These pages contain further analyses of data, additional figures, and a mathematical appendix for the Planta paper.

Relationships between growth, morphology and wall stress in the stalk of Acetabularia acetabulum,

by Michelangelo von Dassow, Garret Odell, and Dina F. Mandoli.

Part one describes the relationship between growth and circumferential curvature in the stalk apex, and the presence of slow growth along the flanks of the apex.

Part two describes further analyses of the cell wall thickness, shape and stress.

Part three describes where the whorl-hairs initiate relative to the stalk tip.

Part four describes an alternate method of analysis of data on the relationship between curvature and growth in the cut-interwhorl.

Part five presents data on the anisotropy of growth in the cut-interwhorl.

Part Six is a summary Figure.

Part Seven is an appendix describing more fully the mathematics used here and in our paper. Since this equation-laden appendix does not display in html, we provide an adobe acrobat pdf format file for it. This appendix includes original figure 5.


Part One: Apex growth

Because the tip grew fastest (when expressed either in relative or absolute units: Fig. 1), and was the narrowest part of the apex (von Dassow, et al), we wondered if growth rate was inversely proportional to stalk width. Indeed when initial stalk width was <100 µm, relative growth in width declined as the initial width increased (Fig. 2). For initial stalk width >100 µm, there was little relative growth in width over 1 h. This analysis complements von Dassow, et al, in which we measured relative growth as a function of position apex but did not address the role of curvature and growth in the apex.

Fig. 1 Absolute growth in stalk width at one landmark versus the initial distance along the meridian from the tip to the landmark. (Original figure 3C)

Fig. 2. Growth over 1 h in the new apex was correlated with the width (n = 11). Relative growth in stalk width versus the initial stalk width. Diagrams indicate how measurements for each axis were made. (Original figure 4)


To determine if the flanks of the apex also grew, which is not apparent over 1 hr (von Dassow, et al's Fig. 3) we tracked carbon particles positioned >110 µm along the meridian from the tip over a longer (8 h) interval. To distinguish growth along the flanks of the apex from that in the cut-interwhorl, we only analyzed plants that grew for 8 h without initiating a new whorl of hairs (n = 5). Hence, this analysis may exclude faster growing apices. All regions of the flanks of the apex grew both in width and meridional length. Significant relative growth in width did occur beyond 100 µm from the tip (0.25 ± 0.019 between 110-250 µm from the tip 0.057 ± 0.010 between 250-600 µm from the tip). Relative growth along the meridian between landmarks initially >110 µm from the tip occurred over 8 h (0.061 ± 0.007). Together, these results mean that the flanks of the apex grew slowly both in width and along the meridian over 8 h even though they did not appear to be growing at all over 1-2 h (von Dassow, et al).




Part Two: Wall thickness, stress and growth

To compare the cell wall thickness of regions that differed in relative growth, we compared a region 0- 30 µm from the tip to a region 70-100 µm from the tip. In live plants, the domed top was 50-62 µm in width so that the region 0-30 µm was at the domed top whereas the region 70-100 µm was on the "cylindrical" flanks (von Dassow, et al). Data of individual plants were pooled within each region (von Dassow, et al's Fig. 4). We compared means in each plant for each region for the population (Fig.3A). Finally, we took the ratio of the means for each individual plant and then analyzed the ratios within the population (Fig. 3B). Our choice of regions minimized the effects of the discontinuity inherent in our approximation of shape.

The mean ratio of cell wall thickness at 70-100 µm to that 0-30 µm from the tip was 1.99 ± 0.21 (Fig. 3), the minimum value required if our hypothesis were true (the median ratio was 1.68 (range: 1.28 to 4.05; Fig. 3)). Given the optical artifacts inherent in our methods, these results are consistent with the hypothesis that growth rate was solely a function of the cell wall stress.


Fig. 3 A, B. Cell wall thickness in wild type A. acetabulum (n = 14). Measurements within each region of the same image of the same cell wall ghost were averaged to compare the cell wall thickness within 30 µm of the tip to that 70-100 µm from the tip. Average cell wall thickness measurements from different images of the same cell wall ghost were then averaged to get one number per cell wall ghost. The exact position and number of measurements varied from plant-to-plant and image-to-image so that most measurements did not cover the full range from 70-100 µm. Panel A. Cell wall thickness 0-30 µm and 70-100 µm from the tip along the meridian. Panel B: The ratio between cell wall thickness from 70-100 µm from the tip to the cell wall thickness 0-30 µm from the tip. In these box plots, the top and bottom lines and the line through the box mark the 75th (top quartile), 25th (bottom quartile) and 50th (median) respectively. The whiskers on the bottom and top of the box extend to the 10th to the 90th percentile (the outer deciles). The symbols in the box mark the means. Diagrams indicate how measurements for each axis were made. (Original figure 8)



Several methods can be used to calculate curvature from the actual shape [Silk, 1978 #66; Hejnowicz, 1977 #67]. We assessed tension as a function of position by calculating the curvature at different positions on a single cell wall ghost. We traced the outer boundary of the cell wall using an "NIH Image" macro, and fit a curve to this boundary using "Mathematica." We then used the resulting curve and the data on cell wall thickness from that one plant to calculate stresses. While meridional stress was nearly constant from 0-100 mm, circumferential stress rose steeply across the hemispherical top of the stalk apex, i.e. in the first 25 mm back from the tip (Fig. 9B). Note that circumferential and meridional stresses were similar at the tip. These results differ from those obtained using a simplified geometry (von Dassow, et al). However, the cell wall ghosts were clearly more pointed near the tip than live plants. The change in apex shape induced by bleaching may account for the difference in results obtained by these two methods of analysis.


Fig. 4 Stress versus distance along the stalk midline in a single cell wall ghost. We calculated stress from measurements of curvature (as determined from a polynomial spline fitted to the actual shape of this cell wall ghost) and measurements of cell wall thickness. Calculated for a pressure of 2.25 Bar [Wendler, 1983 #70]. The dashed line indicates the profile of the apex of the plant used. (Original figure 9B)




Part Three: Hair initiation occurs near the tip.

Both kinds of whorls, rings of hairs or gametophores, initiate near the stalk tip in this species [Schmid, 1987 #45; Kratz, 1999 #48]. In plants not coated with carbon particles, nascent hairs initiated 39 ± 2.6 µm from the tip after the domed top flattened (n = 7). In plants coated with carbon particles, nascent hairs initiated 50 ± 2.2 µm from the tip (n=16), about 20% further away from the tip than in uncoated plants. The difference between these values is statistically significant (P=0.05, two-sided Wilcoxon Rank Sum Test). Initiating whorls of hairs were about as wide as the region that grew rapidly during apex extension (von Dassow, et al), situated approximately where the cell wall began to thicken in younger apices (von Dassow, et al).




Part Four: Curvature and growth

In comparing circumferential curvature to growth in width, we used initial width to calculate both circumferential curvature and the relative growth in width (von Dassow, et al). It is preferable to use a dependent variable that does not include the independent variable as a component [e.g. \Niklas, 1994 #50]. The differential equation described above is also solved if the width after some time interval, W, is a linear function of the width at the beginning of that time interval, W0 (Appendix). The curve described by this function has a positive y-intercept and a slope between 0 and 1, and so will intersect the curve given by W = W0, i.e. stalk width asymptotically approaches some maximum. As predicted by the model, when we plotted initial width versus width after 8 h for positions along the cut-interwhorl and versus the width over consecutive 4 h intervals at the whorl, both sets of data fell on straight lines with slopes between 0 and 1 (Fig. 5). A linear least squares regression of these data (Table 1) had positive y-intercepts and slopes between 0 and 1. We concluded from these data that at each moment, relative growth rate in width was a linear function of circumferential curvature.

This relationship predicts the width that the stalk approaches as it grows. Using this regression analysis comparing initial width to the width at some later time, final stalk width equals a point on the regression line where width no longer changes. Predicted final stalk width was 369 µm (one position, consecutive time intervals) or 391 µm (multiple positions, one time interval) (Table 1). This is similar to the stalk width measured in reproductive and vegetative plants (Table 1 in von Dassow, et al) [Nishimura, 1992 #2]. The results of this analysis are consistent with those in von Dassow, et al, and the hypothesis that the curvature of the cell wall regulates growth.

Fig. 5 A, B. The relation between the initial stalk width (diameter) and stalk width after a time interval. Panel A: Stalk width at different landmarks along the cut-interwhorl after 8 h versus the initial stalk width at those landmarks (n=11). Panel B: Stalk width at the base of the new whorl of hairs after consecutive 4 h intervals versus the stalk width at the base of the whorl of hairs at the beginning of each 4 h interval (n=8). The dashed line indicates the line of no growth. A linear least-squares regression of all data for all plants has been plotted. The intersection of these two lines (arrow) indicates the maximum width the stalk approaches. The diagrams indicate the how measurements were made for each axis. (Original figure 12)

Table 2: Regression parameters for the growth in width versus the circumferential curvature.


"A", µm/hr


"Wf", µm

Along the interwhorl

(over 8 h, n = 11 )

9.93 ± 0.395

0.982 ± 0.00582

391 ± 18.8

At the whorl over time

(over 4 hours, n = 8 )

16.62 ± 2.56

0.962 ± 0.0177

369 ± 69.8

The parameters obtained from individual plants were pooled to produce these means ± standard errors of the means.

Based on linear least squares regression of width versus initial width at that point in individual plants.


Part Five: Anisotropy

Growth in the cut-interwhorl appeared to be anisotropic (Fig. 11). The anisotropy can be calculated from the data on relative growth (Appendix). We calculated that the ratio of the relative growth rate in width to that along the meridian was 1.42 (population median of medians for data from individual plants; 1.31 and 1.52 first and third quartiles).




Part Six: Summary Figure

Fig. 6. Summary of growth patterns in the new apex and in the cut-interwhorl of A. acetabulum documented here and in von Dassow, et al. Growth along the meridian and in width are indicated by gray and black arrows respectively. Numbers associated with the arrows are approximate relative growth rates expressed in %/h for each position and give the range of values at each position. The asterisk marks the real position of the measurement indicated by the arrow and the associated numbers directly below it: only this arrow and these numbers were displaced vertically to avoid obscuring the shading on the apex. (Original figure 13)