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Proc. R. Soc. A (2007) 463, 881–896
doi:10.1098/rspa.2006.1803
Published online 9 January 2007
Osmosis in semi-permeable pores:
an examination of the basic flow equations
based on an experimental and molecular
dynamics study
B Y I. S. D AVIS 1 , B. S HACHAR -H ILL 1 , M. R. C URRY 2 , K. S. K IM 3 ,
T. J. P EDLEY 4 AND A. E. H ILL 1, *
1
Physiological Laboratory, University of Cambridge, Cambridge CB2 3EG, UK
2
Biological Sciences, University of Lincoln, Riseholme LN2 2LG, UK
3
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
4
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge CB3 0WA, UK
Classically ‘semi-permeable’ pores are generally considered to mediate osmotic flow at a
rate dependent upon the hydraulic conductance of the pore and the difference in water
potential. The shape or size of the solute molecules is not considered to exert a first-order
effect on the flow rate nor is the hydraulic conductance thought to be solute dependent.
By the experimental measurement of osmosis in the biological pore AQP (aquaporin)
and hard-sphere molecular dynamics simulation of a model pore, we show here that the
solute radius can have a profound effect on the osmotic flow rate, causing it to decline
steeply with decreasing solute radius.
Using a simple non-equilibrium thermodynamic theory, we propose that an additional
‘osmotic flow coefficient’ is required to describe flows in semi-permeable structures such
as AQPs, and that the fall in flow rate with radius represents a conversion from hydraulic
to diffusive water flow due to increasing penetration of the pore by the solute. The
interaction between the pore geometry and the solute size cannot, therefore, be
overlooked, although for every solute the system obeys the criterion for semipermeability required by basic thermodynamics. The osmotic pore theory therefore
reveals a novel and potentially rich structure that remains to be explored in full.
Keywords: osmotic permeability; molecular dynamics; water transport;
pore structure; aquaporins; reflexion coefficients
1. Pore osmosis: background and aims
When solute gradients are applied across an impermeable porous membrane, it is
widely accepted that the osmotic flow rate will be equal to that created by an
equivalent pressure gradient, and that it will be independent of the shape or the
size of the solute. The equations for osmosis in pores have been formulated on
two implicit assumptions: (i) that the pore geometry is regular and the solute can
* Author for correspondence ([email protected]).
Received 25 July 2006
Accepted 30 November 2006
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either permeate or be totally excluded, and (ii) that in small pores, the
mechanism of water transfer can be considered similar for all driving forces—
osmotic, hydraulic and diffusive. These have resulted in the formulation of the
problem by Kedem & Katchalsky (1958), known as the KK equations.
The first assumption is not true for aquaporins, which have a straight pore
middle section with flared atria at each end. Solute molecules used as osmolytes
can therefore penetrate part of the atrial region, if not the core, and it is the
central section that renders most AQPs impermeable to solutes. The
permeability and selectivity of AQPs has been reviewed in relation to their
conserved core but variable surfaces (Engel & Stahlberg 2002). The main point of
concern is that AQP1 in the red cell is impermeable to the small molecules and
ions used in the experiments described here, although AQP1 shows a partially
induced permeability to ions (Yu et al. 2006), CO2 (Hub & de Groot 2006) and
probably to other very small solutes. The second assumption is not supported by
theory unless the pore is so narrow that water transfer is strictly unidirectional
(Longuet-Higgins & Austin 1966; Finkelstein 1987). Single-file osmosis has been
treated earlier as a filing phenomenon (Levitt 1974, 1975) and more recently as a
linear site occupancy model (Chou 1999).
Molecular dynamics (MD) studies of osmosis through pores are comparatively
rare. In a recent MD study of water transport driven by an osmotic salt gradient
through carbon nanotube (CNT) arrays (Kalra et al. 2003), it appears that the
water is truly confined to a continuous file, a hydrogen-bonded ‘water wire’, but
it should be noted that this may not always be continuous and the nanotube
system may be empty for some of the time (Hummer et al. 2001; Maibaum &
Chandler 2003). As the transitions within the pore seem to have a low activation
energy, the rate of osmosis is dominated by the entry and exit rates at the pore
mouths. Thus, the overall conductance is a weak function of the pore length
because the end-processes dominate. From a hard-sphere MD study of hindered
diffusion in cylindrical pores (Suh et al. 1993), it appears that under specular
conditions at the single-file limit, the solvent can behave like a mechanical rod
that begins to slip through the pore transferring volume faster than by
unrestricted diffusion. Therefore, the behaviour seen in CNT can be reproduced
by hard-sphere MD, but this is not the general case and depends upon the forces
at the wall (Beckstein & Sansom 2004). Recent MD studies of osmosis in the
aquaporins GlpF (Zhu et al. 2002) and AQP1 (Zhu et al. 2004a,b) have yielded a
conductance similar to that found in experiments.
Assumption (ii) has been challenged for narrow, as opposed to very narrow,
pores (Hill 1994, 1995) and recently an MD simulation with hard spheres,
simulating osmotic and hydraulic flow in straight pores, has shown that both
viscous and diffusive modes of water transfer can exist with different rate
coefficients in the same pore (Kim et al. 2005). In these simulations, the osmotic
flow is determined from the volume flow at zero pressure difference. When a
solute can enter a pore section, water flow during osmosis is diffusive in nature.
Therefore, in the atria, a region where a solute can partially gain access, there
should be diffusive and viscous flow in series. Diffusive flow is slower than
viscous flow, so that any solute which can penetrate more deeply by virtue of
its smaller size may be expected to reduce the overall osmotic flow rate through
the structure.
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In the fundamental equation for the osmotic volume flow Jv across a
membrane element,
Jv Z Pos Dp C Pf Dp;
ð1:1Þ
where Dp and Dp are the osmotic and hydrostatic pressure differences, and Pos
and Pf are the osmotic and hydraulic permeabilities related by the reflexion
coefficient
P
s Z os ;
ð1:2Þ
Pf
where the value of Pos should be equal to Pf for all impermeant solutes, i.e. s lies
between 0 and 1. In all studies with biological membranes where pressure
differences cannot be applied, a large impermeant solute is used to measure Pf on
the assumption that PfZPos. The reflexion coefficient obtained in this way is a
ratio of two osmotic permeabilities. In this paper, we have taken care to label this
as srel to distinguish it from the s of equation (1.2) for reasons we discuss below
and which are central to this paper. Where the ‘reflexion coefficient’ of a pore has
been reported in the literature, it is virtually always srel and not the coefficient s.
It has been generally assumed, however, that Pos must be constant for any
semi-permeable pore, but when the pore structure allows a solute to partially
enter, there exists the possibility that the driving forces can be modified within
the pore, in particular the balance between hydrostatic and osmotic pressure
gradients, resulting in changes in Pos. If s is to remain at a maximum value of 1
for a pore structure that is semi-permeable overall, but Pos is to vary, then Pf
must also vary. This may be difficult to envisage because Pf is normally
considered to be the hydraulic conductivity of the pore independent of the
presence of a solute, i.e. the conductivity in pure solvent and a constant. If we
designate the hydraulic conductivity of the pore by the older symbol Lp and
consider that
Pf Z
vJv
;
vp
ð1:3Þ
for any applied gradient of a particular solute, we then have other potential
relations in the semi-permeable case such as
Pf
Pos
Z max
% 1:
Lp
Pos
ð1:4Þ
In the case where PfsLp, the difference can only be due to the interaction of
osmotic and hydrostatic gradients within the pore. In this paper, we show that
there is clear evidence for behaviour as given by equation (1.4) and we provide a
simple derivation of this in terms of solute incursion depth. This obviously
requires a novel departure from the ‘classical’ concept of osmotic flow in which Pf
is constant.
In an earlier study, it was shown that for AQP1 in the red cell, the value of srel for
methylurea, a small osmolyte that cannot traverse AQP1, is substantially lower
than unity (Curry et al. 2001) and close to 0.5 (measured against NaCl), although
the permeability of AQP1 to this and other small organic solutes is close to zero
(Van Hoek & Verkman 1992; Whittembury et al. 1997; Coury et al. 1998). The
traditional thermodynamic explanation for any srel less than unity is that the pore is
permeable to the solute, in which case water and solute fluxes are sharing the same
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I. S. Davis et al.
channel, i.e. water–solute friction or flow interaction is responsible for slowing the
water transfer. This explanation cannot be sustained in this case and, in addition,
the difference cannot be due to second-order effects, such as a solute-specific
interaction with water in only part of the channel where the solute is stationary (i.e.
the net solute flux is zero), because such an effect is nonlinear, while experimentally
(Curry et al. 2001) and in the MD results shown here, the osmosis is very linear with
driving force down to zero solute concentration (this point is dealt with below where
such interactive effects are ruled out). This finding also rules out the involvement of
substantial ‘unstirred-layer’ effects, which show up immediately as a nonlinearity
between the osmotic flux and the solute concentration difference.
With the intention of building up a more comprehensive picture, we measured
the osmotic permeability of several solutes as a function of osmolyte cylindrical
radius, using them to drive osmotic flows across AQP1 in the red cell membrane.
The cylindrical radius, the smallest radius of a cylinder that can contain the
solute, is taken to be the radius controlling the maximum incursion of a solute
into a pore section, as first shown clearly by Soll (1967).
We also investigated the phenomenon by using hard-sphere MD in a system
comprising a range of spherical solutes and a solute-impermeable pore. This
simulation is free of any particular molecular interactions as would have been
present in studies with a real system in which the rates of osmotic flow are
conditioned by specific intermolecular interactions. While atomistic simulations
are clearly revealing the rates of osmotic water transport in real systems such as
AQPs and the effects of specific architecture and interactions on these rates
(de Groot & Grubmuller 2001; Zhu et al. 2002, 2004a,b; Jensen & Mouritsen
2006) and similar insights are appearing with CNT simulations (Hummer et al.
2001; Kalra et al. 2003; Zhu et al. 2004a,b), general osmotic principles can be
revealed with simulations that omit specific interactions (Murad & Powles 1993;
Murad et al. 1993; Murad 1996; Maibaum & Chandler 2003; Beckstein & Sansom
2004; Kim et al. 2005). Hard-sphere MD is therefore a heuristic tool in this case
and reveals fundamental behaviour, such as the interaction between the pore
geometry and the solute size, although it is of limited applicability to the
simulation of biological systems.
We simulated osmotic flow in a pore with flared conical ends that has similarities
to an AQP pore—the ‘hour glass’ structure—which has been determined by
electron cryoscopy (Walz et al. 1997; Murata et al. 2000; Sui et al. 2001) and X-ray
crystallography (Sui et al. 2001) that is supported by atomistic MD simulations
(de Groot & Grubmuller 2001; Zhu et al. 2002, 2004a,b). We also present MD results
from a straight pore, and thus a comparison can be made to show the effect of the
atria. This is the region where solutes can enter to differing depths depending upon
their radii and which would be the site of any mechanism that gives rise to different
osmotic flow rates, the rest of the pore being only solvent-filled.
2. Experiments
(a ) Osmotic flow measurements
Osmotic permeabilities, Pos, were measured as the volume flow per unit area induced
by an osmotic gradient at zero pressure difference, according to equation (1.1).
The techniques used in this paper are identical to those used in a previous study
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Osmotic flow across semi-permeable pores
inward flow
outward flow
AQP1 reflexion coefficient, srel
1.0
0.8
0.6
0.4
0.2
0.0
0
1.0
2.0
3.0
4.0
solute radius (Å)
Figure 1. Experimental relative reflexion coefficients of the red cell AQP1 as a function of solute
cylindrical radius (bidirectional).
(Curry et al. 2001). Human AQP1 (aquaporin-1) is a 28 kDa membrane protein
comprising four identical water channels in a tetramer. At present, there is no
evidence for interaction between the water channels. The configuration of the
molecular complex has approximately conical atria with a straight middle section of
approximately 2 Å radius, which is permeable to water molecules in a no-pass file.
Volume flows were measured without disturbing the molecule from its natural bilayer
(the human red cell ghost). The tetramer has a dyadic axis in the membrane, but
the bilayer (vesicular) system is asymmetric, so flows were induced in both directions,
by reversing the osmotic gradient, to confirm the validity of the techniques.
Eight solutes, to which the AQP1 channels are impermeable (Van Hoek &
Verkman 1992; Toon & Solomon 1996; Whittembury et al. 1997; Coury et al. 1998),
were used to apply osmotic gradients. To determine the solute cylindrical radii, they
were built using desktop molecular modeller software and energy-minimized to
include charge, bond length and angle. van der Waals representations were then
rotated and sized on a grid to determine the minimum containing circle using an
oxygen radius of 1.40 Å as a calibration. These cylindrical radii showed excellent
agreement with those determined from molecular models (Sha’afi et al. 1970), as did
the molecular volumes. They were as follows: (i) glucose (3.69 Å), (ii) 1,2-propanediol
(2.76 Å), (iii) propionamide (2.73 Å), (iv) 1,4-butanediol (2.73 Å), (v) methylurea
(2.70 Å), (vi) acetamide (2.64 Å), (vii) urea (2.62 Å), and (viii) formamide (2.15 Å).
(b ) Results
In figure 1, osmotic permeabilities are plotted as a function of their cylindrical
radius, where it can be seen that there is a clear trend, amounting to a steep fall
in srel over quite a small change in radius. This spectrum of values indicates the
difficulty in obtaining a perfect ranking of the data, because the relationship is so
steep in the intermediate range that small variations in either srel or radius can
have an effect on the ordering. Nevertheless, it will become apparent that this
pattern is common to other systems that have been studied.
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I. S. Davis et al.
(a)
AQP1 reflexion coefficient, srel
1.0
0.8
0.6
0.4
0.2
0
AQP3 reflexion coefficient, srel
(b) 1.0
0.8
0.6
0.4
0.2
0
1.0
2.0
3.0
4.0
solute radius (Å)
Figure 2. (a) Experimental data of Toon & Solomon (1996) (circles) and Rich et al. (1967)
(squares) for human AQP1 plotted as a function of solute cylindrical radius. (b) Experimental data
of Zeuthen & Klaerke (1999) for AQP3 plotted as a function of solute cylindrical radius.
The points in figure 1 are the means of all experiments comprising outflow
(shrinkage) and inflow (swelling). It can be seen that they superimpose very closely
indeed. This indicates that the osmotic flow responses are symmetrical, a finding
which confirms the flow techniques in the context of the symmetrical AQP pore.
(c ) Comparison with other measurements
Similar experiments have been made in the past with AQP1 of the red cell,
which showed a similar pattern of results (Rich et al. 1967; Toon & Solomon
1996). However, these experiments were not performed to test the possible
incursion effect of solutes and the structure of the molecular pore responsible
(AQP1) was not known at that time. As a result of this, we have collected data
from these sources and plotted them in figure 2a.
In figure 2b, we have also plotted numerical data from the literature in which
srel of another similar pore, AQP3, was measured by a somewhat different
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Osmotic flow across semi-permeable pores
p
887
p
s
Figure 3. MD simulation. A snapshot of the pore system initially containing 20 solvent particles per
reservoir with a solute particle entering the right-hand atrium. The pore, of total length 16 Å from
p to p (dotted lines), comprises a central cylindrical core of radius 1.2 Å and flared conical atria
opening to cuboidal reservoirs of dimension 6 Å.
technique (Zeuthen & Klaerke 1999). It can be seen that the osmotic behaviour
of Pos with the solute radius rs is very similar to the results reported here and we
shall argue that a common mechanism is at work.
3. Molecular dynamics simulations
(a ) Procedures
The details of the MD, which are hard-sphere simulations, are similar to those
given in a previous paper (Kim et al. 2005), where the methods and the physical
assumptions are discussed in detail. Hard-sphere simulations evade the specific
molecular interactions, and the intention here is to reveal only the effects of the
solute size and the pore geometry. The solvent fluxes during osmosis were
number counts at constant pressure. The solute mole fraction, Xs, in its reservoir
was initially 1/31. For comparison, the mole fraction of a 1 molal solution in
water is 1/56, so the concentrations used were high. At the larger solute radii,
not more than one solute particle could be accommodated in the solute reservoir.
Although the molar concentration differences were therefore constant across the
pore, the mole fractions were different at steady state for each size of the solute
and this was reflected in the differences in the osmotic pressure.
The following two sets of simulations were used: (i) a straight cylindrical pore
and (ii) a flared pore with two conical atria. The latter allowed solute molecules
of smaller size to penetrate to a greater depth, which is a feature of the AQP pore
used in the flow experiments, and the comparison with a straight pore can reveal
this effect. A snapshot of the flared pore during a simulation with solute and
solvent molecules is shown in figure 3.
(b ) Thermodynamic conversions
Osmotic flows, and the accompanying osmotic theory, are concerned with
volume flows driven by osmotic pressure differences, but the MD used here
measures the net flux of water particles moving per unit time between the
reservoirs at constant pressure (for details see Kim et al. 2005) and osmotic
concentrations as solute and water particles occupying unit volume. Conversion
is thus required for the results to be meaningful, and if the solute size is changing
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I. S. Davis et al.
between the simulations, then the conversion factors are also affected by the
change in solvent molar volume. The molecular volume of solvent, Vw, is the
number of solvent particles distributed over the available volume of two
reservoirs and the pore,
ð2Vr C Vp KVs Þ
Vw Z
;
ð3:1Þ
Nw
where Vr and Vp are the available volumes of reservoir and pore, and Vs is the
solute volume of a sphere of radius (rsCrw), where rs and rw are the radii of the
solute and the solvent particles, respectively. These volumes have to be
calculated for the pores used here as an exercise in Euclidean geometry and
the details are not shown. If Nw is the total number of solvent particles in the
system and Nws is the number of solvent particles in the solute reservoir, then the
mole fraction of the solvent is
Nw ðVr KVs Þ
Nws Z
:
ð3:2Þ
ð2Vr C Vp KVs Þ
Fluxes of solvent are measured in MD as a number flux Jn, related to the volume
flux by
ð3:3Þ
J v Z J n Vw :
The osmotic pressure p of an ideal solution is a function of the solvent mole fraction
KRT ln Xw
pZ
;
ð3:4Þ
Vw
and so we can define the osmotic conductance of the pore, i.e. the osmotic volume flow
per unit osmotic pressure difference, as
KJn Vw2
Pos Z
;
ð3:5Þ
RT ln Xw
which is a general expression that holds for an ideal solution, independent of
concentration, and is therefore the relation we require for the hard-sphere system
employed here. The factor Vw2 =RT ln Xw has been used for converting a number flux
of solvent Jn, generated by the simulations, into osmotic permeabilities.
(c ) Results with the flared pore
The measurement of solvent particle flow Jn as a function of Dp is shown in
figure 4 for a simulation with the solute present. All the simulations with the
solute present are similar in form and highly linear. The gradient is equivalent to
the hydraulic conductivity of the pore. The osmotic flow component, Jn(os), is the
flux intercept at DpZ0, and the effective osmotic pressure Dp due to the presence
of the solute is given by the intercept at JnZ0. Thus, the slope of the line
represents both the osmotic conductance PosZJn(os)/Dp and the hydraulic
conductance PfZdJn/dDp. Linearity is therefore a demonstration that PosZPf
and is an expected consequence of equation (1.1), indicating that sZ1 in
conformity with equation (1.2). We may call this the basic criterion for semipermeability. If it were not true, then the situation where osmotic and
hydrostatic pressures are in balance at equilibrium, required by thermodynamics,
would be violated. The linearity of the results for each solute therefore
guarantees that although changes in Pos may occur, classical semi-permeability
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Osmotic flow across semi-permeable pores
160
water flux, Jn
120
80
Jn ( o s )
40
∆p
0
–1.5
–1.0
– 0.5
0
0.5
pressure difference, ∆ p
1.0
1.5
Figure 4. MD simulation. Linearity of the solvent particle flux as a function of pressure in the
presence of an osmotic gradient. The pure osmotic component is the flux intercept at DpZ0 and
the osmotic pressure is the intercept at JnZ0.
is still preserved. Surprisingly, it indicates that in the presence of the solute, as
the solute radius changes the hydraulic conductivity Pf (as defined by equation
(1.3)) also changes, in tandem with the osmotic permeability Pos. The Lp of the
purely water-filled pore must be unchanged (see equation (1.4)).
In figure 5a, the number flows of solvent at zero hydrostatic pressure difference
of all the data similar to that of figure 4 have been converted to Pos-values and
plotted as a function of the solute radius rs. It is apparent that the osmotic
conductance is falling with decreasing radius in a nonlinear fashion. This is
essentially a confirmation of the experimental data of figure 1. While the
experimental data show an apparently sigmoid relation to solute radius (s can
neither rise above 1.0, nor fall below 0), the points of figure 5a may be regarded
as the lower part of such a relationship, although this is speculative: the full
sigmoid nature of the curve would be apparent only if higher radii were included,
but these cannot be used here owing to the finite size of the reservoirs. If the
incursion depth of the solute into the pore is really the governing factor, then the effect
will plateau at high radii as the incursion depth will become asymptotic to zero.
(d ) Results with the straight pore
Straight pore results for Pos are also shown in figure 5a, where it can be seen
that the steep fall with solute radius is not present over the same range as in the
flared pore—the straight cylindrical pore results are comparably flat. The two
pores have different core radii (due to the history of the simulations), but their
Pos-values, though similar at higher radii, cannot be meaningfully compared due
to the fact that their overall geometry is quite different. But why is there a fall at
all in this case? The most probable explanation is that although the pore has a
uniform cross-section, decreasing the solute radius down to the permeable limit
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(a) 10
core radius,
flared pore
Pos (m3 Pa–1 s–1 /10 –24)
8
core radius,
straight pore
6
4
2
0
1.2
1.4
1.6
1.8
2.0
2.2
solute radius, rs (Å)
(b) 8
Pos (m3 Pa–1s–1 /10 –24)
flared pore
straight pore
6
4
2
0
5
4
3
2
axial incursion depth (Å)
1
0
Figure 5. (a) MD simulation. Osmotic permeability of the two pores as a function of solute radius.
The radius of the straight pore was 1.75 Å and the core radius of the flared pore was 1.2 Å. The
pore was impermeable to all the solutes (filled circles, flared pore; open circles, straight pore).
(b) The osmotic permeabilities of the straight and flared pores as a function of the axial incursion
depth of the solutes (filled circles, flared pore; open circles, straight pore).
(1.75 Å) allows a limited incursion i(rs) into the channel, which is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i Z rs K rs2 Krp2 ;
ð3:6Þ
where rp is the pore radius.
In figure 5b, Pos is shown as a function of the incursion depths for both pores: the
straight pore from equation (3.6) and the flared pore from a similar expression
derived from the conical atrium shown in figure 3. The comparison indicates that
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Osmotic flow across semi-permeable pores
891
the pore shape profoundly affects the flow rate, and these effects can only be
related to the nature of the forces set-up to drive the solvent through the pore core.
The nature of these forces will obviously depend on the molecular fluctuations in
the atria associated with the entry of solutes into this region that, as a time
average, must create changes in the ratio of hydrostatic and osmotic forces.
4. Theoretical considerations
(a ) The possibility of obstruction by a solute
We can use elementary non-equilibrium thermodynamics to predict the effect of
solute particles, whose mean molar velocity is zero owing to their overall
impermeability, on the flux of the solvent through the pore. This may be called
the ‘solute obstruction’ effect. In terms of frictional theory in solution
(Katchalsky & Curran 1965; Kim et al. 2005), the hydraulic permeability Pf
relating Jv to Dp is given by 1
Jv Z
Dp;
ð4:1Þ
fwm C ðcs =c w Þfsw
where fwm and fsw are the solvent–membrane and solute–solvent molar frictional
coefficients (the inverse of mobilities), respectively, within the pore, as opposed
to free solution, which means that fsw includes mutual obstruction effects. cs and
c w are the local membrane (here intra-pore) concentrations of the solute and the
solvent, respectively. As c w[cs, then cs/c w/Xs, the solute mole fraction, and
we may write
1
Pos Z
:
ð4:2Þ
fwm C Xs fsw
Xs is not less than 1/50 in the experiments and 1/30 in the simulations, so unless
fsw is much larger than fwm, Pos will be very insensitive to changes in solute
obstruction effects in the atria. An increase in solute radius rs will increase fsw,
but the mole fraction Xs in the channel will decrease due to solute exclusion from
the atria (or other available volumes) and the effects will tend to cancel. Osmotic
experiments have been run in this system with varying concentrations of the
solute (Curry et al. 2001), when the relationship between Jv and Dp was highly
linear with an intercept insignificantly different from zero. Thus, whatever the
value of Xs at any point within the pore, changing it has no effect upon Pos, which
is effectively constant. Thus, the term Xs fsw in equation (4.2) is small enough to
be neglected, presumably because fsw/fwm.
In the MD studies, there is the possibility that smaller solutes might have a
longer residence time in the atrium, effectively increasing Xs, but this is not
really born out by analysis of the MD and would be difficult to explain for a hardsphere system. It appears that solute obstruction is negligible and cannot explain
the effects of changes in solute radius.
(b ) The incursion of the solute and the bimodal effect
In contrast to the situation where viscous flow is driven by hydrostatic pressure
gradients, when osmotic gradients are present in a pore section they drive water by
a process of diffusive flow. These two modes of water flow have been incorporated
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side 1
pore
side 2
p2
p1
p3
p2
p=0
l1
l2
l0
Figure 6. Pressure and osmotic gradients in a pore with incursion zones for a solute.
to a ‘bimodal’ theory of pore flow, in which the prevailing gradient determines
both the rate and the mechanism of flow at any position. Essentially, the further a
solute molecule can penetrate into the pore, the larger the length over which
diffusive flow predominates. Diffusive flow is slower than viscous (hydraulic) flow
and displays different dimensional dependencies: locally, within a pore section of
width d, the former is proportional to d 2 while the latter to d 4 (Mauro 1957).
In straight cylindrical pores, the differences in incursion depth between
molecules of different radius will be very small, but when the pore has conical
atria or entrances that widen towards the ends, the incursion depth cannot differ
substantially between solutes. Differences in flow rates for a single solute with
different incursion depths have been demonstrated in a previous MD study (Kim
et al. 2005), using a pore designed with a constriction at one end only, and in this
case, the pore shows osmotic rectification. In neither direction can the solute pass
completely through the pore, which is therefore semi-permeable. The
rectification ratio at different pore radii was similar in magnitude and trend to
the measured viscous: diffusive ratio, approaching 2. This provides strong
evidence for the operation of the bimodal concept.
In figure 6, a pore is shown with different incursion depths at each end (Hill
1972). These incursion lengths are determined by the interaction of the solute
and the pore geometry and vary with solute size. Although the pore is straight
cylindrical, it is possible to describe the fall in solute concentration within these
sections in the most general way (here shown as linear), and whatever the shape
of the gradients, they must obey the fundamental relation that in any pore
section during the steady water flow Jw,
Pw
dp dp
K
dx dx
Z Jw ;
ð4:3Þ
i.e. for any water permeability Pw, there is continuity of water flow at every
point. In such a pore, the solute incursions are restricted to a zone in each pore
entrance of length l 1 and l 2, with l 0 being the central water-filled section. Here,
the pressure falls, but it is the overall gradient Dp that drives the osmotic flow
through the pore. In addition, there is an imposed pressure gradient Dp with
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Osmotic flow across semi-permeable pores
893
three pressure drops pd1, pd0 and pd2 over the respective pore sections. Consider
the general case with both osmotic and hydrostatic gradients imposed across the
pore when p1Opd1 and p2!pd2, where the three zones have conductivities l1, l0
and l2 and the pore length LZl 1Cl 0Cl 2. Then, the volume flows are given by
J21 Z
l1 ðp1 Kpd1 Þ l0 pd0
l ðp Kp2 Þ
Z
Z 2 d2
;
l1
l2
l0
ð4:4Þ
and
Dp Z pd1 C pd0 C pd2
;
Dp Z p1 Kp2 :
ð4:5Þ
Substitution then gives
J21 Z
DpKDp
:
ðl 1 =l1 Þ C ðl 0 =l0 Þ C ðl 2 =l2 Þ
ð4:6Þ
If the three zones have identical conductances, then
l0 ðDpKDpÞ
;
ð4:7Þ
L
which represents the flow across a semi-permeable pore by the conventional
description, where l0 is the viscous or hydraulic coefficient of the water-filled
section. When there are solute gradients present in a zone, the mechanism of flow
determines a priori the value of the conductance at that radius. In the zone l 1,
the osmotic gradient drives the flow against the pressure gradient and this
imposes a regime where l1 is diffusive. In l 2, the reverse occurs and the
conductances l0 and l2 are viscous and therefore higher. Setting l2Zl0 in
equation (4.6) and using the viscous: diffusive ratio qZl0/l1 and the hydraulic
conductance of the pore LpZl0/L, we obtain
L
ð4:8Þ
J21 Z
L ðDpKDpÞ Z 4Lp ðDpKDpÞ;
l 1 q C ðLKl 1 Þ p
J21 Z
where the initial expression in brackets is a coefficient that may be called the
‘osmotic flow coefficient’ and which is designated here as 4. It applies to a semipermeable pore, and it is clear that there are identical coefficients 4Lp for Dp and
Dp, which are not independent as (1.1) would imply. Furthermore, the pore can
osmotically rectify (when l 1sl 2). When the solute is completely excluded from
the pore, l 1Z0 and the flow is maximal, corresponding to 4Z1. This model can
be adapted to the more complex case where the middle section is a single file of
water molecules.
What is novel about (4.8) is that the osmotic and the pressure coefficients
behave identically and also decrease with increasing incursion length, although
the pore is still impermeable to solute. In addition, the reflexion coefficient as
defined by equation (1.2) is always equal to 1, because the osmotic and the
pressure flow coefficients are always equal for a semi-permeable pore. This is very
satisfying and accords with the generally accepted description in which s falls
only when the pore becomes permeable to solute, but now it allows for a decrease
in flow rates in the semi-permeable case too. For example, when qZ3 and the
incursion length is a quarter of the pore length, the osmotic flow rate has fallen to
0.7. This is comparable to that in the MD results, but with qZ12–13 calculated
for AQP1 (Mathai et al. 1996; Zhu et al. 2004a,b), it approaches the steeper fall
seen in the AQP1 experiments here. It must be clear, however, that l 1 and q have
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894
I. S. Davis et al.
no simple interpretation for a flared pore such as AQP1, whose channel has a
varying radius, and in this case, bearing in mind that qZ12–13 is a value too
high to be attained from the occupancy of the single-file section of an AQP pore,
it must include contributions from the atrial water where the effective pore
dimensions are greater and q would be higher.
5. Summary
When solutes are completely in the central channel section of the pore, we
propose the following mechanism to account for the osmotic flow data of figures 1
and 2. When excluded from the atria, the osmotic pressure of the solution will
translate to a tension in the solute-inaccessible regions of the pore, which is a
natural result of the exchange of osmotic for hydrostatic pressure when the solute
is excluded from a system close to equilibrium,
Dmw Z Dp C Dp x0;
ð5:1Þ
and this difference in pressure across the central channel drives a single-file water flux
with the maximum or ‘hydraulic’ conductivity. The ability of smaller solutes to enter
the atria will result in a collapse of the pressure regime in the incursion zone (part or
the whole of the atrium), which leads to a diffusive regime for water transfer across the
atria and eventually, for the smaller solutes, an absence of any pressure difference
across the central water channel. Water is then driven through this channel by a
concentration gradient of water alone and there is no pressure difference acting as a
body force on the water file. The fall in flow rate is governed by the size, shape and
properties of the atria in governing solute entry of progressively smaller radii.
If this is correct, and the narrow solvent-filled middle section is acting as a
rate-limiting pore section (compared with the atria, which are wider and
shorter), the water flow should fall from a hydraulic to a diffusive value, whose
ratio is q. For a q-value of approximately 12–13, this would involve a fall to less
than 8% of the maximum osmotic flow rate and this appears consistent with the
flow data. The osmotic flow coefficient 4 that governs the rate for this semipermeable structure thus represents the transition from a form of viscous to
diffusive water flow.
A.E.H. and M.R.C. would like to thank the Wellcome Trust for their generous support of the
experimental work on the red cell AQP1 (Project grant). A.E.H. and T.J.P. would like to thank the
Wellcome Trust for their generous support of the MD work over many years (Project grant with
extension). K.S.K. would like to acknowledge the support by the US DOE (University of California
Center for Computational Biology grant).
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