Journal of Experimental Botany, Vol. 53, No. 373, pp. 1411–1419, June 2002
Revisiting the Münch pressure–flow hypothesis for
long-distance transport of carbohydrates: modelling the
dynamics of solute transport inside a semipermeable tube
S.M. Henton1, A.J. Greaves, G.J. Piller and P.E.H. Minchin
Horticultural Research, Batchelor Research Centre, Private Bag 11030, Palmerston North, New Zealand
Received 5 September 2001; Accepted 25 January 2002
Abstract
A mathematical model of the Münch pressure–flow
hypothesis for long-distance transport of carbohydrates via sieve tubes is constructed using the
Navier–Stokes equation for the motion of a viscous
fluid and the van’t Hoff equation for osmotic pressure. Assuming spatial dimensions that are appropriate for a sieve tube and ensuring suitable initial
profiles of the solute concentration and solution
velocity lets the model become mathematically
tractable and concise. In the steady-state case, it is
shown via an analytical expression that the solute
flux is diffusion-like with the apparent diffusivity
coefficient being proportional to the local solute concentration and around seven orders of magnitude
greater than a diffusivity coefficient for sucrose in
water. It is also shown that, in the steady-state case,
the hydraulic conductivity over one metre can be
calculated explicitly from the tube radius and physical
constants and so can be compared with experimentally determined values. In the time-dependent
case, it is shown via numerical simulations that the
solute (or water) can simultaneously travel in opposite directions at different locations along the tube
and, similarly, change direction of travel over time at
a particular location along the tube.
Key words: Münch pressure –flow hypothesis, Navier–Stokes
equation, phloem transport, van’t Hoff equation.
Introduction
Long-distance transport of carbohydrate, the major
substrate for plant growth, occurs within the phloem
1
vascular tissue. Phloem vasculature consists of sieve
elements that are approximately 20–30 mm in diameter
and 100–500 mm in length and are aligned end-to-end
(20–100 sieve elements per cm) to form a continuous,
membrane lined conduit (MacRobbie, 1971; Canny,
1973). The total length of a file of connected sieve
elements does not appear to have been measured, but is
thought to extend from the leaves to all carbohydrate
sinks of annual plants. Large perennial species, such as
trees, require lengths of at least an order of magnitude
greater than annuals which is possibly achieved by
phloem unloading from one file and reloading into
another (Lang, 1979), although this does not appear to
have been demonstrated. Movement of material from one
vascular trace to another appears to occur commonly
(Eschrich, 1975; Thorpe and Minchin, 1996) so
extremely long open files of conducting sieve tubes are
probably unnecessary.
A number of qualitative experimental observations
have been thought to be in conflict with existing
mathematical models of the Münch pressure–flow
hypothesis. Firstly, the long-distance transport of carbohydrates via the phloem appears diffusion-like in some
respects (Mason and Maskell, 1928; MacRobbie, 1971),
although molecular diffusion itself has been discarded as
a transport mechanism. Both the estimated mass transfer
rates (Milburn, 1975) and the observed dynamics of
radioactive tracer profiles (Minchin and Troughton,
1980) have shown that molecular diffusion of carbohydrates is far too slow. Secondly, a radial flux of
water across the sieve tube walls has been shown to be
an important feature of the long-distance transport of
carbohydrates (Minchin and Thorpe, 1982; Van Bel,
1990; Baker and Milburn, 1994). This flux of water is
found to be, amongst other things, correlated with the
concentration and movement of carbohydrates within
To whom correspondence should be addressed. Fax: q64 6 354 6731. E-mail: [email protected]
ß Society for Experimental Biology 2002
1412
Henton et al.
the sieve tubes. Thirdly, it has been shown that
carbohydrates can simultaneously travel in opposite
directions at different locations inside a sieve tube
(Trip and Gorham, 1968; Eschrich, 1975) and can
change their direction of travel at one particular location inside a sieve tube over a period of time (Geiger,
1987; Turgeon, 1989). Such directionality observations
could be attributed to the spatial and temporal dynamics
of the competing sources and sinks of carbohydrates.
It is the aim of this paper to construct a mathematical
model based on the Münch pressure–flow hypothesis
that can reproduce these qualitative observations of
phloem transport.
This paper presents a model based on the Münch
pressure–flow hypothesis constructed from the Navier–
Stokes equation for the motion of a viscous fluid and
the van’t Hoff equation for osmotic pressure. The derivation is for a solution inside a long, narrow, rigid tube
with no radial fluxes of solute through the tube walls.
Spatial dimensions that are appropriate for a sieve tube
and suitable initial profiles of the solute concentration
and solution velocity are adopted to make the model
more mathematically tractable and concise. Firstly, the
solute flux, the longitudinal water flux, and the radial
water flux for the steady-state case are calculated. The
expression for the solute flux is then shown to be similar
to Fick’s law of diffusion and an expression for the
hydraulic conductivity is calculated from the tube radius
and physical constants and then compared to experimentally determined values. Secondly, two numerical
simulations are used to illustrate some key temporal and
spatial dynamics of the solute and water fluxes that
arise from the relaxation of a non-uniform solute
concentration profile.
The mathematical model
Governing equations
The aim of this section is to derive a set of expressions
to describe, analyse, and simulate the dynamics of solute
transport inside a long, narrow, rigid tube immersed in
a water reservoir that stays at a constant hydrodynamic
pressure (c.f. the models of Christy and Ferrier, 1973;
Tyree and Dainty, 1975; Sheehy et al., 1995). A schematic
diagram of the modelled tube is shown in Fig. 1. It is
firstly assumed that the radial component of the velocity
of the solution inside the tube can be regarded as
negligible in comparison with the longitudinal component of the velocity of the solution. It is secondly
assumed that, due to the narrowness of the tube, the
radial component of molecular diffusion smoothes out
radial variations in the solute concentration (Tyree and
Dainty, 1975). The model presented here is focused on
interpreting the mechanism and resulting dynamics of
the solute per se so it will be further assumed that there
is no radial flux of solute through the tube walls.
Although the loading and unloading of carbohydrates
from sieve tubes are of fundamental importance in
plants, it is, for the intents and purposes of this paper,
considered more appropriate to construct and then
interpret a model of long-distance transport that does
not take these fluxes into account.
The longitudinal component of the velocity of the
solution (i.e. the solute and water) inside the tube is
denoted by v(r, z, t) and the solute concentration is
denoted by C(z, t) where r and z are the radial coordinate
and longitudinal coordinate of a cylindrical coordinate
system and t represents time. The Navier–Stokes equation
describes the motion of a viscous fluid and is typically
Fig. 1. Diagram of a long, narrow tube containing a solution which is immersed in a water reservoir that stays at a constant hydrodynamic pressure.
The radial coordinate r, the longitudinal coordinate z, and a volume element are shown.
Revisiting the Münch pressure–flow hypothesis 1413
regarded as the starting point for formulating a timedependent fluid dynamics problem (Landau and Lifschitz,
1987; Phillips and Dungan, 1993). The one-dimensional
form of this partial differential equation in a cylindrical
coordinate system is
2
Ev
Ev
EP
E v 1 Ev E2 v
r
þv
[m
[
[
¼Y
(1)
Et
Ez
Ez
Er2
r Er Ez2
where P is the hydrodynamic pressure, r is the density of
the solution, and m is the viscosity of the solution. The
terms on the left-hand-side account for the acceleration
of the solution, whereas the terms on the right-hand-side
account for the hydrodynamic pressure gradient and
viscous nature of the solution. For simplicity, the solution
viscosity will be treated as a constant and the force due
to gravity has not been taken into account.
A convenient assumption for formulating the Münch
hypothesis is that the hydrodynamic pressure P inside
the tube is always maintained at an osmotic equilibrium
with respect to the surrounding water reservoir (Thornley
and Johnson, 1990; Minchin et al., 1993). The dynamic
equilibrium thereby results in a local pressure difference
across the tube walls and so the hydrodynamic pressure
inside the tube can be expressed algebraically as
P ¼ P [ P0
(2)
where P is the osmotic pressure and P0 is the constant
hydrodynamic pressure of the water surrounding the
tube. It is noted that the effect of radial water fluxes on
the local hydrodynamic pressure surrounding the tube
is neglected. Assuming a dynamic equilibrium implies
that the resistance to the radial flux of water due to the
permeability of the tube wall is neglected (i.e. the
hydraulic conductivity is infinite) and so, as shown later
in the derivation, the accompanying radial flux of water
across the tube wall can then be calculated independently.
Including the effect of the permeability of the tube wall
increases the complexity of the model and thereby makes
it much more difficult to interpret both analytically and
via numerical simulations (Goeschl et al., 1976; Smith
et al., 1980; Phillips and Dungan, 1993). The van’t Hoff
equation for osmotic pressure of a dilute solution can be
written as
RTC
(3)
M
where R is the gas constant, T is the absolute temperature,
and M is the molar mass of the solute. In the model the
temperature T is taken to be constant along the tube.
Substituting equation (3) into the right-hand side of
equation (1) gives
2
Ev
Ev
RT EC
E v 1 Ev E2 v
[v
[m
þ
r
[
¼Y
(4)
Et
Ez
M Ez
Er2
r Er Ez2
P¼
Qualitative arguments are now put forward as a means
of justifying dropping some of the terms in this
equation. Firstly, it will be assumed that the tube has
dimensions that are comparable to a sieve tube as shown
in Table 1. Secondly, it will be assumed that there are
no large, localized variations in the solution velocity
along the tube, which means that the longitudinal solute
velocity profile is regarded as smooth. Hence the two
terms involving derivatives of v along the tube (i.e. the
derivatives of v with respect to z) in equation (4) can
be treated as insignificant when compared to the other
more dominant terms (e.g. the pressure term or the
viscosity term) and can be dropped. Thus equation (4)
can be approximated as
2
Ev
RT EC
E v 1 Ev
[m
[
(5)
r ¼Y
Et
M Ez
Er2
r Er
The dynamics of the solute concentration inside the
tube can be expressed by way of a conservation-of-mass
equation for the solute:
EC
E
¼ Y ðvC Þ
Et
Ez
(6)
where the longitudinal diffusion of the solute has been
ignored for further simplicity.
Individual flux expressions
The volume flux passing over the cross-sectional area
at a particular location along the tube Qvolume(z, t) is
given by
Z r¼a
Qvolume ¼
2prv dr
(7)
r¼0
Table 1. Physical constants (Lide, 1999) and typical values
(MacRobbie, 1971)
Definition
Symbol
Value
Gas constant
Absolute temperature
Molar mass of sucrose
Partial molar volume of
sucrose
Density of water
Diffusivity coefficient of
sucrose solution
Typical density of solution
Typical solution speed
Typical radius of sieve tube
Typical length of sieve tube
Typical concentration of
sucrose solution
Typical viscosity coefficient
of sucrose solution
R
T
M
V
8.315 J K1 mol1
293 K
0.3423 kg mol1
6.25 3 104 kg1 m3
rwater
k
1.0 3 103 kg m3
4.0 3 1010 m2 s1
r0
v0
a
l
C0
1.3 3 103 kg m3
1.0 3 103 m s1
1.0 3 105 m
3.0 3 101 m
3.0 3 102 kg m3
2.6 3 103 J m3 s
m
a2 RT
8M
Lumped transport coefficient
h¼
Typical apparent diffusivity
coefficient
hC0 ¼
Tube conductivity for 1 m
L¼
a2
8
a2 RT
8M
3.4 3 105 kg1 m5 s1
C0
1.0 3 102 m2 s1
4.8 3 109 m5 J1 s1
1414
Henton et al.
where a is the radius of the tube. The total flux of solute
passing over the cross-sectional area at a particular
location along the tube Jsolute(z, t) is given by
Jsolute ¼ CQvolume
(8)
while Fick’s law of molecular diffusion inside an identical
tube with stationary water is
Jsolute ¼ Y kpa2
EC
Ez
(9)
where k is the diffusivity coefficient of the solute in
stationary water. The total longitudinal flux of water
passing over a cross-sectional area at a particular location along the tube Jlong(z, t) can be calculated by
considering the volume-balance for the total volume
(i.e. splitting the total volume flux up into the water and
solute components) passing over this area (Fig. 1). This
relationship can be expressed as
Jlong ¼ rwater ð1 Y VCÞQvolume
(10)
where rwater is the density of water and V is the partial
molar volume of the solute. Thus the solute flux and
longitudinal water flux are in the same direction at
any given location along the tube. The radial flux of
water passing through a unit length of the tube wall
Jradial(z, t) can be calculated by considering the volume
flux balance for a stationary volume element and then
taking the limit (Fig. 1). This relationship can be
expressed as
EQvolume
Jradial ¼ rwater
Ez
(11)
The sign of the first-order derivative on the right-handside of this equation determines the direction of the radial
flux of water at a particular location along the tube.
Namely, if the sign is negative then Jradial(z, t) is positive
and so water will be incoming at that location and,
conversely, if the sign is positive then Jradial(z, t) is
negative and so water will be outgoing at that location.
Similarly, if the derivative itself is zero then Jradial(z, t) is
also zero.
Results
The steady-state case
The steady-state solution for the situation where the
solute concentration is held constant at the two ends of
the tubes can be expressed analytically. If the concentration at the left-hand end (i.e. z ¼ 0) is C0 and the
concentration at the right-hand end (i.e. z ¼ l ) is Cl then
the steady-state form of equation (5) can be integrated
twice to give
EC
RT 2
(12a)
v¼
r Y a2
4mM
Ez
which then can be substituted into the steady-state form
of equation (6) to give
z
(12b)
C 2 ¼ C02 [ Cl2 C02
l
and so both v and C vary non-linearly along the tube.
The flux of solute along the tube is
4 pa
RT
EC
C
Jsolute ¼ Y
M
Ez
8m
(13)
EC
a2 RT
2
¼ constant; h ¼
¼ Y pa hC
Ez
8mM
where h is a lumped parameter. In a mathematical
sense, this steady-state expression for the solute flux is
diffusion-like with the apparent diffusivity coefficient
(i.e. hC ) being proportional to the local solute concentration. Assuming a value of the sucrose concentration inside a sieve tube is 300 kg m3 (around
876 mol m3u0.876 mol l1), the radius of the tube a is
1 3 105 m, the length of the tube l is 3 3 101 m (c.f.
Christy and Ferrier, 1973; Goeschl et al., 1976), implies
that the apparent diffusivity coefficient is around seven
orders of magnitude greater than the diffusivity coefficient for sucrose in stationary water. Meanwhile, the
respective water fluxes are
4 pa
RT
EC
ð14aÞ
Jlong ¼ rwater
ð1 Y VC Þ ;
M
Ez
8m
Jradial ¼ rwater
pa4
8m
2
RT E C
M Ez2
ð14bÞ
where rwater is the density of water.
Using solute concentration values of C0 ¼ 300 kg m3
(around 876 mol m3u0.876 mol l1) and Cl ¼ 150 kg m3
(around 438 mol m3u0.438 mol l1) produces a maximum value of the longitudinal component of the solution velocity at around 5 3 102 m s1 which is much
greater than the typical solution speed of 1 3 103 m s1
(MacRobbie, 1971). On a similar note, it has been
estimated from experimental data that the apparent
diffusivity coefficient is approximately four orders of
magnitude greater than the diffusivity coefficient for
sucrose in water (MacRobbie, 1971). Hence the apparent
diffusivity coefficient for the model developed here is
around three orders of magnitude greater than values
that have been estimated from experimental data. These
differences serve to highlight the simplifications made
during the model development.
It is known that there are sieve plates inside the sieve
tubes (MacRobbie, 1971; Spanner, 1978) which are likely
to provide an additional resistance to the translocation
of carbohydrates along a sieve tube. The conductivity
for the flux of solution between adjacent sieve elements
within the steady-state models developed previously
Revisiting the Münch pressure–flow hypothesis 1415
(Christy et al., 1973; Goeschl et al., 1976) is assigned a
value of 1.02 3 106 m4 J1 s1. Using the values given
in Table 1 of Goeschl et al. (Goeschl et al., 1976), the
conductivity of the tube for 1 m is
L ¼ (1:02 3 10 Y 6 m4 J Y1 s Y1 )(2:0 3 10 Y 4 m)
¼ 2:0 3 10 Y 10 m5 J Y1 s Y1
which is significantly smaller than
L¼
a2
¼ 4:8 3 109 m5 J Y1 s Y1
8m
which is defined explicitly in the steady-state case.
Hence the models of Christy et al. and Goeschl et al.
(Christy et al., 1973; Goeschl et al., 1976) produce slower
solute and water fluxes because of the comparatively
small value given to the hydraulic conductivity of the
tube (Milburn, 1975).
A possible modification could be made to the model
developed above by judiciously introducing a damping
multiplier, e say, in front of the viscosity term in the
Navier–Stokes equation. The mathematical analysis
thereafter would essentially remain unchanged except
that the lumped parameters h and L would now have e
appearing in the denominators. Specifying a value of
e ¼ (4:8 3 10 Y9 u2:0 3 10 Y10 ) ¼ 2:4 3 101
would provide a reconciliation of the model developed
above with the steady-state models (Christy et al., 1973;
Goeschl et al., 1976) and would also reduce both
the maximum longitudinal component of the solution
velocity and the apparent diffusivity coefficient.
The time-dependent case
Analytical solutions cannot be obtained in the timedependent situation so numerical simulations will be
used to help illustrate some key features of the spatial
and temporal dynamics of the model. The solute is
sucrose, the tube radius a is 1 3 105 m, the tube length
l is 3 3 101 m, and the other constants are given in
Table 1. Non-uniform profiles of the initial solute concentration inside the tube are then specified and key
features of the relaxation toward a steady-state are
observed. These profiles must be smooth and contain no
large, localized variations in the gradient of the solute
concentration profile (i.e. smooth profiles) so that
approximations used in the model development are
satisfied. It is noted that the damping coefficient
defined earlier has not been included in these simulations. The inclusion of the damping factor defined in
the steady-state case given earlier would slow down the
relaxation process by between one to two orders of
magnitude.
A numerical code based on the exact time-dependent
equations using a Crank–Nicholson scheme (Smith, 1985;
Press et al., 1988) is constructed in order to simulate
the dynamics of the solution velocity and solute concentration. The tube is resolved with a large number of
elements and the solute concentration for each element
is then evaluated using equations (5) and (6), respectively.
The total longitudinal water flux and the radial water flux
per unit length for each element can be evaluated using
discrete forms of equations (10) and (11), respectively.
Here the concentration gradient at both ends is fixed
at zero so that neither the solute nor the water enters
or exits the tube across these boundaries. Since solute
cannot enter or exit the tube, the steady-state situation
will have a uniform solute concentration along the tube
and both the accompanying water fluxes will be zero.
Simulation 1
The first simulation illustrates the diffusion-like characteristics of the solute flux and key features of the
accompanying water fluxes. To achieve this result, an
initial concentration profile is chosen such that the
left-hand side of the tube starts off as a source-like
region (i.e. a region of high concentration) and the righthand side of the tube starts off as a sink-like region
(i.e. a region of low concentration) as seen in Fig. 2a.
In this case the initial profile has a minimum concentration of 150 kg m3 (around 438 mol m3u0.438 mol l1)
at the right-hand end the tube and a maximum concentration of 300 kg m3 (around 876 mol m3u0.876 mol l1)
at the left-hand end of the tube although there is no
large localization in the solute concentration along
the tube.
Overlaid profiles of the solute concentration inside
the tube at various times are shown in Fig. 2a. The
dynamics of the solute inside the tube behaves qualitatively similarly to diffusion, the initial solute concentration
profile gets smoothed out until the concentration becomes
uniform along the tube. Hence a net result is that solute
travels from regions of high concentration to regions of
low concentration.
Overlaid profiles of the total longitudinal water flux
over a cross-sectional area inside the tube at various
times are shown in Fig. 2b. Positive values for this flux
signify that the water is travelling from left to right. The
longitudinal flux of water inside the tube is also down
gradients in the local solute concentration and decreases
in magnitude as the solute concentration profile becomes
more uniform. The flux of water is zero across the two
ends of the tube (because the solute gradient has been
set to zero at these boundaries) and has a local maximum
near the centre of the tube.
Overlaid profiles of the radial water flux per unit
length of tube wall at various times are shown in Fig. 2c.
There is an influx of water in the regions of high solute
concentration, i.e. the source-like region, and an outflux
1416
Henton et al.
(a)
to the profile in Fig. 2a. Again the magnitude of the flux
of water decreases as the solute concentration becomes
uniform inside the tube.
Simulation 2
(b)
(c)
Fig. 2. Diagram of (a) the solute concentration inside the tube C(z, t),
(b) the total longitudinal water flux inside the tube Jlong(z, t), and (c) the
radial water flux per unit length of the tube wall Jradial(z, t). The overlaid
profiles are taken at t ¼ 0 s (solid line), t ¼ 5 3 102 s (dotted line),
t ¼ 1.5 3 101 s (dashed line), and t ¼ 5 3 101 s (dash-dot line).
of water in the regions of low solute concentration, i.e. the
sink-like region. The maximum influx is at the left-hand
end of the tube, the maximum outflux is at the right-hand
end of the tube, and the flux is zero near the centre of the
tube. Hence the profile in Fig. 2c is qualitatively similar
The second simulation illustrates the different aspects
of the directionality of the solute and water fluxes. To
achieve this result, a concentration profile that is initially twin-peaked is chosen to reproduce the desired
directionality features (Fig. 3a). The starting profile is
symmetrical and has a minimum concentration of
150 kg m3 (around 438 mol m3u0.438 mol l1) at the
two respective ends of the tube and a maximum concentration of approximately 300 kg m3 (around
876 mol m3u0.876 mol l1)
about
one-third
and
two-thirds of the way along the tube, respectively.
Overlaid profiles of the solute concentration inside
the tube at various times are shown in Fig. 3a. The two
superimposed peaks first coalesce into a single peak
and then this aggregated peak gets smoothed out until
the solute concentration is uniform. As the solute gradient determines the direction of movement, the flux of
the solute inside the tube displays several directionality
features with this particular initial profile. Firstly, the
solute flux near the left-hand end of the tube is always
directed to the left and the flux near the right-hand end
of the tube is always directed to the right. Hence the
solute is able to travel in different directions at different
locations along the tube at the same time. Secondly, the
solute flux at a given location along the tube that is
slightly to the left of the tube centre is initially directed
to the right, eventually reaches zero, and then is directed
to the left thereafter. Hence the flux of the solute can
change direction at a specific location along the tube
over time. Due to the symmetry of the starting profile,
the difference in the solute concentration between the
two ends of the tube is always zero. The solute flux
occurs because it is the local gradients in the solute
concentration along the tube which drive the dynamics
inside the tube.
Overlaid profiles of the total longitudinal water flux
over a cross-sectional area inside the tube at various
times are shown in Fig. 3b. The longitudinal flux of
water inside the tube has similar directionality features
with the solute flux in Fig. 3a. Firstly, the water flux can
be in different directions at different locations along
the tube at the same time. Secondly, the water flux can
change direction at a given location along the tube
over time.
Overlaid profiles of the radial water flux per unit
length of the tube wall at various times are shown in
Fig. 3c. The radial flux of the water across the tube walls
also has several directionality features of interest with
this simulation. Firstly, the first profile shows that water
Revisiting the Münch pressure–flow hypothesis 1417
(a)
overlaid profiles show that the water flux at the centre
of the tube is initially incoming, reaches zero, and then
is outgoing. Hence the radial water flux change can
change direction at a specific location along the tube
over time.
Discussion
(b)
(c)
Fig. 3. Diagram of (a) the solute concentration inside the tube C(z, t),
(b) the total longitudinal water flux inside the tube Jlong(z, t), and (c) the
radial water flux per unit length of the tube wall Jradial(z, t). The overlaid
profiles are taken at t ¼ 0 s (solid line), t ¼ 5 3 103 s (dotted line),
t ¼ 2.5 3 102 s (dashed line), and t ¼ 1 3 101 s (dash-dot line).
flux around one-third the way along the tube is initially
incoming and the water flux around two-thirds the
way along the tube is initially outgoing. Hence there are
locations where the flux is incoming, the flux is zero,
and the flux is outgoing at the same time. Secondly, the
A mathematical model of the Münch pressure–flow
hypothesis for long-distance transport of carbohydrates
via sieve tubes has been constructed from the Navier–
Stokes equation for the motion of a viscous fluid and
the van’t Hoff equation for osmotic pressure. Assuming
spatial dimensions that are appropriate for a sieve tube
and ensuring suitable initial profiles of the solute concentration and solution velocity lets the model become
more mathematically tractable and concise. Thus several
key features of the solute flux and accompanying water
fluxes can be interpreted directly from the resulting
equations themselves without requiring simulations,
although simulations are necessary to determine the
model dynamics explicitly. These key features can
be compared to known qualitative observations of the
long-distance transport of carbohydrates inside the
phloem.
In the steady-state case, the flux of solute inside
the tube is given by an expression that is similar to
Fick’s law of molecular diffusion. This relationship with
Fick’s law is implicit in several other phloem transport
models (Thornley and Johnson, 1990; Minchin et al.,
1993; Sheehy et al., 1995), but has not been derived
explicitly. Observations of diffusion-like behaviour for
carbohydrate transport in the phloem tissue date back
to early work (Mason and Maskell, 1928) and several
papers since (Canny and Phillips, 1963; Passioura and
Ashford, 1974). Here the apparent diffusivity coefficient
is proportional to the local solute concentration and
also depends on the radius of the tube and a set of
physical constants. Using a realistic value for the sucrose
concentration of 300 kg m3 (around 876 mol m3u
0.876 mol l1), a tube radius of 1 3 105 m, and a tube
length of 3 3 101 m, the apparent diffusivity coefficient is
around seven orders of magnitude greater than the
diffusivity coefficient for sucrose in stationary water. It
has been estimated from experimental data that the
apparent diffusivity coefficient is around four orders of
magnitude greater than the diffusivity coefficient for
sucrose (MacRobbie, 1971).
The time-dependent model is able to demonstrate
that it is possible to have a difference in solute
concentration between two locations, but have no translocation of solution between these two locations. The flux
of solute at a given location is driven by the local gradient
in the solute concentration at that location. Therefore,
1418
Henton et al.
a difference in the local solute concentration between two
separate regions is not necessarily indicative of a net
solute translocation between these two regions (Minchin
et al., 1993). This feature can be readily seen in the second
simulation given in this paper where the initial solute
concentration profile along the tube is specified to be a
twin-peaked, symmetrical function. Although the concentration at any location in a given half of the tube is
always greater than the concentration at the opposing
end, the solution in this half never travels toward this end.
Hence there is no translocation of solution between the
two halves of the modelled tube.
In the time-dependent case, a radial flux of water
across the tube walls is intrinsically coupled with the
longitudinal fluxes of solute and water all the way along
the tube. Firstly, it was demonstrated with the aid of a
numerical simulation that the water flux at a specific
location along the tube can change from being incoming,
zero, or outgoing over a period of time. Secondly, it was
demonstrated with the aid of a numerical simulation
that the water flux can be incoming, be zero, or be outgoing at different locations along the tube at a particular
stage in time. The phloem transport literature suggests
that a radial flux of water is an important feature of
the long-distance transport of carbohydrates inside the
sieve tubes (Wardlaw, 1969; Minchin and Thorpe, 1982;
Van Bel, 1990). Hence it has been proposed that the long
distance transport would ‘stall’ (Baker and Milburn,
1994) if significant radial flux of water did not take place
with the result being that carbohydrate demands of
distant sinks would not be met.
There has been historical interest in the concept of
‘bidirectional’ transport of both carbohydrates and
water as a means of experimentally testing the various
proposed mechanisms of phloem transport, some mechanisms would allow a bidirectional movement within a
single sieve element and other mechanisms would not
(e.g. flow en masse) (reviewed by Eschrich, 1975). It needs
to be emphasized at this stage that there are two feasible
interpretations of ‘bidirectional’ transport within a single
sieve element or file of sieve tubes. The first is the transport of carbohydrate anduor water in opposite directions
at different locations along a sieve tube (Trip and
Gorham, 1968; Eschrich et al., 1972) whereas the second
is the transport of carbohydrates anduor water in opposite
directions at a particular location along the sieve tube
(Chen, 1951; Biddulph and Cory, 1957; Eschrich, 1975). It
has yet to be demonstrated experimentally that bidirectional transport of carbohydrate occurs at a particular
location within a single file of sieve tubes, although
it has been demonstrated within a stem and also within
the same vascular bundle (Eschrich, 1975; Thorpe and
Minchin, 1996).
In the time-dependent model presented here only
bidirectional transport of the first interpretation is
possible. Firstly, it was demonstrated that the solute
(and water) can simultaneously travel in opposite
directions at different locations along the model tube at
a given time, i.e. arising from competing sources
and sinks (Eschrich et al., 1972; Minchin et al., 1993).
Secondly, it was demonstrated that the solute (and
water) can change the direction of travel at a given
location along the model tube over time, i.e. comparable to a sinkusource transition (Geiger, 1987; Turgeon,
1989). Thus it can be concluded from the simulations
given in this paper that these bidirectionality results
for the transport of solute within a single tube are
still consistent with the original Münch pressure–flow
hypothesis.
The model presented in this paper does not take
into account any flux of solute through the tube walls.
The focus of this work was on the translocation of
solute and the accompanying fluxes of water inside the
tube which can be successfully modelled independently
of any radial solute fluxes. Here the relaxation of a
suitable non-uniform profile within the tube helped
provide both a rapid and sufficient means of illustrating and interpreting the translocation process. Loading
(Ho and Baker, 1982; Komor et al., 1996) or unloading
(Oparka, 1990; Patrick, 1990) of solute from the tube
could be incorporated by including radial flux terms
into the derivation of the presented model. Such flux
terms have been included in previous mathematical
models of solute transport within a tube (Christy and
Ferrier, 1973; Tyree and Dainty, 1975; Goeschl et al.,
1976). However, a disadvantage of these models is that
they are less amenable to being studied analytically and
therefore more reliant on numerical simulations for
making interpretations. The advantage of the analytical
model developed in this paper is that key features of
the model dynamics can be easily distinguished from the
governing equations themselves and are less reliant on
numerical simulations.
The model presented in this paper does not take into
account the presence of sieve plates (MacRobbie, 1971;
Spanner, 1978) or transcellular strands (MacRobbie,
1971; Johnson et al., 1976). The analysis presented here
assumes that the tube is strictly one-dimensional with a
constant radius and contains only the solute and water.
As briefly discussed in the text, introducing a damping
factor into the viscosity term in the Navier–Stokes
equation is one possible way of incorporating the effect
of the sieve plates into the proposed model. Similarly
the elasticity of the sieve tube walls is not taken into
account (Milburn, 1970; Lee, 1981). The analysis
presented here assumes that the tube walls are always
rigid. Thus a time-dependent model of solute transport
that incorporates the effect of the sieve plates and the
elastic nature of the tube walls could be viewed as
future work.
Revisiting the Münch pressure–flow hypothesis 1419
Acknowledgements
We would like to acknowledge the comments and advice kindly
given by Alistair Hall, Alfred Sneyd, Ian Craig, Michael Spink,
Michael Thorpe, Martin Hunt, and Adam Matich. Anonymous
referees are thanked for pointing out several weaknesses in
earlier manuscripts. We also gratefully acknowledge the service
provided by our librarians, Sarah Nation, Ann Ainscough and
Steven Northover. This work was funded by the PGSF CO6806.
References
Baker DA, Milburn JA. 1994. Photoassimilate transport. In:
Basra AS, ed. Mechanisms of plant growth and improved
productivity—modern approaches. New York: Marcel
Dekker Inc.
Biddulph O, Cory R. 1957. An analysis of translocation in the
phloem of the bean plant using THO, P32, C14,12. Plant
Physiology 32, 608–619.
Canny MJ. 1973. Phloem translocation. Cambridge: Cambridge
University Press.
Canny MJ, Phillips OM. 1963. Quantitative aspects of a theory
of translocation. Annals of Botany 27, 379– 402.
Chen SL. 1951. Simultaneous movement of phosphorus-32 and
carbon-14 in opposite directions in phloem tissue. American
Journal of Botany 38, 203–211.
Christy AL, Ferrier JM. 1973. A mathematical treatment of
Münch’s pressure–flow hypothesis of phloem transport. Plant
Physiology 52, 531–538.
Eschrich W. 1975. Bidirectional transport. In: Zimmermann
MH, Milburn JA, eds. Transport in plants. I. Phloem
transport. Encyclopedia of plant physiology, New series,
Vol. 1. Berlin, Heidelberg: Springer-Verlag.
Eschrich W, Evert R, Young JH. 1972. Solution flow in tubular
semipermeable membranes. Planta 107, 279–300.
Geiger DR. 1987. Understanding interactions of source and
sink regions of plants. Plant Physiology and Biochemistry
5, 659–666.
Goeschl JD, Magnuson CE, DeMichele DW, Sharpe PJH. 1976.
Concentration-dependent unloading as a necessary assumption for a closed form mathematical model of osmotically
driven pressure flow in phloem. Plant Physiology 58, 556–562.
Ho LC, Baker DA. 1982. Regulation of loading and unloading
in long-distance transport systems. Physiologia Plantarum
56, 225–230.
Johnson RPC, Freundlich A, Barclay GF. 1976. Transcellular
strands in sieve tubes; what are they? Journal of Experimental
Botany 27, 1117–1136.
Komor E, Orlich G, Weig A, Kockenberger W. 1996. Phloem
loading—not metaphysical, only complex: towards a unified
model of phloem loading. Journal of Experimental Botany
47, 1155–1164.
Landau LD, Lifshitz EM. 1987. Fluid dynamics, 2nd edn.
England: Pergamon Books Ltd.
Lang A. 1979. A relay mechanism for phloem transportation.
Annals of Botany 44, 141–145.
Lee DR. 1981. Elasticity of phloem tissue. Journal of
Experimental Botany 32, 251–260.
Lide DR. 1999. Editor-in-chief, CRC handbook of chemistry and
physics, 80th edn. Florida: CRC Press.
MacRobbie EAC. 1971. Phloem translocation, facts and
mechanisms: a comparative survey. Biological Reviews
46, 429–481.
Mason TG, Maskell EJ. 1928. Studies on the transport of
carbohydrates in the cotton plant. II. The factors determining
the rate and direction of movement of sugars. Annals of
Botany 42, 571–636.
Milburn JA. 1970. Phloem exudation from castor bean:
induction by massage. Planta 95, 272–276.
Milburn JA. 1975. Pressure flow. In: Zimmermann MH,
Milburn JA, eds. Transport in plants. I. Phloem transport.
Encyclopedia of plant physiology, New series, Vol. 1. Berlin,
Heidelberg: Springer-Verlag.
Minchin PEH, Thorpe MR. 1982. Evidence for a flow of water
into sieve tubes associated with phloem loading. Journal of
Experimental Botany 133, 233–240.
Minchin PEH, Thorpe MR, Farrar JF. 1993. A simple
mechanistic model of phloem transport which explains sink
priority. Journal of Experimental Botany 44, 947–955.
Minchin PEH, Troughton JH. 1980. Quantitative interpretation
of phloem translocation data. Annual Review of Plant
Physiology 31, 191–215.
Oparka KJ. 1990. What is phloem loading? Plant Physiology
94, 393–396.
Passioura JB, Ashford AE. 1974. Rapid translocation in the
phloem of wheat roots. Australian Journal of Plant Physiology
1, 521–527.
Patrick JW. 1990. Sieve element unloading: cellular
pathway, mechanism and control. Physiologia Plantarum
78, 298–308.
Phillips RJ, Dungan SR. 1993. Asymptotic analysis of flow in
sieve tubes with semi-permeable walls. Journal of Theoretical
Biology 162, 465–485.
Press W, Flannery B, Teukolsky S, Vetterlins W. 1988.
Numerical recipes in C: the art of scientific computing.
Cambridge University Press.
Sheehy JE, Mitchell PL, Durand J-L, Gastal F, Woodward FI.
1995. Calculation of translocation coefficients from
phloem anatomy for use in crop models. Annals of Botany
76, 263–269.
Smith GD. 1985. Numerical solution of partial differential
equations—finite difference method, 3rd edn. Claredon Press.
Smith KC, Magnuson CE, Goeschl JD, DeMichele DW. 1980.
A time-dependent mathematical model of the Münch hypothesis of phloem transport. Journal of Theoretical Biology
86, 493–505.
Spanner DC. 1978. Sieve-plate pores, open or occluded?
A critical review. Plant, Cell and Environment 1, 7–20.
Thornley JHM, Johnson IR. 1990. Plant and crop
modelling—a mathematical approach to plant and crop
physiology. Oxford University Press.
Thorpe MR, Minchin PEH. 1996. Mechanisms of long- and
short-distance transport from sources to sinks. In: Zamski E,
Schaffer AA, eds. Photo-assimilate distribution in plants and
crops—source–sink relationships. New York: Marcel-Dekker
Publishers.
Trip P, Gorham PR. 1968. Bidirectional translocation of
sugars in sieve tubes of squash plants. Plant Physiology
43, 877–882.
Turgeon R. 1989. The source–sink transition in leaves. Annual
Review of Plant Physiology and Plant Molecular Biology
40, 119–138.
Tyree MT, Dainty J. 1975. Theoretical considerations.
In: Zimmermann MH, Milburn JA, eds. Transport in
plants. I. Phloem transport. Encyclopedia of plant physiology,
New series, Vol. 1. Berlin, Heidelberg: Springer-Verlag.
Van Bel AJE. 1990. Xylem–phloem exchange via the rays: the
undervalued route of transport. Journal of Experimental
Botany 41, 631–644.
Wardlaw IF. 1969. The control and pattern of movement of
carbohydrates in plants. Botanical Reviews 34, 79–105.