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(American Journal of Botany. 2009;96:531-536.) doi: 10.3732/ajb.0800250 © 2009 Botanical Society of America, Inc. |
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2 Department of Plant Biology, Cornell University, Ithaca, New York 14853 USA 3 Institüt für Biologie III, Universität Freiburg, Freiburg D-79104, Germany
Received for publication 21 July 2008. Accepted for publication 9 October 2008.
ABSTRACT
The manner in which increases in leaf surface area S scale with respect to increases in leaf dry mass Mt within and across species has important implications to understanding the ability of plants to harvest sunlight, grow, and ultimately reproduce. Thus far, no mechanistic explanation has been advanced to explain why prior work shows that the scaling exponent governing the S to Mt relationship is generally significantly less than one (i.e., S 
Mt
< 1.0) such that increases in Mt yield diminishing returns with respect to increases in S across most species. Here, we show analytically why this phenomenon occurs and present equations that predict trends observed in the numerical values of scaling exponents for the S vs. Mt relationships observed across dicot tree species and two aquatic vascular plant species.
Key Words: diminishing returns hypothesis • Hagen–Poiseuille equation • Lemna minor • Myriophyllum heterophyllum • plant allometry • scaling relationships
Prior work has demonstrated empirically that increases in the surface area of the leaf lamina fail to keep pace with increases in lamina dry mass, both at the level of individual mature leaves drawn from a broad spectrum of vascular plant species and at the level of total leaf area and total leaf mass per individual plant across dicot tree species (Milla and Reich, 2007
; Niklas et al., 2007; Niklas and Cobb, 2008). This phenomenon, which has been called "diminishing returns," is important for a variety of reasons. For example, it provides one explanation for why the allometry of total plant growth with respect to total mass is governed by a scaling exponent significantly less than one (Niklas and Enquist, 2001). However, with the exception of a general model based on the allometry of resource distribution networks (Price and Enquist, 2007), no mechanistic explanation has been advanced to account for the diminishing returns allometry of leaves, although both theory and observation suggest that the accrual of hydraulic or mechanical tissues with increases in overall leaf size is likely responsible for the diminishing returns phenomenon (Pearcy et al., 2005; Niinemets et al., 2006).
The Price and Enquist (2007) or PE model is an extension of the West, Brown, and Enquist or WBE model but differs from it in three significant ways (West et al., 1997, 1999, 2000). Specifically, leaf vascular networks are assumed to be closer to two-dimensional than to three-dimensional objects, the minimization of energy or hydraulic resistance is assumed not to hold true, and the vascular delivery system in leaves are treated as "leaky" rather than "closed pipes." The PE model therefore relaxes the area-preserving and volume-filling constraints originally required by the WBE model. It also makes additional critical assumptions that redirect the mathematics of the WBE model to specifically treat leaf geometry. By so doing, the PE model predicts discrete numerical values for the scaling exponent governing the relationship between leaf area and mass (e.g., 0.66, 0.75, > 0.857, and >1.0; see Price and Enquist, 2007, table 1), although the authors themselves "expect that no single...exponent(s) will describe scaling relationships across all leaves" (Price and Enquist, 2007, p. 1139). The strength of this model lies in its ability to make numerous predictions about leaf morphometry including petiole size. One of its weaknesses is a reliance on what may be unreliable expectations about vascular tissue architecture and physiology.
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DERIVATIONS
Standardized major axis regression analyses of data reported for 668 leaves from over 100 diverse seed plant species (see Niklas et al., 2007) indicate that the scaling exponent () governing the relationship between laminar dry and fresh mass across species is statistically indistinguishable from unity (i.e., = 1.02, r2 = 0.953; 95% CIs = 1.00, 1.03). Also, for individual species for which sample sizes equal or exceed 30 randomly selected leaves, the scaling exponents and their 95% CIs are statistically indistinguishable from unity, e.g., Alnus rhombifolia (r2 = 0.994), Betula jacquemonti (r2 = 0.996), Eucalyptus ficifolia (r2 = 0.991), E. pauciflora (r2 = 0.978), Ginkgo biloba (r2 = 0.990), Populus nigra (r2 = 0.999), and Salix babylonica (r2 = 0.965). Therefore, in the following derivations, we make no distinction when using "mass" to refer to dry or fresh laminar mass because the scaling exponents predicted to govern the relationship between lamina surface area and mass are statistically indifferent to how laminar mass is measured.
Hydraulic argument
Referring to Fig. 1A, the total mass of any portion of the lamina Mt equals the mass of its hydraulic tissues Mh (represented by a cylinder of uniform radius r and length L) plus the mass of its nonhydraulic tissues Mnh (represented by a rectangular block with thickness a, breadth b, and length L). Accordingly, Mh = (
r2L
h)n and Mnh = (ab – r2)Lnh, where h and nh are the bulk tissue densities of Mh and Mnh, respectively, and n is the number of hydraulic tissue strands per unit lamina area. It therefore follows that Mt is given by the formula
 | (1) |
 | (2) |
|
h – nh) = 0. However, this isometric relationship is lost when four hydraulic conditions are met: (1) the rate of water flowing into any portion of the lamina, dV/dt, equals the product of its tissue hydraulic conductivity, Lp, and the applied pressure gradient, dP/dl, i.e., dV/dt = Lp (dP/dl), (2) the hydraulic conductivity of the conducting tissues scales according the Hagen–Poiseuille equation, i.e., Lp = n (/8
)r4 = nk1r4, where is viscosity, r is the average cell radius in the hydraulic tissue, and k1 has units s•m–1•kg–1, (3) dP/dl is relatively constant throughout the lifetime of a leaf, or, at the least, can be averaged, and (4) the rate of water loss from S due to transpiration, Tr, scales as a simple linear function of S, i.e., Tr = k2S = Lp, where k2 has units s•m•kg–1. [Note: The stipulations that Lp r4 and that dP/dl is relatively constant are gross approximations for a hydraulic strand composed of conducting cells differing in radii or possessing nonperforate end walls and for leaves experiencing water deprivation. However, the general form of Lp r4 is retained if r denotes an average value. Likewise, the assumption that dP/dl is invariant can be substituted with the assumption that dP/dl is a time averaged parameter.]
When conditions (1) – (4) are met, r = (k2/nk1)1/4S1/4 and, because n = k0b = k0S/L, we see that r = k31/4L1/4 such that Eq. 2 takes the form
 | (3) |
 | (4) |
, will range between 0.80 
1.0 depending on the numerical values of anh and k0(k3k4)1/2(h – nh). This equation also predicts that the numerical value of will shift from the upper to the lower limit of this range as the bulk density of the hydraulic tissue increases with respect to that of the nonhydraulic tissues.
This limit changes numerically as the diameter of the hydraulic strands, 2r, approaches, equals, or eventually exceeds leaf thickness, a (Fig. 1B). Specifically, as 2r 
a 2 (k2/nk1)1/4 S1/4, it follows that 2r = 2k31/4L1/4 = 2(k3 k4)1/4S1/8. Inserting this last expression into Eq. 4 yields
 | (5) |
0.89 depending, once again, on the numerical values of (k3k4)1/4nh and k0(k3k4)1/2(h – nh).
Mechanical argument
An anisometric scaling relationship also emerges from an alternative (but not mutually exclusive) mechanical argument based on the observation that the bending moment Mb experienced by a leaf lamina with mass Mt supported by n strands of mechanical tissue is given by the formula
 | (6) |
 | (7) |
 | (8) |
This equation predicts that the scaling exponent for the relationship between the lamina area and mass should have a range of 0.67 < 1.0 depending on the numerical values of anh and k10(h – nh). However, as the radius of mechanical tissue strands approaches the thickness of the lamina a, we see that a 2k91/2 S1/4 such that Eq. 8 becomes
 | (9) |
This last equation predicts that 0.67 < 0.80. As in the prior hydraulic argument, this mechanical argument obtains an anisometric relationship governed by a scaling exponent that is predicted to numerically decrease as the volume fraction of mechanical tissue in the lamina increases.
Summary of predictions
The forgoing indicates that the numerical value of the exponent governing the area to mass scaling relationship depends (1) on whether a hydraulic or mechanical tissue component is present or absent, (2) on the quantity of this component relative to that of the nonhydraulic or nonmechanical tissue component, (3) whether the bulk density of the hydraulic or mechanical tissue component equals or exceeds that of the nonhydraulic or nonmechanical tissue, and (4) whether hydraulic or mechanical constraints affect leaf architecture (Fig. 2). The numerical value of the exponent also depends on the numerical values of two generalized allometric constants, denoted here by k11 and k12, which in turn depend on which equation is judged to be the most appropriate to use and which are free to vary ontogenetically or phyletically.
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 | (10) |
Inspection of this formula shows that the scaling exponent for the area to mass relationship, denoted here by , will equal 1 when (h – nh) = 0. For example, if hydraulic constraints dominate and (h – nh) = 0, Eq. 10 takes the form Mt = k11nh S1, where 1 = 9/8, which indicates that surface area will scale as the 8/9 power of lamina dry mass (see Eq. 5). Likewise, if mechanical constraints dominate and (h – nh) = 0, Eq. 10 becomes Mt = k11nh S1, where 1 = 5/4, which indicates that area will scale as the 4/5 power of mass (see Eq. 9). However, when (h – nh) 
0, the scaling exponent for S vs. Mt is a complex function of 1 and 2 that is free to vary over the limit set by the values of 1 and 2 depending on the numerical values of k11 and k12. The scaling exponent for S vs. Mt is also specific to the size range of leaves examined, e.g., numerical values of predicted by Eqs. 2, 4, 5, 8, and 9 are provided in Table 1 based on numerical simulations using the leaf size ranges reported by Niklas and Cobb (2008) and Niklas et al. (2007).
Finally, we note that the scaling exponent for the S vs. Mt relationship will exceed the numerical value of 1.0 if lamina thickness a changes with overall leaf size, as measured for example by length L. This prediction quickly emerges if, for convenience, we neglect differences in the tissues densities for any set of geometrically self-similar leaves (i.e., S L2). Under these circumstances, Mt abL al2 aS. Consider now three cases regarding the scaling of a with respect to L:
L0, then Mt S1 such that S Mt1. L1, then Mt S3/2 such that S Mt2/3. L–β, then Mt S1 –β/2 such that S Mt1/(1 –β/2).
In the last case, we see that = 1 / (1 – β/2) > 1.0 whenever a increases with decreasing leaf size (i.e., when the exponent for a vs. L is negative).
PREDICTIONS AND OBSERVATIONS
Limitations
Three factors limit the extent to which our derivations shed light on the allometry of foliar surface area and mass. First, average values for leaf thickness a, the number of vascular strands per unit lamina area n, and the bulk tissue densities of hydraulic/mechanical tissues and nonhydraulic/nonmechanical tissues, h and nh, can be determined empirically, but only with considerable effort. Second, the numerical values for the allometric constants used in our derivations (i.e., k0, k1... k12) are not reliably determined empirically without extensive time-averaged physiological measurements (e.g., the relationship between vascular strand hydraulic conductivity and the rate of transpiration), which are impractical across the large number of species required to determine broad allometric relationships. Third and perhaps most important, we are currently unable to distinguish a priori whether the scaling of leaf area to mass relationships is driven exclusively or predominantly by hydraulic or mechanical constraints on foliar architecture. A number of factors contribute to this uncertainty. For example, numerical ranges of the scaling exponents predicted by the hydraulic and by the mechanical argument theoretically overlap significantly in part because both hydraulic conductivity and the second moment of area scale as the fourth power of radius, i.e., Lp I r4. Also, as the bulk densities of different tissue components converge numerically, Eqs. 3, 4, and 8 yield the same prediction, i.e., S is predicted to scale one-to-one with Mt. Another serious limitation to our (or any allometric) approach when dealing with interspecific scaling relationships is that allometric "constants" can vary across species in ways that elevate best fit regression curves. When this happens, scaling exponents greater than one can be obtained. As a consequence, the utility of our derivations is currently largely confined to examining whether broad predictions about trends in the numerical values of scaling exponents comply with general trends observed within selected species or species groupings.
Observations
Despite the aforementioned limitations, the predictions of our equations comply reasonably well with the numerical range of -values reported by Niklas and Cobb (2008) for 23 dicot tree species represented by 46 specimens and by Niklas et al (2007) for 1943 species represented by >5000 specimens differing in life form. Specifically, across all 46 dicot tree specimens, Niklas and Cobb (2008) report that total leaf area per specimen scales as the 0.75 (95% CI = 0.69, 0.81) power of total leaf dry mass. The numerical value of this interspecific scaling exponent and its confidence intervals fit comfortably within the range of values predicted by Eqs. 4, 5, 8, and 9 assuming that the differences in the bulk densities of vascular/mechanical tissues and foliar ground tissues differ (i.e., = 0.67–0.81; see Table 1). Indeed, based on the numerical ranges for the scaling exponent predicted by our equations (see Table 1), Eq. 9 appears to provide the best fit (Fig. 3), suggesting that mechanical rather than hydraulic constraints may dominate the leaf architecture of the "average" dicot leaf in this data set.
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reported that surface area scales as the 0.98 power of dry mass (95% CI = 0.97, 0.99). Over the size range represented by these data, this scaling exponent is most consistent with the prediction of Eq. 2 (see Table 1), which suggests neither hydraulic nor mechanical constraints operate in a detectable or consistent manner across all species. However, when these data are segregated into different functional species groups (e.g., vines and shrubs), statistically significant differences among species groups are evident. For example, the scaling exponent for the surface area to mass relationship reported for vines is 0.84 (95% CI = 0.79, 0.88), which is consistent with the prediction of Eq. 5 for hydraulic constraint, whereas the scaling exponent for shrubs is 0.978 (95% CI = 0.958, 0.999), which converges on the null hypothesis that neither hydraulic nor mechanical factors limit the scaling of leaf area with respect to dry mass. We cannot over-stress, however, that a scaling exponent of one is predicted by Eqs. 3, 4, and 8 if the bulk densities of different leaf tissue components converge on similar values. Likewise, we once again draw attention to the possibility that allometric "constants" may vary in ways that elevate interspecific regression curves and thus inflate the numerical values of scaling exponents artificially. This variation is nowhere better illustrated when the influence of leaf thickness on the numerical values of the scaling exponents governing lamina surface area vs. mass relationships is examined. Specifically, our model predicts scaling exponents in excess of unity when leaf thickness a increases with decreasing leaf size (see case 3). Although leaf thickness correlates poorly with leaf size across all species, analysis of the large data set used by Niklas et al. (2007) shows that leaf thickness increases with increasing leaf size across all species. But thickness decreases with increasing leaf size across the conspecifics of some species (e.g., A. rhombifolia, G. biloba, and P. nigra) and, for these species, the numerical value of the scaling exponent for the S vs. Mt relationship concomitantly increases to exceed unity as predicted (data for examples of this phenomenology are available upon request). Similar trends can be inferred from the data reported by others (e.g., Koch et al., 2004).
In light of all the variables that influence interspecific and some intraspecific trends, we tested the general prediction that surface area will scale nearly one-to-one with leaf mass when neither hydraulic or mechanical constraints play a large role in leaf architecture by examining the leaf surface area and mass scaling relationships of Lemna minor and Myriophyllum heterophyllum growing under natural conditions. These two aquatic species were selected because it is reasonable to assume that hydraulic and mechanical constraints are more relaxed for floating and submerged leaves as opposed to the leaves of dicot tree species. Myriophyllum heterophyllum was also selected because its aerial shoots produce leaves differing in size, shape, and anatomy from their submerged counterparts (Fassett, 1957, see fig. 41, p. 265), which permits a direct comparison between different leaf architectures produced by the same stem. The scaling exponent for the L. minor surface area to mass relationship was 0.90 with 95% CI = 0.88, 0.93 (r2 = 0.92, N = 21). These values fall outside the confidence intervals of the scaling exponent reported for dicot leaves and are reasonably close to the predicted value of 0.88 for a hydraulic constraint (see Eq. 5). The scaling exponents for M. heterophyllum submerged and aerial leaves were 0.93 (95% CI = 0.88, 0.99; r2 = 0.97, N = 17) and 0.84 (95% CI = 0.79, 0.88; r2 = 0.91, N = 19), respectively. Although the 95% confidence intervals of these two exponents overlap marginally, we judge that the scaling exponents of submerged emergent leaves differ and that the surface area to mass scaling relationship of the former comes close to the predictions of Eq. 2, whereas the scaling relationship of emergent leaves appears to reflect either a hydraulic or a mechanical constraint (see Table 1).
An alternative model
Similar numerical values for the scaling exponents of these two hydrophytes have been reported by Price and Enquist (2007) who present a model (denoted as the PE model) for leaf scaling relationships based on the principles of network geometry developed by West, Brown, and Enquist (1997, 1999, 2000). However, the PE model differs from that of West, Brown, and Enquist in terms of seven assumptions: (1) leaf vascular networks are assumed to be closer to two-dimensional objects then they are to three-dimensional ones, (2) the minimization of energy or hydraulic resistance likely does not hold, (3) vascular delivery systems behave like "leaky" rather than "closed pipes," (4) leaves typically grow in two dimensions with relatively little investment in thickness, (5) foliar vascular networks are evolutionarily selected to deliver water and nutrients both laterally and terminally and to slow the delivery of substances to facilitate mass exchange, (6) species differences in leaf form and function mainly result from differential tissue investments, and (7) the dimensions and volume flow of any vascular branch-like element are proportional to those of its subtending vascular branch. The PE model makes additional predictions regarding leaf shape as well as foliar allometry (including that of the petiole). However, using these seven basic assumptions (and taking biologically reasonable values for the parameters required to define the allometry of lamina area with respect to leaf dry mass), the PE model predicts that the scaling exponent governing the S vs. Mt relationship can take on discrete values of 0.66, 0.75, >0.857, and >1.0 for specific combinations of parameter values (see Price and Enquist, 2007, table 1), although the authors note that the exponent is free to vary numerically in many ways if some model parameters are altered.
In contrast, the hydraulic and mechanical variants of our model are based on only five assumptions: (1) the rate of water flowing into any portion of the lamina equals the product of its tissue hydraulic conductivity and the applied pressure gradient, (2) the hydraulic conductivity of the conducting tissues scales according the Hagen–Poiseuille equation, (3) the hydraulic pressure gradient is relatively constant throughout a leaf’s lifetime, (4) the rate of water loss due to transpiration scales as a linear function of lamina surface area, and (5) leaves, on average, manifest geometric self-similarity. Assumptions (1) and (2) reflect a fundamental physical principle that cannot be violated. Assumption (3) is biologically reasonable because the pressure gradient in any structure is a time-averaged property over the structures functional life span. Assumption (4) is biologically reasonable especially for conspecifics because stomatal density is a species-specific functional trait. Assumption (5) is the most precarious because the scaling exponent governing the relationship between leaf area and length can differ significantly depending on the species selected for analyses. For example, Price and Enquist (2007, table 2) report a scaling exponent that significantly exceeds 2.0 for the 21 species examined in their study. In contrast, analyses of the data drawn from 25 species examined by Niklas and Cobb (2008) show that the scaling exponent for leaf area vs. length is 2.07 (r2 = 0.973). Accordingly, the assumption that leaf lamina manifest geometric self-similarity is uncertain, although clearly not wrong for some species. We believe our model passes the test of Ockham’s razor more successfully than does the PE model. However, it is not our intention to place these two models in negative apposition. Both models rely on similar albeit not identical physical principles that either directly or indirectly draw attention to the influence of hydraulic and mechanical phenomena on leaf morphometry. Thus, both provide alternative world-views that require further empirical study.
Conclusions
The hypothesis that the phenomenology of "diminishing returns" is the result of hydraulic/mechanical constraints on leaf architecture is consistent with prior theoretical and empirical studies (e.g., Pearcy et al., 2005; Niinemets et al., 2006). Price and Enquist (2007) have proposed an extension of a resource distribution network model (see West et al., 1999) to account for the scaling of leaf area and mass. However, the analytical approach taken here is, to the best of our knowledge, the first of its kind to exclusively apply simple yet fundamental physical laws to predict this allometry analytically. Our approach highlights the fact that the scaling exponent observed for a seemingly simple relationship can be the emergent property of two (or more) scaling exponents as well as the numerical values of their allometric constants (see Eq. 10).
A comparison of the numerical values of observed with predicted scaling exponents suggests this approach provides qualitatively and, in some cases, quantitatively good predictions, even in the absence of knowing the numerical values of allometric constants (which are free to vary in as yet unexplored ways). In theory, therefore, our derivations provide predictions about the scaling exponents governing the surface area to mass scaling relationships of any appendicular photosynthetic organ. Additional insights, however, require far more accurate knowledge about the physiological, mechanical, and physical properties of different foliar tissue- types than is currently available, particularly the bulk densities of different tissue types. Our equations advance the field by showing analytically that these properties dictate the numerical values of allometric constants and thus have a profound affect on the numerical values of scaling exponents. In turn, our formulas advance the field by showing that species-specific differences in the scaling of leaf area to mass are as much a consequence of interspecific (and biologically intrinsic) variation in these properties as they are a consequence of physical constraints on the construction of light harvesting organs.
FOOTNOTES
1 The authors thank two anonymous reviewers for suggestions to improve this paper. Funding from the College of Agriculture and Life Sciences, Cornell University, is gratefully acknowledged. 
4 Author for correspondence (e-mail: kjn2{at}cornell.edu)
LITERATURE CITED
Fassett, N. C. 1957. A manual of aquatic plants. University of Wisconsin Press, Madison, Wisconsin, USA.
Koch, G. W., S. C. Sillet, G. M. Jennings, AND S. D. Davis. 2004. The limits to tree height. Nature 428: 851–854.[CrossRef][Medline]
Milla, R., AND P. B. Reich. 2007. The scaling of leaf area and mass: The cost of light interception increases with leaf size. Proceedings of the Royal Society of London, B, Biological Sciences 274: 2109–2114.
Niinemets, Ü., A. Portsmuth, AND M. Tobias. 2006. Leaf size modifies support biomass distribution among stems, petioles and mid-ribs in temperate plants. New Phytologist 171: 91–104.[CrossRef][Web of Science][Medline]
Niklas, K. J., AND E. D. Cobb. 2008. Evidence for "diminishing returns" from the scaling of stem diameter and specific leaf area. American Journal of Botany 95: 549–557.
Niklas, K. J., E. D. Cobb, Ü. Niinemets, P. B. Reich, A. Sellin, A. B. Shipley, AND I. J. Wright. 2007. "Diminishing returns" in the scaling of functional leaf traits across and within species groups. Proceedings of the National Academy of Sciences, USA 104: 8891–8896.
Niklas, K. J., AND B. J. Enquist. 2001. Invariant scaling relationships for interspecific plant biomass production rates and body size. Proceedings of the National Academy of Sciences, USA 98: 2922–2927.
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Price, C. D., AND B. J. Enquist. 2007. Scaling mass and morphology in leaves: An extension of the WBE model. Ecology 88: 1132–1141.[CrossRef][Web of Science][Medline]
West, G. B., J. H. Brown, AND B. J. Enquist. 1997. A general model for the origin of allometric scaling laws in biology. Science 276: 122–126.
West, G. B., J. H. Brown, AND B. J. Enquist. 1999. A general model for the structure and allometry of plant vascular systems. Nature 400: 664–667.[CrossRef][Web of Science]
West, G. B., J. H. Brown, AND B. J. Enquist. 2000. The origin of universal scaling laws in biology. In J. H. Brown, and G. B. West [eds.], Scaling in biology, 87–112. Oxford University Press, Oxford, UK.
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