•Plant light interception efficiency is a crucial determinant of carbon uptake by individual plants and by vegetation. Our aim was to identify whole-plant variables that summarize complex crown architecture, which can be used to predict light interception efficiency.
•We gathered the largest database of digitized plants to date (1831 plants of 124 species), and estimated a measure of light interception efficiency with a detailed three-dimensional model. Light interception efficiency was defined as the ratio of the hemispherically averaged displayed to total leaf area. A simple model was developed that uses only two variables, crown density (the ratio of leaf area to total crown surface area) and leaf dispersion (a measure of the degree of aggregation of leaves).
•The model explained 85% of variation in the observed light interception efficiency across the digitized plants. Both whole-plant variables varied across species, with differences in leaf dispersion related to leaf size. Within species, light interception efficiency decreased with total leaf number. This was a result of changes in leaf dispersion, while crown density remained constant.
•These results provide the basis for a more general understanding of the role of plant architecture in determining the efficiency of light harvesting.
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The crown architecture of plants is highly diverse, but it remains unclear how this diversity affects light interception and growth across species. Leaf display depends on a multitude of morphological traits (Halléet al., 1978; Barthélémy & Caraglio, 2007; Valladares & Niinemets, 2007), but it is difficult to generalize how these traits influence the light interception efficiency of individual plants. Moreover, different crown traits may result in similar light interception efficiencies (light interception per unit leaf area), suggesting functional equivalence of different architectural layouts (Valladares et al., 2002). In this paper, we develop a coherent theoretical framework that allows the diversity in crown architectures seen among individual plants to be compared, in terms of their light interception efficiency. As plant productivity over long time-scales is approximately proportional to intercepted light (Monteith, 1977; Cannell et al., 1987), it is hoped that such a framework can provide a basis for understanding differences in productivity between plants and vegetation types.
Although plants vary in a myriad of qualitative architectural properties, models predicting light interception typically focus on a small number of quantitative features of the canopy, such as total leaf area, leaf angle distribution, and leaf dispersion (clumped, random, or regular) (Campbell & Norman, 2000), as these can be reliably quantified for different species and vegetation types. The well-known Lambert–Beer model estimates light interception by horizontally homogeneous canopies, assuming that leaves are randomly distributed in space (Monsi & Saeki, 1953, 2005):
Qint, intercepted photosynthetic photon flux density (PPFD); Q0, incident PPFD above the canopy; L, the leaf area index; k, an extinction coefficient.) However, random leaf spacing is clearly a poor assumption for most real canopies, where foliage is clumped at shoot and whole-plant levels. As a result, model errors can be very large (Baldocchi et al., 1985; Whitehead et al., 1990; Cescatti, 1998). For this reason, a leaf dispersion parameter is frequently introduced in Eqn 1 (Nilson, 1971; Ross, 1981). The leaf dispersion parameter is typically estimated from inversion of PPFD transmission measurements in canopies (Nilson, 1971; Cescatti & Zorer, 2003), and is rarely related to direct measurements of canopy structure. A notable exception is a shoot clumping parameter developed for conifers (Oker-Blom & Smolander, 1988; Stenberg, 1996), but it is not clear how this parameter could be applied to other plant architectures. To this end, Sinoquet et al. (2007) developed a method based on the spatial variance of foliage in tree crowns, but this method seems to be difficult to apply to field measurements. The lack of a simple operational method to account for grouping of foliage is thus limiting our ability to link canopy light interception to plant and canopy structure.
A few attempts have been made to provide simplified models that account for grouping of foliage at the whole-plant scale (Jackson & Palmer, 1979; Kucharik et al., 1999; Chen et al., 2008; Ni-Meister et al., 2010). However, the resulting models are often complex, with many species-specific parameters that are time-consuming to obtain. Furthermore, available models do not provide estimates of light interception by individual plants, limiting their usage to stand-scale applications. Estimates of light interception on an individual plant level are needed in individual-based models of vegetation dynamics (e.g. Moorcroft et al., 2001; Falster et al., 2011). A different field of study has avoided simplification altogether by developing highly detailed three-dimensional plant models with spatially explicit representation of leaves and stems. These studies have provided valuable insights into the details of plant architecture and how it affects plant performance in specific environments (see reviews by Valladares & Pearcy, 1999; Pearcy et al., 2005; Vos et al., 2010). However, detailed architecture studies are yet to discover which aspects of canopy structure most influence whole-plant light interception, in part because of the limited sample sizes that necessarily result when using such methods.
Generally speaking, plant crowns are defined by the number, size, shape, three-dimensional distribution and orientation of their leaves. Together, these variables determine the size of the crown, the arrangement of leaves inside the crown, and the average leaf overlap (‘self-shading’) when viewed from a given direction. We define light interception efficiency as the ratio of displayed (i.e. exposed) to total leaf area, averaged over the entire sky hemisphere () (Farque et al., 2001; Delagrange et al., 2006) (see Table 1 for a list of symbols). relates directly to the amount of diffuse radiation intercepted by the plant, which can play an important role in determining total carbon uptake (Ackerly & Bazzaz, 1995; Roderick et al., 2001). Direct light interception also scales with the sunlit leaf area fraction (Campbell & Norman, 2000), which is probably correlated with . Simple indices of self-shading – such as have also been shown to predict total carbon uptake. For example, comparing branches from 38 perennial species, Falster & Westoby (2003) found that > 90% of variation in whole-branch CO2 assimilation rate expressed per unit leaf area (excluding differences in leaf photosynthetic capacity) was accounted for by an index of self-shading. has also been used in other applications to rank light interception efficiency of whole plants (Delagrange et al., 2006; Sinoquet et al., 2007) and shoots (Niinemets et al., 2005), suggesting that it provides a reliable indicator of plant performance.
Table 1. Symbols, their definitions and units
Leaf area index
Crown surface area – total surface area of 3D convex hull wrapped around the leaf cloud
Total plant leaf area
Displayed leaf area from angle Ω
Crown silhouette area from angle Ω
Crown porosity from angle Ω
Displayed leaf area averaged over all angles
Mean leaf area of individual leaves
Total plant leaf number
Leaf projection coefficient averaged over all viewing angles
Extinction coefficient for a homogenous canopy
Leaf projection coefficient from angle Ω
Silhouette to total area ratio, averaged over all viewing angles
Leaf dispersion parameter
Viewing angle (elevation, azimuth pair)
Weighting function for AD,Ω
Observed average distance to five nearest leaves
Expected average distance to five nearest leaves for a random distribution
As possible predictors of , we propose two simple whole-plant variables: crown density and leaf dispersion. These variables were selected based on a statistical model predicting shading within the crown, based on similar models independently developed by Sinoquet et al. (2007) and Duursma & Mäkelä (2007), but with a new leaf dispersion parameter. The model is derived (see the Materials and Methods section) by viewing the plant first from one direction, and estimating the leaf area displayed in that direction from the silhouette of the crown, the number of leaves, the mean leaf area, and the mean leaf angle. After making simplifying assumptions, this leads to an approximation for the average leaf area displayed in all directions:
(AL/AC, the crown density (ratio of plant leaf area, AL, to crown surface area, AC); β, a leaf dispersion parameter; ε and φ, empirical parameters.) The ‘extinction coefficient’K is constant because we integrate over the entire hemisphere (Stenberg, 2006) (see the Materials and Methods section). The crown surface area is defined as the total surface area of a crown, that is, the area of a sheet wrapped around the crown. It is calculated as the surface area of the three-dimensional shape constructed by joining all outlying points of the plant crown, so that the shape is convex (i.e. there are no indentations in the three-dimensional shape). The remarkable aspect of the approximation (Eqn 2) is that only two plant variables, crown density (AL/AC) and leaf dispersion (β), are needed to estimate , in addition to the two (constant) parameters. In this paper we test the hypothesis that, across plants of diverse architecture, size, and growth environments, variation in can be explained by crown density and leaf dispersion. To do so, we used a database of 1831 virtual plants, reconstructed from precise digitization of the position and orientation of leaves and stems of real plants, to estimate (cf. Farque et al., 2001). We then compared these empirical estimates to those given by the simplified model. Finally, we tested the hypothesis that declines with increasing plant size for a given species (Farque et al., 2001; Niinemets et al., 2005; Delagrange et al., 2006; Sinoquet et al., 2007), and whether this decline is related to changes in leaf dispersion or crown density.
Materials and Methods
Model of light interception efficiency
Here, we derive a model for the light interception efficiency of individual plants with optically black – that is, nonreflecting and nontransmitting – leaves. We first consider the general case of light intercepted from an arbitrary direction. We then outline an approximation for the average light interception efficiency. Our derivation is similar to that provided by Duursma & Mäkelä (2007) and Sinoquet et al. (2007).
Consider a plant with N leaves, each with area aL, and total leaf area . First consider the case where the sun is in the sky at a fixed position Ω (defined by zenith and azimuth angle). For a plant with optically black leaves, light interception is proportional to the leaf area displayed in the direction of Ω (AD,Ω) (Ross, 1981), otherwise known as the ‘silhouette area’. Displayed leaf area can then be written as:
HΩ, the outline area of the whole crown when viewed from Ω; PΩ, the porosity of that outline (Sinoquet et al., 2007). HΩ is calculated from a two-dimensional convex hull wrapped around the leaf cloud and typically depends on Ω and the crown shape. PΩ is the fraction of the outline not obscured by leaves, or the probability that a solar ray passes through the crown. Note that the probability that a light ray is not intercepted by a single leaf facing in the direction of Ω is . The probability that a light ray passes through a crown of N randomly distributed leaves, each with an average projected area of in the direction Ω, is then
This equation is similar to a binomial model employed by Nilson (1971), with the difference that this approach does not require estimates of the path length of solar rays in the crown, but instead uses the outline area of the crown (HΩ; cf. Sinoquet et al., 2007).
Taking the log of both sides and using the fact that tends to ex for large N, Eqn 4 can be reorganized into a form similar to the familiar Lambert–Beer law (Monsi & Saeki, 1953, 2005):
Substituting the result from Eqn 5 into Eqn 3 gives an expression for AD, Ω:
Next, we derive an approximation for the hemispherically averaged AD. By definition, AD averaged over all angles Ω is given by
Here, fΩ is a weighting function whose integral over Ω sums to unity. The integral in Eqn 7 is difficult to solve, because all variables depend on Ω in complex ways. We hypothesize that Eqn 7 can be approximated by using effective averages for HΩ and kΩ. This simplification yields
where the angled brackets denote an effective average. The result in Eqn 8 is not exact, and it can be shown that the error of this approximation depends, among other factors, on the covariance of KΩ and HΩ along different viewing angles Ω (in other words, the degree to which KΩ and HΩ are correlated).
The model can be completed by using the fact that HΩ averaged over all viewing angles is related to the lateral surface area of the convex hull (AC) through one of Cauchy’s theorems:
where fΩ is a uniform weighting function. This result is general, and does not depend on the actual shape of the crown, as long as it is convex (see Lang (1991) for more details, and the section ‘Crown surface area and leaf dispersion’ for more clarification of AC). While Cauchy’s theorem was originally derived for a spherical integration, Eqn 9 still holds for a hemispherical integration because the silhouette area is the same when viewed from either direction.
Similarly, the mean leaf projection function, , is well approximated by a function that depends on the mean leaf angle (Sinoquet et al., 2007), or if the weighting function is uniform (i.e. all angles have equal weight), it is equal to 0.5 (Stenberg, 2006). We replace <K> in Eqn 8 with K, which could be a function of the leaf angle distribution (but we leave the exact dependence for a further study).
Next, we replace <H> in Eqn 8 with from Eqn 9, and use an empirical parameter φ in place of the ¼ to account for the fact that this rearrangement is not exact. This yields the final result for :
Light interception efficiency is defined here by the ratio of to total leaf area (AL), commonly known as (‘silhouette to total area ratio’), that is, :
The result in Eqn 11 follows from the assumption that leaves are randomly distributed within the crown envelope (Eqn 4). We introduce a simple modification to account for a more clumped or more regular leaf dispersion. We use a dispersion parameter, β, defined so that if β = 1, leaves are randomly distributed. If β < 1, leaves are more clumped, and for β > 1 they are more regular. We multiply AL in Eqn 11 by this dispersion parameter, and include an empirical exponent ε that is to be estimated from data. The final result is Eqn 2. It should be stressed that the modification for nonrandom leaf dispersion is a heuristic solution, and does not follow rigorous derivations such as those provided by, for example, Nilson (1971). The purpose of this study was to find a simple useful model of that can be evaluated against measurements.
Virtual plant material
We collected existing input files for the 3D model YPLANT (see the next paragraph in this section) from published and unpublished sources. These plants were usually digitized with a 3D magnetic device (see Pearcy et al., 2011), or sometimes measured by hand. For each virtual plant, all spatial locations of leaves were measured, as well as their orientation (see Fig. 1 for examples of digitized plants). Leaf shape was determined by x, y coordinates of leaf margins of a representative leaf, but leaf size is allowed to vary within plants. Stem sections, and their diameters, were also included (with the exception of two subdatasets; see Supporting Information Table S1). The collection includes plants, from widely differing environments (understorey, glasshouse and plantation), of 124 different species from different taxonomic families, and from all across the globe (Valladares et al., 2000, 2002, 2005; Falster & Westoby, 2003; Pearcy et al., 2004; Lusk et al., 2006, 2011; Reich et al., 2009; and a number of unpublished studies; see Table S1 for more details). We excluded a few plants from our final analysis: plants with leaves that were folded across the midrib (Falster & Westoby, 2003) and plants with fewer than six leaves, in both cases because the leaf dispersion parameter could not be calculated correctly, leaving 1831 plants in the database. Species of two genera dominated the collection: Psychotria (644 plants of 19 species) and Eucalyptus (295 plants of five species). The rest of the collection had an average of nine individuals per species.
We used the 3D plant model YPLANT (Pearcy & Yang, 1996; Pearcy et al., 2011) to estimate the amount of leaf area displayed in different directions (AD). YPLANT accounts for the detailed 3D distribution of leaf elements. Shading by stems was also accounted for (except in two of the data sets; see Table S1), but was only a small source of variation in AD (not shown). YPLANT does not account for penumbral effects, but the penumbra does not affect total light interception (Stenberg, 1998). In YPLANT, AD is calculated for the centroids of each of 160 sectors of the hemisphere, corresponding to eight classes of azimuth and 20 solar elevation (α) classes (Pearcy & Yang, 1996). We calculated a hemispherically averaged STAR from these individual estimates of AD, by weighting each AD by the relative area of each sector. First, we averaged STAR (AD/AL) across all azimuth values for a given α. We then weighted these STARα values by the area of the zone (the area of a section of the hemisphere between α1 and α2). This area is proportional to sin(α2) – sin(α1), where α2 is the larger of the two angles that define the zone. This yielded an estimate of for each plant that is consistent with its use in the summary model. An alternative method to finding is provided by the VegeSTAR software package (Adam et al., 2002).
Crown surface area and leaf dispersion
In the derivation of Eqn 2, we used the fact that the average two-dimensional projected area of any convex three-dimensional shape is a quarter of its surface area (Eqn 9); a famous theorem by Cauchy (Lang, 1991). For example, for a sphere, the projection is a circle (area = πr2), which is one quarter of the total spherical surface area (area = 4πr2). While this theorem is useful in determining canopy extinction coefficients (Stenberg, 2006), it also implies that the correct definition of crown size with respect to light interception is that defined by its three-dimensional convex hull which contains all the leaves (Duursma & Mäkelä, 2007). We used the quickhull algorithm (Barber et al., 1996), in the geometry package in R (R Development Core Team, 2010) to find the convex hull (and its surface area) spanning the leaves, using all coordinates of the leaf edges. This algorithm finds the smallest set of x, y, z points that defines the convex hull, that is, the polyhedral surface that contains all other points, and is convex (see Fig. 1 for examples of convex hulls).
As an index of leaf dispersion, we used a nearest-neighbour approach to quantify deviation from a random distribution of leaf positions within the convex hull. For each leaf, we calculated the mean distance between the midpoint of that leaf to the midpoint of the five nearest leaves. This mean distance was calculated for each leaf, and averaged over all leaves. This yielded the observed mean distance to the five nearest leaves (O5). We compared O5 to the expected distance if the leaf locations were completely random within the convex hull (E5) (see Fig. 3a). This expected value was calculated numerically (for 10 replicates), for a square box with the same volume and number of leaves as the plant crown to account for edge effects. This edge-effect-corrected E5 was not much different from an estimate where we did not account for the edge effect (Fig. 3a), so more precise methods that account for the actual shape of the plant crown were not applied. The leaf dispersion was defined as O5/E5. Other choices for the number of neighbouring leaves were also tested (between two and 25), without large effects on the goodness of fit or any other of the main results. Values of unity indicate a random distribution, while values of less than unity indicate a more clumped distribution, and values larger than unity a distribution that is more regular than random.
Both the crown surface area (AC) and leaf dispersion (β) were estimated for the digitized plants, but it is possible to measure both variables on plants in the field. There is a close linear correlation between AC and the product of crown width and length, with the constant of proportionality depending on the crown shape (e.g. cone vs ellipse). Both crown width and length are easily measured in the field, and can thus be used to provide estimates of AC (Duursma et al., 2010). For β, it is necessary to measure the distance to the five nearest leaves (O5), whereas the expected distance to the five nearest leaves (E5) is readily obtained from simulation (using the R package YpTools; see Pearcy et al., 2011). For our virtual plants, we measured O5 for all leaves, but sufficiently accurate estimates can be obtained based on a random sample of leaves. A random table may be used for small plants, where all leaves can be assigned a number, or for larger plants a stratified sampling strategy may be employed (by first selecting branches at random, and then leaves within branches).
The two variables in the model (AL/AC and β) were determined for each plant. The K parameter was set to 0.5, and the remaining two empirical parameters in Eqn 1, φ and ε, were estimated by nonlinear regression (using the ‘nls’ function in R), that is, the two empirical parameters are the same for all plants. The model R2 was then calculated as , where is the variance of the model residual, and is the variance of the dependent variable across single plants () (Kvålseth, 1985). We analysed the residuals for patterns with a number of plant structure variables, such as AL, AC, β, number of leaves, and crown shape.
As with any data-pooling exercise, our data set is potentially confounded by multiple factors, such as experimental setting, growth environment, and unmeasured traits conserved within genera, within families or at a higher taxonomic level. We adopted two approaches to determine whether taking account of clade membership improves predictions of light interception efficiency. First, we refitted the model by family, to determine whether this resulted in divergent parameter sets. Secondly, we used a linear mixed-effects model to partition the residual variance into division (Angiosperm or Conifer), family, and species (using nested random effects), and the light environment (shaded, intermediate or exposed, using a fixed effect) (with the nlme package in R).
The scaling of AL with AC, and the correlation between and total leaf number were tested with standardized major axis (SMA) following Warton et al. (2006), for each species separately, using a subset of the data where individual species had at least five replicates.
The data set included plants with a wide range in leaf area and displayed area (Fig. 2a), and a wide range in as estimated by the YPLANT model (Fig. 2b). Note that the limit value of (calculated for the whole hemisphere) is 0.5, obtained when there is no overlap between leaves. However, plants with more than one leaf cannot actually achieve this value when is calculated for the whole hemisphere. As indicated by the measure of leaf dispersion (β), 608 of the 1831 plants showed a clumped arrangement in crowns (Fig. 3), while 460 plants tended to a more regular display, leaving 763 plants where the leaf dispersion was not significantly different from a random distribution.
The simple model (Eqn 2) incorporating both crown density and leaf dispersion explained 85% of the variation in across the 1831 virtual plants (Figs 4, 5). As shown in Fig. 5, declined consistently with increasing AL/AC at a given value of β, and increased with β at a given value of AL/AC. By itself, AL/AC explained only 43% of the variation in (by refitting the model after setting β to a constant value), and β explained only 33% (see Table 2 for details). We performed a residual analysis to determine if variables not included in the model explained additional variation in . This analysis did not reveal any systematic over- or underestimation related to the leaf angle distribution or total plant leaf area, and only a weak relationship with crown shape (giving a slight overestimation for flat crowns) (Fig. S1). It is not surprising that we found no correlation between the residuals and the leaf angle distribution, as we used the hemispherically averaged in our model, but a correlation might arise if the leaf angle distribution was related to unmeasured traits that affect .
Table 2. Diagnostics of the nonlinear regression fit of the simple model of (Eqn 2) to the whole data set
RMSE (m2 m−2)
Parameter values were φ = 0.254 (SE = 0.0019). and ε = −2.28 (SE = 0.034), with K set to 0.5. Shown are the R2 and RMSE (root mean squared error) for the full model, and for two simplified models where dispersion was excluded (‘AL/AC only’) or AL/AC was excluded from the model fit (‘dispersion only’) (and instead set to the average value for the whole data set).
Variation between and within species dwarfed the influences of family and division membership on differences in light interception efficiency. The model performed equally well – or better – when fitted to each plant family separately, with one exception (Myrtaceae) (Fig. S2). However, the estimated parameter values (φ and ε) did not vary greatly between families. These results show that the excellent fit of the model to the full data set was not confounded by taxonomic relatedness, and that the same model can predict for a variety of taxa. However, there was some indication of taxa-specific effects that are not accounted for in the simple model. Partitioning of the residual variance from the full model showed that 14.1% was related to division (Gymnosperm or Angiosperm), 3.7% to family, 36.9% to species, and the remaining 45.6% to individual within species. Thus, an additional 8.25% of total variance in (55% of residual variance) could be attributed to species, family, or division, in addition to the 85% of total variance already accounted for by crown density and leaf dispersion.
Within species, the leaf dispersion parameter was closely related to total leaf number (N). There was a strong tendency for plants with more leaves to exhibit a more clumped leaf dispersion (Fig. 6a). This observed correlation was directly related to N, because at a given AL, β increased with mean leaf size (i.e. plants with smaller leaves have more clumped foliage) (Fig. 6b). Secondly, at a given leaf size, the dispersion parameter (β) decreased with AL (i.e. larger plants had more clumped foliage) (Fig. 6c). These two patterns are confounded when leaf dispersion is plotted against either leaf size or leaf area, because larger leaved species tended to have greater total area in our data set. To illustrate this, we fit a simple regression model to leaf dispersion (β) as a function of total plant leaf area (AL) and mean leaf size (aL), as β = bALc×aLd (see Fig. 6), which can also be rearranged as β = aNc*aL(c+d). Because c ≈ −0.15 and d ≈ 0.15, the leaf size component cancels, leaving a dependence of β on number of leaves (with exponent ≈ −0.15).
Finally, we tested the hypothesis that declines with plant size. We found a strong relationship between and total leaf number (N) (Fig. 7a). Out of the 65 species in the data set that had more than five replicates, 50 showed a significant negative correlation between and N. To assess whether this decrease in with increasing N was attributable to leaf dispersion (Fig. 6) or crown density (AL/AC), we tested for size-related trends in AL/AC by assessing the scaling of AL as AL = aACb (Fig. 7b). If the exponent b >1, this would result in an increase in AL/AC with AC, and would lead to decreased light interception efficiency with increasing crown size. Out of the 65 species, 38 showed an exponent not significantly different from unity (the mean exponent estimated by SMA was 1.05; SD = 0.25). For the remainder of the data set, 18 species had an exponent > 1, and only nine had an exponent < 1. There is, then, no general evidence that AL/AC increased with plant size in this data set, and the decrease in with N is entirely attributable to a more clumped leaf dispersion in larger plants.
We found that two simple variables, summarizing the display of leaves in plant crowns, explained most of the variation in plant light interception efficiency () for 1831 plants of 124 species across a variety of environments. The first variable, crown density, is given by the ratio of leaf area to crown hull surface area. The second variable, leaf dispersion, describes the arrangement of leaves inside the crown. Moreover, our model provides a robust framework for ranking individual plants in terms of , and thus for understanding trade-offs in structure and function of plant crowns. Carbon uptake is closely linked to plant light interception; therefore, our results will also help improve ecosystem models aimed at quantifying plant structure and productivity.
Determinants of light interception efficiency
Although several other traits have been shown to affect plant light interception (see review by Valladares & Niinemets, 2007), our study implies that these variables are either summarized by the two simple variables used here or play a relatively small part in determining light interception efficiency. For example, phyllotaxy (arrangement of leaves on plant stems) (Brites & Valladares, 2005), petiole twisting (Gálvez & Pearcy, 2003), and petiole lengthening (Takenaka, 1994) will affect the leaf dispersion parameter, and the crown density if crown size is affected. The crown shape (length to width ratio) affects crown density, unless crown hull surface area is held constant – explaining at least some of the theoretical relationships between crown shape and light interception (see Duursma & Mäkelä, 2007). We expect that previously published interspecific comparisons in (Valladares et al., 2002; Pearcy et al., 2004) could be partly explained in terms of crown density and leaf dispersion (see also Planchais & Sinoquet, 1998; Cescatti & Zorer, 2003; Massonnet et al., 2008). Therefore, our analysis provides a more robust mechanistic framework for interpreting interspecific comparisons.
We found that the clumping of leaves within crowns increased with increasing plant size and with decreasing leaf size. A previous study argued that the centroids of smaller leaves are necessarily closer to plant stems and to each other (Falster & Westoby, 2003), leading to a more clumped distribution. Our results support this interpretation. The connection between clumping and plant size, however, is less widely recognized. A more clumped distribution is less costly in terms of supporting biomass as plants grow larger, because fewer branches are needed to support a given total leaf area when leaves are closer together. Thus, increased clumping may be favoured, despite the reductions in light interception. Similarly, gradual shedding of leaves inside crowns leads to a more clumped distribution because newer leaves are borne at the tips of branches. In any case, this novel finding points to a connection between crown structure and function that is yet to be explored.
In a simplistic viewpoint, plants with high should outperform those with lower because they intercept more light per unit leaf area, which would lead to higher whole-plant carbon uptake rates. Nonetheless, we observed a wide range in (Fig. 2b), even among co-occurring plants. We found that declined with increasing total leaf number for the majority of the species in our data set, and that this decline was attributable to a more clumped leaf dispersion in larger plants (Figs 6, 7a), while crown density (AL/AC) remained roughly constant (Fig. 7b). Other studies have also found a decline in (or related measures) with increasing plant size. Niinemets et al. (2005) found decreased light interception efficiency of shoots in Agathis australis with increasing tree size, while several other studies (Farque et al., 2001; Delagrange et al., 2006; Sinoquet et al., 2007; Lusk et al., 2011) have found a decrease in with increasing AL in a variety of species. Our results generalize and extend these previous findings.
Variation in of plants grown in a common light environment can probably be understood through a cost–benefit analysis of plant architecture (cf. Küppers, 1989; Sterck & Schieving, 2007): maintaining a high light interception efficiency incurs a substantial cost via the construction and maintenance of the woody structures needed for support and water supply. Taking into account the balance between biomass cost and gain from photosynthesis, Pearcy et al. (2005) showed that the optimum internode length (and therefore crown density) was shorter than that required for maximal light interception. Similarly, others (Mäkelä & Sievänen, 1992; Duursma et al., 2010) have found that it is beneficial for plants to slightly decrease (by increasing AL/AC) with increasing plant size because the construction cost increases more than the benefit from higher light interception per unit leaf area. However, here we found approximately isometric scaling between AL and AC, which would lead to a roughly constant AL/AC with plant size (Fig. 7b). Taken together, these results provide a link between light interception efficiency and the cost for leaf display as a plant grows, because total plant woody biomass scales closely with AC (Mäkelä, 1997; Valentine & Mäkelä, 2005; Duursma et al., 2010). The strong influence of leaf dispersion on found in this study suggests that a cost–benefit analysis of leaf dispersion is worth investigating.
We have so far ignored variation in leaf angle, although leaf angle distribution affects the function of real plants (Valladares & Pearcy, 1998; Falster & Westoby, 2003). Benefits of steep leaf angles include avoidance of photo-damage caused by excess sunlight in hot midday conditions (Valladares et al., 2005), and avoidance of high leaf temperatures (Valiente-Banuet et al., 2010). The model for was based on the assumption that diffuse skylight has no angular dependence, so that K =0.5 for any leaf angle distribution. It has previously been shown that K can be a simple function of the mean leaf angle if light does have a strong angular dependence (Sinoquet et al., 2007). Such dependence will be stronger for plants growing beneath small canopy gaps. Clearly, the trade-off between diffuse and direct light will affect plant architecture, as well as the correct definition of . In support of the importance of diffuse light, Ackerly & Bazzaz (1995) found that the leaf orientation of seedlings in the field was influenced by the hemispherical distribution of diffuse (rather than direct) light sources.
Relation to other radiative transfer models
Radiative transfer modelling in plant canopies has shown that complex canopy structure can be summarized with measures of canopy density (leaf area per unit volume) and a leaf dispersion parameter (Nilson, 1971; Ross, 1981). Our approach is similar to an aggregated turbid medium model (Ross, 1981) but applied to single isolated plant crowns instead of horizontally extended canopies. A similar approach was used by Cescatti & Zorer (2003) and Niinemets et al. (2005) for conifer shoots, but their approach requires information on the path length of rays through the shoot envelope. The novelty of our approach is that, instead of testing the complex integral of the displayed leaf area over all viewing angles directly against measurements (Eqn 7), we hypothesized a simple approximation that yields a compact model. The final model is essentially equivalent to that proposed by Sinoquet et al. (2007) and Duursma & Mäkelä (2007) but with a different form of the leaf dispersion parameter. Our nearest-neighbour approach to the leaf dispersion parameter is novel; previous approaches have used the variance of the contact number (Nilson, 1971), the variance in leaf area density within crown envelopes (Sinoquet et al., 2005), and of conifer shoots (Stenberg, 1996).
Relation to stand-level models of light interception
The importance of leaf dispersion (i.e. clumped vs regular leaf dispersion) has long been recognized (Nilson, 1971; Baldocchi et al., 1985; Whitehead et al., 1990) but some ecosystem models continue to assume that foliage is randomly distributed. Nilson (1999) showed that the problem of canopy transmission in heterogeneous canopies can be simplified when separating within-plant from between-plant shading. Canopy transmission is then a function of self-shading, which is a function of crown density and leaf dispersion as we have shown, and the spatial distribution of neighbouring plants. In the simplest case of identical plants that are randomly distributed in space, the fraction of light intercepted by a canopy (fAPAR) approximately follows Lambert–Beer’s law (Eqn 1) with the extinction coefficient depending on of the individual plants (see Methods S1):
The interpretation of this equation is that, at a given total leaf area index L, a canopy consisting of crowns with a low will intercept a lower fraction of available light than a canopy consisting of plants with high . Adjustments for nonrandom distribution of plants can also be introduced (Nilson, 1999). It would be worthwhile to test this equation for real stands, or against a detailed canopy-scale model that takes into account grouping of foliage in tree crowns.
The simple canopy-scale model of light interception (Eqn 12) used plant , which describes light interception from all angles. However, for individual plants in canopies, light can be strongly directional. Duursma & Mäkelä (2007) compared this model to a detailed stand-level model of light interception, and found good agreement, but the leaf angle was kept constant in those simulations. For our data set, we have also calculated STAR averaged over a 30° canopy gap (STAR30) centered at the zenith (results not shown). Our model for explained only 52% of the variance in STAR30, but when a function of leaf angle was added to the model (in place of K in Eqn 2), the R2 increased to 81%. The effect of crown shape was also significant, but of minor importance compared with the strong effect of leaf angle. A further generalization of the simple model presented here to plants growing under gaps or in stands of plants will be the subject of a future contribution.
We identified two simple variables that together explain 85% of variation in silhouette-to-area ratio of foliage (), a measure of light interception efficiency, across a diverse collection of plants. We think it is feasible to develop methods for field sampling of both these variables (crown density and leaf dispersion), potentially obviating the need for detailed models to predict light interception efficiency if a basic index of light interception efficiency is warranted. A simple model of that incorporates these two variables can be readily incorporated in canopy models (cf. Duursma & Mäkelä, 2007). We found that was negatively correlated with total plant leaf number for most of the species in the data set, as a result of a more clumped leaf dispersion for larger plants. These results allow a more general interpretation of the wide diversity in plant architecture, and its connection to plant and canopy function. We thus provide a new framework for understanding how the diversity found in crown architecture contributes to plant convergence or divergence in light interception efficiency, both within and across plant communities.
B.E.M. and D.S.E. acknowledge the support of the Australian Research Council (ARC) (grant DP0881221), and D.T.T. received support from ARC grant DP0879531. R.A.D. was partly supported by a NSW Government Climate Action Grant (NSW T07/CAG/016). B.J.A. thanks Aderajew Haddis for assistance with collection of eucalypt data.