Journal of Experimental Botany, Vol. 59, No. 6, pp. 1137–1147, 2008
doi:10.1093/jxb/ern054 Advance Access publication 28 March, 2008
OPINION PAPER
On the mechanism of C4 photosynthesis intermediate
exchange between Kranz mesophyll and bundle sheath
cells in grasses
Paweł Sowiński1,2,*, Jarosław Szczepanik1 and Peter E. H. Minchin3
1
University of Warsaw, Institute of Plant Experimental Biology, Department of Plant Growth and Development,
Miecznikowa 1, 02-096 Warszawa, Poland
2
Plant Breeding and Acclimatization Institute, Plant Biochemistry and Physiology
Department, Radzików, 05-870 Błonie, Poland
The Horticulture and Food Research Institute of New Zealand Ltd, 412 No 1 Road, RD 2 Te Puke 3182,
New Zealand
3
Received 10 October 2007; Revised 4 February 2008; Accepted 5 February 2008
Abstract
C4 photosynthesis involves cell-to-cell exchange of
photosynthetic intermediates between the Kranz mesophyll (KMS) and bundle sheath (BS) cells. This was
believed to occur by simple diffusion through plentiful
plasmodesmatal (PD) connections between these cell
types. The model of C4 intermediates’ transport was
elaborated over 30 years ago and was based on
experimental data derived from measurements at the
time. The model assumed that plasmodesmata occupied about 3% of the interface between the KMS and
BS cells and that the plasmodesmata structure did not
restrict metabolite movement. Recent advances in the
knowledge of plasmodesmatal structure put these
assumptions into doubt, so a new model is presented
here taking the new anatomical details into account. If
one assumes simple diffusion as the sole driving
force, then calculations based on the experimental
data obtained for C4 grasses show that the gradients
expected of C4 intermediates between KMS and BS
cells are about three orders of magnitude higher than
experimentally estimated. In addition, if one takes into
account that the plasmodesmata microchannel diameter might constrict the movement of C4 intermediates
of comparable Stokes’ radii, the differences in concentration of photosynthetic intermediates between KMS
and BS cells should be further increased. We believe
that simple diffusion-driven transport of C4 intermediates between KMS and BS cells through the
plasmodesmatal microchannels is not adequate to explain the C4 metabolite exchange during C4 photosynthesis. Alternative mechanisms are proposed, involving
the participation of desmotubule and/or active mechanisms as either apoplasmic or vesicular transport.
Key words: C4 photosynthesis, grasses,
plasmodesmata, symplasmic transport.
modelling,
C4 photosynthesis
The C4 carbon cycle involved in carbon dioxide trapping
prior to photosynthesis has been well researched since its
discovery in the late 1960s. This process involves
morphological and physiological adaptations, so it has
been studied by anatomists, biochemists, and physiologists. This pathway enables carbon dioxide to be concentrated at the site of Rubisco action, reducing
photorespiration and enhancing water use efficiency.
Primary carbon assimilation (PCA) takes place in the
Kranz mesophyll (KMS) cells. The product of phosphoenolpyruvate (PEP) carboxylation, i.e. oxalacetate is
converted to either malate or aspartate. C4 acids are
exported to the bundle sheath (BS) cells where they are
decarboxylated. The released CO2 is incorporated into the
Calvin cycle for primary carbon reduction (PCR). The
route of decarboxylation depends on the sub-type of C4
photosynthesis: NADP-malic enzyme (NADP-ME),
* To whom correspondence should be addressed. E-mail: [email protected]
ª The Author [2008]. Published by Oxford University Press [on behalf of the Society for Experimental Biology]. All rights reserved.
For Permissions, please e-mail: [email protected]
1138 Sowiński et al.
NAD-malic enzyme (NAD-ME), and PEP-carboxykinase
(PEP-CK). After reduction, a fraction of the assimilated
carbon moves back from the BS to the KMS cells as
pyruvate, where it is regenerated into PEP. Phosphoglyceride (PGA) and triosephosphate (TP) are also
shuttled to the KMS (Furbank and Foyer, 1988).
The architectural arrangement of the cells involved in
photosynthesis and photosynthate export optimizes this
cell-to-cell exchange. According to Gamalei’s (1991)
classification based upon the route of phloem loading, the
veins in C4 plants represent a type 2c ultrastructure,
specific for many C4 and crassulacean acid plants. In
plants with this vein ultrastructure type, Kranz mesophyll
layer(s) surround the bundle sheath layer, and are interconnected by numerous plasmodesmata, while the number
of plasmodesmata between companion cell/sieve tube
complex and adjoining cells is limited.
In C4 grasses, symplasmic continuity exists between the
Kranz mesophyll, the bundle sheath, and the vascular
parenchyma (VP). In some species (Botha, 1992) or subspecies (Sowiński et al., 2001), symplasmic continuity
occurs between bundle sheath cells and companion cells,
but this is rare. In grasses, sieve tubes in small and
intermediate vascular bundles are of two types: thinwalled sieve tubes connected to companion cells, and
thick-walled sieve tubes connected to vascular parenchyma cells. The role of the thick-walled sieve tubes is
still unknown, while the companion cell/thin-walled sieve
tube complex is responsible for phloem loading (Fritz
et al., 1983). There are some anatomical differences
among C4 photosynthesis sub-types, manifested mostly in
the distribution of BS chloroplasts, located centrifugally in
NADP-ME, PEP-CK, and PCK-like NAD-ME species
and centripetally in the classical NAD-ME species
(Ohsugi and Murata, 1986; Dengler et al., 1994; Giussani
et al., 2001; Ueno et al., 2006). There is general
agreement that exchange of C4 photosynthetic intermediates between KMS and BS cells is solely through
plasmodesmata (Hattersley and Browning, 1981; Hattersley,
1987, but see Eastman et al., 1988a, b). The role of
plasmodesmata in C4 photosynthesis is supported by the
positive correlation between the number of plasmodesmata and the net photosynthesis rate found in several C4
grasses (Botha, 1992; Sowiński et al., 2007). In species
that synthesize sucrose in KMS, it is symplastically
transported through at least three cells: KMS–BSC–VP,
before being loaded into the phloem. The crucial role of
plasmodesmata in the export of photosynthates from
leaves finds strong support in studies of a maize mutant,
SXD-1 (Russin et al., 1996), in which plasmodesmata at
the BSC/VP interface were occluded by callose (Botha
et al., 2000), resulting in the arrest of sucrose export. All
these data support the conclusion that the rates of C4
photosynthesis and photosynthate export depend on the
number and conductivity of plasmodesmata.
Plasmodesmata linking KMS and BS cells in C4 grasses
differ in ultrastructure and dimensions (Botha et al., 2005,
and literature cited herein). In some species, sphincters
may occur on one or both cell sides (Evert et al., 1977;
Robinson-Beers and Evert, 1991; Botha et al., 2005).
KMS/BS plasmodesmata diameter is of approximately
100 nm, however, if suberin lamellae are present
plasmodesmata diameter might be restricted down to
approximately 40 nm (Robinson-Beers and Evert, 1991;
Botha et al., 2005). Even if plasmodesmata do not cross
suberin lamellae (NAD-ME sub-type), they show constriction at the neck regions down to approximately 40 nm
(Valle et al., 1989; Sowiński et al., 2007). The diameter
of plasmodesmata at the KMS/BS interface in the dicotyledonous C4 plant Salsola kali L. was approximately
50 nm (Olesen, 1975).
Mechanism of C4 intermediate transport between
KMS and BS cells
It has been proposed that C4 photosynthesis intermediates
were transported between KMS and BS cells by means of
diffusion, driven by a concentration gradient (Leegood,
2000, and citations therein). This was supported by
estimations of concentration differences of the main
photosynthetic metabolites in maize (Leegood, 1985; Stitt
and Heldt, 1985) that were in agreement with values
obtained by modelling transport of the C4 intermediates
(Osmond, 1970; Hatch and Osmond, 1976). The model,
elaborated over 30 years ago, was based on the experimental data of Tyree (1970). Authors assumed that
plasmodesmata occupied about 3% of the interface
between the KMS and BS cells and that the plasmodesmata structure did not constrict metabolite movement.
Recent advances in knowledge of plasmodesmatal structure throw doubt on these assumptions, so these are
revised, taking into account the new anatomical details.
The number of plasmodesmata linking KMS and BS
cells in C4 plants is well documented (Botha, 1992; Cooke
et al., 1996; Sowiński et al., 2007) and it is agreed that
this number is higher in C4 than in C3 plants (Botha,
1992; Cooke et al., 1996), with C4 plants having
approximately 6 plasmodesmata lm2 of KMS/BS interface (Table 1). With a plasmodesma diameter of 40 nm,
the total plasmodesmatal cross-section occupies approximately 0.8% of the cellular interfaces. However, according to present knowledge of plasmodesmata ultrastructure,
part of the cross-section is occupied by the desmotubule
and transport takes place within the 7–9 microchannels
(Overall et al., 1982; Ding et al., 1992), each with
a diameter of 2.5–4 nm (Overall et al., 1982; Roberts and
Oparka, 2003). Therefore the cross-section open for
transport would constitute only ;0.07% of total KMS/BS
interface area, i.e. two orders of magnitude less than is
assumed for models postulated 30 years ago. With this
limitation on the area available for exchange, simple
Mechanism of C4 photosynthesis transport 1139
Table 1. Properties of plasmodesmata (PD) at the Kranz mesophyll (KMS) and bundle sheath (BS) interface and assumptions for the
model on symplasmic transport between photosynthetic cells in C4 grasses
Feature
Values or phenomenon reported
by other authors or possible
in vivo situation
Length of a PD between
150–250 nma
KMS and BS cells
Diameter of single microchannel 2.5–4.0 nmb
Value or phenomenon
assumed in the current
model
Implications of assumed value
The shorter transport channel, the more
efficient diffusion
4.0 nm
The wider transport channel, the more
efficient diffusion
Straight
High tortuousity factor lowers diffusion
Tortuousity of a microchannel
Tortuousb
coefficient inside a transport channel
9
The higher number of channels, the more
Number of microchannels per PD 7–9b
efficient diffusion
PD shape in longitudinal view
Constriction at neck regions or at A cylinder of equal thickness Neck regions act as bottlenecks, limiting
crossing of suberin lamella
diffusion coefficient
BS circumference
KMS/BS interface contributes to
KMS/BS interface contributes Shorter cell interface raises the volume
20–50% of the BS circumference to 50% of the BS
fraction of plasmodesmatal transport channels,
circumference
that, in turn, raises the diffusion coefficient
The presence of other molecules inside
Molecules transported
Various types of small molecules Only C4 metabolites, triose
phosphates and sucrose
microchannels could affect diffusion of C4
through KMS/BS PD
and probably proteins and RNAs
metabolites
Interactions between
Bidirectional transport of molecules Absent
Interactions between transported molecules
transported molecules
lower diffusion coefficients of the molecules
Interactions between transported Transport of molecules of size
Absent
Interactions between transported molecules
molecules and microchannel
compared to microchannel
and the channel wall lower diffusion
diameter
coefficients of the molecules
Molecule shape
Transported molecules differ
Each transported molecule is
in shape
a rigid sphere of radius rST,
equal to the Stokes’ radius
Polarity of transported
Negatively charged molecules
Molecules have no
Charged molecules interact with structural
metabolites
electric charge
proteins of PD and other molecules
transported through microchannelsc
Hydration spheres around
Present
Absent
Hydration spheres raise molecular radii of
transported molecules
transported metabolites that results in a
decreased diffusion coefficient
a
b
c
150 nm
Botha et al. (2005); Botha et al. (1993); Botha et al. (1982); Botha and Evert (1988); Robinson-Beers and Evert (1990); Sowiński et al. (2007).
Ding et al. (1992); Overall et al. (1982).
For details, see Tyree (1970) and Woermann (1976).
diffusion would not seem to be sufficient to account for
the volume of metabolites being transported. This problem
has led us to propose new calculations. Our calculations
are based on the experimental data obtained for C4
grasses, representing all three C4 sub-types (Botha et al.,
1982; Ohsugi and Murata, 1986; Botha and Evert, 1988;
Valle et al., 1989; Botha, 1992; Soros and Dengler, 1998;
Ueno et al., 2006; Sowiński et al., 2007) and show that
the expected gradients between KMS and BS cells of C4
intermediates are much higher than experimentally estimated. These calculations confirm that diffusion-driven
transport of C4 intermediates between KMS and BS cells
through the plasmodesmatal microchannels is not adequate to explain the observed concentration differences.
An alternative mechanism is proposed.
Simple diffusion: first approximation
Diffusion through the plasmodesmatal microchannels in
the cell wall can be treated as diffusion within a porous
membrane, with microchannels acting as the pores. Then,
the transport rate through such membranes will be affected
in two ways: by the frequency of pores in a membrane and
by the pore size. The importance of the porosity factor on
the diffusion coefficient is rather obvious—the more pores
within a membrane, the larger the space for diffusion
(Bret-Harte and Silk, 1994; Patrick, 1997).
We are aware that plasmodesmatal microchannels are
not simple tubes, but complex and irregular structures
with many fjord-like structures branching out from the
channel’s lumen. Such channel architecture might be
thought to impede metabolite flux. However, while
surface roughness does affect diffusivity of a single molecule, it has no effect on transport diffusivity. This difference is of great significance when the channel is rough
even at the molecular level (Malek and Coppens, 2003),
as in plasmodesmatal microchannels, which have diameters similar to the size of the transported metabolites (the
diameters of photosynthesis intermediates are calculated
further).
1140 Sowiński et al.
The other factor concerning plasmodesma architecture is
its possible helical arrangement (Overall et al., 1982; Ding
et al., 1992; Roberts, 2005). Diffusion will be most
efficient, if transport channels are straight cylinders. If
they are tortuous, the diffusion pathway inside a channel
will increase. This may be the case of plasmodesmatal
microchannels. Anyway, without detailed knowledge on
plasmodesmata ultrastructure, a helical arrangement of
microchannels could not be taken into account.
Assumptions for the model
Assuming that diffusion is the only mechanism involved
in transport of metabolites between KMS and BS cells and
that this is through the microchannels of the plasmodesmata, then there must be a sufficient concentration
gradient of each metabolite to sustain diffusion flow given
by Fick’s law:
J ¼ D=c
ð1Þ
@c
where =c ¼ @x
denotes concentration gradient, and D is
the diffusion coefficient for the specific metabolite.
To calculate the concentration gradient necessary to
sustain diffusion between KMS and BS cells, we start
with several assumptions, most of them intentionally
chosen as favourable for diffusion. The assumptions are
shown in Table 1.
Biometric data and cross-section of transport channels
To calculate the symplasmic flow of photosynthates
between KMS and BS cells, experimental data of six C4
grasses have been considered: Zea mays (NADP-ME),
Digitaria sanguinalis (NADP-ME), Themeda triandra
(NADP-ME), Panicum miliaceum (classical NAD-ME),
Eragrostis plana (classical NAD-ME), and Panicum
maximum (PEP-CK). Panicum miliaceum and Eragrostis
plana will be further referred to as NAD-ME species. All
biometric and carbon flux data (Table 2) have been taken,
or calculated, from published data (Botha et al., 1982;
Oshugi and Murata, 1986; Botha, 1992; Soros and
Dengler, 1998; Ogle, 2003; Ueno et al., 2006; Sowiński
et al., 2007).
The area of KMS/BS cell walls (SW) mm2 of leaf area
was calculated as:
SW ¼ 1000IBS CBS nV
ð2Þ
where 1000 is a conversion factor, since 1 mm ¼ 1000
lm, IBS is equal to 0.5 and allows for the contribution of
intercellular spaces to BS circumference (Table 1), CBS is
a circumference of BS cells (Table 2), nV gives the
number of veins in leaf segment of 1 mm2 (Table 3).
Total cross-sectional area of microchannels (TSK) is
given by:
TSK ¼ 9fPD SK
ð3Þ
for nine microchannels per plasmodesma (Table 1), fPD is
the number of plasmodesmata mm2 of leaf area (Table 3),
and SK is the cross-sectional area of a single microchannel
(SK¼12.56 nm2).
On the basis of equations (2) and (3), u¼TSK/SW is
defined as the surface fraction of plasmodesmatal microchannels in the KMS/BS cell walls mm2 of leaf area.
The results of calculations of u made for different C4
photosynthesis sub-types are shown in Table 3. The
surface fraction of microchannels in NADP-ME and PCK
sub-types is just about 0.06% and in NAD-ME species
having the highest u values, it is only 0.3%.
Diffusion coefficients for transported metabolites
Diffusion coefficients in water for each metabolite were
calculated using the Stokes–Einstein formula:
DðiÞ ¼
kT
6pgrST
ð4Þ
where D(i) is the diffusion coefficient for metabolite i of
Stokes’ radius rST in solution of viscosity g and
temperature T¼298.15 K and k¼Boltzman’s constant
(1.3831023 J K1).
Table 2. CO2 assimilation rates and biometric parameters for different C4 photosynthetic sub-types in grasses
Mean values are shown in parenthesis. PD, plasmodesmata; KMS, Kranz mesophyll cell(s); BS, bundle sheath cell(s).
Photosynthetic sub-type
NADP-MEa
NAD-MEa
PEPCKa
CO2 assimilation (lmol m2 s1)
13.00–23.00
(19.30) a, b
86.30–123.60
(109.88) a, b, c, d, e, f, g
153.20–211.50
(182.29) a, b, g
284.92–559.00
(440.31) a, b
3.13–6.23
(4.74) a, b, h
23.00–27.00
(25.00) a, b
149.79–213.60
(171.76) a, b, c, d, e, f
240.40–246.70
(243.55) a, b
2587.35–3045.00
(2816.17) a, b
20.97–25.33
(23.13) a, b
22.00–23.00
(22.50) a, b
116. 61–148.20
(131.75) a, b, c, d, e
263.80–292.70
(278.25) a, b
574.00–857.41
(715.70) a, b
3.92–6.53
(5.22) a, b
IVD, Interveinal distance (lm)
CBS, Circumference of BS cells (lm)
nPD, number of PD per lm of vein (lm1)
PD per lm2 KMS/BS interface (lm2)
a
(a) Data from Sowiński et al. (2007); (b) data from Botha (1992); (c) calculated from Ogle (2003); (d) values from Ohsugi and Murata (1986);
(e) data from Ueno et al. (2006); (f) data from Soros and Dengler (1998); (g) calculated from Botha et al. (1982); (h) data from Cooke et al. (1996).
Mechanism of C4 photosynthesis transport 1141
Stokes’ radii for transported metabolites (Table 2) were
determined using HyperChem 7.5 Student software
(www.HyperChem.com), with all metabolites assumed to
have no hydration spheres around them. The results of
calculations are shown in Table 4
There is no agreement on the viscosity of the cytoplasm.
The mobility of BCECF (fluorescein derivate, MW 520)
in cytoplasm using spot photobleaching was a quarter of
that in water (Verkman, 2002), while in vivo measurements of GFP (27 kDa) movement in Escherichia coli was
one-tenth of that in water (Sear, 2005, and references
therein). However, in our calculations, the lowest reported
value (1.2 mPa s) was used for the viscosity of the
cytoplasm’s aqueous phase (Fushimi and Verkman, 1991).
The calculated diffusion coefficients for all considered
metabolites are given in Table 4. The calculated data are
comparable to values assumed by other authors (Hatch
and Osmond, 1976).
Metabolite fluxes and concentration differences
required to sustain diffusion between KMS and BS
cells
be built into triose phosphates (C3-P), which in turn were
completely used for sucrose synthesis. It was assumed that
for NADP-ME species sucrose was synthesized in KMS
cells only, while for NAD-ME and PEP-CK species only
half the sucrose was produced in KMS cells (Ohsugi and
Huber, 1987; Usuda and Edwards, 1980). For all species
examined, 60% of synthesized sucrose was assumed to be
exported to the phloem (Sowiński et al., 2007).
Metabolite fluxes (J) were expressed here as a number
of a given metabolite molecules [nM, (moles)] transported
through 1 nm2 of single channel’s cross-section
(SK¼12.56 nm2) in 1 s, using the following equation:
nM
ð5Þ
J¼
9fPD SK
Calculated metabolite fluxes are given in Table 5.
The required concentration differences (@c) between
KMS and BS cells to give the estimated flow rates for
each metabolite was calculated using the transformed
equation (1):
@c ¼ J
The stoichiometry between carbon assimilation and C4
metabolites transported between KMS and BS cells is
shown in Fig. 1. All the assimilated CO2 was assumed to
@x
D
ð6Þ
where J is given by equation (5), @x equals length of
plasmodesma (150 nm, Table 1), and D is the
Table 3. Biometric parameters used in the model, calculated on the basis of mean values from Table 2
NADP-ME
2
2
nV, number of veins in 1 mm of leaf blade (mm )
2
9.10
2
PEPCK
NAD-ME
5.82
6
Calculation formula
nv¼ 1000
IVD
7.59
6
6
fPD¼1000nVnPD
fPD, number of KMS/BS PD in 1 mm of leaf blade (mm )
4.01310
TSK, total cross-sectional area of plasmodesmatal microchannels
in cell walls between Kranz mesophyll (KMS) and bundle
sheath (BS) cells (nm2)
SW, total area of KMS/BS cell walls (nm2)
0.453109
1.853109
0.613109
TSK¼9fPDSKa
0.8331012
0.7131012
1.0631012
SW¼1000IBSCBSnVb
3
u, surface fraction of microchannels in KMS/BS cell
walls of 1 mm2 leaf blade
a
b
16.39310
3
0.55310
2.62310
5.43310
0.58310
3
K
u ¼ TS
Sw
SK area of single microchannel’s cross-section (12.56 nm2).
See text for details.
Table 4. Molecular size, diffusion coefficients and confinement factors for metabolites transported between Kranz mesophyll and
bundle sheath cells
Metabolite
Malate
Pyruvate
Alanine
Aspartate
PEP
C3-P
Sucrose
a
rST: Stokes’
radius (nm)
0.27
0.26
0.26
0.34
0.35
0.35
0.44
D, diffusion
coefficient in
water (m2 s1)
8.1231010
8.4031010
8.4031010
6.4031010
6.3031010
6.3031010
5.0031010
Dcyt, diffusion
coefficient in
cytoplasma (m2 s1)
6.7731010
7.0031010
7.0031010
5.3331010
5.2531010
5.2531010
4.1731010
Confinement factor inside
microchannel with
diameter of 4 nm (Kc)
0.56
0.57
0.57
0.47
0.47
0.47
0.38
Assuming 1.2 times higher viscosity for cytoplasm than for water (Fushimi and Verkman, 1991).
Confinement factor (Kc) inside
desmotubule of diameter
15 nm
25 nm
35 nm
0.86
0.87
0.87
0.83
0.83
0.82
0.78
0.92
0.92
0.92
0.89
0.89
0.89
0.87
0.94
0.94
0.94
0.92
0.92
0.92
0.90
1142 Sowiński et al.
Fig. 1. Schematic diagram of transport processes during C4 photosynthesis. (A) NADP-ME sub-type, (B) NAD-ME sub-type, (C) PEP-CK sub-type.
Abbreviations: BS, bundle sheath cell; KMS, Kranz mesophyll cell; PD, plasmodesmata; C3-P, triose phosphates. Numbers in parentheses show how
many molecules of C4 metabolites must be transported through PD for each one molecule of CO2 assimilated. For amounts of C3-P and sucrose being
transported, see ‘Metabolite fluxes’. PD are drawn without desmotobules, since microchannels in cytoplasmatic sleeve were assumed to be the only
transport pathway.
plasmodesmatal diffusion coefficient
be proportional to the surface fraction
(u, Table 3) in KMS/BS cell walls for
1 mm2 and cytoplasmatic diffusion
Table 4):
DPD ðiÞ ¼ Dcyt ðiÞu
(DPD) taken to
of microchannels
a leaf segment of
coefficient (Dcyt,
species, characterized by similar values of @c. These
discrepancies reflect different plasmodesmatal frequency
(Table 2).
Conclusions to the simple diffusion approximation
ð7Þ
The values of @c obtained in our model are shown in
Table 5. The calculated data have been compared with
experimental data obtained by Stitt and Heldt (1985).
Their data concerned concentrations of C4 intermediates
in KMS and BS cells of maize. These authors obtained
concentrations of C4 metabolites as high as a few
hundreds of nanomoles per mg of chlorophyll (mg Chl).
They assumed that the chlorophyll was equally distributed
between KMS and BS cells and that the combined volume
of chloroplasts and cytoplasm was 40 ll per mg Chl. As
a result, the estimated concentration of each metabolite
between KMS and BS cells was a few nanomoles per ll,
equal to a few milimoles per litre. Comparison of the data
of Stitt and Heldt (1985) with the values of @c we
calculated using the different approaches, are shown in
Table 3.
In addition, the values of @c proposed by Weiner et al.
(1988) are shown in Table 5. Authors assumed a concentration gradient of 1 mM to describe the rate of diffusion
of particular photosynthetic metabolite into BS cells.
Calculated concentration differences of metabolites
required to sustain diffusion are higher by about three
orders of magnitude, as compared to experimental data
(Stitt and Heldt, 1985; Weiner et al., 1988). Moreover,
differences between photosynthetic types were observed:
in NAD-ME species, concentration differences were about
three times lower than in NADP-ME and PEP-CK
The estimated concentration differences required assuming transport by diffusion through microchannels, were
very high. These concentration differences, being tens of
moles, seem unrealistic given that in the species studied to
date, concentrations of C4 metabolites and triose phosphates were in the order of a few tens of millimoles
(Hatch and Osmond, 1976; Leegood, 1985, 2000; Stitt
and Heldt, 1985). Similar discrepancies were noticed by
Bret-Harte and Silk (1994), when they estimated solute
deposition rates and corresponding fluxes in growing root
of Zea mays, assuming that diffusion was the only
mechanism for metabolite transport. Diffusion coefficients
and concentration gradients calculated by these authors
were a few orders of magnitude higher than expected. Our
calculations of DPD and @c made using Bret-Harte and
Silk’s model gave values similar to the approach assuming
transport through microchannels (data not shown).
Simple diffusion: second approximation
All C4 metabolites considered were of similar size and
therefore had similar diffusion coefficients (Table 4). All
had low molecular weight, compared with the plasmodesmata exclusion limit of about 0.9 kDa, but their Stokes’
radii (rST) were quite high compared with the microchannel radius (rK¼2 nm). This observation raises queries
of our assumption (Table 1) that microchannel diameter
does not affect metabolite movement.
Mechanism of C4 photosynthesis transport 1143
2
Table 5. Metabolite fluxes (mol nm s ) and concentration differences of photosynthetic metabolites (mol dm3) between Kranz
mesophyll and bundle sheath cells in C4 grasses
Photosynthetic
sub-type
C4 metabolite
1
Metabolite fluxes
through single
microchannel)
Concentration differencies of metabolites between Kranz mesophyll and bundle sheath cells
Experimental
data
No constrictions
from channel size, all
metabolites move
through palsmodesmatal
microchannels
Constrictions from
channel size present,
all metabolites move
through plasmodesmatal
microchannels
Metabolites passing
from KMS to BS
move inside
desmotubule of diameter
15 nm
NADP-ME
NAD-ME
PEP-CK
a
b
Malate
Pyruvate
C3-P
Sucrose
Alanine
Aspartate
C3-P
Sucrose
Malate
Pyruvate
Aspartate
PEP
C3-P
Sucrose
4.2631020
4.2631020
1.4231020
0.2131020
1.3531020
1.3531020
0.2231020
0.0331020
1.8331020
1.8331020
1.8331020
1.8331020
0.6131020
0.0931020
0.018a
0.005a
0.010a
0.001b
0.001b
0.001b
0.001b
–
–
–
–
17.16
17.16
7.51
1.40
1.10
1.45
0.25
0.05
6.99
6.76
8.88
9.01
3.00
0.56
30.64
30.10
15.98
3.68
1.93
3.08
0.53
0.13
12.48
11.85
18.89
19.17
6.38
1.47
8.20
30.10
15.98
0.73
1.93
0.72
0.53
0.02
3.33
11.85
4.38
19.17
6.38
0.30
25 nm
35 nm
1.00
0.26
0.09
0.02
0.09
0.02
0.003
0.40
0.001
0.10
0.52
0.13
0.03
0.01
Data from Hatch and Osmond (1976).
Values based on Weiner et al. (1988).
The dependence of diffusion coefficient on pore diameter is not simple. If the pore diameter is over ten times
that of the transported molecule, then diffusion through
the pore will equal that of the bulk fluid, while with a pore
diameter smaller than two molecular diameters, single file
diffusion will occur (Cui, 2005). For a ratio of pore width
to molecular diameter of 2–10 (relevant to the metabolites
we considered, where this was 4.54 to 7.69) pore
diffusivity falls between those two extremes (Liu et al.,
2005). Taking this into account, transport through a microchannel is the result of at least three types of diffusion:
continuous diffusion (diffusion in bulk fluid), Knudsen’s
diffusion, and surface diffusion, the latter two reducing
transport (Gudmundsson, 2003; Valiullin et al., 2004; Liu
et al., 2005). However, for the simplicity of current
approximation, the diffusion through the microchannels
was assumed to be continuous, and to vary with the ratio
of the channel’s diameter to the molecular size, according
to the confinement factor (Kc), defined as:
rst 4
ð8Þ
Kc ¼ 1 rK
which is a simplification of Renkin’s (1954) approach,
valid for rST/rK > 0.01. Equation (7) is now generalized
to:
DPD ðiÞ ¼ Dcyt ðiÞuKc
ð9Þ
The confinement factor varied from 0.38 to 0.54 (Table
4), results in a 2–3-fold slow down of diffusion inside the
microchannel in relation to bulk fluid conditions, and
correspondingly in approximately twice higher concentration differences of photosynthetic intermediates between
KMS and BS cells necessary to sustain the transport of
photosynthates between KMS and BS cells, as compared
to the data obtained in the first approximation (Table 5).
Simple diffusion model: the need for the
third approximation?
The model presented here is a highly simplified version of
the situation encountered in planta. However, it shows,
that even under assumptions favouring diffusion, the
concentration differences of transported metabolites between KMS and BS cells necessary for maintaining the
current net photosynthetic rates are high and hardly
possible in living cells. If this model is to be valid in
describing transport processes in vivo, several additional
assumptions, neglected here, must be taken into consideration. The most important constriction to the model is that
C4 photosynthesis, because of its nature, needs exchange
of metabolites between cells, i.e. simultaneous movement
of some intermediates from KMS to BS, and others from
BS to KMS. As it is stated above, Stokes’ radii of
photosynthetic metabolites are comparable to the microchannel radius. So the assumption (Table 1), that two
streams of molecules moving in opposite directions in
narrow channels do not disturb each other, is improbable.
In addition, transport of other compounds simultaneously
with the transport of photosynthetic intermediates; the
existence of hydration spheres around polar molecules
1144 Sowiński et al.
increasing the Stokes’ radius of a molecule; the specificity
of diffusion inside micropores cannot be disregarded.
Therefore, one must be aware that taking these processes
into consideration will result in further increase of the
concentration differences required to sustain diffusion.
Clearly C4 plants do transport a large amount of
photosynthates. Photosynthesis in C4 plants, which might
even excess 40 lmol CO2 m2 s1, produces a significant
amount of assimilates exchanged between KMS and BS
cells symplasmically. Thus: (i) other diffusion pathways
apart from the plasmodesmal microchannels are involved;
and (ii) another transport mechanism is involved in
metabolic exchange between KMS and BS cells. These
possibilities are considered below.
Simple diffusion model: combined two-way metabolite
exchange utilizing desmotubule and microchannels
of plasmadesmata
If combined two-way metabolic exchange is assumed,
then the second route remains to be found. This would
result in spatial separation of the transport from KMS to
BS cells from that of the flux in the opposite direction.
The desmotubule seems to be an ideal candidate. The role
of desmotubules as a transport pathway was postulated
many years ago, also in C4 plants (Evert et al., 1977).
Recently, this idea has been restated (Waigmann et al.,
1997; Cantrill et al., 1999). One should underline,
however, that there are strong arguments for the opinion
that the desmotubule is a static, appressed structure at the
centre of PD, not available for transport processes and
acting as a structural component, often referred to as
a central rod (Gunning and Overall, 1983; Tilney et al.,
1991; Botha et al., 1993, Overall and Blackmann, 1996;
Ding, 1998).
In this approach, it is assumed that metabolites moving
from the KMS to the BS cells are transported inside
desmotubules, while photosynthetic intermediates move
from BS to KMS in plasmodesmatal microchannels.
Various possible desmotubule sizes (15, 25, and 35 nm in
diameter) have been considered, with the resulting
confinement factors (see Table 4) taken into consideration.
The metabolite fluxes and the desmotubule diffusion
coefficients were calculated as described in the section on
‘Metabolite fluxes and concentration differences required
to sustain diffusion between KMS and BS cells’ and the
concentration differences between photosynthetic cells,
necessary to maintain the current net photosynthesis rates,
were estimated (Table 5).
With the desmotubule assumed to be the additional
transport pathway for diffusion, the required concentration
differences between KMS and BS cells decreased significantly (Table 5). For NAD-ME species, when the widest
desmotubule was taken into account, the differences
were similar to the metabolite concentrations estimated
experimentally. For other C4 sub-types, these values were
higher, but the difference was reduced to one order of
magnitude only.
Participation of desmotubules in cell-to-cell transport
was postulated by Waigmann et al. (1997) for cotton
extrafloral nectary trichomes expelling large amounts of
nectar. Desmotubules have also been postulated as a transport route in the symplasmic phloem loading mechanism
(Gamalei et al., 1994). This sort of phloem-loading
mechanism is related to so-called open (type 1) vein
ultrastructure (Gamalei, 1991), where companion cells are
connected to adjoining mesophyll cells by numerous
plasmodesmata (more than 10 PD per lm2 of the cell
interface). Symplasmic phloem loading was postulated to
be powered by polymer trapping mechanism (Turgeon,
1996), however, even in plants showing abundant plasmodesmata linking companion cells and mesophyll
cells, for the transport of carbohydrates from photosynthetic cells to companion cells/sieve tube complex other
mechanisms have been postulated as mass flow (Voitsekhovskaja et al., 2006) or even apoplasmic transport
(Turgeon and Medville, 2004).
In C4 plants, metabolite concentration differences
between KMS and BS cells obtained when desmotubular
transport was assumed to occur were more realistic than
those from other approximations. Thus, desmotubule
involvement as a transport pathway in C4 photosynthesis
seems reasonable. However, it has been assumed that this
pathway is available only for metabolites moving in one
direction (i.e. from KMS to BS cells). Transport in the
opposite direction remains a problem as there are only the
microchannels available, and these require high values of
@c (Table 5). This implies the involvement of transport
mechanisms other than simple diffusion.
Alternative mechanisms
Apoplasmic transport is an alternative to symplasmic
transport. However, in the case of exchange of metabolites
between KMS and BS cells, apoplasmic transport may be
questioned for two reasons. One is the suberin lamella
within the KMS/BS walls of many C4 plants, which nearly
precludes apoplasmic transport of solutes (Hattersley, 1987;
Hatterlsey and Browning, 1981). It has also been shown
that PCMBS, an inhibitor of the proton pump, has no
distinct effect on photosynthesis in maize, a C4 plant
(Bourquin et al., 1990; Sowiński 1998), which clearly
demonstrates that apoplasmic transport is not involved in
the photosynthate transport in that species. Unfortunately,
such studies have not been performed with other C4
plants.
There are two possible alternatives to simple diffusion,
the first being mass flow, postulated as an efficient means
of cell-to-cell transport (Anisimov and Egorov, 2002;
Voitsekhovskaja et al., 2006), and the second being
Mechanism of C4 photosynthesis transport 1145
vesicular transport (Bil’ et al., 1976; Karpilov et al., 1976;
Evert et al., 1977), similar to that postulated for proteins
and other high molecular weight molecules and viruses
(Chen and Kim, 2006). Vesicles could accumulate solutes
to very high concentrations, using transporters located in
the vesicle membrane. Vesicles could be unloaded at the
plasmodesma neck region in a manner similar to the
vesicle-mediated secreting transport system involving the
vacuole and plasmalemma (Echeveria, 2000). The idea of
a vacuole–desmotubule–vacuole continuum (Gamalei,
1996; Rinne et al., 2001) is intriguing in this context.
However, we await experimental data to support these
mechanisms.
Conclusions
In light of current knowledge on plasmodesmata ultrastructure, the conventional model of C4 photosynthetic
intermediate exchange between KMS and BS cells based
only on simple diffusion is not satisfactory, since the
concentration differences for photosynthetic intermediates
in KMS and BS cells seem unrealistic.
Theoretically, participation of desmotubules could improve transport efficiency in C4 grasses, however, only in
one direction (e.g. from KMS to BS cells). Transport in
the opposite direction remains a problem as there are only
the microchannels available, and these require high values
of @c to enable a simple diffusion model to hold.
A more effective mechanism than simple diffusion is
needed for cell-to-cell exchange of photosynthetic intermediates in C4 plants. Two plausible mechanisms have
been proposed here, but there is no experimental evidence
to confirm that either of them is operative in C4 plants.
Appendix
c(i) concentration of metabolite i (mol dm3)
CBS circumference of BS cells (lm)
D(i) diffusion coefficient for metabolite i in water (m2 s1)
Dcyt(i) diffusion coefficient for metabolite i in cytoplasm (m2
s1)
DPD(i) plasmodesmatal diffusion coefficient for metabolite i
(m2 s1)
fPD plasmodesmatal frequency per 1 mm2 (mm2)
u surface fraction of plasmodesmatal microchannels in leaf
segment of 1 mm2
IBS contribution of intercellular spaces to BS circumference
IVD interveinal distance (lm)
J(i) flux of metabolite i (mol nm2 s1)
k Boltzman’s constant: 1.3831023 J K1
Kc confinement factor inside plasmodesmatal microchannel
nM(i) number of moles of metabolite i transported between KMS
and BS cells (mol)
nV number of veins in a given volume of the leaf
g viscosity of water: 1 mPa s
rK radius of plasmodesmatal microchannel: 2 nm
rST(i) Stokes’ radius for metabolite i (nm)
SK cross-sectional area of single microchannel (nm2)
SW total area of KMS/BS cell walls in leaf segment of 1 mm2
(nm2)
T temperature (K)
TSK total area of plasmodesmatal microchannels cross-section
in KMS/BS cell walls in leaf segment of 1 mm2 (nm2)
x diffusion pathway [nm]
Acknowledgements
The authors wish to thank anonymous reviewers for suggestions
that have greatly improved the clarity of the paper.
References
Anisimov AV, Egorov AG. 2002. Plasmodesmata as a modulator of
osmotic water fluxes in plants. Russian Journal of Plant
Physiology 5, 758–766.
Bil’ KYa, Belobrodskaya LK, Karpilov YuS. 1976. Localization
of ATPases in cellular structures of assimilating tissues in
amaranth leaves. Doklady Akademii nauk SSSR; English translation in Botanical Science 226, 24–27.
Botha CEJ. 1992. Plasmodesmatal distribution, structure and
frequency in relation to assimilation in C3 and C4 grasses in
southern Africa. Planta 187, 348–358.
Botha CEJ, Cross RHM, Liu L. 2005. Comparative structures of
specialised monocotyledonous leaf blade plasmodesmata. In: Oparka
KJ, ed. Plasmodesmata. India: Blackwell Publishing Ltd, 73–89.
Botha CEJ, Cross RHM, van Bel AJE, Peter CI. 2000. Phloem
loading in the sucrose-export-defective (SXD-1) mutant maize is
limited by callose deposition at plasmodesmata in bundle sheathvascular parenchyma interface. Protoplasma 214, 65–72.
Botha CEJ, Evert RF. 1988. Plasmodesmatal distribution and
frequency in vacular bundles and contiguous tissues of the leaf of
Themeda triandra. Planta 173, 433–441.
Botha CEJ, Evert RF, Cross RHM, Marshall DJ. 1982.
Comparative study of mature Themeda triandra Forsk. leaf
blades: a correlated light and electron microscope study. Journal
of South African Botany 48, 311–328.
Botha CEJ, Hartley BJ, Cross RHM. 1993. The ultrastructure
and computer-enhanced digital image analysis of plasmodesmata
at the Kranz mesophyll–bundle sheath interface of Themeda
triandra var. imberbis (Retz) A. Camus in conventionally-fixed
leaf blades. Annals of Botany 72, 255–261.
Bourquin S, Bonnemain JL, Delror S. 1990. Inhibition of loading
of 14C-assimilates by p-chlormercuribenzenesulfonic acid. Plant
Physiology 92, 97–102.
Bret-Harte SM, Silk WK. 1994. Fluxes and deposition rates of
solutes in growing roots of Zea mays. Journal of Experimental
Botany 45, 1733–1742.
Cantrill LC, Overall RL, Goodwin PB. 1999. Cell–to–cell
communication via plant endomembranes. Cell Biology International 23, 653–661.
Chen XY, Kim JY. 2006. Transport of macromolecules through
plasmodesmata and the phloem. Physiologia Plantarum 126,
560–571.
Cooke TJ, Tilney MS, Tilney LG. 1996. Plasmodesmatal networks in apical meristems and mature structures: geometric
evidence for both primary and secondary formation of plasmodesmata. In: Smallwood M, Knox JP, Bowles DJ, eds. Membranes: specialized functions in plants. Oxford: BIOS Scientific
Publishers, 471–488.
Cui ST. 2005. Molecular self-diffusion in nanoscale cylindrical
pores and classical Fick’s law predictions. Journal of Chemical
Physics 123, 054706.
1146 Sowiński et al.
Dengler NG, Dengler RE, Donnelly PM, Hattersley PW. 1994.
Quantitative leaf anatomy of C3 and C4 grasses (Poaceae): bundle
sheath and mesophyll surface area relationships. Annals of Botany
73, 241–255.
Ding B. 1998. Intercellular protein trafficking through plasmodesmata. Plant Molecular Biology 38, 279–310.
Ding B, Turgeon R, Parthasarathy MV. 1992. Substructure of
freeze-substituted plasmodesmata. Protoplasma 169, 28–41.
Eastman PAK, Dengler NG, Peterson CA. 1988a. Suberized
bundle sheaths in grasses (Poaceae) of different photosynthetic
types. I. Anatomy, ultrastructure and histochemistry. Protoplasma
142, 92–111.
Eastman PAK, Peterson CA, Dengler NG. 1988b. Suberized
bundle sheaths in grasses (Poaceae) of different photosynthetic
types. II. Apoplastic permeability. Protoplasma 142, 112–126.
Echeveria E. 2000. Vesicle-mediated solute transport between the
vacuole and the plasma membrane. Plant Physiology 123, 1217–
1226.
Evert RF, Eschrich W, Heyser W. 1977. Distribution and
structure of the plasmodesmata in mesophyll and bundle-sheath
cells of Zea mays L. Planta 136, 77–89.
Fritz E, Evert RF, Heyser W. 1983. Microautographic studies of
phloem loading and transport in the leaf of Zea mays L. Planta
159, 193–206.
Furbank RT, Foyer CH. 1988. C4 plants as valuable model
experimental systems for the study of photosynthesis. New
Phytologist 109, 265–277.
Fushimi K, Verkman AS. 1991. Low viscosity in the aqueous
domain of cell cytoplasm measured by picosecond polarization
microfluorometry. Journal of Cell Biology 112, 719–725.
Gamalei YV. 1991. Phloem loading and its development related to
plant evolution from trees to herbs. Trees 5, 50–64.
Gamalei YV. 1996. Assimilate export under natural and experimental conditions. Russian Journal of Plant Physiology 45, 614–
631.
Gamalei YV, van Bel AJE, Pakhomova MV, Sjutkina AV. 1994.
Effects of temperature on the conformation of the endoplasmic
reticulum and on starch accumulation in leaves with the
symplasmic minor-vein configuration. Planta 194, 443–453.
Giussani LM, Cota-Sanchez JH, Zuloaga FO, Kellogg E. 2001.
A molecular phylogeny of the grass subfamily Panicoidae
(Poaceae) shows multiple origins of C4 photosynthesis. American
Journal of Botany 88, 1993–2012.
Gudmundsson K. 2003. An approach to determining the water
vapour transport properties of building materials. Nordic Journal
of Building Physics 3, 1–10.
Gunning BES, Overall RL. 1983. Plasmodesmata and cell to cell
transport in plants. BioScience 33, 260–265.
Hatch MD, Osmond CB. 1976. Compartmentation and transport in
C4 photosynthesis. In: Stocking CR, Heber U, eds. Encyclopaedia
of plant physiology, Vol. III. Springer-Verlag, 144–184.
Hattersley PW. 1987. Variations in photosynthetic pathway. In:
Soderstrom TM, Hilu KW, Campell CS, Barkworth ME, eds.
Grass systematics and evolution. Washington: Smithsonian Institution Press, 49–64.
Hattersley PW, Browning AJ. 1981. Occurrence of the suberized
lamella in leaves of grasses of different photosynthetic types in
parenchymateous bundle sheaths and PCR (‘Kranz’) sheaths.
Protoplasma 109, 371–401.
Karpilov YuS, Bil’ KYa. 1976. Transport of intermediate photosynthesis products through the cytoplasm of cells assimilation
tissues in C4 plants. Doklady Akademii nauk SSSR; English
translation in Botanical Science 226, 16–19.
Leegood RC. 1985. The intercellular compartmentation of metabolites in leaves of Zea mays L. Planta 164, 163–171.
Leegood RC. 2000. Transport during C4 photosynthesis. In:
Leegood RC, Sharkey TD, von Caemmerer S, eds. Photosynthesis: physiology and metabolism. Dodrecht: Kluwer, 459–469.
Liu YC, Wang Q, Li XF. 2005. A diffusion model for fluids
confined in micropores. Journal of Chemical Physics 122,
044714.
Malek K, Coppens MO. 2003. Knudsen self- and Fickian
diffusion in rough nanoporous media. Journal of Chemical
Physics 119, 2801–2811.
Ogle K. 2003. Implications of interveinal distance for quantum
yield in C4 grasses: a modelling and meta-analysis. Oecologia
136, 532–542.
Ohsugi R, Huber SC. 1987. Light modulation and localization of
sucrose phosphate synthase activity between mesophyll cells and
bundle sheath cells in C4 species. Plant Physiology 84, 1096–
1101.
Ohsugi R, Murata T. 1986. Variations in the leaf anatomy among
some C4 Panicum species. Annals of Botany 58, 443–453.
Olesen P. 1975. Plasmodesmata between mesophyll and bundle
sheath cells in relation to the exchange of C4-acids. Planta (Berl).
123, 199–202.
Osmond CB. 1970. Metabolite transport in C4 photosynthesis.
Australian Journal of Biological Science 24, 159–163.
Overall RL, Blackman LM. 1996. A model of the macromolecular structure of plasmodesmata. Trends in Plant Science 1, 307–
311.
Overall RL, Wolfe J, Gunning BES. 1982. Intercellular communication in Azolla roots: I. Ultrastructure of plasmodesmata.
Protoplasma 111, 134–150.
Patrick JW. 1997. Phloem unloading: sieve element unloading and
post-sieve element transport. Annual Review of Plant Physiology
and Plant Molecular Biology 48, 191–222.
Renkin EM. 1954. Filtration, diffusion, and molecular sieving
through porous cellulose membranes. Journal of General Physiology 38, 225–243.
Rinne PLH, Kaikuranta PM, van der Schoot C. 2001. The shoot
apical meristem restores its symplasmic organization during
chilling-induced release from dormancy. The Plant Journal 26,
249–264.
Roberts AG. 2005. Plasmodesmal structure and development. In:
Oparka KJ, ed. Plasmodesmata. India: Blackwell Publishing Ltd,
1–32.
Roberts AG, Oparka KJ. 2003. Plasmodesmata and the control of
symplastic transport. Plant, Cell and Environment 26, 103–124.
Robinson-Beers K, Evert RF. 1991. Fine structure of plasmodesmata in mature leaves of sugarcane. Planta 184, 307–318.
Russin WA, Evert RF, Vanderveer PJ, Sharkey TD, Briggs SP.
1996. Modification of a specific class of plasmodesmata and loss
of sucrose export ability in the sucrose export defective1 maize
mutant. The Plant Cell 8, 645–658.
Sear RP. 2005. The cytoplasm of living cells: a functional mixture
of thousand of components. Journal of Physics: Condensed
Matter 17, 3587–3595.
Soros CL, Dengler NG. 1998. Quantitative leaf anatomy of C3 and
C4 Cyperaceae and comparisions with the Poaceae. International
Journal of Plant Science 159, 480–491.
Sowiński P. 1998. The effect of irradiance, p-chloromercuribenzenesulphonic acid and fusicoccin on the long distance
transport in Zea mays L. seedlings. Acta Physiologiae Plantarum
20, 79–84.
Sowiński P, Bilska A, Barańska K, Fronk J, Kobus P. 2007.
Plasmodesmata density in vascular bundles in leaves of C4
grasses grown at different light conditions in respect to
photosynthesis and photosynthate export efficiency. Enviromental
and Experimental Botany 61, 74–84.
Mechanism of C4 photosynthesis transport 1147
Sowiński P, Rudzińska-Langwald A, Dalbiak A, Sowińska A.
2001. Assimilate export from leaves of chilling-treated seedlings
of maize. The path to vein. Plant Physiology and Biochemistry
39, 881–889.
Stitt M, Heldt HW. 1985. Generation and maintenance of
concentration gradients between the mesophyll and bundle sheath
in maize leaves. Biochimica et Biophysica Acta 808, 400–414.
Tilney LG, Cooke TJ, Connelly PS, Tilney MS. 1991. The
structure of plasmodesmata as revealed by plasmolysis, detergent
extraction and protease digestion. Journal of Cell Biology 112,
739–747.
Turgeon R. 1996. Phloem loading and plasmodesmata. Trends in
Plant Science 1, 418–423.
Turgeon R, Medville R. 2004. Phloem loading. A reevaluation of
the relationship between plasmodesmatal frequencies and loading
strategies. Plant Physiology 136, 3795–3803.
Tyree MT. 1970. The symplast concept. A general theory of
symplastic transport according to the thermodynamics of irreversible processes. Journal of Theoretical Biology 26, 181–214.
Ueno O, Kawano Y, Wakayama M, Takeda T. 2006. Leaf
vascular system of C3 and C4 grasses: a two dimensional analysis.
Annals of Botany 91, 611–621.
Usuda H, Edwards GE. 1980. Localization of glycerate kinase and
some enzymes for sucrose synthesis in C3 and C4 plants. Plant
Physiology 65, 1017–1022.
Valiullin R, Kortunov P, Kärger J, Timoshenko V. 2004.
Concentration-dependent self-diffusion of liquids in nanopores:
a nuclear magnetic resonance study. Journal of Chemical Physics
120, 11804–11814.
Verkman AS. 2002. Solute and macromolecule diffusion in cellular
aqueous compartments. Trends in Biochemical Sciences 27, 27–
33.
Voitsekhovskaja OV, Koroleva OA, Batashev DR, Knop C,
Tomos D, Gamalei YV, Heldt HW, Lohaus G. 2006. Phloem
loading in two scrophulariaceae species. What can drive symplastic flow via plasmodesmata. Plant Physiology 140, 383–395.
Valle EM, Craig S, Hatch MD, Heldt HW. 1989. Permeability
and ultrastructure of bundle sheath cells isolated from C4 plants:
structure–function studies and the role of plamsodesmata.
Botanica Acta 102, 276–282.
Waigmann E, Turner A, Peart J, Roberts K, Zambryski P.
1997. Ultrastructural analysis of leaf trichome plasmodesmata
reveals major differences from mesophyll plasmodesmata. Planta
203, 75–84.
Weiner H, Burnell JN, Woodraw IE, Heldt HW, Hatch MD.
1988. Metabolite diffusion into bundle sheath cells from C4
plants. Plant Physiology 88, 815–822.
Woermann D. 1976. Mass transport across membranes. In:
Stocking CR, Heber U, eds. Encyclopaedia of plant physiology,
Vol. III. Springer Verlag, 419–464.