Critical review: incorporating the arrangement of mitochondria and chloroplasts into models of photosynthesis and carbon isotope discrimination

Abstract

The arrangement of mitochondria and chloroplasts, together with the relative resistances of cell wall and chloroplast, determine the path of diffusion out of the leaf for (photo)respired CO₂. Traditional photosynthesis models have assumed a tight arrangement of chloroplasts packed together against the cell wall with mitochondria located behind the chloroplasts, deep inside the cytosol. Accordingly, all (photo)respired CO₂ must cross the chloroplast before diffusing out of the leaf. Different arrangements have recently been considered, where all or part of the (photo)respired CO₂ diffuses through the cytosol without ever entering the chloroplast. Assumptions about the path for the (photo)respiratory flux are particularly relevant for the calculation of mesophyll conductance (g_m). If (photo)respired CO₂ can diffuse elsewhere besides the chloroplast, apparent g_m is no longer a mere physical resistance but a flux-weighted variable sensitive to the ratio of (photo)respiration to net CO₂ assimilation. We discuss existing photosynthesis models in conjunction with their treatment of the (photo)respiratory flux and present new equations applicable to the generalized case where (photo)respired CO₂ can diffuse both into the chloroplast and through the cytosol. Additionally, we present a new generalized Δ¹³C model that incorporates this dual diffusion pathway. We assess how assumptions about the fate of (photo)respired CO₂ affect the interpretation of photosynthetic data and the challenges it poses for the application of different models.

Appendices

Appendix 1: generalized C₃ photosynthesis model for the dual diffusion pathway

Quadratic equations to calculate C _c in function of C _m for 0 < γ ≤ 1

Photorespiration rate can be calculated (Farquhar et al. 1980):

$$F=\frac{{{\Gamma ^*}}}{{{C_{\text{c}}}}}{V_{\text{c}}},$$

(33)

where Γ* (Pa) is the CO₂ compensation point in the absence of mitochondrial respiration, C_c is the pCO₂ in the chloroplast (Pa) and V_c (µmol CO₂ m⁻² s⁻¹), is the rate of carboxylation by rubisco calculated with Eq. 2. Substituting V_c in Eq. 2 for $\frac{{F{C_{\text{c}}}}}{{{\Gamma ^*}}}$ (solved from Eq. 33) results in the following expressions for F:

$$F=\left\{ {\begin{array}{*{20}{l}} {({\text{a}})~F=\frac{{{\Gamma ^*}{V_{{\text{cmax}}}}}}{{{C_{\text{c}}}+{K_{\text{c}}}\left( {1+\frac{O}{{{K_{\text{o}}}}}} \right)}},~~{\text{RuBP-saturated}}} \\ {({\text{b}})~F=\frac{{{\Gamma ^*}J}}{{4{C_{\text{c}}}+8{\Gamma ^*}}},~~{\text{electron~transport~limited}}} \end{array}} \right.,$$

(34)

where V_cmax is the maximal rubisco carboxylation rate (µmol CO₂ m⁻² s⁻¹), K_c and K_o (Pa) are the Michaelis–Menten constants of rubisco for CO₂ and O₂, respectively, and J (µmol electrons m⁻² s⁻¹) is the potential electron transport rate.

From Eq. 12, the photosynthetic rate, A (µmol CO₂ m⁻² s⁻¹) can be solved as:

$$A=\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{r_{\text{c}}}}} - \gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right)$$

(35)

where C_m (Pa) is mesophyll pCO₂, r_c is chloroplast resistance (m² s Pa µmol⁻¹) and ${\mathcal{R}_{\text{d}}}$ (µmol CO₂ m⁻² s⁻¹) is the mitochondrial respiration in the light.

Substituting F in Eq. 35 for its calculation given in Eq. 34, and combining the resulting formulations with the FvCB calculation for the RuBP-saturated (A_c) and electron transport limited (A_j) photosynthetic rates presented in Eqs. 4 and 5, results in the following quadratic expressions to calculate C_c for a given C_m for any value of 0 < γ ≤ 1:

RuBP-saturated

$$C_{{\text{c}}}^{2}+{C_{\text{c}}}\left( { - {C_{\text{m}}}+{r_{\text{c}}}{V_{{\text{cmax}}}}+{K_{\text{c}}}\left( {1+\frac{O}{{{K_{\text{o}}}}}} \right) - {\mathcal{R}_{\text{d}}}{r_{\text{c}}}\left( {1 - \gamma } \right)} \right) - {C_{\text{m}}}{K_{\text{c}}}\left( {1+\frac{O}{{{K_{\text{o}}}}}} \right) - {K_{\text{c}}}\left( {1+\frac{O}{{{K_{\text{o}}}}}} \right){\mathcal{R}_{\text{d}}}{r_{\text{c}}}\left( {1 - \gamma } \right) - {r_{\text{c}}}{V_{{\text{cmax}}}}{{{\Gamma}}^*}\left( {1 - \gamma } \right)=0$$

(36)

RuBP-limited

$$C_{{\text{c}}}^{2}+{C_{\text{c}}}\left( { - {C_{\text{m}}}+\frac{{{r_{\text{c}}}J}}{4}+2{\Gamma ^*} - {\mathcal{R}_{\text{d}}}{r_{\text{c}}}\left( {1 - \gamma } \right)} \right) - {C_{\text{m}}}2{\Gamma ^*} - 2{\mathcal{R}_{\text{d}}}{r_{\text{c}}}{\Gamma ^*}\left( {1 - \gamma } \right) - \frac{{{\Gamma ^*}{r_{\text{c}}}J\left( {1 - \gamma } \right)}}{4}=0$$

(37)

When γ = 1, Eqs. 36 and 37 are identical to Eqs. A28 and A29 in von Caemmerer (2013), respectively.

Calculation of Γ_c, Γ, $C_{{\text{m}}}^{*}$, and $C_{{\text{i}}}^{*}$

${\Gamma _{\text{c}}}$ (pCO₂ in the chloroplast at which A = 0) is obtained from setting A = 0 in Eqs. 4 and 5 and solving for C_c (Eqs. A30 and A32 in von Caemmerer 2013):

$${\Gamma _c}=\left\{ {\begin{array}{*{20}{l}} {({\text{a}})~{\Gamma _{\text{c}}}=\frac{{{\Gamma ^*}+\frac{{{\mathcal{R}_{\text{d}}}}}{{{V_{{\text{cmax}}}}}}{K_{\text{c}}}\left( {1+\frac{O}{{{K_{\text{o}}}}}} \right)}}{{1 - ~\frac{{{\mathcal{R}_{\text{d}}}}}{{{V_{{\text{cmax}}}}}}}},~~{\text{RuBP-saturated}}~} \\ {({\text{b}})~{\Gamma _{\text{c}}}=\frac{{{\Gamma ^*}\left( {J+8{\mathcal{R}_{\text{d}}}} \right)}}{{J - 4{\mathcal{R}_{\text{d}}}}},~~{\text{electron~transport~limited}}} \end{array}} \right.$$

(38)

Γ (pCO₂ in the leaf intercellular spaces at which A = 0) is calculated by solving for C_m in Eq. 35 setting A = 0 and C_c = Γ_c, which results in ${C_{\text{m}}}={\Gamma _{\text{c}}}+\gamma {r_{\text{c}}}{\mathcal{R}_{\text{d}}}+\gamma {r_{\text{c}}}F$. In this expression F can be substituted by its calculation given in Eq. 34, which results in:

$$\Gamma =\left\{ {\begin{array}{*{20}{l}} {({\text{a}})~\Gamma ={{{{\Gamma}}}_{\text{c}}}+\gamma {r_{\text{c}}}{\mathcal{R}_{\text{d}}}+\gamma {r_{\text{c}}}\frac{{{\Gamma ^*}{V_{{\text{cmax}}}}}}{{{\Gamma _{\text{c}}}+{K_{\text{c}}}\left( {1+\frac{O}{{{K_{\text{o}}}}}} \right)}},~~{\text{RuBP-saturated}}~} \\ {({\text{b}})~\Gamma ={\Gamma _{\text{c}}}+\gamma {r_{\text{c}}}{\mathcal{R}_{\text{d}}}+\gamma {r_{\text{c}}}\frac{{{\Gamma ^*}J}}{{4{\Gamma _{\text{c}}}+8{\Gamma ^*}}},~~{\text{electron~transport~limited}}} \end{array}} \right.$$

(39)

When γ = 1, Eq. 39 is the same as Eqs. A31 and A33 in von Caemmerer (2013), with the exception that there was a typographical error in her Eq. A31 that we have corrected here.

$C_{{\text{m}}}^{*}$ (pCO₂ in the mesophyll cytosol at which ${\text{A=}} - {\mathcal{R}_{\text{d}}}$) is derived by solving C_m from Eq. 35, assuming that ${\text{A=}} - {\mathcal{R}_{\text{d}}}$ and C_c = Γ*, which results in $C_{{\text{m}}}^{*}=~{\Gamma ^*}+\gamma {r_{\text{c}}}F+{\mathcal{R}_{\text{d}}}{r_{\text{c}}}\left( {\gamma - 1} \right)$. Then substituting F for its expression given in Eq. 34 results in:

$$C_{\text{m}}^{*}=\left\{ {\begin{array}{*{20}{c}} {({\text{a}})~C_{{\text{m}}}^{*}={\Gamma ^*}+\gamma {r_{\text{c}}}\frac{{{\Gamma ^*}{V_{{\text{cmax}}}}}}{{{\Gamma ^*}+{K_{\text{c}}}\left( {1+\frac{O}{{{K_{\text{o}}}}}} \right)}} - {\mathcal{R}_{\text{d}}}{r_{\text{c}}}\left( {1 - \gamma } \right),~~{\text{RuBP-saturated}}~} \\ {({\text{b}})~C_{{\text{m}}}^{*}={\Gamma ^*}+\gamma {r_{\text{c}}}\frac{J}{{12}} - {\mathcal{R}_{\text{d}}}{r_{\text{c}}}\left( {1 - \gamma } \right),~~{\text{electron~transport~limited}}} \end{array}} \right.$$

(40)

When γ = 1, Eq. 40 is identical to Eqs. A35 and A38 in von Caemmerer (2013).

For γ > 0, $C_{{\text{i}}}^{*}$ (pCO₂ in the leaf intercellular spaces at which ${\text{A=}} - {\mathcal{R}_{\text{d}}}$) is derived by solving C_i from Eq. 11$\left( {{C_{\text{i}}}={C_{\text{m}}}+{r_{\text{w}}}A} \right)$, setting ${\text{A=}} - {\mathcal{R}_{\text{d}}}$, and ${C_{\text{m}}}=C_{{\text{m}}}^{*}$:

$$C_{\text{i}}^{*}=C_{\text{m}}^{*} - {\mathcal{R}_{\text{d}}}{r_{\text{w}}},$$

(41)

with different calculations for the RuBP-saturated and electron transport limited cases given by the different expressions for $C_{{\text{m}}}^{*}$ (Eq. 40). When γ = 1, Eq. 41 is identical to Eqs. A34 and A37 in von Caemmerer (2013).

When γ = 0, $C_{{\text{i}}}^{*}$ is derived by solving C_i from Eq. 8$\left( {{C_{\text{i}}}={C_{\text{m}}}+{r_{\text{m}}}A} \right)$, setting ${\text{A=}} - {\mathcal{R}_{\text{d}}}$, and ${C_{\text{m}}}=C_{\text{m}}^{*}={\Gamma ^*}$, which results in:

$$C_{{\text{i}}}^{*}={\Gamma ^*} - {\mathcal{R}_{\text{d}}}{r_{\text{m}}}$$

(42)

Correspondence of our model with Yin and Struik (2017) two-resistance model

For the comparison, we use the most comprehensive case scenario (Case 4, Fig. 1b) in Yin and Struik (2017), where the coverage of chloroplasts is discontinuous and mitochondria are located both in the inner and outer layers of cytosol. In this case, Yin and Struik (2017) stablished the following mass balance equation (Eq. 12 in Yin and Struik 2017) :

$${C_{\text{m}}} - {C_{\text{c}}}={r_{\text{c}}}\left[ {{V_{\text{c}}} - k\lambda ({\mathcal{R}_{\text{d}}}+F)} \right]$$

(43)

Because ${V_{\text{c}}}=A+~{\mathcal{R}_{\text{d}}}+F$, and σ = kλ, Eq. 43 can be re-written as:

$${C_{\text{m}}} - {C_{\text{c}}}={r_{\text{c}}}\left[ {A+~{\mathcal{R}_{\text{d}}}+{F_{\text{c}}} - \sigma \left( {{\mathcal{R}_{\text{d}}}+F} \right)} \right]={r_{\text{c}}}\left[ {A+\left( {{\mathcal{R}_{\text{d}}}+F} \right)\left( {1 - \sigma } \right)} \right],$$

(44)

and solving ${r_{\text{c}}}$ from Eq. 44 results in:

$${r_{\text{c}}}=\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{A+(1 - \sigma )\left( {{\mathcal{R}_{\text{d}}}+F} \right)}}$$

(45)

Equation 45 is identical to Eq. 12 with the equivalence $\gamma =1 - \sigma$. From Eq. 10${g_{\text{m}}}$ can be calculated as:

$${g_{\text{m}}}=\frac{1}{{{r_{\text{w}}}+{r_{\text{c}}}\left( {1+\gamma \frac{{{R_{\text{d}}}+F}}{A}} \right)}}$$

(46)

Defining ${r_{\text{w}}}+{r_{\text{c}}}={r_{{\text{m,diff}}}},$${r_{\text{c}}}=\omega {r_{{\text{m,diff}}}}$ and ${r_{\text{w}}}=(1 - \omega ){r_{{\text{m,diff}}}}$ (Yin and Struik 2017), Eq. 46 results in Eq. 15, which is identical to Eq. 15 in Yin and Struik (2017) with the equivalence $\gamma =1 - \sigma$.

Appendix 2: generalized Δ¹³C model for the dual diffusion pathway

Δ¹³C model for 0 ≤ γ ≤ 1

Expression for the carbon isotope ratio of assimilation rate $\left( {{R_{\text{A}}}} \right)$

From Eq. 11, the rates of assimilation of ¹²CO₂ (A) and ¹³CO₂ (A′) are:

$$A=\frac{{{C_i} - {C_{\text{m}}}}}{{{r_{\text{w}}}}}$$

(47)

$$A^{\prime}=\frac{{{R_{\text{i}}}{C_{\text{i}}} - {R_{\text{m}}}{C_{\text{m}}}}}{{{r_{\text{w}}}{\alpha _{\text{m}}}}}$$

(48)

where ${C_{\text{i}}}$ (Pa) and ${C_{\text{m}}}$ (Pa) are the pCO₂ inside the leaf intercellular spaces, and the mesophyll cytosol, respectively. The ${r_{\text{w}}}$ (Pa m² s µmol⁻¹) is the wall resistance to CO₂ diffusion. ${R_{\text{i}}}$ and ${R_{\text{m}}}$ are the C-isotope ratios of CO₂ inside the leaf intercellular spaces, and in the mesophyll cytosol, respectively. The ${\alpha _{\text{m}}}=1+{a_{\text{m}}}$, where a_m (= e_s + a₁ = 1.8‰), is the ¹²C/¹³C fractionation associated with dissolution of CO₂ (e_s = 1.1‰) and its subsequent diffusion (a₁ = 0.7‰).

The R_A is calculated as:

$${R_{\text{A}}}=\frac{{A^{\prime}}}{A}=\frac{{{R_{\text{i}}}{C_{\text{i}}} - {R_{\text{m}}}{C_{\text{m}}}}}{{{\alpha _{\text{m}}}({C_{\text{i}}} - {C_{\text{m}}})}}.$$

(49)

The term ${R_{\text{m}}}{C_{\text{m}}}$ can be solved from Eq. 49 as:

$${R_{\text{m}}}{C_{\text{m}}}={R_{\text{i}}}{C_{\text{i}}} - {R_{\text{A}}}{\alpha _{\text{m}}}\left( {{C_{\text{i}}} - {C_{\text{m}}}} \right).$$

(50)

The ternary corrected equation relating R_A to ${R_{\text{a}}}$ (isotope ratio of the CO₂ in the ambient air) and ${R_{\text{i}}}$ is (Eq. 6 in “Appendix 2” section in Farquhar and Cernusak 2012):

$${R_{\text{A}}}=\frac{{{R_{\text{a}}}{C_{\text{a}}} - {R_{\text{i}}}{C_{\text{i}}} - t({R_{\text{a}}}{C_{\text{a}}}+{R_{\text{i}}}{C_{\text{i}}})}}{{{\alpha _{{\text{ac}}}}\left( {{C_{\text{a}}} - {C_{\text{i}}}} \right) - t\left( {{C_{\text{a}}}+{C_{\text{i}}}} \right)}},$$

(51)

where $C_{\text{a}}$ (Pa) is the pCO₂ mole fraction in ambient air, t is the ternary correction factor:

$$t=\frac{{{\alpha _{{\text{ac}}}}E}}{{2{g_{{\text{ac}}}}}},$$

(52)

where E (mol m⁻² s⁻¹) is transpiration rate, ${g_{{\text{ac}}}}$ (mol m⁻² s⁻¹) is the combined boundary layer and stomatal conductance to CO₂, and ${\alpha _{{\text{ac}}}}$ is the fractionation factor for the isotopologues of CO₂ diffusing in air, ${\alpha _{{\text{ac}}}}=1+\bar {a}$, where $\bar {a}$ is the weighted fractionation across the boundary layer and stomata in series:

$$\bar {a}=\frac{{{a_{\text{b}}}\left( {{C_{\text{a}}} - {C_{\text{s}}}} \right)+{a_{\text{s}}}\left( {{C_{\text{s}}} - {C_{\text{i}}}} \right)}}{{{C_{\text{a}}} - {C_{\text{i}}}}},$$

(53)

where C_s (Pa) is the pCO₂ at the leaf surface, and a_b and a_s are the ¹²C/¹³C fractionations for CO₂ diffusion in the boundary layer (2.9‰) and in air (4.4‰), respectively.

Expression for the carbon isotope ratio of CO₂ at the sites of carboxylation $\left( {{R_{\text{c}}}} \right)$

A relationship for the situation whether CO₂ enters the stroma from the front, sides or back of the chloroplasts is presented in Farquhar and Cernusak (2012):

$${R_{\text{c}}}={R_{\text{A}}}\left( {{\alpha _{\text{b}}} - \frac{{{\alpha _{\text{b}}}{\Gamma ^*}f}}{{{\alpha _{\text{f}}}{C_{\text{c}}}}} - \frac{{{\alpha _{\text{b}}}{\mathcal{R}_{\text{d}}}e^{\prime}}}{{{\alpha _{\text{e}}}{V_{\text{c}}}}}} \right)$$

(54)

where ${\alpha _{\text{b}}}=1+{b^{\prime}_3}$, ${\alpha _{\text{f}}}=1+f$, and ${\alpha _{\text{e}}}=1+e^{\prime}$, with ${b^{\prime}_3}$, $e^{\prime}$ and f (‰) being the ¹²C/¹³C fractionations during carboxylation by rubisco, decarboxylation and photorespiration, respectively. The ${{\text{C}}_\text{c}}$ (Pa) is the pCO₂ at the sites of carboxylation in the chloroplast, Γ* (Pa) is the CO₂ compensation point in absence of mitochondrial respiration, and V_c (µmol m⁻² s⁻¹) is the rubisco carboxylation rate.

Write Eq. 12 in ratios

Equation 12 links conditions in the cytosol with those in the chloroplast stroma and its analog for ¹³CO₂ is:

$${R_{\text{m}}}{C_{\text{m}}} - {R_{\text{c}}}{C_{\text{c}}}=\left( {{R_{\text{A}}}A+\gamma \left( {{{\mathcal{R}^{\prime}}_{\text{d}}}+F^{\prime}} \right)} \right) \, {r_{\text{c}}}{\alpha _{\text{c}}},$$

(55)

where ${\alpha _{\text{c}}}=1+{a_{\text{c}}}$ and ${a_{\text{c}}}$ is the ¹²C/¹³C fractionation that occurs as CO₂ travels from the cytosol to the sites of carboxylation in the chloroplast. We used a_c = 1.8‰, and a detailed justification for this value is presented at the end of this subsection. The ${\mathcal{R}^{\prime}_{\text{d}}}=\frac{{{\mathcal{R}_{\text{d}}}{R_{\text{A}}}}}{{{\alpha _{\text{e}}}}}$ and $F^{\prime}=\frac{{F{R_{\text{A}}}}}{{{\alpha _{\text{f}}}}}$. Because $\frac{1}{{{\alpha _{\text{x}}}}}=1 - \frac{x}{{{\alpha _{\text{x}}}}}$, then the ${\mathcal{R}^{\prime}_{\text{d}}}$ and $F^{\prime}$ can be written as:

$${\mathcal{R}^{\prime}_{\text{d}}}={\mathcal{R}_{\text{d}}}{R_{\text{A}}} - \frac{{{\mathcal{R}_{\text{d}}}{R_{\text{A}}}e^{\prime}}}{{{\alpha _{\text{e}}}}}$$

(56)

$$F^{\prime}=F{R_{\text{A}}} - \frac{{F{R_{\text{A}}}f}}{{{\alpha _{\text{f}}}}}.$$

(57)

Using Eqs. 56 and 57, rearranging terms and substituting ${r_{\text{c}}}$ for $\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{A+\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right)}}$ (Eq. 12), the right-hand term of Eq. 55 can be written as:

$$\left( {{R_{\text{A}}}A+\gamma \left( {{{\mathcal{R}^{\prime}}_{\text{d}}}+F^{\prime}} \right)} \right) \, {r_{\text{c}}}{\alpha _{\text{c}}}={R_{\text{A}}}\left( {{C_{\text{m}}} - {C_{\text{c}}}} \right){\alpha _{\text{c}}} - \gamma \frac{{{\alpha _{\text{c}}}\left( {\frac{{e^{\prime}{\mathcal{R}_{\text{d}}}}}{{{\alpha _{\text{e}}}}}+\frac{{fF}}{{{\alpha _{\text{f}}}}}} \right)}}{{A+\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right)}}{R_{\text{A}}}\left( {{C_{\text{m}}} - {C_{\text{c}}}} \right)$$

(58)

We define:

$${\gamma _{\text{m}}}=\gamma \frac{{{\alpha _{\text{c}}}\left( {\frac{{e^{\prime}{\mathcal{R}_{\text{d}}}}}{{{\alpha _{\text{e}}}}}+\frac{{fF}}{{{\alpha _{\text{f}}}}}} \right)}}{{A+\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right)}}.$$

(59)

In Eq. 59 if F is substituted by $\frac{{{\Gamma ^*}\left( {A+{\mathcal{R}_{\text{d}}}} \right)}}{{{C_{\text{c}}} - {\Gamma ^*}}}$ results in the expression for ${\gamma _{\text{m}}}$ presented in Eq. 20.

Substituting the right-hand term in Eq. 55 by the right-hand term in Eq. 58 results in:

$${R_{\text{m}}}{C_{\text{m}}} - {R_{\text{c}}}{C_{\text{c}}}={R_{\text{A}}}\left( {{C_{\text{m}}} - {C_{\text{c}}}} \right)\left( {{\alpha _{\text{c}}} - {\gamma _{\text{m}}}} \right).$$

(60)

Operate Eq. 60 to derive an expression for ${R_{\text{i}}}{C_{\text{i}}}$

In Eq. 60 substitute ${R_{\text{m}}}{C_{\text{m}}}\;{\text{and}}\;{R_{\text{c}}}$ for their calculations given in Eqs. 50 and 54, respectively:

$${R_{\text{i}}}{C_{\text{i}}} - {R_{\text{A}}}{\alpha _{\text{m}}}\left( {{C_{\text{i}}} - {C_{\text{m}}}} \right) - {R_{\text{A}}}\left( {{\alpha _{\text{b}}} - \frac{{{\alpha _{\text{b}}}{\Gamma ^*}f}}{{{\alpha _{\text{f}}}{C_{\text{c}}}}} - \frac{{{\alpha _{\text{b}}}{\mathcal{R}_{\text{d}}}e^{\prime}}}{{{\alpha _{\text{e}}}{V_{\text{c}}}}}} \right){C_{\text{c}}}={R_{\text{A}}}\left( {{C_{\text{m}}} - {C_{\text{c}}}} \right)\left( {{\alpha _{\text{c}}} - {\gamma _{\text{m}}}} \right),$$

(61)

and solve ${R_{\text{i}}}{C_{\text{i}}}$ from Eq. 61:

$${R_{\text{i}}}{C_{\text{i}}}={R_{\text{A}}}\left[ {{\alpha _{\text{m}}}\left( {{C_{\text{i}}} - {C_{\text{m}}}} \right)+\left( {{\alpha _{\text{c}}} - {\gamma _{\text{m}}}} \right)\left( {{C_{\text{m}}} - {C_{\text{c}}}} \right)+\left( {{\alpha _{\text{b}}} - \frac{{{\alpha _{\text{b}}}{\Gamma ^*}f}}{{{\alpha _{\text{f}}}{C_{\text{c}}}}} - \frac{{{\alpha _{\text{b}}}{\mathcal{R}_{\text{d}}}e^{\prime}}}{{{\alpha _{\text{e}}}{V_{\text{c}}}}}} \right){C_{\text{c}}}} \right]$$

(62)

Derive an expression for ${{{R_{\text{a}}}} \mathord{\left/ {\vphantom {{{R_{\text{a}}}} {{R_{\text{A}}}}}} \right. \kern-0pt} {{R_{\text{A}}}}}$

Substituting in Eq. 51, the term ${R_{\text{i}}}{C_{\text{i}}}$ for its calculation in Eq. 62 results in an equation with only two isotope ratios, ${R_{\text{A}}}$ and ${R_{\text{a}}}$ which separate as:

$${R_{\text{A}}}\left\{ {{\alpha _{{\text{ac}}}}\left( {{C_{\text{a}}} - {C_{\text{i}}}} \right) - t\left( {{C_{\text{a}}}+{C_{\text{i}}}} \right)+\left( {1+t} \right)\left[ {{\alpha _{\text{m}}}\left( {{C_{\text{i}}} - {C_{\text{m}}}} \right)+({\alpha _{\text{c}}} - {\gamma _{\text{m}}})\left( {{C_{\text{m}}} - {C_{\text{c}}}} \right)+{C_{\text{c}}}\left( {{\alpha _{\text{b}}} - \frac{{{\alpha _{\text{b}}}{\Gamma ^*}f}}{{{\alpha _{\text{f}}}{C_{\text{c}}}}} - \frac{{{\alpha _{\text{b}}}{\mathcal{R}_{\text{d}}}e^{\prime}}}{{{\alpha _{\text{e}}}{V_{\text{c}}}}}} \right)} \right]} \right\}={R_{\text{a}}}{C_{\text{a}}}(1 - t).$$

(63)

Equation 63 can be rearranged to give ${{{R_{\text{a}}}} \mathord{\left/ {\vphantom {{{R_{\text{a}}}} {{R_{\text{A}}}}}} \right. \kern-0pt} {{R_{\text{A}}}}}$ as:

$$\frac{{{R_{\text{a}}}}}{{{R_{\text{A}}}}}=\frac{1}{{1 - t}}\frac{{{\alpha _{{\text{ac}}}}\left( {{C_{\text{a}}} - {C_{\text{i}}}} \right)}}{{{C_{\text{a}}}}} - \frac{t}{{1 - t}}\left( {1+\frac{{{C_{\text{i}}}}}{{{C_{\text{a}}}}}} \right)+\frac{{1+t}}{{1 - t}}\frac{{{\alpha _{\text{m}}}\left( {{C_{\text{i}}} - {C_{\text{m}}}} \right)+({\alpha _{\text{c}}} - {\gamma _{\text{m}}})\left( {{C_{\text{m}}} - {C_{\text{c}}}} \right)+{C_{\text{c}}}\left( {{\alpha _{\text{b}}}~ - ~\frac{{{\alpha _{\text{b}}}{\Gamma ^*}f}}{{{\alpha _{\text{f}}}{C_{\text{c}}}}}~ - ~\frac{{{\alpha _{\text{b}}}{\mathcal{R}_{\text{d}}}e^{\prime}}}{{{\alpha _{\text{e}}}{V_{\text{c}}}}}} \right)}}{{{C_{\text{a}}}}}.$$

(64)

Apply the definition of discrimination

By definition $\alpha =1+\Delta ~={{{R_{\text{a}}}} \mathord{\left/ {\vphantom {{{R_a}} {{R_A}}}} \right. \kern-0pt} {{R_\text{A}}}}$. Substituting in Eq. 64${\alpha _{{\text{ac}}}}=1+\bar {a}$, ${\alpha _{\text{m}}}=1+{a_{\text{m}}}$, ${\alpha _{\text{b}}}=1+{b^{\prime}_3}~$ and ${\alpha _{\text{c}}}=1+{a_{\text{c}}}$, and a few lines of rearrangement, yields the formulation for ${\Delta ^{13}}{\text{C}}$ for any value of γ:

$${\Delta ^{13}}{\text{C}}=\frac{{\bar {a}}}{{1 - t}}\frac{{{C_{\text{a}}} - {C_{\text{i}}}}}{{{C_{\text{a}}}}}+\frac{{1+t}}{{1 - t}}\left[ {{a_{\text{m}}}\frac{{{C_{\text{i}}} - {C_{\text{m}}}}}{{{C_{\text{a}}}}}+({a_{\text{c}}} - {\gamma _{\text{m}}})\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{C_{\text{a}}}}}+{b^{\prime}_{3}}\frac{{{C_{\text{c}}}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{{V_{\text{c}}}}}\frac{{{C_{\text{c}}}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{f}}}}}f\frac{{{\Gamma ^*}}}{{{c_{\text{a}}}}}} \right].$$

(65)

When γ = 0, Eq. 65 transforms into the traditional Farquhar and Cernusak (2012) model. An equivalent expression for Eq. 65 can be obtained by replacing $\frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{{V_{\text{c}}}}}\frac{{{C_{\text{c}}}}}{{{C_{\text{a}}}}}$ with the equivalent $\frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{{\text{c}} - }}{\Gamma ^*}}}{{{c_{\text{a}}}}}$ (obtained by substituting ${V_{\text{c}}}=\frac{{\left( {A+{R_{\text{d}}}} \right){C_{\text{c}}}}}{{{C_{\text{c}}} - {\Gamma ^*}}}$), which results in Eq. 19.

The fractionation factor ${a_{\text{c}}}$ is first introduced in Eq. 55 to describe the fractionation experienced by CO₂ as it travels from the cytosol to the sites of carboxylation in the chloroplast. This CO₂ can originate both in the ambient (A) and in the mitochondria $\left( {\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right)} \right)$. As CO₂ travels from the ambient air into the cytosol, it experiences fractionations when it dissolves in water (e_s = 1.1‰) and diffuses through water (a_l = 0.7‰). However, CO₂ evolved from the mitochondria into the cytosol only experiences ${a_{\text{l}}}$ because no dissolution fractionation can occur when all the evolved CO₂ dissolves in water. Subsequently, the flux $A+\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right)$ enters the chloroplast and diffuses to the sites of rubisco. The substrate for rubisco is gas CO₂ (note that the value for rubisco fractionation, ${b^{\prime}_3}$, is valid for a gas-phase context); thus, dissolved CO₂ has to come out of solution to be carboxylated, and at this step there will be a dissolution fractionation. Accordingly, the total fractionation experienced by CO₂ diffusing from the cytosol to the sites of carboxylation is a_c = e_s + a_l = 1.8‰. With this definition, ${a_{\text{c}}}={a_{\text{m}}}$, so we could simply have used ${a_{\text{m}}}$. Nevertheless, we choose to retain ${a_{\text{c}}}$ to conceptually differentiate the diffusion steps from ambient into the mesophyll and from the mesophyll into the chloroplast.

One could derive an equation considering that the flux $\gamma ({\mathcal{R}_{\text{d}}}+F)$ is only affected by ${a_{\text{l}}}$, meanwhile A is affected by both ${a_{\text{l}}}$ and ${e_{\text{s}}}$. In this case Eq. 55 is replaced by ${R_{\text{m}}}{C_{\text{m}}} - {R_{\text{c}}}{C_{\text{c}}}=\left( {{R_{\text{A}}}A+\frac{\gamma }{{{\alpha _{\text{s}}}}}\left( {{{\mathcal{R}^{\prime}}_{\text{d}}}+F^{\prime}} \right)} \right) \, {r_{\text{c}}}{\alpha _{\text{m}}}$, where ${\alpha _{\text{s}}}=1+{e_{\text{s}}}$, ${\alpha _{\text{m}}}=1+{a_{\text{m}}}$, and $\frac{{{\alpha _{\text{m}}}}}{{{\alpha _{\text{s}}}}}={\alpha _{\text{l}}}=\frac{{1.0018}}{{1.0011}}=1.0007$. The resulting Δ¹³C model is identical to that in Eq. 65, with the exception that the term ${a_{\text{c}}} - {\gamma _{\text{m}}}$ is replaced by ${\gamma _{\text{l}}}=\frac{{{a_{\text{m}}}A+{a_{\text{l}}}\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right) - {\alpha _{\text{l}}}\gamma \left( {\frac{{{\mathcal{R}_{\text{d}}}e^{\prime}}}{{{\alpha _{\text{e}}}}}+\frac{{fF}}{{{\alpha _{\text{f}}}}}} \right)}}{{A+\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right)}}$. The difference between ${a_{\text{c}}} - {\gamma _{\text{m}}}$ and ${\gamma _{\text{l}}}$ values is very small (see Supplemental Fig. 1). In turn, this demonstrates that even if we believe that a fractionation ${a_{\text{c}}}$ is more appropriate than ${a_{\text{l}}}$ for the flux $\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right),$ the actual difference between both solutions is almost unnoticeable.

Deriving r _m from Δ¹³C

The predicted Δ¹³C when ${g_{\text{m}}}$ is infinite, Δ_i, is (Farquhar and Cernusak 2012):

$${\Delta _{\text{i}}}=\frac{{\bar {a}}}{{1 - t}}\frac{{{C_{\text{a}}} - {C_{\text{i}}}}}{{{C_a}}}+\frac{{1+t}}{{1 - t}}\left[ {b^{\prime}_{3}\frac{{{C_{\text{i}}}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{\text{i}}} - {\Gamma ^*}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{f}}}}}f\frac{{{\Gamma ^*}}}{{{C_{\text{a}}}}}} \right]$$

(66)

Subtracting Δ¹³C (Eq. 19) from Δ_i (Eq. 66) results in:

$${\Delta _{\text{i}}} - {\Delta ^{13}}{\text{C}}=\frac{{1+t}}{{1 - t}}\left[ {\left( {b^{\prime}_{3} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}} \right)\frac{{{C_{\text{i}}} - {C_{\text{c}}}}}{{{C_{\text{a}}}}} - {a_{\text{m}}}\frac{{{C_{\text{i}}} - {C_{\text{m}}}}}{{{C_{\text{a}}}}} - \left( {{a_{\text{c}}} - {\gamma _{\text{m}}}} \right)\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{C_{\text{a}}}}}} \right].$$

(67)

Equation 67 can be written as a function of resistances as:

$${\Delta _{\text{i}}} - {\Delta ^{13}}{\text{C}}=\frac{{1+t}}{{1 - t}}\left[ {\left( {b^{\prime}_{3}- \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}} \right)\frac{A}{{{C_{\text{a}}}}}{r_{\text{m}}} - {a_{\text{m}}}\frac{A}{{{C_{\text{a}}}}}{r_{\text{w}}} - \left( {{a_{\text{c}}} - {\gamma _{\text{m}}}} \right)\frac{{\left[ {A+\gamma ({R_{\text{d}}}+F)} \right]}}{{{C_{\text{a}}}}}{r_{\text{c}}}} \right]$$

(68)

where ${r_{\text{m}}},\;{r_{\text{w}}}\;{\text{and}}\;{r_{\text{c}}}$ are mesophyll, wall and chloroplast resistances, respectively, in units of µmol m⁻² s⁻¹ Pa⁻¹. The ${r_{\text{m}}}$ can be solved from Eq. 68 as:

$${r_{\text{m}}}=\frac{{\frac{{1 - t}}{{1+t}}\frac{{{C_{\text{a}}}}}{A}\left( {{\Delta _{\text{i}}} - {\Delta ^{13}}{\text{C}}} \right)+{a_{\text{m}}}{r_{\text{w}}}+\left( {{a_{\text{c}}} - {\gamma _{\text{m}}}} \right){r_{\text{c}}}\left[ {1+\frac{{\gamma ({R_{\text{d}}}+F)}}{A}} \right]}}{{{{b^{\prime}_{3}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}}},$$

(69)

and the inverse of Eq. 69 is the calculation of ${g_{{\text{m}}13}}$ provided in Eq. 32. In Eq. 69, the term γ_m depends on C_c (see Eq. 20). Alternatively, C_c can be directly solved from Eq. 19, substituting in this equation γ_m for its calculation given in Eq. 20. This results in a quadratic equation with solution:

$${C_{\text{c}}}=\frac{{ - II+\sqrt {I{I^2} - 4 \times I \times III} }}{{2 \times I}},$$

(70)

where:

$$I=\left( {b^{\prime}_{3} - {a_{\text{c}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}} \right)\left( {A+\gamma {\mathcal{R}_{\text{d}}}} \right)+\gamma \frac{{{\alpha _{\text{c}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}{\mathcal{R}_{\text{d}}}$$

(71)

$$II=\frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_d}}}{{A+{\mathcal{R}_{\text{d}}}}}\left[ {2{\Gamma ^*}A+\gamma {\Gamma ^*}\left( {{\mathcal{R}_{\text{d}}} - A} \right)} \right] - \frac{{{\alpha _{\text{c}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}{\mathcal{R}_{\text{d}}}\gamma \left( {{C_{\text{m}}}+{\Gamma ^*}} \right)+\gamma \frac{{{\alpha _{\text{c}}}}}{{{\alpha _{\text{f}}}}}{\Gamma ^*}f\left( {A+{\mathcal{R}_{\text{d}}}} \right)+\left( {{a_{\text{c}}} - {{b^{\prime}_{3}}}} \right){\Gamma ^*}A\left( {1 - \gamma } \right) - \left( {A+\gamma {\mathcal{R}_{\text{d}}}} \right)\left[ {\frac{{1 - t}}{{1+t}}{\Delta _{{\text{obs}}}}{C_{\text{a}}} - \frac{{\bar {a}}}{{1+t}}\left( {{C_{\text{a}}} - {C_{\text{i}}}} \right) - {a_{\text{m}}}\left( {{C_{\text{i}}} - {C_{\text{m}}}} \right) - {a_{\text{c}}}{C_{\text{m}}}+{\Gamma ^*}f\frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{f}}}}}} \right]$$

(72)

$$III={C_{\text{m}}}\gamma {\Gamma ^*}\left( {\frac{{{\alpha _{\text{c}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}{\mathcal{R}_{\text{d}}} - \frac{{{\alpha _{\text{c}}}}}{{{\alpha _{\text{f}}}}}f(A+{\mathcal{R}_{\text{d}}})} \right)+{\Gamma ^*}A\left( {1 - \gamma } \right)\left[ {\frac{{1 - t}}{{1+t}}{\Delta _{{\text{obs}}}}{C_{\text{a}}} - \frac{{\bar {a}}}{{1+t}}\left( {{C_{\text{a}}} - {C_{\text{i}}}} \right) - {a_{\text{m}}}\left( {{C_{\text{i}}} - {C_{\text{m}}}} \right)+\frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{f}}}}}{\Gamma ^*}f - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}{\Gamma ^*} - {a_{\text{c}}}{C_{\text{m}}}} \right]$$

(73)

where Δ_obs (‰) is the observed ¹³C photosynthetic discrimination.

Partition Δ¹³C in component discriminations

Substituting $\bar {a}=\frac{{{a_{\text{b}}}\left( {{C_{\text{a}}} - {C_{\text{s}}}} \right)+{a_{\text{s}}}\left( {{C_{\text{s}}} - {C_{\text{i}}}} \right)}}{{{C_{\text{a}}} - {C_{\text{i}}}}}$ in Eq. 19 and rearranging terms results in yet another equivalent expression for Δ¹³C:

$${\Delta ^{13}}{\text{C}}=\frac{{1+t}}{{1 - t}}\left[ {b^{\prime}_{3} - \left( {b^{\prime}_{3} - \frac{{{a_{\text{b}}}}}{{1+t}}} \right)\frac{{{C_{\text{a}}} - {C_{\text{s}}}}}{{{C_{\text{a}}}}} - \left( {b^{\prime}_{3} - \frac{{{a_{\text{s}}}}}{{1+t}}} \right)\frac{{{C_{\text{s}}} - {C_{\text{i}}}}}{{{C_{\text{a}}}}} - \left( {b^{\prime}_{3} - {a_{\text{m}}}} \right)\frac{{{C_{\text{i}}} - {C_{\text{m}}}}}{{{C_{\text{a}}}}} - \left( {b^{\prime}_{3} - {a_{\text{c}}}+{\gamma _{\text{m}}}} \right)\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{\text{c}}} - {{{{\Gamma}}}^*}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{f}}}}}f\frac{{{{{{\Gamma}}}^*}}}{{{C_{\text{a}}}}}} \right].$$

(74)

Equation 74 can be written in the notation of Ubierna and Farquhar (2014) as:

$${\Delta ^{13}}{\text{C}}={\Delta _{\text{b}}} - {\Delta _{{\text{rb}}}} - {\Delta _{{\text{rs~}}}} - {\Delta _{{\text{rw}}}} - {\Delta _{{\text{rc}}}} - {\Delta _{\text{e}}} - {\Delta _{\text{f}}},$$

(75)

where the calculation of ${\Delta _{\text{b}}},\;{\Delta _{{\text{rb}}}},\;{\Delta _{{\text{rs}}}}\;{\text{and}}\;{\Delta _{\text{f}}}$ is given in the Eqs. 22–24 and Eq. 28 and:

$${\Delta _{{\text{rw}}}}=\frac{{1+t}}{{1 - t}} \, \left[ {b^{\prime}_{3} - {a_{\text{m}}}} \right]\frac{{{C_{\text{i}}} - {C_{\text{m}}}}}{{{C_{\text{a}}}}}$$

(76)

$${\Delta _{{\text{rc}}}}=\frac{{1+t}}{{1 - t}} \, \left[ {b^{\prime}_{3} - {a_{\text{c}}}+{\gamma _{\text{m}}}} \right]\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{C_{\text{a}}}}}$$

(77)

$${\Delta _{\text{e}}}=\frac{{1+t}}{{1 - t}}\frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{{\text{c}} }}-{\Gamma ^*}}}{{{C_{\text{a}}}}}$$

(78)

The ${\Delta _{\text{r}}}$ terms are so denoted, rather than as the ${\Delta _{\text{g}}}$ terminology used in Ubierna and Farquhar (2014), because their magnitudes are proportional to the magnitudes of the resistances involved. In Eq. 78, if C_c is substituted by ${C_{\text{m}}} - {r_{\text{c}}}\left[ {A+\gamma \left( {{R_{\text{d}}}+F} \right)} \right]$ (Eq. 12), and subsequently C_m is substituted by ${C_{\text{i}}} - {r_{\text{w}}}A$ (Eq. 11), the following result is obtained:

$${\Delta _{\text{e}}}=\frac{{1+t}}{{1 - t}}\left[ {\frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{{\text{i}} }} -{\Gamma ^*}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{r_{\text{c}}}\left[ {A+\gamma \left( {{\mathcal{R}_{\text{d}}}+F} \right)} \right]}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{r_{\text{w}}}A}}{{{C_{\text{a}}}}}} \right].$$

(79)

Because ${C_{\text{m}}} - {C_{\text{c}}}={r_{\text{c}}}\left[ {A+\gamma \left( {{R_{\text{d}}}+F} \right)} \right]$ and ${C_{\text{i}}} - {C_{\text{m}}}={r_{\text{w}}}A$, Eq. 79 can be written as:

$${\Delta _{\text{e}}}=\frac{{1+t}}{{1 - t}}\left[ {\frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{{\text{i}} }}- {\Gamma ^*}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{C_{\text{a}}}}} - \frac{{{\alpha _{\text{b}}}}}{{{\alpha _{\text{e}}}}}e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{\text{i}}} - {C_{\text{m}}}}}{{{C_{\text{a}}}}}} \right].$$

(80)

Now, in Eq. 80 the terms that are function of $\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{C_{\text{a}}}}}$ and $\frac{{{C_{\text{i}}} - {C_{\text{m}}}}}{{{C_{\text{a}}}}}$ can be moved and placed with the component fractionations that are a function of these same gradients, which are ${\Delta _{{\text{rc}}}}$ and ${\Delta _{{\text{rw}}}}$, respectively, which results in the formulations of ${\Delta _{{\text{r}}{{\text{w}}^*}}}$ and $~{\Delta _{{\text{r}}{{\text{c}}^*}}}$ presented in the main text. Thus ${\Delta _{{\text{rw}}}}$, ${\Delta _{{\text{rc}}}}$ and ${\Delta _{\text{e}}}$ are replaced by ${\Delta _{{\text{r}}{{\text{w}}^{\text{*}}}}},~{\Delta _{{\text{r}}{{\text{c}}^{\text{*}}}}},{\text{and}}\;{\Delta _{{{\text{e}}^*}}}$ given in Eqs. 25–27. Note that ${\Delta _{{\text{rw}}}}+{\Delta _{{\text{rc}}}}+{\Delta _{\text{e}}}$ (Eqs. 76–78) is the same as ${\Delta _{{\text{r}}{{\text{w}}^{\text{*}}}}}+{\Delta _{{\text{r}}{{\text{c}}^{\text{*}}}}}+{\Delta _{{{\text{e}}^{\text{*}}}}}$ (Eqs. 25–27), they are just two different approaches of splitting the different components. Also note that in Eq. 77 and Eq. 26${\gamma _{\text{m}}}$ depends on C_c.

Compare Eq. 19 with the expression given in Evans and von Caemmerer (2013)

The expression for ${\Delta ^{13}}{\text{C}}$ derived by Evans and von Caemmerer (2013) for the case γ = 1 is:

$${\Delta ^{13}}{\text{C}}=\frac{{\bar {a}}}{{1 - t}}+\frac{1}{{1 - t}}\left[ {\left( {1+t} \right){b^{\prime}_{3}} - \bar {a}} \right]\frac{{{C_{\text{i}}}}}{{{C_{\text{a}}}}} - \frac{{1+t}}{{1 - t}}\left[ {\left( {b^{\prime}_{3} - {a_{\text{m}}}} \right)\frac{{{r_{\text{w}}}A}}{{{C_{\text{a}}}}}+\frac{{f{{{{\Gamma}}}^*}+\frac{{e^{\prime}{\mathcal{R}_{\text{d}}}}}{k}}}{{{C_a}}}+\left( {\left( {b^{\prime}_{3} - {a_{\text{m}}}} \right)+\frac{{fF+e{\mathcal{R}_{\text{d}}}}}{{{V_{\text{c}}}}}} \right)\frac{{{r_{\text{c}}}{V_{\text{c}}}}}{{{C_{\text{a}}}}}} \right],$$

(81)

where ${r_{\text{w}}}=\frac{{{C_{\text{i}}} - {C_{\text{m}}}}}{A}$ and ${r_{\text{c}}}=\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{V_{\text{c}}}}}$ and k = V_c/C_c.

There are a few terms in Eq. 81 that can be substituted for equivalent expressions:

• 1. The term ${r_{\text{w}}}A$ can be replaced by ${C_{\text{i}}} - {C_{\text{m}}}$ (Eq. 11).

• 2. The term ${r_{\text{c}}}{V_{\text{c}}}$ can be replaced by ${C_{\text{m}}} - {C_{\text{c}}}$.

• 3. The term $\frac{{e^{\prime}{\mathcal{R}_{\text{d}}}}}{k}$ can be replaced by $\frac{{e^{\prime}{\mathcal{R}_{\text{d}}}({C_{\text{c}}} - {\Gamma ^*})}}{{A+{\mathcal{R}_{\text{d}}}}}$.

• 4. The term $\frac{{fF+e^{\prime}{\mathcal{R}_{\text{d}}}}}{{{V_{\text{c}}}}}$ can be replaced by $\frac{{f{\Gamma ^*}}}{{{C_{\text{c}}}}}+\frac{{e^{\prime}{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\left( {1 - \frac{{{\Gamma ^*}}}{{{C_{\text{c}}}}}} \right)$, by substituting $F=\frac{{{\Gamma ^*}\left( {A+{R_{\text{d}}}} \right)}}{{{C_{\text{c}}} - {\Gamma ^*}}}$ and ${V_{\text{c}}}=A+{\mathcal{R}_{\text{d}}}+F$.

With these substitutions, Eq. 81 rearranges to:

$${\Delta ^{13}}{\text{C}}=\frac{{\bar {a}}}{{1 - t}}\frac{{{C_{\text{a}}} - {C_{\text{i}}}}}{{{C_{\text{a}}}}}+\frac{{1+t}}{{1 - t}}\left[ {{a_{\text{m}}}\frac{{{C_i} - {C_{\text{m}}}}}{{{C_{\text{a}}}}}+\left( {{a_{\text{m}}} - \frac{{f{\Gamma ^*}}}{{{C_{\text{c}}}}} - \frac{{e^{\prime}{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\left( {1 - \frac{{{\Gamma ^*}}}{{{C_{\text{c}}}}}} \right)} \right)\frac{{{C_{\text{m}}} - {C_{\text{c}}}}}{{{C_{\text{a}}}}}+{b^{\prime}_{3}}\frac{{{C_{\text{c}}}}}{{{C_{\text{a}}}}} - e^{\prime}\frac{{{\mathcal{R}_{\text{d}}}}}{{A+{\mathcal{R}_{\text{d}}}}}\frac{{{C_{\text{c}}} - {\Gamma ^*}}}{{{C_{\text{a}}}}} - f\frac{{{\Gamma ^*}}}{{{C_{\text{a}}}}}} \right],$$

(82)

which is equivalent to our derivation in Eq. 19 when γ = 1, approximating the “alpha” terms to 1 and substituting a_c by a_m.