Photosynthesis biological demand

Governing equations

The net photosynthetic uptake of CO\(_2\), \(A_n\) [\(\mu\)mol CO\(_2\) m\(^{-2}\) s\(^{-1}\)], is given as

\[\begin{eqnarray} \label{eqn_an_bio_demand} A_n &=& \min(A_c, A_j, A_p) - R_d \nonumber \\ &=& A_g - R_d \end{eqnarray}\]

where \(A_c\) [\(\mu\)mol CO\(_2\) m\(^{-2}\) s\(^{-1}\)] is the Rubisco-limited CO\(_2\) assimilation, \(A_j\) [\(\mu\)mol CO\(_2\) m\(^{-2}\) s\(^{-1}\)] is the light-limited CO\(_2\) assimilation, \(A_p\) [\(\mu\)mol CO\(_2\) m\(^{-2}\) s\(^{-1}\)] is the PEP carboxylase-limited CO\(_2\) assimilation, \(A_g\) [\(\mu\)mol CO\(_2\) m\(^{-2}\) s\(^{-1}\)] is the co-limited gross CO\(_2\) assimilation, and \(R_d\) [\(\mu\)mol CO\(_2\) m\(^{-2}\) s\(^{-1}\)] is mitochondrial respiration. \(A_g\) is given as the smaller of two quadratic roots

\[\begin{eqnarray} 0.98A^2_i - (A_c + A_j) + A_c A_j &=& 0 \\ 0.98A^2_g - (A_i + A_p) + A_i A_p &=& 0 \end{eqnarray}\]

The assimilation fluxes (\(A_c\), \(A_j\), and \(A_p\)) for C3 and C4 photosynthesis pathway are described next.

C3 Photosynthesis

The CO\(_2\) assimilation fluxes for C3 photosynthesis are

\[\begin{eqnarray} A_c &=& \frac{V_{cmax} (c_i - \Gamma^*)}{c_i + K_c(1 - o_i/K_o)}\\[0.4em] A_j &=& \frac{J}{4}\left( \frac{c_i - \Gamma^*}{c_i + 2 \Gamma^*}\right) \\[0.4em] A_p &=& 0 \end{eqnarray}\]

where \(V_{cmax}\) [\(\mu\)mol CO\(_2\) m\(^{-2}\) s\(^{-1}\)] maximum rate of carboxylation, \(c_i\) [\(\mu\)mol CO\(_2\) mol\(^{-1}\)] is the intercellular CO\(_2\), \(o_i\) [\(\mu\)mol O\(_2\) mol\(^{-1}\)] is the intercellular O\(_2\), \(K_c\) [\(\mu\)mol CO\(_2\) mol\(^{-1}\)] is the Michaelist-Mention constant for CO\(_2\), \(K_o\) [\(\mu\)mol O\(_2\) mol\(^{-1}\)] is the Michaelist-Mention constant for O\(_2\), \(J\) [\(\mu\)mol CO\(_2\) mol\(^{-1}\)] is the electron transport rate, and \(\Gamma^*\) [\(\mu\)mol CO\(_2\) mol\(^{-1}\)] is the compensation point defined as the \(c_i\) at which no net CO\(_2\) update occurs.

The rate of electron transport is related to photosynthetically active radiation and is given as the smaller root of the following quadratic equation.

\[\begin{equation} \Theta_j J^2 - (I_{PSII} + J_{max}) + I_{PSII} J_{max} = 0 \end{equation}\]

where \(I_{PSII}\) [\(\mu\)mol CO\(_2\) mol\(^{-1}\)] is the amount of light utilized in photosynthesis II, \(J_{max}\) [\(\mu\)mol CO\(_2\) mol\(^{-1}\)] is the maximum transport rate, and \(\Theta_j = 0.9\) is the curvature parameter. The amount of light utilized in photosynthesis II is

\[\begin{equation} I_{PSII} = \frac{\Phi_{PSII}}{2} \alpha_\ell \overrightarrow{I}_{PAR} \end{equation}\]

where \(\Phi_{PSII} = 0.7\) [mol mol\(^{-1}\)] is the quantum yield of photosystem II, \(\alpha_\ell = 1\) is the leaf absorptance, \(\overrightarrow{I}_{PAR}\) [\(\mu\)mol photon m\(^{-2}\) s\(^{-1}\)] is the absorbed photosynthetically active radiation.

The parameters \(K_c\), \(K_o\), \(\Gamma^*\), \(V_{cmax}\), \(J_{max}\), and \(R_d\) vary from their values at 25\(^0\)C as function of leaf temperature, \(T_\ell\), that are given as

\[\begin{eqnarray} K_{c} &=& K_{c25} f(T_\ell) \\[0.4em] K_{o} &=& K_{o25} f(T_\ell) \\[0.4em] \Gamma^* &=& \Gamma^* f(T_\ell) \\[0.4em] V_{cmax} &=& V_{cmax25} f(T_\ell) f_H(T_\ell)\\[0.4em] J_{max} &=& J_{cmax25} f(T_\ell) f_H(T_\ell)\\[0.4em] R_{d} &=& R_{d25} f(T_\ell) f_H(T_\ell) \end{eqnarray}\]

where

\[\begin{eqnarray} f(T_\ell) &=& \exp \left[ \frac{\Delta H_a}{298.15 \mathcal{R}} \left( 1 - \frac{298.15}{T_\ell} \right) \right] \\[0.4em] f_H(T_\ell) &=& \left[1 + \exp \left( \frac{298.15\Delta S - \Delta H_d}{298.15\mathcal{R}} \right) \right] \left[1 + \exp \left( \frac{\Delta S T_\ell - \Delta H_d}{\mathcal{R} T_\ell} \right)\right]^{-1} \end{eqnarray}\]

Lastly, the \(J_{max}\) and \(R_{d}\) at 25\(^0\)C are given as

\[\begin{eqnarray} J_{max25} &=& 1.67 V_{cmax25} \\ R_{d25} &=& 0.015 V_{cmax25} \end{eqnarray}\]

C4 Photosynthesis

The CO\(_2\) assimilation fluxes for C4 photosynthesis are

\[\begin{eqnarray} A_c &=& V_{cmax} \\ A_j &=& \alpha_\ell \overrightarrow{I}_{PAR} E\\ A_p &=& k_p c_i \end{eqnarray}\]

where \(E = 0.05\) [mol mol\(^{-2}\)] is the quantum yield and \(k_p\) [mol m\(^{-2}\) s\(^{-1}\)] is the initial slope of the CO\(_2\) response curve. The temperature dependence of \(V_{cmax}\), \(R_d\), and \(k_p\) are given as

\[\begin{eqnarray} V_{cmax} &=& V_{cmax25} Q_{10}^{(T_\ell - 298.16)/10} \left( 1 + \exp[s_1(T_\ell - s_2)] \right)^{-1} \left( 1 + \exp[s_3(s_4 - T_\ell)] \right)^{-1} \\[0.4em] R_d &=& R_{d25} Q_{10}^{(T_\ell - 298.16)/10} \left( 1 + \exp[s_5(s_6 - T_\ell)] \right)^{-1} \\[0.4em] k_p &=& k_{p25} Q_{10}^{(T_\ell - 298.16)/10} \end{eqnarray}\]

where \(Q_{10} = 2\), \(s_1 = 0.3\) [K\(^{-1}\)], \(s_2 = 313.15\) [K], \(s_3 = 0.2\) [K\(^{-1}\)], \(s_4 = 288.15\) [K], \(s_5 = 1.3\) [K\(^{-1}\)], and \(s_6 = 328.15\) [K].

The max \(R_{d}\) and \(k_{p}\) at 25\(^0\)C are given as

\[\begin{eqnarray} R_{d25} &=& 0.025 V_{cmax25} \\ k_{p25} &=& 0.02 V_{cmax25} \end{eqnarray}\]