determine the mechanical basis of expansion anisotropy, we compared measurements of wall expansion with expansion patterns predicted from wall structural models that were either fully isotropic, transversely isotropic, or fully anisotropic. Our results indicate that a model based on a transversely isotropic wall structure can provide a good fit of the data although a fully anisotropic model offers the best fit overall. We discuss how such mechanical properties could be controlled at the microstructural level.
Tip growth is the primary mode of morphogenesis for a wide range of cells including some prokaryotes (e.g. Streptomycetes), fungal hyphae, certain unicellular algae, pollen tubes, and root hairs (see articles in Heath, 1990  ). The shape of tip-growing cells is characterized by a long cylinder capped with a prolate dome. Evidence that cell wall expansion is limited to the dome was first obtained by Haberlandt (1887) and Reinhardt (1892) who, respectively, used starch grains and red lead particles to mark the cell surface. Since then, refined marking protocols and improved microscopic techniques have established two fundamental features of wall expansion during tip growth (Castle, 1958; Green, 1965; Chen, 1973; Hejnowicz et al., 1977; Kataoka, 1982; Shaw et al., 2000; von Dassow et al., 2001). First, the rate of wall expansion is distributed in a meridional gradient with a maximum at or near the pole of the dome and minimum at the equator. Wall expansion is therefore inhomogeneous. Second, over most of the dome surface wall expansion in the circumferential direction exceeds expansion in the meridional direction. Wall expansion is therefore anisotropic.
To explain these observations, we must consider the cellular control of wall expansion. Wall expansion is commonly interpreted as a mechanical deformation (Heyn, 1940
Ever since Reinhardt's work (1892), it is known that the meridional gradient of wall expansion observed in tip-growing cells does not reflect a similar gradient of wall stresses. In fact, the wall stresses predicted for the dome of the cell follow a gradient opposite to the observed gradient of expansion (Reinhardt, 1892
The control of expansion anisotropy was studied extensively in diffusely growing cells such as the Nitella internodal cell but not in tip-growing cells. In the cylindrical Nitella cell, the axial rate of wall extension is four times the circumferential rate of extension, whereas the axial stress is only one-half the circumferential stress (Green, 1963
We can ask whether the anisotropy of wall expansion in tip-growing cells depends, as for diffusely growing cells, on structural anisotropy in the cell wall. Studies of wall structure at the dome of various tip-growing cells have all reported a random orientation of cellulose microfibrils (Houwink and Roelofsen, 1954 ; O'Kelley and Carr, 1954; Belford and Preston, 1961; Newcomb and Bonnett, 1965; Chen, 1980; Kataoka, 1982). The lack of specific microfibril orientation suggests that the mechanical properties are isotropic in the plane of the cell wall. One implication of this observation is that expansion anisotropy in tip-growing cells may arise from stress anisotropy and not from directional reinforcement of the cell wall as observed in diffusely growing cells (Fig. 1A).
A recent investigation of tip growth in Medicago truncatula root hairs (Shaw et al., 2000
Our results are based on a reanalysis of data collected on four actively growing root hairs of the legume Medicago truncatula (Shaw et al., 2000 ). Cell numbers 1 to 4 in this paper correspond to cells D, C, A, and B in Figure 4 of Shaw et al. (2000). Using time-lapse imaging of root hairs labeled with fluorescent microspheres, it was possible to record the geometry of the cell and the expansion of the cell surface over a period of 30 to 45 min. Figures 2A and 3A illustrate the raw data on which this work is based.
A mechanical analysis of tip growth must begin with measurements of two types of variables: the relative elemental rates of wall expansion or strain rates (  ) and the turgor-induced wall stresses ( ). The strain rates and stresses are defined along three principal directions (s,  , and n) corresponding to the meridional, circumferential, and normal directions on a dome (Fig. 1B). Our analysis focuses on the strain rates and stresses in the meridional and circumferential directions only because current experimental protocols do not allow us to measure these variables in the normal direction. Inspection of the time-lapse sequences in Figures 2A and 3A reveals that the cell geometry and the rate of elongation are relatively constant. Given these two features, it is possible to pool the data collected over the entire growth sequence and to express the stresses and strain rates associated with tip growth in terms of only two independent variables: the meridional curvature ( s) of the cell and the meridional velocity of material points (v; see Eqs. 1 and 2, and 4 and 5 in "Materials and Methods" section).
The fiducial points marking the cell outlines (Figs. 2A and 3A) were used to compute
The wall stresses computed from the cell geometry (Eqs. 1 and 2) are shown in Figure 4, A and B. The meridional and circumferential stresses are identical at the pole of the cell. While the meridional stress shows only small deviations from its value at the pole, the circumferential stress first decreases slightly below the meridional stress and then increases steadily to reach, at the equator, a value that is nearly twice that of the meridional stress. In the cylindrical portion of the cell, where
The displacement of fluorescent microspheres was used to compute the v as shown in Figures 2C and 3C. These data were fitted with a function of the form
Strain rates were computed from Equations 4 and 5 and our fit of the velocity data for n = 3 (Fig. 4, C and D). The inhomogeneity and anisotropy of wall expansion mentioned in the introduction are here quantified. Near the pole of the cell, the two rates of wall extension are approximately 8%/min, while 5 microns away from the pole the circumferential and meridional rates have, respectively, dropped to 5% and 2%/min. An alternative, but equivalent, statement is that the apical region within 5 microns of the pole contributes 55% to 60% of the total increase in wall surface area. Wall expansion anisotropy also varies on the dome. The meridional extension rate can exceed the circumferential rate near the pole of the cell, although the most salient feature of the strain anisotropy are the long shoulders where wall extension in the circumferential direction is dominant.
The mechanical properties of the cell wall can be inferred from a comparison of the strain rates and wall stresses. We first note that the mean in-plane stress increases away from the pole, while the strain rates show an overall decline. This discrepancy suggests that there is a meridional gradient of mechanical properties. An estimate of wall extensibility is given by the ratio
The mechanical anisotropy that may be present in the cell wall defies any simple measurement and is best evaluated by comparison with explicit models of wall structure. We consider three different wall structures and their corresponding set of mechanical properties (Fig. 6A; Table II). The first model is a wall with randomly oriented cellulose microfibrils. Given the lack of structure in such a cell wall, we may expect isotropic mechanical properties between all three principal directions. The second model is inspired by observations of wall structure in tip-growing cells where cellulose microfibrils are organized into layers parallel to the cell surface but have no preferential orientation within a layer (Fig. 6A). This structure emerges naturally from wall assembly since cellulose microfibrils are synthesized by enzyme complexes that travel along the cell membrane. This wall organization suggests that the mechanical properties can be isotropic in the cell wall plane but that they are likely to differ from the mechanical properties in the direction normal to the cell surface. We shall refer to this model as transverse isotropy. Finally, we consider also a fully anisotropic cell wall with different mechanical properties in the three principal directions. This type of cell wall is characteristic of diffusely growing cells such as the Nitella internodal cell and reflects a highly organized deposition of cellulose microfibrils within the plane of the wall (Fig. 6A).
The models impose specific constraints on the pattern of strains that can be observed. To visualize these constraints, we must first introduce two ratios: the strain anisotropy,  and the stress anisotropy,  These two ratios define a space of possible anisotropy (Fig. 6B). As indicated in Table II, the strain and stress anisotropy for an isotropic cell wall must lie on a straight line in the anisotropy space. Strain and stress anisotropy for a transversely isotropic cell wall can cover a larger subset of the anisotropy space (shaded area in Fig. 6B). Finally, an anisotropic cell wall can lead to any combination of stress and strain anisotropy. The observed strain and stress anisotropy for cell number 1 and cell number 2 were plotted in the anisotropy space (Fig. 6B). The data deviate from the line corresponding to a fully isotropic model and also slightly from the region accessible to a transversely isotropic model. The deviation near the origin indicates that meridional stiffening is present in the plane of the cell wall. For comparison, we also included stress and strain anisotropy measurements for the Nitella internodal cell and Hydrodictyon, both of which show strong structural anisotropy in their wall. The measurements lie squarely outside the subspace accessible to an isotropic or transversely isotropic model confirming that these simpler structural models offer a poor description of the mechanical anisotropy present in the wall of these cells. To go beyond the qualitative conclusions that emerge from inspection of the anisotropy space, we also fitted the velocity data directly with functions compatible with the three structural models (see "Materials and Methods" section). The simplest model that offers a good quantitative fit can be retained as the best candidate for the wall structure. Goodness-of-fit was measured with the SSR associated with each model (Table III). As already hinted by the plot of Figure 6B, a fully anisotropic model offers the best fit overall. Moreover, F-tests indicate that the improvement over the transversely isotropic model is statistically significant for all cells. We note that the fit for an anisotropic model is the same as the free fit used in the previous section. The strain rates of Figure 4, C and D thus correspond to the anisotropic structural model. We have also determined the strain rates for a transversely isotropic structural model (Fig. 4, E and F). Comparison of the strain rates for the two models indicates that all of the key features of the expansion patterns are retained in the simpler, transversely isotropic model.
The data presented in Table III can also be used to determine the fraction of the total strain anisotropy explained by a transversely isotropic model. This model has no structural anisotropy in the plane of the wall and thus only stresses can contribute to anisotropy in the expansion of the cell surface. The strain anisotropy present in our data is the amount of variation that cannot be explained by a model with isotropic strains minus the variation that cannot be explained at all (i.e. the variation due to noise). Consequently, a measure of the total strain anisotropy is the SSR for a model with isotropic strains (i.e.  ) minus the SSR for the fully anisotropic model that corresponds to the best fit of the data. The latter SSR value is the variation, or noise, not accounted for by any reasonable model. Taking cell number 1 as example, we find that the amount of variation that can be imputed to strain anisotropy is 0.528 – 0.141 = 0.387 (µm/min)2. On the other hand, the reduction in SSR for the transversely isotropic model is 0.528 – 0.202 = 0.32 (µm/min)2. Thus, we can infer that a transversely isotropic model accounts for 83% of the total strain anisotropy. Similar calculations for cells number 2, number 3, and number 4 give values of 99%, 73%, and 89%, respectively. We conclude that a transversely isotropic model offers a good qualitative and quantitative fit of our data, although the anisotropic model provides the best fit overall.
The strain pattern observed in M. truncatula root hairs is similar to that reported for the Chara and Nitella rhizoids (Chen, 1973 ; Hejnowicz et al., 1977) and for Phycomyces (Castle, 1958). These distantly related organisms show the same predominance of meridional strain rate in the most apical region of the dome with a wider subapical region where the circumferential strain rate exceeds the meridional strain rate. The wide range of cells sharing this strain pattern suggests that it is a fundamental feature of the tip growth process, one that transcends known differences in the chemical composition of the cell wall. With our analysis, we can propose a basis for the wall expansion pattern observed in M. truncatula.
Root hairs and other tip-growing cells show a strong meridional gradient of wall expansion. Using measurements of strain rates and stresses, we were able to infer the wall extensibility necessary to maintain tip growth (Fig. 5). Gradients of wall extensibility have often been postulated but their profile was never quantified. Our measure of wall extensibility, although not definitive, is indicative of the type of mechanical inhomogeneities that can be expected in the wall of tip-growing cells. In particular, one might ask whether the sharp transition between regions of high and low wall extensibility mimics the distribution of wall secretion or some wall components involved in regulating expansion. Such spatial correlations between mechanical properties and wall composition are a key first step toward elucidating the control of wall expansion at the molecular level.
Wall expansion anisotropy can arise from only two sources: anisotropy in the stresses driving wall expansion or anisotropy in the mechanical properties of the cell wall. Green and King (1966)
The fits with different structural models confirms that a wall with transversely isotropic mechanical properties can provide a good fit of our data (Table III). However, even if the fit is in some cases very good (e.g. cell no. 2), this does not prove that the mechanical properties are transversely isotropic since many fully anisotropic models could have generated the same data. We have also found evidence for a nonnegligible amount of mechanical anisotropy in the plane of the cell wall. For all the cells studied, the pattern of wall expansion is suggestive of meridional reinforcement (Fig. 6B). Studies of wall texture at the dome of tip-growing cells did not report such reinforcement (O'Kelley and Carr, 1954 Table III indicates that a transversely isotropic model offers a substantially better match for the data than a fully isotropic model. Since the only difference between these two models is the possibility of anisotropy between the plane directions and the normal direction, one may wonder why this type of anisotropy affects in any way the performance of these models in fitting the strain anisotropy present in the plane of the cell wall. The reason is that the deformations along the three principal directions are coupled. This coupling, known as a Poisson's ratio effect for elastic materials, can be demonstrated by stretching a material along one direction leading invariably to contraction in the transverse directions. Therefore, a mechanical change that alters wall deformation in the normal direction will also affect, via this coupling, deformation in the plane directions. It is interesting to note that a type of anisotropy that emerges naturally from the deposition of microfibrils into layers may in itself affect expansion anisotropy in the plane of the cell wall.
Our findings suggest some important differences between cell morphogenesis by tip growth and diffuse growth. First, tip growth requires a strong gradient of mechanical properties, while such gradients are not necessary for diffuse growth. Gradients imply polarity and thus one key aspect of tip growth is the maintenance of cell polarity (Bibikova et al., 1999
Microscopic Techniques and Numerical Analysis
All observations were made on actively growing root hairs of the legume Medicago truncatula Gaernt. cv Jemalong. Each cell was imaged at 1-min intervals using differential interference contrast (DIC) and fluorescence microscopy as detailed in Shaw et al. (2000) The numerical analysis of the data was based on algorithms written with Matlab (version 5.2; The MathWorks, Natick, MA). All programs are available from J. Dumais (jdumais{at}oeb.harvard.edu). The final versions of graphs and other illustrations were prepared with Adobe Illustrator (Adobe Systems, San Jose, CA).
The outline of the cell was determined from DIC images by selecting 17 to 40 fiducial points on the cell boundary. Of these fiducial points, at least 15 were located within ±15 µm of the pole where nearly all of the wall expansion takes place. Therefore, on average, these points were spaced by 2 µm. The
The mean cell geometry for the whole sequence was determined by fitting the curvature recorded at the fiducial points with the function To obtain continuous cell outlines, the fiducial points were interpolated with smoothing splines (function "csaps" in Matlab). The splines were then used to specify 1000 evenly spaced points around the cell. Figures 2A and 3A show these outlines at a 3-min time interval. The osculating circle at a given location on the cell outline was used to define the surface normal at that location. Finally, the growth axis was defined as the curve, among the family of curves orthogonal to the cell outlines, that best followed the approximate pole of the cell over the entire sequence (Figs. 2A and 3A).
The stresses acting on an axisymmetric surface loaded by internal pressure are given by the following equations (Hejnowicz et al., 1977
 is the wall thickness, and s and are the meridional and circumferential curvatures, respectively. Since an axisymmetric dome is fully determined by one of its meridians, it is possible to write as a function of s; that is,
 is a dummy variable of integration. The stresses were computed using a wall thickness () of 0.1 µm (Schröter and Sievers, 1971) and a turgor pressure (P) of 0.7 MPa (Ekdahl, 1953; Lew, 1996).
For the computation of the meridional velocity, we selected microspheres that were close to the cell outline (mid-plane of the cell). Nevertheless, the displacement of some microspheres did not follow exactly the outline of the cell, indicating that the surface markers were either slightly above or below the mid-plane. Before microspheres could be used as material points, their positions had to be projected onto the cell outline (Figs. 2A and 3A). The projection was done by first finding the position of the microspheres on the first and last cell outlines, here defined as the point on these outlines closest to the first and last microsphere positions, respectively. The original microsphere trajectories were then fitted with a polynomial and "stretched" between the two new endpoints. The intersections of the new trajectories with the cell outlines were taken as the microsphere positions. For each microsphere, the meridional distance from the pole was computed and the derivative of this distance with respect to time gave the meridional velocity (Figs. 2C and 3C).
Inspection of Figures 2C and 3C reveals that the meridional velocity approximates a sine function. We can thus suggest as basic functional form to fit the velocity data the function:
To determine the appropriate degree of the polynomial Pn, we fitted the velocity data for increasing values of n. For each fit, we computed the sum of the squared residuals with the equation:
 is the significance level (here = 0.01), and the degrees of freedom, df, equal the number of data points (N) minus the number of free parameters (p) in the full model. A conservative estimates of the critical F value is F1,100,0.01 = 6.90.
For a steady rate of elongation, the strain rates can be expressed directly in terms of
  and 
Three structural models are considered: isotropy, transverse isotropy, and anisotropy (Fig. 6A). To test the compatibility of these structural models with our data we need first to adopt a set of constitutive equations that relates the wall strain rates, stresses, and mechanical properties. Several constitutive models have been suggested for the plant cell wall (Lockhart, 1965a
The matrix on the left-hand side of Equation 6 represents the wall strain rates measured in the two principal directions. The strain rates are equal to the product of a matrix of stresses (
To illustrate how the three structural models constrain the pattern of strain rates we define two ratios: the strain rate anisotropy, The strain anisotropy and stress anisotropy reported in Figure 6B correspond to global fits of the velocity and curvature data (i.e. the left and right sides of the cell were not fitted independently). This approach was adopted because the strain anisotropy is sensitive to fluctuations when the strain rates approach zero such that small variations in the measured strain rates can lead to large variations in the calculated strain anisotropy. A global fit increases the precision of our strain measurements by including more data points into a single fit but has the disadvantage of hiding some of the variation present in the data. Therefore, the global fit was used only for illustration within Figure 6B, while the quantitative data presented in Table III were based on independent fits for the left and right sides of the cell (see below).
This section describes how the velocity data can be fitted such that the resulting strain rates are compatible with one of the three structural models described above. The fitting function must be such that the constitutive constraints in Table II can be enforced. We selected a fitting function based on the exact solution for an isotropic wall material. The isotropic solution can be found by substituting isotropic mechanical properties in Equation 6 (i.e. ass = a
 is a function of tip geometry only, and is a dummy variable of integration. A Taylor series expansion of the exponential term in Eq. 10 gives a polynomial in  :
. We are looking for a fitting function that, under its most constrained form, reduces to Equation 11 and in its most general form represents an arbitrary function. We can thus suggest as basic functional form to fit the velocity data the function: 
In the solution for the isotropic model, the coefficients ci are all determined by the derivatives of G(
Finally, the three models above are compared to a reference model that has no strain anisotropy. Equating the meridional and circumferential strain rates leads to the equation:
A Taylor series expansion of the exponential term reduces this equation to the generic fitting function given above. The statistical analysis of the structural models used Equation 3 and the approach described above to obtain the best estimate of the strain rates. Received March 29, 2004; returned for revision July 15, 2004; accepted August 4, 2004.
1 This work was supported by the Center for Computational Genetics and Biological Modeling, Stanford University (studentship to J.D.), by Dr. L. Mahadevan at the University of Cambridge (a research fellowship to J.D.), and by the U.S. Department of Energy (grant no. DE–FG03–90ER20010 to S.R.L.).  Article, publication date, and citation information can be found at www.plantphysiol.org/cgi/doi/10.1104/pp.104.043752. * Corresponding author; e-mail jdumais{at}oeb.harvard.edu; fax 617–496–5854.
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in 
where 
is a polynomial whose degree n should be selected carefully. Increasing the number of terms in 
improves the fit of the data and can ultimately lead to a perfect fit if the number of free terms equals the number of data points. This fit would, however, be meaningless given the noise in the data. Therefore, we need to determine the maximum power of the polynomial that has statistical significance. This was done by recording the sum of squared residuals (SSR) associated with different least squares fits of the data (
4) with significant F values are associated with noise in the data. 
As illustrated in 
= 3
(dotted line), while those for a transversely isotropic material must lie in the shaded area. The stress and strain anisotropy for a fully anisotropic material can occupy any point in the anisotropy space. The measured stress and strain anisotropy for cell number 1 and cell number 2 are shown (solid lines). The approximate meridional positions where the strain rates and stresses were measured are indicated. It can be seen that the data deviate from the region of transverse isotropy near to origin. The strain and stress anisotropy for the transversely isotropic model that best fit the data are shown (dashed lines). The strain and stress anisotropy observed in Nitella and Hydrodictyon (Green, 1963
where 
is the straight distance between pi–1 and pi+1 and 
is the interior angle between the vectors joining pi to pi–1 and pi+1. 
where 
/2, two basic geometrical constraints for the tip geometry. The curvatures on the "right" and "left" sides of the cell were fitted independently to emphasize potential asymmetries in the cell geometry such as the slight turn for cell number 2 (
where 
Further justification for the form of the fitting function is given in the next section. The least squares fits of the velocity data were performed for the two sides of the cell independently to reveal variations within the cell. However, two constraints had to be imposed to ensure that the resulting fits were physically sound. First, the value of c0 on either side of the cell must be the same to yield compatible strain rates at the pole. Second, the coefficient c1 must be zero so that the strain rates at the pole are either maxima or local minima. 
where vi is the ith velocity measurement and 
is the corresponding velocity predicted by the fitting function. According to Snedecor and Cochran (1967)
and the stress anisotropy, 
)/(
and 
must be equal when 
where 
and since 
Substitution of the fitting function in this equation gives the following constraint for a transversely isotropic material: 
