Photosynthesis

Modelling of crop growth follows a generic approach used with the SUCROS model (van Keulen et al., 1982). Daily net dry matter production by photosynthesis and respiration is driven by radiation and temperature. Gross CO₂ assimilation is calculated by estimating the sky cover duration:

\[A_g = O_r \cdot A_0 + (1-O_r) \cdot A_c\]

\(A_g\) Gross CO₂ assimilation rate \([kg\ , CO_2 \, ha^{-1} \, h^{-1}]\)
\(O_r\) Relative sky cover duration \([d \, d^{-1}]\)
\(A_0\) CO₂ assimilation under clouded sky \([kg\ , CO_2 \, ha^{-1} \, h^{-1}]\)
\(A_c\) CO₂ assimilation under clear sky \([kg\ , CO_2 \, ha^{-1} \, h^{-1}]\)

where

\[O_r = \frac{R_c - (0.5 \cdot R_s \cdot 10^9)} {0.8 \cdot R_c}\]

\(O_r\) Relative sky cover duration \([d \, d^{-1}]\)
\(R_s\) Global radiation \([MJ \, m^2 \, d^{-1}]\)
\(R_c\) Irradiation under clear sky \([J \, m^2]\)

Gross CO₂ assimilation includes assimilation under clouded and under clear sky conditions:

\[A_c = \begin{cases} PHCL & LAI < 5 \\ PHCH & LAI \geq 5 \end{cases}\]

\[A_O = \begin{cases} PHOL & LAI < 5 \\ PHOH & LAI \geq 5 \end{cases}\]

\(A_O\) CO₂ assimilation under clouded sky \([kg \, CO_2 \, ha^{-1} \, h^{-1}]\)
\(A_C\) CO₂ assimilation under clear sky \([kg \, CO_2 \, ha^{-1} \, h^{-1}]\)
\(LAI\) Leaf area index \([m^2 \, m^{-2}]\)

The following auxiliary algorithms were used:

\[PHCL = \begin{cases} PHC3 \cdot \left( 1-e^{\frac{PHC4}{PHC3}} \right) & PHC3 < PHC4 \\ PHC4 \cdot \left( 1-e^{\frac{PHC3}{PHC4}}\right) & PHC3 \geq PHC4 \end{cases}\]

\[PHCH = 0.95 \cdot (PHCH1 + PHCH2) + 20.5\]

\[PHOL = \begin{cases} PHO3 \cdot \left( 1-e^{\frac{PHC4}{PHO3}} \right) & PHO3 < PHC4 \\ PHC4 \cdot \left( 1-e^{\frac{PHO3}{PHC4}}\right) & PHO3 \geq PHC4 \end{cases}\]

\[PHOH = 0.9935 \cdot PHOH1 + 1.1\]

where

\[PHC3 = PHCH \cdot (1-e^{-0.8 \cdot LAI})\]

\[PHC4 = N_{atmo} \cdot LAI \cdot A\]

\(N_{atmo}\) Atmospheric day length \([h]\)
\(LAI\) Leaf area index \([m^2 \, m^{-2}]\)
\(A\) CO₂ assimilation rate \([kg \, CO_2 \, ha^{-1} \, h^{1}]\)

\[PHO3 = PHOH \cdot (1-e{-0.8 \cdot LAI})\]

\[PHCH1 = h_p \cdot A \cdot N_{eff} \cdot \frac{X}{1+X}\]

\(LAI\) Leaf area index \([m^2 \, m^{-2}]\)
\(h_p\) Sun’s culmination (vertical projection) \([^{\circ}]\)
\(A\) CO₂ assimilation rate \([kg \, CO_2 \, ha^{-1} \, h^{1}]\)
\(N_{eff}\) Effective day length \([h]\)

\[X = log \left( \frac{1+0.45 \cdot R_c} {N_{eff} \cdot 3600} \right) \cdot \frac{\varepsilon_N}{h_p \cdot A}\]

\(R_c\) Irradiation under clear sky \([J \, m^{-2}]\)
\(\varepsilon_N\) Net radiation use efficiency of CO₂ assimilation \([kg \, CO_2 \, ha^{-1} \, h^{1}]\)
\(N_{eff}\) Effective day length \([h]\)
\(h_p\) Sun’s culmination (vertical projection) \([^{\circ}]\)
\(A\) CO₂ assimilation rate \([kg \, CO_2 \, ha^{-1} \, h^{1}]\)

\[PHCH_2 = (5-h_p) \cdot A \cdot N_{eff} \cdot \frac{Y}{1+Y}\]

\(h_p\) Sun’s culmination (vertical projection) \([^{\circ}]\)
\(A\) CO₂ assimilation rate \([kg \, CO_2 \, ha^{-1} \, h^{-1}]\)
\(N_{eff}\) Effective day length \([h]\)

\[Y=log \left( \frac{1+0.55 \cdot R_c}{N_{eff} \cdot 3600} \right) \cdot \frac{\varepsilon_N}{(5-h_p) \cdot A}\]

\(A\) CO₂ assimilation rate \([kg \, CO_2 \, ha^{-1} \, h^{-1}]\)
\(\varepsilon_N\) Net radiation use efficiency of CO₂ assimilation \([kg \, CO_2 \, J^{-1} \, ha^{-1} \, h^{-1}]\)
\(R_c\) Irradiation under clear sky \([J \, m^{-2}]\)
\(h_p\) Sun’s culmination (vertical projection) \([^{\circ}]\)

\[PHOH1 = 5 \cdot A \cdot \varepsilon_N \cdot \frac{Z}{1+Z}\]

\(A\) CO₂ assimilation rate \([kg \, CO_2 \, ha^{-1} \, h^{-1}]\)
\(\varepsilon_N\) Net radiation use efficiency of CO₂ assimilation \([kg \, CO_2 \, J^{-1} \, ha^{-1} \, h^{-1}]\)

\[Z = \frac{R_0}{N_{eff} \cdot 3600} \cdot \frac{\varepsilon_N}{5\cdot A}\]

\(Z\)
\(N_{eff}\) Effective day length \([h]\)
\(\varepsilon_N\) Net radiation use efficiency of CO₂ assimilation \([kg \, CO_2 \, J^{-1} \, ha^{-1} \, h^{-1}]\)
\(A\) CO₂ assimilation rate \([kg \, CO_2 \, J^{-1} \, ha^{-1} \, h^{-1}]\)

\[h_p = \sin(90 + \delta + \varphi) \cdot \left( \frac{\pi}{180}\right)\]

\(h_p\) Sun’s culmination (vertical projection) \([rad]\)
\(\delta\) Solar declination \([^{\circ}]\)
\(\varphi\) Latitude \([^{\circ}]\)

\[\varepsilon_N = (1-\alpha_c) \cdot \varepsilon_L\]

\(\varepsilon_N\) Net radiation use efficiency of CO₂ assimilation \([kg \, CO_2 \, J^{-1} \, ha^{-1} \, h^{-1}]\)
\(\alpha_c\) Crop albedo
\(\varepsilon_L\) Radiation use efficiency of CO₂ assimilation \([kg \, CO_2 \, J^{-1} \, ha^{-1} \, h^{-1}]\)

\[N_{atmo} = 12 \cdot \frac{ \left( \pi + 2 \cdot \arcsin \left( \frac{\delta_{sin}}{\delta{cos}} \right) \right)}{\pi}\]

\(N_{atmo}\) Atmospheric day length \([h]\)

\[N_{eff} = 12 \cdot \frac{\left(\pi + 2 \cdot \arcsin \left( \frac{(-\sin(\frac{8 \cdot \pi}{180}) + \delta{sin})}{\delta_{cos}} \right) \right)}{\pi}\]

\(N_{eff}\) Effective day length \([h]\)

\[N_{photo} = 12 \cdot \frac{\left(\pi + 2 \cdot \arcsin \left( \frac{(-\sin(\frac{-6 \cdot \pi}{180}) + \delta_{sin})} {\delta_{cos}} \right)\right)}{\pi}\]

\(N_{photo}\) Photoperiodic day length \([h]\)

\[R_c = 0.5 \cdot 1300 \cdot \bar{R}_{photo} \cdot e^{\frac{-0.14}{\bar{R}_{photo}}}\]

\(R_c\) Irradiation under clear sky \([J \, m^{-2}]\)

\[R_o = 0.2 \cdot R_c\]

\(R_o\) Irradiation under clouded sky \([J \, m^{-2}]\)

\[\bar{R}_{photo} = 3600 \cdot \left( \delta_{sin} \cdot N_{astro} + \frac{24}{\pi} \cdot \delta_{cos} \cdot \sqrt{\left(1-\left(\frac{\delta_{sin}}{\delta_{cos}}\right)^2\right)} \,\,\,\right)\]

\(\bar{R}_{photo}\) Mean photosynthetically active radiation \([J \, m^{-2}]\)

CO₂ has an impact on the crop’s photosynthesis rate and stomata resistance, which in turn influences transpiration (Nendel et al., 2009). Mitchell et al. (1995) presented a set of algorithms for the calculation of the maximum photosynthesis rate, based on ideas of Farquhar and von Caemmerer (1982) and Long (1991):

\[A = \frac{( C_i - \Gamma^{*}) \cdot V_{c_{max}}}{C_i + K_c \cdot \left( 1 + \frac{O_i}{K_o} \right)}\]

\(A\) CO₂ assimilation rate \([kg \, CO_2 \, ha^{-1} \, d^{-1}]\)
\(C_i\) Inter-cellular CO₂ concentration \([\mu mol \, mol^{-1}]\)
\(\Gamma^{*}\) Compensation point of photosynthesis, related to \(C_i\) in absence of dark respiration \([\mu mol \, mol^{-1}]\)
\(O_i\) Inter-cellular O₂ concentration \([\mu mol \, mol^{-1}]\)
\(V_{c_{max}}\) Maximum saturated Rubisco carboxylation rate \([\mu mol \, m^{-2} \, s^{-1}]\)
\(K_c\) Michaelis-Menten constant for CO₂ \([\mu mol \, mol^{-1}]\)
\(K_o\) Michaelis-Menten constant for O₂ \([\mu mol \, mol^{-1}]\)

Temperature dependencies of \(C_i\), \(O_i\), \(K_c\), \(K_0\) and \(V_{c_{max}}\) and its parameters were described by Long (1991). Accordingly, \(C_i\) is calculated from atmospheric CO₂ concentration \(C_a\) as:

\[C_i = C_a \cdot 0.7 \cdot \frac{(1.674 - 6.1294 \cdot 10^{-2} \cdot T + 1.1688 \cdot 10^{-3} \cdot T^2 - 8.8741 \cdot 10^{-7} \cdot T^3 )}{0.73547}\]

\(C_i\) Inter-cellular CO₂ concentration \([\mu mol \, mol^{-1}]\)
\(T\) Daily mean air temperature in 2 m height \([^{\circ} C]\)

Respectively, \(O_i\) is calculated by:

\[O_i = 210 + \frac {( 0.047 - 1.3087 \cdot 10^{-4} \cdot T + 2.5603 \cdot 10^{-6} \cdot T^2 - 2.1441 \cdot 10^{-8} \cdot T^3 )} {2.6934 \cdot 10^{-2}}\]

\(O_i\) Inter-cellular O₂ concentration \([\mu mol \, mol^{-1}]\)
\(T\) Daily mean air temperature in 2 m height \([^{\circ} C]\)

The seasonal dynamic of the atmospheric CO₂ concentration from 1958 until today is described using:

\[C_a = 222 + e ^{0.0119 \cdot (t_{dec} - 1580)} + 2.5 \cdot \sin \left( \frac{t_{dec} - 0.5} {0.1592} \right)\]

\(C_a\) Atmospheric CO₂ concentration \([\mu mol \, mol^{-1}]\)
\(t_{dec}\) Date in decimal form

The algorithm used for light intensities below saturation which Mitchell et al. (1995) presented, was not applied. Instead, \(A_{max}\) is adapted to light interception according to Goudriaan and van Laar (1978). Mitchell et al. (1995) proposed the following algorithm for the transition between photosynthetic quantum use efficiency and light-saturated photosynthesis:

\[\varepsilon_L = \frac{0.37 \cdot (C_i - \Gamma^{*})} {4.5 \cdot C_i + 10.5 \cdot \Gamma^{*}}\]

\(\varepsilon_L\) Radiation use efficiency of CO₂ assimilation \([kg \, CO_2 \, J^{-1} \, ha^{-1} \, h^{-1}]\)
\(C_i\) Inter-cellular CO₂ concentration \([\mu mol \, mol^{-1}]\)
\(\Gamma^{*}\) Compensation point of photosynthesis, related to \(C_i\) in absence of dark respiration \([\mu mol \, mol^{-1}]\)

The compensation point of photosynthesis is obtained from:

\[\Gamma^{*} = \frac{0.5 \cdot 0.21 \cdot V_{c_{max}} \cdot O_i} {V_{c_{max}} \cdot K_o}\]

\(C_i\) Inter-cellular CO₂ concentration \([\mu mol \, mol^{-1}]\)
\(\Gamma^{*}\) Compensation point of photosynthesis, related to \(C_i\) in absence of dark respiration \([\mu mol \, mol^{-1}]\)
\(V_{c_{max}}\) Maximum saturated Rubisco carboxylation rate \([\mu mol \, m^{-2} \, s^{-1}]\)
\(K_o\) Michaelis-Menten constant for O₂ \([\mu mol \, mol^{-1}]\)

The maximum saturated Rubisco carboxylation rate \(V_{c_{max}}\) is calculated from:

\[V_{c_{max}} = 98 \cdot \frac{A_{max}} {34.668} \cdot k(T)_{v_{cmax}}\]

\(V_{c_{max}}\) Maximum saturated Rubisco carboxylation rate \([\mu mol \, m^{-2} \, s^{-1}]\)
\(A_{max}\) Plant-specific maximum CO₂ assimilation rate \([kg \, CO_2 \, ha^{-1} \, d^{-1}]\)
\(k(T)_{v_{cmax}}\) Temperature function for \(V_{c_{max}}\)

Figure 1: Temperature function for \(V_{c_{max}}\) (Long, 1991).

For crops using the C4 metabolism, no direct impact of the atmospheric CO₂ concentration on photosynthesis is assumed. The crop-specific maximum CO₂ assimilation rate is merely modified by a simple temperature function.

Figure 2: Temperature function for the CO₂ assimilation rate of C4 crops (Sage & Kubien 2007).

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References

• Farquhar, G. D. & von Caemmerer, S. (1982). Modelling of photosynthetic response to environmental conditions. In: Encyclopedia of plant physiology. New series. Volume 12B. Physiological plant ecology. II. Water relations and carbon assimilation. (Eds O. L. Lange, P. S. Nobel, C. B. Osmond, & H. Ziegler), pp. 549-587. Berlin: Springer.

• Goudriaan, J. & van Laar, H. H. (1978). Relations between leaf resistance, CO2 concentration and CO2 assimilation in maize, beans, lalang grass and sunflower. Photosynthetica 12 (3), 241-249.

• Long, S. P. (1991). Modification of the response of photosynthetic productivity to rising temperature by atmospheric CO2 concentrations - Has its importance been underestimated. Plant Cell and Environment 14 (8), 729-739.

• Mitchell, R. A. C., Lawlor, D. W., Mitchell, V. J., Gibbard, C. L., White, E. M., & Porter, J. R. (1995). Effects of elevated CO2 concentration and increased temperature on winter-wheat - Test of ARCWHEAT1 simulation model. Plant Cell and Environment 18 (7), 736-748.

• Nendel, C., Kersebaum, K. C., Mirschel, W., Manderscheid, R., Weigel, H. J., & Wenkel, K.-O. (2009). Testing different CO2 response algorithms against a FACE crop rotation experiment. NJAS - Wageningen Journal of Life Sciences 57 (1), 17-25.

• Sage, R. F. & Kubien, D. S. (2007). The temperature response of C3 and C4 photosynthesis. Plant Cell and Environment 30 (9), 1086-1106.

• van Keulen, H., Penning de Vries, F. W. T., & Drees, E. M. (1982). A summary model for crop growth. In: Simulation of plant growth and crop production (Eds F. W. T. Penning de Vries & H. H. van Laar), pp. 87-97. Wageningen: PUDOC.