Optimal crop canopy architecture to maximise canopy photosynthetic CO₂ uptake under elevated CO₂ – a theoretical study using a mechanistic model of canopy photosynthesis

Additional keywords: canopy architecture, elevated CO₂, photosynthesis, ray tracing, sunlit–shaded model.

Improving photosynthetic capacity is one of the main approaches to further enhance crop productivity (Long et al. 2006b; Zhu et al. 2010), since canopy rather than leaf photosynthesis is closely related to crop yields. One of the major mechanisms underlying the improved crop yields during the ‘green revolution’ has been improved canopy architectures. In particular, selection of cultivars with more erect leaves, especially at the top of the canopy, has led to improved light environments inside a canopy and hence, improved canopy photosynthetic CO₂ uptake rate (A_c) (Long et al. 2006b). Though there is still some controversy regarding whether improving photosynthesis can lead to increased crop yields, this may be more related to using leaf rather than canopy photosynthesis in deriving the relationship (Zelitch 1982). The challenge now is how to identify the ideal canopy architectural and leaf metabolic features to breed or engineer for increased canopy photosynthesis for current, and more importantly, for future elevated CO₂ conditions. This is a challenge, partially due to the lack of efficient methods to measure A_c, although many efforts have been devoted to developing canopy photosynthesis chambers (Reicosky and Peters 1977; Steduto et al. 2002). Given the logistic difficulty of estimating canopy photosynthesis in the field, mathematical modelling has typically been used to estimate A_c.

Many canopy photosynthesis models with different levels of details have been developed to date. Depending on the level of complexity, these models can be roughly divided into three categories: (i) the big-leaf model (Running and Coughlan 1988; Thornley and Johnson 1990; Sellers et al. 1992; Amthor 1994; Kull and Jarvis 1995; Lloyd et al. 1995); (ii) the sunlit–shaded model (de Pury and Farquhar 1997; Wang and Leuning 1998; Dai et al. 2004); (iii) and the multi-layer model (deWit 1965; Duncan et al. 1967; Lemon et al. 1971; Norman 1979). The big-leaf model assumes that the leaf nitrogen is optimally distributed to match the light environments inside a canopy (Sellers et al. 1992; Amthor 1994; Lloyd et al. 1995). Farquhar (Farquhar 1989) demonstrated that equations describing leaf photosynthesis are the same as those for chloroplast photosynthesis as long as (a) the chloroplast photosynthetic capacity is proportional to the irradiance it absorbs; and (b) the responses of the photosynthetic CO₂ uptake to irradiance are identical among different chloroplasts. This principle was extended to the canopy and formed the basis for the big-leaf canopy photosynthesis models. However, inside a canopy, the light distribution is highly heterogeneous both spatially and temporarily, mostly due to the heterogeneity of leaf angles and the sunflecks deep in the canopy (de Pury and Farquhar 1997). Such heterogeneities are ignored in current big-leaf, sunlit–shaded and multilayer models. Given this inaccuracy, some model parameters – namely the photosynthetic capacity (Lloyd et al. 1995) and the smoothness of the transition from light-limited to light-saturated photosynthesis (Sellers et al. 1992; Amthor 1994), i.e. the curvature factor – can be adjusted in the big leaf model. This improves the prediction accuracy of the big-leaf models; however, it renders big-leaf models essentially unscaleable in the sense that these parameters, e.g. curvature factors, need to be modified dependent on the leaf area index (LAI), leaf nitrogen content and the proportion of diffuse light (de Pury and Farquhar 1997). This also makes direct interpretation of the parameters used in the big-leaf model very difficult.

The other extreme of the canopy photosynthesis model is the multi-layer model, where the whole canopy is divided into many categories of leaves, each with different light levels. Most of these models are based on two assumptions: (i) that radiation attenuation inside canopy can be predicted by Beer’s Law (Monsi and Saeki 2005); and (ii) that the light inside a canopy can be divided into two categories, i.e. direct and diffuse light, dependent on the different attenuation in canopies (Goudriaan 1977). In these models the predicted light environment for each leaf category is combined with leaf photosynthesis models to predict A_c. These multi-layer models have the flexibility of incorporating the heterogeneity in both environmental and physiological parameters. So far, these models are considered as the most accurate canopy photosynthesis models, with the major drawback of being complex and requiring longer computation time.

The intermediate class of models are the sunlit–shade canopy photosynthesis models. The sunlit–shaded canopy photosynthesis model simplifies the multi-layer model by dividing the whole canopy into two categories of leaves: (i) sunlit; and (ii) shaded leaves (Sinclair et al. 1976; Norman 1980). The key for the prediction accuracy relies on the capacity to predict (a) the proportion of the sunlit leaf area and shaded leaf area; and (b) the light levels in the sunlit and shaded leaves. Modelling comparison has demonstrated that the sunlit-shaded model predicted similar A_c as the multi-layer canopy models (de Pury and Farquhar 1997). Besides sunlit–shaded models, another approach to simplify the multi-layer model is to derive analytical solutions of canopy photosynthesis (Acock et al. 1978; Johnson and Thornley 1984). However, the predictions using this approach are not as accurate as the sunlit–shaded models (Boote and Loomis 1991).

The sunlit–shaded model has been used in various applications. First, it has been used to identify options to engineer for higher canopy CO₂ uptake rates (Zhu et al. 2004a; Ort et al. 2011).It has also been extended to include the interaction of stomatal conductance and photosynthesis and even the response of stomatal conductance to water vapour deficit and soil water content, which enabled prediction of net photosynthesis, latent and sensible heat flux of a canopy under a wide range of soil water availability and meteorological conditions (Wang and Leuning 1998). The sunlit–shaded model was also applied in forest studies, for example, to calculate the forest primary production (Kotchenova et al. 2004), CO₂ uptake (Catovsky et al. 2002) and the effect of urban forest on air pollutant dry deposition (Hirabayashi et al. 2011). Recently, it was used to model the growth and production of the bioenergy crop Miscanthus × giganteus (Miguez et al. 2009). The sunlit–shaded canopy photosynthesis model was also used as a major tool in the agro-ecology, e.g. in the study of canopy transpiration (Tuzet et al. 2003; Yang et al. 2009; Chen et al. 2011) and evapotranspiration of winter wheat (Mo and Liu 2001), effect of elevated CO₂ on plant (Reynolds et al. 1992) and the effects of clouds and atmospheric particles on plant (Roderick et al. 2001; Gu et al. 2002).

However, all these above-mentioned models are unable to predict the precise light environment inside a canopy with defined canopy architecture and therefore, cannot precisely predict the spatial and temporal heterogeneities of light inside a canopy. However, such heterogeneities of the light environment is extremely important to gain an accurate prediction of A_c, ignoring this heterogeneity can lead to an overestimate of A_c for a canopy (Zhu et al. 2012). As a result, the current used sunlit–shaded canopy photosynthesis model cannot predict the influence of modifying canopy architecture, such as tiller number, leaf shape or leaf angle on A_c. With regards to this, a few models representing 3D canopy architectures for different crops including maize (España et al. 1999; Guo et al. 2006; Zheng et al. 2008), rice (Watanabe et al. 2005; Zheng et al. 2008) and wheat (Evers et al. 2007) have been developed in recent years.

This paper describes a model of canopy photosynthesis in which the canopy architecture and light environment inside a canopy is simulated in detail: these two components are then combined with the steady-state biochemical model of C₃ photosynthesis (Farquhar et al. 1980) to predict A_c. With this model, we studied the influence of canopy architecture on canopy photosynthesis and explored the theoretically impact of elevated CO₂ on optimal canopy architectures required to maximise A_c.

In this study, we used the architectural features of the indica rice Teqing (Oryza sativa L. subsp. indica). The cv Teqing was planted in Beijing (39.92°N and 116.46°E) on 1 June (day 150) in 2009. To define the plant structure, several parameters were measured, including tiller number, leaf number, leaf base height, leaf length, leaf width, leaf angle and leaf curvature. The parameters used in the model are illustrated in Fig. 1b, c including: leaf base height (distance between the leaf base and the ground); leaf length (maximal length of a leaf when stretched to be straight); leaf width (maximum width of each leaf); leaf angle (angle between leaf blade and stem); and leaf curvature radius (leaf curvature along its longitudinal axis was assumed to correspond to an arc of a circle, the radius of which was termed leaf curvature radius).

Fig. 1.  (a) A single rice tiller with alternative leaf arrangement. (b) The leaf and canopy architectural features, which were used to construct the 3D canopy model in the mCanopy script (leaf angle ß, leaf curvature radius and leaf base height) and (c) (leaf length and leaf width). Leaf angle is the angle between leaf base and stem: x is the horizontal distance between leaf tip and stem and y is the distance between leaf tip and ground surface. The leaf curvature radius is calculated based on x and y assuming that the leaf blade represents an arc section of a circle.

Features of the canopy architecture of rice at the grain filling stage were obtained by taking average architectural parameters from three rice plants. To illustrate the procedure used to obtain average parameters, we assumed that each plant has m tillers and each tiller has n leaves. We first ordered and labelled all tillers in a plant according to its tiller height and all leaves in a tiller according to the leaf base heights. Then each leaf can be uniquely labelled as leaf (i, m, n), where i is the plant number (i.e. 1, or 2 or 3), m and n are the tiller number and leaf number. The average of a canopy architectural parameter is then calculated as the average value of the parameter for a leaf with the same index m and n.

A rice plant constructed with the averaged features was used to develop a canopy model (Fig. 2). Here the canopy included 64 rice plants (eight rows by eight columns) with row and column distances both being 25 cm. Each plant in the model contained 13 tillers and each tiller contained 3–5 leaves with alternate leaf arrangement (Fig. 1a). A leaf was represented as a rectangle with measured leaf width, leaf length, leaf angle and curvature, which was divided into 5-cm long segments and each segment was divided into two parts by the major vein of the leaf. The individual section of a leaf surface, defined by four edges or four points, was defined as a facet. Leaves on a tiller were arranged alternatively with a random orientation angle and measured distance from leaf base to the base of the tiller. Tillers in a plant were arranged symmetrically so that taller tillers formed an inner circle and shorter tillers formed an outer circle. Many individual plants were generated individually and then combined to form a canopy. We randomly oriented each plant in the canopy. A MatLab script (MathWorks, Natick, MA, USA), mCanopy, was developed to construct the 3D model (code available from authors upon request). The canopy model can be adjusted by varying parameters of the canopy architectures.

Fig. 2.  (a) A 3D plant with 14 tillers and a rice canopy re-constructed with the mCanopy program.

We developed a java program (fastTracer; PICB, Shanghai, China) that uses canopy architecture model as input to simulate the light distribution inside a canopy with a forward ray tracing algorithm. The fastTracer can predict the light environment not only for a canopy with defined architecture, but also for a canopy constructed based on 3D co-ordinate data from 3D digital equipment (e.g. FASTSCAN http://www.fastscan3d.com/, accessed 4 December 2012).

The basic unit of a leaf surface was assumed to be facet, defined by four boundaries or points. The model is able to simulate three categories of light, i.e. direct light, diffuse light and scattered light (Fig. 3a–c). The direction of direct light is determined by the solar elevation angle, which further depends on the time and location for each particular simulation (Appendix 2, Eqn A26–29). The direction of the diffuse light is randomly distributed with equal probability in each direction. Scattered light is generated once a direct or diffuse light ray hits a leaf surface with the model of Cook-Torrance bidirectional reflectance distribution function (BRDF) (Torrance and Sparrow 1967; Cook and Torrance 1981) and Lambert bidirectional transmittance distribution function (BTDF) models (Grant 1987). The directions of scattered light were simulated with a Monte Carlo method (Tucker and Garratt 1977). The leaf absorbance is different under different incident angles (Brodersen and Vogelmann 2010). As a simplification, this model assumed an average leaf absorbance of 0.85.

Fig. 3.  Schematic representation of the ray tracing algorithm used to simulate the light distribution in a plant canopy. Direct light is represented with parallel lines (a), diffuse light with lines with random directions (b) and scattered light (c). When a light ray hits the boundary of the model geometry (a big cuboid), it was moved to a symmetrical position on the opposite side of the cuboid (d). The cuboid was divided into many cubes and light was traced from one cube to a neighbouring cube (e).

A forward ray tracing algorithm simulates the path of a light ray from its source until it is finally absorbed completely by its illuminated objective. The ray tracing process in the model is shown in Fig. S1, available as Supplementary Material to this paper. In this study, a cuboid (length 1 m, width 1 m and height 2.8 m) in the centre of the reconstructed canopy, which had a length >2 m, a width >2 m and a height <2.8 m, was chosen and only leaves inside the cuboid were used in the simulation to avoid boundary effects. The light source used in the simulation is mimicked by a cluster of light ray with a density (η) of 10⁶ light rays per square meter above the cuboid (Fig. 3e). The cuboid is divided into many cubes and the light is traced from one cube to another (Fig. 3e). The facets in the cube were checked to determine whether it is hit by the light ray. In our algorithm, when a light ray hits the boundary of the cuboid, this light ray is moved to the opposite side of the cuboid (Fig. 3d). For direct light, the directions of the light rays are parallel to each other; while for diffuse light, the directions of these light rays are randomised. The value of photosynthetic photon flux density (PPFD) on each facet (I_f) is calculated by dividing the absorbed light energy (e_i is the energy of each ray, N is the number of rays) intercepted by a facet with area s.

Each facet ABCD (a rectangle) has four boundaries, i.e. AB, BC, CD and DA. In our forward ray tracing algorithm, the intersection point, P(x₁, y₁, z₁), of the light ray with the plane ABCD is first calculated. Then, it is determined whether point P is inside ABCD or not. To do this, the rectangle ABCD is divided into two triangles ABC and ACD to check whether the point falls into ABC or ACD. If the sum of the areas of PAB, PBC and PCA equals the area of ABC (Eqn 3), point P is inside the triangle ABC.

The reflected and transmitted light rays are generated once a light ray hits a leaf surface. The energy of reflected light ray (e_r) or transmitted light ray (e_t) were assumed to be 7.5% of energy of incident light ray (e_i) and their directions were determined with a Monte Carlo method (Halton 1970) as in work by Lao et al. (2005).

The light environment above the canopy is influenced by atmospheric transmittance. During a growing season of a particular crop, the air transmittance varies. In this study, we assumed an atmospheric transmittance of 0.7. The directions and PPFD of direct light and diffuse light are needed to run the forward ray tracing algorithm. The PPFD of incoming solar radiation can be provided either by measured values or through prediction using mechanistic physical models. In the current study, we used a macroclimate model (Humphries and Long 1995) (Appendix 2, Eqns A22–31), which has been used in several previous studies of canopy photosynthesis e.g. (Zhu et al. 2004a, 2004b). Solar elevation angle, azimuth angle and PPFD of direct and diffuse light are calculated following Eqns A26–31 in Appendix 2. The predicted light environment inside the canopy using the forward ray tracing algorithm is used in (Eqns 1, 2) to predict the total canopy CO₂ uptake rate.

The photosynthetic CO₂ uptake rates (A) for each facet in a canopy is predicted by combining the steady-state biochemical model of leaf photosynthesis (Farquhar et al. 1980) with the predicted PPFD. Canopy photosynthetic CO₂ uptake rate (A_c) is calculated by dividing the total photosynthetic CO₂ uptake for all the leaves above a ground area S_ground by the ground area.

The daily canopy photosynthetic CO₂ uptake rate (A_c′) is the integral of A_c during a whole day. In the calculation, A_c is numerically integrated.

where A_c,t is the total photosynthetic CO₂ uptake for a unit ground area during a particular hour; A_c,u represents the total canopy photosynthetic CO₂ uptake rate at the middle of an hour, i.e. at 30th minute of each hour.

We examined the influences on daily total canopy CO₂ uptake by several canopy architectural parameters including stem height, leaf width and leaf angle. We further studied the effects of elevated CO₂ on the theoretically optimal canopy architectural parameters to gain maximal A_c. In the study of the influence of elevated atmospheric CO₂ concentration (C_a) on A_c, we used three scenarios: (i) current temperature (T) and current C_a of 380 μbar; (ii) elevated temperature (T + 1.5°C) and C_a of 550 μbar, mimicking the scenario of 2050; and (iii) elevated temperature (T + 3°C) and a C_a of 760 μbar, mimicking the scenario of 2100 based on work by Parry et al. (2007). The temperature dependence of the parameters for leaf photosynthesis and respiration are detailed in Eqn A7–15 in Appendix 1. Following work by Davey et al. (2004), we assumed that dark respiration (i.e. R_d) under elevated CO₂ is 13% higher than that under the ambient CO₂ level.

In addition to label photosynthetic CO₂ uptake as limited by either RuBP-limited photosynthesis or Rubisco-limited photosynthesis following the Farquhar et al. (1980) model, in this work, we also label photosynthesis as either light-limited or light-saturated. To do this, we first calculated the slope of the light response curve at each light level. The point on the light response curve where the slope is 0.005 was marked as the transition point between light-limited and light-saturated photosynthesis.

Fig. 4 shows PPFD of the predicted direct, diffuse and scattered light at every facet in the plant canopy. When fitting the data with Beer’s Law, the coefficients of determination, R², are 0.24 for direct light, 0.90 for diffuse light and 0.41 for all light. From Fig. 4c the distribution of scattered light differs from those of direct and the diffuse light in the canopy. From the top to the bottom layers of the canopy, the PPFD of scattered light first increases and then decreases. Fig. 4d shows the gradient of total PPFD with depth on every facet in the canopy.

Fig. 4.  Predicted distributions of direct, diffuse and scattered light at different depths of a canopy at 1130 hours on day 253 at a latitude of 39.92°N. Each point represents the photosynthetic photon flux density (PPFD) of light absorbed by a facet on a leaf in the canopy.

To demonstrate the impacts of using average instead of the detailed PPFD distribution on estimating A_c, we compared the A_c calculated from average and detailed PPFDs. For this analysis, the canopy was divided into 15 layers with the height of each layer being 5.49 cm. In each layer, PPFD of all facets were averaged and then used to calculate A_c. Fig. 5 shows that A_c calculated with averaged PPFD is higher than that with the detailed PPFD at each time point, especially at midday when PPFD is high. Integrating A_c for a whole day calculated with average PPFD was 25% higher than that predicted with the detailed PPFD on each facet (Fig. 5).

Fig. 5.  The diurnal canopy CO₂ uptake rate (A_c) calculated with average photosynthetic photon flux density (PPFD) at different layers of a canopy (average light) compared with A_c calculated using the detailed PPFD of each individual facet in the canopy (detailed light).

The leaf light response curves (Fig. 6) for the three combinations of CO₂ and temperature were predicted with the Farquhar et al. (1980) model with the temperature response functions of photosynthetic parameters following work by Bernacchi et al. (2001). Fig. 7d shows the proportion of leaf area in a canopy performing light-limited or light-saturated photosynthesis based on the scenarios (with the current air temperature and CO₂ concentration). More than 71% of the leaf area was predicted to perform light-limited photosynthesis throughout the day. At a C_a of 380 μbar, leaves performing light-limited photosynthesis absorb 32% (Fig. 7b) of total incident solar energy and contributed ~47% of A_c (Fig. 7a). With an increase in C_a, the proportion of leaf area conducting light-limited photosynthesis gradually increases (Fig. 7c). Furthermore, with increasing C_a, the proportion of RuBP-limited photosynthesis forms a greater proportion of total A_c (Fig. S2). Because in the current simulation, we assumed that plants had no photosynthesis acclimation (i.e. no changes in both the maximum rate of carboxylation at RuBP and CO₂ saturation (V_cmax) and the light saturated potential rate of whole chain electron transport through PSII (J_max) under elevated CO₂, representing a scenario where plenty of nitrogen is available (Long et al. 2004). Under this assumption, at C_a of 550 and 700 μbar, photosynthesis is completely RuBP-limited (Fig. S2).

Fig. 6.  The predicted light response curves under three different combinations of atmospheric CO₂ concentration (C_a) and air temperature (T_air).

Fig. 7.  The proportion of A_c contributed by light-limited photosynthesis and light-saturated photosynthesis (a); the proportion of total absorbed light energy by leaves conducting light-limited photosynthesis and light-saturated photosynthesis (b); the proportion of leaf area conducting light-limited photosynthesis and light-saturated photosynthesis (c). All these simulations were done under three CO₂ and temperature combinations. (d) The proportion of leaf area conducting light-limited or light-saturated photosynthesis at different times during a day at a CO₂ concentration of 380 μbar.

The PPFD of each facet inside a canopy was predicted and used to theoretically study the influences of different plant architectural features on A_c. Given that the canopy structure constructed was not symmetrical, we generated many canopies with varying leaf orientations but keep the overall canopy architectural features, e.g. stem height, leaf length, leaf width, leaf angle and leaf curvature, constant. Fig. 8 shows that A_c′ was not significantly affected by an increase of the stem height of the Teqing cultivar, but A_c′ gradually decreased when the stem height decreased below than 60% of the current height. Fig. 9 shows the influence of leaf angle on A_c′ of two plant canopies with LAI 4.8 (Fig. 9a–c) and 7.68 (Fig. 9d–f) under different combinations of C_a and T_air. With an increase in leaf angle, A_c′ gradually increased when the LAI was 4.8. However, A_c′ gradually decreased with leaf angle when LAI was 7.68. The increase in atmospheric CO₂ concentration does not change the relationship between A_c′ and leaf angle. Fig. 10 shows the influence of leaf width on A_c′. In our simulations, LAI changed proportionally with leaf width when other parameters were kept constant. Results show that the optimal leaf width was around 85% of the current leaf width under current C_a (Fig. 10).

Fig. 8.  The predicted influence of stem height on canopy photosynthetic CO₂ uptake rate under three different combinations of CO₂ and temperature. The points show mean ±s.d. of nine simulations.

Fig. 9.  The predicted influence of the leaf angle on canopy photosynthetic CO₂ uptake rate (A_c) under three different combinations of CO₂ and temperature. (a) C_a = 380 μbar, current T_air (T); (b) C_a = 550 μbar, elevated T_air (T + 1.5°C); (c) C_a = 760 μbar, elevated T_air (T + 3°C); (d) C_a = 380 μbar, current T_air (T); (e) C_a = 550 μbar, elevated T_air (T + 1.5°C); (f) C_a = 760 μbar, elevated T_air (T + 3°C). (a–c) LAI = 4.8; (d–f) LAI = 7.68. The points show mean ±s.d. of nine simulations.

Fig. 10.  The predicted influence of leaf width on canopy photosynthetic CO₂ uptake rate (A_c) under three different combinations of CO₂ and temperature. (a) C_a = 380 μbar, current T_air (T); (b) C_a = 550 μbar, elevated T_air (T + 1.5°C); (c) C_a = 760 μbar, elevated T_air (T + 3°C). The points show mean ± s.d. of nine simulations.

Crop yields are related to photosynthesis of the whole canopy instead of the photosynthetic capacities of only the top leaves (Zhu et al. 2010, 2012). This has been demonstrated in different crops, e.g. in cotton (Wells et al. 1986) and soybean (Harrison and Ashley 1980). The challenge is to identify architectural and biochemical properties that can be modified to gain increased canopy photosynthetic CO₂ uptake rates. In particular, although historically canopy architecture has been a major target of breeding for most crops, it is unknown whether further improving canopy architecture will increase crop yields for current and future CO₂ conditions. The model developed here aimed to tackle this challenge through incorporation of three features. First, canopy architecture was abstracted into several properties, which enabled an easy construction of a diverse set of canopy architectures for different crops, not only crops with relatively simple architecture, e.g. maize (Fournier and Andrieu 1999), rice (Watanabe et al. 2005) and wheat (Fournier et al. 2003), but also crops with relatively complex architecture, e.g. soybean and tomato. This contrasts with previous models where canopy architecture is used as input and correspondingly cannot be easily modified (Zheng et al. 2008). Second, the model uses a forward ray tracing algorithm to predict the detailed PPFD of every point inside a canopy. In addition, similar to previous efforts (Lao et al. 2005), the transmitted and reflected light rays in this model are traced using a Monte Carlo approach (Tucker and Garratt 1977), which enables exploration of the effects of different leaf optical properties, i.e. leaf reflectance, transmittance and absorbance, on light environments inside a canopy. Third, the model combines the detailed light environment in a canopy with the steady-state biochemical model of C₃ photosynthesis (Farquhar et al. 1980) to simulate the daily total canopy CO₂ uptake rate (A_c′). With these three features, the current model enables an accurate estimate of the impacts of different canopy architectural parameters on total canopy CO₂ uptake rate. Notably, the heterogeneous light environment inside a canopy has also been combined with the Farquhar et al. (1980) model in the Y-Plant system (Pearcy and Yang 1996; Pearcy and Yang 1998; Pearcy et al. 2004) and has been used to estimate the influence of different canopy features on canopy photosynthesis, revealing that leaf angle and self-shading help ameliorate photoinhibition (Pearcy et al. 2004).

With the current model, we first demonstrated that there can be a substantial difference between the estimated A_c dependent on whether the light environments on each point inside a canopy or an ‘average’ light level were used in the calculations of the photosynthetic rate. The attenuation of PPFD has been typically assumed to follow Beer’s Law (Monsi and Saeki 2005). Fig. 4b shows that PPFD of the diffuse light at different depths in a canopy is consistent with Beer’s Law (R² = 0.90), but the distribution of direct light cannot be correctly represented by using Beer’s Law. Furthermore, even if the extinction coefficient can be estimated accurately, such models can only be used to predict the average PPFDs in different layers of a canopy (Ledent 1977). Because A becomes light saturated around 30% of the total PPFD at solar noon, leaves receiving direct light dissipate much of its absorbed solar energy as heat (Fig. 6). This, together with the non-linearity of the response of A to PPFD, results in an overestimate total CO₂ uptake for models that use an ‘average’ light intensity. For example, a canopy shown in this study using averaged light led to a 25% higher A_c compared with the A_c predicted with the PPFD at individual points in the canopy. Though theoretically the estimated canopy photosynthetic CO₂ uptake using PPFD of each point would be more accurate compared with prediction using an ‘average’ light level, experimental measurements of canopy photosynthetic CO₂ uptake rates using canopy chambers are needed to ultimately judge the validity of these two approaches.

The canopy photosynthesis model was used to demonstrate that photosynthetic CO₂ uptake of the shaded leaves can be more than 47% of the total A_c′. The canopy photosynthesis includes both light-limited and light-saturated photosynthesis. For the chosen LAI, V_cmax, J_max and plant architecture used in this study, more than 71% of the total leaf area conducting light-limited photosynthesis (Fig. 7d). The light-limited photosynthesis accounted for ~47% of A_c′ (Fig. 7a). This demonstrates that improving photosynthesis of shaded leaves and not only that of the sunlit leaves at the top is important in terms of gaining a higher canopy photosynthetic rate. The large contribution of light-limited photosynthesis to A_c may also contribute to the lower than expected increase in A_c under elevated CO₂ (Long et al. 2006a). This is because when C_a is elevated to be around 550 μbar and 760 μbar, a higher proportion of the total leaf area will perform light-limited photosynthesis, which is less responsive to elevated CO₂ (Zhu et al. 2010). Some approaches to increase photosynthesis at lower layers of the canopy have been proposed recently, e.g. decreasing the leaf chlorophyll content (Ort et al. 2011), expanding the light spectrum (Chen and Blankenship 2011) and use Rubisco of increased specificity in the lower layer of the canopy (Zhu et al. 2004a; Long et al. 2006b).

Identification of optimal canopy architectural features can expedite ideal type breeding for higher crop yield potential (Peng et al. 2008; Parry et al. 2011). This is especially relevant given the currently increasing demand for food production (Tilman et al. 2011) and the recent progress in elucidating the molecular mechanisms controlling plant architecture (Wang and Li 2008). With the new mechanistic model of canopy photosynthesis, we examined the optimality of several genetically modifiable canopy architectural features, i.e. leaf width, stem height and leaf angle, for both current and future elevated CO₂ conditions. First, when leaf width and correspondingly LAI increase, the A_c′ was predicted to first increase and then decrease (Fig. 10). This pattern is caused by trade-off between increased A_c and the unavoidable increase in respiratory cost under elevated LAI. The optimal leaf width for this rice cultivar is predicted to be ~85% of that in the current cultivar (Fig. 10), suggesting a potential target for genetic engineering.

With an increase in stem height, A_c was predicted to gradually increase until it plateaued (Fig. 8). This pattern held under all three CO₂ concentrations (C_a) used in this study. We note that the optimal stem height is ~80% of that of the Teqing cultivar. This indicates that even though the rice height has been dramatically decreased during the ‘green revolution’, there is still space for further decrease, which can potentially improve rice yields, given that decreased stem height contributes to logging resistance and can decrease respiratory cost for building and maintaining the extra stem tissue (Gale et al. 1985; Evans 1996; Peng et al. 1999). It is worth noting that when the stem is too short, A_c will decrease because leaves with internode lengths below a minimum threshold will create higher mutual shading and decrease the total absorbed solar energy (Table A1 in Appendix 4).

Leaf angles in different rice cultivars differ dramatically (Mohanty and Gangopadhyay 1982). Under a relatively lower LAI of 4.8, under current and elevated C_a (Fig. 9a), the A_c was predicted to gradually increase with increase of leaf angle as a result of the decreased light penetrating the canopy and falling to the ground and correspondingly increased light absorbance. Under a relative higher LAI of 7.68, an increased leaf angle decreases A_c under current and elevated C_a because erect leaves improve light distribution in a canopy. The optimal leaf angle varies for canopies with different LAIs as those features are interdependent.

In this study, the canopy features, i.e. stem height, leaf length, leaf width, leaf angle and leaf curvature were independently adjusted to examine its optimal value. However, those features interact with each other. For example, tiller number influences are related to planting density (Fagade and Dedatta 1971). Planting density can also influence the red/far-red ratio, which can further modify the leaf length and width (Franklin et al. 2003). Thus far, molecular mechanisms underlining these inter-dependencies are still not well understood. In the current model, as a simplification, we did not include these inter-dependencies. In addition to this simplification, the model also included several other simplifications that will need to be improved in the future. These include (i) photosynthetic properties in different layers of the canopy are assumed to be the same, (ii) perturbation of leaf position by wind is ignored and (iii) the CO₂, temperature and humidity profiles in the canopy were assumed to be uniform.

Canopy microclimatic parameters, e.g. PPFD, leaf temperature, CO₂ and water vapour concentrations, differ at different depths in a canopy. Furthermore, many leaf traits, e.g. chlorophyll concentration, V_cmax and J_max, also vary with leaf age and position of a leaf inside a canopy. These environmental and physiological heterogeneities can influence the estimate of A_c and correspondingly, the choice of the optimal parameter values to maximise canopy photosynthesis. For example, the assumed constant chlorophyll concentration inside a canopy might potentially under-estimate PPFD in the lower layer of a canopy because chlorophyll concentration of leaves in lower layer is usually higher than that of leaves in top layer (Ciganda et al. 2008). Furthermore, using an average constant V_cmax and J_max values instead of using actual gradients of V_cmax and J_max values in a canopy can also potentially underestimate A_c. So far it has been a major challenge to measure these leaf physiological parameters with a throughput required to effectively parameterise a canopy photosynthesis model. In this regard, it is encouraging that a spectroscopic method (Serbin et al. 2012) based on leaf reflective spectra can estimate a range of leaf physiological parameters including chlorophyll content, leaf specific area, V_cmax and J_max, with relatively high levels of accuracy.

In summary, we developed a new model of canopy photosynthesis, the aim of which was to provide a direct linkage between canopy architectural parameters with a detailed light distribution inside a canopy and correspondingly total canopy photosynthetic CO₂ uptake rate. Although several areas in the model still need to be further developed, the model already provides a useful framework for designing ideal crop architectures to optimise light distribution and enhance photosynthetic CO₂ uptake for different crops under different light and CO₂ conditions. One major challenge now is to link the model predictions with experimentally-measured canopy light environments and canopy photosynthetic CO₂ uptake rates. Only after such a close linkage is fully established can the potential of such modelling approach in guiding crop improvements be fully realised.