The mechanistic basis of internal conductance: a theoretical analysis of mesophyll cell photosynthesis and CO2 diffusion.

Photosynthesis is limited by the conductance of carbon dioxide (CO(2)) from intercellular spaces to the sites of carboxylation. Although the concept of internal conductance (g(i)) has been known for over 50 years, shortcomings in the theoretical description of this process may have resulted in a limited understanding of the underlying mechanisms. To tackle this issue, we developed a three-dimensional reaction-diffusion model of photosynthesis in a typical C(3) mesophyll cell that includes all major components of the CO(2) diffusion pathway and associated reactions. Using this novel systems model, we systematically and quantitatively examined the mechanisms underlying g(i). Our results identify the resistances of the cell wall and chloroplast envelope as the most significant limitations to photosynthesis. In addition, the concentration of carbonic anhydrase in the stroma may also be limiting for the photosynthetic rate. Our analysis demonstrated that higher levels of photorespiration increase the apparent resistance to CO(2) diffusion, an effect that has thus far been ignored when determining g(i). Finally, we show that outward bicarbonate leakage through the chloroplast envelope could contribute to the observed decrease in g(i) under elevated CO(2). Our analysis suggests that physiological and anatomical features associated with g(i) have been evolutionarily fine-tuned to benefit CO(2) diffusion and photosynthesis. The model presented here provides a novel theoretical framework to further analyze the mechanisms underlying diffusion processes in the mesophyll.


Introduction
Environmental stress may decrease the rate of photosynthesis not only because of detrimental effects on cell biochemistry, but also because of changes in the diffusion of carbon dioxide (CO 2 ) from the atmosphere to the site of carboxylation (Flexas et al. 2008;Warren 2008a). For example, limited CO 2 diffusion is thought to be the main cause of lower rates of photosynthesis under drought conditions (Grassi and Magnani 2005;Keenan et al. 2010). Even under normal conditions, the slow diffusion of CO 2 to the site of carboxylation in the chloroplast can significantly limit photosynthesis (Flexas et al. 2008;Warren 2008a;Zhu et al. 2010). The diffusion of CO 2 between intercellular airspaces and the Rubisco enzyme is usually described by a parameter called internal or mesophyll conductance (g i ; see Table I for a definition of all symbols used in this work).
Different species and even different genotypes of the same species show substantial variation in g i and the limiting effect of g i on photosynthesis is close to that of stomatal conductance (Flexas et al. 2008;Warren 2008a;Barbour et al. 2010). As a result, there is significant potential to increase photosynthesis and growth through breeding for a higher g i (Warren 2008a; Barbour et al. 2010).
Though the concept of g i has been known since the pioneering work of Gaastra (1959), its physical basis remains poorly understood (Evans et al. 2009). After entering through the stomata, CO 2 diffuses through airspaces, cell walls, cytosol, and chloroplast envelopes, and finally reaches the chloroplast stroma, where it is fixed by Rubisco (Evans and Von Caemmerer 1996;Evans et al. 2009). CO 2 from photorespiration and mitochondrial respiration diffuses through the mitochondrial envelope, cytosol and chloroplast membranes into the stroma and forms an additional source of CO 2 for photosynthesis. When we consider the relative contribution of several diffusion barriers, it is convenient to consider the reciprocal of g i , i.e. the internal resistance (r i ). The r i is 5 60 (Farquhar et al. 2001). One possible approach is to describe diffusion processes inside a structure using partial differential equations. Several such reaction-diffusion models have been presented, mainly for analyzing airspace conductance (Gross 1981;Parkhurst 1994;Vesala et al. 1996;Aalto and Juurola 2002;Galloët and Herbin 2000).
Because the liquid phase resistance is currently thought to be the most limiting part of the total internal resistance (Flexas et al. 2008;Evans et al. 2009), several authors have included a simplified liquid phase component within whole leaf diffusion models (Vesala et al. 1996;Aalto and Juurola 2002). However, the complex intracellular structure and variation in the diffusion coefficients in different cellular components were not considered. Cowan (1986) developed a more detailed diffusion system for the chloroplast stroma, considering both the facilitating effect of carbonic anhydrase (CA) and the fact that (photo)respiratory CO 2 is spatially separated from the influx of intercellular CO 2 . This model demonstrated the influence of CA and chloroplast thickness on photosynthesis, yet it was limited by a one dimensional description of the diffusion pathway inside a single chloroplast stroma.
If g i is determined in a way that considers all the physical resistances between a CO 2 source and sink, it can be expected that leaves would have constant g i unless the physical properties of the pathway change. Indeed, g i was originally assumed constant throughout a leaf's lifespan (Evans and Von Caemmerer 1996), but more recent evidence has called this assumption into question. For example, g i can respond rapidly to changes in leaf temperature (Bernacchi et al. 2002;Yamori et al. 2006) or intercellular CO 2 concentration (Flexas et al. 2007). In addition, g i decreases under soil water stress (Miyazawa et al. 2008;Warren 2008b) and salinity stress (Delfine et al. 1999;Centritto et al. 2003). There have been various attempts to explain the mechanistic basis of these responses. For example, based on the Q 10 of the response of g i to temperature, Bernacchi et al. (2002) suggested that the diffusion may be an enzymatically facilitated process.
Facilitation by CA and aquaporins has been proposed to explain some of the observed changes of g i in response to the environment (Flexas et al. 2008;Warren 2008a;Evans et al. 2009;Terashima et al. 2011). However, the mechanisms underlying such responses are still far from clear.
One possible reason for our limited progress in elucidating the mechanisms that determine g i may be the lack of a proper method to evaluate the contribution of both anatomical and biochemical components. Experimentally testing the contribution of each factor to g i is challenging, because it is extremely difficult to manipulate only one factor of the system without influencing others. The development of mathematical models combined with a sensitivity analysis on a number of factors may provide a useful tool to describe the mechanistic basis of g i . This work will present a novel mechanistic description of mesophyll diffusion and biochemical reactions that allows for a spatial analysis of the C3 photosynthesis model developed by Farquhar et al. (1980). Using this model, we will explore the key limiting factors for diffusion in a mesophyll cell, and analyze the possibilities for increasing photosynthesis by minimizing diffusion limitations. We will show that biochemical processes play an important role in the liquid phase diffusion. In addition, we will propose several new mechanisms that might explain the response of g i to environmental perturbations.

Theory
We assumed a leaf type in which airspace resistance is negligible and the apoplastic CO 2 concentration (C i ) is in complete equilibrium with the partial pressure of CO 2 in intercellular airspaces (p i ). Given these assumptions, we represented a typical leaf as having a number of identical spherical mesophyll cells. The volumes of the various components were representative for a typical C 3 leaf (Winter et al. 1994;Pyankov et al. 1999). Because diffusion through liquid is relatively slow, most of the cytosolic CO 2 flux is expected to go through the small layer of cytoplasm between cell walls adjacent to intercellular airspaces and the chloroplast membrane facing this cell wall. In this case, the surface area of the chloroplast membrane facing intercellular airspaces should correlate with g i , and this has been experimentally confirmed (Terashima et al. 2006). In the present study, we did not examine this well-known effect of the chloroplast surface area and kept it constant at 11.4 m 2 m -2 .
A 3D mesh of a simplified spherical mesophyll cell was constructed and is shown in Figure 1A. In this structure, several diffusional fluxes and biochemical reactions can be defined, as described in Figure 1B. The diameter of the cell was 20 µm and contained 96 chloroplasts and 96 mitochondria. The subcellular structure was further defined by the following parameters: chloroplast-wall distance (variable, 0.1 µm by default), maximum chloroplast thickness (2 µm), chloroplast diameter (3.5 µm), chloroplast-mitochondria distance (0.2 µm), diameter of the mitochondria (1.4 µm), chloroplast-vacuole distance (1.1 µm), wall-vacuole distance (3.1 µm).
Analogous to the equations given by Cowan (1986), the diffusive flux of CO 2 through the liquid phase is described by: for CO 2 , η is a dimensionless factor representing the relative viscosity of the compartment in which the diffusion takes place, C is the CO 2 concentration [mol m -3 ] at a given x, y and z coordinate, f is the rate of RuBP carboxylation, h is the net hydration rate of CO 2 to bicarbonate (HCO 3 -) , r d is respiration rate and r p is the rate of photorespiratory CO 2 release (all rates are in mol m -3 s -1 ). The reaction terms are derived from a biochemical model of C3 photosynthesis (Farquhar et al. 1980, see Materials & Methods). These terms differ in different cellular compartments, i.e. f=0 in mitochondria, and r d =r p =0 inside chloroplasts. The equation for HCO 3diffusion is:

Results
The CO 2 and HCO 3concentrations in the liquid phase The model was solved using default parameter values given in Table I. In Supplemental Fig. S1, the resulting CO 2 and HCO 3concentrations at a p i of 30 Pa are shown. Except for the small region between chloroplast and cell membrane, the CO 2 concentrations in the cytosol were close to that in the apoplast. The average CO 2 concentration in the stroma was only 5.3% lower than in the cytosol. Because the pH of the stroma was 0.7 units higher than that of the cytosol, the Henderson-Hasselbalch equation (Terashima et al. 2011) predicts that at equilibrium, the HCO 3concentration in the stroma would be 4.80 times that of the cytosol. In good agreement with this, the reaction-diffusion model predicted an average HCO 3concentration in the stroma 4.74 times that of the cytosol.

A comparison with the C3 biochemical model
In Figure 2 the results of the reaction-diffusion model are compared with the commonly used biochemical model of photosynthesis described by Farquhar et al. (1980), which does not consider diffusional limitations. This leads to an overestimation of photosynthesis at all but the highest CO 2 concentrations. The figure also shows that changing a model parameter such as cell wall conductance can have a significant effect on the shape of the curve. In the subsequent paragraphs we will analyze the sensitivity of the reaction-diffusion model to different parameters at three CO 2 partial pressures: 15, 30 and 60 Pa.

The components of the diffusion pathway
Commonly used descriptions of the diffusion pathway use a global resistance or conductance parameter that implies a one-dimensional diffusion between a single source and sink. However, such descriptions obscure some of the complexity inherent in 3D diffusion systems. Therefore, instead of analyzing the effect of a given model parameter on internal conductance, Parkhurst (1994) proposed to test the effect of a parameter by calculating its relative effect on the net rate of photosynthesis. We followed this suggestion and tested the effect of different parameters or model structures on the rate of photosynthesis. In addition, to allow comparison of our results with experimental results using conductances, we estimated g i from the simulation results as the conductance between C i and the average CO 2 concentration in the chloroplast (Eq. 11).
CO 2 dissolves in the water-filled pores of the cell wall and diffuses through the plasmalemma into the cytosol. The plasmalemma is usually closely associated with the cell wall, and for simplicity we combined the resistances of these two components. Figures 3A and B show the effects of a change in the permeability of the wall and plasmalemma (G wall ) on the total internal conductance and on the relative rate of photosynthesis. The figure shows a saturating response; at ambient CO 2 , 90% of the maximum rate of photosynthesis could be reached with a G wall of 0.02 mol (m wall) -2 s -1 .
Interestingly, photosynthesis changed little in the range of biologically relevant values for G wall (indicated by the white area in Figure 3A) at elevated CO 2 levels (p i =60 Pa).
CO 2 subsequently diffuses through a thin layer of cytosol, where it merges with CO 2 from photorespiration and respiration. In Figures 3C and D the effects of increasing the distance between chloroplast and plasmalemma (distance a in Figure 1) are shown.
Although increasing this distance resulted in a lower g i , the predicted decrease in photosynthesis over this range remained below 3%. Figure 4 shows that decreasing the permeability (P CO2 ) of the chloroplast envelope to CO 2 had an effect similar to that of decreasing G wall . However, when P CO2 becomes unrealistically low, the simulation indicated that CO 2 fixation becomes nearly zero and a significant amount of (photo)respiratory CO 2 leaks out of the cell, resulting in a negative net rate of photosynthesis. Interestingly, an increased permeability of the chloroplast membrane to HCO 3decreased photosynthetic rates (Fig. 4C). This occurred because the stromal HCO 3concentration is almost 5 times higher than that of the cytosol and when the membrane permeability is high, such a large gradient results in a significant leakage of HCO 3from the stroma into the cytosol.
We examined whether CA plays a significant role in the facilitation of CO 2 transport across the chloroplast stroma. Removing all CA from the chloroplast stroma decreased photosynthesis (at p i =30 Pa) a little over 7% compared to at default conditions (Fig. 5A).
Note that over the same range of CA-concentrations, g i changed by more than 40%. . Surprisingly, even although none of the individual diffusion resistances in the model changed in response to CO 2 , the predicted g i increased rapidly with an increase of p i from 0 to about 10 Pa, but decreased when p i was further increased (Fig. 6F). The large increase in g i at low p i was not observed when the simulation was performed at 1% O 2 (Fig. S2). Varying the permeability of the chloroplast envelope to HCO 3affected the decrease of g i at high p i , which suggests that this effect could be attributed to an increased leakage of HCO 3from the chloroplast stroma into cytosol.

A 3D reaction-diffusion model enables a systematic study of the internal conductance
Here we have presented an integrated three-dimensional reaction-diffusion model that  (Table II).
The main differences between the resistance approach and the reaction-diffusion model are the calculated resistances of the chloroplast membrane and stroma (Table II). The discrepancy in the estimated resistance for the chloroplast membranes is mainly the result of necessary simplifications in the resistance-based model, such as the assumption that the diffusion between wall and chloroplast envelope only occurs across the small cytoplasmic volume where the chloroplast is directly adjacent to the intercellular airspace. In reality, diffusion occurs through three dimensions and even the back-side of the chloroplast contributes somewhat to the total conductance (and receives CO 2 from respiration and photorespiration). Since the resistance in both calculations was expressed per unit surface area exposed to intercellular airspaces, the reaction-diffusion model predicts somewhat lower values for the resistance of the chloroplast membranes.

The role of CA in facilitating CO2 diffusion through the liquid phase
The resistance of the chloroplast stroma is influenced by the average path length for diffusion in the stroma, and the amount of CA in the stroma that facilitates the diffusion of CO 2 . The effective diffusion path length for CO 2 in a system with a distributed uptake depends not only on the geometry, but also on the concentration and uptake rate of the diffusing substances (Parkhurst, 1994). Thus, diffusion through the stroma cannot be accurately estimated based on the diffusion constant and stroma thickness alone. For further calculations using the resistance-approach we will assume here that the effective diffusion path length in the stroma is 50% of the stroma thickness. The total carbon flux . This is only about two times higher than the facilitated stroma resistance estimated by Cowan´s model (7 m 2 s bar mol -1 ), giving a facilitation factor of 1.9.
In Table II  (X a,s =0.27 mM) the facilitation factor was close to 7, suggesting that this CA concentration is somewhat limiting diffusion. When we take this facilitation factor into account, the stroma resistance calculated by a resistance-approach is still two times higher compared to those resulting from the reaction-diffusion model. This suggests that the effective diffusion path length in the modeled stroma at p i =30 Pa was only around 25% of the total stroma thickness, and not 50% as we initially assumed. Estimating the effective path length in advance is difficult as it depends on any factor that influences the assimilation rate (see Parkhurst 1994, page 458-459 for further discussion).
In our model, the chloroplast stroma accounted for 17% of the total diffusion resistance.
However, this contribution would be higher without the facilitating effect of CA as discussed above. We found that removing all CA from the stroma reduced g i by 44% compared to the default values. Nevertheless, the effect on the net rate of photosynthesis at a p i of 30 Pa was only around 7%. This limited effect of CA on photosynthesis is in agreement with experimental data presented by Price et al. (1994). In addition, Cowan (1986) estimated that an optimal partitioning of nitrogen between Rubisco and CA would increase photosynthesis by only 7.5% compared to if the same amount of nitrogen were invested in Rubisco alone (see also Fig. S4).
Because of the lower pH in the cytosol compared to the chloroplast stroma, cytosolic  . 1977). Thus, after entering the cytosol, CO 2 is rapidly converted into HCO 3 -, increasing the gradient in CO 2 concentrations (Fig. 1B). Close to the chloroplast envelope, there is net dehydration of HCO 3into CO 2 that subsequently diffuses into the chloroplast stroma. Thus, this process can maintain a high CO 2 flux through the chloroplast membrane. Because a certain amount of volume is required for significant hydration and dehydration of CO 2 , a very small distance between chloroplast and cell wall results in a slight decrease in the internal conductance (Fig. 3C). Although this facilitating effect of CA for transport across the membranes may offer an explanation for the presence of CA in the cytosol, the modeled effect of cytosolic CA on photosynthesis at ambient CO 2 was less then 3% (data not shown).

Photorespiratory fluxes need to be considered when calculating internal conductance
A notable difference between our model and previous resistance-based approaches is that we took the (photo)respiratory flux from the mitochondria into account when estimating diffusion resistances. The chloroplast CO 2 concentration is determined by both the net CO 2 influx and the CO 2 flux originating in the mitochondria, but g i is usually calculated using only the net photosynthetic flux. An increase in photorespiration reduces net photosynthesis but may maintain a similar chloroplast CO 2 concentration. Thus even although C c is similar, the apparent g i calculated using Eq. 12 would be lower. Such an additional "resistance" resulting from the effect of respiration and photorespiration has thus far been neglected when estimating g i . We include an explanation of this phenomenon using the traditional resistance analogy in Supplemental  6B) indicate that at low p i , the rate of photorespiration would need to double to explain a 50% reduction in g i . However, photorespiration was reported to be only 40% higher in the mutants, suggesting that this mutation has additional effects that result in a lower g i .
The model predicted that at p i =30 Pa, g i increases by 11% when the O 2 concentration rises from 2% to 21% (Fig. 6B). The magnitude of this effect depends strongly on the ratio between the wall resistance and the resistance of the chloroplast envelope and stroma. We suggest that the response of g i to environmental variables such as drought and temperature must be carefully reexamined, taking into account that some variation in g i may be attributable to differences in photorespiratory fluxes.

Leakage of HCO 3across the chloroplast membrane may explain the decrease of g i with increasing CO 2
A number of studies (Flexas et al. 2007;Hassiotou et al. 2009;Yin et al. 2009;Vrábl et al. 2009) have shown a similar response to CO 2 as that shown in Figure 6D conditions where the HCO 3permeability of the chloroplast envelopes is above 10 -7 m s -1 (Fig 6D). This suggests that HCO 3leakage may be responsible for the decrease in g i at high CO 2 , as is explained below.
The large pH gradient between the cytosol and chloroplast stroma results in a HCO 3concentration that is about 5 times higher in the stroma compared to the cytosol (Werdan and Heldt 1972). However, when the external CO 2 concentration was increased, a decrease of the concentration difference across the membrane was observed, suggesting a possible loss of carbon from the stroma or a decrease of stroma pH (Werdan and Heldt 1972;Sicher 1984). In addition, Heber and Purczeld (1978) showed that even though the permeability of the chloroplast membrane to HCO 3is very low, some leakage of HCO 3may occur, especially at high CO 2 concentrations. Such a leakage from stroma to cytosol reduces the concentration gradient and leads to a decrease in g i that is bigger at high CO 2 concentrations. Our simulation showed that if the chloroplast membrane permeability to HCO 3was lowered, this decrease in g i could be eliminated (Fig. 6F). If P HCO3-varies between different species, this could account for the absence of a significant reduction in g i in some species (Tazoe et al. 2009(Tazoe et al. , 2011. The modeled decrease was much lower than that estimated by fluorescence-based methods (up to 90%, Fig. 6D, Flexas et al.

Implications for experimental approaches that estimate internal conductance
Our determination of g i is based on the average chloroplast CO 2 concentration given by the model solution. In practice, several methods have been used to estimate this average C c (Pons et al. 2009). To study the relation between g i and CO 2 , measurements of gas exchange combined with chlorophyll fluorescence or isotopic discrimination against 13 CO 2 have been used. If chlorophyll fluorescence can be used to get an accurate estimation of the average electron transport rate, and if the NADPH supply is limiting RuBP regeneration, equation 13 can be used to reliably estimate C c . Pons et al. (2009) and Evans (2009) have recently examined these assumptions and concluded that even small errors in these parameters may cause substantial differences in the estimated g i . By contrast, the isotopic method is often seen as a more reliable way to determine g i (Terashima et al. 2006;Pons et al. 2009). The ratio between carbon isotopes in air coming out of the leaf is related to g i because diffusion through the liquid phase discriminates against 13 C. The isotopic method was developed based on a linear model of resistances (Farquhar et al. 1982) and may therefore be somewhat inaccurate for the determination of the total resistance in a 3D system. In addition, fractionation by hydration and dehydration is assumed to have no net effect on the total discrimination against 13 C in C3 leaves. Leakage of bicarbonate over the membrane would effectively lengthen the diffusion pathway and be measurable as an increased discrimination against 13 C. However the dehydration of this bicarbonate in the cytosol results in a discrimination against 12 C, which will partly mask this effect. Using a reaction-diffusion model to analyze the diffusion of 13 CO 2 and 12 CO 2 independently would allow for the analysis of the assumptions underlying the isotopic method and will be the focus of future research.

The importance of different physical components to photosynthetic CO 2 uptake rates
We summarized the effects of varying different model parameters on photosynthetic rates in Table III. Chloroplast membrane conductance had the biggest effect on photosynthesis. However, there is a large range of variation in the estimates for membrane permeability (Evans et al., 2009). Some reported values of this quantity are unusually low, leading to a negative rate of photosynthesis in the model (Fig. 4A). In Table III,  comm.). If chloroplast envelope permeability is indeed at the low end of the range used in Table III, photosynthetic rates could be drastically improved by increasing this permeability. One possible way of achieving such a goal is by targeted overexpression of cooporins in the chloroplast envelope. Previous work has already shown that a constitutive overexpression of cooporins can indeed cause an increase in photosynthesis up to 20% in both rice and tobacco (Hanba et al. 2004;Flexas et al. 2006).
The cell wall conductance had the second largest effect on photosynthesis. Previous analyses have reached a similar conclusion (Evans et al. 2009;Terashima et al. 2011).
However, we showed that this effect is much less pronounced at elevated CO (Fig. 3A,   B). Thus, an increased cell wall thickness that is sometimes observed at elevated CO 2 (Teng et al. 2006) may have a negligible effect on the rate of photosynthesis under such conditions.
The width of the cytosolic layer between the cell wall and the chloroplast envelope had a minor effect on photosynthesis and g i (Figs. 3C,D, Table III). Sharkey et al. (1991) reported that the distance between cell wall and chloroplast more than doubled in transgenic plants with excess phytochrome, when compared to wild type tobacco. In contrast to our results, those authors reported a 43% decrease in assimilation rates at ambient CO 2 and a 27% reduction in g i . However, additional pleiotropic effects on g i in such transgenic plants are hard to exclude. For example, the surface area of chloroplasts exposed to intercellular airspace was not determined, but it might have been able to explain part of the observed decreases in photosynthesis and g i . In this respect, it is interesting to consider that g i correlated linearly with the amount of surface area exposed to intercellular airspaces in a mutant with an altered chloroplast location (chup1); no additional effect of the increased distance between cell wall and chloroplasts was found (Tholen et al. 2008). Increasing g i has been suggested as a method for optimizing photosynthetic efficiency in crop species (Barbour et al. 2010;Zhu et al. 2010). Figures 3, 4 and Table III demonstrate that current estimations of the maximum permeability of several cellular components are close to values giving optimal photosynthetic rates. This suggests that these parameters might have already been fine-tuned during evolution. The lower estimates for the wall conductance (Figs. 3A, B), are often observed for woody species (Gillon and Yakir 2000), and can clearly form a significant limitation for photosynthesis.

Physiological and anatomical features associated with g i may have been fine-tuned through natural selection to benefit CO 2 diffusion and photosynthesis
In addition, the lower end of the estimates for the membrane CO 2 permeability would result in a severe limitation of photosynthesis, but it should be stressed that there is a large uncertainty surrounding the these values (Evans et al. 2009).
Care must be taken with the use and interpretation of the parameter g i ; a large increase in g i does not necessarily result in a significant increase of photosynthesis. For example, increasing the wall conductance from 0.1 mol m -2 s -1 to 1.0 mol m -2 s -1 would increase g i by 33%, but the corresponding change in photosynthesis would be only 2.3% (Fig. 3).
Therefore, comparing different genotypes or treatments based on the relative inhibition to photosynthesis (Table III) seems to be a more useful approach to identify diffusional limitations to photosynthesis.

Conclusions
This work presents a spatial model of the liquid phase diffusion pathway in the mesophyll, which can be used to explore the influence of physical and structural characteristics of the leaf on photosynthesis and g i . With this model, we systematically examined factors influencing CO 2 diffusion, and gained new insights into mechanisms that affect this process. We demonstrated that photorespiration has an important effect on the apparent g i and that the decrease in internal conductance at high CO 2 concentrations can be explained, at least partially, by a leakage of HCO 3from the 23 540 545 550 555 stroma to the cytosol. This work also demonstrated that the incorporation of diffusion and anatomical features can dramatically increase the potential of biochemical models for their use in systems biology.

Materials and Methods
The CO 2 fixation in the chloroplast was calculated (Farquhar et al. 1980;Von Caemmerer 2000) as:  Table I for an overview of the parameters used in this work). Since detailed information about variation in biochemistry is lacking, we set all biochemical parameters to be constant throughout the leaf. The volumetric electron transport rate in the chloroplast stroma was calculated from the total electron transport rate per unit leaf area (J [mol m -2 s -1 ]) as: where S m [m 2 m -2 ] is the mesophyll surface area per unit leaf area and V s [m 3 m -2 ] is the stroma volume per unit of mesophyll wall surface area. Similarly, the volumetric rate of respiration was calculated as: where V m [m 3 m -2 ] is the mitochondrial volume per unit of mesophyll wall surface area.
Note that the the respiration rate per unit leaf area, R d [mol m -2 s -1 ], was assumed to be constant and independent from the rate of photosynthesis.
The rate of RuBP oxygenation by Rubisco in the chloroplast can be calculated according to the Farquhar model. Half this rate corresponds to the expected CO 2 release in the mitochondria (Von Caemmerer, 2000). We integrated this rate over the chloroplast volume (chl) to get the total photorespiratory CO 2 production in a given cell. This was then divided by the volume of the mitochondria (mit) present in that cell to yield the local volumetric rate of photorespiration (r p ): where f is the local CO 2 fixation rate calculated from equation 3. Eq. 9 We considered the wall and plasmalemma to be impermeable to HCO 3 -. In addition to these external boundaries, the chloroplast envelope, mitochondrial envelope and the tonoplast were described as thin boundary layers having a fixed permeability: Eq. 10 The left side of this equation represents the normal diffusive flux across the boundary, P CO2 is the permeability of the boundary to CO 2 , and C 1 and C 2 [mol m -3 ] are CO 2 concentrations on the two sides of the boundary. A corresponding boundary was also defined for HCO 3 -.
The average CO 2 concentration in the chloroplast was calculated by integrating the local CO 2 concentrations in the stroma divided by the total chloroplast volume: The g i [mol m -2 s -1 ] was calculated by: Estimates of internal conductance by the variable-J method (Harley et al. 1992;Loreto et al. 1992) rely on the assumption that the average CO 2 concentration in the stroma can be calculated from the electron transport rate (assuming NADPH is limiting for RuBP regeneration): For the model, the values obtained by this expression were equal to those calculated by equation 11.

Parametrization of the model
There exist large variations in biochemical leaf properties. For the purpose of this model, we assumed a C3 leaf with chloroplasts containing 1.25 mM Rubisco sites, an electron transport capacity of 90 µmol m -2 s -1 , a chloroplast CO 2 compensation point of 4.1 Pa, and a dark respiration rate of 0.8 µmol m -2 s -1 (Table I). The concentration of CA was based on in vitro and in vivo activity measurements (Atkins et al. 1972a,b, Price et al. 1994, Peltier et al. 1995, Gillon and Yakir 2000. Atkins et al. (1972a) determined the total CA activity in extracts from a range of species. Using a specific activity of 23900 units per mg (Atkins et al. 1972b), a molecular weight of 28 kDa per active site, and a chloroplast volume of 30 µL mg -1 chl, it can be calculated that the CA concentration in these species was between 0.04 and 0.69 mol active sites m -3 . Alternatively, the CA concentration can be estimated from measurements of in vivo CA activity (Price et al. 1994, Peltier et al. 1995, Gillon and Yakir 2000. This yields a CA concentration between 0.14 and 0.67 mol m -3 CA for a number of herbaceous species, corresponding well with the earlier work of Atkins et al. (1972a,b). These calculations assumed that all CA activity is confined to the stroma. However, Rumeau et al. (1996) suggested that up to one-third of the total CA may be present in the cytosol. In our model, we assumed a default stromal CA concentration based on the value for tobacco by Gillon and Yakir Although there is probably less variation in physical parameters, few have been measured in plant leaves. For example, the diffusion coefficient through the liquid phase has often been considered to be close to that in water (Aalto and Juurola 2002;Evans et al. 2009). However, the mitochondrial matrix and chloroplast stroma contain relatively high concentrations of salts, and protein concentrations of up to 400 mg ml -1 (Robinson and Walker 1981), which dramatically increase the viscosity and limit diffusion (Ogawa et al. 1995). According to the Stokes-Einstein law, the diffusion coefficient is inversely related to the kinematic viscosity of the solvent (Einstein 1905). Ogawa et al. (1995) estimated the viscosity of the chloroplast stroma to be 69 times that of water, based on their measurement of the viscosity of a 40% w/v protein solution. Scalettar et al. (1991) found that the effective viscosity in the mitochondrial matrix was 25 times, and that of cytosol 2-5 times larger than that of dilute buffer. Köhler et al. (2000) determined the diffusion coefficient of green-fluorescent protein in tobacco cells. Their results indicate that diffusion through the stroma is about 100 and through cytosol 2-3 times slower compared to that through an aqueous solution. These high values are in contrast to an analysis of diffusion coefficients for CO 2 in different human tissues, where a protein concentration of 40% corresponds to an apparent diffusion coefficient 70% lower than that of CO 2 through water (Gros and Moll 1971). Such a limited effect of high protein concentrations on the apparent diffusion coefficient for small molecules may be caused by the presence of low-viscosity water channels between the protein molecules. We tentatively used a relative (to water) viscosity of 10 for the mitochondrial matrix and chloroplast stroma, and a relative viscosity of 2 for the cytosol (Table I). The modeled response of photosynthesis and g i to differences in stromal viscosity are shown in Fig.   S6.
µm. Leaf level photosynthesis was calculated by multiplying the parameters per unit cell wall with estimations of the total mesophyll surface area exposed to intercellular airspaces per unit area (S m = 15 m 2 m -2 ).

Numerical simulation
The finite element method in a specialized software platform (COMSOL Multiphysics ® version 3.5a) was used to solve the steady state diffusion equation for CO 2 and HCO 3diffusion in three dimensions. The geometry described in Figure 1 was created and subdomain and boundary equations were added as explained above. The source code of the default model used in our simulations is given as supplemental data.

Plant Material
Seeds of Arabidopsis thaliana (Columbia) were placed in sealed Petri dishes on moistened filter paper and put in a growth room (short-   Table I, and expressed per unit exposed chloroplast surface. Note that in the chloroplast, an effective diffusion path length 50% of the stroma thickness was assumed. For the reaction-diffusion model, resistances were calculated by dividing the average concentration difference across a component by the average flux through that component, and again expressed per unit exposed chloroplast surface. Liquid phase resistance (m chl 2 s bar mol -1 ) Resistance model Reaction-diffusion model (saturating CA)  Figure 1: A) The 3D structure of a mesophyll cell with a diameter of 20 µm and containing 96 chloroplasts and mitochondria. The structure was further defined by the following parameters: a) The chloroplastwall distance (variable, 0.1 µm by default), b) maximum chloroplast thickness (2 µm), c) chloroplastmitochondria distance (0.2 µm), d) diameter of the mitochondria (1.4 µm), e) chloroplast-vacuole distance (1.1 µm). f) wall-vacuole distance (3.1 µm). B) Schematic representatation of the reactions and fluxes in the model. CO 2 enters through the cell wall into the cytosol (A) where it is partially converted into bicarbonate (h C,1 ). Both CO 2 and HCO − 3 diffuse towards the chloroplast (with diffusivities d C,1 and d B,1 ), but only CO 2 can easily enter the chloroplast stroma (with diffusivity d C,2 ). This results in dehydration (h C,2 ) close to the chloroplast envelope, in order to maintain the equilibrium between CO 2 and HCO − 3 . Although the chloroplast membrane is relatively impermeable to [HCO − 3 ], some leakage (d B,2 ) may occur because the high chloroplast pH results in a much larger  in the stroma compared to that in the cytosol. Rapid dehydration of HCO − 3 in the center of the chloroplast stroma prevents a large decrease in [CO 2 ], even although CO 2 is fixed by the Rubisco enzyme (f ). An additional source of CO 2 is the respiration (r d ) and photorespiration (r p ) in the mitochondria.  Figure 2: An example of a typical CO 2 response curve (continuous line) as predicted by the C3 biochemical model (Farquhar et al. 1980). The 3D reaction-diffusion model is an extension of this biochemical model that adds a description of the diffusion pathway through the mesophyll. Using the default parameters of the model (Table I), the reaction-diffusion model predicts somewhat lower rates of photosynthesis (dotted line). The dashed line gives an example of the expected rates of photosynthesis when the resistance of the cell wall is 10 times higher than the default value. Vertical lines indicate three different CO 2 concentrations at which the sensitivity of the model to different parameters was tested.  Table I Figure 6: In A and B, the modeled dependency of the net photosynthesis and of internal conductance on the oxygen partial pressure at three different CO 2 levels are shown. Measurements of these parameters at three different oxygen concentrations are included in the same graphs. The Γ * values used for the calculation of internal conductance were determined at the three oxygen levels and were 1.7 Pa, 3.8 Pa and 7.6 Pa. The measured responses of photosynthesis and internal conductance as a function of CO 2 partial pressure are shown in C and D. In (E) and (F) the modeled results are shown, calculated at three different membrane HCO − 3 permeabilities to estimate the effect of HCO − 3 leakage. For all measurements, averages of four Arabidopsis plants are shown ± SE.