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Keywords:

  • allometry;
  • biological networks;
  • branching;
  • fractal;
  • leaves;
  • plants;
  • scaling

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References

Size is one of the most important axes of variation among plants. As such, plant biologists have long searched for unifying principles that can explain how matter and energy flux and organ partitioning scale with plant size. Several recent models have proposed a universal biophysical basis for numerous scaling phenomena in plants based on vascular network geometry. Here, we review statistical analyses of several large-scale plant datasets that demonstrate that a true hallmark of plant form variability is systematic covariation among traits. This covariation is constrained by allometries that combine and trade off with one another, rather than any single universal allometric scaling exponent for a trait or suite of traits. Further, we show that covariation can be successfully modeled using network approaches that allow for species-specific designs in plants and geometric approaches that constrain relationships among economic traits in leaves. Finally, we report large-scale efforts utilizing semi-automated software tools that quantify physical networks and can inform our attempts to link vascular network structure to plant form and function. Collectively, this work highlights how the linking of morphology, biomass partitioning and the structure of physical distribution networks can improve our empirical and theoretical understanding of important drivers of plant functional diversity.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References

Interest in the relationship between plant form and function has intensified during the last several decades as a result, in part, of the introduction of several prominent and controversial scaling models (West et al., 1997, 1999; Gillooly et al., 2001). These models, known collectively as the Metabolic Theory of Ecology (MTE), invoke the geometry of physical distribution networks as the principal determinants of a suite of scaling relationships (Brown et al., 2004; Price et al., 2010a). MTE’s theoretical foundation, laid out in a series of papers by West, Brown and Enquist (hereafter WBE), rests in the idea that natural selection has optimized physical distribution networks such that the power required to distribute resources is minimized. WBE predicts a ‘fractal-like’ distribution network, that is, an idealized organismal form inspired by the biology of mammalian cardiovascular networks (West et al., 1997) and extended to the case of vascular networks in plants (West et al., 1999).

These models have generated significant interest and the broader MTE has been described as one of the most important recent ideas in ecology (Whitfield, 2004). However, the models have also been quite controversial and have been challenged on both theoretical and empirical grounds (i.e. Dodds et al., 2001; McCulloh et al., 2003). For plants, WBE’s numerous predictions rely primarily on two central assumptions regarding network geometry: first, that the branching network is ‘volume filling’ and, second, that the sum of the cross-sectional areas across branching generations remains constant across branching levels (McMahon & Kronauer, 1976). These assumptions then lead to a number of predictions regarding morphological allometry, biomass partitioning and whole-plant metabolic rates (West et al., 1999). Detailed studies of various aspects of the assumptions, scope and state of the theory as applied to plants have been published elsewhere (Niklas, 2004; Savage et al., 2008, 2010; Price et al., 2010a). The fractal model is an idealized abstraction of real plants and clearly no plant exactly fits its predictions or assumptions (Savage et al., 2008; Price et al., 2010a). The more interesting questions are: do plants exhibit scaling behavior that is more or less consistent with a universal design (constant allometric exponents)?; and are the statistical properties of plant networks consistent with the expectations for fractal networks or some other form of network?

Allometric covariation in botanical form and function

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References

WBE and many of its derivative models predict universal scaling relationships for plant form and biomass partitioning, such that the relationships between plant biomass (M), height (H), basal stem diameter (D) and leaf mass (ML) follow: inline image, inline image, inline image, inline image at both inter- and intraspecific levels, where c1–c4 are prefactors (a.k.a. intercepts) (West et al., 1999; Enquist & Niklas, 2002). As these scaling relationships are inherently multiplicative, they are typically explored on log–log axes where the exponent becomes the slope, for example, inline image (Kerkhoff & Enquist, 2009).

Although preliminary analyses have provided support for some of these universal relationships at the interspecific level (Enquist & Niklas, 2002), subsequent investigations have found significant variability in the intraspecific scaling exponents for both whole-plant morphological allometry and the allometry of biomass partitioning (Price et al., 2007, 2009). Moreover, as discussed below, the mean intraspecific exponents across these groups differ from WBE model predictions (Price et al., 2009).

Variability in plant morphological allometry can be understood by considering specific growth forms or functional groups separately. Two recent studies illustrate this point. First, the morphological scaling of succulent species is better described by a structural model that retains a highly branched internal network without assuming a complex external branching network (Price & Enquist, 2006). Second, a consideration of variation in leaf venation network structure helps to predict the scaling of leaf mass, with morphological parameters such as petiole diameter, length or surface area (Price & Enquist, 2007).

These two studies have provided the impetus to replace WBE’s prediction of a single optimal plant (West et al., 1999) with a ‘fractal continuum’ model that can account for species-level variation in network geometry (Price et al., 2007). The model formulation predicts that the intraspecific allometry of plant form, function and biomass partitioning should fall along a continuum of variation that contains a spectrum of possible hierarchical fractal networks, rather than a single optimal network to which every plant will conform (Fig. 1). The fractal continuum model relies on two parameters: the effective dimension filled by a given plant network, and the extent to which the network bifurcations are area preserving (inline image a.k.a. DaVinci’s rule) or area increasing (e.g. inline image a.k.a. Murray’s law), where rd and rp refer to the radius of daughter and parent branches, respectively. The model predicts covariation among allometric exponents, and these parameters could be used to describe species-level scaling relationships (Price et al., 2007), or scaling relationships for clades or functional groups that are similar in form (Price & Enquist, 2006).

image

Figure 1. Allometric covariation of morphological scaling exponents for gymnosperm and angiosperm leaves, herbaceous annuals and perennials, succulent and woody plant species. (a) Radius vs mass exponent plotted against length vs radius exponent. (b) Length vs mass exponent plotted against length vs radius exponent. (c) Radius vs mass exponent plotted against length vs mass exponent. The exponents for whole plants tend to fall at the end of the fractal continuum, which is consistent with plants that are volume filling (three dimensional). By contrast, gymnosperm leaves (needles) fall towards the end of the continuum, associated with one-dimensional objects (lines), and angiosperm leaves fall in between, which is consistent with the expectations for two-dimensional objects. It should be noted that the groups and group means (oversized circles) segregate consistently into different regions of the allometric covariation space. The expectations for the stress similarity (red circle), West, Brown and Enquist (WBE)/elastic similarity (red diamond) and geometric similarity (red triangle) models all fall along the fractal continuum. For a detailed explanation, see Price et al. (2007).

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In allometric covariation, the exponents for the aforementioned scaling relationships covary with one another in systematic ways (Fig. 1). This approach can also be used to evaluate morphological constraints based on the range of values the parameters are likely to take (Price & Enquist, 2006; Price et al., 2007). For example, the fractal dimension should vary from one to three. The radius branching parameter is theoretically bounded only by zero, but most values are expected to cluster between the expectations for area-preserving branching (DaVinci’s rule) and the expectation arising from Murray’s law formulation (Murray, 1926; West et al., 1999).

Empirical data agree well with the fractal continuum formulation (Fig. 1; Price & Enquist, 2007; Price et al., 2007). In addition, four previously proposed universal scaling models all fall along the fractal continuum (Fig. 1). These models, which are based on biomechanical considerations (elastic and stress similarity; McMahon & Kronauer, 1976), the expectations for simple geometric shapes (geometric similarity; Galilei, 1638) or fractal geometries (West et al., 1999), have all been suggested as explanations for broad-scale interspecific allometric patterns, and serve as potential alternatives to one another. Therefore, as we show next, a more informative approach is to evaluate several scaling models simultaneously.

Evaluation of scaling models in plant biology

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References

One important aspect of the aforementioned body of theory is that both the original WBE model (West et al., 1999) and the fractal continuum extension (Price et al., 2007) predict scaling exponents for a suite of relationships describing aspects of plant form, biomass partitioning and material flux. This is one of the most attractive features of this modeling framework, and the breadth of areas on which the broader theory (MTE) touches is unparalleled in ecology (Brown et al., 2004; Enquist et al., 2007a,b; Price et al., 2010a). Unfortunately, the overwhelming majority of empirical tests of both WBE and MTE address a single prediction at a time, ignoring its synthetic potential.

To address this, Price et al. (2009) evaluated six different scaling models and their specific predictions for six different intraspecific scaling relationships (Fig. 2) using three large plant allometric datasets. The scaling models evaluated fell into three general classes: ‘universal’ models that predict a single scaling exponent for each relationship, including elastic, stress, fractal (WBE) and geometric similarity models (Galilei, 1638; McMahon & Kronauer, 1976; West et al., 1999); a ‘constrained’ model, the fractal continuum model (Price et al., 2007); and an ‘unconstrained’ model which has no biological basis and assumes that species-level variability is governed only by algebraic relationships among scaling exponents. The analyses were performed using a hierarchical Bayesian framework which simultaneously evaluates multiple scaling relationships and accounts for uncertainty in all measured traits (Dietze et al., 2008; Ogle & Barber, 2008).

image

Figure 2. Posterior distributions from a hierarchical Bayesian analysis for the global exponents for three different plant allometric datasets: two for whole plants (Sonoran plants and Cannell trees) and one for leaves. The empirical exponents are for the allometry of plant height (or leaf length), plant stem diameter (or leaf petiole diameter) or plant (or leaf) mass with one another as in Fig. 1. The vertical strings of symbols (circles, stars and diamonds) represent exponent values predicted by the universal models as denoted in the legend (West, Brown and Enquist (WBE) and elastic similarity make identical predictions). The Sonoran plants dataset lacks surface area estimates, and thus there are only two posterior distributions in (c). It should be noted that none of the universal models enjoys strong support across all datasets. Moreover, the predictions for the geometric and Metabolic Theory of Ecology (MTE) models are identical in (c) (the stress similarity model does not predict the surface area). A detailed explanation of the data, results, methodology and caveats (e.g. the WBE model was not intended to be applied to leaves) can be found in Price et al. (2009).

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Several important findings emerged from these analyses. First, although central tendencies exist for each scaling relationship, these were not adequately captured by any of the universal scaling models (Fig. 2). Second, of the biologically inspired models, the fractal continuum (Price et al., 2007) enjoyed the strongest support, outperforming the universal models, even after accounting for increases in model complexity. Third, despite its increase in parameters, the ‘unconstrained’ model received the strongest support overall. This suggests that the scaling of plant form and function can and should be explored using models that incorporate greater variability (more parameters, e.g. Purves et al., 2008).

Allometric covariation in specific leaf area (SLA)

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References

Systematic allometric covariation is a hallmark of plant form and function and, as we show, is also a hallmark of leaf form and functional traits. Functional trait analyses have identified SLA as one of the principal leaf traits influencing plant growth rates, reproductive strategies and lifespan (Wright et al., 2004; Poorter et al., 2009). A global compilation of leaf traits has identified a suite of functional tradeoffs related to SLA, known as the leaf economics spectrum (Wright et al., 2004). The spectrum is characterized by leaves that fall along a continuum, from those with rapid growth and return on investment, but with short lifespan and high resource requirements, to those with slow growth and return on investment, but long lifespan and lower resource requirements. Recent analyses have also shown that SLA is often allometric with changes in leaf size, both intra- and interspecifically (Niklas et al., 2007; Price & Enquist, 2007), although comparisons within sites can exhibit no relationship (Pickup et al., 2005). However, despite intense interest, a mechanistic understanding of the origin of this spectrum, and its links to variation in SLA, remains incomplete.

One possible mechanism is that differential allocation to leaf thickness and density strongly influences leaf SLA (Niinemets, 1999); however, relatively few studies exist in which these traits have been collected simultaneously (Poorter et al., 2009). To address this, we measured all of these traits in 900 leaves from 44 broad-leaf angiosperm species covering a broad range of sizes for each species (Price & Weitz, 2010). To account for size-dependent variability in SLA, we proposed a zero-sum allometric framework, where zero-sum refers to the inevitable tradeoff in investment that occurs when allocation to one trait takes away from allocation to another. For example, leaf mass can be approximated as the product of the surface area, average thickness and average density, inline image. These three components, leaf thickness (T), surface area (AS) and density (ρ), often exhibit an allometric dependence on mass (M): inline image, inline image and inline image. By substituting these scaling relationships into the equation for leaf mass, we arrive at the following zero-sum function, inline image, which constrains the product of the scaling intercepts, inline image, and the sum of the scaling exponents inline image, to both equal unity (for a detailed explanation, see: Price & Weitz, 2010). According to this model, leaves that increase their allocation to thickness, either via a change in scaling exponent or via a change in scaling intercept, must do so at the expense of allocation to leaf density and/or surface area (Fig. 3a; Price & Weitz, 2010). For example, for leaves in which density decreases more slowly than thickness increases, the exponent describing the allometry of leaf area to mass will be < 1, known as ‘diminishing returns’ (Niklas et al., 2007). By contrast, if density decreases more rapidly than thickness increases, the exponent describing the allometry of leaf area to mass will be > 1, or ‘increasing returns’ (Fig. 3a). Moreover, species that tend to have a higher leaf area to mass scaling exponent also tend to have a higher scaling intercept (Fig. 3b), which indicates that leaves that exhibit ‘increasing returns’ tend to have higher SLA values to begin with. Such constraints are the hallmark of allometric covariation, which helps us to understand variability in species-level allocation strategies and serves to complement more traditional interspecific approaches (Wright et al., 2004; Pickup et al., 2005).

image

Figure 3. (a) The surface area vs mass relationship exponent as a function of the sum of the density vs mass and thickness vs mass exponents for 44 species of angiosperm leaves for both fresh and dry leaves (for a detailed description, see Price & Weitz, 2010). It should be noted that the exponent values covary systematically. When the sum on the x-axis is close to zero, the y-axis value is close to unity (dashed vertical and horizontal lines, respectively). In the lower right quadrant (sum of exponents > 0), density is decreasing more slowly than thickness is increasing. Similarly, in the upper left quadrant (sum of exponents < 0), thickness is increasing more slowly than density is decreasing. The open circles represent the few cases in which the density vs mass and thickness vs mass exponents are both positive or both negative. (b) Positive correlation between the surface area vs mass exponent and the surface area vs mass intercept, indicating that the leaves that exhibit ‘increasing returns’ tend to have higher specific leaf area (SLA) values to begin with. As described in Price & Weitz (2010), surface area was necessarily measured on fresh leaves, yet regressions were calculated for both fresh (blue circles) and dry (red diamonds) mass, which results in a higher intercept for the dry mass regressions (same area for less mass). This explains the offset between the two regression functions depicted in (b).

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Quantification of the dimensions of physical networks: leaves

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References

The aforementioned work on WBE, the fractal continuum, plant and leaf allometric covariation, the leaf economics spectrum and zero-sum SLA modeling highlights the need for an improved understanding of the role played by vascular networks in influencing morphology, nutrient flux and patterns of biomass partitioning, particularly in leaves. Leaf vein networks display remarkable variety. They have been shown to influence whole-leaf conductance and photosynthetic rates (Sack & Holbrook, 2006; Brodribb et al., 2010), and have been linked to species diversification rates (Brodribb et al., 2010). However, as a result of the small size and high abundance of veins in leaves, almost no exhaustive descriptions of their geometry exist.

As a step towards the quantification of the role of vascular networks in leaf economics, Price et al. (2010b) have developed a series of image segmentation and network extraction algorithms that measure the geometry of leaf veins and the areoles they surround, and have bundled these algorithms into software available as a free graphical user interface (http://www.leafgui.org; Price et al., 2010b). Before the release of this software, these measures were largely inaccessible for biologists, and thus the software addresses a long-standing need in plant science. This work complements semi-automated efforts to estimate conduit dimensions within cross-sections of above-ground branches (e.g. Weitz et al., 2006; Mencuccini & Holtta, 2007; Savage et al., 2010).

We used the LEAF GUI software to measure the dimensions and connectivity of the vein networks from 353 leaves representing 72 angiosperm families (Price et al., 2011). We found that, although vein diameter distributions exhibit a power law behavior consistent with a scale-free (fractal-like) design, many aspects of leaf network geometry, in particular vein length distributions, are more consistent with a characteristic scale (exponential distribution; Fig. 4a,b). These results agree with previous suggestions that these aspects of leaf network geometry should be driven by bulk flow constraints and diffusion limitations, respectively (Sack & Holbrook, 2006; Noblin et al., 2008; Brodribb et al., 2010). Moreover, several other aspects of leaf form are consistent with a characteristic scale, namely the mean vein length, the mean distance within areoles to the nearest vein and the vein density (Fig. 4c,d), and agree in many respects with expectations resulting from a null model of an isometric honeycomb lattice. These results highlight the importance of the empirical validation of the actual geometry of physical networks in plants, and suggest a need to revisit optimal theories of network form (e.g. Savage et al., 2010).

image

Figure 4. Statistical properties of leaf vein networks. (a) and (b) represent the frequency distributions for vein lengths and diameters, respectively, for a single leaf (Acer rubrum L.). It should be noted that the distribution in (a) is approximately linear on a log–linear axis, consistent with an exponential distribution (and characteristic scale; see Price et al., 2011), whereas the distribution in (b) is linear on a log–log axis, consistent with a power law. Thus, leaves possess a mixture of distributions underlying their geometry, similar to other planar networks, such as river networks. (c) and (d) represent the mean distance from any areole point to the nearest vein and the network density (vein length/leaf area) for entire leaves (each point represents a single leaf), respectively. Over almost two orders of magnitude in leaf size, we see no trend in either of these measures, suggesting that diffusion limitations constrain leaf networks to a characteristic scale.

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Conclusions and future directions

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References

A unifying principle underlying the universal models mentioned is that they all invoke natural selection, acting on one or two optimization criteria, as the mechanism shaping allometric functions within and across species. These include the need to dissipate heat in geometrically shaped mammals (Rubner, 1883), the need to maintain biomechanical similarity in hierarchical branching networks (McMahon & Kronauer, 1976), the need to minimize the costs of network construction and blood delivery (Murray, 1926) and the need to minimize the power required to distribute resources in mammals (West et al., 1997) or plants (West et al., 1999). Although the scope, the mechanisms they invoke and the values of the scaling exponents they predict typically differ between these models, they all share the fact that they predict a single numerical value for any given scaling relationship (e.g. Fig. 1).

It is easy to imagine that natural selection favors more efficient organisms, and thus optimization is an intuitively satisfying approach. However, it is also easy to imagine that organisms must do many things well and that these things may trade off against one another to yield a spectrum of evolved structures (Niklas, 1994). For example, in the absence of external disturbance, an optimal leaf would be one that had a high photosynthetic rate per unit leaf mass, low construction costs and remained on the plant for a very long time. However, such leaves do not exist because a long leaf lifespan requires investment in structural integrity and/or defense against disturbances, such as herbivory, wind damage or high evaporative demand (Reich et al., 1999; Wright et al., 2004).

By exploring scaling relationships at the level of the species or functional group, approaches that explore the covariation among allometric exponents inherently acknowledge that plants must do many things well, and that different environments, or different life history strategies within the same environment, may favor different solutions. For example, as demonstrated in the aforementioned zero-sum model for SLA allometry, the consideration of simple mass conservation constraints demonstrates that investments in leaf thickness, density or area necessarily trade off against one another, in both the scaling exponents and prefactors (Fig. 3; Price & Weitz, 2010). A challenge for future work is to understand the environmental drivers that select for certain combinations of allometric functions.

An understanding of why different plant functional types occupy certain regions of the allometric covariation space is also a work in progress (Fig. 1). At a coarse scale, the segregation into different regions may simply reflect the overall shape of the organism or leaf. For example, the scaling exponents for prostrate herbaceous taxa, such as Sonoran sandmat (Chamaesyce micromera (Boiss. ex Engelm.) Woot. & Standl), that invest most of their growth in two dimensions, overlap more strongly with the scaling exponents for leaves than for other herbaceous plants (Price et al., 2007). Similarly, when considering species that vary in their degree of branching, highly branched species, such as velvet mesquite (Prosopis velutina Woot.), tend to have lower exponents and higher intercepts for the scaling of height to mass when compared with intermediately branched species, such as buckhorn cholla (Opuntia acanthocarpa Engelm. & J.M. Bigelow), or minimally branched species, such as giant saguaro (Carnegia gigantea (Engelm.) Britton & Rose). Moreover, the central tendencies for different functional groups segregate consistently into different regions of the covariation space. For example, looking at Fig. 1(a–c), we see that the functional group means are generally arrayed in the following order: woody, succulent, herbaceous perennial, herbaceous annual, angiosperm leaves and gymnosperm leaves. Broadly, this ordering represents trends of decreasing size, degree of branching, bulk tissue density and whether growth is predominantly three, two or one dimensional. However, given the large amount of work required to generate these empirical relationships, the sample sizes within these groups are small and our statistical power to segregate these groups is low; thus, these trends require further empirical validation.

We have reviewed evidence that allometric covariation is a hallmark of scaling behavior within and across plants and leaves. The work described here demonstrates that patterns of whole-plant morphological allometry and biomass partitioning (Price et al., 2007), leaf morphological allometry (Price & Enquist, 2007) and the allometry of SLA (Price & Weitz, 2010) can be better understood when considered within the context of the covariation of species-level scaling exponents. Moreover, this covariation can be predicted using network modeling approaches that account for species-specific variation, as well as geometric approaches that constrain allometric variability to tradeoffs among scaling exponents. Such approaches acknowledge that multiple environmental constraints act on plants and that it is unlikely that a single network optimization approach will work for all plant species (Price et al., 2009). Finally, we have demonstrated how direct measurements of leaf networks and, ultimately, whole-plant networks, will help to improve links between resource distribution networks and organismal form and function (Blonder et al., 2010; Price et al., 2010b, 2011; Savage et al., 2010).

In sum, we acknowledge that the introduction and evolution of MTE has had a significant impact on ecology. No current ecological theory shares its breadth, and it serves as a benchmark for attempts to model the complexity of ecological phenomena across scales. That said, MTE, and the underlying WBE model, fail to embrace several important selective constraints acting on plants, for example, the heterogeneous availability of light, water and nutrients, or the many factors influencing metabolic expenditures other than resistance to flow in vascular networks. Moreover, any general model that attempts to predict scaling phenomena across plants should recognize the strong and systematic covariation among intraspecific allometric exponents that exists in empirical datasets. Such covariation reflects the results of environmentally driven selection on plant hydrodynamics, biomass partitioning and form. We encourage the use and evaluation of allometric covariation as an alternative means to characterize plant morphological and functional diversity, particularly in the continued absence of strong evidence for a single, universal model of plant form and function.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References

Joshua S. Weitz, PhD, holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund. The authors thank Cathy Angel, Michael Renton, David Ackerly and two anonymous reviewers for helpful comments on the manuscript. They would also like to thank their many collaborators who have contributed to this work.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Allometric covariation in botanical form and function
  5. Evaluation of scaling models in plant biology
  6. Allometric covariation in specific leaf area (SLA)
  7. Quantification of the dimensions of physical networks: leaves
  8. Conclusions and future directions
  9. Acknowledgements
  10. References