The slow reversibility of photosystem II thermal energy dissipation on transfer from high to low light may cause large losses in carbon gain by crop canopies: a theoretical analysis

Abstract

Regulated thermal dissipation of absorbed light energy within the photosystem II antenna system helps protect photosystem II from damage in excess light. This reversible photoprotective process decreases the maximum quantum yield of photosystem II (F_v/F_m) and CO₂ assimilation (Φ_CO2), and decreases the convexity of the non‐rectangular hyperbola describing the response of leaf CO₂ assimilation to photon flux (θ). At high light, a decrease in Φ_CO2 has minimal impact on carbon gain, while high thermal energy dissipation protects PSII against oxidative damage. Light in leaf canopies in the field is continually fluctuating and a finite period of time is required for recovery of Φ_CO2 and θ when light drops below excess levels. Low Φ_CO2 and θ can limit the rate of photosynthetic carbon assimilation on transfer to low light, an effect prolonged by low temperature. What is the cost of this delayed reversal of thermal energy dissipation and Φ_CO2 recovery to potential CO₂ uptake by a canopy in the field? To address this question a reverse ray‐tracing algorithm for predicting the light dynamics of 120 randomly selected individual points in a model canopy was used to describe the discontinuity and heterogeneity of light flux within the canopy. Because photoprotection is at the level of the cell, not the leaf, light was simulated for small points of 10⁴ µm rather than as an average for a leaf. The predicted light dynamics were combined with empirical equations simulating the dynamics of the light‐dependent decrease and recovery of Φ_CO2 and θ and their effects on the integrated daily canopy carbon uptake (A′_c). The simulation was for a model canopy of leaf area index 3 with random inclination and orientation of foliage, on a clear sky day (latitude 44° N, 120th day of the year). The delay in recovery of photoprotection was predicted to decrease A′_c by 17% at 30 °C and 32% at 10 °C for a chilling‐susceptible species, and by 12.8% at 30 °C and 24% at 10 °C for a chilling‐tolerant species. These predictions suggest that the selection, or engineering, of genotypes capable of more rapid recovery from the photoprotected state would substantially increase carbon uptake by crop canopies in the field.

Leaf canopies, photoprotection, photosynthesis, reverse ray‐tracing, yield loss.

Received 2 December 2003; Accepted 2 March 2004

Introduction

Light is the source of energy for photosynthesis, but on most days plants encounter light fluxes that exceed their photosynthetic capacity. As light levels increase, a process that operates within the antenna ensemble of photosystem II (PSII) is progressively engaged which harmlessly discharges a portion of photon flux energy as heat. Thermal dissipation of absorbed light helps protect the photosynthetic apparatus from damage, particularly by controlling the rate of damage to the D1 protein of PSII (Long et al., 1994). Although photodamage has been documented in crops grown outside their ancestral geographic range, the vast majority of plants in native habitats, and most crops under cultivation, deal successfully with excess light avoiding photodamage even under daunting environmental challenges (Ort, 2001). The process of photoprotection has been extensively reviewed (Aro, 1999; Long et al., 1994; Ort, 2001). Despite the intensity of study at the molecular to leaf level, remarkably little is known about the quantitative impact of photoprotection on carbon gain at the whole plant level and at the canopy level; i.e. is it relevant to crop production in the field?

The increased thermal dissipation due to photoprotection lowers the maximum quantum yield of PSII (maximum Φ_PSII^‡, indicated by F_v/F_m), which in turn results in a lower maximum quantum yield of CO₂ assimilation (Φ_CO2), i.e. a reduced initial slope in the response of photosynthetic CO₂ assimilation rate (A) to photosynthetic photon flux density (Q) (Long et al., 1994). This competition for excitation energy between thermal dissipation and photochemistry not only decreases Φ_CO2, but also the convexity (θ) of the non‐rectangular hyperbolic response of A to Q (Leverenz et al., 1990). θ reflects the transition from the initial slope of the light response curve (Φ_CO2) and asymptote where the maximum assimilation rate (A_sat) is achieved. A θ near 1 represents an abrupt transition in the influence of Φ_CO2 and A_sat on A with respect to Q, while low θ (approaching 0) represents a long transition in which both Φ_CO2 and A_sat determine A across the full range of Q (Leverenz et al., 1990). The decrease in θ coupled with a decrease in Φ_CO2 that occurs during a photoprotective response is significant because it increases the light level at which A will be depressed by any decrease in Φ_CO2 (Long et al., 1994). At light‐saturation, when photoprotection of PSII is most likely to be fully engaged, the resulting decrease in Φ_CO2 will by definition have no effect on CO₂ assimilation. When light levels decline, Φ_CO2 and θ recover, but recovery is not instantaneous. When a leaf is transferred to lower light, Φ_CO2 and θ will determine photosynthetic rate, and A will remain below the potential value for that light level until the readjustment of the photoprotective state appropriate for that light level is complete. How much loss of carbon gain does this lag in recovery cause in a canopy in the field?

Photon flux within a canopy of leaves in the field is highly discontinuous and heterogeneous in space and time (Pearcy, 1990; Pearcy et al., 1994). Photon flux at any point within a canopy will fluctuate during the course of the day, not only because of intermittent cloud cover, but also because of transient shading by the overlap of leaves of different canopy layers. As solar angle changes, individual mesophyll cells of the leaf may pass from full sunlight to shade within a second. In this varying light environment, photoprotection, expressed as decreased F_v/F_m and Φ_CO2, may be predicted from light history and temperature (Long et al., 1994; Ögren and Sjöström, 1990). Canopy microclimate models have been developed to predict the diurnal course of the average Q at different levels within canopies (dePury and Farquhar, 1997; Forseth and Norman, 1993), which have, in turn, been used to predict the effect of photoprotection on daily total carbon assimilation (Long et al., 1994; Ögren and Sjöström, 1990; Werner et al., 2001). These studies concluded that the effect of photoprotection on daily total carbon assimilation is small, but nevertheless significant. However, these studies did not consider the spatial heterogeneity of Q within leaf layers of canopies. Averaging removes the abrupt transitions in light that will occur at the level of the photosynthetic cells and may underestimate losses. These earlier studies are extended here by capturing and quantifying this spatial variability due to the changing solar angle over the course of the day. This was achieved by determining the diurnal Q at a large number of randomly selected individual points within a model canopy. In turn, the light dynamics were used to estimate decrease and recovery of Φ_CO2 and θ at each point, and then daily carbon assimilation. By summing the effects at a large number of randomly selected points, the effect of the slow dynamics of photoprotection readjustment/recovery on daily carbon assimilation by a whole canopy was estimated. This simulation takes full account of the dynamic and highly heterogeneous Q within a canopy due to solar angle and the dynamic nature of the photoprotective state as indicated by decline and recovery of Φ_CO2 and θ. These simulations demonstrate that the lag in recovery from the photoprotective state can have a very substantial effect on daily canopy carbon assimilation, compared with the hypothetical situation of instantaneous recovery.

Materials and methods

Any prediction of potential carbon loss in a canopy due to photoprotection requires three sub‐models: (1) a phenomenological model of photoprotection in response to light history to predict how defined change in light affects Φ_CO2 and θ at different temperatures; (2) a reverse ray tracing algorithm to predict the dynamics of photon flux at any specific point within a three‐dimensional canopy of leaves; and (3) a dynamic model of the effect of photoprotection on canopy carbon assimilation linking (1) and (2) to predict canopy photosynthesis.

Phenomenological model of photoprotection

Empirical studies have shown that the response of CO₂ uptake (A) to Q can be described effectively for higher plants by a non‐rectangular hyperbola (equation 1) (Long et al., 1994). The parameters of this relationship are Φ_CO2, θ, and A_sat; where A_sat was assumed to be a constant as 25 µmol m^–2 s^–1 in this study, which is an approximate average for healthy C₃ leaves (Long, 1985). Diurnal fluctuation in photoprotection represents a dynamic balance between decrease and recovery of Φ_CO2 and θ. For a given point on a leaf, change in Φ_PSII over the day is calculated from a cumulative weighted light dosage (I_int) over the past 24 h (equation 2.1). Equation 2.1 weights light such that the effect of a high light event diminishes with the time which has elapsed since that event (Long et al., 1994). Equation 2 determines F_v/F_m from I_int using the empirically derived constant f_h and T_f. Given the same I_int, the decrease of F_v/F_m is less as temperature (T) increases. This is simulated using an empirical factor T_f (equation 2.2; Fig. 2) based on data on maize (Aguilera et al., 1999) and willow (Ögren and Sjöström, 1990). The value of f_h for cold‐tolerant species was parameterized based on a photoinhibition experiment on willow (Ögren and Sjöström, 1990). To do this, the diurnal clear‐sky Q for the 199th and the 200th day of the year in Umeå, Sweden was predicted based on the Sun–Earth geometry and atmospheric transmittance after Campbell (1977); the diurnal I_int for the 200th day of the year was simulated following equation 2.1. The value of f_h was then estimated based on the maximal I_int in the 200th day, the reported maximal percentage decrease of F_v/F_m (20%), and the air temperature at the time of the maximum decrease of F_v/F_m (Ögren and Sjöström, 1990). The f_h for cold‐susceptible species was determined similarly based on the percentage decrease of F_v/F_m and the applied light and temperature conditions in a photoinhibition experiment on maize (Aguilera et al., 1999). Figure 3 shows an example of predicted F_v/F_m for a cold‐tolerant species given a dynamic diurnal light fluxes.

F_v/F_m is the maximum quantum yield of PSII photochemistry for quanta absorbed by the pigments associated with PSII complexes. Variation in the maximum quantum yield of CO₂ fixation for quanta absorbed by leaves showed a strong linear and positive relationship with variation in F_v/F_m(Björkman and Demmig, 1987; Genty et al., 1989; Long et al., 1994). Such an empirical relationship between F_v/F_mand Φ_CO2 (equation 3) (Demmig and Björkman, 1987) was used in this study to predict Φ_CO2 from simulated F_v/F_m. The coupled decrease in θ and Φ_CO2 is simulated using equation 4, where f_c is the ratio of the decrease of θ to the decrease of Φ_CO2 with I_int; f_c is assumed to be 1 in this study after Ögren and Sjöström (1990).

θA² – (Φ_CO2Q + A_sat)A – Φ_CO2QA_sat = 0(1)

T_f = 0.0033T² – 0.1795T + 3.4257(2.2)

Φ_CO2 = 0.1367(F_v/F_m) – 0.0106(3)

Reverse ray‐tracing algorithm

A model crop canopy of 1 m height was divided into 12 layers of equal depth (8.3 cm). Each layer was assumed to have the same leaf area index of 0.25 (Fig. 1). The top layer was assigned as the first layer and the bottom as the 12th layer. All leaves were assumed to be circular (1 cm diameter) and randomly distributed in each layer, with the limitation that, within a layer, one leaf could not overlap another. Ten points on leaves were chosen by random coordinates within each layer. A reverse ray‐tracing algorithm was used to calculate Q for each point over 24 h at 1 s intervals; described as follows.

f_t = 0.262(t – t_o)(6)

A_l = arcsin(sinδ cosλ + cosλ cosδ cosf_t)(7)

Z_i = arccos(sin δ sin λ + cos δ cos λ cos f_t)(9)

To determine photosynthetic photon flux density (Q) of incident light on one of the selected points (P) in a given layer (n), one reverse ray is issued from P towards the overlying layer (n–1) along the reverse direction of the solar beam, which is determined by Sun–Earth geometry for a specific time and latitude (Campbell, 1977) (equations 5–9, 16). The intersection of the reverse ray with the plane of layer (n–1) is predicted based on the coordinates of P, the direction of the reverse ray and the plane of layer (n–1) (equations 10–12). If the intersection fell within any leaf in layer (n–1), the reverse ray ends and P is assumed to be in shade at that time; otherwise, the reverse ray continues travelling upwards to the next layer (n–2) and so on until the intersection falls within a leaf in one of the upper layers (case 1) or the reverse ray leaves the canopy without intercepting any leaf (case 2) (Fig. 1). In case 1, P receives diffuse light (I_diff(n)) and transmitted light (I_trans(n)). In case 2, point P receives direct sunlight (I_direct) together with I_diff(n) and I_trans(n) (equation 13), i.e. point P is in a sunfleck. It was assumed that leaves absorb 90% of the total intercepted light energy and the remaining 10% was transmitted, emerging in the form of diffuse light from the leaf lower surface to the immediate lower layer (equation 14). This absorbance is an average for healthy leaves as measured with an integrating sphere (Long and Hällgren, 1993). The proportion of leaf area receiving direct sunlight (p_(n)) is determined by the ratio of the number of sunlit points to the total number of all points. I_diff and I_trans were assumed to be uniform within a given leaf layer (equations 13.1, 13.2).

Q_(p,n) = I_direct + I_trans(n) + I_diff(n)(13)

where

I_trans(n) = I_direct × p_(n–1) × LAI_(n–1) × 0.1 + (I_trans(n–1) + I_diff(n–1)) × LAI_(n–1) × 0.1(13.1)

I_diff(n) = (1 – LAI_(n–1)) × (I_diff(n–1) + I_trans(n–1))(13.2)

Q_A = Q_(p,n) × 0.9(14)

I_direct and diffuse sunlight above the canopy (I_diff(1)) were determined (equations 15–19) from Sun–Earth geometry and atmospheric transmittance after Campbell (1977).

I_direct = I_p × sin θ(15)

sin θ = sin λ sin δ + cos λ cos δ cos 15(t – t₀)(16)

I_p = a^mI_p0(17)

Dynamic model of effects of photoprotection on canopy carbon assimilation

The total daily canopy photosynthetic carbon assimilation (A′_c) was calculated by integrating A over the day at each randomly sampled point in the canopy and then summing the average for each of the 12 layers (equation 20). The diurnal I_diff(1) and I_direct incident above the canopy was simulated for the 120th day of the year, at latitude 44° N. To calculate A′_c, the reverse ray‐tracing algorithm (equations 5–19) was first used to predict diurnal Q for 10 randomly chosen points at each of 12 canopy layers. I_int for a point at any given time was then calculated based on Q of the past 24 h for that point (equation 2.1). Φ_CO2 and θ were determined from I_int and T (equations 2–4). Photosynthetic rate (A) at each point was then determined by substituting values of I, Φ_CO2, and θ, as calculated above, into equation 1.

A_c represents total daily photosynthetic carbon assimilation of the whole canopy assuming no decrease in F_v/F_m, and correspondingly Φ_CO2 and θ, i.e., assuming Φ_CO2 = Φ_max and θ=θ_max throughout the day. The loss of total carbon assimilation due to photoprotection was calculated as decrease in A′_c relative to A_c (equation 21).

Results

Great spatial and temporal heterogeneity in Q were predicted within the hypothetical model canopy (Fig. 4). Points on leaves in the upper layers of the canopy received more direct sunlight than lower layers of the canopy and most sunflecks were predicted to occur in clusters (Fig. 4). The proportion of daily total light incident as direct light (Q_direct/Q_total) progressively decreases with depth into the canopy (Fig. 5). At layer 1, Q_direct /Q_total is 0.87; at layer 3, it is about 0.8 and at layer 10, just 0.1 (Fig. 5).

At 20 °C simulated F_v/F_m of the uppermost canopy layer of a chilling‐tolerant species over the diurnal period varied from 0.815 at 06.00 h to 0.63 at 14.00 h, but from 0.815 to 0.5 for a chilling‐susceptible species over the same time interval (Fig. 6). The decrease in F_v/F_m was progressively less with depth into the canopy for both chilling‐tolerant and chilling‐susceptible species (Fig. 6A, B).

Great variation in photosynthetic rates was predicted as a result of the temporal and spatial variation in Q at different layers of the canopy (Fig. 7). The decrease in daily total canopy carbon assimilation (A′_c) due to photoprotection was much greater in the upper layers. For example, decreased daily total photosynthetic canopy carbon assimilation in the top four layers of the simulated chilling‐tolerant canopy was 11–24% compared with negligible decreases below layer 8 (Fig. 8).

For a chilling‐tolerant species, the decrease of A′_c due to photoprotection (i.e. by comparison to A_c) was c. 12.8%, 14.6%, and 24% at 30 °C, 20 °C, and 10 °C, respectively (Fig. 9). For the hypothetical chilling‐susceptible species, decrease in A′_c was c. 17.3%, 19.7%, and 32% at 30 °C, 20 °C, and 10 °C, respectively (Fig. 9; Table 1). These results suggest that if full account is taken of spatial heterogeneity, loss of photosynthetic efficiency due to photoprotection could cost a canopy 12–30% of A′_c over a diurnal cycle simply due to loss of efficiency and delay in recovery when diurnal changes in light and dynamic shading cause sudden decreases in photon flux (Fig. 9).

Discussion

The simulation shows that decreased maximum efficiency of photosynthesis in high light results in a loss of potential carbon gain when change in solar angle places a point on one leaf in the shade of another leaf. This loss continues until thermal dissipation is readjusted to the level appropriate for a lower light level. Accumulated across a canopy such losses amount to between 12.8% and 30% of total potential carbon gain. The simulation suggests that considerable gain in carbon assimilation in crop canopies in the field could be achieved if these decreases in efficiency could be avoided by more rapid or instantaneous readjustments to thermal dissipation. As noted in the Introduction, these decreases in efficiency fulfil a necessary function of decreasing the probability of oxidative damage to the D1 protein, which would lower photosynthetic efficiency, termed photodamage, and require repair and replacement of the protein before efficiency could be restored. In the longer term a continued excess of excitation energy would lead to irreversible photo‐oxidation (reviewed by Long et al., 1994). Could the loss found here be decreased without the risk of photodamage and photo‐oxidation? Falkowski and Dubindky (1981) identified algae associated with corals that could withstand 1.5× full sunlight without evidence of loss of maximum photosynthetic efficiency or photoinhibition, showing that the loss of efficiency is not an intrinsic requirement of the photosynthetic apparatus, although this tolerance to high light could reside in an enhanced ability to repair and replace photodamaged protein D1 rapidly. In higher plants, Wang et al. (2002) have shown a close correlation between increased rate of recovery from the photoprotected state and increased biomass production in the ‘super‐high yield’ rice cultivars.

This theoretical analysis uses a hypothetical canopy with uniform leaf size and division area between layers. The direction of the solar beam and canopy structure parameters, i.e. leaf orientation, leaf inclination, and leaf area index, are primary determinants of sunfleck patterns (Barradas et al., 1998, 1999; Denicola et al., 1992). Canopies with different structural parameters show different sunfleck patterns (Barradas et al., 1998, 1999; Chazdon, 1988; Pearcy, 1990; Pearcy et al., 1990). Is discontinuity in light levels with time in the hypothetical canopy realistic? The predicted diurnal light dynamics simulated here with a reverse ray‐tracing algorithm showed two major characteristics: (1) sunflecks are clustered in time (Fig. 4); (2) Q_direct/Q_total was greater in the upper layers of the canopy than in the lower layers (Fig. 5). These predictions are qualitatively consistent with field measurements (Pearcy et al., 1990).

Although the operating quantum efficiency of PSII (F′_m) is different and might not necessarily reflect changes in F_v/F_m, this study used the predicted F_v/F_m to infer Φ_CO2, based on the linear relationship between F_v/F_m and Φ_CO2 as suggested previously (Björkman and Demmig, 1987; Genty et al., 1989; Long et al., 1994). In the model, Φ_CO2 will only determine A directly at very low light, when F′_v/F′_m approaches or equals F_v/F_m. Several different empirical models have been developed to simulate the decrease of F_v/F_m for a given light history using different formulae to calculate a weighted light dose (I_int) (Long et al., 1994; Ögren and Sjöström, 1990; Valladares and Pearcy, 1999; Werner et al., 2001). These different formulae were developed for different crop or tree species and for different environmental conditions. In the current study, the empirical model of Long et al. (1994) describing the decrease of F_v/F_m due to photoprotection was used, except that the light history used was extended to the past 24 h. The reasoning was that recovery of Φ_CO2from the photoprotective state may take 2–3 d in cool conditions (Farage and Long, 1986).

Werner et al. (2001) found, through simulation, a daily carbon loss of 7.5–8.5% of potential carbon gain in upper sunlit canopy layers and a 3% decrease in lower layers of the canopy, and an overall loss of 6.1% for a Mediterranean evergreen oak Quercus coccifera under climate conditions which cause mild photoprotection. Long et al. (1994) estimated a daily total carbon loss of 9% through simulation for a similar climate and a generic C₃ canopy. However, these studies did not predict the rapid decrease in light that occurs at individual points within the canopy. When the transient photosynthetic rates under dynamic light conditions are considered, this simulation showed that photoprotection caused a loss in daily total photosynthetic carbon uptake of 12.8–24% due to delayed recovery for a chilling‐tolerant species on the 120th day of the year at 44° N (Fig. 9). The latitude used in the simulation was 44° N and was chosen as intersecting many of the major agricultural production areas of the northern hemisphere. Temperature and LAI also approximate to mean conditions that are likely in early summer. In this study, light conditions were predicted for a clear sky. Temporal fluctuations in Q due to intermittent cloud cover were not incorporated, but would almost certainly increase the losses predicted here, by further increasing the frequency of abrupt transitions in Q.

There is inter‐ and intra‐specific variation in rates of decrease and recovery of F_v/F_m during and following exposure to excess light (Long et al., 1994). The combination of chilling temperature and high light often leads to a more severe and prolonged photoprotective response (Long et al., 1994). The impact of photoprotection on carbon gain in plants with different chilling tolerance was simulated using different f_h values. As expected, photoprotection was predicted to cause much greater carbon loss for cold‐susceptible species compared with cold‐tolerant species (Fig. 9). Furthermore, photoprotection was predicted to cause much greater carbon loss at low temperature, especially for chilling‐susceptible species. For a chilling‐susceptible species on the 120th day of the year at 44° N, photoprotection would decrease daily total photosynthetic carbon assimilation by 17.3% at 30 °C, but 32% at 10 °C. Lower temperatures cause a greater decrease of F_v/F_m (Fig. 2) and by assumption, lower Φ_CO2 and θ (equations 2, 4) explaining the simulated larger carbon loss.

In conclusion, simulating the spatial and temporal heterogeneity of light in canopies and incorporating the cost of delayed recovery in photosynthetic efficiency on transfer from high to low light may provide a more accurate prediction of the loss of diurnal carbon gain due to photoprotection. The decrease in diurnal total carbon uptake is between 12.8–32% depending on light conditions, temperature, and chilling tolerance of plant species. These are almost certainly conservative estimates as these analyses only considered the spatial heterogeneity of Q within leaf layers of canopies caused by changes in solar angle. Nastic and tropic movements of leaves, movement caused by air currents, and intermittent cloud will cause dynamic heterogeneities perhaps as significant as those due to changing solar angle. These results suggest that the selection or engineering of genotypes better able to recover more rapidly from the photoprotective state, more tolerant to photodamage, and thereby able to function with smaller photoprotective decreases in F_v/F_m, could substantially increase carbon uptake by crop canopies. The occurrence of organisms, which can resist any significant reduction in F_v/F_m in high light, suggest that such changes are possible.

Acknowledgements

This work was supported by a grant to SPL and JW from the National Center for Supercomputing Applications (NCSA) and by a fellowship from the Physiological and Plant Molecular Biology training program of the University of Illinois to X‐GZ.

Fig. 1. Principle of reverse ray tracing. A virtual canopy of 1 m in height was divided into 12 layers, with a leaf area index of 0.25 in each layer. Circular leaves of 1 cm diameter were randomly distributed in each layer. To determine photosynthetic photon flux density of point P in a given layer n, one reverse ray was issued from P upward to layer (n–1) along the reverse direction of a direct solar beam. If the intersection of the reverse ray with any overlying layer (n–1, n–2, …, 1) fell within any leaf, the reverse ray ended and point P was therefore shaded (Case 1); otherwise, the reverse ray left the canopy without intercepting any overlying leaves, then the leaf was sunlit (Case 2).

Fig. 2. The empirical temperature factor relating the relative decrease of F_v/F_m to temperature at the same I_int. The empirical relationship, T_f=0.0033T²–0.1795T+3.4257 (R²=0.98), was based on data for willow as a chilling‐tolerant plant (Ögren and Sjöström, 1990) and for maize as a chilling‐intolerant plant (Aguilera et al., 1999).

Fig. 3. Predicted diurnal photosynthetic photon flux density (Q), maximum quantum yield of PSII (represented by F_v/F_m) and photoprotection coefficient, defined as the decrease of Φ_PSII in proportion to Φ_max, at a random point in the third layer of the hypothetical canopy described in Fig. 1.

Fig. 4. Examples of predicted diurnal Q of one random point in each of six different layers within the hypothetical model canopy. Photon flux above the canopy is simulated from Sun–Earth geometry on 120th day of the year at 44° N, assuming a clear sky.

Fig. 5. Percentage of light incident in the form of direct light as a proportion of total (Q_direct /Q_total) at different layers in the canopy on the 120th Julian day, 44° N latitude. Each point represents the average value ±1 SE of the Q at ten randomly chosen points.

Fig. 6. Predicted diurnal maximal quantum yield of PSII (represented by F_v/F_m) of one randomly chosen point in five different layers of the canopy. The light dynamics were as Fig. 4. Temperature was kept constant at 20 °C. (A) Chilling‐tolerant species; (B) chilling‐susceptible species.

Fig. 7. Predicted diurnal photosynthetic rates in different layers of the canopy. The simulation was for a chilling tolerant species. The values of Q and temperature were as in Fig. 6.

Fig. 8. Predicted percentage decrease of the total daily integral of photosynthetic carbon assimilation in different canopy layers. The values of Q were as in Fig. 6. (A) Chilling‐tolerant species; (B) chilling‐susceptible species.

Fig. 9. Percentage decrease of the daily integral of total canopy photosynthetic carbon assimilation A′_c due to photoprotection for chilling‐tolerant and chilling‐susceptible species at different temperatures. The values of Q and temperature were as in Fig. 6.

Table 1.

Predicted photosynthetic CO₂ uptake of different layers. Simulations were done for a chilling‐susceptible species on 120th Julian day, 44° N latitude assuming constant canopy temperature of 20 °C. A_c is the canopy carbon gain assuming no photoprotective reduction in θ and Φ_CO2; A′_c is the canopy carbon gain with photoprotective reduction in θ and Φ_CO2.

	Ac	A′c
	(mol m–2 d–1)	(mol m–2 d–1)
LAI 0–0.5	2.25	1.57
LAI 0.5–1	1.49	1.21
LAI 1–1.5	0.87	0.78
LAI 1.5–2	0.46	0.44
LAI 2–2.5	0.24	0.24
LAI 2.5–3	0.13	0.13
Total canopy	5.45	4.38
Percentage decrease	–	19.7

Appendix I.

List of variables, constants, parameters

Abbreviation	Full name	Unit
A	Rate of photosynthetic leaf CO2 uptake	µmol m–2 s–1
a	Atmospheric transmittance	Dimensionless
Ac	Daily total photosynthetic canopy CO2 uptake without photoprotection	mol m–2 d–1
A′c	Daily total photosynthetic canopy CO2 uptake with photoprotection	mol m–2 d–1
Asat	A at saturating light, a constant in this study assumed as 25	µmol m–2 s–1
Az	Azimuth angle, angle of the deviation from north of the line connecting the point of observation and the point that sun projects on earth vertically (northern hemisphere)	Radians
D	Day of year	d
fc	Ratio of relative change of θ to relative change of ΦCO2	Dimensionless
fh	Empirical constant in determining the decrease of Fv/Fm for a given Iint. For cold‐susceptible species, fh is 5.1×105; for cold‐tolerant species, fh is 7×105	Dimensionless
ft	Hour angle, the angular distance from local median, Greenwich UK	Radians
F′v/F′m	Maximum quantum yield pf PSII	Dimensionless	Idiff(n)	The average photosynthetic photon flux density in the form of diffuse light at layer n	µmol m–2 s–1
Idirect	Photosynthetic photon flux density of direct sunlight	µmol m–2 s–1
Iint	Weighted light dose at point P at time t	Dimensionless
Ip	Direct solar radiation on a surface perpendicular to the beam	µmol m–2 s–1
Ipo	Solar constant at the top of the atmosphere	µmol m–2 s–1
Itrans(n)	Photosynthetic photon flux density of transmitted light at layer n from leaves of the immediate upper layer	µmol m–2 s–1
LAI	Leaf area index	Dimensionless
m	Optical airmass number	Dimensionless
n	Layer number, n=1, 2, …12	Dimensionless
P	Randomly selected point on a leaf in a certain canopy layer	Dimensionless
Q	Photosynthetic photon flux density	µmol m–2 s–1
QA	Photosynthetic photon flux density absorbed by leaf	µmol m–2 s–1
Q(p,n)	Photosynthetic photon flux density at point P at layer n
Qi	Photosynthetic photon flux density at ith min counting backward from the current time for which Iint is calculated	µmol m–2 s–1
Qdirect	The daily total photosynthetic photon flux as direct light in a given layer	µmol m–2 d–1
Qtotal	The daily total photosynthetic photon flux in a given layer	µmol m–2 d–1
Pr	Air pressure at a given location (105)	Pa
Pro	Air pressure at sea level (105)	Pa
Pd	Proportionate decrease of canopy assimilation due to photoprotection	Dimensionless
p(n)	Probability that a leaf is in direct sunlight at layer n	Dimensionless
t	Time in hours (0∼24)	h
T	Temperature	°C
Tf	Empirical factor relating the relative decrease of Fv/Fm to temperature	Dimensionless
to	Time of solar noon (12.00 h)	h
X(n)	X coordinate of intersection point of reverse ray with upper layer	Dimensionless
Y(n)	Y coordinate of intersection point of reverse ray with upper layer	Dimensionless
Z(n)	Z coordinate of intersection point of reverse ray with upper layer	Dimensionless
Zi	Zenith angle	Radians
δ	Solar declination angle, the angle between light ray and the equatorial plane of the earth	Radians
ϕ	Solar elevation angle	Radians
ΦCO2	Initial and maximum slope of non‐rectangular hyperbolic response of A to Q	Dimensionless
ΦPSII	Quantum yield of PSII	Dimensionless
θ	The convexity of non‐rectangular hyperbolic response of A to Q	Dimensionless
θmax	Maximum θ (0.95)	Dimensionless
Φmax	Maximum ΦCO2(0.1)	Dimensionless	λ	Latitude (44° N)	°

Note: Values in parenthesis represent values used in this study.