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Anatomy and Morphology |
2Department of Plant Biology, Cornell University, Ithaca, New York 14853-5908 USA; 3Instituto de Ecologia UNAM, Apartado Postal 1354, Hermosillo, Sonora CP83000, Mexico
Received for publication September 27, 2002. Accepted for publication December 10, 2002.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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Key Words: biomechanics • Cactaceae • flexural rigidity • procumbent stem growth • stem bending • Stenocereus
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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, 1989
; Gibson and Nobel, 1986
) (Fig. 1A–B). This study was motivated by the suggestion that S. eruca is a recent evolutionary derivative species of S. gummosus (Gibson, 1973
, 1989
) and by our field observations, which indicate that decurrent branches of S. gummosus often touch the ground, develop adventitious roots, and may rarely detach to establish an independent existence. Thus, the difference in growth habit between these two species may have evolved recently and could reflect comparatively minor genomic changes affecting physiological or developmental modifications (e.g., hormone-mediated plagiotropic growth) that altered the ability of stems to grow vertically.
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, p. 106) contrasts the "rigid woody cylinder in the young stems of S. gummosus (Gibson, 1971)" with the "almost total absence of xylem strength in even the oldest stems of S. eruca (Gibson and Nobel, 1986
)" and further postulates that the adventitious roots of S. eruca are contractile and "eventually pull the heavy raised [distal portion of the] shoot to the ground" (italics added; see Gibson, 1989
, p. 105). Thus, although hormone-mediated developmental features may account for the procumbent posture of S. eruca stems, Gibson postulates that anatomical factors, including the mechanical properties of stem tissues and adventitious roots, are proximate causes of the differences in the growth habits of these two species. Here, we examine this hypothesis and show that, with the exception of very old portions of S. gummosus stems, the principal stem stiffening agent for both species is a peripheral (collenchymatous) tissue complex that is mechanically insufficient to support the weight of S. eruca stems because of its low breaking stress and extensibility. We also show that the volume fraction and mechanical properties of wood are not germane to the issue of the mechanical stability. Additionally, we find no evidence for contractile adventitious roots for S. eruca. Nonetheless, our observations are consistent with Gibson's hypothesis that the procumbent growth habit of S. eruca is the result of numerous developmental modifications, among which some affected the mechanical properties of stem tissues.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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) and occurs primarily on sandy flats and dunes near the Pacific Ocean west of Villa Constitución (24–26° N latitude), Baja California Sur (see Gibson, 1989
, p. 104; Turner et al., 1995
). Three plants were collected for anatomical and mechanical study from each of two populations selected for their contrasting habitats: one population growing in unconsolidated sand dunes, located in the coastal region of Estero Salinas (24°36.18' N, 111°46.68' W) and another growing in consolidated sandy soils, located approximately 20 km from the coast near San Carlos (24°54.68' N, 111°59.47' W) (see Clark-Tapia, 2000
). A sample of 173 stems from an additional 91 plants (43 from Estero Salinas and 48 from San Carlos) were also examined morphometrically. Within each population, groups of 3–5 healthy plants were systematically sampled in a random pattern. For each plant, we counted stem number, diameter, length, angle of elevation, height of the elevated apex, and number of ribs (Fig. 2). Stem diameters were measured to the closest millimeter using a caliper at three locations along the length of each stem: basal, mid-length, and 10 cm below the apex. Stem length (accounting for curvature) was measured to the closest millimeter with a flexible pocket rod. The number of ribs was the actual number seen in the area near the apex.
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, p. 104; Turner et al., 1995
). We selected two populations for study: San Carlos (24°54.68' N, 111°59.47' W) in Baja California and Punta Onah (29°05.10' N, 112°09.47' W) in Sonora. Three specimens of S. gummosus from Punta Onah were collected for mechanical and anatomical examination. We also examined 54 stems from 16 plants from San Carlos to contrast their morphometry with those used previously to construct a data set for Punta Onah (see Molina-Freaner et al., 1998
). In each case, S. gummosus plants were measured for stem length, stem diameter, and inclination using the same protocols as are described for S. eruca plants.
Mechanical tests
The objective of these tests was to determine the flexural rigidity, EI, of intact stems and surgically isolated anatomical regions. Here, E is Young's elastic modulus, a measure of the ability of a material to resist bending forces, and I is the second moment of area, a measure of the contribution of cross-sectional area, geometry, and shape to the ability to resist bending forces (Niklas, 1992
).
Each of the six stems was cantilevered horizontally or placed in three-point bending and deformed to the breaking point using an Instron 4502 Testing Machine fitted with a blunt-edge loading point (Fig. 1C–E). Individual stem segments (measuring between 15 and 25 cm in length) were also removed in a distal to proximal sequence along the length of each stem and tested in three-point bending to their breaking point (Fig. 1F).
The cross sections at breakage points were photographed digitally to provide measurements of the areas and locations of the different anatomical regions. The I of stems was calculated based on measurements (from digital photographs) using a computer program designed for this purpose (see below). Breaking stresses 
B were calculated using the formula B = MD/2I, where M is the bending moment and D is stem diameter; E was calculated using the formula E = mL3/3I, where m is the slope of the force vs. deformation graph and L is the effective bending length.
Each stem segment was then cut into a minimum of five beams of length L for additional bending or tensile tests (Fig. 2B). Beam side dimensions were between 5 and 20 mm such that the slenderness ratio 
15. Beams were vertically supported at each end and loaded at their mid-length with a deformation rate of 15 mm/min. Beams removed from stem perimeters were tested in bending such that their external surface was first placed in compression and then in tension. Beams removed from the cortex or from the central woody stem core were not reoriented once tested.
Beam testing terminated in the elastic range of tissue mechanical behavior, i.e., no permanent deformations or breaks were noticed. Each beam was subsequently labeled and fixed in FAA for anatomical study (1 part formalin, 1 part glacial acetic acid, 18 parts 50% or 70% ethanol). The E for each beam was calculated using the formula E = mL3/(4wh3), where m is the slope of the secant line (the best fit regression curve) of the force vs. displacement graph, L is the unsupported length of each beam (see Niklas, 1992
; Beer et al., 2001
), and w and h are the beam transverse dimensions.
Strips of peripheral tissues, beams through the stele and cortex, and S. eruca adventitious roots were also vertically tested in tension to failure by clamping both ends of each specimen covered in a fine-grain sandpaper to prevent slippage. The E and B of each specimen were calculated using the formulas E = FL/
A and B = F/A, where F is the tensile force, L is the free length of the strip, is deformation, and A is cross-sectional area. Data are reported only for specimens that broke near their mid-length and for which no slippage occurred (as gauged by visual inspection of the force vs. displacement graph and the condition of clamped ends after testing).
For purposes of comparison, tissue samples from potato, celery, and cucumber were tested in bending or tension to determine their E as described above.
Anatomical studies
Materials for anatomical studies were those tested mechanically and fixed in FAA. Slides were prepared from sections cut by hand, by sliding microtome, or in a cryostat, depending on their purpose, and viewed with or without brief staining in toluidine blue at acid pH (OBrien and McCully, 1981
). Estimates of percentage wall area represented in sections of collenchyma required thin sections because of the irregular shapes of cell lumens, and these were cut at 10 µm in the cryostat. Quantitative evaluations on wood development and on the characteristics of collenchyma were made on one stem each of ca. 1 m length for S. eruca and S. gummosus so that we could compare anatomy of the two species at equivalent distances from the shoot tip. The trends in the differences were confirmed by qualitative examinations of additional specimens.
Photographic records were obtained using an image capture board and image processing program under the control of a microcomputer. Quantitative measurement of percentage area represented by walls were made by weighing separately the walls vs. lumens cut from plain-paper prints of collenchyma cells at 750x magnification. The error contributed by ink to positive images was evaluated at <1% by comparing images with reversed contrast. The percentage of the axial system represented by fibers vs. vessel elements and parenchyma was calculated similarly. Collenchyma thickness was measured on images of 120x final magnification, validated by confirming some of the measurements at twice this magnification. The choice of the lower magnification for broad comparisons was a compromise necessitated by the extensive lengths of section surface that had to be surveyed with nearly continuous sampling to determine how thickness varied with distribution over the ribbed surface of the shoot. These surveys were made with measurements at 0.1 mm intervals for distances up to 14.0 mm.
Lignification of tissues was judged using phloroglucinol-HCl. Statistical comparisons of wall fractions were made using simple t tests. Comparisons of the thickness of collenchyma were made using the Tukey-Kramer HSD.
Computer simulations
To determine the contributions made by different anatomical regions to the total (bulk) stem E, we developed a computer program written in MATLAB (MathWorks, Natick, Massachusetts, USA) (see Appendix). The program was conceptually similar to the computational protocols developed by Speck and colleagues (e.g., Speck et al., 1990
; Speck, 1994
).
The location and quantity of anatomical regions were based on measurements taken on stem surfaces broken in bending. The E of different anatomical regions was the mean of each region determined for each stem. Values of predicted E and I were then compared to those observed for each bending test to assess the reliability of the program.
A series of computer simulations were subsequently used to determine the influence of different anatomical regions (ribs, wood, and cortex) on the capacity of intact stems to resist bending. In these analyses, a range of geometries and absolute sizes for stem ribs and the cortical region was used; the E for each anatomical region was held constant.
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RESULTS |
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, Gibson and Horak (1978)
, and Gibson and Nobel (1986)
for shoots of Stenocereus held in general for the present materials. The shoot of either species attains its bulk by primary thickening behind the apical meristem. The primary plant body includes a broad parenchymatous pith, a vascular cylinder of collateral bundles, a very broad parenchymatous cortex, a collenchymatous hypodermis, and a generally simple epidermis. Together the hypodermis and epidermis form a peripheral complex that can be thought of as the "skin" of the axis. We concerned ourselves with a strategic comparison of the vascular cylinder and skin in S. eruca vs. S. gummosus because anatomical variations in these two histologically complex regions were likely to be significant for biomechanical differences between the two species.
Vascular cylinder
In transverse sections vascular bundles were arranged in a ring separated into eight major groups by periodic wide parenchymatous gaps (Fig. 3A). At 10 cm from the tip of the plant, the vascular tissue was in the early stages of secondary growth (Fig. 3A), and the degree of development of xylem was essentially the same in the two species. But the bundles of S. gummosus were more compactly arranged within groups than was the case in S. eruca, many being in contact at the metaxylem region of the bundles. In S. eruca, phloem fibers were maturing in the primary phloem opposite a minority of the protoxylem points. Mature protophloem fibers were not a prominent feature of the sections taken at 10 cm in S. gummosus.
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At 100 cm from the shoot tip xylem fibers remained the minor component of area of axial system exposed in section in S. eruca (12%; Fig. 3B), and the major component in S. gummosus (73%; Fig. 3C, F). Vessel elements of S. gummosus had greater coverage of primary wall by secondary wall than those in S. eruca. Protophloem fibers were a regular occurrence opposite protoxylem poles in both species. The radial extent of accumulation of secondary xylem in S. gummosus was approximately two times that of S. eruca. At this distance in S. gummosus the large gaps between the original groups of bundles were mostly closed (Fig. 3D), whereas the wide gaps in S. eruka remained essentially unchanged (Fig. 3E). In S. gummosus, ray cells were lignified in all rays, whereas ray cells were not lignified in any rays of S. eruca.
In summary, the vascular tissue system in S. gummosus engaged in more vigorous secondary growth with much greater development of woodiness than was found in S. eruca. The secondary xylem was very rigid at and near the base of the shoot of S. gummosus, but remained flexible throughout the length of the shoot in S. eruca.
Epidermis and collenchyma
In both S. gummosus and S. eruca the epidermis was uniformly one cell layer in thickness and the underlying collenchyma was two or more cell layers in thickness. Cells of the collenchyma lacked the chloroplasts found in abundance in the thin-walled chlorenchyma of the cortex. The latter were elongated radially, except that a more or less continuous layer of cells of short radial dimension formed a boundary with the collenchyma. While these cells typically matured as chlorenchyma, it appeared that, at locations where the collenchyma was especially thick, cells of this layer matured as collenchyma. The typical collenchyma cells were somewhat irregular in shape in paradermal view, with no preferred axis of elongation among a population of cells except where they were wrapped around the substomatal cavities.
We did not detect differences in the epidermis between species. In the collenchyma we found differences in the wall fraction and the thickness of the collenchyma layer. At 25 cm from the shoot tip, where the collenchyma was "young" but mature, the percentage cell wall area observed in transverse section tended to be higher in S. gummosus than in S. eruca (79.6 ± 1.3% vs. 74.3 ± 0.9% dorsal side and 76.3 ± 1.2% ventral side, means ± 1 SE). In S. eruca at 75 cm from the shoot tip, the wall fractions on dorsal and ventral sides did not differ significantly from those found at 25 cm.
The number of cell layers within the collenchyma and the absolute thickness of the collenchyma layer varied in a complex fashion. Again at 25 mm from the shoot tip, in both S. gummosus and S. eruca, collenchyma was several cell layers thick at the centerline of the sinus (Fig. 3G). In S. gummosus, the thickness at this location (0.148 mm) was less than that in S. eruca, where the dimensions varied from dorsal to ventral sides of the stem (0.213 vs. 0.175 mm, respectively).
Over a short distance from the centerline, the collenchyma layer thinned out rapidly to (typically) three cell layers in S. gummosus (Fig. 3H) and two cell layers in S. eruca (Fig. 3G). At the crest of the ribs, the collenchyma remained thin in S. eruca (Fig. 3I) but, in S. gummosus, the collenchyma was again thicker (Fig. 3J), forming a substantial reinforcement over much of the crest of each rib.
In S. eruca, the values for absolute thickness of collenchyma along the flanks of the ribs and at the crest of the ribs (on both dorsal and ventral sides of the stem) did not vary significantly from an average of 0.064 mm. In contrast, in S. gummosus, collenchyma was thicker at the crests of the ribs than along the flanks (0.156 vs. 0.096 mm). Both these values were significantly higher than the comparable values in S. eruca. Measurements at older parts of the axis showed that the collenchyma layer in both species to be thinner at comparable points along the transect (as in Fig. 3J–L), except at the centerline of the sinus.
Young and old collenhyma in S. eruca showed a tendency to separate between cell layers along the middle lamella (Fig. 3M). Although the separations may have occurred during sectioning, this phenomenon indicated a weakness in S. eruca collenchyma that was not detected in S. gummosus.
In summary, S. gummosus appears to have a somewhat higher wall fraction in the collenchyma than does S. eruca. Furthermore, in terms of absolute thickness, the collenchyma of S. gummosus is strategically reinforced along the flanks and crests of the ribs (by 1.5x and 2.4x, respectively), whereas S. eruca has thicker collenchyma than S. gummosus only along the centerline of the sinus. With distance along the axis, the collenchyma layer thins in both species, except along the centerline of the sinus. However, the wall fraction of the collenchyma is maintained.
Secondary dermal tissue
Over most of its surface in the secondary state of growth, the epidermis formed a persistent cork cambium. Little phelloderm was formed to the interior of the cambium, and the external derivatives matured as alternating layers of thin-walled cork and tabular sclereids (Fig. 3N). Near the aerioles, and sometimes elsewhere, cork formation occurred within and beneath the collenchyma layer differentiated during primary growth. Thus, depending on location, the collenchyma experienced little disturbance during secondary growth of the dermal tissue system, or rarely it was partially to completely disrupted. In a few instances we found evidence of a collenchymatous redifferentiation of parenchyma beneath corky layers that disrupted the original collenchyma.
In S. eruca the formation of cork appeared to be influenced by growth habit, with generalized cork formation occurring closer to the shoot tip on the ventral side than on the dorsal side of the axis. Generalized cork formation occurred more basipetally along the axis of S. gummosus, and evenly around the stem axis.
Allometry
Reduced major axis regression of log10-transformed data indicated that S. eruca stem length scales as the 5.5-power of stem diameter measured at stem mid-length (r2 = 0.42, N = 171, F = 122.8, P < 0.0001) and that S. gummosus stem length scales as the 4.4-power of stem diameter measured near the base (r2 = 0.67, N = 134, F = 271.7, P < 0.0001) (Fig. 4). These scaling exponents were statistically indistinguishable based on 95% confidence intervals, i.e., for S. eruca and S. gummosus, 95% CI = 2.94–4.21 and 3.17–4.04, respectively. Similarly, the y-intercepts of the regression curves for stem length vs. diameter were statistically indistinguishable, i.e., for S. eruca and S. gummosus, 95% CI = 4.36–6.42 and 4.50–5.73, respectively.
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Significant differences in stem cross-sectional geometry and thus I were observed. For example, the mean ± 1 SE number of S. eruca and S. gummosus stem ribs was 14.9 ± 0.13 and 9.4 ± 0.12, respectively. Rib elevation and cortical apothems (i.e., the distance from the stem center to the rib base) also differed, i.e., 0.54 cm and 3.3 cm for S. eruca, and 0.73 cm and 2.8 cm for S. gummosus, respectively.
Bending tests
Stenocereus eruca peripheral stem tissues were less stiff and extensible in bending than S. eruca wood or S. gummosus peripheral stem tissues. The stiffness of S. gummosus peripheral stem tissues was equivalent to that of S. gummosus wood (Fig. 5). The beams removed from either species were more stiff when their external surface was placed in tension than in compression.
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The longitudinal trends in the stiffness of core samples were attributed to the ontogenetic trends in the wood development of both species. The transverse heterogeneity of S. eruca peripheral tissue samples was attributed to differences in the amounts of hypodermal collenchyma (see Results: Anatomy).
Tensile tests
The tensile stiffness, breaking stress, and extensibility of S. eruca peripheral tissue samples were an order of magnitude lower than those of S. gummosus and statistically comparable to those of cortical tissue samples from either species (Fig. 6A). These properties also differed by an order of magnitude for tissue samples removed from the woody regions of both species (Fig. 6B). Across all anatomical regions, the stiffness and breaking stress were statistically significantly correlated; a log-log linear relationship was observed, i.e., ordinary least squares regression slope = 0.905, r2 = 0.80, F = 196.8, P < 0.0001 (Fig. 6C). There was a statistically significant relationship between tissue breaking strain, stiffness, and breaking stress, i.e., peripheral tissue extensibility increased in proportion to tissue stiffness and breaking stress.
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; Reyneke and Van Der Schijff, 1974
; Ruzin, 1979
; Jernstedt, 1984
). The protoxylem ran straight with the axis of the root, and annular thickenings were widely separated. No sign of longitudinal compression was detected in the walls of any cell layers.
Computer simulations
Comparisons between predicted and observed values for EI and E for S. eruca and S. gummosus stems indicated acceptable percent errors, especially in light of the longitudinal variations in anatomical and mechanical features (Table 1).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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B, and extensibilities. The large EI and weight per unit length are related features because the large second moments of area I reflects large amounts of structurally weak, hydrated tissues. The stems of S. eruca are not in mechanical jeopardy because they are supported along their length and thus do not experience bending forces. However, if unsupported, these fleshy stems would incur bending moments sufficient to rupture peripheral (principally collenchymatous) tissues. In contrast, the stems of S. gummosus are capable of self-support because their peripheral tissues have a higher elastic modulus, breaking stress, and extensibility as well as a lower weight per unit length.
The relationships among these features can be expressed mathematically. The maximum bending stresses max in any cross section through a stem experiencing a bending moment M is given by the formula max = My/I, where y is the distance from the neutral axis of the stem to the farthest surface of a cross section (see Speck et al., 1990
; Niklas, 1992
). However, the maximum bending moment and thus the maximum bending stress anywhere along the length of a stem is at the base. For a horizontally cantilevered stem, this maximum bending stress is given by the formula max = MLy/I, where M is stem weight and L is stem length. Noting that the factor of safety FS for such a stem may be expressed as the quotient of the breaking stress and maximum stress, B/max, we see that FS = BI/MLy. Noting further that the minimum factor of safety equals unity, it follows that the maximum length a cantilevered S. eruca or S. gummosus stem can achieve before it breaks is given by the formula L = BI/My. Based on averaged values of B, I, M, and y for each species, the critical lengths for S. eruca and S. gummosus stems are 2.88 and 5.73 m, respectively. Imperfections in peripheral portions of stems as well as additional static or dynamic loadings (e.g., lateral branch weight or wind-induced drag forces) would reduce these lengths dramatically, but the proportional difference (i.e., 2.88/5.73 
0.50) is a reasonable estimate for the stems of both species experiencing the same loading conditions.
Estimates of critical stem lengths are also dependent on stem turgor pressures, since the basic mechanical "architecture" of S. eruca and S. gummosus stems is similar to that of a core-rind hydrostatic model (see Niklas, 1989
, 1992
), wherein the fleshy stem tissues collectively serve as an inflatable core that when turgid places the stiffer peripheral stem tissues (rind) in tension. Since mechanical stresses are additive, the factor of safety against peripheral tissue rupture is expected to decrease as a function of stem turgidity. This phenomenology is consistent with our laboratory observations, which indicate that the stems of both species are more prone to breaking when turgid.
The anatomical data support the finding that the collenchyma layer of S. eruca is weaker than that of S. gummosus, if one assumes that the wall chemistry is the same in both species. We caution, however, that in addition to the lower wall content, lesser abundance of collenchyma at mechanically important peripheral locations, and the tendency of the collenchyma to delaminate, there may be differences in cell wall chemistry that could explain why the peripheral tissues of S. eruca are weaker than those of S. gummosus.
Contrary to expectation, our analyses show that, with the exception of old portions of S. gummosus stems, the wood of both species is comparatively flexible, relatively weak, fails to accumulate in sufficient quantity, and develops too close to the neutral axis (where the magnitudes of tensile and compressive bending stresses converge on zero) to mechanically contribute significantly. Indeed, in comparison to other plant tissues, the wood in young portions of S. gummosus and S. eruca is remarkably weak, being comparable to that of apple, corn, or potato parenchyma (measured in three-point bending), and significantly lower than that of carrot root secondary tissues or potato parenchyma/periderm (measured in three-point bending or in tension) (Fig. 10).
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Growth forms in an evolutionary context
Our field and laboratory observations are consistent with the hypothesis that S. eruca may be a comparatively recent evolutionary derivative of an S. gummosus-like ancestor (see Gibson, 1989
). Parallels certainly exist in terms of stem growth patterns. For example, with increasing growth in length, the lateral branches of S. gummosus bend and become increasingly decurrent (see Fig. 1B). These lateral branches often touch the ground where they produce adventitious roots and rarely detach from their parent stems to assume an independent existence. The appearance of these S. gummosus ramets is much like that of S. eruca.
It is thus not unreasonable to speculate that S. eruca may have evolved as a consequence of one or more genetic alterations that, in addition to reducing the strength of stem tissues, diminished the capacity to produce viable seedlings under natural conditions and altered hormone-mediated growth responses to the gravity vector. Seedlings of S. eruca are exceedingly rare and most plants are reported to be the result of vegetative propagation through fragmentation. In the context of plagiotropic growth, we note that branches emerging from the sides of parent S. eruca axes are inclined between 30° and 50° angles with respect to the vertical, whereas branches produced on the dorsal surfaces of S. eruca stems may initially grow vertically at angles close to 90° but nevertheless fail to establish a central root system and subsequently assume a procumbent orientation. This growth habit may be ecologically favorable in the wind-shifted sandy dunes of the Magdalena region, e.g., the slight upward curvature of the distal ends of S. eruca stems may permit the reemergence of plants that are periodically buried during wind storms.
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APPENDIX |
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Script file: stiffness.m
Purpose:
This program calculates the moment of inertia of a cross-section composed of two circular regions and a ribbed region. As outputs it gives:
1. The second moment of area (of the entire cross-section).
2. The flexural rigidity (EI) of the cross-section as well as the EI of each part.
3. A composite Young's modulus E, calculated from EI/I.
Variables defined
disp (‘Please remember to keep consistent with units of measure.’);
The user enters the measured value for the radius of the pith, as well as the E of the pith. The moment of inertia as well as EI are calculated for the pith in this portion.
r1 = input(‘Enter the radius of the circle forming the pith, in meters:’);
ipith = (pi *(r1^4))/4;
Epith = input(‘Enter the calculated Young's Modulus for the pith, in Megapascals:’);
pithstiff = ipith*Epith;
The user enters the measured value for the radius of the wood, as well, as the E of the wood. The 2nd moment of area as well as EI are calculated for the wood in this portion of the stem.
r2 = input(‘Enter the radius of the circle forming the wood-pith combined, in meters:’);
iwood = (pi*(r2^4-r1^4))/4;
Ewood = input(‘Enter the calculated Young's Modulus for the wood, in Megapascals:’);
woodstiff = iwood*Ewood;
The user inputs the number of ribs, which is also the number of sides of the polygon as well as the apothem of the polygon forming the cortex, and the Young's Modulus of the cortex. The 2nd moment of area is calculated.
n = input(‘Enter the number of ribs in the stem cross-section:’);
apothem = input(‘Enter the distance from the center of the cross-section to the center of the base of a rib, in meters:’);
icortex = (n*(apothem^4)*(tan(pi/n)^3)/12)*((3/(sin(pi/n)^2))-2)-iwood-ipith;
Ecortex = input(‘Enter the calculated Young's Modulus for the cortex, in Megapascals:’);
cortexstiff = Ecortex*icortex;
side = input(‘Enter the average length from the base to the tip of a rib, in meters:’);
m = 1;
icribs = 0;
while m< = n
icribs = icribs + ((32*apothem*tan(pi/n)*side^3 +
112*apothem^3*(tan(pi/n)^3)*side)/105) +
((32*apothem*tan(pi/n)*side^3 –
112*apothem^3*(tan(pi/n)^3)*side)*cos(4*pi*m/n)/105);
m = m + 1;
end
Eribs = input(‘Enter the calculated Young's Modulus for the ribs, in Megapascals:’);
x = 1;
arsq = 0;
while x< = n
arsq = arsq + ((4*(apothem^3)*side*tan(pi/n)/3)*(sin(2*x*pi/n)^2));
x = x + 1;
end
Both portions of the rib moment of inertia are added, then multiplied by the Young's Modulus to give the flexural rigidity
iribs = arsq + icribs;
ribstiff = iribs*Eribs;
% The total 2nd moment of area, flexural rigidity and Composite Young's Modulus are calculated
EItotal = ribstiff + cortexstiff + woodstiff + pithstiff;
Itotal = (iribs + icortex + iwood + ipith)*(100^4);
Ecomp = EItotal*(100^4)/Itotal;
For displaying in Newton meters, the flexural rigidities are multiplied by a conversion factor
pithstiff2 = pithstiff*1000000;
woodstiff2 = woodstiff*1000000;
cortexstiff2 = cortexstiff*1000000;
ribstiff2 = ribstiff*1000000;
EItotal2 = EItotal*1000000;
Prints a space, then displays all the output, a reminder about units, which have been normalized, is given.
fprintf (‘\n’);
fprintf (‘The total 2nd moment of area is, in cm^4, is: %f\n’, Itotal);
fprintf (‘The composite Youngs Modulus, in MPa, is: %f\n’, Ecomp);
fprintf (‘\n’);
disp (‘All remaining outputs are in Newtons-meter-squared’);
fprintf (‘The flexural rigidity of the pith is: %f\n’,pithstiff2);
fprintf (‘The flexural rigidity of the wood is: %f\n’,woodstiff2);
fprintf (‘The flexural rigidity of the cortex is: %f\n’, cortexstiff2);
fprintf (‘The flexural rigidity of the ribs is: %f\n’,ribstiff2);
fprintf (‘The total flexural rigidity of the cross-section is: %f\n’,EItotal2);
end of program
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FOOTNOTES |
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4 kjn2{at}cornell.edu
; phone: 607-255-8727, FAX: 607-255-5407
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LITERATURE CITED |
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