| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
Ecology |
2Department of Plant Biology, Cornell University, Ithaca, New York 14853 USA; 3Institut für Biologie III, Universität Freiburg, Freiburg D-79104, Germany
Received for publication January 31, 2006. Accepted for publication March 6, 2006.
ABSTRACT
Recent allometric theory has postulated that standing leaf mass will scale as the 3/4 power of stem mass and as the 3/4 power of root mass such that stem mass scales isometrically with respect to root mass across very large vascular plant species with self-supporting stems. We show that the isometric scaling of stem mass with respect to root mass (i.e., MS 
MR) can be derived directly from mechanical theory, specifically from the requirement that wind-induced bending moments acting at the base of stems must be balanced by a counter-resisting moment provided by the root system to prevent uprooting. This derivation provides indirect verification of the allometric theory. It also draws attention to the fact that leaf, stem, and root biomass partitioning patterns must accommodate the simultaneous performance of manifold functional obligations.
Key Words: allometry • bending moments • biomechanics • plant biomass partitioning • root biomass • scaling rules • stem biomass • wind drag
Recent and rapid theoretical developments in the area of a theory for plant allometry have far-reaching implications to ecology and evolution (Enquist and Niklas, 2001
, 2002
; Niklas and Enquist, 2001
, 2002a
, b; Niklas, 2003
, 2004
). These developments have considerably over-reached the available data with which to currently test them empirically. Although a few large data bases exist for this purpose (see Niklas, 2003
, 2004
), each carries the risk of affecting statistical results by virtue of "phyletic effects," i.e., differences in the phylogenetic composition of even large data bases can substantially alter the numerical values of allometric constants and scaling exponents (Niklas, 1994
). An alternative approach to dealing with the need to verify the claims of allometric theory is to determine whether they are mathematically and conceptually consistent with physical "first principles."
Here, we consider the juxtaposition of allometric theory with the mechanical theory treating wind-induced bending moments and the manner in which vascular plant biomass is partitioned annually in constructing leaves, stems, and roots. To remain mechanically stable when dynamically loaded by the wind, the bending moment generated at the stem base by any drag force (designated as MOMB) must be balanced by a counter-resisting moment generated by the root system (designated by MOMCR) (Wainwright et al., 1976
; Niklas, 1992
; Ennos, 1993
). Failure to do so will result in uprooting and eventual death (Coutts, 1983
; see also Dupuy et al., 2005
). Uprooting and stem breakage of healthy trees are observed only under very high wind speeds (e.g., Putz et al., 1983
). It is logical therefore to assume that natural selection has resulted in biomass partitioning patterns that achieve, on average, a balance between bending and counter-resisting moments.
Nevertheless, it is unclear whether the mathematics underwriting this fundamental physical requirement analytically provide support for the explicit theoretical claim that standing leaf mass will scale as the 3/4 power of stem mass and as the 3/4 power of root mass such that stem mass scales isometrically with respect to root mass across most vascular plant species with self-supporting stems, i.e., in simple proportional terms, ML MS3/4 MR3/4 such that MS MR (where M denotes dry mass and the subscripts L, S, and R refer to leaves, stems, and roots, respectively) (Enquist and Niklas, 2002
). Because leaf and stem mass provide obstructions to airflow (and thus generate drag forces) and because stem mass is correlated with plant height (and thus with the lever arm of wind-induced bending moments; see Vogel, 1981
; Niklas, 1995
, 2003
; Niklas and Spatz, 2004
), it is reasonable to assume that MS provides a gauge of the magnitude of the wind-induced bending moments plants experience. Likewise, the counter-resisting moments generated by root systems must be related in some way to root mass (Ennos, 1993
). Therefore, if the proportional relationship MS MR emerges directly from the physical requirement that MOMB = MOMCR, it indirectly tests whether the theory for plant allometry is correct.
Here, we show analytically that this is the case. Specifically, we show that the isometric relationship MS MR can be derived directly from the fundamental physical stipulation that MOMB = MOMCR based on a few biologically reasonable assumptions regarding plant hydraulic requirements, which are themselves size dependent. This demonstration provides indirect support for the assumptions embedded in the allometric theory asserting that MS MR holds true across ecologically and phylogenetically diverse species. Equally important, we believe this demonstration provides evidence that the manner in which biomass is annually partitioned to produce new leaf, stem, and root tissues reflects an evolutionary optimization solution to the simultaneous performance of multiple biological tasks required for growth, survival, and reproductive success (e.g., mechanical stability, transpiration, elevation, and the display of photosynthetic leaves).
DERIVING MS MR FROM MECHANICAL THEORY
As noted, wind-induced bending moments at the plant stem base must be balanced by the counter-resisting moments generated by the root systems (Fig. 1), i.e.,
|
| (1) |
), i.e.,
|
| (2) |
), these effects cannot be modeled canonically (see Discussion). Neglecting the bending forces that are generated by the weight of stems and leaves (Spatz and Brüchert, 2000
), the wind-induced bending moment must equal the product of the wind-induced drag force DF exerted at mid-canopy height H and the lever arm, which equals H, i.e.,
|
| (3) |
ASuH2CD, where is the density of air, AS is the canopy sail area, uH is the wind speed measured at H, and CD is the drag coefficient (Vogel, 1981
). Provided that CD numerically varies little as a function of uH, eq. 3 takes the form
|
| (4) |
CD) has dimensions m5. kg–1. s–2. Note that under real biological conditions, CD can vary in a manner that causes drag to scale to a power of ambient wind speed as low as 3/2 (Wood, 1995
). However, substituting DF uH3/2 for DF uH2 only modestly changes the predicted numerical value of the scaling exponent governing the relationship of stem with respect to root mass (see Discussion). Thus, the assumption that CD varies little is not critical.
|
; Niklas and Spatz, 2000
), i.e., AS = ß3rL2 and uH2 = ß4H. Under these circumstances, eq. 4 can be restated as
|
| (5) |
In passing, we also note that many tree crown silhouettes are better approximated by an ellipse (with a major axis either vertically or horizontally aligned with ground level) such that AS is given by the formula 
ab, where a and b are the major and minor semi-axes. However, inserting this variant expression for AS into the subsequent steps of our derivation does not alter the outcome of the derivation in any substantive way.
The derivation of MS MR concludes by noting from theory and observation that the hydrodynamic requirement to provide water through stems to leaves requires leaf mass to scale roughly as the square of basal stem radius rS, i.e., ML = ß6rS2 (see Niklas and Spatz, 2004
). Provided that ML is proportional to the cube of canopy radius (i.e., ML = ß7rL3), it follows that ß7rL3 = ß6rS2 such that, across ecologically diverse species, rL2 = (ß6 / ß7)2/3rS4/3 = ß8rS4/3. Inserting this last expression into eq. 5 gives the formula
|
| (6) |
). Therefore, eq. 6 can be recast as
|
| (7) |
|
| (8) |
|
| (9) |
DISCUSSION
The preceding shows how the biomass partitioning pattern adduced from recent allometric theory (i.e., MS MR) can be derived directly from the theory of mechanics treating wind-induced stem bending moments. This derivation employs five "conjectures," i.e., (1) the root counter-resisting moment is isometrically proportional to root mass, (2) the numerical value of the drag coefficient varies little over a broad range of tree geometries and wind speeds, (3) wind speed measured at canopy height is well approximated by a square root vertical profile, (4) leaf mass scales as the cube of the canopy radius and the square of basal stem diameter, (5) plant height, and thus mid-canopy height H, scales as the 2/3 power of basal stem diameter, and following from conjectures 4 and 5, stem mass scales as the 8/3 power of basal stem diameter. Many of these conjectures are empirically demonstrable (for examples, see Figs. 2–3). However, here, for the sake of analysis, we show that each of these conjectures is justifiable on theoretical grounds as well.
|
|
; Ennos, 2000
and references therein; Niklas et al., 2002
). In the case of taproots, failure can occur by rotating the root through the soil. Ennos (1993)
has shown that the moment necessary is proportional to the mass of the root system, with or without horizontal fibrous roots coming from the bottom of the taproot. Ennos (1993)
has also shown that for plate systems the resistance of the leeward hinge to bending, the resistance of windward roots and sinkers to being pulled out and the resistance of the soil below the plate to tensile failure will all be proportional to the mass of the roots. The resistance against overturning that originates from the root–soil plate acting as a counterweight scales with a slightly higher scaling power than unity. However, unless the soil is extremely wet, this source of resistance should be small as compared to the other three.
Ennos' (1993) derivations are based on the theory of earth piles (Broms, 1964
), which shows that, for a broad spectrum of root-like morphologies, the root counter-resisting moment scales isometrically with respect to the product of average root length LR and the square of average root radius RR (i.e., MOMCR LRRR2), or as some function of the cube of RR3, or as some function of LR2RR. If for the majority of roots, even in very old root systems, an isometric scaling relationship exists for root length with respect to root radius (i.e., LR RR; see Ennos, 1993
; Schenk and Jackson, 2002
) and because the mass of an average root is proportional to the product of root length and the square of root radius (i.e., MR LRRR2), it follows that the root counter-resisting moment will scale one-to-one with respect to root mass (i.e., MOMCR LRRR2 MR, or MOMCR RR3 MR). Alternative morphological models for root systems do not change the conclusion that the counter-resisting moment is proportional to root mass.
Conjecture (2)
The conjecture that the numerical value of the drag coefficient (i.e., CD) varies little across a large range of wind speeds rests on the work of Rouse (1946)
, Fraser (1962)
, and Mayhead (1973)
. Rouse (1946
; see also Campbell, 1977
) provides the numerical values for CD for geometrically two- and three-dimensional objects differing in size and variously oriented toward oncoming airflow. These values for CD range between 0.17 and 0.95 (for a hemisphere with its convex surface oriented upwind and an infinitely long rectangular plate, respectively). More germane biologically, using a wind tunnel, Fraser (1962)
determined the drag forces and drag coefficients for a series of pine trees differing in size subjected to wind speeds ranging between 9 and 26 m/s and reported little substantive numerical variation in CD. This conclusion was subsequently confirmed by Mayhead (1973)
whose reexamination of the data indicated that CD varied between 0.11 and 0.43 across a range of wind speed that varied by a factor of four (39 and 10 m/s, respectively). Likewise, more recent studies have confirmed that drag forces, particularly those exerted on broadleaf tree species, can decrease with increasing wind speeds by virtue of a reduction in canopy sail area resulting from leaf curling and twig flexure (Vogel, 1992
, 1995
). Collectively, these and other theoretical and empirical studies indicate that the magnitude of CD is confined across a range of normally experienced wind speed such that it may be submerged into an appropriate allometric "constant", e.g., eq. 4.
Importantly, the assumption that CD can be approximated as a constant is not critical for our derivation to work. Empirically, drag can scale as the 3/2 (rather than as the square) power of ambient wind speed, i.e., DF uH3/2 rather than DF uH2 (see eq. 4). This scaling relationship likely emerges as a result of the systematic reduction of CD with increasing uH as a consequence of tree crown "streamlining." Inserting this 3/2 relationship into our derivation obtains the prediction that MS MR16/15 = 1.066 which differs little from MS MR (see eq. 9).
Curiously, prior empirical studies of the allometry of stem mass with respect to root mass report that the scaling exponent exceeds unity (Enquist and Niklas, 2002
), which differs from the theoretical prediction of isometry. This discrepancy has been rationalized away by arguing among other things that the root systems of progressively larger trees become disproportionately more difficult to excavate such that root mass is systematically underestimated for larger tree-sized plants (e.g., Niklas, 2004
). An equally plausible explanation is that the exponent governing stem vs. root mass truly exceeds unity. If so, then our derivation using the proportional relationship DF uH3/2 accords well with observation (and proposition) that biomass allocation patterns favor mechanical stability under windy conditions.
Conjecture (3)
A number of alternative wind speed profiles exist for the purpose of modeling wind-induced bending moments (Bussinger, 1975
; Campbell, 1977
; Herbig et al., 1988
; Spatz and Speck, 1994
; Spatz and Brüchert, 2000
). Most common is the use of a logarithmic profile (Peltola et al., 1993
; Niklas and Spatz, 2000
). However, allometric theory cannot deal with log-log nonlinear power relationships. The conjecture that wind speed measured at canopy mid-height can be approximated by a square root vertical profile is particularly realistic for plants growing in open terrain (Gardiner et al., 1997
; Niklas and Spatz, 2000
). Specifically, the square root wind speed profile is simulated by the formula ui /U = (xi /Href)1/2, where ui is wind speed measured anywhere along the length of the plant, U is the wind speed measured at a reference height Href, and xi is the vertical distance measured from ground level (Niklas and Spatz, 2000
). Thus, ui2 will scale as mid-canopy height H as asserted in eq. 5.
Conjecture (4)
The assertion that leaf mass scales as the square of stem radius receives substantial statistical support (Fig. 2A; see Niklas 2003
, fig. 1A). But we believe that this "conjecture" is justified theoretically based on fluid as well as mechanical theory. Specifically, it is consistent with the observation that xylem water flux per day, denoted here by 
(in units of kg . m–2. h–1), is reported to scale as the 3/4 power of aboveground mass across 37 tree species (Enquist et al., 1998
; for additional confirmation, see Meinzer, 2003
). Noting that aboveground mass is the sum of stem and leaf mass (MS and ML, respectively), this scaling relationship is expressed by the formula = ß15(MS + ML)3/4, which can be recast exclusively in terms of basal stem radius rS, because for large plants, MS = ß13rS8/3 and ML = ß6rS2 (Fig. 2B; see Niklas and Spatz, 2004
). Inserting these relationships into the equation for yields = ß15(ß13rS8/3 + ß6 rS2)3/4. Because MS >> ML, the second term becomes numerically trivial such that ~ ß16rS2, which predicts that xylem water flux per day per plant scales, on average, as the square of the basal stem diameter. Equally important, because it is reasonable to believe that scales isometrically with respect to leaf mass, it follows that ML rS2.
Conjecture (5)
The suppositions that plant height, and thus mid-canopy height H, scales as the 2/3 power of basal stem radius and that stem mass scales as the 8/3 power of basal stem diameter also receives substantial empirical support based on the analyses of large data bases and have been previously derived analytically (McMahon, 1973;
McMahon and Kronauer, 1976;
Niklas and Spatz, 2004
). Specifically, we have shown elsewhere that across all plant sizes, L = ß15rS2/3 – ß16 and MS = ß17rS8/3 – ß18rS6/3, where L is plant height (Niklas and Spatz, 2004
). These formulas indicate that L will scale as the 2/3 power of r when ß15rS2/3 >> ß16 and that MS will scale as the 8/3 power of rS, when ß17rS2/3 >> ß18. Prior work indicates that these conditions hold true for moderate- to large-sized plants such that L rS2/3 and MS rS8/3 (as alleged in eqs. 8 and 9).
Conclusions
That an isometric or near-isometric scaling relationship exists between stem and root mass is empirically evident (Fig. 3; Niklas, 2004
, fig. 2C). That it can be shown to emerge analytically from the first principles of mechanical theory provides indirect confirmation of an allometric approach to predict plant biomass partitioning patterns across ecologically and phylogenetically diverse species with self-supporting stems and large foliage leaves.
Clearly, this demonstration rests on a number of assumptions, some of which are empirically justified and others of which require direct tests in the field. However, we believe that our derivation fosters a deeper appreciation of how the "scaling rules" epitomized by plant biomass partitioning patterns are capable of reconciling the simultaneous performance of manifold biological obligations, which have potentially antagonistic "design" specifications, such as the optimal display of foliage leaves to sunlight, the mechanical instabilities resulting from externally applied mechanical forces, and the constraints on the translocation of water from roots, through stems, to leaf mesophyll (see McCulloh and Sperry, 2005
). Because none of these functional obligations has priority over the others in terms of growth and survival, the performance of all must be reconciled and thus optimized in accommodating ways. When viewed in this way, we should be able to derive analytically the key elements identified in the biomass partitioning pattern recently identified by allometric theory from a few basic physical principles.
FOOTNOTES
1 Support from the College of Agriculture and Life Sciences, Cornell University is gratefully acknowledged. 
4 Author for correspondence (e-mail: kjn2{at}cornell.edu
), fax: +1 607-255-5407
LITERATURE CITED
Broms B. B.. 1964. Lateral resistance of piles in cohesive soils. Mechanical Foundations, Division of the American Society of Civil Engineers SM2 90: 27-63.
Bussinger J. A.. 1975. Aerodynamics of vegetated surfaces. In D. A. DeVries and N. H. Afgan [eds.] Heat and mass transfer in the biosphere 104-119 John Wiley, New York, New York, USA.
Campbell G. S.. 1977. An introduction to environmental biophysics Springer-Verlag, New York, New York, USA.
Coutts M. P.. 1983. Root architecture and tree stability. Plant and Soil 71: 171-188.
Dupuy L. Fourcaud T. Stokes A.. 2005. A numerical investigation into the influence of soil type and root architecture on tree anchorage. Plant and Soil 278: 119-134.[CrossRef][Web of Science]
Ennos A. R.. 1993. The scaling of root anchorage. Journal of Theoretical Biology 161: 61-75.[CrossRef]
Ennos A. R.. 2000. The mechanics of root anchorage. Advances in Botanical Research Incorporating Advances in Plant Pathology 33: 133-157.[Web of Science]
Enquist B. J. Brown J. H. West G. B.. 1998. Allometric scaling of plant energetics and population density. Nature 395: 163-165.[CrossRef][Web of Science]
Enquist B. J. Niklas K. J.. 2001. Invariant scaling relations across tree-dominated communities. Nature 410: 655-660.[CrossRef][Medline]
Enquist B. J. Niklas K. J.. 2002. Global allocation rules for patterns of biomass partitioning across seed plants. Science 295: 1517-1520.
Fraser A. I.. 1962. Wind tunnel studies of the forces acting on the crowns of small trees. Report Forestry Research (UK) 1962: 178-183.
Gardiner B. A. Stacey G. R. Belcher R. E. Wood C. J.. 1997. Field and wind tunnel assessments of the implications of respacing and thinning for tree stability. Forestry 70: 233-252.
Herbig A. Sinn G. Wessolly L.. 1988. Zur Standsicherheit von Bäumen im städtischen Bereich. Natürliche Konstruktionen. Leichtbau in Architektur und Nature Mitteilung, SFB 230: 39-57.
Mayhead G. J.. 1973. Some drag coefficients for British forest trees derived from wind tunnel studies. Agricultural Meteorology 12: 123-130.
McCulloh K. A. Sperry J. S.. 2005. Patterns in hydraulic architecture and their implications for transport efficiency. Tree Physiology 25: 257-267.
McMahon T. A.. 1973. The mechanical design of trees. Science 233: 92-102.
McMahon T. A. Kronauer R. E.. 1976. Tree structures: deducing the principle of mechanical design. Journal of Theoretical Biology 59: 443-466.[CrossRef][Web of Science][Medline]
Meinzer F. C.. 2003. Functional convergence in plant responses to the environment. Oecologia 134: 1-11.[CrossRef][Web of Science][Medline]
Niklas K. J.. 1992. Plant biomechanics: an engineering approach to plant form and function University of Chicago Press, Chicago, Illinois, USA.
Niklas K. J.. 1994. Plant allometry: the scaling of form and process University of Chicago Press, Chicago, Illinois, USA.
Niklas K. J.. 1995. Size-dependent allometry of tree height, diameter and trunk-taper. Annals of Botany 75: 217-227.
Niklas K. J.. 2003. Reexamination of a canonical model for plant organ biomass partitioning. American Journal of Botany 90: 250-254.
Niklas K. J.. 2004. Plant allometry: is there a grand unifying theory?. Biological Review 79: 871-889.[CrossRef]
Niklas K. J. Enquist B. J.. 2001. Invariant scaling relationships for interspecific plant biomass production rates and body size. Proceedings of the National Academy of Sciences, USA 98: 2922-2927.
Niklas K. J. Enquist B. J.. 2002a. Canonical rules for plant organ biomass partitioning and growth allocation. American Journal of Botany 89: 812-819.
Niklas K. J. Enquist B. J.. 2002b. On the vegetative biomass partitioning of seed plant leaves, stems, and roots. American Naturalist 159: 482-497.[CrossRef][Web of Science][Medline]
Niklas K. J. Molina-Freaner F. Tinoco-Ojanguren C. Paolillo D. J. Jr.. 2002. The biomechanics of Pachycereus pringlei root systems. American Journal of Botany 89: 12-21.
Niklas K. J. Spatz H.-C.. 2000. Wind-induced stresses in cherry trees: evidence against the hypothesis of constant stress levels. Trees 14: 230-237.[CrossRef]
Niklas K. J. Spatz H.-C.. 2004. Growth and hydraulic (not mechanical) constraints govern the scaling of tree height and mass. Proceedings of the National Academy of Sciences, USA 101: 15661-15663.
Peltola H. S. Kellomäki A. S Hassinen M Lemettinen Aho J.. 1993. Swaying of trees as caused by wind: analysis of field measurements. Silva Fennica 27: 113-126.
Putz F. E. Coley P. D. Lu K. Montalvo A. Aiello A.. 1983. Uprooting and snapping of trees: structural determinants and ecological consequences. Canadian Journal of Botany 13: 1011-1020.
Rouse H.. 1946. Elementary mechanics of fluids John Wiley, New York, New York, USA.
Schenk H. J. Jackson R. B.. 2002. Rooting depths, lateral root spreads and below-ground/above-ground allometries of plants in water-limited ecosystems. Journal of Ecology 90: 480-494.[CrossRef][Web of Science]
Spatz H.-C. Speck T.. 1994. Mechanische Eigenschaften von Hohlrohren am Beispiel von Gräsern. In Nachtigall W. BIONA-report 9. Technische Biologie und Bionik 2 91-132 Akad. Wiss. u. Lit. Mainz. Fischer, Stuttgart, Germany.
Spatz H.-C. Brüchert F.. 2000. Basic biomechanics of self-supporting plants: wind loads and gravitational loads on a Norway spruce tree. Forest Ecology and Management 135: 33-44.[CrossRef][Web of Science]
Vogel S.. 1981. Life in moving fluids: the physical biology of flow Willard Grant Press, Boston, Massachusetts, USA.
Vogel S.. 1992. Twist-to-bend ratios and cross-sectional shapes of petioles and stems. Journal of Experimental Botany 43: 1527-1532.
Vogel S.. 1995. Twist-to-bend ratios of woody structures. Journal of Experimental Botany 46: 981-985.
Wainwright S. A. Biggs W. D. Currey J. D. Gosline J. M.. 1976. Mechanical design in organisms John Wiley, New York, New York, USA.
Wood C. J.. 1995. Understanding wind forces on trees. In M. P. Coutts and J. Grace [eds.] Wind and trees 133-164 Cambridge University Press, Cambridge, UK.
This article has been cited by other articles:
|
|
 |
|
  M. Philippe, V. Daviero-Gomez, and V. Suteethorn Silhouette and palaeoecology of Mesozoic trees in Thailand Geological Society, London, Special Publications, January 1, 2009; 315(1): 85 - 96. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
