Components of mesophyll resistance and their environmental responses: A theoretical modelling analysis

2.1 An analytical model of mesophyll resistance

Mesophyll resistance represents the total resistance of CO₂ diffusion from the substomatal cavity to chloroplast stroma (Figure 1a). Mesophyll resistance (r_m) is divided into a gaseous phase (r_ias) and a liquid phase (r_liq). CO₂ first diffuses through intercellular air space, with CO₂ concentration ([CO₂]) decrease from gas-phase [CO₂] C_i in the substomatal cavity to liquid-phase [CO₂] C_{w_o}, that is, [CO₂] at the outer surface of the cell wall. Throughout the text, the unit of [CO₂] is bars, and the unit of all resistance is per mole per square metre second bar. It is important to note that an ideal gas equation and Henry's law are applied respectively to convert the unit of [CO₂] in gaseous and liquid phases. Therefore, r_m is calculated as a composite resistance of gaseous and liquid components as (Niinemets & Reichstein, 2003; Tomás et al., 2013)

(1)

where H_c is the Henry law constant (bar m³ mol⁻¹) for CO₂, R is the gas constant (bar m³ K⁻¹ mol⁻¹), and T is the absolute temperature (K; Niinemets & Reichstein, 2003; Tomás et al., 2013). A dimensionless factor (RT)/H_c is needed to convert r_ias to its corresponding liquid-phase equivalent resistance. In terms of CO₂, this is equivalent to replacing the gas-phase C_i with its equivalent liquid-phase C_i′ = C_i · (RT)/H_c (Niinemets & Reichstein, 2003; Tomás et al., 2013).

The gas-phase resistance is modelled the same as (Tomás et al., 2013)

(2)

where ΔL_ias (m) is the average gas-phase thickness, which is taken as half of the mesophyll thickness here; ς (m m⁻¹) is the tortuosity of the diffusion path, suggested to be 1.57 (Niinemets & Reichstein, 2003; Tomás et al., 2013); D_a (m² s⁻¹) is the diffusion coefficient of CO₂ in the gas phase; and f_ias is the fraction of leaf intercellular air space.

In the liquid phase of diffusion, CO₂ first diffuses through the cell wall and membrane, during which [CO₂] decreases from C_{w_o} to C_{w_i}, that is, [CO₂] at the inner surface of the cell wall. Next, CO₂ diffuses through the cytosol to reach the surface of the chloroplast envelope, with the [CO₂] further decreasing to C_{se_o}, that is, [CO₂] at the outer surface of the chloroplast envelope facing the cell wall. Then the chloroplast envelope forms another barrier, where [CO₂] further decreases to C_{se_i}, that is, [CO₂] at the inner surface of the chloroplast envelope. Finally, CO₂ diffuses inside the chloroplast stroma and is fixed by Rubisco, resulting in C_c, that is, the average [CO₂] in the stroma (Figure 1a, extended based on Tholen et al., 2012).

In Figure 1a, the photorespired and respired CO₂ from mitochondria (F + R_d) are separated into two fluxes with different diffusion paths and further refixed by Rubisco, which has been reported to be influenced by the relative position of mitochondria and chloroplast (Busch et al., 2013). Although in Figure 1a, all the (photo)respired flux F + R_d finally enters the chloroplast, in a real leaf, part of F + R_d can leak to the air (Figure 1b), which can be detected by isotope signals. However, there is no difference in the calculation of all resistances between the frameworks of Figure 1a,b because resistance merely relies on net flux across the membrane.

The r_liq therefore is mathematically divided into four items corresponding to different barriers (Equation 3), including resistance of the cell wall and membrane (r_{I_wall}), resistance of the cytosol between chloroplast and the cell wall (r_{II_cyto}), resistance of the chloroplast envelope (r_{III_se}), and resistance of the stroma (r_{IV_chlo}), that is,

(3)

where A_N is the net photosynthesis rate per leaf area. On the basis of this partitioning, a one-dimensional (1D) approximation of the R-D processes of these four barriers is presented in Figure 1c and used as the basis to derive an analytical model.

2.2 Resistance of the cell wall and plasma membrane

The resistance of the cell wall and membrane (r_{I_wall}) is defined as the difference of [CO₂] on two sides of the barrier divided by A_N (Equation 4). Because we assume that there are no other fluxes across this barrier influencing the CO₂ gradient C_{w_o}–C_{w_i} (Figure 1c), it is expected that the r_{I_wall} is equal to the physical resistance of the cell wall and plasma membrane (Equation 4, Tomás et al., 2013).

(4)

where S_m (m²) is the area of cell wall that is exposed to the intercellular air space per leaf area. The scale factor S_m here is introduced due to the increase in area available for CO₂ diffusion. The thickness of cell wall is denoted by d_w (m), the aqueous phase diffusion coefficient of CO₂ by D_c (m² s⁻¹), the decrease of diffusion coefficient compared to free diffusion in water by r_f,w, and the effective porosity of the membrane by p (Evans et al., 2009; Tomás et al., 2013).

2.3 Resistance of cytosol

Resistance of the cytosol (r_{II_cyto}) is defined as the gradient of [CO₂] between the inner cell wall (C_{w_i}) and the outer surface of the chloroplast envelope (C_{se_o}) divided by A_N. Inside the cytosol, there are three sources of CO₂: (a) CO₂ from the intercellular air space, (b) CO₂ generated from dehydration of HCO₃⁻, and (c) CO₂ released from mitochondria by photorespiration and respiration. These three sources of CO₂ together form the CO₂ gradient in the cytosol and determine the CO₂ diffusion flux across the cytosol (Figure 1c). In our approximation, the influence of the (photo)respiratory flux in the cytosol is not considered for simplicity. In other words, only the effect of CO₂ diffusion and the process of hydration are included in the 1D approximation.

The expression of r_{II_cyto} is then derived and simplified as the sum of two parts (Equation 5, see Appendix S2 for derivations in detail).

(5)

The first part matches the formula by Evans et al. (2009; Equation 6), where they calculated the resistance of the cytosol, assuming there is no limitation of carbonic anhydrase (CA). We designated this item to be r_{II_cyto_CAfree}. Here, S_c (m² m⁻²) is the area of chloroplast exposed to the intercellular air space. The scale factor S_c here actually represents the area of parallel 1D diffusion in a 3D leaf from the outer cell wall to the chloroplast (Evans et al., 2009; Tomás et al., 2013). The thickness of the cytosol between the chloroplast and cell wall is represented by d_c (m); D_c,c and D_b,c represent D_c × r_f,c and D_b × r_f,c, respectively, meaning the effective diffusion coefficients of CO₂ and HCO₃⁻ in the cytosol; K_eq (mol m⁻³) is the equilibrium constant; and H (mol m⁻³) is the proton concentration.

(6)

Furthermore, our calculation shows that the limitation from hydration catalysed by CA actually formed another item, which we denote as r_{II_cyto_CA}:

(7)

where α is a function of D_c,c, D_b,c, and biochemical constants of the hydration reaction (see Appendix S2 for the expression in detail). The leakage rate of HCO₃⁻ from the chloroplast to the cytosol per leaf area is denoted by lk_b (mol m⁻² s⁻¹). It is worth noticing that under different light and CO₂ conditions, the expression of r_{II_cyto} solved is simplified to be independent of the input parameter C_{w_i}. It becomes merely a function of the ratio lk_b/A_N in addition to physical parameters of diffusion and biochemical parameters of hydration (Equations 6 and 7).

2.4 Resistance of the chloroplast envelope

The effect of relative position between the chloroplast and mitochondria is represented by a parameter φ (Figure 1). That is, a portion of the (photo)respiratory flux, φ(F + R_d), influences the CO₂ gradient between the cell wall and chloroplast envelope, which will increase the net flux across the chloroplast envelope facing the cell wall from A_N to F_x = A_N + φ(F + R_d); see Figure 1a,b. The rest of the (photo)respiratory flux F_y = (1 − φ)(F + R_d) diffuses through the abaxial side of chloroplast.

Notice that, on the chloroplast envelope, there is an efflux in the form of HCO₃⁻ (lk_b) due to the difference of pH values between the cytosol and stroma. The influx of carbon in the form of CO₂ should equal F_x + lk_b following the mass conservation of carbon (Figure 1c). For the derivation of r_{III_se} (Equation 8), CO₂ diffusion through the envelope is treated as two resistances in parallel. In one of the fluxes, CO₂ directly diffuses through the envelope. In another flux, CO₂ is first hydrated to HCO₃⁻ and diffuses through the envelope and enters the stroma where HCO₃⁻ is dehydrated back to CO₂. F_x was naturally introduced to the numerator and denominator considering that the real flux corresponding to the CO₂ gradient C_{se_o}–C_{se_i} across the chloroplast envelope is F_x rather than A.

(8)

Here, r_{se_c} is the resistance of the chloroplast envelope for CO₂.

2.5 Resistance of the chloroplast stroma

R-D processes in the stroma are similar to those in the cytosol except the additional carboxylation of CO₂ (Figure 1c, and see Equations S12 and S18 in Appendix S2). Similar to the case for the chloroplast envelope, the flux corresponding to the CO₂ gradient C_{se_i}–C_c should be F_x (Figure 1c), so the expression of r_{IV_chlo} is written and solved as

(9)

where D_c,s and D_b,s equal D_c × r_f,s and D_b × r_f,s, respectively, representing the effective diffusion coefficients of CO₂ and HCO₃⁻ in the stroma, d_s (m) is the thickness of the chloroplast, V_c (mol m⁻² s⁻¹) is the carboxylation rate per leaf area, and β is a complex function of constants of diffusion, hydration, and carboxylation processes (see Appendix S2 for expressions and derivations in detail).

Finally, by combining these different components of r_liq (Equations 4, 5, 8, and 9), we derive a complete formula for r_liq. Further combining this expression of r_liq with r_ias (Equations 1 and 2), we obtain an analytical model of the whole r_m.

7.1 Comparison between mesophyll resistance of the liquid phase predicted by the analytical model and three-dimensional cell model

On the basis of the mathematical decomposition of mesophyll resistance (r_m; Equations 1 and 3), r_m is divided into resistance of the gaseous phase (r_ias) and resistance of the liquid phase (r_liq), whereas r_liq is divided into four components, namely, resistance of the cell wall and membrane (r_{I_wall}), resistance of the cytosol (r_{II_cyto}), resistance of the chloroplast envelope (r_{III_se}), and resistance of the interior stroma (r_{IV_chlo}). Here, a new analytical model of r_liq is derived by using explicit formulas for each of these four barriers (Table 1, Spreadsheet S1). The outputs of this analytical model are r_liq and its components. Its input parameters for this model include (a) biophysical parameters such as permeability of the cell wall and diffusion properties of CO₂ and HCO₃⁻ in different components, (b) anatomical parameters such as distance between the cell wall and chloroplast and thickness of the chloroplast, (c) biochemical parameters describing CA activities, and (d) other variables such as environmental CO₂ conditions C_i, net photosynthesis rate A_N, HCO₃⁻ leakage across the chloroplast envelope lk_b, (photo)respiration rate, and variable φ characterizing the relative position between mitochondria and chloroplasts (Table 1, Spreadsheet S1).

To evaluate the robustness of the new model, we first compared its predictions with a 3D R-D model of a rice MSC. Similar responses of r_liq to light and CO₂ were predicted by the cell model (Figure 4a) and the analytical model (Figure 4b). The ratio of r_liq between the analytical model and cell model was predicted to be around 1.32 under most light and CO₂ conditions. Under low light (150 μmol m⁻² s⁻¹), this ratio was slightly higher at about 1.4, whereas under low CO₂ (180 μbar), this ratio became approximately 1.2 (Figure 4c). The ratio of C_c between the analytical model and cell model was predicted to be around 0.98 under different light and CO₂ levels (Figure 4d).

Moreover, we evaluated the robustness of the analytical model to predict individual components of r_liq by comparing its prediction against those from the 3D cell model. We found that r_{I_wall} calculated from the cell model was almost constant under different light and CO₂ conditions, except under extremely low C_i (Figure 5a). The ratios between r_{I_wall} calculated from the analytical model (Equation 4) and from the cell model were around 0.98 under most light and C_i levels (Figure 5e). Similar levels of deviation were found between predictions of r_{II_cyto} from the analytical model (Equation 5) and the cell model under most conditions (Figure 5b,f). Even under extremely low light, the deviation was still less than 7% (Figure 5f). The deviation of the estimated resistance of the chloroplast envelope (r_{III_se}) between the analytical formula (Equation 8) and the cell model was less than 5% (Figure 5c,g). However, a much higher deviation was observed between the calculated resistance of the stroma (r_{IV_chlo}) between the analytical model and the cell model (Figure 5d,h).

7.2 Influence of the relative position of mitochondria and chloroplasts on mesophyll conductance

Previously, the g_m–C_i curve simulated using an R-D model of a spherical MSC predicted an increasing trend of g_m with C_i under relatively low C_i followed by a decreasing trend (Tholen & Zhu, 2011), and the initial increase was interpreted as the influence of the (photo)respiratory flux (Tholen et al., 2012; Tholen & Zhu, 2011). However, the CO₂ response of r_liq simulated by our cell model showed a monotonic decreasing trend with C_i (Figure 4a). We hypothesized that this difference was because mitochondria in our rice cell model were located completely at the inner side of the chloroplast layer.

To test this hypothesis, we built two types of cell models with four different relative positions between mitochondria and chloroplasts (Figure 3a–d). Simulation shows that the increase of g_liq with C_i was directly linked to the location of mitochondria (Figure 6). When mitochondria were totally at the inner side of the chloroplast layer (Figure 3d), the simulated g_liq–C_i curve showed a monotonic decreasing trend, similar to the simulated g_liq–C_i curve with the default cell model in Figure 2b. When mitochondria were located in the gap between chloroplasts (Figure 3a–c), which was mimicked by holes on the modelled chloroplast membrane layer, the g_liq–C_i curve gradually exhibited an increasing trend under low C_i. With a closer distance between mitochondria and the cell wall, the increasing trend became more significant and the C_i value corresponding to the peak in the g_liq–C_i curve became larger (Figure 6). In our analytical model, we set a variable φ to represent this relative location of mitochondria and chloroplasts, which enables the analytical model to predict similar g_liq–C_i responses (lower panel in Figure 6) as predicted by 3D cell models (Figure 3 and upper panel in Figure 6).

7.3 Comparison between the whole mesophyll resistance predicted by the analytical model and three-dimensional leaf model

Scale factor S_m is applied in the analytical formula and the 3D cell model to convert resistances expressed on the basis of exposed cell wall surface area to resistances expressed on the basis of leaf area. This implicitly assumes that the photosynthetic statuses of all MSCs are uniform across the leaf, that is, acting as a single big MSC. However, the light and CO₂ environments inside the leaf are nonuniform (Supporting Information Figures S1 and S2a). With a 3D R-D model of a rice leaf, we also compared the whole r_m predicted by our analytical model and the 3D leaf model. Figure 7a shows the r_m predicted by the leaf model under different light and CO₂ conditions, and Figure 7b shows that the ratio of r_m predicted by the analytical model and the leaf model varies between 1.03 and 1.18 under different light and CO₂ levels. Specifically, Figure 7c shows the calculated r_ias by the 3D leaf model. The predicted r_ias by the analytical formula (Equation 2) underestimated the value predicted by the leaf model by around 37% (Figure 7d).

7.4 Biochemical factors influencing the mesophyll resistance

Mesophyll resistance is analogous to the resistance in Ohm's law. The resistance in electricity is not influenced by either voltage or currency; therefore, it is expected that r_m is not influenced by CO₂ conditions or the photosynthetic rate. We explored the potential biochemical factors leading to this variable r_m under different conditions. Specifically, we sequentially eliminated four potential biochemical factors in the 3D leaf model. First, the effect of nonuniform distribution of light was eliminated by manually assigning the leaf electron transport rate (J) into all chloroplasts uniformly (Figure 8a and Supporting Information Figure S2b). Further, the potential impact of photorespiration and respiration fluxes on r_m was eliminated by setting the Г^* to be zero so that no CO₂ will be released from the mitochondria (Figure 8b). Moreover, the bicarbonate leakage at the chloroplast envelope was blocked by manually setting the permeability of bicarbonate through the envelope to be zero (Figure 8c). Finally, (de)hydration activities in cytosol and stroma were removed in the R-D equation (Figure 8d). The corresponding r_m of each model under different light and CO₂ levels was predicted. We found that these four factors all contributed to the variation of r_m under different light and CO₂ conditions. When all these different biochemical components were eliminated, we obtained a constant r_m under different light and relatively high CO₂ levels (Figure 8d). Under this situation, there are only CO₂ diffusion and carboxylation in the 3D leaf. When the leaf was not CO₂ saturated, an increase of r_m with C_i was still observed due to the change of effective path length under the 3D nature of diffusion (Figure 8d).

8.1 A new analytical formula representing mesophyll resistance as a property of a complex biophysical, biochemical, and anatomical system

Mesophyll resistance (r_m) is an integrative leaf parameter jointly controlled by a large array of complex biophysical, biochemical, and 3D anatomical factors. Tholen et al. (2012) developed an analytical model of r_m, which incorporated the effects of (photo)respiratory fluxes. A number of studies have measured anatomical features and r_m of different species and used them to parameterize r_m models (Berghuijs et al., 2015; Peguero-Pina et al., 2012; Tomás et al., 2013; Tosens et al., 2012). In these models, resistances of different barriers (cell wall, cytosol, chloroplast envelope, and chloroplast stroma) were calculated based on (a) the measured thickness of the cell wall, cytosol, and chloroplast and (b) assumptions about diffusive properties such as effective diffusive path length, porosity, and viscosity (Tomás et al., 2013). Recently, Berghuijs et al. (2015) combined the frameworks of Tholen et al. and Tomás et al. (2013) to produce a new r_m model, with which the impacts of (photo)respiratory flux and leaf anatomical factors on r_m can be explored.

Here, with a mechanistic 3D R-D model to set as the ground truth, we developed a new analytical formula of r_m, which explicitly incorporates all known anatomical and biochemical components of r_m. Values set in the 3D R-D model such as porosity, viscosity, Г^*, and R_d can be directly used in the analytical formula, which enables us to directly evaluate the accuracy of analytical models to predict not only the whole leaf r_m but also its components. Here, we compare the prediction from our new analytical model with earlier r_m model predictions by Tomás et al. (2013) and Tholen et al. (2012). The formula of resistance of intercellular air space (r_ias, Equation 2) and the formula of resistance of the cell wall and plasma membrane (r_{I_wall}, Equation 4) are the same between our new model and the model by Tomás et al. Compared to r_ias predicted from the 3D leaf model, r_ias was underestimated by this formula (Figure 7d), which suggests that the accuracy of this formula may be influenced by different structures of intercellular air space. In other words, tortuosity under different leaf anatomies needs to be incorporated later into this formula, as suggested by Parkhurst (1994).

For the resistance of cytosol, Tomás et al. (2013) predicted a resistance equal to 0.6 mol⁻¹ m⁻² s bar without considering the hydration process. Our formula (Equation 5), which explicitly considers the hydration process, predicted that under a C_i of 30 μbar and a photosynthetic photon flux density of 800 μmol m⁻² s⁻¹, the total cytosolic resistance (r_{II_cyto}) was 0.34 mol⁻¹ m⁻² s bar (Figure 5b,f), whereas among this, the resistance due to limitation from hydration catalysed by CA (r_{II_cyto_CA}) was 0.26 mol⁻¹ m⁻² s bar (76% of r_{II_cyto}). Therefore, although the facilitation of CO₂ diffusion by hydration decreased the resistance of cytosol significantly, the hydration rate catalysed by CA also contributed to r_{II_cyto} substantially, which was earlier attributed to the nonequilibrium status of the hydration reaction in cytosol (Tholen & Zhu, 2011). For the resistance of the chloroplast envelope, Tomás et al. calculated it to be 0.39 mol⁻¹ m⁻² s bar, whereas our analytical formula (Equation 8) and 3D cell model predicted this resistance to be mostly around 0.5 to 0.8 mol⁻¹ m⁻² s bar under various conditions (Figure 5c,g). This difference between predictions using our new analytical model and the Tomás et al. model is caused by the HCO₃⁻ efflux on the chloroplast envelope in our analytical formula. Theoretically, this leakage of HCO₃⁻ will be influenced by pH, [CO₂], activity of CA in the cytosol and stroma, and the permeability of the envelope to HCO₃⁻ (Tholen & Zhu, 2011). Our new analytical model provides an opportunity to explore these different factors.

For the resistance of the stroma (r_{IV_chlo}), Tomás et al. (2013) calculated it to be as high as 7.56 mol⁻¹ m⁻² s bar, which is equal to the ratio between half the chloroplast thickness and CO₂ diffusion coefficient in the stroma. However, the r_{IV_chlo} predicted from our 3D cell model was mostly less than 0.5 mol⁻¹ m⁻² s bar (Figure 5d). This much lower r_{IV_chlo} is attributed to two factors: (a) the facilitated diffusion by CA and (b) carboxylation occurring throughout the stroma instead of being at the midpoint of the stroma as assumed by Tomás et al. Our analytical formula considers the effect of hydration; therefore, the relative error for r_{IV_chlo} is smaller but still as high as around 200% (Figure 5h), which is the main reason leading to the relative error of 32% for the total r_liq (Figure 4c). Two factors contributed to this high relative error. First, as a result of the diffusion into an ever-decreasing volume inside a 3D chloroplast, the concentration and fluxes of CO₂ and HCO₃⁻ on a unit volume basis gradually increase with depth into the chloroplast. This is analogous to the situation when CO₂ diffuses from the stomata to intercellular air space, which is essentially diffusion in an ever-increasing volume in 3D (Parkhurst, 1994). This 3D nature of diffusion is ignored in the analytical model. Second, r_{IV_chlo} is calculated as the ratio of C_{se_i}–C_c and net photosynthesis rate, where C_c is an average value of [CO₂] in the whole stroma and C_{se_i} is the [CO₂] in the inner boundary of the chloroplast envelope. An accurate estimate of C_c requires an accurate prediction of the distribution of [CO₂] in the stroma, which is difficult to achieve in the 1D analytical model. Furthermore, because r_{IV_chlo} is the last component of the r_m, C_c is numerically close to C_{se_i} in the R-D model, and the difference between C_c and C_{se_i} is very small; as a result, even a 3% deviation in the calculated C_c between the analytical model and 3D cell model can generate an over 200% deviation in the calculated r_{IV_chlo}, as compared to the 3D cell model.

Tholen et al. (2012) developed Equation 10 to account for the effect of (photo)respiration on r_m under different CO₂ conditions,

(10)

where r_w represents the resistance of the cell wall and plasma membrane and r_ch represents the resistance of chloroplast envelope and stroma. An equivalent form of r_liq in our analytical model (Equation 3) is written as follows:

(11)

where F_x equals A_N + φ(F + R_d). The F_x is used here to make the flux on the denominator correspond to the CO₂ gradient on the numerator. To degenerate from our analytical model (Equation 11) to Tholen's model (Equation 10), three conditions need to be satisfied: (a) r_{II_cyto} is much less than r_{I_wall} in Equation 11; (b) F_x equals A_N + F + R_d, which means φ in our analytical model is 1; (c) C_{se_o} − C_c/F_x equals a constant r_ch, which means the influence of carboxylation, hydration, and bicarbonate leakage to resistance of the chloroplast envelope and stroma is negligible (Equations 8 and 9); thus, C_{se_o} − C_c/F_x approximates a constant. This third simplification makes Tholen's model (Equation 10) unable to predict the decreasing trend of g_m at high C_i. Both our analytical model and the 3D R-D models in this study (Figure 6) and of Tholen and Zhu (2011) can predict this decreasing trend.

During the deduction of our analytical model, the fluxes A_N, F + R_d, and lk_b were preserved in the formula of r_{II_cyto}, r_{III_se}, and r_{IV_chlo} (Equations 5, 8, and 9), and those fluxes cannot be eliminated like C_{w_i}, C_{se_o}, and C_{se_i}. Essentially, it is because r_m is a parameter defined on an indivisible complex system. Although it seems that we can divide the r_liq into four components by Equation 3, any changes in the resistance of one barrier would affect the resistances of the other three barriers. This coupling between different components of r_m is manifested through fluxes A_N, F + R_d, and lk_b in the analytical model.

8.2 Practical considerations during the application of the new analytical model

Comparison studies show that our analytical model can predict the r_m with a high level of accuracy (Figure 5). However, during applications on real leaf, the calculated r_m using the analytical model can potentially deviate from the r_m experimentally measured due to a number of reasons. First, the formulas of r_{I_wall} may still possibly deviate, although it seems to have a very high accuracy compared with the 3D model (Figure 5e), as effective porosity of the cell wall in the formula of r_{I_wall} may vary between different species. Different leaves have different structures of intercellular air space, which also affects the accuracy of the formula of r_ias without changing the tortuosity.

Second, during the application of our analytical model (Spreadsheet S1), A_N and F + R_d per leaf area can be experimentally measured. So far however there is no method developed for the measurement of lk_b. In this study, we estimated lk_b under different light and CO₂ levels from the 3D R-D models. Then the estimated lk_b was applied in the analytical model to correct the effect of bicarbonate leakage to r_m. Simulations with our leaf model show that the ratio of lk_b to A was around 0.27 to 3 under most light and CO₂ conditions in our model (Supporting Information Figure S3). Under ambient CO₂ and relative high light (e.g., photosynthetic photon flux density > 400 μmol m⁻² s⁻¹), this ratio was predicted to be between 0.27 and 0.54 (data attached in Spreadsheet S1). This ratio can potentially be different under different CA concentrations and pH in cytosol and stroma, which is at this point not considered in the current analytical model. More work is needed still to measure lk_b to better use our analytical approach.

Furthermore, the factor φ in our analytical model, which represents the effect of relative special location between mitochondria and chloroplast on r_m, is also difficult to estimate directly from the anatomical measurements of leaf. For plants such as rice, which has a very high chloroplast coverage, we can approximate φ to be 0. However, for other leaf anatomies, the best way to infer φ is possibly through numerical estimation based on sophisticated 3D modelling. Especially for leaf with a low chloroplast coverage, the r_m is influenced not only by the scale factor S_c but also because of the effect of φ. Moreover, it is worth noting that the value of φ is not necessarily a constant under different light and CO₂ conditions, considering the relative changes in A_N and F + R_d with the change of environmental conditions.

In Table 2, we list all these contributing factors that can potentially decrease the accuracy of the analytical model and hence are potential areas of future research. Briefly, deviation between r_m calculated by our analytical model and r_m measured on a real leaf is introduced during three steps, that is, during simplification of the 3D cell model to the analytical model, during simplification of a 3D leaf model to a 3D cell model, and during representation of a real leaf into a 3D leaf model.

8.3 Does photorespiration and respiration influence the mesophyll resistance?

Tholen and Zhu (2011, 2012) suggested that photorespiration and respiration led to the intrinsic decrease of r_m under low C_i. Evans and von Caemmerer (2013) measured the response of Δ¹³C signal to oxygen concentration and observed that fractionation of photorespiration (Δ_f) responded in a linear way to the oxygen concentration; therefore, they suggested a higher factor f (fractionation factor of photorespiration) to explain the decrease of r_m under low C_i. Evans and von Caemmerer (2013) explained the results of Tholen et al. (2012) by artefacts with the equation of Δ¹³C signal and biased f used in the data analysis (Griffiths & Helliker, 2013). However, due to the unknown real value of f, we still cannot rule out the possibility that (photo)respiration influences r_m. In other words, (photo)respiration may still affect the intrinsic r_m and then affect the measured r_m by the Δ¹³C method. Meanwhile, it may also affect the factor f therefore involved in the interpretation of the Δ¹³C signal and the calculated r_m (Griffiths & Helliker, 2013).

In our simulation, we showed that the position of mitochondria affected the response of r_m with C_i (Figures 3 and 6), and a factor φ was introduced to represent this effect, which means only part of the (photo)respiratory flux, φ(F + R_d), influenced the CO₂ flux between the cell wall and chloroplast. In this sense, theoretically, φ would influence both the value of f and the degree of the effect of (photo)respiration on r_m. The interaction between the position of mitochondria, f, and the (photo)respiratory flux was not considered in previous experiments and simulations. A new model of the R-D process of ¹³CO₂/¹²CO₂ in the future can potentially form a unified theoretical framework to simulate the measured Δ¹³C signal and to disentangle the mechanism(s) contributing to the variation of r_m under different environments.