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2Section of Plant Biology, Cornell University, Ithaca, New York 14853; and 3Instituto de Ecologia UNAM, Apartado Postal 1354, Hermosillo, Sonora CP 83000, Mexico
Received for publication August 21, 1998. Accepted for publication December 21, 1998.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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Key Words: biomechanics • Cactaceae • mechanical stability • plants • stems • tissue density • tissue stiffness • wood • Young's modulus
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; Carlquist, 1961
, 1969
, 1975
; Wainwright et al., 1976
; Niklas, 1992
; Speck, 1994
; Spatz et al., 1995
). A number of factors influence the mechanical properties of each tissue type, but prior studies suggest that stiffness is positively correlated with the volume fraction of cell wall materials (and thus specific gravity and density), especially among the secondary tissues (Forsaith, 1926
; Record, 1934
; Seibt, 1964
; see Esau, 1967
), and with water content and thus turgor, especially among the primary tissues (Falk, Hertz, and Virgin, 1958
; Niklas, 1992
). However, comparatively few studies have addressed these relationships experimentally in quantitative terms.
The stems of columnar cacti are ideal organs with which to empirically study the relationship between anatomy and tissue stiffness because of their tall, comparatively sparsely branched and woody growth habit, the persistence of pith and cortex in older portions of stems, and the presence of vascular strands that, despite the accumulation of secondary tissues, remain slender and largely unconnected to neighboring strands (Gibson, 1978
; Mauseth, 1988
; Mauseth and Plemons, 1995
; Mauseth et al., 1995
). These and other features provide an opportunity to remove nearly untapered cylindrical samples of xylem and other tissue types from different locations along the length of stems, test these samples in bending to determine their stiffness, assess whether longitudinal gradients in this property exist, and evaluate whether observed gradients correlate with anatomical variations.
In this paper we present the first phase of this research agenda by reporting the longitudinal variations in the stiffness of tissue types surgically removed from different locations along the length of Pachycereus pringlei stems. This species is the most massive plant in the Sonoran Desert, reaching heights of 15–20 m and producing stems up to 1.5 m in diameter (Turner, Bawers, and Burgess, 1995
). It was selected for study because of the availability of specimens differing in size and thus presumably age. It was also selected because of its growth habit, which permits a comparatively straightforward biomechanical interpretation. The cactus ranges from sea level to 950 m and grows mainly in areas dominated by warm-season rainfall, with the exception of central Baja California where it can be found in areas of mainly winter rainfall (Shreve, 1964
; Turner, Bowers, and Burgess, 1995
). Since individual stems are unbranched and since plants are severely damaged or killed by frost, the biomechanics of P. pringlei is not likely to be adapted to transient snow or ice loadings. Finally, although the xylem strands of old plants become interconnected near the base of stems, they are only modestly laterally interconnected to adjoining strands along much of the length of even very massive and tall stems. These xylem strands are thus easily removed from ground tissues, and segments differing in position with respect to stem height can be tested in bending to determine how stiffness varies along stem length.
In this paper we show that the stiffness of the xylem increases in a basipetal direction toward the base of young and old stems but sharply decreases 
1 m above ground level to a level comparable to that found just below the stem tip. In terms of their per unit volume contribution to the ability of stems to resist bending forces, the xylem strands of P. pringlei are ill equipped to cope with the potentially large bending forces that can occur in the base of stems. However, we demonstrate that the geometric contribution made by the xylem to the ability of stems to resist bending forces increases sharply at the base of plants. Consequently, the amount of xylem at the base of stems more than compensates for its low stiffness. We also demonstrate that the stem ribs running much of the length of even old and tall stems contribute significantly to mechanical stability, by virtue of their location and the stiffness of their tissues. The mechanical principles underlying the architecture of P. pringlei and presumably other columnar cactus species are reviewed and discussed in light of the data presented here. The relationship between tissue stiffness and anatomy is discussed in terms of our preliminary findings, but will be treated in greater detail in subsequent publications.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; Felger and Moser, 1985
). The study site was located on a west-facing bajada of the Sierra Seri. Plants of Pachycereus pringlei (S. Watson) Britton & Rose were studied at Rancho El Sacrificio (29°05.82' N, 112°08.00' W). The population density at the study site was 58.0 ± 10.8 plants/ha (mean ± SD). Individuals in this population ranged from 0.012 to 1.02 m in basal diameter and from 0.086 to 12.62 m in height. Young plants had a single vertical stem with 10–15 vertical ribs that can expand and contract depending on the availability of water (Moran, 1968
). Older plants, which may be 300 yr old, had lateral stems that rise near the trunk base at acute angles and may surpass the main stem in height. Two plants were selected for study from this locality because of their size and healthy appearance. These plants measured 3.69 and 5.89 m in height and are denoted throughout this paper as plants 1 and 2, respectively (Fig. 1A, B). Additional specimens were examined during the course of our field investigations to determine patterns of self loading, especially on lateral branches.
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0.93 m in length. Stem segments were dissected either in the field or laboratory to remove their xylem strands, which contained secondary tissues in all cases, from surrounding ground tissues. During this process, the smaller lateral strands that interconnected the main strands were purposely broken and removed to produce beam-like specimens for mechanical tests (see Fig. 1D–E). On average, 12 of these specimens in excess of 1 m long were successfully removed from each of the three stem segments of plant 1 and tested in bending. Between 7 and 15 beam-like specimens in excess of 1 m long were removed from each of the five stem segments of plant 2 and tested in bending. A smaller number of vascular bundle specimens was sampled from segment A of plant 2 because the xylem strands in the base of this segment were laterally fused together in various combinations such that a beam-like sample for each xylem strand could not be obtained. Finally, between 9 and 15 xylem strand specimens were removed from the lateral branch of plant 2 reflecting the fact that some of the branch strands broke during the process of dislodging the branch from its main stem. We present the data from the strands that were successfully removed and tested from this branch even though they reveal no statistically legitimate trend in tissue stiffness.
Portions of the stem ribs (consisting of epi- and hypodermal tissues), pith, and inner cortex were removed from plant 2 to determine their stiffness. Three stem ribs with triangular cross sections were removed with the aid of a sharp knife from each of segments B–E (segment A had a well–defined periderm and lacked evident external ribs). Three pith samples were removed with the aid of a cork borer or a sharp knife from the base of each of segments B–E; pith samples were also removed from the distal part of segment A (no pith was found at the base of this stem segment). Three cortex samples were removed from segments A and B and successfully tested in bending; cortical samples removed from segments C–D could not be tested in bending because they sagged under their own weight when suspended horizontally.
Bending tests were used to determine the material stiffness (Young's modulus) E and the flexural rigidity EI of tissue samples (Fig. 1F). (Young's modulus is a measure of the ability of a material to resist bending; flexural rigidity is a measure of a structure's ability to resist bending. The latter is important because the second moment of area I is a measure of the contribution made by the transverse geometry and size of a material to the ability of a structure to resist bending. Measuring E and I is necessary therefore because stems can rely on either the stiffness or the quantity of different tissues for mechanical support.) Tissue samples from each stem segment were placed between two vertical supports and then loaded by placing bags of sand varying in weight attached at their mid-lengths (Fig. 1F). A horizontally oriented needle was sighted against a metric ruler to measure the mid-length vertical deflection 
resulting from the externally applied load. The Young's modulus of each sample was computed from the formula E = P
3/ 48I, where P is the mass-force of the load and is the free length of the sample between the two vertical supports.
Second moments of area were computed on the basis of morphometric measurements of transections taken at the bottom, middle, and upper thirds of each sample. Different formulas for I were used depending on the transverse geometry of tissue samples and on the orientation of the sample's cross section with respect to the plane of bending. The transverse geometry of the xylem strands varied as a function of location with respect to the stem tip (Fig. 1D and F); most of the strands examined had an elliptical cross section near the stem tip and irregular cross section near the stem base. Different formulas for I were also necessary because each vascular tissue sample was tested in bending by loading it in the radial and tangential direction with respect to stem transverse anatomy to determine the radial and tangential stiffness (designated as ER and ET, respectively) and because the geometry and dimensions of strands viewed in these opposing directions differed. Finally, we note that each stem rib was tested in bending such that its epidermis was located on the side of the specimen experiencing tension. This orientation was selected because it crudely conformed to when stems dilate due to water storage and place their epidermis in tension. The reverse orientation with regard to bending forces would also have underestimated the stiffness of this portion of stems.
The transverse geometry and size of tissues were not uniform along the length of some vascular bundle segments (Fig. 1F). For these segments, the relationship E = P
3/ 48I was invalid because this formula is predicated on the assumption that beams have uniform I. For these "irregular" bundle segments, which occurred near the stem base, we used another formula adapted for flexed conical beams (see Niklas, 1992
, p. 336–337). The geometry of each conical beam was determined on the basis of three sets of morphometric measurements taken at the middle and ends of these irregular strands. Sensitivity analyses indicated that the alternative formula consistently overestimated E. We consider this error unimportant because it is biased in favor of a conservative interpretation of the sharp drop in E observed at the base of stems for each stem (see Results).
ANOVA and Model Type I and II regression analyses were used to determine whether differences in the E or EI of tissues varied significantly and predictably as a function of distance from shoot tips. Statistical comparisons were also drawn among the different tissue types to estimate their respective contributions to the ability of intact stems to cope with their static (self-weight) loadings. All statistical analyses were performed using the software package JMP© (SAS Institute Inc.) using a Power Macintosh 8100/80.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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In terms of other tissues and portions of the stems, a significant difference in cortex stiffness was observed between samples successfully removed and tested in bending from stem segments A and B (5.30 and 0.549 GN/m2, respectively). The stiffness of the cortex in the more distal segments of the stem was assumed to be <<0.549 GN/m2 because these samples failed to support their own weight when suspended horizontally. A statistically significant basipetal increase in pith stiffness was observed (Fig. 5A). Although no strong statistical trend was evident for longitudinal variations in the bulk stiffness of stem rib tissues (Fig. 5B), the stem ribs removed from the upper two-fifths of stems were stiffer than their corresponding xylem strands (Fig. 6).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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; Nobel and Meyer, 1991
). Thus, the morphology and anatomy of these stems should not be interpreted exclusively in the context of a single biological function but are more profitably discussed in terms of how they successfully cope with performing all three functions simultaneously. For example, engineering shows that the best location for the principal stiffening agent in any vertical support member is at the surface because bending and twisting stresses reach their maximum intensities at this location (Timoshenko and Gere, 1961
). The optimal location for photosynthetic tissues is likewise at the surface of the columnar stems lacking foliage leaves because this location favors gas exchange and light interception. Since the stiffest known plant tissues are ill equipped for photosynthesis (sclerenchyma and wood), it is clear that the requirements for mechanical support and photosynthesis must be reconciled. In the case of columnar cacti, this reconciliation is achieved in the form of a fluted transverse stem geometry whose peripheral tissues are sufficiently stiff to provide mechanical support, yet are living and thus photosynthetically competent. Our data show that the bulk stiffness of P. pringlei stem rib tissues is equivalent to that of the vascular tissues. The peripheral stem tissues of this and presumably similar species can thus serve as an important stiffening agent especially in the upper portions of tall and presumably old plants where the stiffness of the rib tissues can exceed the stiffness of the vascular tissues by nearly an order of magnitude. The fact that this fluted geometry can simultaneously cope with stem dilation when water is stored in stem ground tissues must not escape attention (Moran, 1968
; Mauseth, 1988
).
Owing to their location, the geometric contribution (the second moment of area) made by stem ribs to the overall flexural rigidity of stems is substantial and is shown easily with the aid of a few simple calculations. Each fluted transverse stem geometry (Fig. 8A) can be approximated by a series of triangles (each representing the transverse geometry of a stem rib) with radial and tangential thickness h and b surrounding a regular polygon with n sides (representing the remaining portions of the stem cross section). Noting that the number of triangles and their tangential thickness define the number and length of the polygon's sides, we see that the total transverse area of all the triangles (i.e., the sum of all the rib cross sections) AR equals nbh/2 and that the transverse area of the polygon AP equals (nb2/4) cot (180°/n). It also follows that the second moment of area of all the triangles IR and the second moment of the polygonal portion of the stem IP must equal the second moments of area of circular cross sections with equivalent cross sectional areas AR and AP, respectively. (Note that this is equivalent to taking the average radius of the triangular sections and that of the polygon section, respectively.) We now denote the unit radius of the circular cross section with area AP as ri and the unit radius of the stem as a whole (due to the average radial thickness added by the stem ribs) as ro. Solving for the flexural rigidity contributed by the polygon gives 0.7854r4iEP , where EP is the bulk material stiffness of the tissues comprising this portion of the stem. Solving for the flexural rigidity contributed by all of the triangular ribs gives 0.7854(r4o – r4i)ER , where ER is the bulk material stiffness of the stem rib tissues. Setting these two formulas equal to one another reveals when the stem ribs contribute as much to the flexural rigidity of the stem as the central portion of the stem, and solving this identity for the dimensionless quotient h/b gives the formula
= ER/EP (the dimensionless quotient of the bulk stiffness of rib tissues and the bulk stiffness of the remaining stem tissues). The demonstration that the ribs are structurally important is completed by plotting the aspect ratio h/b against the number of ribs n on a stem for different values of = ER/EP. This graphic device allows us to predict the rib aspect ratio for any stem with n number of ribs that obtains a flexural rigidity equal to that of the remaining portion of the stem (Fig. 8B). In the case of plant 2, n = 15 and, for the upper third of stem length, ER 
EP (i.e., = 1). Inserting these values into the formula gives h/b = 0.974 in contrast to h/b = 1.02 ± 0.5, which was observed based on 30 measurements of 15 ribs. Taken at face value, the agreement between the predicted and observed rib aspect ratio argues that, by virtue of their location as well as their bulk tissue stiffness, the ribs of P. pringlei contribute significantly to the ability of stems to cope with bending forces.
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Despite the mechanical importance of the stem ribs in the younger portions of stems, the longitudinal variations in the material stiffness and cross-sectional area of the xylem cannot be neglected. Despite the sharp drop in the stiffness of the xylem at the base of stems, the geometric contribution made by this tissue (i.e., the axial second moment of area of the xylem) to the flexural stiffness of the stem as a whole increases significantly toward the stem base because of the basipetal increase in the xylem transverse area. Specifically, engineering theory shows that this geometric contribution increases as a function of the fourth power of the radial thickness of the xylem (see Wainwright et al., 1976;
Niklas, 1992
). Thus what the xylem may lack in material stiffness is more than compensated for by its basipetal increase in cross-sectional area. The xylem is also mechanically anisotropic since it is much stiffer when bent in the radial than in the tangential direction with respect to stem length. This anisotropy may be functionally important since the xylem strands are arranged in a circle when seen in stem transection, such that an almost equivalent number will bend in the radial and in the tangential directions regardless of the direction of a bending load with respect to the stem as a whole.
Under any circumstances, the mechanical anisotropy of the xylem is entirely compatible with the anatomy of this tissue. Preliminary anatomical studies reveal that P. pringleii xylem consists of very tall rays composed of living thin-walled cells that, when viewed in either the tangential or transverse plane of section, are approximately as wide as the intervening layers of axial cell types. The xylem strand of P. pringleii can be thus crudely approximated as a beam composed of alternating vertical plate-like layers of two materials differing in stiffness (Fig. 9A). Designating the stiffness of these two materials as E1 and E2, it follows from the basic theory of composite materials (Hollister and Thomas, 1966;
Wainwright et al., 1976;
Niklas, 1992
) that the bulk stiffness of the strand as a whole measured in the radial and tangential directions ER and ET are given by the formulas ER = (E1V1 + E2 V2) and 1/ET = (V1/E1) + (V2/E2), respectively, where V1 and V2 are the volume fractions of the two component materials such that V1 + V2 = 1. Taking the quotient of the radial and tangential stiffness, we see that ER/ET = (
V1 -V1 + 1)[(V1/) - V1 + 1], where = E1 / E2. Plotting the quotient ER/ET against the volume fraction of either of the two materials (i.e., V = V1 or V2) shows that ER 
ET regardless of the numerical value of E1/E2. That is, the stiffness of the composite material (i.e., the bulk stiffness of the vascular strand) measured in the radial direction must always equal or exceed the stiffness measured in the tangential direction (Fig. 9B).
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cf. figs. 41–42). The anatomical trends observed for this lobelioid species show that the volume fraction of cell wall materials (and perhaps the stiffness) of secondary vascular tissues is not invariably correlated with the age and location of the cambium with respect to the stem apex. It is clear that detailed anatomical studies are required to determine why the stiffness of P. pringlei vascular tissues changes abruptly near the stem base. These studies are currently underway and will be reported in subsequent publications.
Comparatively few studies have devoted attention to the biomechanics of cacti (see Nobel and Meyer, 1991;
Niklas and Buchmann, 1994;
Cornejo and Simpson, 1997;
Molina-Freaner, Tinoco-Ojanguren, and Niklas, 1998
). But there is little doubt that these organisms provide exciting opportunities to study how their manifold biological functions are anatomically resolved in what are generally considered difficult environmental conditions. The cacti also provide numerous opportunities to identify the extent to which anatomical features correlate with tissue stiffness and strength and how these mechanical properties are affected by ecological specialization—a research agenda tracing its first explicit exposition to the seminal work of Sherwin Carlquist (1961,
1975
) to whom we dedicate this paper.
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FOOTNOTES |
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LITERATURE CITED |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONLITERATURE CITED |
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———. 1969 Wood anatomy of Lobelioideae (Campanulaceae). Biotropica 1: 47 47–72.
———. 1975 Ecological strategies of xylem evolution. University of California Press, Berkeley, CA.
Cornejo, D. O., and B. B. Simpson. 1997 Analysis of form and function in North American columnar cacti (Tribe Pachycereeae). American Journal of Botany 84: 1482–1501.[Abstract]
Esau, K. 1967 Plant anatomy, 2nd ed. John Wiley, New York, NY.
Falk, S., H. Hertz, and H. Virgin. 1958 On the relation between turgor pressure and tissue rigidity. I. Experiments on resonance frequency and tissue rigidity. Physiologia Plantarum 11: 802–817.[CrossRef]
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Forsaith, C. C. 1926 The technology of New York State timbers. New York State College of Forestry, Syracuse University, Technical Publication 18, vol. 26, 1–213.
Gibson, A. C. 1978 Architectural designs of wood skeletons in cacti. Cactus and Succulent Journal (Great Britain) 40: 73–80.
———, and P. S. Nobel. 1986 The cactus primer. Harvard University Press, Cambridge, MA.
Hollister, G. S., and C. Thomas. 1966 Fibre reinforced materials. Elsevier, Amsterdam.
Mauseth, J. D. 1988 Plant anatomy. Benjamin Cummings, New York, NY.
———, and B. J. Plemons. 1995 Developmental variable, polymorphic woods in cacti. American Journal of Botany 82: 1199–1205.[CrossRef][ISI]
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Niklas, K. J. 1992 Plant biomechanics: an engineering approach to plant form and function. University of Chicago Press, Chicago, IL.
———, and S. L. Buchmann. 1994 The allometry of saguaro height. American Journal of Botany 81: 1161–1168.[CrossRef][ISI]
Nobel, P. S, and R. W. Meyer. 1991 Biomechanics of cladodes and cladode-cladode junctions for Opuntia ficus-indica (Cactaceae). American Journal of Botany 78: 1252–1259.[CrossRef][ISI]
Record, S. J. 1934 Identification of the timbers of temperate North America. John Wiley & Sons, New York, NY.
Seibt, G. 1964 Über die unterscheidliches raumdichte und sahrringbreite des wurzelholzes von nadelbaümen in einem meliorationsversuch. Allgemeine Forst und Jagdzeitung 135: 66–74.
Shreve, F. 1964 Vegetation of the Sonoran Desert. In F. Shreve and I. L. Wiggins [eds.], Vegetation and flora of the Sonoran Desert, 3–186. Stanford University Press, Stanford, CA.
Spatz H.-C., H. Beismann, A. Emanns, and T. Speck. 1995 Mechanical anisotropy and inhomogeneity in the tissues comprising the hollow stem of the giant reed Arundo donax. Biomimetics 3: 141–155.
Speck, T. 1994 Bending stability of plant stems: Ontological, ecological, and physiological aspects. Biomimetics 2: 109–128.
Schwendener, S. 1874 Das Mechanische Prinzip im Anatomischen Bau der Monocotylen. Verlag W. Engelmann, Leipzig.
Timoshenko, S. P, and J. M. Gere. 1961 Theory of elastic stability. McGraw-Hill, New York, NY.
Turner, R. M., J. E. Bowers, and T. L. Burgess. 1995 Sonoran desert plants: an ecological atlas. University of Arizona Press, Tucson, AZ.
Wainwright, S. A., W. D. Biggs, J. D. Currey, and J. M. Gosline. 1976 Mechanical design in organisms. John Wiley & Sons, New York, NY.
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of samples removed from plants 1 and 2 plotted against the relative location of samples with respect to stem length. (A) Bulk density of xylem and stem rib tissue samples removed from plant 2. Solid horizontal line denotes mean bulk density of pith samples. (B) Bulk density of xylem tissue samples removed from plant 1 (no stem rib tissues were examined for this plant). Dark vertical lines are standard errors of means; thin vertical lines are standard deviations of means.
