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(American Journal of Botany. 2001;88:1266-1278.)
© 2001 Botanical Society of America, Inc.

Ecology

Evolutionary trends in safety factors against wind-induced stem failure1

Karl J. Niklas2,4 and Thomas Speck3

2Department of Plant Biology, Cornell University, Ithaca, New York 14853-5853 USA 3Botanischer Garten der Albert-Ludwigs-Universität, Freiburg i. Br., D-79104 Germany

Received for publication August 15, 2000. Accepted for publication November 7, 2000.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 LITERATURE CITED
 
We explore the hypothesis that the safety factor against wind-induced stem failure remained high during early land plant evolution despite an evolutionary increase in height with concomitant increases in wind-induced drag forces, bending stresses, and moments. This hypothesis was examined for 17 Paleozoic plant species assuming that each (1) existed in a densely packed community of conspecifics with equivalent height, (2) coped with the same wind profile (where ambient wind speed decreased toward ground level), but (3) had different within-canopy wind speeds depending on plant height and general morphology. Drag forces, stresses, and moments were computed, and a safety factor was calculated for each taxon using the quotient of its stem-tissue breaking stress and maximum wind-induced bending stress.

The highest factors of safety were calculated among the most ancient rhyniophyte and zosterophyllophyte species examined (e.g., Rhynia and Gosslingia), and, on average, decreased among the taller and geologically younger species. The tallest species examined (e.g., Archaeopteris and Diaphorodendron) had safety factors equal to or higher than those of some of their presumed ancestors (e.g., Psilophyton and Leclercqia). These trends were statistically more robust among rhyniophytes and their presumed descendants.

Even though the results comply with the hypothesis, numerous limitations of our protocol exist (e.g., the requirement for reliable whole-plant reconstructions). These are discussed in terms of our theory. Nonetheless, we believe our theory and protocol afford a reasonable opportunity to explore the effects of wind on early plant evolution.

Key Words: biomechanics • factors of safety • fossil plants • mechanical stability • wind-loading


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 LITERATURE CITED
 
The most dramatic morphological radiation of vascular embryophytes occurred between late Silurian and late Devonian times (Banks, 1968 ; Chaloner and Sheerin, 1979 ; Gensel and Andrews, 1984 ; Stewart and Rothwell, 1993 ; Taylor and Taylor, 1993 ; Niklas, 1997 ). During this time interval, the morphological repertoire of the sporophyte generation expanded from the comparatively simple open-dichotomous branching patterns characterizing the leafless axes of the earliest vascular plants to include anisotomous, pseudomonopodial, and monopodial branching patterns supporting micro- and megaphylls as well as a variety of different kinds of reproductive organs (Banks, 1968 ; Chaloner and Sheerin, 1979 ; Gensel and Andrews, 1984 ).

This period of morphological diversification was attended by an equally impressive anatomical diversification (Banks, 1968 ; Chaloner and Sheerin, 1979 ; Gensel and Andrews, 1984 ; Stewart and Rothwell, 1993 ; Taylor and Taylor, 1993 ). The vegetative axes of most, if not all, Silurian vascular plants were apparently mechanically supported by an undifferentiated parenchymatous cortex surrounding a centrally located strand of primary vascular tissues (Speck and Vogellehner, 1988, 1994 ; Niklas, 1990, 1992 ). By middle Devonian times, many vascular plants relied on stiffer and stronger primary tissues (e.g., collenchyma and sclerenchyma) or secondary tissues for mechanical support. This anatomical diversification permitted an increase in plant stature (Chaloner and Sheerin, 1979 ; Niklas, 1997 ). The most ancient sporophytes measured only a few centimeters in height, whereas the height of many late Devonian species rivaled that of some of tallest plants today (Mosbrugger, 1990 ; Niklas, 1992, 1997 ; Taylor and Taylor, 1993 ; Bateman et al., 1998). Considerable attention has been paid to the evolution of plant biomechanics, particularly with regard to the capacity for self-support. Studies of fossil and living plants indicate that the anatomical evolution of progressively stiffer and stronger primary tissues increased the ability of stems to resist the bending forces induced by gravity and thus permitted an increase in plant height, which benefits photosynthesis and long-distance spore dispersal (Niklas, 1992, 1993, 1994 ; Speck, 1994 ; Speck and Vogellehner, 1994 ; Rowe and Speck, 1998 ; Speck and Rowe, 1999a ). Studies likewise show that the acquisition of secondary growth permitted a basipetal increase in the cross-sectional area of stems. This resulted in a reduction in the magnitudes of bending stresses, as well as an increase the axial second moment of area, both of which enhance the ability of a stem to support its mass and that of attached organs (Niklas, 1992, 1997 ; Speck, 1994 ; Speck and Vogellehner, 1994 ). There is ample evidence that the early evolution of plant anatomy was as much a consequence of natural selection acting on the mechanical behavior of stems as on their physiology (Niklas, 1992 ; Bateman et al., 1998 ).

In contrast, comparatively few studies have addressed the evolutionary response of plants to the mechanical effects of wind-loadings (see, however, Niklas, 2000 ; Niklas and Spatz, 2000 ), even though simple calculations show that the drag forces exerted on aerial stems by wind pressure typically exceed those generated by gravity (Alexander, 1971 ; Vogel, 1981 ). For example, the bending moment due to wind-induced drag Md acting at the base of an untapered cylindrical stem with length L and radius r is given by the formula Md = 0.5 {rho}a u2 Sp CD L, where {rho}a is the density of air, u is ambient wind speed, Sp is projected area, and CD is the drag coefficient. Since the maximum projected surface area of a cylinder is given by the formula Sp = 2 r L, it follows that the maximum moment due to wind-loading equals Md = {rho}a u2 r L2 CD. For the same stem oriented at any angle {phi} with respect to the vertical, the bending moment resulting from self-loading Ms acting on the base is given by the formula Ms = [({pi} r2 L2/2) ({rho} – {rho}a) g sin {phi}], where {rho} is the bulk density of the stem and g is the acceleration due to gravity (=9.81 m/sec2). Since the density of most plant tissues is 1000 times that of air, a rough approximation is Ms ~ {pi} {rho} r2 L2 g sin {phi}/2. The quotient of these two moments Md/Ms equals 2 {rho}a u2 CD/{pi} {rho} r g sin {phi}, which shows that the moment due to wind-loading exceeds that of self-loading even for comparatively modest wind speeds (e.g., calculations indicate that Md/Ms ~ 2 when u = 3 m/sec and {phi} = 2°, assuming that {rho}a = 1.2 kg/m3, {rho} = 1000 kg/m3, CD = 1.0, and r = 0.01 m). Similar calculations for entire trees indicate that the stresses produced in stems by wind exceed those caused by the self-mass of stems for wind speeds between 1 and 5 m/sec, depending on the geometrical parameters used to model the tree (Speck, Spatz, and Vogellehner, 1990 ).

Since the bending moments created by gravity and wind are additive, stems that are more than sufficient to support their own mass may mechanically fail when subjected to modest to large wind-induced drag. For example, the maximum bending stress {sigma}M that develops in a cylindrical stem with length L and radius r is given by the formula {sigma}M = 4 MT /{pi} r3, where MT = Md + Ms ~ {rho}a u2 r L2 CD + ({pi} {rho} r2 L2 g sin {phi}) /2 = r L2 [{rho}a u2 CD + ({pi} {rho} r g sin {phi})/2]. Thus, {sigma}max = (4 L2/{pi} r2) [{rho}a u2 CD + ({pi} {rho} r g sin {phi})/2]. Assuming that {rho}a = 1.2 kg/m3, {rho} = 1000 kg/m3, CD = 1.0, and r = 0.01 m and assuming a tissue-breaking stress {sigma}b of 10 MN/m2, calculations indicate that a horizontally cantilevered stem measuring one meter in length will mechanically fail by tissue rupture when u = 25 m/sec (i.e., {sigma}M > {sigma}b). It is thus reasonable to believe that early land plant evolution was as much influenced by the effects of wind as by the effects of gravity on stems and other aerial organs.

We explore this hypothesis by calculating the maximum bending stresses {sigma}M produced by drag forces acting on the aerial portions of a constellation of early Paleozoic plants and by using the magnitudes of these stresses in tandem with those of empirically determined tissue-breaking stresses {sigma}b of anatomically comparable, modern-day plants to estimate the factors of safety SF ("mechanical reliability") against wind-induced stem failure (i.e., SF = {sigma}b/{sigma}M). For each plant, we mathematically simulate a vertical wind speed profile, determine the plant surface area projected toward the wind as a function of height above ground, calculate the drag forces and the resulting bending moments and stresses acting on different portions of the plant body, and then compute the factors of safety based on the breaking stress of the tissue believed to serve as the principal stiffening agent at the base of each species (see Niklas, 2000 ; Niklas and Spatz, 2000) . Our objective is to evaluate whether the mechanical reliability of fossil plants adaptively changed during the early phases of land plant evolution.

This research agenda requires information about plant community structure (to estimate within-canopy speed profiles), the morphology of each species (to calculate projected areas), and stem anatomy (to select the appropriate tissue-breaking stress to compute a factor of safety). Clearly, for many fossil species, much uncertainty surrounds each of these requirements. Many aspects of fossil community structure are often poorly preserved; whole-plant reconstructions are always subject to revision as new information comes to light; and the basal anatomy of some species is either unknown or the subject of continued debate.

However, recent studies indicate that estimates of the drag-induced bending stresses acting in the base of plants are far more sensitive to variations in plant size and morphology than to the "geometry" of wind-speed profiles (Niklas, 2000 ; Niklas and Spatz, 2000) . By the same token, differences in the breaking stresses of unlignified primary plant tissues (e.g., parenchyma and collenchyma) are comparatively small and differ significantly from those of thick-walled, lignified primary or secondary tissues (e.g., sclerenchyma and secondary xylem) (Niklas, 1992 ). Thus, errors in judgement concerning which wind-speed profile or tissue-breaking stress should be used in the analysis of wind-induced stem failure have far less effect on the conclusions drawn than errors resulting from the use of morphologically unrepresentative whole-plant reconstructions.

Nonetheless, we are aware of the limitations imposed by the fossil record on the protocol used in this study. The following is presented more as a theoretical than as an empirical evaluation of whether wind-induced bending stresses played a significant role in shaping early land plant morphology and anatomy.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 LITERATURE CITED
 
Three criteria were used to select species for this study: (1) the availability of a reliable or generally accepted whole-plant reconstruction, (2) reasonable preservation of anatomy at the plant base, and (3) the extent to which each species is morphologically representative of its particular lineage or higher taxon. The first of these criteria was required in order to compute the drag forces exerted by wind along different portions of the plant body. The second criterion was required in order to compute the factor of safety against wind-loadings based on the breaking stress of mechanically supportive tissue. The third criterion was necessary since our objective was to determine whether evolutionary trends in factors of safety are evident. Based on these three criteria, a total of 17 species, representing two major lines of plant evolution (the rhyniophytes and their presumed descendants and the zosterophyllophytes and their descendants, the lycopods), were found suitable for study (Table 1).


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  		Table 1. Morphometric data (in meters), geological age (in Myr), and primary literature references for 17 Paleozoic plant fossils. D = maximum basal stem diameter; Hp = published estimated height; HSF = estimated height assuming a 2.5 factor of safety; HA = estimated height assuming an allometric relationship; Refs. = studies from which data for D and H were obtained. See text for further details

 
Surface area determinations
Whole-plant reconstructions were used to compute the surface area projected by the plant body toward the wind at different distances above ground level.

The silhouette of each reconstruction was darkened by hand and then scanned into the memory of a desktop computer equipped with the program WinDrag (developed by KJN; unpublished program) (see Fig. 1 for examples). The size of each "element" of the reconstruction (i.e., sporangia, axes, and leaves, if present) was then scaled with respect to the maximum height published in the literature (designated by HP) and the maximum stem diameter (designated here as D) reported in the literature for each species. The projected surface area Sp of each element i with scaled diameter di and length {ell}i was automatically computed by WinDrag based on the quotient of the number of pixels creating the image of the element and the total number of pixels falling in a portion of a grid system that was superimposed on the entire scanned image (Fig. 2). WinDrag also calculated one or more stem "path lengths" (i.e., the distance dt of each element i from the tip of an interconnected series of branches) for each species based on points of attachment ("nodes") entered into the computer's memory by the operator. The locations of nodes were determined by visual inspection of each reconstruction.



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  		Fig. 1. Silhouettes of reconstructions of representative fossil plant species scanned into the computer program WinDrag to calculate projected surface areas and drag forces (also see Fig. 2 ). (A) Horneophyton lignieri (horizontal line denotes inferred location of stem base with respect to ground level). (B) Steganotheca striata. (C) Rhynia gwynne-vaughanii. (D) Diaphorodendron vasculare. (E) Archaeopteris sp. (F) Pleuromeia longicaulis with inferred leaf orientations with no wind (left) and oncoming wind (right)

 


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  		Fig. 2. Protocol used to calculate projected surface areas of different portions of the plant body illustrated for Pleuromeia longicaulis. Whole-plant reconstruction (A) is scanned into computer memory, outlines of each element (e.g., leaves and stems) are darkened, and maximum basal stem diameter reported in literature is used to scale the size of other body parts (B). The orientations of flexible elements (e.g., leaves) with respect to oncoming wind are inferred based on the behavior of modern-day analogues before a grid system is superimposed over the image (C). WinDrag computes the projected areas of all elements located in each portion of the grid system (D)

 
Plant size determinations
All calculations were dependent on estimates of maximum plant size as measured by either stem diameter or height (see below). A number of factors jeopardize the reliability of height estimates as reported in the primary literature (i.e., HP). For example, the basal portions of some species are unknown (e.g., Cooksonia and Psilophyton spp.). Thus, the heights of these plants are likely to be underestimated. Also, different authors estimate maximum plant height using different techniques. Thus, a certain degree of inconsistency among authors is to be expected.

For these and other reasons, we scaled the absolute size of each fossil plant reconstruction using two different techniques to estimate plant height as well as using the maximum plant height reported in the literature by different authors. The first of these two techniques used the empirically determined allometric relationships between height and basal stem diameters observed for a broad spectrum of extant plant species with nonwoody and woody axes or stems (e.g., moss sporophytes, pteridophytes, and arborescent palm, dicot, and gymnosperm species). This "allometric" technique has successfully predicted the height (= length) of representative intact fossil stems based on their basal stem diameters (see Niklas, 1994 ). Allometrically determined plant heights are designated as HA throughout this paper.

The second technique estimated the height of fossil plant species by calculating the critical buckling height Hcrit (see Greenhill, 1881) of stems based on their reported maximum basal stem diameter and dividing this height by a factor of safety SF against mechanical failure due to self-loading equal to 2.5. This "safety factor" height for self-loading is designated as HSF throughout this paper (i.e., HSF = Hcrit/SF = Hcrit/2.5; see Speck and Vogellehner, 1992, 1994 ; Speck, 1994 ; Speck and Rowe, 1999b ). This technique was successfully "tested" by calculating the known heights of extant herbaceous species based on their basal stem diameters. We note in passing that the use of Hcrit calculated in this fashion is not subject to the criticism of "circular logic," since these estimates of plant height are based on static loading conditions, whereas the factors of safety we calculate based on wind-induced bending moments and stresses are for dynamic loading conditions.

These two techniques were used to introduce a degree of internal consistency among all of the estimated plant heights used in our study, and as the basis of a "sensitivity" analysis that could highlight "problem taxa," that is, species for which height estimates vary widely depending on the technique used.

Drag-force and bending moment computations
The magnitude of the drag force Df exerted on each element i of a whole-plant reconstruction was calculated based on the formula Df = 0.5 {rho} Ui2 Sp CD, where {rho} is the density of air (taken at 1.2 kg/m3), Ui is the wind speed measured for each element i, CD is the drag coefficient (taken as ~1.0), and Sp = di {ell}i.

The drag-induced bending moment acting on any series of elements i connected by nodes was computed using the formula Md = {Sigma} (0.5 {rho}a Ui2 Sp CD) dt = {Sigma} (0.5 {rho}a Ui2 di {ell}i CD) dt. This formula gives a "running sum," such that the bending moment increases basipetally toward the base of each series of interconnected elements and reaches its maximum value at the base of the plant.

Wind-speed profiles
The wind speeds used to calculate drag forces were computed based on a maximum wind speed of 10 m/sec at 10 m above ground. The wind speed at any distance h above ground for an open terrain was computed by means of the formula Uh = 10 m/sec (h/10 m)0.5. Within-canopy wind speeds Ui were then calculated based on the formula Ui = UH exp [a (hi/H – 1)], where UH is the wind speed at the top of a canopy with height H (i.e., UH = Uh when H = h), a is a scaling factor that increases from ~1 to 5 as the number of plants in a community increases, and hi is the distance above ground measured anywhere within the canopy (Fig. 3). Thus, regardless of their different canopy heights and geological ages, this protocol assures each species "sees" the same above-canopy wind speed profile (i.e., the speed at the top of each canopy was scaled in the same manner with respect to absolute plant height). For the purposes of this study, each plant species was assumed to have grown in a densely packed community composed of plants of equivalent height (i.e., a = 5).



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  		Fig. 3. Wind speeds above and within plant canopies (Uh and Ui, respectively) plotted as a function of height above ground h. Each fossil plant (diagrammed as a shaded vertical rectangle in diagram to the right of the graph) is assumed to exist in a densely packed community of conspecifics with equivalent height H growing in an otherwise open terrain. Wind speeds above the canopy are calculated assuming that U = 10 m/sec at h = 10 m (see formula in lower right of graph and concave descending curve). Wind speeds within the canopy of each community are calculated assuming a densely packed community (i.e., a = 5, see formula in the upper left of the graph). The effect of the numerical value of a on within-canopy wind speeds is illustrated by three convex descending curves. For further details, see text

 
Bending stresses
The bending stresses {sigma} for any element i of radius r reach their maximum intensities at the element's surface and are given by the formula {sigma} = Md r/I, where I is the axial second moment of area. For a terete cylindrical element, this moment of area is given by the formula I = {pi} r4/4 such that {sigma} = 4 Md/{pi} r3 ~ 1.273 Md/r3. However, since the maximum bending stresses along the entire length of any stem occur at the base, where the maximum bending moment MM occurs, the maximum bending stresses {sigma}M for an entire plant is given by the formula {sigma}M = (32/{pi} D3) {Sigma} (0.5 {rho}a Ui2 Sp CD) H ~ (10/D3) {Sigma} (0.5 {rho}a Ui2 Sp CD) H, where D is maximum (basal) stem diameter and H is estimated plant height (i.e., either HP or HSF).

This formula emphasizes that estimates of overall plant size (D and H) exert a powerful influence on estimates of stress, since D and H are used to scale the size of all elements i to compute projected surface area and thus drag forces and since D is raised to the third power to calculate stresses.

Factors of safety
The factor of safety was computed using the formula {sigma}b/{sigma}M, where {sigma}b is the breaking stress of the plant tissue believed to serve as the principal stiffening agent at the base of each fossil species. For the purposes of our analyses, no distinction was made between tissue-breaking stress and the yield stress because any loading condition that exceeds the breaking stress will result in catastrophic mechanical failure and because any loading condition that results in tissue yielding will produce permanent plastic deformation, either of which can severely impair or kill a plant.

The values for the tissue-breaking stresses used to calculate the factor of safety were taken either from the primary literature or determined experimentally based on bending tests of tissue samples surgically removed from a variety of living plants (data are available upon request). The specific values used for {sigma}b are as follows: parenchyma = 5 MN/m2, collenchyma = 7 MN/m2, primary tracheids = 25 MN/m2, sclerenchyma = 75 MN/m2, and secondary xylem = 80 MN/m2. These values are conservative in that they are neither the highest breaking stresses reported in the literature nor the highest determined by us experimentally for each of these tissues. For example, the breaking stresses of the hypodermal sterome in the basal parts of Equisetum giganteum and E. hyemale aerial stems, which consists either of collenchyma or nonlignified sclerenchyma, range between 30 MN/m2 and 80 MN/m2 (Spatz, Köhler, and Speck, 1998 ; Speck et al., 1998 ). Likewise, subepidermal lignified sclerenchyma isolated from the leaf stalk or midrib of Musa textilis has tensile breaking stresses that range between 35 MN/m2 and 80 MN/m2 depending on their location within the leaf.

We studied and analyzed thin sections and peels of many of the plants treated in this study (Speck, 1994 ; Speck and Vogellehner, 1994 ) and, in addition, surveyed the literature to determine the tissue composition of each fossil treated in this study. In most cases, exceptionally well preserved axes provided unambiguous anatomical information about which breaking stress should be used to compute the factor of safety (e.g., the cortex of Aglaophyton major and Rhynia gwynne-vaughanii is composed of parenchyma). However, in some cases, stem anatomy is either questionable or unknown. For example, it is not clear whether the outer cortex of Drepanophycus spinaeformis is composed of collenchyma or parenchyma, whereas the cortical tissues of Cooksonia spp. are essentially unknown. In these cases, the lower breaking stress among those of candidate tissues was used (e.g., the breaking stress of parenchyma was used to calculate the factor of safety for D. spinaeformis and Cooksonia spp.). For some other species, though having well preserved axes, it was impossible to decide in the fossil material whether the hypodermal sterome was built of collenchyma or sclerenchyma fibres (see Speck and Vogellehner, 1994 ). In these cases, the lower breaking stress among the possible tissues was used (e.g., the breaking stress of collenchyma was used to calculate the factor of safety for Zosterophyllum llanoveranum, Gosslingia breconensis, and Psilophyton dawsonii). For this reason, the factors of safety reported here are likely to be conservative estimates of mechanical reliability.

Within-plant (longitudinal) variation in factors of safety
In addition to determining the factor of safety at the base of each plant fossil, we also determined the extent to which the factors of safety varied as a function of distance above ground along the length of each plant fossil reconstruction. Points of stem attachment were entered manually into computer memory and WinDrag computed the bending moments and stresses as well as the factor of safety at each attachment site based on estimates of local wind speed and scaled stem diameter and length (see above). A representative "path length" for each plant was then selected (i.e., a series of representative attached stems ascending from the base to the most elevated and distal portion of each reconstruction), and the factors of safety were plotted as a function of normalized (relative) plant height HN (where HN = 1.0 and 0.0 at the top and base of the plant, respectively).

Factors of safety were computed for different parts of the plant body assuming either a uniform tissue-breaking stress in the case of species lacking the capacity for secondary growth or different tissue-breaking stresses as a function of stem location in the case of species known to produce secondary tissues. For example, the breaking stress of parenchyma was used to compute the factors of safety for all portions of the branching infrastructure of Rhynia major and Asteroxylon mackiei, and for the more distal stems of Diaphorodendron vasculare and Archaeopteris, whereas the breaking stresses of sclerenchyma and wood were used to compute the factors of safety for the older, more proximal portions of the latter two species. The selection of which tissue-breaking stresses should be assigned to which portions of a woody plant body was, once again, conservative and based on anatomical inspection of well-preserved stems differing in size (and thus location).


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 LITERATURE CITED
 
Plant height estimates
Regression analyses indicated that both of the techniques used to estimate overall plant height (each based on maximum basal stem diameter), on average, obtain estimates that are statistically indistinguishable from the estimated plant heights reported in the primary literature (Fig. 4A). Although a near-isometric relationship exists among all three estimates, the allometric technique, on average, predicted larger plant heights compared to the technique based on the assumption that stems have a 2.5 factor of safety for self-loading. Specifically, the slope of the ordinary least squares regression curve for height estimates using the allometric technique plotted against published height was 1.1 (r2 = 0.90), whereas that of the regression curve for height based on a 2.5 factor of safety plotted against published height was 0.93 (r2 = 0.94). With the exception of Pleuromeia longicaulis and Drepanophycus spinaeformis, there was considerable agreement among published plant height and height estimated on the basis of the allometric and self-loading safety factor techniques (Fig. 4B).



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  		Fig. 4. Estimated and published heights (HP) for 17 fossil species. Estimated plant heights are based on either the empirically determined allometric relationship between height and stem diameter, designated as HA (see Niklas, 1994 ), or based on a 2.5 factor of safety against mechanical failure resulting from self-loading, designated as HSF (see Speck and Vogellehner, 1992, 1994 ; Speck, 1994 ; Speck and Rowe, 1998 ). (A) HA and HSF plotted as a function of HP. No published height estimates exist for Psilophyton dawsonii. Solid lines denote ordinary least squares regression curves for each data set; the line with the larger slope is the regression curve for HA. (B) Estimated plant heights (HP, HA, and HSF) plotted as a function of maximum basal stem diameters D reported in the literature. Thick curved line is the ordinary least squares regression curve for HA vs. D; thin lines denote the 95% confidence intervals of this curve. The heights of two taxa based on the assumption of a 2.5 factor of safety against self-loading failure (i.e., HSF) fall below the 95% confidence intervals (i.e., Drepanophycus spinaeformis and Pleuromeia longicaulis)

 
Based on these analyses, plant height estimates based on both the allometric and the 2.5 self-loading safety factor techniques were used to provide a comparison between conservative and nonconservative estimates of wind speeds, bending moments and stresses, and wind-loading factors of safety for each species.

Plant size and wind speeds
A statistically significant relationship was found between estimated above-canopy wind speed UH and maximum stem diameter D for the 17 species, regardless of the technique used to estimate plant height. Regression of the log10-transformed data for UH (computed on the basis of HSF) against basal stem diameter gave log UH = 1.2 + 0.40 log D – 0.002 (log D)2 (r2 = 0.85, P < 0.0001). Regression of the log10-transformed data for UH (computed on the basis of HA) against basal stem diameter gave log UH = 1.5 + 0.30 log D – 0.24 (log D)2 (r2 = 1.0) (Fig. 5). Both of these relationships indicate that, on average, above-canopy wind speed is not expected to increase in direct proportion to an increase in basal stem diameter, especially for the tallest species in the data set (e.g., Archaeopteris, Diaphorodendron, and Psaronius). We interpret these relationships to indicate that an evolutionary increase in plant height was attended by a disproportionate increase in basal stem diameter, presumably as a mechanical adaptation to increases in above-canopy wind speeds resulting from an evolutionary increase in plant height.



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  		Fig. 5. Above-canopy wind speeds UH (see Fig. 3 ) plotted against maximum basal stem diameter D reported in the literature for 17 Paleozoic plant fossils. Two UH are plotted for each species, one for each of the two techniques used to estimate height (see Fig. 4B ). Curvilinear line denotes ordinary least squares regression curve for wind speeds computed on the basis of allometric estimates of plant height (HA)

 
Maximum bending stresses
When the data for all 17 species were log10-tranformed, wind-induced bending maximum stresses {sigma}M varied curvilinearly or linearly as a function of estimated plant height depending on whether plant height was estimated allometrically, HA, or on the basis of a 2.5 safety factor against self-loading failure, HSF. However, regardless of the technique used to estimate plant height, bending stresses were predicted to increase among nonwoody species with increased height and to decrease relative to plant height among the taller species in the data set. Specifically, ordinary least squares regression analysis indicated that a second-order polynomial curve best fit the data when height was estimated on the basis of a 2.5 factor of safety, log {sigma}M = 4.2 + 0.80 log HSF – 0.24 (log HSF)2 (r2 = 0.64, P < 0.0007), indicating that taller species in the data set experienced smaller bending stresses than their shorter counterparts relative to their absolute height (Fig. 6A). Likewise, regression analysis obtained a linear best fit between the maximum bending stress and plant height estimated allometrically, log {sigma}M = 4.1 + 0.73 log HA (r2 = 0.76, P < 0.0001), indicating that the magnitude of the maximum bending stress decreased relative to plant height as absolute plant height increased across the species examined (Fig. 6B). Owing to the smaller sample size, similar regression analyses for each of the two species groups (rhyniophytes and related species, n = 9, and zosterophyllophytes and lycopods, n = 8) indicated weaker statistical correlations between the maximum bending stress and HA or HSF (0.36 ≤ r2 ≤ 0.59). However, in each case, the trend observed within each of the two groups agreed with the general trend observed when the two species groups were pooled.



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  		Fig. 6. Maximum bending stresses {sigma}M estimated for rhyniophyte and presumed related species (R-T) and zosterophyllophytes and lycopods (Z-L) plotted against estimated plant height H. (A) Maximum stresses calculated for plant height assuming a 2.5 factor of safety against self-loading (i.e., HSF). Curved line denotes ordinary least squares regression curve for both species groups (pooled data). (B) Maximum stresses calculated on the basis of plant height assuming an allometric relationship for plant height (i.e., HA). Solid line denotes ordinary least squares regression curve for the pooled data

 
A linear or curvilinear statistical relationship was found when maximum bending stresses were regressed against maximum stem diameters published in the literature. Regardless of the method used to compute the plant heights required for calculating bending stresses, the wind-induced bending stresses obtained for species with larger stem diameters were smaller for the taller species relative to those calculated for the shorter species. Specifically, when the data from the 17 species were log10-transformed and pooled, ordinary least squares regression of maximum bending stresses computed on the basis of HSF gave log {sigma}M = 5.5 + 0.76 log D (r2 = 0.75, P < 0.0001), indicating that {sigma}M tends to decrease relative to D as plant height increases across species (Fig. 7A), whereas regression of the maximum bending stresses computed on the basis of HA gave log {sigma}M = 5.3 + 0.37 log D – 0.12 (log D)2 (r2 = 0.77, P < 0.0001), indicating, once again, that {sigma}M tends to be smaller relative to D for the taller species in the data set (Fig. 7B). Due to the smaller sample sizes, similar regression analyses for the two species groups (rhyniophytes and related species, n = 9, and zosterophyllophytes and lycopods, n = 8) indicated weaker statistical correlations between the maximum bending stress (computed on the basis of HSF or HA) and maximum stem diameter (0.21 ≤ r2 ≤ 0.62). In each case, however, the trend observed for each of the two species groups agreed with the trend observed for the pooled species (n = 17).



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  		Fig. 7. Maximum bending stresses {sigma}M estimated for rhyniophytes and presumed related species (R-T) and zosterophyllophytes and lycopods (Z-L) plotted against maximum basal stem diameter D reported in literature. (A) Maximum stresses calculated on the basis of plant heights assuming a 2.5 factor of safety against self-loading (i.e., HSF). Solid line denotes ordinary least squares regression curve for both species groups (pooled data). (B) Maximum stresses calculated on the basis of plant heights assuming an allometric relationship between height and stem diameter (i.e., HA). Solid line denote ordinary least squares regression curve for the pooled data

 
Factors of safety
Regardless of how plant height was estimated, the factor of safety against wind-induced mechanical failure {sigma}b/{sigma}M decreased among progressively taller nonwoody species and then increased among progressively taller woody plant species. Based on HSF, ordinary least squares regression of the log10-transformed data for the 17 species gave log ({sigma}b/{sigma}M) = 2.7 – 0.26 log HSF + 0.55 (log HSF)2 (r2 = 0.35). Although not statistically robust (P < 0.048), the trend across species indicated that the tallest species had larger factors of safety than many species other than their smallest counterparts (Fig. 8A). Based on HA, regression of the log10-transformed data gave log ({sigma}b/{sigma}M) = 2.8 – 0.35 log HA + 0.25 (log HA)2 (r2 = 0.52), which is statistically significant (P < 0.0056) (Fig. 8B). Although regression analyses of the data from the two species groups failed to identify highly statistically significant relationships between the factor of safety and plant height, the trends observed for each of the two groups complied with that observed for the pooled data.



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  		Fig. 8. Factors of safety (i.e., {sigma}b/{sigma}M) for rhyniophytes and presumed related species (R-T) and zosterophyllophytes and lycopods (Z-L) plotted against estimated plant height. (A) Maximum stresses calculated on the basis of a 2.5 factor of safety against self-loading (i.e., HSF). Solid line denotes ordinary least squares regression curve for both species groups (pooled data). (B) Maximum stresses calculated on the basis of an allometric relationship between plant height and basal stem diameter (i.e., HA). Solid line denote ordinary least squares regression curve for the pooled data

 
No statistically significant correlation was found for the pooled data when the factor of safety (computed on the basis of HSF) was plotted against maximum basal stem diameter D (Fig. 9A). In contrast, the factor of safety against wind-induced mechanical failure (computed on the basis of HA) was significantly correlated with stem diameter: log ({sigma}b/{sigma}M) = 2.9 + 0.62 log D + 0.30 (log D)2 (r2 = 0.50, P < 0.008), indicating that the safety factor decreased among progressively larger nonwoody species and then increased among the larger species in the data set (Fig. 9B). This correlation was a consequence of the data for rhyniophytes and related species, since regression gave log ({sigma}b/{sigma}M) = 2.8 + 0.61 log D + 0.30 (log D)2 (r2 = 0.54, P < 0.095) for these species. Nonetheless, the general trend in the data indicated that the factors of safety for the largest among the species examined were comparable to those of many of the smaller species in the data set. For example, the factor of safety (computed on the basis of a 2.5 factor of safety against self-loading failure) for Archaeopteris was comparable to that of Rhynia and not significantly above Zosterophyllum llandoveranum (see Fig. 9A).



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  		Fig. 9. Factors of safety (i.e., {sigma}b/{sigma}M) for rhyniophytes and presumed related species (R-T) and zosterophyllophytes and lycopods (Z-L) plotted against maximum basal stem diameter D reported in the primary literature. (A) Maximum stresses calculated on the basis of plant height assuming a 2.5 factor of safety against self-loading (i.e., HSF). (B) Maximum stresses calculated on the basis of plant height assuming an allometric relationship between height and stem diameter (i.e., HA). Solid line denotes ordinary least squares regression curve for both species groups (pooled data)

 
The factor of safety varied within the branching structure of individual species. Two general patterns of variation were observed, one found among nonwoody species in which the safety factor decreased acropetally in a step-wise pattern along stem path lengths from base to the tip of the plant (Fig. 10A, B) and another pattern found in woody species in which the safety factor decreased along the length of the trunk and then varied among lateral branches (Fig. 10C, D). The step-wise pattern was attributed to the dichotomous branching patterns of the nonwoody species examined, since each step-wise decrease in the factor of safety computed along a path length of interconnecting stems was associated with the junction of two nearly equivalent sized stems. The pattern observed among the stems of woody species was attributed to the effects of heterogeneity in the tissue providing the principal stiffening agent in stems differing in size (diameter) and to the population of branches that imposes an additive bending moment at the top of trunks and lateral branches experiencing different bending moments depending on their location (in the wind-speed profile) and size.



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  		Fig. 10. Variations in the factor of safety (i.e., {sigma}b/{sigma}M) along representative stem path lengths plotted as a function of normalized plant height HN for two nonwoody species (A and B) and two woody species (C and D). Maximum bending stresses were computed on the basis of a 2.5 safety factor against self-loading (i.e., HSF). (A) Rhynia major. (B) Asteroxylon mackiei. (C) Archaeopteris. (D) Diaphorodendron vasculare

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
LITERATURE CITED
 
As an empirical study, our attempt to calculate factors of safety against wind-induced mechanical failure for fossil plant species is subject to many sources of error. We have already drawn attention to the requirements for reliable whole-plant reconstructions as well as difficulties in estimating absolute plant height from fragmented fossil remains. Our conceptualization of the growth habit of even such well-known plants as Archaeopteris continues to change as new information comes to light (see Meyer-Berthaud, Scheckler, and Bousquet, 2000) . Likewise, much remains to be learned concerning the community structure of many fossil plants, especially the spacing among neighboring conspecifics, which influences estimates of within canopy wind speeds (see Remy, Remy, and Hass, 1997 ). For these reasons, attempts to quantify the wind speeds profiles and the bending moments and stresses resulting that influence early land plant evolution must be viewed with as much or more skepticism than we have tried to convey. However, as a theoretical exercise, we believe the approach presented here has merit, since it uses the same mathematical tools that have been successfully applied to study the effects of wind on living plants, which are consistent with those reported here. Even so, the following discussion is presented largely in the context of a theoretical treatment of the affects of wind-loading on early Paleozoic land plant evolution.

Taken at face value, the results of this study are consistent with the hypothesis that the factor of safety against wind-induced mechanical failure decreased as plant height increased during the early stages of land plant evolution only to increase with the subsequent evolutionary origin of secondary growth among the rhyniophytes and their presumed descendant species. The data for the zosterophyllophytes and lycopods are less clear, since only very weak correlations were found between the factor of safety and plant height or stem diameter for the species of this group represented in our data set. Nonetheless, the largest lycopod in our data set, Diaphorodendron, is estimated to have had a substantial factor of safety compared to some of its herbaceous antecedents (e.g., Gosslingia), which is once again consistent with the hypothesis that arborescent species were adapted to substantial wind-loadings.

However, the data set used to evaluate whether factors of safety against wind-induced mechanical failure evidenced an evolutionary trend was heavily skewed in terms of sampling Devonian taxa. Only two Carboniferous taxa (Diaphorodendron and Psaronius) and only one Triassic species (Pleuromeia longicaulis) were represented among the 17 species examined by us. This bias in the age distribution of species, which was in part due to the availability of reliable whole-plant reconstructions, prevented us from drawing statistically meaningful conclusions regarding an evolutionary trend in the factors of safety. Qualitatively, however, an inverse relationship between plant size (measured either in terms of maximum stem diameter or estimated plant height) and the factor of safety (computed either on the basis of the bending stresses associated with HSF or HA) was observed (Fig. 11). This relationship accorded with the hypothesis that the evolutionary advent of the arborescent growth habit and the attending increase in plant size across species nonetheless either maintained or increased the factor of safety against wind-induced mechanical failure.



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  		Fig. 11. Plant size and factors of safety for 17 species plotted as a function of geological age. (A) Plant size (measured in terms of maximum [reported] basal stem diameter D, published maximum height HP, allometric estimates of height HA, and height estimated on the basis of a 2.5 factor of safety against self-loading HSF) plotted against age. (B) The factor of safety against wind-induced mechanical failure (estimated on the basis of HSF) plotted against age. (C) The factor of safety against wind-induced mechanical failure (estimated on the basis of HA) plotted against age. See text for further details

 
Among the nonwoody species examined, plant size, on average, increased during much of the Devonian (Fig. 11A), whereas the factor of safety, on average, decreased among these species, regardless of which height was used to estimate bending stresses (Fig. 11B and C). By Carboniferous times, plant size increased by two orders of magnitude among the species examined, whereas the factor of safety computed for these arborescent species was either equal to or greater than that of some antecedent nonwoody species. The short unbranched but leafy pachycaulis growth habit of the Triassic species Pleuromeia longicaulis was associated with a factor of safety equal to that of geologically much older nonwoody species of comparable height but characterized by profuse branching, indicating that a number of morphological and anatomical factors other than arborescence and woody tissues can influence the ability to sustain wind-loadings.

Analyses of the factors of safety within the branched architecture of fossil arborescent taxa, such as Archaeopteris and Diaphorodendron, are also consistent with the hypothesis that plant evolution has responded to the effects of wind- as well as self-loading. Our analyses indicate that the factor of safety varies along the lengths of woody plants and reaches equivalently local minima among a constellation of smaller, more distal branches (see also Niklas, 2000) . Similar patterns in the longitudinal variation of the factor of safety are reported for extant arborescent species (see Niklas and Spatz, 2000) . Since peripheral portions of the arborescent plant body are the most susceptible to mechanical failure and since failure at the base of a tree is arguably more devastating, it is reasonable to suggest that some of the distant and most recent descendants of the earliest land plants have evolved adaptations to progressively larger wind-loadings as plant size increased such that they "self-prune" when subject to exceptionally high wind speeds. The loss of some younger stems can reduce drag and thus bending moments and stresses and in this way reduce the likelihood of trunk failure.

In contrast, our analyses of nonwoody early Paleozoic fossil plants indicates that their factors of safety decreased in a step-wise manner from the base to the top of the plant body. This pattern is unquestionably the consequence of their iso-dichotomous branching geometry and anatomical homogeneity, since each step-wise decrease in the factor of safety corresponds to the location of a bifurcation in the branching geometry at which point the wind-induced bending stresses experienced by the two more distal stems are additively transmitted to the subtending stem sharing much the same anatomical configuration and size. The resulting step-wise pattern of safety factors cannot be maintained indefinitely as overall plant size increases, since the most distal elements of the plant body, which support sporangia in the case of many species, would have safety factors approaching or dropping below unity. It is thus reasonable to conclude that the changes in the branching geometry attending the evolutionary transformation of nonwoody to woody plants were functionally adaptive in terms of coping with the larger drag forces taller plants typically experience.

From first principles, we know that the drag forces exerted by wind on the aboveground portions of plants increase as the density of neighboring plants decreases or as plant height, projected surface area, or wind speed increases. Previous studies also indicate that estimates of the drag forces acting on plants are far less sensitive to the shape of the wind-speed profile (and thus, to a limited degree, on estimates of community density) than to plant morphology (e.g., the frequency of branching and the manner in which stems taper). For this reason, we believe that precise knowledge of community structure and the shape of the wind speed profile is less essential than reliable morphological information when attempting to estimate the relative differences among the drag forces acting on different fossil plants.

Clearly, however, the absolute magnitudes of wind speeds are critical to estimating drag forces, since these forces are computed on the basis of the square of ambient wind speeds. In this regard, community structure and the locations of individuals are critical, since closely space conspecifics will reduce ambient wind speeds that will vary as a function of location within the same community. Individuals growing at the perimeter of a dense community are likely to experience higher wind speeds than those sheltered by others within the community as a whole. It is notable, however, that most extant plant species are thigmomorphogenetic such that individuals that experience chronic mechanical perturbation are reduced in size and produce more flexible organs compared to those that are sheltered from the wind. This response to wind reduces the drag forces exerted on individuals exposed to higher wind speeds (and thus increases the factor of safety for wind-loading) by reducing the surface area projected toward the wind, since absolute size is reduced and since organs can bend in the wind and thereby reduce their projected areas. Unfortunately, even though there is no reason to believe that thigmomorphogenesis did not evolve by early Paleozoic times, we have little information from the fossil record regarding the degree to which different species were capable of adjusting their size and the material properties of their tissues or organs in response to wind-induced mechanical perturbations.

From a theoretical perspective, it is clear that plant stature cannot increase without evoking large deformations or buckling unless stem diameter or tissue stiffness increases. This conclusion holds true for wind- as well as self-loading, because the magnitudes of bending forces increase nonlinearly and rapidly as a function of distance above ground due to an increase in wind speeds or the additive mass of the plant body. Even a small increase in stem diameter will dramatically increase the mechanical stability of a vertical plant stem, because the axial second moment of area (which is a measure of the contributions that absolute size, shape, and geometry make to the ability of a beam or column to resist bending) increases exponentially as a function of diameter. Likewise, for any force F, the magnitude of the resulting stress {sigma} is reduced with even a modest increase in the diameter d of a terete stem, since {sigma} = 4 F/{pi} d2. The mechanical advantages of increased stem diameter are thus obvious and help to explain why plant height correlates so well with basal stem diameter across a broad taxonomic spectrum of extant plant species.

It is equally obvious that any increase in stem diameter increases the magnitude of self- and wind-loadings by increasing the mass and projected area elevated above ground. The benefits of evolving stiffer tissues and locating these agents at or just below the perimeter of the stem cross section are clear (since stiffer tissues can sustain higher bending stresses before they yield or break and because bending stresses reach their highest intensities at the surface of a flexed stem). Our data agree with this supposition. Taken at face value, they indicate that the most ancient vascular land plants had extremely large factors of safety. Their small stature and comparatively sparsely branched morphologies exposed them to comparatively low wind speeds and drag forces, which was more than adequate to compensate for the comparatively low breaking stresses of their parenchymatous or collenchymatous cortical tissues and the absence of secondary growth. However, with increasing height and branching, the factor of safety, on average, steadily declined (albeit not to potentially dangerous levels) despite the appearance of more stiff and strong cortical tissues (e.g., sclerenchyma). This trend of decreasing mechanical reliability was reversed with the appearance of species capable of manufacturing wood or of deploying and accumulating sclerified organs around the base of their stems (e.g., the adventitious roots of Psaronius), either of which could produce a broad stem base.

We believe that the protocol presented here is useful and holds the potential to shed light on the consequences of morphological and anatomical elaboration on the mechanical stability of terrestrial plants.


   FOOTNOTES
 
1 The authors thank Prof. Dr. Hanns-Christof Spatz (Institute of Biology III, Freiburg Universität) who supervised the review process and served as Editor-in-Chief for this manuscript, and two anonymous referees who provided many helpful comments. This research was supported by Hatch Act Award 185-403 and an Alexander von Humboldt Stiftung prize (to KJN). Back

4 Author for correspondence (Phone: 607-255-8727; FAX: 607-255-5407; kjn2{at}cornell.edu) Back


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 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
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