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On the economy and safety of hollow non-septate peduncles¹

Department of Plant Biology, Cornell University, Ithaca, New York 14853-5908 USA

Received for publication August 27, 2002. Accepted for publication October 25, 2002.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

 
We examined the allometry and mechanical properties of the peduncles of Hieracium pilosella and other species in the Asteraceae (H. aurantiacum, Taraxicum officinale, Tragopogon pratensi) to evaluate the hypothesis that tapered, tubular, and non-septate peduncles optimize the trade-offs among stem biomass allocation, elevating flowers and thus their wind-dispersed fruits, and the requirement for a factor of safety against mechanical failure. This hypothesis was evaluated by comparing peduncle morphometry (e.g., biomass M_s, floral biomass M_f, and length L) and mechanical properties (e.g., bending rigidity EI) for populations growing in windy and wind-sheltered sites as well as transplants between sites. Regardless of ambient wind speeds, M_s L^4/3 and EI L^11/4 M_s^5/3, whereas M_f /M_s L^–1.15, i.e., peduncles disproportionately increase in their biomass as they increase in length, but mechanically support a disproportionately smaller floral biomass relative to their biomass. Calculations show that the tall peduncles from wind-sheltered sites have a larger fruit dispersal range and a lower factor of safety than the shorter peduncles produced in open sites. These and other observations are interpreted to indicate that tubular peduncles enhance relative fitness in terms of propagule dispersal (but not propagule number per stem) while maintaining a sufficient factor of safety against mechanical failure.

Key Words: allometry • Asteraceae • Brazier buckling • flexural stiffness • fruit dispersal • Hieracium pilosella • plant biomechanics • winding loading

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

 
In theory, the relative fitness of short herbaceous plants with wind-dispersed fruits is increased by maximizing the height and number of fruits supported by individual stems (Okubo and Levin, 1989 ). Hollow non-septate stems are functionally adaptive in this context, because they can elevate flowers or fruits farther above ground than a solid counterpart with equivalent external diameter and mass (Niklas, 1992 ). However, there are trade-offs among peduncle economy, dispersal efficacy, and safety, because hollow tubes, regardless of their mode of taper, are susceptible to mechanical failure as a result of their static (mass) or wind-induced drag forces (Wainwright et al., 1976 ; Speck et al., 1990a , b ; Niklas, 1992 ; Spatz and Speck, 1994 ).

These trade-offs may be evaluated theoretically in terms of the allometric (scaling) relationships among stem diameter D, biomass (mass) M, and flexural stiffness EI with respect to length L, because, for any stem biomass investment, mechanical theory predicts very specific scaling exponents (Table 1). For example, three models for stem taper (geometric, elastic, and stress self-similarity; see Appendix), one of which is equivalent to the Euler buckling formula (elastic self-similarity; see McMahon, 1973 ), predict very different exponents for the relationship between stem diameter and length (1, 3/2, and 2, respectively), stem mass (proportional to D²L) and length (3, 4, and 5, respectively), and stem axial second moment of area (proportional to D⁴) and length (4, 6, and 8, respectively).

Hollow beams with very thin walls are also susceptible to Brazier buckling, i.e., localized ovalization and crimping (Niklas, 1989 , 1990 , 1992 ; Spatz et al., 1997 ). Engineering theory shows that the critical Brazier bending moment M_B is proportional to D³ for thin-walled tubes. For wind-loaded stems, this bending moment is proportional to the product of drag and stem length, i.e., FL D³. Thus, if wind speeds scale as the 1/2- or 1-power of stem length, then stem diameter is predicted to scale as the 3/2- or 2-power of length (Table 1).

These theoretical expectations are based on a number of assumptions predicated on the supposition that self-loading and wind-induced drag forces govern the allometry of peduncles. Nevertheless, empirically determined scaling exponents, significantly lower or higher than those predicted for the relationships among stem diameter, biomass, and second moment of area, would provide circumstantial evidence that trade-offs among economy, dispersal, and safety have occurred.

This study was designed to examine these trade-offs empirically by comparing the morphometry and mechanical allometry of the vertical flower stalks (peduncles) of Hieracium pilosella drawn from populations found in windy and sheltered sites against the scaling relationships predicted by mechanical allometric theory. Hieracium pilosella was selected because its peduncles are hollow and non-septate, its aggregated (composite) flowers are borne distally, and its dry fruits (achenes) are wind dispersed. To develop a sense of the interspecific allometric trends, additional species in the Asteraceae with similar peduncle morphology were studied (i.e., H. aurantiacum, Taraxicum officinale, Tragopogon pratensis).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

 
Field work
Three populations of H. pilosella were selected for study based on differences in ambient wind speeds (see below). Two populations (designated pop. 1 and pop. 2) were sheltered from the wind under a Quercus canopy with an herbaceous understory. The third population (pop. 3) was located in an open windy field.

Reciprocal transplants were made between pop. 1 and pop. 3 (in May of the year preceding this study to maximize transplant acclimation to new environmental conditions) to determine whether genotypic or environmental differences affected peduncle morphometry or mechanical properties. Twenty plants of comparable size (rosette diameters) from pop. 1 and pop. 3 were randomly selected along a 6-m linear transect, removed with as much of their roots as possible, and randomly planted among members of the other population. Transplants were watered for 2 wk and then left undisturbed until the following year during which the herbaceous vegetation surrounding pops. 1–2 and pop. 3 was regularly mowed or hand clipped to an average height of 10 cm and 2 cm, respectively. All transplants from pop. 1 to pop. 3 failed to flower or died for unknown reasons. Therefore, data are reported for plants transplanted from pop. 3 to pop. 1 (designated as Tr. 3-1; see Table 3).

A total of 251 peduncles was collected. Each peduncle was cut at its base at ground level and placed in water to reduce wilting. All morphometric measurements and mechanical tests were performed within 24 h of collection. Hieracium pilosella peduncles (n = 195) were assigned an identification number that did not identify the population from which it was collected to provide double-blind comparisons of peduncle morphometry and mechanical properties among pops. 1–3.

The flowers of each peduncle were removed and weighed. The remaining stem was segmented to determine its outer and inner diameters along its length using a handheld microcaliper. Segments were also weighed to determine fresh mass per unit length. The inner and outer diameters of segments were used to calculate peduncle wall thickness. The peduncles of some species were branched. The dimensions and masses of branches were also measured and recorded (n = 100). The mass of lateral branches was added to that of the main stems to calculate loading conditions at the base of each stem.

Mechanical tests
The composite-tissue elastic modulus E of the basal 8 cm long segment of representative H. pilosella peduncles was determined by means of three-point-bending tests using an Instron 4502 testing machine. Before each bending test, the outer and inner diameters of each specimen were measured at both ends using a handheld electronic microcaliper to determine an average axial second moment of area I using the formula I = (I_b + I_a)/ 2 where I = (d_o⁴ – d_i⁴)/64, subscripts b and a refer to the basal and apical portions of the peduncle, and d_o and d_i are averaged outer and inner diameters (measured at two locations), respectively. Alternative methods for calculating I were explored; the method used was least sensitive to error.

Each basal segment was then suspended horizontally between two vertical supports 5 cm apart and bent by applying a blunt-edge loading point at its mid-length at a strain rate of 25 mm/min. No transverse compression at the supported ends of peduncles was observed during loading experiments. In each case, the length to outer average diameter of each specimen 20. Ten data points per second were recorded continuously during each test to construct a force vs. displacement graph. A bending test was terminated once the applied (maximum) force F_max exceeded the proportional limit of the peduncle; F_max was determined by visual inspection of the force vs. displacement graph. The yield stress _y (the onset of plastic deformation) at this stage of testing was calculated using the formula _y = 16 Ld_oF_max/[ (d_o⁴ – d_i⁴)]. The composite-tissue elastic modulus of each basal segment was calculated using the formula E = 4mL³/[3 (d_o⁴ – d_i⁴)], where m is the slope of the initial (linear) portion of the force vs. displacement graph and L is the distance between the two vertical supports (=5 cm).

Simulated wind speed profiles and estimated fruit dispersal distances
Fruit dispersal distance was estimated based on the average wind speed u_top recorded at 0.5 m above ground within pop. 1 and pop. 3 when flower mass was maximum (June). Wind speeds were simultaneously measured in the field at 0.5-h intervals for 48 h using Sierra-Misco model WSD335 anemometers and wind vanes and Omnidata model EL824-MS data loggers. Each wind vane was located 0.5 m above ground within each population.

A logarithmic vertical wind speed profile was simulated using the formula u_i = [u_top/ln(h/z_o)]ln[(h – x_i)/z_o], where u_i is the estimated wind speed at any distance x_i from where u_top was measured, h equals 0.5 m, and z_o is the roughness parameter (which was taken as 0.05 times the height of surrounding vegetation; see Niklas and Spatz, 2000 ). The surrounding vegetation of pop. 1 and pop. 3 was 0.20 m and 0.10 m, respectively; therefore, z_o was taken as 0.01 and 0.005, respectively.

Fruit dispersal distance y was estimated using the formula y = Lu_ia/u_T, where L is the average peduncle length for each population, u_ia is the averaged wind speed between L and ground level (where u_i = 0), and u_T is the terminal settling velocity of fruits (see Okubo and Levin, 1989 ; Niklas, 1994 ). Values for u_ia were determined from simulated wind speed profiles for pop. 1 and pop. 3 (see Fig. 6). Values for u_T were determined by dropping 10 representative fruits from pop. 1 and 3 into a cylinder measuring 1.5 m in length and 0.20 m in diameter fitted with two orthogonally positioned holes located 0.20 m above the cylinder base to provide access for a stroboscopic light and a camera lens. Photographic prints of descending stroboscopically illuminated fruits (500 flashes/s) were measured to calculate rates of descent (see Niklas, 1992 , pp. 459–461).

View larger version (15K): [in this window] [in a new window]   Fig. 6. Factors of safety for Hieracium pilosella peduncles produced in populations growing in wind-sheltered sites (pops. 1–2), an open windy site (pop. 3), and transplants from pop. 3 to pop. 1 (Tr. 3-1). (A) Factor of safety computed as the quotient of the Euler critical buckling mass Pcr and inflorescence biomass Mf. (B) Factor of safety computed as the quotient of stem tissue yield stress y (measured at the base of peduncle) and stem compressive stress c (i.e., peduncle biomass divided by basal cross-sectional area). (C) Factor of safety computed as the quotient of yield stress and the wind-induced bending stress at the base of the stem W. See text for details

Statistical analyses
Model Type II (reduced major axis) regression analysis was used to determine the scaling exponents and allometric constants (regression slopes and Y-intercepts, _RMA and ß_RMA, respectively) of pair-wise comparisons of log₁₀-transformed data. This protocol is recommended when functional rather than predictive relationships are sought among variables that are biologically interdependent and subject to unknown measurement error (Sokal and Rohlf, 1981 ; Niklas, 1994 ).

All statistical analyses used the formulas log Y₂ = log ß_RMA + _RMAlogY₁, where Y₂ and Y₁ are interdependent variables (e.g., reproductive and standing leaf biomass per plant), _RMA = _OLS/r, where _OLS and r are the slope and correlation coefficient determined from ordinary least squares (Model Type I) regression analysis, and log ß_RMA = log(cap Y₂) – _RMAlog(cap Y₁), where (cap Y) denotes the mean of variable Y. The 95% confidence intervals for ß_RMA values were computed based on the corresponding 95% CI values of _RMA (Sokal and Rohlf, 1981 ).

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

 
Peduncle taper
Peduncles were tapered acropetally and basipetally in their outer diameter and wall thickness. Stem biomass per unit length, cross-sectional area, and axial second moments of area thus varied along peduncle length (Fig. 1). On average, minimum outer diameter and wall thickness occurred beneath inflorescences, whereas maximum outer diameter and wall thickness occurred between 0.6 and 0.7 normalized length (i.e., distance from flowers divided by total peduncle length). This pattern was observed for each species studied. Thus, the stereotypical peduncle geometry is a double-tapered tubular truncated cone with maximum cross-sectional area and axial second moment of area at approximately 0.65 normalized length (Fig. 1).

View larger version (23K): [in this window] [in a new window]   Fig. 1. Taper of representative peduncles of Hieracium pilosella and Taraxacum officinale. (A) Normalized outer diameter (local diameter divided by maximum diameter) plotted against normalized length (distance measured from distal end of peduncle divided by total peduncle length). (B) Normalized wall thickness (local wall thickness divided by maximum wall thickness) plotted against normalized length. (C) Peduncle biomass per unit length (units of kilograms of fresh mass per meter) plotted against normalized length. (D) Highly exaggerated three-dimensional geometry of taper (distal end at left; inflorescence not shown)

Mechanical properties
The axial second moments of area I at the base of H. pilosella peduncles were statistically significantly correlated with peduncle length L, whereas the composite-tissue elastic modulus E was not (Fig. 2A–B). Nevertheless, the flexural rigidity EI and thus the ability to resist bending forces at the base of peduncles correlated significantly with L (Fig. 2C). Across all peduncles, I scaled roughly as the 2.23-power of L, whereas EI scaled as the 2.75-power of L and as the 1.66-power of peduncle biomass M_s (Table 2). No statistically significant differences were evident among H. pilosella growing in different populations.

View larger version (21K): [in this window] [in a new window]   Fig. 2. Bivariant plots of log10-tranformed data for mechanical properties of Hieracium pilosella peduncles from plants growing in wind-sheltered sites (pops. 1–2) and an open, windy site (pop. 3) and transplants from pop. 3 to pop. 1 (denoted as Tr. 3-1). (A) Second moments of area I (measured at base of peduncles) vs. peduncle length L. (B) Composite-tissue elastic modulus E (measured at base of peduncles) vs. peduncle length L. (C) Flexural stiffness EI (measured at base of peduncles) vs. peduncle length L. (D) Composite-tissue elastic modulus E (measured at base of peduncles) vs. yield stress y

Allometric relations
Hieracium pilosella peduncle biomass M_S scaled as the 1.36-power of peduncle length L. Based on comparisons of the 95% confidence intervals of the scaling exponents (_RMA), this relationship held for transplants and non-transplants (Fig. 3A; Table 3). Across all species, including H. pilosella, M_S scaled as the 1.56-power of L (r² = 0.893, n = 335, F = 2794, P < 0.0001) (Fig. 3B).

View larger version (29K): [in this window] [in a new window]   Fig. 3. Allometry of log10-tranformed data for peduncle biomass Ms and peduncle length L for Hieracium pilosella (growing in wind-sheltered sites [pops. 1–2], an open windy site [pop. 3], and transplants from pop. 3 to pop. 1 [denoted as Tr. 3-1]) (A) and other related taxa (B). Solid lines are reduced major axis regression curves for pops. 1 and 3. See text for additional details

Hieracium pilosella inflorescence biomass relative to peduncle biomass, i.e., M_f /M_S, decreased with increasing peduncle length (Fig. 4A). M_f /M_S scaled as the –1.15-power of L (Table 3). The scaling exponent and Y-intercept for this relationship were statistically invariant across pops. 1–3, and, based on the allometry of transplants, not affected by wind conditions (Table 3). Because H. pilosella inflorescence biomass was poorly correlated with peduncle length (0.14 r² 0.22), we interpret these data to indicate that taller and more massive peduncles do not provide an advantage in terms of supporting a larger number of progeny per stem.

View larger version (29K): [in this window] [in a new window]   Fig. 4. Allometry of log10-tranformed data for the quotient of peduncle biomass Ms and floral biomass Mf with respect to peduncle length L of Hieracium pilosella (growing in wind-sheltered sites [pops. 1–2], an open windy site [pop. 3], and transplants from pop. 3 to pop. 1 [denoted as Tr. 3-1]) (A) and other related taxa (B). Solid lines are reduced major axis regression curves for pops. 1–3. See text for additional details

Wind speeds and estimated fruit dispersal distances
The mean ± SE wind speed measured at 0.5 above ground for pop. 1 and pop. 3 was 0.288 ± 0.043 m/s and 0.518 ± 0.023 m/s, respectively; maximum (minimum) wind speeds measured were 1.1 (0.01) m/s and 2.2 (0.01) m/s, respectively. Based on a logarithmic vertical wind speed profile and the average peduncle lengths, wind speeds at the top of peduncles in pop. 1 and pop. 3 were 0.26 m and 0.24 m/s, and 0.12 m and 0. 36 m/s, respectively (Fig. 5). The average wind speed from the top of peduncles to ground level in pop. 1 and pop. 3 (i.e., u_ia) was 0.18 m/s ad 0.27 m/s, respectively. The average terminal settling velocity of mature fruits did not differ between the two populations (u_T = 0.013 ± 0.006 m/s).

View larger version (29K): [in this window] [in a new window]   Fig. 5. Relationship between aboveground height h and estimated local wind speeds ui in Hieracium pilosella pop. 1 (wind-sheltered site) and pop. 3 (windy site) based on wind speeds utop measured in the field at h = 0.5 m. Logarithmic vertical wind speed profiles predict that ui = 0.24 m/s at tops of peduncles of pop. 1 plants (h = 0.26 m) and ui = 0.36 m/s at tops of peduncles of pop. 3 plants (h = 0.12 m). Average wind speeds from tops of peduncles to groundlevel uia (within shaded areas) for pop. 1 and pop. 3 equal 0.18 m/s and 0.27 m/s, respectively

Factors of safety
Three criteria for stem mechanical failure were selected to calculate factors of safety for stem mechanical failure: the quotient of the Euler critical buckling biomass and floral biomass, P_cr /M_f , the quotient of the stem tissue yield stress and the stem compressive stress, _y /_c, and the quotient of the stem tissue yield stress and the wind-induced bending stress at the base of the stem, _y /_W. For each of these criteria, all peduncles had factors of safety greater than unity (Fig. 6). On average, the peduncles of plants growing in windy sites had significantly higher factors of safety than those produced by wind-sheltered plants (P < 0.001). The lowest factors of safety were those sheltered peduncles computed for _y/_W.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

 
The allometry and mechanics of hollow, non-septate peduncles point to an adaptive trade-off between elevating developing flowers to maximize fruit dispersal distance and the need for a factor of safety against stem failure during flowering. This interpretation is based on five observations: (1) peduncles produced in wind-sheltered populations are taller with respect to their biomass than those sampled from open, windy sites, (2) taller peduncles do not support a larger floral biomass (and thus do not disperse a larger number of fruits) than their shorter counterparts, (3) regardless of wind conditions, all peduncles have a high factor of safety against mechanical failure, (4) sheltered peduncles have significantly lower factors of safety against wind-induced bending moments than their shorter counterparts growing in windy habitats, and (5) taller peduncles in the wind-sheltered population can theoretically disperse fruits a greater distance than the shorter peduncles produced in populations growing in windy sites.

In terms of elevating reproductive organs, it is easy to show that tapered stems can elevate flowers and fruits higher than their untapered counterparts with an equivalent biomass, basal diameter, and bulk material (tissue) density. For example, if L_stc denotes the length of the solid truncated conical stem with apical and basal diameters D_a and D_b and if L_cy is the length of its cylindrical counterpart with an equivalent D_b and bulk density, then L_stc will be 2.4 times longer than L_cy when D_a /D_b = 0.2 (see Appendix). Hollow, tapered peduncles are even more cost-effective. For example, if L_htc denotes the length of a hollow truncated conical stem with wall thickness t, then L_htc will be 17 times longer than L_stc when t/D_b = 0.01 (see Appendix).

However, Euler's formula for elastic buckling and the formula for the critical Brazier bending moment show that trade-offs exist between the potential for mechanical failure and the extent to which a distal load can be elevated (see Wainwright et al., 1976 ; Niklas, 1992 ; Speck et al., 1990a , b ). Euler's formula states that the critical buckling load P_cr of a cylinder with flexural rigidity EI, diameter D, and length L is proportional to EI/L². Because I is proportional to D⁴ and E is constant for any tissue construction, we see that P_cr L²/r⁴, where r = L/D, i.e., for any biomass investment, the critical buckling load increases rapidly as a stem elevates a distal load. Likewise, the critical Brazier buckling moment M_br for a hollow tube is proportional to EDt², where t is wall thickness (see Ades, 1957 ). Because M_br is proportional to the product of a bending force F and some function of tube length L and because the mass of a tube is proportional to DLt, we see that F Et/L², i.e., for any mass investment, the force required to locally crimp a thin-walled, tubular stem decreases dramatically with increasing length and decreasing wall thickness.

The susceptibility of tubular peduncles to mechanical failure is exacerbated under windy conditions because plant aerial organs experience wind-induced drag forces in addition to the bending forces incurred by their own mass and the mass of the flowers they support (see Vogel, 1981 ; Niklas and Spatz, 2000 ; Spatz and Brüchert, 2000 ). Because mechanical forces are additive, either stem length must be reduced or wall thickness must be increased to maintain a sufficient factor of safety against failure. The allometry of H. pilosella peduncles complies with this expectation. Plants growing in open sites produce shorter peduncles with significantly higher factors of safety than those produced in wind-sheltered habitats.

Nonetheless, the factor of safety steadily erodes with increasing peduncle length and increasing fruit dispersal range, which is mechanically tenable for two reasons. First, taller peduncles do not support more massive inflorescences than their shorter counterparts, and, second, calculations indicate that the taller peduncles produced in wind-sheltered sites experience lower average wind speeds and thus smaller drag forces. This phenomenology may not hold true for all populations. Our measurements of ambient wind conditions are limited and thus idiosyncratic. However, for the habitats studied here, our data indicate that the potential for wind-induced mechanical stem failure is balanced against the benefits of propagule dispersal. Thus, the allometry and mechanics of hollow non-septate peduncles reflect a number of trade-offs that are reconciled in ways that are likely to be reproductively beneficial.

Finally, we draw attention to the radical departures between the scaling exponents predicted by simple mechanical theory for stems (Table 1) and those empirically observed for peduncles (Tables 2–3). For example, both elastic self-similarity (which follows directly from Euler's formula) and Brazier buckling theory (for wind-loaded peduncles) predict that stem biomass and second moment of area should scale as the 4- and 6-power of length, respectively. Likewise, a constant stress model for self- or wind-loaded peduncles predicts that stem biomass and second moment of area should scale as the 5- and 8-power of stem length, respectively. In contrast, we observe that M L^1.36 and I L^2.75. These and other significant differences between observed and predicted allometric scaling exponents indicate that the morphometry of peduncles is far more efficient with respect to elevating reproductive organs than would be expected based on naive expectations about self-loading or wind-induced drag. Although mechanical theory can be used effectively to postulate form-function relationships, it must always be challenged empirically. Plants cannot obviate physical principles, but they have evolved adaptively in often unexpected ways.

	APPENDIX

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

 
Scaling exponents of geometric, stress, and elastic self-similarity
Here, we derive the scaling exponents for these three models (see Table 1). The derivation for the governing geometric self-similarity is mathematically trivial, because the model stipulates that radius R measured anywhere along length L is proportional to L, i.e., R L¹.

To determine the stress self-similarity , we must find the numerical value of for R = kL such that stress measured anywhere along L is constant. Noting that the maximum stress _max in any cross section is given by the formula _max = EKR_max, where E is the elastic modulus, K is curvature, and R_max is the maximum distance from the centroid axis, the bending moment at any distant x from the tip of a leaning stem M_x is given by the formula M_x = EKI_x. Because _max = M_xR_x/I_x, where I_x is the second moment of area measured at x, it follows that M_x = F_xL_x = (/3) V_xL_x = R_x²L_x² = k₁L_x²⁺², where F_x is the self-generated bending force, is bulk tissue density, V_x is stem volume measured at x, and k is a constant of proportionality. Because I_x = (/4)R_x⁴ = k₂R_x⁴, we see that _max = M_xR_x/I_x = (k₁L_x²⁺²kL/k₂R_x⁴) = k₃L_x³⁺²/kL⁴ = k₅ L_x^3+2–4. If _max is constant anywhere along L_x, then 3 + 2 – 4 = 0. Therefore, = 2, i.e., R = kL². This derivation does not hold true for perfectly vertical stems, because _x is a compressive stress and thus proportional only to the mass of tissues distal to x. Assuming that R does not vary as a function of x, it follows that _x L.

The scaling exponent for the elastic self-similarity model follows directly from Euler's formula for the critical buckling load P_cr of a vertical columnar support member with length L and uniform radius R, i.e., P_cr = kEI/L². Noting that P_cr V R²L, we see that R²L I/L². Because I R⁴, it follows that R L^3/2.

Volume and length relations
Peduncle geometry conforms to a hollow, double-tapered, truncated cone (see Fig. 1). However, for simplicity, peduncle geometry is here approximated as that of a hollow single tapered truncated cone to draw heuristic comparisons with solid cones and cylinders. The objective is to estimate the ability of hollow peduncles to extend in length beyond that of their solid counterparts with equivalent volume and bulk material (tissue) densities, i.e., equivalent mass. The volume of a solid cylinder V_sc with diameter D and length L equals D²L/4. The volume of a solid truncated cone V_stc with length L_stc and basal and apical diameters D_a and D_b equals L_stc(D_b² + D_bD_a + D_a²)/12, or L_stc(1 + k + k²) D_b²/12, where k = D_a /D_b. Assuming both geometries are composed of the same materials with equivalent bulk densities and that both have equivalent basal diameters (i.e., D = D_b), it follows that L_stc/L_sc = 3/(1 + k + k²) such that, when k = 0.2, L_stc = 2.419L_sc.

The volume of a hollow truncated cone with length L_htc, outer and inner basal diameters D_b and d_b, and outer and inner apical diameters D_a and d_a is given by the formula V_htc = L_htc [(1 + k + k²)D_b² – (d_b² + d_bd_a + d_b²)]/12, where k = D_a/D_b. Assuming that the structure has a uniform wall thickness t, we see that d_b = D_b – 2t and d_a = D_a – 2t = kD_b – 2t. Therefore, V_htc = L_htc {(1 + k + k²)D_b² – [(D_b – 2t)² + (D_b – 2t)(kD_b – 2t) + (kD_b – 2t)²]}/12 = L_htc(6D_bt + 6 kD_bt – 12t²)/12.

Provided that a hollow and a solid truncated cone have the same volume and material bulk density, L_htc(6D_bt + 6kD_bt – 12 t²) = L_stc(1 + k + k²) D_b². Denoting t/D_b as (where 0.5 and k 2), we see that L_htc[6(1 + k – 2 )] = L_stc(1 + k + k²) such that L_htc/L_stc = (1 + k + k²)/[6(1 + k – 2)]. If k = 2 (no inner apical diameter, i.e., D_a = 2 t), then L_htc/L_stc = (1 + 2 + 4²)/6. Therefore, when = 0.01, L_htc = 17L_stc.

	FOOTNOTES

 
¹ The authors thank Prof. James A. Bartsch (Biological and Environmental Engineering, Cornell University) for providing access to an Instron testing machine, Prof. Dr. Hanns-Christof Spatz (Institut fur Biologie III, Universität Freiburg), who as an Associate Editor of the American Journal of Botany, supervised the review process and served as the acting Editor-in-Chief, two thoughtful reviewers for suggestions to improve the manuscript, and the College of Agricultural and Life Sciences, Cornell University, for providing financial support.

² Author for reprint requests (kjn2{at}cornell.edu )

	LITERATURE CITED

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

 
Ades C. S. 1957 Bending strength of tubing in the plastic range. Journal of Aeronautical Science 24: 605-610

McMahon T. A. 1973 Size and shape in biology. Science 179: 1201-1204[Abstract/Free Full Text]

Monteith J. L. 1973 Principles of environmental physics. Elsevier, New York, New York, USA

Niklas K. J. 1989 Nodal septa and the rigidity of aerial shoots of Equisetum hyemale. American Journal of Botany 76: 521-531[CrossRef][ISI]

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