A generalised dynamic model of leaf-level C₃ photosynthesis combining light and dark reactions with stomatal behaviour

Abstract

Global food demand is rising, impelling us to develop strategies for improving the efficiency of photosynthesis. Classical photosynthesis models based on steady-state assumptions are inherently unsuitable for assessing biochemical and stomatal responses to rapid variations in environmental drivers. To identify strategies to increase photosynthetic efficiency, we need models that account for the timing of CO₂ assimilation responses to dynamic environmental stimuli. Herein, I present a dynamic process-based photosynthetic model for C₃ leaves. The model incorporates both light and dark reactions, coupled with a hydro-mechanical model of stomatal behaviour. The model achieved a stable and realistic rate of light-saturated CO₂ assimilation and stomatal conductance. Additionally, it replicated complete typical assimilatory response curves (stepwise change in CO₂ and light intensity at different oxygen levels) featuring both short lag times and full photosynthetic acclimation. The model also successfully replicated transient responses to changes in light intensity (light flecks), CO₂ concentration, and atmospheric oxygen concentration. This dynamic model is suitable for detailed ecophysiological studies and has potential for superseding the long-dominant steady-state approach to photosynthesis modelling. The model runs as a stand-alone workbook in Microsoft® Excel® and is freely available to download along with a video tutorial.

Appendix: model details

Flows

A submodel for light reactions of photosynthetic CO₂ assimilation in C₃ leaves: potential ATP and NADPH production rate

The submodel calculates I₁, I₂, J₁, J₂, J_ATP and J_NADPH when f_Cyc, f_Pseudocyc, f_Q, f_NDH, Y(II)_LL, s, h, α_V, V_0V and θ_V are known. I₁ and I₂ are the light absorbed by PSI and PSII, respectively. J₁ and J₂ are the electron flow though PSI and PSII, respectively. J_ATP and J_NADPH are the steady-state rates of ATP and NADPH production, respectively. f_Cyc is the proportion of electron flow at PSI which follows CEF, f_Pseudocyc is the fraction of J₁ used by alternative electron sinks (APX cycle and nitrate reduction), f_Q is the level of Q-cycle engagement, and f_NDH is the fraction of f_Cyc flowing through NDH. The Y(II)_LL is the initial yield of PSII extrapolated under zero PPFD, s is the combined energy partitioning coefficient described in Yin et al. (2009), and h is the number of protons required to synthesise each ATP. α_V, V_0V and θ_V, define the slope, the offset and the curvature of the function f′(PPFD), expressing the PPFD dependence of Y(II).

When f_Cyc = 0, I₁, I₂, J₁ and J₂ take the values I_1,0, I_2,0, J_1,0 and J_2,0, respectively, and J_1,0 = J_2,0. Then I_2,0 and I_1,0 can be expressed as (Yin et al. 2004, 2009):

$${I_{2,0}}~={\text{PPFD}} \times s,$$

(1)

$${I_{1,0}}=\frac{{{I_{2,0}}~Y{{({\text{II}})}_{{\text{LL}}}}}}{{Y{{({\text{I}})}_{{\text{LL}}}}}}.$$

(2)

The total light absorbed by both PSI and PSII is I = I_1,0 + I_2,0 and I < PPFD.

When CEF is engaged, I₁ increases by a quantity χ (Yin et al. 2004):

$${I_{1~}}=\left( {1+\chi } \right)~{I_{1,0}},$$

(3)

where χ is calculated as a function of f_Cyc as (Yin et al. 2004):

$$\chi =\frac{{{f_{{\text{Cyc}}}}}}{{~1+{f_{{\text{Cyc}}}}+~Y{{({\text{II}})}_{{\text{LL}}}}}}.$$

(4a)

I simulate the rate of cyclic electron flow through a tentative function as:

$${f_{{\text{Cyc}}}}=\hbox{max} \left( {0,~ - 1+{{15}^{~~\frac{{v{\text{ATP}}~}}{{{J_{{\text{ATP}}}}}} - ~\frac{{v{\text{NADPH}}}}{{{J_{{\text{NADPH}}}}}}}}} \right).$$

(4b)

When the ratio of actual ATP production relative to the potential $\left( {\frac{{{v_{{\text{ATP}}}}~}}{{{J_{{\text{ATP}}}}}}} \right)$ is greater than the ratio of actual NADPH production relative to the potential $\left( {\frac{{{v_{{\text{NADPH}}}}}}{{{J_{{\text{NADPH}}}}}}} \right)$, indicating ATP demand greater than NADPH demand, f_Cyc will be greater than zero. Equation 4b yields values very close to zero in C₃ plants, and further testing will be necessary before application with other photosynthetic types.

If I is constant, I₂, J₂ and J₁ are calculated as (Yin et al. 2004):

$${I_{2~}}=\left( {\frac{1}{{~Y{{({\text{II}})}_{{\text{LL}}}}}} - \chi } \right){I_{2,0}}~~Y{({\text{II}})_{{\text{LL}}}},$$

(5)

$${J_{2~}}={I_{2~}}~Y({\text{II}}),$$

(6)

$${J_{1~}}=\frac{{{J_{2~}}}}{{1 - {f_{{\text{Cyc}}}}}},$$

(7)

where Y(II) is the yield of photosystem II, which depends on PPFD and feedbacks from dark reactions through the novel process-based function:

$$Y({\text{II}})=Y{({\text{II}})_{{\text{LL}}}}~\frac{{v{\text{ATP}}}}{{{J_{{\text{ATP}}}}}}~\frac{{v{\text{NADPH}}}}{{{J_{{\text{NADPH}}}}}}~\left( {1 - \hbox{max} \left[ {0,~f^{\prime}\left( {{\text{PPFD}}} \right)} \right]} \right).$$

(8)

The rationale of Eq. 8 is that Y(II) has a maximum operational value, Y(II)_LL, and is quenched by three distinct factors (Müller et al. 2001): (1) the slowing down of ATP synthesis caused by limiting availability of phosphate or ADP (described by $\frac{{{v_{{\text{ATP}}}}}}{{{J_{{\text{ATP}}}}}}$); (2) the reduction of the plastoquinone pool (described by $\frac{{{v_{{\text{NADPH}}}}}}{{{J_{{\text{NADPH}}}}}}$); (3) reaching the maximum capacity for electron transport (described by f′(PPFD), which responds to PPFD as a non-rectangular hyperbola). f′(PPFD) is calculated with Eq. 20, below, but with different parameterisation, see Table S2. Parameter values were adjusted by fitting modelled assimilation against A/PPFD response curves (Fig. 3, Table S2).

The proton flow to the lumen includes: one proton per electron from water oxidation, one proton from electron flow through the cytochromes (J_Cyt), two protons from the electron flow through the Q-cycle (J_Q, Yin et al. 2004) and two protons from the electron flow through NDH (J_NDH, Kramer and Evans 2011). The rate of ATP production is:

$${J_{{\text{ATP~}}}}=\frac{{{J_2}~+~{J_{{\text{Cyt}}}}+~2{J_{Q~}}+~2{J_{{\text{NDH}}}}}}{h},$$

(9)

where h is the number of protons required to synthesise each ATP molecule, the flow through the Q-cycle is J_Q = f_QJ₁; the complement, directly flowing to the b₆f complex, is J_Cyt = (1 − f_Q) J₁; and the flow through the NDH complex is J_NDH = f_Cycf_NDHJ₁.

The total NADPH production can be expressed as (Yin et al. 2004):

$${J_{{\text{NADPH}}}}={J_{1~}}\frac{{1 - {f_{{\text{Cyc}}}} - {f_{{\text{Pseudocyc}}}}}}{2},$$

(10a)

The alternative electron sinks include nitrogen metabolism (chiefly reduction) as well as the APX cycle. The APX cycle is known to depend on O₂ concentration and the availability of PSI acceptors (Miyake and Yokota 2000; Schreiber and Neubauer 1990). I describe f_Pseudocyc as a linear function of O₂ concentration and $\frac{{{v_{{\text{NADPH}}}}}}{{{J_{{\text{NADPH}}}}}}$ as:

$${f_{{\text{Pseudocyc}}}}={f_{{\text{Pseudocyc~NR}}}}+4~\left[ {{{\text{O}}_2}} \right]~\left( {1 - \frac{{v{\text{NADPH}}}}{{{J_{{\text{NADPH}}}}}}} \right),$$

(10b)

where the coefficient 4 was fitted empirically to yield a value of f_Pseudocyc ≈ 0.1 under ordinary ambient conditions, and the fraction of f_Pseudocyc partitioned to nitrate reduction (f_{Pseudocyc NR}) was set at ≈ 0 for simplicity.

The time dependence of J_ATP and J_NADPH was modelled after Bellasio et al. (2017) as:

$$\left\{ {\begin{array}{*{20}{l}} {{J_{{\text{ATP~or~NADPH~}}~t+{\text{d}}t}}={J_{{\text{ATP~or~NADPH}}~t}}~+~\frac{{{J_{{\text{ATP~or~NADPH}}~}} - {J_{{\text{ATP~or~NADPH}}~t}}}}{{{K_{J{~_{{\text{ATP~or~NADPH}}}}}}}}~~{\text{d}}t}&{~{\text{if}}~{J_{{\text{ATP~or~NADPH}}~t}}<{J_{{\text{ATP~or~NADPH}}}}} \\ {{J_{{\text{ATP~or~NADPH }}t}}={J_{{\text{ATP~or~NADPH}}}}}&{{\text{else}}} \end{array}} \right.,$$

(11)

where K_{J ATP or NADPH} is the time constant for an increase in J_ATP or J_NADPH, J_{ATP or NADPH t+dt} and J_{ATP or NADPH t} are the values at the time step t + dt or at the previous step t, respectively; J_{ATP or NADPH} are the steady-state values (Eqs. 9 and 10a).

Actual rates of ATP and NADPH production

The actual rates of ATP (v_ATP) and NADPH (v_NADPH) production are calculated after Wang et al. (2014b):

$${v_{{\text{ATP}}}}=\frac{{{J_{{\text{ATP}}}}~\left( {\left[ {{\text{ATP}}} \right]\left[ {{P_{\text{i}}}} \right] - \frac{{\left[ {{\text{ADP}}} \right]}}{{{K_{{\text{e}}~}}}}} \right)}}{{{K_{{\text{m~ADP}}}}{K_{{\text{m~}}{P_{\text{i}}}}}\left( {1+\frac{{\left[ {{\text{ADP}}} \right]}}{{{K_{{\text{m~ADP}}}}}}+\frac{{\left[ {{\text{ATP}}} \right]}}{{{K_{{\text{m~ATP}}}}}}+\frac{{\left[ {{P_{\text{i}}}} \right]}}{{{K_{{\text{m~}}{P_{\text{i}}}}}}}+\frac{{\left[ {{\text{ATP}}} \right]\left[ {{P_{\text{i}}}} \right]}}{{{K_{{\text{m~ADP}}}}{K_{{\text{m~}}{{\text{P}}_{\text{i}}}}}}}} \right)}},$$

(12)

$${v_{{\text{NADPH}}}}=\frac{{{J_{{\text{NADPH}}}}~\left( {\left[ {{\text{NADP}}} \right] - \frac{{\left[ {{\text{NADPH}}} \right]}}{{{K_{{\text{e}}~}}}}} \right)}}{{{K_{{\text{m~NADP}}}}\left( {1+\frac{{\left[ {{\text{NADP}}} \right]}}{{{K_{{\text{m~NADP}}}}}}+\frac{{\left[ {{\text{NADPH}}} \right]}}{{{K_{{\text{m~NADPH}}}}}}} \right)}},$$

(13)

where square brackets indicate metabolite concentration, K_m represents the Michaelis–Menten constant for a given metabolite, and K_e is the equilibrium constant of the reaction (Table S1).

CO₂ diffusion, dissolution and hydration

The rate of CO₂ diffusion through the stomata is:

$${\text{C}}{{\text{O}}_2}~{\text{stomatal~diffusion}}=\frac{{{g_{\text{S}}}~\left( {{C_{\text{a}}} - {C_{\text{i}}}} \right)}}{{1000}},$$

(14)

where g_S is stomatal conductance to CO₂ (mol m⁻² s⁻¹); C_a and C_i are the CO₂ concentrations (µmol mol⁻¹) external to the leaf and in the intercellular space, respectively; and the 1000 is used to convert the units from micromoles to mmol m⁻² s⁻¹.

The rate of CO₂ dissolution in aqueous media within the leaf is:

$${\text{C}}{{\text{O}}_2}~{\text{dissolution}}=\frac{{{g_{\text{M}}}~\left( {{C_{\text{i}}} - \left[ {{\text{C}}{{\text{O}}_2}} \right]{K_{{\text{hC}}{{\text{O}}_2}~}}} \right)}}{{1000}},$$

(15)

where g_M is mesophyll conductance to CO₂ diffusion (mol m⁻² s⁻¹), [CO₂] is the CO₂ concentration in mesophyll cells (mM), which is assumed to be spatially uniform, ${K_{{\text{hC}}{{\text{O}}_2}}}$ is CO₂ volatility (the reciprocal of solubility) (µbar mM⁻¹) and 1000 is used to convert the units into mmol m⁻² s⁻¹, the unit of all subsequent rates.

The rate of CO₂ hydration to bicarbonate is (Wang et al. 2014b):

$${\text{CA}}=\frac{{{V_{{\text{MAX~CA}}}}~\left( {\left[ {{\text{C}}{{\text{O}}_2}} \right] - \frac{{\left[ {{\text{HC}}{{\text{O}}_3}^{ - }} \right]\left[ {{{\text{H}}^+}} \right]}}{{{K_{{\text{e~}}}}}}} \right)}}{{{K_{{\text{m~C}}{{\text{O}}_2}}}\left( {1+\frac{{\left[ {{\text{C}}{{\text{O}}_2}} \right]}}{{{K_{{\text{m~C}}{{\text{O}}_2}}}}}+\frac{{\left[ {{\text{HC}}{{\text{O}}_3}^{ - }} \right]}}{{{K_{{\text{m~HC}}{{\text{O}}_3}}}}}} \right)}},$$

(16)

where V_{MAX CA} is the maximum hydration rate.

Reaction rates

The rate of Rubisco carboxylation (V_C) was modified from Wang et al. (2014b) as:

$${V_{\text{C}}}=\frac{{{V_{{\text{C~MAX}}}}~~{R_{{\text{act}}}}~~f\left( {{\text{RuBP}}} \right)~\left[ {{\text{RuBP}}} \right]\left[ {{\text{C}}{{\text{O}}_2}} \right]}}{{\left( {K_{{{\text{m~C}}{{\text{O}}_2}}}^{'}+\left[ {{\text{C}}{{\text{O}}_2}} \right]} \right)\left( {K_{{{\text{m~RuBP}}}}^{'}+\left[ {{\text{RuBP}}} \right]} \right)}},$$

(17)

where V_{C MAX} is the maximum carboxylation rate. In the V_{C MAX} used in Farquhar et al. (1980), Rubisco is assumed fully activated and also fully RuBP saturated in the ‘enzyme-limited’ case. Here, V_{C MAX} is more closely comparable to the in vitro rate. R_act is the Rubisco activation state, a time-dependent variable calculated as:

$${R_{{\text{act~}}t{\text{+d}}t}}={R_{{\text{act}}~t}}+\left\{ {\begin{array}{*{20}{l}} {\frac{{{R_{{\text{act~eq}}}} - ~{R_{{\text{act}}~t}}}}{{{\tau _{\text{i}}}}}~{\text{d}}t}&{i{\text{f}}~{R_{{\text{act}}~t}}<{R_{{\text{act~eq}}}}} \\ {\frac{{{R_{{\text{act~eq}}}} - ~{R_{{\text{act~}}t}}}}{{{\tau _{\text{d}}}}}~{\text{d}}t}&{{\text{else}}} \end{array}} \right.,$$

(18)

where τ_i and τ_d are the time constants for Rubisco induction and deactivation, respectively (Seemann et al. 1988), and the steady state R_act value is:

$${R_{{\text{act~}}}}=f\left( {{\text{PPFD}}} \right)~~~f\left( {\left[ {{\text{C}}{{\text{O}}_2}} \right]} \right),$$

(19)

where $f\left( {{\text{PPFD}}} \right)$ simulates activation state of Rubisco independently of CO₂ concentration, and I included f([CO₂]) to capture the inactivation of Rubisco observed in vivo at low CO₂. The $f\left( {{\text{PPFD}}} \right)$ and f([CO₂]) were modelled with non-rectangular hyperbolas (Gross et al. 1991):

$$f\left( {{\text{PPFD}}} \right)~=~{V_0}+\frac{{{\alpha _{\text{V}}}~{\text{PPFD}}+1 - {V_0}~ - \sqrt {{{\left( {{\alpha _{\text{V}}}~{\text{PPFD}}+1 - {V_0}} \right)}^2} - 4{\alpha _{\text{V}}}~{\text{PPFD}}~{\theta _{\text{V}}}} }}{{2~{\theta _{\text{V}}}}},$$

(20)

$$f\left( {\left[ {{\text{C}}{{\text{O}}_2}} \right]} \right)~=~{V_{0{\text{C}}}}+\frac{{{\alpha _{\text{C}}}~\left[ {{\text{C}}{{\text{O}}_2}} \right]+1 - {V_{0{\text{C}}}}~ - \sqrt {{{\left( {{\alpha _{\text{V}}}~\left[ {{\text{C}}{{\text{O}}_2}} \right]+1 - {V_{0{\text{C}}}}} \right)}^2} - 4{\alpha _{\text{C}}}~\left[ {{\text{C}}{{\text{O}}_2}} \right]~{\theta _{\text{C}}}} }}{{2~{\theta _{\text{C}}}}},$$

(21)

where V₀, α_V, and θ_V are empirical parameters of the hyperbola for f(PPFD) defining the initial activity in the dark, the slope of the dependency and the curvature, respectively; V_0C, α_C, and θ_C are the equivalent parameters for f([CO₂]).

The f(RuBP) is a function of RuBP concentration, relative to the concentration of Rubisco active sites, which was modelled using a non-rectangular hyperbola after Farquhar et al. (1980):

$$f\left( {{\text{RuBP}}} \right)~=~\frac{{{E_{\text{T}}}~+~K_{{{\text{m RuBP}}}}^{'}+\left[ {{\text{RuBP}}} \right] - \sqrt {{{\left( {{E_{\text{T}}}~+~K_{{{\text{m RuBP}}}}^{'}+\left[ {{\text{RuBP}}} \right]} \right)}^2} - 4\left[ {{\text{RuBP}}} \right]{E_{\text{T}}}} }}{{2{E_{\text{T}}}}}~,$$

(22)

where E_T is the total concentration of Rubisco catalytic sites, calculated from V_{C MAX} and turnover rate after Wang et al. (2014b). The Michaelis–Menten constant for RuBP and CO₂ are:

$$K_{{{\text{m RuBP}}}}^{\prime }={K_{{\text{m RuBP}}}}\left( {1+\frac{{\left[ {{\text{PGA}}} \right]}}{{{K_{{\text{m~PGA}}}}}}+\frac{{\left[ {{\text{NADP}}} \right]}}{{{K_{{\text{i~NADP}}}}}}+\frac{{\left[ {{\text{ADP}}} \right]}}{{{K_{{\text{i~ADP}}~}}}}+\frac{{\left[ {{P_{\text{i}}}} \right]}}{{{K_{{\text{i~}}{P_{\text{i}}}{\text{~}}}}}}} \right),$$

(23)

$$K_{{{\text{m~C}}{{\text{O}}_2}}}^{\prime }={K_{{\text{m~C}}{{\text{O}}_2}}}\left( {1+\frac{{\left[ {{{\text{O}}_2}} \right]}}{{{K_{{\text{m~}}{{\text{O}}_2}}}}}} \right),$$

(24)

where K_i are the constants for the competitive inhibition.

The rate of Rubisco oxygenation (V_O) was calculated after Farquhar et al. (1980) as:

$${V_{\text{O}}}={V_{\text{C}}}~2{\gamma ^*}~\frac{{\left[ {{{\text{O}}_2}} \right]}}{{\left[ {C{{\text{O}}_2}} \right]}},$$

(25)

where γ* is half the reciprocal Rubisco specificity, calculated in the liquid phase (von Caemmerer 2000) using constants from Sander (2015) and Warneck and Williams (2012). In this model, the glycine decarboxylase (GDC) decarboxylation rate equals V_O; for a justification and possible stoichiometric variants, see Bellasio (2017).

The rate of RuP phosphorylation was modified from Wang et al. (2014b) as:

$${\text{Ru}}{{\text{P}}_{{\text{Phosp}}}}=\frac{{{V_{{\text{MAX~}}}}~\left[ {{\text{ATP}}} \right]\left[ {{\text{RuP}}} \right]~ - \frac{{\left[ {{\text{ATP}}} \right]\left[ {{\text{RuBP}}} \right]}}{{{K_{{\text{e~}}}}}}}}{{\left( {\left[ {{\text{ATP}}} \right]~+~{K_{{\text{m~ATP}}}}\left( {1+~\frac{{\left[ {{\text{ADP}}} \right]}}{{{K_{{\text{i~ADP}}}}}}} \right)} \right)\left( {\left[ {{\text{RuP}}} \right]+{K_{{\text{m~RuP}}}}\left( {1+~\frac{{\left[ {{\text{PGA}}} \right]}}{{{K_{{\text{i~PGA}}}}}}~+~\frac{{\left[ {{\text{RuBP}}} \right]}}{{{K_{{\text{i~RuBP}}}}}}~+~\frac{{\left[ {{P_{\text{i}}}} \right]}}{{{K_{{\text{i~}}{P_{\text{i}}}}}}}} \right)} \right)}}.$$

(26)

The reducing phase of the reductive pentose phosphate pathway was modelled as a single-step pseudoreaction. The rate of PGA reduction (PR) was calculated by fusing the rates of PGA phosphorylation and DPGA reduction from Wang et al. (2014b) as:

$${\text{PR}}=\frac{{{V_{{\text{MAX~}}}}~\left[ {{\text{ATP}}} \right]\left[ {{\text{PGA}}} \right]\left[ {{\text{NADPH}}} \right]~}}{{\left( {\left[ {{\text{PGA}}} \right]~+~{K_{{\text{m~PGA}}}}\left( {1+~\frac{{\left[ {{\text{ADP}}} \right]}}{{{K_{{\text{i~ADP}}}}}}} \right)} \right)\left( {\left[ {{\text{ATP}}} \right]~+~{K_{{\text{m~ATP}}}}\left( {1+~\frac{{\left[ {{\text{ADP}}} \right]}}{{{K_{{\text{i~ADP}}}}}}} \right)} \right)\left( {\left[ {{\text{NADPH}}} \right]~+~{K_{{\text{m~NADPH}}}}\left( {1+~\frac{{\left[ {{\text{ADP}}} \right]}}{{{K_{{\text{i~ADP}}}}}}} \right)} \right)}}.$$

(27)

The constants in Eq. 27 were adapted from the original separate reactions in Wang et al. (2014b) to maintain physiologically realistic concentrations of product and substrates. This was necessary to account for neglecting phosphorylated intermediates of the RPP.

Carbohydrate synthesis was assumed to be a single-step reaction, and the rate was calculated by simplifying the combined processes of starch and sucrose synthesis from Wang et al. (2014b) as:

$${\text{CS}}=\frac{{{V_{{\text{MAX~}}}}~\left( {\left[ {{\text{DHAP}}} \right] - 0.4} \right)\left( {1 - \frac{{\left| {{\text{PR}}} \right|~\left[ {{P_{\text{i}}}} \right]}}{{{K_{{\text{e~}}}}}}} \right)}}{{\left( {\left[ {{\text{DHAP}}} \right]~+~{K_{{\text{m~DHAP}}}}\left( {1+~\frac{{\left[ {{\text{ADP}}} \right]}}{{{K_{{\text{i~ADP}}}}}}} \right)} \right)}}.$$

(28a)

With Eq. 28a carbohydrates are synthesised when [DHAP] > 0.4 mM following a saturating Michaelis–Menten kinetics, inhibited by ADP. Concentration of sucrose, starch and their precursor are not calculated. To capture the reversible nature of the original Wang et al. (2014b) formulation, I use the quantity $\frac{{\left| {{\text{PR}}} \right|~\left[ {{P_{\text{i}}}} \right]}}{{{K_{{\text{e~}}}}}}$, where $\left| {{\text{PR}}} \right|$ ‘senses’ the concentration of sucrose and starch, hypothesised to be proportional to the rate of DHAP synthesis.

The interconversion phase of the RPP was modelled as a single-step pseudoreaction through a generic Michaelis Menten equation for equilibrium reaction (Zhu et al. 2007) as:

$${\text{RPP=}}\frac{{{V_{{\text{MAX~}}}}{\text{~}}\left[ {{\text{DHAP}}} \right]{\text{~}}\left( {{\text{1}} - \frac{{\left| {{\text{RuP}}} \right|}}{{{K_{{\text{e~}}}}}}} \right)}}{{\left( {\left[ {{\text{DHAP}}} \right]{\text{~+~}}{K_{{\text{m~DHAP}}}}} \right)}}{\text{.}}$$

(28b)

The constants in Eq. 28b were adapted from the original separate reactions in Wang et al. (2014b) to operate and maintain physiologically realistic concentrations of substrates.

Stocks

Change in metabolite concentrations

The change in metabolite concentrations in time $\frac{{{\text{d}}\left[ {} \right]}}{{{\text{d}}t}}~$ was described by a set of ordinary differential equations based on the stoichiometry of Bellasio (2017) informed with the reaction rates described above and converted from variation in leaf-level pool to variation in concentration using the mesophyll volume as described in Wang et al. (2014b).

The rates of change in concentrations of CO₂, bicarbonate (HCO₃⁻), RuBP, PGA, DHAP, ATP and NADPH were calculated as:

$$\frac{{{\text{d}}\left[ {{\text{C}}{{\text{O}}_2}} \right]}}{{{\text{d}}t}}{V_{\text{M}}}={\text{C}}{{\text{O}}_2}~{\text{Dissolution}}+{R_{{\text{LIGHT}}}} - {V_{\text{C}}}+0.5~{\text{GDC}} - {\text{CA,}}$$

(29)

$$\frac{{{\text{d}}\left[ {{\text{HC}}{{\text{O}}_3}^{ - }} \right]}}{{{\text{d}}t}}{V_{\text{M}}}={\text{CA,}}$$

(30)

$$\frac{{{\text{d}}\left[ {{\text{RuBP}}} \right]}}{{{\text{d}}t}}{V_{\text{M}}}={\text{Ru}}{{\text{P}}_{{\text{Phosp}}}} - ~{V_{\text{C}}} - {V_{\text{O}}},$$

(31)

$$\frac{{{\text{d}}\left[ {{\text{PGA}}} \right]}}{{{\text{d}}t}}{V_{\text{M}}}=2~{V_{\text{C}}}+{V_{\text{O}}}+0.5{\text{~GDC}} - {\text{PR}} - \frac{1}{3}{R_{{\text{LIGHT}}}},$$

(32)

$$\frac{{{\text{d}}\left[ {{\text{DHAP}}} \right]}}{{{\text{d}}t}}{V_{\text{M}}}={\text{PR}} - {\text{CS}} - \frac{5}{3}{\text{Ru}}{{\text{P}}_{{\text{Phosp}}}},$$

(33)

$$\frac{{{\text{d}}\left[ {{\text{ATP}}} \right]}}{{{\text{d}}t}}{V_{\text{M}}}={v_{{\text{ATP}}}} - {\text{Ru}}{{\text{P}}_{{\text{Phosp}}}} - {V_{\text{O}}} - {\text{~PR}} - 0.5~{\text{CS,}}$$

(34)

$$\frac{{{\text{d}}\left[ {{\text{NADPH}}} \right]}}{{{\text{d}}t}}{V_{\text{M}}}={v_{{\text{NADPH}}}} - ~{\text{PR}} - 0.5~{V_{\text{O}}},$$

(35)

$$\frac{{{\text{d}}\left[ {{\text{RuP}}} \right]}}{{{\text{d}}t}}{V_{\text{M}}}={\text{RPP}} - ~{\text{Ru}}{{\text{P}}_{{\text{Phosp}}}},$$

(36)

where V_M is mesophyll volume per meter square of leaf (L m⁻²) calculated after considering the leaf half-full of mesophyll (Lawlor 1993), R_LIGHT is light respiration and is input to the model as described in Bellasio (2017), and all the other flux rates have been previously described: CO₂ dissolution (Eq. 15), CA (Eq. 16), RuP_Phosph (Eq. 26), V_C (Eq. 17), V_O (= GDC, Eq. 25), PR (Eq. 27), CS (Eq. 28a), v_ATP (Eq. 12), v_NADPH (Eq. 13) and RPP (Eq. 28b). Equations 29–30 were derived in this study and Eqs. 31–36 are modified from (Bellasio 2017).

Concentrations determined from total metabolite pools

The concentrations of ADP, NADP⁺ and phosphate ($\left[ {{P_{\text{i}}}} \right]$) are calculated simply by subtraction from a total pool:

$$\left[ {{\text{ADP}}} \right]={A_{{\text{Tot}}}} - \left[ {{\text{ATP}}} \right],$$

(37)

$$\left[ {{\text{NAD}}{{\text{P}}^+}} \right]={N_{{\text{Tot}}}} - \left[ {{\text{NADPH}}} \right],$$

(38)

$$\left[ {{P_{\text{i}}}} \right]={P_{{\text{iTot}}}} - \left[ {{\text{PGA}}} \right]~ - \left[ {{\text{DHAP}}} \right] - \left[ {{\text{RuP}}} \right] - 2\left[ {{\text{RuBP}}} \right]~ - \left[ {{\text{ATP}}} \right],$$

(39)

where A_Tot, N_Tot, and P_iTot are the total pools of adenylates, nicotinamides and phosphate, respectively.

The hydro-mechanical model of stomatal behaviour

The model calculates g_S after Bellasio et al. (2017) as:

$${g_{\text{S}}}=\hbox{max} \left( {{g_{{\text{S}}~0}},~\frac{{\chi ~\beta ~\tau ~\left( {{\Psi _{{\text{Soil}}}}+{\pi _{\text{e}}}} \right)}}{{1+\chi ~\beta ~\tau ~{R_{\text{h}}}~{D_{\text{S}}}}}} \right),$$

(40)

where χβ is a combined parameter scaling turgor-to-conductance and the hydro-mechanical-to-biochemical response; τ is the sensor of biochemical forcing; Ψ_Soil is soil water potential; π_e is epidermal osmotic pressure; R_h is the effective hydraulic resistance to the epidermis, calculated as 1/K_h, the corresponding hydraulic conductance; and D_S is the leaf-to-boundary layer H₂O mole fraction gradient, a measure of vapour pressure deficit, VPD. The parameter τ encompasses the biochemical components of the model and is calculated from f(RuBP) as:

$$\tau ={\tau _0}+f\left( {{\text{RuBP}}} \right),$$

(41)

where τ₀, the basal level of τ, was manually assigned. Stomatal dynamics were accounted for by describing the time dependence of g_S with a set of recursive equations (Bellasio et al. 2017):

$${g_{{\text{S}}~t+{\text{dt}}}}={g_{{\text{S}}~t}}+\left\{ {\begin{array}{*{20}{l}} {\frac{{{g_{{\text{S}}~}} - {g_{{\text{S}}~t}}}}{{{K_{\text{i}}}}}~{\text{d}}t}&{i{\text{f}}~{g_{{\text{S}}~t}}<{g_{\text{S}}}} \\ {\frac{{{g_{{\text{S}}~}} - {g_{{\text{S}}~t}}}}{{{K_{\text{d}}}}}~{\text{d}}t~}&{{\text{else}}} \end{array}} \right.,$$

(42)

where g_St+dt and g_St are the g_S values at the time step t + dt or at the previous step t, respectively; g_S is the steady-state value (Eq. 40), K_i and K_d are the time constants for an increase and decrease in g_S, respectively.