Electronic Supplementary Material ( ESM ) : The limiting factors and regulatory processes that control the environmental responses of C 3 , C 3C 4 intermediate , and C 4 photosynthesis

The expressions in Appendices I-V are given for idealized C3, Type I C3-C4, and NADP-ME C4 cases. These cases allow us to focus on developing an explicit representation of chloroplast electron transport, while maintaining an implicit representation of mitochondrial electron transport. Appendix I gives expressions that describe the whole-leaf net assimilation rate (A), in relation to net CO2 assimilation in the mesophyll (Am) and the bundle sheath (As). These expressions are relevant under all environmental conditions. Appendices II, III, and IV give additional expressions that describe the dependence of Am and As on light, pCO2, and pO2. Appendix II gives expressions for a pure light-limited state, and Appendix III gives expressions for a pure light-saturated state. Appendix IV gives expressions for mixed light-limited/light-saturated states, and Appendix V gives the solution to the model. For the analyses in this paper, the equations in Appendices I-V were implemented in MATLAB (2020b, The MathWorks, Natick, MA, USA). The Symbolic Math Toolbox was used to generate analytical solutions to the systems of equations specified in Appendix V, and the Global Optimization and Parallel Computing Toolboxes were used to fit the model to data. 1Department of Global Ecology, Carnegie Institution for Science, 260 Panama Street, Stanford CA 94305 USA 2Woods Institute for the Environment, Stanford University, 473 Via Ortega, Stanford CA 94305 USA Correspondence: Jennifer E. Johnson, Email: jjohnson@carnegiescience.edu, ORCID ID: 0000-0003-2181-8402 2 https://doi.org/10.1007/s00442-021-05062-y Appendix I: Governing rate equations Steady-state electron transport We begin by describing a metabolic state in which the supplies of Fd, NADPH, and ATP from chloroplast electron transport are dynamically coordinated with the energetic demands from carbon metabolism. For the C3 and the Type I C3-C4 mesophyll, the rates of electron transport through PS II and PS I depend on the activity of the photosynthetic carbon reduction (PCR) and photosynthetic carbon oxidation (PCO) cycles: JP680m = 4 ·Vcm +4 ·Vom (1a) JP700m = (3 ·Vcm +3.5 ·Vom− JP680m ·nL)/nC + JP680m (1b) where JP680m is the total rate of linear electron flow (LEF) through mesophyll PS II (mol em-2 s-1), JP700m is the total rate of LEF and cyclic electron flow (CEF1) through mesophyll PS I (mol em-2 s-1), Vcm and Vom are the carboxylation and oxygenation rates of mesophyll Rubisco (mol CO2 or O2 m-2 s-1), and nL and nC are the ATP coupling efficiencies of LEF and CEF1, respectively (mol ATP mol-1 e-). For the Type I C3-C4 bundle sheath, the analogous expressions are given by: JP680s = 4 ·Vcs +4 ·Vos (1c) JP700s = (3 ·Vcs +3.5 ·Vos− JP680s ·nL)/nC + JP680s (1d) where JP680s is the total rate of LEF through bundle sheath PS II (mol em-2 s-1), JP700s is the total rate of LEF and CEF1 through bundle sheath PS I (mol em-2 s-1), and Vcs and Vos are the carboxylation and oxygenation rates of bundle sheath Rubisco (mol CO2 or O2 m-2 s-1). For the C4 mesophyll, the corresponding expressions are: JP680m = 2 ·Vpm (2a) JP700m = 2 ·Vpm · (1−nL/nC +1/nC) (2b) where Vpm is the mesophyll activity of PEPC (mol CO2 m-2 s-1). In Eqns. 2a-2b, the partitioning between LEF and CEF1 in the mesophyll reflects the requirement for 2 ATP and 1 NADPH to drive the NADP-ME type C4 cycle. For the C4 bundle sheath, the corresponding expressions are: JP680s = 4 ·Vcs +4 ·Vos−2 ·Vds (2c) JP700s = (3 ·Vcs +3.5 ·Vos− JP680s ·nL)/nC + JP680s (2d) where Vds is the bundle sheath activity of NADP-ME (mol CO2 m-2 s-1). In Eqns. 2c-2d, the partitioning between LEF and CEF1 in the bundle sheath reflects the energetic requirements to drive the PCR and PCO cycles as well as the partial suppression of LEF due to NADPH production by NADPME. Steady-state gas-exchange We continue by defining expressions that link the activity of carbon metabolism to gas-exchange. For a C3, Type I C3-C4, or C4 leaf, a general expression for the net assimilation rate (A) is given by the sum of net CO2 assimilation in the mesophyll (Am) and the bundle sheath (As): A = Am +As (3a) Am =Vcm−0.5 ·Vom−Rm (3b) As =Vg +Vp−L (3c) where A, Am, and As can have units of mol assimilated C m-2 s-1 or mol CO2 m-2 s-1. In Eqn. 3c, Vg is the net rate at which the glycine shuttle transfers assimilated C from the mesophyll into the bundle sheath (mol C m-2 s-1), Vp is the net rate at which the malate shuttle transfers assimilated C from the mesophyll into the bundle sheath (mol C m-2 s-1), and L is the rate at which CO2 leaks out of the bundle sheath via diffusion (mol CO2 m-2 s-1). The rate of the glycine shuttle is determined by mesophyll Rubisco oxygenase activity that is coupled to bundle sheath GDC activity, and the rate of the malate shuttle is based on mesophyll PEPC activity that is coupled to bundle sheath NADP-ME activity: Vg = 0.5 ·Vom−Vgm (4a) Vp =Vpm−Vdm (4b) where Vgm is the rate of GDC activity in the mesophyll (mol CO2 m-2 s-1), Vpm is the rate of carboxylation via mesophyll PEPC (mol CO2 m-2 s-1) and Vdm is mesophyll C4decarboxylase activity that results in futile cycling (mol CO2 m-2 s-1). Combining Eqns. 3-4 then yields a second expreshttps://doi.org/10.1007/s00442-021-05062-y 3 sion for the overall net CO2 assimilation rate (A): A =Vcm−Vgm +Vpm−Vdm−L−Rm (5) which can represent photosynthesis in C3, any of the various C3-C4, or C4 plants. If all of the Rubisco is localized to the mesophyll, this expression reduces to the regular C3 expression described by Farquhar et al. (1980). If some of the Rubisco and all of the GDC are localized to the bundle sheath, the reduced expression is equivalent to the Type I C3C4 expression described by von Caemmerer (1989). If all of the Rubisco is localized to the bundle sheath and there is no futile C4 cycling, the reduced expression is equivalent to the C4 expression described by Berry and Farquhar (1978). In the dark, the leak rate L is equivalent to the rate of bundle sheath respiration, Rs. Linking electron transport with gas-exchange The electron transport rates can now be linked directly to the net CO2 exchange rates using the relationship: Vo/Vc = 1 S · O C = Kc · ko ·O kc ·Ko ·C (6) where O and C are the pO2 and pCO2 in the chloroplast (bar), S is the specificity of Rubisco for carboxylation relative to oxygenation (mol CO2 mol-1 O2), kc and ko are the catalytic constants of Rubisco for CO2 and O2 (mol CO2 or O2 mol-1 sites s-1, and Kc and Ko are the Michaelis-Menten constants for CO2 and O2 (bar). For the C3 and Type I C3-C4 mesophyll, combining Eqns. 1a, 1b, 5, and 6 gives: JP680m = JP700m 1− nL nC + [ 3+7 ·Om/(2 ·S ·Cm) (4+4 ·Om/(S ·Cm)) ·nC ] (7a) Am = JP680m · [1−Om/(2 ·S ·Cm)] 4 · [1+Om/(S ·Cm)] −Rm (7b) where Om and Cm are the mesophyll pO2 and pCO2 (bar). For the Type I C3-C4 bundle sheath, the analogous expressions are derived by combining Eqns. 1c, 1d, 5, and 6: JP680s = JP700s 1− nL nC + [ 3+7 ·Os/(2 ·S ·Cs) (4+4 ·Os/(S ·Cs)) ·nC ] (8a) As = JP680s · [1−Os/(2 ·S ·Cs)] 4 · [1+Os/(S ·Cs)] −Rs (8b) where Os and Cs are the bundle sheath pO2 and pCO2 (bar). For the C4 mesophyll, the expressions for linear electron flow through PS II and the net CO2 assimilation reduce to: JP680m = JP700m 1− nL nC + 1 nC (9a)


Steady-state electron transport
We begin by describing a metabolic state in which the supplies of Fd, NADPH, and ATP from chloroplast electron transport are dynamically coordinated with the energetic demands from carbon metabolism. For the C 3 and the Type I C 3 -C 4 mesophyll, the rates of electron transport through PS II and PS I depend on the activity of the photosynthetic carbon reduction (PCR) and photosynthetic carbon oxidation (PCO) cycles: where J P680m is the total rate of linear electron flow (LEF) through mesophyll PS II (mol em -2 s -1 ), J P700m is the total rate of LEF and cyclic electron flow (CEF1) through mesophyll PS I (mol em -2 s -1 ), V cm and V om are the carboxylation and oxygenation rates of mesophyll Rubisco (mol CO 2 or O 2 m -2 s -1 ), and n L and n C are the ATP coupling efficiencies of LEF and CEF1, respectively (mol ATP mol -1 e -). For the Type I C 3 -C 4 bundle sheath, the analogous expressions are given by: J P680s = 4 ·V cs + 4 ·V os (1c) J P700s = (3 ·V cs + 3.5 ·V os − J P680s · n L )/n C + J P680s (1d) where J P680s is the total rate of LEF through bundle sheath PS II (mol em -2 s -1 ), J P700s is the total rate of LEF and CEF1 through bundle sheath PS I (mol em -2 s -1 ), and V cs and V os are the carboxylation and oxygenation rates of bundle sheath Rubisco (mol CO 2 or O 2 m -2 s -1 ). For the C 4 mesophyll, the corresponding expressions are: where V pm is the mesophyll activity of PEPC (mol CO 2 m -2 s -1 ). In Eqns. 2a-2b, the partitioning between LEF and CEF1 in the mesophyll reflects the requirement for 2 ATP and 1 NADPH to drive the NADP-ME type C 4 cycle. For the C 4 bundle sheath, the corresponding expressions are: where V ds is the bundle sheath activity of NADP-ME (mol CO 2 m -2 s -1 ). In Eqns. 2c-2d, the partitioning between LEF and CEF1 in the bundle sheath reflects the energetic requirements to drive the PCR and PCO cycles as well as the partial suppression of LEF due to NADPH production by NADP-ME.

Steady-state gas-exchange
We continue by defining expressions that link the activity of carbon metabolism to gas-exchange. For a C 3 , Type I C 3 -C 4 , or C 4 leaf, a general expression for the net assimilation rate (A) is given by the sum of net CO 2 assimilation in the mesophyll (A m ) and the bundle sheath (A s ): where A, A m , and A s can have units of mol assimilated C m -2 s -1 or mol CO 2 m -2 s -1 . In Eqn. 3c, V g is the net rate at which the glycine shuttle transfers assimilated C from the mesophyll into the bundle sheath (mol C m -2 s -1 ), V p is the net rate at which the malate shuttle transfers assimilated C from the mesophyll into the bundle sheath (mol C m -2 s -1 ), and L is the rate at which CO 2 leaks out of the bundle sheath via diffusion (mol CO 2 m -2 s -1 ). The rate of the glycine shuttle is determined by mesophyll Rubisco oxygenase activity that is coupled to bundle sheath GDC activity, and the rate of the malate shuttle is based on mesophyll PEPC activity that is coupled to bundle sheath NADP-ME activity: where V gm is the rate of GDC activity in the mesophyll (mol CO 2 m -2 s -1 ), V pm is the rate of carboxylation via mesophyll PEPC (mol CO 2 m -2 s -1 ) and V dm is mesophyll C 4decarboxylase activity that results in futile cycling (mol CO 2 m -2 s -1 ). Combining Eqns. 3-4 then yields a second expres-sion for the overall net CO 2 assimilation rate (A): which can represent photosynthesis in C 3 , any of the various C 3 -C 4 , or C 4 plants. If all of the Rubisco is localized to the mesophyll, this expression reduces to the regular C 3 expression described by Farquhar et al. (1980). If some of the Rubisco and all of the GDC are localized to the bundle sheath, the reduced expression is equivalent to the Type I C 3 -C 4 expression described by von Caemmerer (1989). If all of the Rubisco is localized to the bundle sheath and there is no futile C 4 cycling, the reduced expression is equivalent to the C 4 expression described by Berry and Farquhar (1978). In the dark, the leak rate L is equivalent to the rate of bundle sheath respiration, R s .

Linking electron transport with gas-exchange
The electron transport rates can now be linked directly to the net CO 2 exchange rates using the relationship: where O and C are the pO 2 and pCO 2 in the chloroplast (bar), S is the specificity of Rubisco for carboxylation relative to oxygenation (mol CO 2 mol -1 O 2 ), k c and k o are the catalytic constants of Rubisco for CO 2 and O 2 (mol CO 2 or O 2 mol -1 sites s -1 , and K c and K o are the Michaelis-Menten constants for CO 2 and O 2 (bar). For the C 3 and Type I C 3 -C 4 mesophyll, combining Eqns. 1a, 1b, 5, and 6 gives: where O m and C m are the mesophyll pO 2 and pCO 2 (bar).
For the Type I C 3 -C 4 bundle sheath, the analogous expressions are derived by combining Eqns. 1c, 1d, 5, and 6: where O s and C s are the bundle sheath pO 2 and pCO 2 (bar).
For the C 4 mesophyll, the expressions for linear electron flow through PS II and the net CO 2 assimilation reduce to: which are derived from Eqns. 2a, 2b, and 5. For the C 4 bundle sheath, the corresponding expressions for the linear electron flow through PS II and the net CO 2 assimilation expand to: which are derived from Eqns. 2c, 2d, 5, and 6. Under different environmental conditions, Eqns. 7-10 can be solved using the expressions in Appendices II, III, IV, and V.
Appendix II: Pure light-limited state

Expressions for net CO 2 assimilation (A mj and A sj )
In a pure light-limited state, the net CO 2 assimilation rates of the mesophyll and bundle sheath (A mj , A sj ) are kinetically controlled by the maximum activity of Cyt b 6 f and the excitation balance of PS I and PS II. For the C 3 , Type I C 3 -C 4 , and C 4 cases, the light-limited rates of PS I electron transport in the mesophyll and bundle sheath (J P700mj , J P700sj ) are given by: where Q is the photosynthetically active radiation incident Expressions for bundle sheath CO 2 (C s ) For the Type I C 3 -C 4 and C 4 cases, the steady-state CO 2 concentration of the bundle sheath is a balance between CO 2 supply (a function of glycine or malate shuttling) and demand (a function of net assimilation and diffusive leakage).
However, the supply of CO 2 to the bundle sheath depends on the state of the mesophyll. For the Type I C 3 -C 4 case, the light-limited rate of glycine shuttling is given by: which is dependent on the energy supply in the mesophyll.
For the C 4 case, the light-limited rate of malate shuttling is given by: which is also dependent on the energy supply in the mesophyll. On the demand side, the rate of diffusive leakage (L) is given by: where g bs is the conductance of the bundle sheath cell wall to CO 2 (mol CO 2 m -2 s -1 ), C s -C m is the CO 2 partial pressure gradient between the bundle sheath and the mesophyll (bar), and P tot is the total atmospheric pressure (bar). Writing a conservation equation and rearranging yields: where C s is represented as a function of A sj .

Expressions for bundle sheath O 2 (O s )
Just as for CO 2 , the steady-state O 2 concentration of the bundle sheath is a balance between O 2 supply (a function of splitting of water at PS II) and demand (a function of Rubisco oxygenase activity, respiration, and diffusive leakage).
For the Type I C 3 -C 4 case, we specify that net O 2 evolution is proportional to net CO 2 assimilation (A s ). For the C 4 case, we specify that net O 2 evolution is proportional to net CO 2 assimilation less the O 2 evolution that is suppressed by NADP-ME activity (A s −V p /2). For the diffusive leak, L o is given by: where g o is the conductance of the bundle sheath cell wall to O 2 (mol O 2 m -2 s -1 ), and O s -O m is the O 2 partial pressure gradient between the bundle sheath and the mesophyll (bar). The conductance of the bundle sheath cell wall to O 2 is linked to the conductance of the bundle sheath cell wall to CO 2 by: where O s is represented as a function of A sj .
Appendix III: Pure light-saturated state

Expressions for net CO 2 assimilation (A mc and A sc )
In a pure light-saturated state, the CO 2 assimilation rates of the mesophyll and bundle sheath (A mc , A sc ) are kinetically controlled by the maximum activity of Rubisco and/or https://doi.org/10.1007/s00442-021-05062-y 5 PEPC. For A mc and A sc , the rate equations are given by: where V mmax and V smax are the maximum carboxylase activities of Rubisco for the mesophyll and bundle sheath (mol CO 2 m -2 s -1 ). For the C 3 and Type I C 3 -C 4 mesophyll, Eqn.
17a can be solved for A mc directly from O m and C m , and then substituted into Eqn. 7a-7b to calculate the corresponding rates of electron flow through mesophyll PS II and PS I (J P680mc , J P700mc ). For the Type I C 3 -C 4 and C 4 bundle sheath, Eqn. 17b can be solved for A sc from O s and C s , and then substituted into Eqn. 8a-8b and Eqn. 10a-10b to calculate the corresponding rates of electron flow through bundle sheath PS II and PS I (J P680sc , J P700sc ). Expressions for calculation of C s and O s are given below.

Expressions for bundle sheath CO 2 and O 2 (C s and O s )
As before, the CO 2 supply depends on the state of the mesophyll. For the Type I C 3 -C 4 case, the light-saturated rate of glycine shuttling (V gc ) is given by: where all of the terms have been defined previously. For the C 4 case, the light-saturated rate of malate shuttling (V pc ) is given by: where V pmax is the maximum mesophyll activity of PEPC (mol CO 2 m -2 s -1 ), and K p is the Michaelis-Menten constant of PEPC for CO 2 (bar).
The expression for the partial pressure of CO 2 in the bundle sheath is then given by: where C s is represented as a function of A sc . The corresponding expression for the partial pressure of O 2 in the bundle sheath is given by: where O s is also represented as a function of A sc .
Appendix IV: Mixed light-limited/light-saturated states For the Type I C 3 -C 4 and C 4 cases, we also define two 'mixed states': one in which the mesophyll is light-limited while the bundle sheath is light-saturated, and the other in which the mesophyll is light-saturated while the bundle sheath is light-limited. In the first mixed state, the CO 2 supply to the bundle sheath is controlled by the light-limited rates of glycine shuttling (V gj ) or malate shuttling (V pj ) while the rate of net CO 2 assimilation in the bundle sheath is controlled by the light-saturated rate of Rubisco activity (A sc ).
In this state, the partial pressures of CO 2 and O 2 in the bundle sheath are given by: which are solved with substitution from Eqns. 12a, 12b, and 17b. In the second mixed state, the CO 2 supply to the bundle sheath is controlled by the light-saturated rates of glycine shuttling (V gc ) or malate shuttling (V pc ) while the rate of net CO 2 assimilation in the bundle sheath is controlled by the light-limited rate of Rubisco activity (A sj ). In this state, the partial pressures of CO 2 and O 2 in the bundle sheath are given by: which are solved with substitution from Eqns. 8a, 8b, 11b, and 18a for the Type I C 3 -C 4 case and with substitution from Eqns. 10a, 10b, 11b, and 18b for the C 4 case.

Appendix V: Solution for pure and mixed states
Once the potential limiting states are defined, the model is solved in three steps. In the first step, the realized net CO 2 where min{} denotes the minimum of the potential lightlimited (A mj ) and light-saturated (A mc ) rates. In the second step, the realized net CO 2 assimilation rate of the bundle sheath is calculated. We first calculate the potential lightlimited net CO 2 assimilation rate using: to account for the pure state where the mesophyll and bundle sheath are both light-limited as well as the mixed state where the mesophyll is light-saturated and the bundle sheath is light-limited. We then calculate the potential light-saturated net CO 2 assimilation rate using: to account for the mixed state where the mesophyll is lightlimited and the bundle sheath is light-saturated as well as the pure state where the mesophyll and bundle sheath are both light-saturated. The realized net CO 2 assimilation rate of the bundle sheath is then given by: where min{} denotes the minimum of the potential lightlimited (A sj ) and light-saturated (A sc ) rates. In the third step, the mesophyll and bundle sheath rates from Eqns. 23 and 24c are combined: which gives the overall rate of net CO 2 assimilation.
Once the limiting states have been identified, the expressions in Johnson and Berry (2021) can be used to derive many other aspects of the internal state of the electron transport system (e.g., the photochemical yields of PS II and PS I, the redox poise of the mobile electron carrier pools in the intersystem chain, the level of non-photochemical quenching of PS II, and the state of photosynthetic control of Cyt b 6 f). These internal states then provide a basis for simulating observable fluorescence-and absorbance-based signals. Example calculations are provided in the model code. Note that for simplicity and clarity, all of these calculations assume that: (i) LEF is always accompanied by the amount of CEF1 required to balance the energy supply with the demands of carbon metabolism; and (ii) the absorption cross-sections of PS II and PS I are static rather than dynamic. See Johnson and Berry (2021) for discussion of alternatives.