The Mechanistic Basis of Internal Conductance: A
Theoretical Analysis of Mesophyll Cell Photosynthesis
and CO2 Diffusion1[W][OA]
Danny Tholen and Xin-Guang Zhu*
Chinese Academy of Sciences and Max Planck Society Partner Institute for Computational Biology, Key
Laboratory of Computational Biology, Shanghai 200031, People’s Republic of China (D.T., X.-G.Z.); and
Institute of Plant Physiology and Ecology, Shanghai Institutes for Biological Sciences, Chinese Academy of
Sciences, Shanghai 200032, People’s Republic of China (X.-G.Z.)
Photosynthesis is limited by the conductance of carbon dioxide (CO2) from intercellular spaces to the sites of carboxylation.
Although the concept of internal conductance (gi) 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 C3 mesophyll cell that includes all major
components of the CO2 diffusion pathway and associated reactions. Using this novel systems model, we systematically and
quantitatively examined the mechanisms underlying gi. 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 CO2 diffusion, an effect that has thus far been ignored when determining gi. Finally, we
show that outward bicarbonate leakage through the chloroplast envelope could contribute to the observed decrease in gi under
elevated CO2. Our analysis suggests that physiological and anatomical features associated with gi have been evolutionarily
fine-tuned to benefit CO2 diffusion and photosynthesis. The model presented here provides a novel theoretical framework to
further analyze the mechanisms underlying diffusion processes in the mesophyll.
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 CO2 from the atmosphere to the site of
carboxylation (Flexas et al., 2008; Warren, 2008a). For
example, limited CO2 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 CO2 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 CO2 between intercellular air spaces
and the Rubisco enzyme is usually described by a
parameter called internal or mesophyll conductance (gi;
for definitions of all symbols used in this work, see
1
This work was supported by a Chinese Academy of Sciences
Fellowship for Young International Scientists (grant no. 2009Y2AS10
to D.T.) and by the National Natural Science Foundation of China
(grant no. 30970213 to X.-G.Z.).
* Corresponding author; e-mail [email protected].
The author responsible for distribution of materials integral to the
findings presented in this article in accordance with the policy
described in the Instructions for Authors (www. plantphysiol.org) is:
Danny Tholen ([email protected]).
[W]
The online version of this article contains Web-only data.
[OA]
Open Access articles can be viewed online without a subscription.
www.plantphysiol.org/cgi/doi/10.1104/pp.111.172346
90
Table I). Different species and even different genotypes
of the same species show substantial variations in gi,
and the limiting effect of gi 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 gi (Warren, 2008a;
Barbour et al., 2010).
Although the concept of gi 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, CO2 diffuses through air
spaces, 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). CO2 from photorespiration and
mitochondrial respiration diffuses through the mitochondrial envelope, cytosol, and chloroplast membranes into the stroma and forms an additional source
of CO2 for photosynthesis. When we consider the
relative contribution of several diffusion barriers, it is
convenient to consider the reciprocal of gi, the internal
resistance (ri). The ri is the sum of all resistances
between the stomata and the Rubisco enzyme. Traditionally, the ri has been divided into two major components (Evans et al., 1994): the intercellular air space
resistance (from substomatal cavities to the cell surface) and the liquid-phase resistance (from the cell
walls to the site of carboxylation). Theoretical and
Plant PhysiologyÒ, May 2011, Vol. 156, pp. 90–105, www.plantphysiol.org Ó 2011 American Society of Plant Biologists
The Mechanistic Basis of Internal Conductance
Table I. Parameters and constants used in the model (at 25°C)
Name
Symbol
Default Value (Range)
Units
22
21
Net assimilation
Flux into the cell per unit of
mesophyll surface area
HCO32 concentration
CO2 concentration
Chloroplast CO2 concentration
Apoplast CO2 concentration
Chloroplast-plasmalemma distance
A
a
Calculated
Calculated
mol m s
mol m22 s21
B
C
Cc
Ci
d
mol
mol
mol
mol
m
Diffusion constant HCO32
Db
Calculated
Calculated
Calculated
Calculated
0.1 3 1026
(5 3 1028–0.7 3 1026)
0.52 3 Dc
Diffusion constant CO2a
CO2 fixation rate
Internal conductance
Wall and plasmalemma conductanceb
Dc
f
gi
Gwall
m2 s21
mol m23 s21
mol m22 s21
mol m22 s21
H+ concentration
Net hydration rate
Volumetric electron transport rate
Electron transport rate per unit of
leaf area
CA turnover rate
H
h
j
J
1.83 3 1029
Calculated
Calculated
0.10
(8.7 3 1023–0.45)
Calculated
Calculated
Calculated
90 3 1026
mol
mol
mol
mol
ka
3 3 105
s21
Rubisco turnover rate
kc
2.84
s21
CA hydration Km
Equilibrium constant CA
CA dehydration Km
Rubisco effective Kmc
Oxygen partial pressure
KCO2
Keq
KHCO3
Km
O
mol
mol
mol
mol
Pa
Air pressure
CO2 partial pressure
P
pi
CO2 membrane permeabilityd
PCO2
Cytosolic pH
HCO32 membrane permeabilityd
pHc
PHCO32
Mitochondrial pH
Stromal pH
Respiration rate
Volumetric respiration rate
Photorespiration rate
Solubility CO2e
pHm
pHs
Rd
rd
rp
sc
1.5
5.6 3 1027
34
18.7 3 1023
21 3 103
(1.0 3 103–40 3 103)
100 3 103
30
(15, 30, 60)
3.5 3 1023
(1 3 1026–16 3 1023)
7.3
5 3 1027
(2 3 1029–1.2 3 1026)
8
8
0.8 3 1026
2.2
Calculated
0.33 3 1023
Mesophyll surface area per unit of
leaf area
Mitochondrial volume per unit of
mesophyll surface area
Stroma volume per unit of mesophyll
surface area
CA concentration cytosol
CA concentration stroma
Sm
15
Vm
Rubisco concentration
CO2 compensation point
Cytosol viscosity
Stroma viscosity
m23
m23
m23
m23
Notes and References
Equation 9
Equation 8
Equation 2
Equation 1
Equation 1
sc 3 pi
Sharkey et al. (1991)
m2 s21
m23
m23 s21
m23 s21
m22 s21
Uchida et al. (1983);
Hoofd et al. (1986)
Hoofd et al. (1986)
Equation 3
Equation 12
Evans et al. (2009)
102pH
Equation 4
Equation 5
Assumed
Pocker and Ng (1973);
Larsson et al. (1997)
Sage (2002);
Eichelmann et al. (2009)
Pocker and Ng (1973)
Pocker and Miksch (1978)
Pocker and Miksch (1978)
von Caemmerer (2000)
Assumed
m23
m23
m23
m23
Pa
Pa
Assumed
Assumed
m s21
Evans et al. (2009)
–
m s21
Felle and Bertl (1986)
See “Materials and Methods”
–
–
mol
mol
mol
mol
m2 m22
Llopis et al. (1998)
Oja et al. (1986)
Assumed
Equation 6
Equation 7
Uchida et al. (1983);
Hoofd et al. (1986)
Slaton and Smith (2002)
0.27 3 1027
m3 m22
See text
Vs
0.87 3 1026
m3 m22
See text
Xa,c
Xa,s
0.5 3 Xa,s
0.27
(0.04–0.69)
mol m23
mol m23
Xc
G*
hc
hs
1.25
1.35 3 1023
2
10
mol m23
mol m23
Relative to water
Relative to water
Rumeau et al. (1996)
Atkins et al. (1972a, 1972b);
Price et al. (1994); Peltier et al.
(1995); Gillon and Yakir (2000)
Assumed
von Caemmerer (2000)
Assumed
Assumed
m22
m23
m23
m23
s21
s21
s21
Pa21
a
b
Assuming a cellular ionic strength of 0.25 M.
Per unit of mesophyll surface. The model has a chloroplast-mesophyll surface ratio of 0.76.
d
Assuming an oxygen concentration of 21 kPa.
Identical permeabilities are assumed for the mitochondrial, chloroplast, and vacuolar
e
Assuming an ionic strength in the apoplast of 20 mM.
membranes.
c
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Tholen and Zhu
experimental results suggested that the resistance of
the intercellular air spaces is negligible in many cases
(Genty et al., 1998; Aalto and Juurola, 2002; Piel et al.,
2002), although it accounts for a significant portion of
ri in thick, hypostomatous leaves (Parkhurst and Mott,
1990; Parkhurst, 1994). The resistances of the cell wall
(25%–50%) and chloroplast envelopes (24%–76%) are
considered to make major contributions to the total ri
(Evans et al., 1994, 2009; Gillon and Yakir, 2000).
Although the parameter gi is widely used to study
diffusion inside a leaf (Flexas et al., 2008; Warren,
2008a; Evans et al., 2009), it represents a simplification
that ignores the three-dimensional (3D) nature of diffusion, and the use of this parameter as an accurate
representation of the diffusional properties of leaves
has been questioned in the past (Jones, 1985; Parkhurst,
1994). The liquid-phase conductance is usually determined from estimates of chloroplast CO2 concentration
and is defined as the total photosynthetic flux divided
by the concentration gradient between the concentration at the point where CO2 enters the liquid-phase (Ci)
and chloroplast CO 2 concentration (C c ). Parkhurst
(1994) elegantly explained that when CO2 diffusion
takes place in a network with distributed sources and
sinks, the apparent conductance becomes dependent
on the assimilation rate. Paradoxically, this could
mean that a decrease in the rate of photosynthesis
results in a lower gi, instead of the other way around.
In addition, the conductance approach usually considers only one dimension and therefore cannot incorporate the effects of the full 3D leaf structure on
diffusion (Parkhurst, 1994; Aalto and Juurola, 2002).
It remains challenging to modify biochemical models
of photosynthesis with an adequate 3D description
of diffusion (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 air space conductance (Gross,
1981; Parkhurst, 1994; Vesala et al., 1996; Aalto and
Juurola, 2002; Galloët and Herbin, 2005).
Because the liquid-phase resistance is currently
thought to be the most limiting part of the total ri
(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 CO2 is spatially separated from the influx of intercellular CO2.
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 gi is determined in a way that considers all the
physical resistances between a CO2 source and sink, it
92
can be expected that leaves would have a constant
gi unless the physical properties of the pathway
change. Indeed, gi was originally assumed to be constant throughout a leaf’s lifespan (Evans and von
Caemmerer, 1996), but more recent evidence has called
this assumption into question. For example, gi can
respond rapidly to changes in leaf temperature
(Bernacchi et al., 2002; Yamori et al., 2006) or intercellular CO2 concentration (Flexas et al., 2007). In addition, gi 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 large increase in gi produced by raising 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 gi 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 gi may be
the lack of a proper method to evaluate the contributions of both anatomical and biochemical components.
Experimentally testing the contribution of each factor
to gi 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 of a number of factors may provide a useful tool to describe the
mechanistic basis of gi. 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 gi to environmental perturbations.
THEORY
We assumed a leaf type in which air space resistance
is negligible and the apoplastic CO2 concentration (Ci) is
in complete equilibrium with the partial pressure of
CO2 in intercellular air spaces (pi). 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 C3 leaf (Winter et al., 1994; Pyankov et al., 1999).
Because diffusion through liquid is relatively slow,
most of the cytosolic CO2 flux is expected to go through
the small layer of cytoplasm between cell walls adjacent
to intercellular air spaces and the chloroplast membrane facing this cell wall. In this case, the surface area
Plant Physiol. Vol. 156, 2011
The Mechanistic Basis of Internal Conductance
of the chloroplast membrane facing intercellular air
spaces should correlate with gi, and this has been experimentally confirmed (Terashima et al., 2006). In this study,
we did not examine this well-known effect of the chloroplast surface area and kept it constant at 11.4 m2 m22.
A 3D geometry 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 mm and contained 96 chloroplasts and 96 mitochondria. The subcellular structure was
further defined by the following parameters: chloroplastwall distance (variable, 0.1 mm by default), maximum
chloroplast thickness (2 mm), chloroplast diameter (3.5
mm), chloroplast-mitochondria distance (0.2 mm), diameter of the mitochondria (1.4 mm), chloroplast-vacuole
distance (1.1 mm), and wall-vacuole distance (3.1 mm).
Analogous to the equations given by Cowan (1986),
the diffusive flux of CO2 through the liquid phase is
described by:
Dc 2
, C ¼ f þ h 2 rd 2 rp
h
2
2
ð1Þ
2
where ,2 C ¼ ddxC2 þ ddyC2 þ ddzC2 , Dc is the liquid-phase
diffusion coefficient (m2 s21) for CO2, h is a dimensionless factor representing the relative viscosity of the
compartment in which the diffusion takes place, C is
the CO2 concentration (mol m23) at a given x, y, and z
coordinate, f is the rate of ribulose 1,5-bisphosphate
(RuBP) carboxylation, h is the net hydration rate of
CO2 to bicarbonate (HCO32), rd is respiration rate, and
rp is the rate of photorespiratory CO2 release (all rates
are in mol m23 s21).
The reaction terms are derived from a biochemical
model of C3 photosynthesis (Farquhar et al., 1980, see
“Materials and Methods”). These terms differ in different cellular compartments (i.e. f = 0 in mitochondria, and rd = rp = 0 inside chloroplasts). The equation
for HCO32 diffusion is:
Figure 1. A, The 3D structure of a mesophyll cell with a diameter of 20 mm and containing 96 chloroplasts and 96 mitochondria.
The structure was further defined by the following parameters: a, chloroplast-wall distance (variable, 0.1 mm by default); b,
maximum chloroplast thickness (2 mm); c, chloroplast-mitochondria distance (0.2 mm); d, diameter of the mitochondria (1.4
mm); e, chloroplast-vacuole distance (1.1 mm); and f, wall-vacuole distance (3.1 mm). B, Schematic representation of the
reactions and fluxes in the model. CO2 enters through the cell wall into the cytosol (A), where it is partially converted into HCO32
(hC,1). Both CO2 and HCO32 diffuse toward the chloroplast (with diffusivities dC,1 and dB,1), but only CO2 can easily enter the
chloroplast stroma (with diffusivity dC,2). This results in dehydration (hC,2) close to the chloroplast envelope, in order to maintain
the equilibrium between CO2 and HCO32. Although the chloroplast membrane is relatively impermeable to HCO32, some
leakage (dB,2) may occur, because the high chloroplast pH results in a much larger [HCO32] in the stroma compared with that in
the cytosol. Rapid dehydration of HCO32 in the center of the chloroplast stroma prevents a large decrease in [CO2], even
although CO2 is fixed by the Rubisco enzyme (f ). An additional source of CO2 is the respiration (rd) and photorespiration (rp) in
the mitochondria.
Plant Physiol. Vol. 156, 2011
93
Tholen and Zhu
Db 2
, B ¼ 2h
h
ð2Þ
where Db is the diffusion coefficient (m2 s21) for
HCO32, and B is the HCO32 concentration (mol m23).
Different subdomains (chloroplasts, cytosol, mitochondria) were defined inside the 3D geometry of the
mesophyll cell, and the equations above were applied
to each subdomain. In addition, boundary resistances
were defined between the different subdomains (see
“Materials and Methods”). The resulting model was
solved using the finite element method (COMSOL
Multiphysics Reference Guide, version 3.5a) and resulted in an estimate of the steady-state CO2 and
HCO32 concentrations in each discrete element of the
model structure.
RESULTS
The CO2 and HCO32 Concentrations in the Liquid Phase
The model was solved using default parameter
values given in Table I. In Supplemental Figure S1,
the resulting CO2 and HCO32 concentrations at a pi of
30 Pa are shown. Except for the small region between
chloroplast and cell membrane, the CO2 concentrations in the cytosol were close to those in the apoplast.
The average CO2 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 HCO32 concentration in the stroma would be 4.80 times that of the
cytosol. In good agreement with this, the reactiondiffusion model predicted an average HCO32 concentration 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 CO2 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 sections, we will
analyze the sensitivity of the reaction-diffusion model to
different parameters at three CO2 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 diffu94
Figure 2. An example of a typical CO2 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 CO2
concentrations at which the sensitivity of the model to different
parameters was tested.
sion systems. Therefore, instead of analyzing the effect
of a given model parameter on internal conductance
(gi), 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 model with experimental
results using conductances, we estimated gi from the
simulation results as the conductance between Ci and
the average CO2 concentration in the chloroplast (see
Eq. 11 below).
CO2 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. Figure
3, A and B, shows the effects of a change in the
permeability of the wall and plasmalemma (Gwall) on
the total internal conductance and on the relative rate
of photosynthesis. The figure shows a saturating
response; at ambient CO2, 90% of the maximum rate
of photosynthesis could be reached with a Gwall of
0.02 mol (m wall)22 s21. Interestingly, photosynthesis
changed little in the range of biologically relevant
values for Gwall (indicated by the white area in Fig. 3A)
at elevated CO2 levels (pi = 60 Pa).
CO2 subsequently diffuses through a thin layer of
cytosol, where it merges with CO2 from photorespiration and respiration. In Figure 3, C and D, the effects of
increasing the distance between chloroplast and plasmalemma (distance a in Fig. 1) are shown. Although
increasing this distance resulted in a lower gi, the
Plant Physiol. Vol. 156, 2011
The Mechanistic Basis of Internal Conductance
Figure 3. The relative rates of photosynthesis (A)
and internal conductance (B) as affected by the
cell wall (and plasmalemma) conductance. The
model was run at three different intercellular CO2
levels. The gray areas indicate wall conductances
outside the biologically relevant range as given in
Table I. Vertical lines indicate default values used
in the model. The effects of a change in the
structure on photosynthesis (C) and internal conductance (D) were examined by increasing the
distance (d ) between the cell wall and the chloroplast envelope. Eight different model structures
(indicated by the points) were analyzed.
predicted decrease in photosynthesis over this range
remained below 3%.
Figure 4 shows that decreasing the permeability
(PCO2) of the chloroplast envelope to CO2 had an effect
similar to that of decreasing Gwall. However, when PCO2
becomes unrealistically low, the simulation indicated
that CO2 fixation becomes nearly zero and a significant
amount of (photo)respiratory CO2 leaks out of the cell,
resulting in a negative net rate of photosynthesis.
Interestingly, an increased permeability of the chloroplast membrane to HCO32 decreased photosynthetic
rates (Fig. 4C). This occurred because the stromal
HCO32 concentration is almost five times higher than
that of the cytosol, and when the membrane permeability is high, such a large gradient results in a significant
leakage of HCO32 from the stroma into the cytosol.
We examined whether CA plays a significant role in
the facilitation of CO2 transport across the chloroplast
stroma. Removing all CA from the chloroplast stroma
decreased photosynthesis (at pi = 30 Pa) a little over 7%
compared with that at default conditions (Fig. 5A).
Note that over the same range of CA concentrations, gi
changed by more than 40% (Fig. 5B).
The Effect of Oxygen and CO2 Concentrations on the
Apparent Conductance
Figure 6, A and B, shows the effect of different
oxygen partial pressures on photosynthesis and internal conductance. It is well known that increasing
oxygen levels inhibit photosynthesis as a result of an
increased photorespiration. However, we found that
Plant Physiol. Vol. 156, 2011
the model also predicted that gi would decrease with
increasing oxygen concentrations (Fig. 6B), especially
when pi was low. Using measurements of chlorophyll
fluorescence and gas exchange, we estimated gi in
Arabidopsis (Arabidopsis thaliana) at three different
oxygen levels (and at an average pi of 18 Pa). Although
the absolute values differ, the measured trend was
similar to the model predictions, with gi being lower at
higher oxygen levels (Fig. 6B). We also included experimental data for the response of photosynthesis
and gi to pi (Fig. 6, C and D) and modeled this relation
with the reaction-diffusion model (Fig. 6, E and F).
Surprisingly, even though none of the individual diffusion resistances in the model changed in response to
CO2, the predicted gi increased rapidly with an increase of pi from 0 to about 10 Pa but decreased when pi
was further increased (Fig. 6F). The large increase in gi
at low pi was not observed when the simulation was
performed at 1% oxygen (Supplemental Fig. S2). Varying the permeability of the chloroplast envelope to
HCO32 affected the decrease of gi at high pi, which
suggests that this effect could be attributed to an
increased leakage of HCO32 from the chloroplast
stroma into the cytosol.
DISCUSSION
A 3D Reaction-Diffusion Model Enables a Systematic
Study of the Internal Conductance
Here, we have presented an integrated 3D reactiondiffusion model that enables a systematic exploration
95
Tholen and Zhu
Figure 4. The effects of chloroplast envelope
permeability to CO2 (A and B) and HCO32 (C
and D) on photosynthesis (A and C) and internal
conductance (B and D). The gray areas indicate
permeabilities outside the biologically relevant
range as given in Table I. The dashed vertical lines
indicate default values used in the model.
of factors that may affect internal conductance and leaf
photosynthesis. Previous studies have estimated the
resistance of a cellular component by dividing the
thickness of the component by its diffusion constant
(Evans et al., 1994, 2009). However, this method cannot
account for differences in the structure and location of
the components or their 3D nature. To better compare
our data with such estimates, we calculated resistances
using the average CO2 concentration gradients and
fluxes through the cell wall, cytosol, chloroplast envelope, and stroma. The resulting resistances were
compared with the results of the resistance-based
approach presented by Evans et al. (2009; 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 resistancebased model, such as the assumption that the diffusion
between the wall and the chloroplast envelope only
occurs across the small cytoplasmic volume where the
chloroplast is directly adjacent to the intercellular air
space. In reality, diffusion occurs through three dimensions, and even the back side of the chloroplast
contributes somewhat to the total conductance (and
receives CO2 from respiration and photorespiration).
Since the resistance in both calculations was expressed
per unit surface area exposed to intercellular air
spaces, the reaction-diffusion model predicts somewhat lower values for the resistance of the chloroplast
membranes.
96
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 by the amount of CA in the stroma that
facilitates the diffusion of CO2. The effective diffusion
path length for CO2 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 through the stroma is influenced
by the amount of CA in the stroma. This enzyme can
improve the diffusion through the stroma by allowing
part of the carbon to diffuse in the form of HCO32.
Diffusion facilitation by CA can be viewed as a way
to maintain a nearly constant CO2 concentration
throughout the stroma that allows a more effective
use of the available Rubisco (Supplemental Fig. S3).
Without stromal CA, the CO2 concentration in the
center of the chloroplast would drop, leading to lower
rates of photosynthesis and increased rates of photorespiration.
Evans et al. (2009) proposed that in the presence
of CA, diffusion increases with a factor of 1 + B/C 3
Db/Dc. Using this equation, they calculated that the
diffusion of CO2 and HCO32 combined would lead to
Plant Physiol. Vol. 156, 2011
The Mechanistic Basis of Internal Conductance
Figure 5. The effects of the stromal CA concentration on photosynthesis (A) and internal conductance (B). The gray areas indicate concentrations that are thought to be outside the biologically relevant range
(Table I). The dashed lines indicate the default values used in the model.
an approximately 26 times higher carbon flux in the
chloroplast stroma (at pH 8, B/C is approximately 50)
than if CO2 were the only diffusing substance. However, this equation implicitly assumes that sufficient
CA is present and HCO32 and CO2 are in complete
equilibrium. From Figure 5, it can be concluded that
such a situation occurs when over 1 mM CA is present
in the stroma. Current estimates of the CA concentration in the stroma of herbaceous plants are between
0.04 and 0.69 mM (see “Materials and Methods”), suggesting that the amount of CA may somewhat limit the
conductance in the stroma. Cowan (1986) used an analytical solution of a one-dimensional reaction-diffusion
model of the chloroplast stroma to describe the interaction between CO2 fixation, (de)hydration, and diffusion. Such an approach is similar to ours and can take
the limiting effect of CA into account. The expected
diffusion resistance without facilitation, using a CO2
diffusion coefficient of 2 3 1029 m2 s21 and an effective
Plant Physiol. Vol. 156, 2011
diffusion path length of 0.85 mm (a stroma thickness
of 1.7 mm), is then 0.85 3 1026/2 3 1029 = 425 s m21 or
13 m2 s bar mol21 . This is only about two times higher
than the facilitated stroma resistance estimated by the
Cowan (1986) model (7 m2 s bar mol21), giving a
facilitation factor of 1.9.
In Table II, we analyzed the diffusion resistance of
the chloroplast stroma with the reaction-diffusion
model under both CA-limited and CA-saturated conditions. The facilitation factor under CA-saturating
conditions, as predicted by the reaction-diffusion
model, was 27, close to that estimated by Evans et al.
(2009). Under default conditions (Xa,s = 0.27 mM), the
facilitation factor was close to 7, suggesting that this
CA concentration is somewhat limiting for diffusion.
When we take this facilitation factor into account, the
stroma resistance calculated by a resistance approach
is still two times higher compared with those resulting
from the reaction-diffusion model. This suggests that
the effective diffusion path length in the modeled
stroma at pi = 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 (for further discussion, see Parkhurst,
1994).
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 gi by 44%
compared with the default values. Nevertheless, the
effect on the net rate of photosynthesis at a pi 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% than if the same amount of
nitrogen was invested in Rubisco alone (Supplemental
Fig. S4).
Because of the lower pH in the cytosol compared
with the chloroplast stroma, cytosolic CA is not expected to play an important role in increasing diffusion through the cytosol, and its function has remained
elusive (Evans et al., 2009). However, CA may help to
increase the flux of CO2 across membranes by steepening the concentration gradients across the membrane and the adjacent unstirred layers (Broun et al.,
1970; Gutknecht et al., 1977). Thus, after entering the
cytosol, CO2 is rapidly converted into HCO32, increasing the gradient in CO2 concentrations (Fig. 1B). Close
to the chloroplast envelope, there is net dehydration of
HCO32 into CO2 that subsequently diffuses into the
chloroplast stroma. This process can maintain a high
CO2 flux through the chloroplast membrane. Because a
certain amount of volume is required for significant
hydration and dehydration of CO2, a very small distance between the chloroplast and the cell wall results
in a slight decrease in the internal conductance (Fig.
97
Tholen and Zhu
Figure 6. A and B, The modeled and measured
dependency of the net photosynthesis and of
internal conductance on the oxygen partial pressure at three different CO2 levels. For the measured data, G* was determined at 10, 21, and 40
kPa oxygen and was 1.7, 3.8, and 7.6 Pa. C and D,
The measured responses of photosynthesis and
internal conductance as a function of CO2 partial
pressure. E and F, The modeled results, calculated
at three different membrane HCO32 permeabilities to estimate the effect of HCO32 leakage. For
all measurements, average values of four Arabidopsis plants are shown 6 SE.
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
CO2 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 CO2 concentration is determined by both
the net CO2 influx and the CO2 flux originating in the
mitochondria, but gi is usually calculated using only
the net photosynthetic flux. An increase in photorespiration reduces net photosynthesis but may maintain
a similar chloroplast CO2 concentration. Thus, even
although Cc is similar, the apparent gi calculated using
Equation 12 (below) would be lower. Such an addi98
tional “resistance” resulting from the effect of respiration and photorespiration has thus far been neglected
when estimating gi. We include an explanation of this
phenomenon using the traditional resistance analogy
in Supplemental Figure S5. Based on the equations
given there, the apparent resistance incurred by respiration and photorespiration can be estimated and is
included in the resistance-based model (Table II). The
results correspond well with the estimations from the
reaction-diffusion model.
The direct effect of photorespiration on gi may
provide an explanation for a number of experimental observations. First, the strong increase in gi after
increasing pi from low to ambient levels (Flexas et al.,
2007; Vrábl et al., 2009; Fig. 6D; Supplemental Fig. S2)
can be explained by the photorespiration flux becoming smaller. Tazoe et al. (2011) observed this trend for a
reduced gi at low CO2 concentrations only at 21%
oxygen but not at 2% oxygen. This agrees well with
our modeled results (Supplemental Fig. S2). Second,
Evans et al. (1994) and Tazoe et al. (2011) found that gi
Plant Physiol. Vol. 156, 2011
The Mechanistic Basis of Internal Conductance
Table II. Partitioning of liquid-phase resistance over the different components of the diffusion pathway
The traditional resistance-based approach is compared with results determined from the reaction-diffusion model. For the resistance model, liquidphase resistance was calculated according to Evans et al. (2009) using the parameters from Table I and expressed per unit of exposed chloroplast
surface. Note that in the chloroplast, an effective diffusion path length of 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 of exposed chloroplast surface.
Liquid-Phase Resistance
Component
Resistance Model
Reaction-Diffusion Model
(Saturating CA)
7.6 (25%)
6.6 (22%)
8.7 (29%)
2.8 (9%)
4.6 (15%)
30.2
7.6 (33%)
6.1 (26%)
5.0 (22%)
1.4 (6.2%)
3.1 (14%)
23.3
Wall and plasmalemma (Gwall = 0.1 mol m22 s21)
Cytosol (thickness = 0.2 mm)
Chloroplast envelope (PCO2 = 0.35 cm s21)
Chloroplast stroma (thickness = 0.9 mm)
(Photo)respiration
Total
mchl2 s bar mol21
is lower at 21% oxygen than at 2% oxygen. Although
a decrease in gi under photorespiratory conditions
has not always been observed (von Caemmerer and
Evans, 1991; Loreto et al., 1992), the variations in such
measurements are often quite substantial, and a small
effect of oxygen on gi may have gone unnoticed.
An additional complicating factor is the uncertainty
surrounding the estimate of the amount of isotope
fractionation associated with photorespiration (Tazoe
et al., 2009, 2011). Third, a reduction in stomatal
conductance is often paralleled by a decrease in gi
(Flexas et al., 2008). This effect may be partially
explained by higher rates of photorespiration after
stomatal closure. Lastly, Priault et al. (2006) showed a
link between photorespiration and internal conductance in a mutant lacking complex I of the mitochondrial electron transport chain. Our results (Fig. 6B)
indicate that at low pi, the rate of photorespiration
would need to double to explain a 50% reduction in gi.
However, photorespiration was reported to be only
40% higher in the mutants, suggesting that this mutation has additional effects that result in a lower gi.
The model predicted that at pi = 30 Pa, gi increases by
11% when the oxygen 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 gi to environmental
variables such as drought and temperature must be
carefully reexamined, taking into account that some
variation in gi may be attributable to differences in
photorespiratory fluxes.
Leakage of HCO32 across the Chloroplast Membrane
May Explain the Decrease of gi with Increasing CO2
A number of studies (Flexas et al., 2007; Hassiotou
et al., 2009; Vrábl et al., 2009; Yin et al., 2009) have
shown a similar response to CO2 as that shown in
Figure 6D (i.e. the internal conductance first increases
rapidly and then slowly decreases as the CO2 concenPlant Physiol. Vol. 156, 2011
Reaction-Diffusion Model
(Default)
7.6 (27%)
6.7 (23%)
5.0 (18%)
4.9 (17%)
4.3 (15%)
28.5
tration increases). Flexas et al. (2007) obtained this
result by using the fluorescence method and confirmed these measurements by using the more reliable
carbon isotope method. However, Tazoe et al. (2009)
demonstrated that the single-point carbon isotope
method used by Flexas et al. (2007) may lead to large
errors in the estimation of gi. Furthermore, Evans
(2009) suggested that estimates of electron transport
rate by the fluorescence method (which are needed
to calculate gi) are taken from the leaf surface and
may not be representative for electron transport rates
deeper in the leaf. However, in this case, it is surprising that large differences in gi between ambient and
saturating CO2 concentrations are found in Arabidopsis (Flexas et al., 2007), which has rather thin leaves
with few differences between fluorescence parameters
measured from the top or bottom (Bunce, 2008). A
recent detailed analysis of the response of gi to CO2 in
tobacco (Nicotiana tabacum) using tunable diode laser
spectroscopy found a significant decrease in gi when
CO2 increased (Tazoe et al., 2011), although this decrease was much smaller than that predicted by fluorescence measurements. The response of gi to CO2 as
measured by Tazoe et al. (2011) is comparable with our
modeled results for gi predicted under conditions
where the HCO32 permeability of the chloroplast
envelopes is above 1027 m s21 (Fig. 6F). This suggests
that HCO32 leakage may be responsible for the decrease in gi at high CO2, as is explained below.
The large pH gradient between the cytosol and
chloroplast stroma results in a HCO32 concentration
that is about five times higher in the stroma compared
with the cytosol (Werdan and Heldt, 1972). However,
when the external CO2 concentration was increased,
the concentration difference was lower than expected,
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 HCO32 is very low, some
99
Tholen and Zhu
leakage of HCO32 may occur, especially at high CO2
concentrations. Such a leakage from stroma to cytosol
reduces the concentration gradient and leads to a
decrease in gi that is bigger at high CO2 concentrations.
Our simulation showed that if the chloroplast membrane permeability to HCO32 was lowered, this
decrease in gi could be eliminated (Fig. 6F). If PHCO32
varies between different species, this could account for
the absence of a significant reduction in gi in some
species (Tazoe et al., 2009, 2011). The modeled decrease
was much lower than that estimated by fluorescencebased methods (up to 90%; Fig. 6D; Flexas et al., 2007)
but comparable with the 26% to 40% decrease determined by the isotopic method (Tazoe et al., 2011).
Implications for Experimental Approaches That Estimate
Internal Conductance
Our determination of gi is based on the average
chloroplast CO2 concentration given by the model
solution. In practice, several methods have been used
to estimate this average Cc (Pons et al., 2009). To study
the relation between gi and CO2, measurements of
gas exchange combined with chlorophyll fluorescence
or isotopic discrimination against 13CO2 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 (below) can be used to
reliably estimate Cc. 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 gi. By
contrast, the isotopic method is often seen as a more
reliable way to determine gi (Terashima et al., 2006;
Pons et al., 2009). The ratio between carbon isotopes in
air coming out of the leaf is related to gi, because
diffusion through the liquid phase discriminates
against 13C. The isotopic method was developed based
on a linear model of resistances (Farquhar et al., 1982)
and therefore may 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 discrim-
ination against 13C in C3 leaves. Leakage of HCO32
over the membrane would effectively lengthen the diffusion pathway and be measurable as an increased
discrimination against 13C. However, the dehydration
of this HCO32 in the cytosol results in a discrimination
against 12C, which will partly mask this effect. Using a
reaction-diffusion model to analyze the diffusion of
13
CO2 and 12CO2 independently would allow for the
analysis of the assumptions underlying the isotopic
method and will be the focus of future research.
The Importance of the Different Physical Components to
Photosynthetic CO2 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, we limited the minimum membrane
conductance to 0.1 mm s21. This value is still five
times higher than that reported by Uehlein et al.
(2008). However, the shortcomings of current methods
for determining permeabilities from isolated vesicles
and the effect of unstirred layers along the membranes
may lead to a large underestimation of in vivo membrane permeabilities (Missner and Pohl, 2009; R.
Kaldenhoff, personal communication). 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 (Oryza sativa) 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.,
Table III. Predicted limitation of the rate of photosynthesis by liquid-phase diffusion limitations (Figs. 3–5) at three different CO2 concentrations
For comparison, the estimated limitation by the stomatal conductance (for a gs between 0.1 and 0.4 mol m22 s21) was included. The relative
limitations were calculated by dividing the rate of photosynthesis when a given component has a finite conductance by the rate of photosynthesis
when the component has infinite conductance (Parkhurst, 1994).
Component
Stomatal conductance (gs = 0.1–0.4 mol m22 s21)
Wall and plasmalemma (G = 0.01–0.4 mol m22 s21)
Cytosol (thickness = 0.2–0.7 mm)
Chloroplast envelope (PCO2 = 0.1–10 mm s21)
CA concentration stroma (Xa,s = 0.04–0.69 mM)
100
Limitation of Photosynthesis
pi = 15 Pa
pi = 30 Pa
pi = 60 Pa
%
11–32
0.9–27
4.0–5.4
1.3–74
0.9–6.9
8.2–24
0.7–21
3.1–4.2
0.8–50
0.5–4.3
1.6–5.1
0.1–5.0
0.6–0.8
0.1–21
0.1–0.8
Plant Physiol. Vol. 156, 2011
The Mechanistic Basis of Internal Conductance
2011). However, we showed that this effect is much
less pronounced at elevated CO2 (Fig. 3A). Thus, the
increased cell wall thickness that is sometimes observed
at elevated CO2 (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 gi (Fig. 3, C and D; Table III).
Sharkey et al. (1991) reported that the distance between the cell wall and the chloroplast more than
doubled in transgenic plants with excess phytochrome
when compared with wild-type tobacco. In contrast to
our results, those authors reported a 43% decrease in
assimilation rates at ambient CO2 and a 27% reduction
in gi. However, additional pleiotropic effects on gi in
such transgenic plants are hard to exclude. For example, the surface area of chloroplasts exposed to intercellular air space was not determined, but it might
have been able to explain part of the observed decreases in photosynthesis and gi. In this respect, it is
interesting to consider that gi correlated linearly with
the amount of surface area exposed to intercellular air
spaces in a mutant with an altered chloroplast location
(chup1); no additional effect of the increased distance
between the cell wall and chloroplasts was found
(Tholen et al., 2008).
Physiological and Anatomical Features Associated with
gi May Have Been Fine-Tuned through Natural Selection
to Benefit CO2 Diffusion and Photosynthesis
Increasing gi has been suggested as a method for
optimizing photosynthetic efficiency in crop species
(Barbour et al., 2010; Zhu et al., 2010). Figures 3 and 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 (Fig. 3, A and B) are
often observed for woody species (Gillon and Yakir,
2000) and can clearly form a significant limitation for
photosynthesis. In addition, the lower end of the
estimates for the membrane CO2 permeability would
result in a severe limitation of photosynthesis, but it
should be stressed that there is a large uncertainty
surrounding these values (Evans et al., 2009).
Care must be taken with the use and interpretation
of the parameter gi; a large increase in gi does not
necessarily result in a significant increase of photosynthesis. For example, increasing the wall conductance from 0.1 to 1.0 mol m22 s21 would increase gi 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.
Plant Physiol. Vol. 156, 2011
CONCLUSION
This work presents a spatial model of the liquidphase diffusion pathway in the mesophyll, which can
be used to explore the influences of physical and
structural characteristics of the leaf on photosynthesis
and gi. With this model, we systematically examined
factors influencing CO2 diffusion and gained new
insights into mechanisms that affect this process. We
demonstrated that photorespiration has an important
effect on the apparent gi and that the decrease in
internal conductance at high CO2 concentrations can
be explained, at least partially, by a leakage of HCO32
from the stroma to the cytosol. This work also demonstrated how 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 CO2 fixation in the chloroplast was calculated (Farquhar et al., 1980;
von Caemmerer, 2000) as:
f ¼ min
kc X c C
jC
;
C þ Km 4C þ 8G
!
ð3Þ
where kc (s21) is the catalytic rate of Rubisco, Xc (mol m23) is the Rubisco
concentration in the chloroplast, C (mol m23) is the CO2 concentration, Km
(mol m23) is the effective Michaelis-Menten constant in the presence of
oxygen, j (mol m23 s21) is the volumetric electron transport rate, and G* (mol
m23) is the compensation point in the absence of mitochondrial respiration
(for an overview of the parameters used in this work, see Table I). Since
detailed information about variation in biochemistry is lacking, we set all
biochemical parameters to be constant throughout the leaf.
The net hydration h (mol m23 s21) of CO2 to HCO32 by CA was modeled
with a first-order approximation (modified from Spalding and Portis, 1985):
h¼
ka Xa ðC 2
BH
Keq
Þ
CO2
ðKCO2 þ KKHCO3
B þ CÞ
ð4Þ
where ka (s21) is the turnover rate of CA, Xa (mol m23) is the CA concentration,
B (mol m23) is the HCO32 concentration, H (mol m23) is the proton concentration derived from the pH value, KCO2 and KHCO3 (mol m23) are the
Michaelis-Menten constants of hydration and dehydration, respectively, and
Keq (mol m23) is the equilibrium constant.
The volumetric electron transport rate in the chloroplast stroma was
calculated from the total electron transport rate per unit of leaf area (J [mol
m22 s21]) as:
j¼
J
Sm V s
ð5Þ
where Sm (m2 m22) is the mesophyll surface area per unit of leaf area, and Vs
(m3 m22) is the stroma volume per unit of mesophyll wall surface area.
Similarly, the volumetric rate of respiration was calculated as:
rd ¼
Rd
Sm Vm
ð6Þ
where Vm (m3 m22) is the mitochondrial volume per unit of mesophyll wall
surface area. Note that the respiration rate per unit of leaf area, Rd (mol m22 s21),
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 model by Farquhar et al. (1980). Half this rate
corresponds to the expected CO2 release in the mitochondria (von Caemmerer,
2000). We integrated this rate over the chloroplast volume (chl) to get the total
photorespiratory CO2 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 (rp):
101
Tholen and Zhu
ÐÐÐ
ðf GC Þ dx dy dz
ÐÐÐ
rp ¼
ð7Þ
chl
dx dy dz
mit
where f is the local CO2 fixation rate calculated from Equation 3.
A number of boundary conditions for the model were defined. CO2 enters
the cell through the cell wall, so analogous to Cowan (1986), we defined a
constant cell wall (including the plasma membrane) conductance, Gwall (mol
m22 s21). The total influx (mol m22 s21) at each point along the cell wall is then
described by:
a¼
Gwall
ðCi 2 Cwall Þ
sc P
23
ð8Þ
21
where P (Pa) is the air pressure, sc (mol m Pa ) is the solubility of CO2, Ci
(mol m23) is the CO2 concentration at the point where CO2 enters apoplastic
water layers, and Cwall is the CO2 concentration in the cytosol on the cell wall
boundary. The average influx into a mesophyll cell can be determined by
integrating an over the cell wall surface (Swall) and dividing by the total surface
area of this cell wall. The net photosynthesis per unit of leaf area (A [mol m22
s21]) can then be calculated by multiplying by Sm:
Ð
a dSwall
S
A ¼ Ðwall
Swall
dSwall
Sm
ð9Þ
We considered the wall and plasmalemma to be impermeable to HCO32. In
addition to these external boundaries, the chloroplast envelope, mitochondrial
envelope, and tonoplast were described as thin boundary layers having a
fixed permeability:
n×ð 2 D,CÞ ¼ PCO2 ðC1 2 C2 Þ
ð10Þ
The left side of this equation represents the normal diffusive flux across the
boundary, PCO2 is the permeability of the boundary to CO2, and C1 and C2 (mol
m23) are CO2 concentrations on the two sides of the boundary. A corresponding boundary was also defined for HCO32.
The average CO2 concentration in the chloroplast was calculated by
integrating the local CO2 concentrations in the stroma divided by the total
chloroplast volume:
ÐÐÐ
Cc ¼
Cc dx dy dz
ÐÐÐ
chl
ð11Þ
dx dy dz
chl
The gi (mol m22 s21) was calculated by:
gi ¼
AP
ðCi 2 Cc Þ=sc
ð12Þ
Estimates of internal conductance by the variable J method (Harley et al.,
1992; Loreto et al., 1992) rely on the assumption that the average CO2
concentration in the stroma can be calculated from the electron transport
rate (assuming that NADPH is limiting for RuBP regeneration):
Cc ¼
G ð8Rd þ J þ 8AÞ
ð J 2 4Rd 2 4AÞ
ð13Þ
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 23,900 units
mg21 (Atkins et al., 1972b), a molecular mass of 28 kD per active site, and a
chloroplast volume of 30 mm mg21 chlorophyll, it can be calculated that the
CA concentration in these species was between 0.04 and 0.69 mol active sites
m23. 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 m23 CA
for a number of herbaceous species, corresponding well with the earlier work
of Atkins et al. (1972a, 1972b). 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
(Nicotiana tabacum) by Gillon and Yakir (0.27 mol m23).
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 mL21 (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 two to five times,
larger than that of dilute buffer. Köhler et al. (2000) determined the diffusion
coefficient of GFP in tobacco cells. Their results indicate that diffusion through
the stroma is about 100, and that through the cytosol is two to three times,
slower compared with that through an aqueous solution. These high values
are in contrast to an analysis of diffusion coefficients for CO2 in different
human tissues, where a protein concentration of 40% corresponds to an
apparent diffusion coefficient 70% lower than that of CO2 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 gi to differences in
stromal viscosity are shown in Supplemental Figure S6.
Another source of uncertainty is the permeability of the chloroplast and
mitochondrial membranes to CO2 and HCO32. Reported values for the
permeability of biological membranes to CO2 vary over 4 orders of magnitude
(Evans et al., 2009). Based on the measured total diffusion resistance, a
permeability of around 0.35 cm s21 is to be expected (Evans et al., 2009). Values
for the permeability to HCO32 are scarce and have been assumed negligible
(Cowan, 1986). However, the lower limit for HCO32 permeability through
artificial phospholipid membranes was estimated to be around 4.3 3 1028 m
s21 (Norris and Powell, 1992). Heber and Purczeld (1978) reported a HCO32
uptake rate by isolated chloroplast between 30 and 100 mmol (mg chlorophyll)21 h21 in the presence of a concentration difference of 100 mM. This
translates into a permeability of around 0.2 3 1028 m s21, assuming a spherical
chloroplast with volume of 25 mm (mg chlorophyll)21 and a diameter of 2 mm.
This value is similar to the maximum HCO32 permeability predicted for a
simple lipid bilayer by Gutknecht et al. (1977) but somewhat lower compared
with the HCO32 permeability reported for the membranes of erythrocytes
(1.2 3 1026 m s21; Chow et al., 1976) and of colon epithelia cells (0.87 3 1026
m s21; Endeward and Gros, 2005). We took a HCO32 membrane permeability
of 5 3 1027 m s21 as a starting point for the simulations (Table I) and tested for
the effect of variation in this parameter (Figs. 4, C and D, and 6, E and F).
For the model, the values obtained by this expression were equal to those
calculated by Equation 11.
Spatial Representation of the Liquid-Phase
Diffusion Path
Parameterization 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 mmol m22 s21, a chloroplast
CO2 compensation point of 4.1 Pa, and a dark respiration rate of 0.8 mmol m22
s21 (Table I). The concentration of CA was based on in vitro and in vivo activity
measurements (Atkins et al., 1972a, 1972b; Price et al., 1994; Peltier et al., 1995;
102
The cell structure shown in Figure 1 contained 96 chloroplasts and 96
mitochondria. The time used to solve the model could be drastically reduced
by making use of the inherent symmetry of the structure. Thus, the solution
was found by using a section of one-eighth of the total volume. This section
consisted of a curved cell wall, 12 chloroplast with a fixed distance to this wall
(0.1 mm in the default model), 12 spherical mitochondria (ø = 1.4 mm) at a fixed
distance from the chloroplast (0.1 mm), and a vacuole at a distance of 1.1 mm
Plant Physiol. Vol. 156, 2011
The Mechanistic Basis of Internal Conductance
from the chloroplasts. The volume-surface area ratio of the chloroplast was 0.5
mm. Leaf level photosynthesis was calculated by multiplying the parameters
per unit of cell wall with estimations of the total mesophyll surface area
exposed to intercellular air spaces per unit of area (Sm = 15 m2 m22).
earlier draft of the manuscript, in particular Vincent Devloo, Wang Yue, and
Carolina Boom for helpful discussions on mesophyll diffusion and Yan Xin
for technical assistance. We thank Martin Wang of CnTech Co., Ltd., for
advice and help with using COMSOL Multiphysics.
Received January 7, 2011; accepted March 20, 2011; published March 25, 2011.
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 CO2 and HCO32 diffusion 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 in Supplemental Data S1.
Plant Material
Seeds of Arabidopsis (Arabidopsis thaliana Columbia) were placed in sealed
petri dishes on moistened filter paper and put in a growth room (short-day
conditions: 8 h of light, a photosynthetic photon flux density [PPFD] of 250 mmol
m22 s21, 25°C and 16 h of darkness, 18°C, relative humidity of 60%). After 5 d,
viable seedlings were transferred to 0.4-dm3 pots with soil in the same room.
Plants were watered with demineralized water twice a week. Starting 4 weeks
after the transfer to pots, 100 mL of half-strength modified Hoagland solution
[2 mM KNO3, 2 mM Ca(NO3)2, 0.75 mM MgSO4, 0.665 mM NaH2PO4, 25 mM FeEDTA, 5 mM MnSO4, 0.5 mM ZnSO4, 0.5 mM CuSO4, 25 mM H3BO4, 0.25 mM
Na2MoO4, 50 mM NaCl, and 0.1 mM CoSO4] was added to each pot once a week.
Ten weeks after germination, plants were used for the subsequent measurements.
Gas Exchange and Internal Conductance
Gas-exchange measurements were performed with an infrared gas analyzer (LI-6400 XT; Li-Cor). Day respiration (Rd) and the apparent compensation point (pi*) were determined according to the method of Laisk (as
described in von Caemmerer, 2000) by measuring photosynthesis at six
different CO2 concentrations and at three different PPFDs. Calibration of the
relationship between chlorophyll fluorescence and rates of electron transport
was carried out using the LI-6400 with an integrated fluorescence chamber
head (LI-6400-40; Li-Cor). Photochemical efficiency of PSII was calculated
following the procedures of Genty et al. (1989). The electron transport rate was
determined by taking the slope of the relationship between photochemical
efficiency of PSII and (A + Rd)/PPFD at different ambient CO2 concentrations
and under nonphotorespiratory conditions. This electron transport rate at
each ambient CO2 concentration was used to calculate Cc according to
Equation 13. Internal conductance was then calculated using Equation 12.
Measurements were done at 21% oxygen and an ambient CO2 concentration of
25 Pa, and were repeated at 10% and 40% oxygen to examine the effect of
oxygen on gi.
Supplemental Data
The following materials are available in the online version of this article.
Supplemental Figure S1. Concentrations and fluxes of CO2 and HCO32 in
a cross section of a mesophyll cell.
Supplemental Figure S2. Response of gi to CO2 at different oxygen levels.
Supplemental Figure S3. CO2 net hydration and fixation rates inside the
stroma.
Supplemental Figure S4. The effect of the relative amount of nitrogen
allocated to CA on the rate of photosynthesis.
Supplemental Figure S5. The effect of photorespiration on internal conductance explained by a resistance-based approach.
Supplemental Figure S6. The effect of stromal viscosity on the rate of
photosynthesis and internal conductance.
Supplemental Data S1. The source code of the default model.
ACKNOWLEDGMENTS
We thank the members of the Plant Systems Biology Group at the Partner
Institute for Computational Biology in Shanghai, China, for comments on an
Plant Physiol. Vol. 156, 2011
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