A dual-pathway ultrastructural model for the tight junction of rat proximal tubule epithelium
Peng Guo,1
Alan M. Weinstein,2 and
Sheldon Weinbaum1
1CUNY Graduate School and New York Center for
Biomedical Engineering, Department of Mechanical Engineering, The City College
of the City University of New York, New York 10031; and
2Department of Physiology and Biophysics, Weill
Medical College of Cornell University, New York, New York 10021
Submitted 16 September 2002
; accepted in final form 10 March 2003
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ABSTRACT
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A dual-pathway model is proposed for transport across the tight junction
(TJ) in rat proximal tubule: large slit breaks formed by infrequent
discontinuities in the TJ complex and numerous small circular pores, with
spacing similar to that of claudin-2. This dual-pathway model is developed in
the context of a proximal tubule model (Weinstein AM. Am J Physiol Renal
Fluid Electrolyte Physiol 247: F848F862, 1984) to provide an
ultrastructural view of solute and water fluxes. Tubule model paramters (TJ
reflection coefficient and water permeability), plus the measured epithelial
NaCl and sucrose permeabilities, provide constraints for the dual-pathway
model, which yields the small-pore radius and spacing and large slit height
and area. For a small-pore spacing of 20.2 nm, comparable to the distance
between adjacent particle pairs in apposing TJ strands, the small-pore radius
is 0.668 nm and the large slit breaks have a height of 19.6 nm, occupying
0.04% of the total TJ length. This pore/slit geometry also satisfies the
measured permeability for mannitol. The numerous small circular pores account
for 91.25% of TJ NaCl permeability but only 5.0% of TJ water permeability. The
infrequent large slit breaks in the TJ account for 95.0% of TJ water
permeability but only 8.7% of TJ NaCl permeability. Sucrose and mannitol (4.6-
and 3.6-Å radius) can pass through both the large slit breaks and the
small pores. For sucrose, 78.3% of the flux is via the slits and 21.7% via the
pores; for mannitol, the flux is split nearly evenly between the two pathways,
50.8 and 49.2%. In this ultrastructural model, the TJ water permeability is
21.2% of the entire transepithelial water permeability and thus an order of
magnitude greater than that predicted by the single-pore/slit theory (Preisig
PA and Berry CA. Am J Physiol Renal Fluid Electrolyte Physiol 249:
F124F131, 1985).
paracellular pathway; water transport; compartment model; reflection coefficient
WATER AND SOLUTES CAN TRAVERSE the proximal tubule epithelium of
mammalian kidney via both transcellular and paracellular routes. The tight
junction (TJ) complex forms the major barrier in the paracellular route, and
its ability to seal the paracellular route is variable. In freeze-fracture
electron micrographs, the TJ appears to be a set of long, parallel, and linear
fibrils that bifurcate to form an interconnected network. These fibrils
consist of junction proteins of the claudin family and occludin
(7,
24,
30,
31). Several species of
claudins interspersed with occludin from one cell may copolymerize to form a
strand in a side-by-side manner
(15). Strands from neighboring
cells form a pair in a head-to-head homotypic or heterotypic interaction
(15,
31). Freeze-fracture electron
microscopic observations show that the TJ of rat proximal tubule consists
typically of a two-strand complex that is shallow (
100 nm) in the
apical-basal direction and that these strands exhibit discontinuities that can
exceed 0.1 µm in length
(25).
Although there are two basic transport routes, transcellular and
paracellular, the relative importance of each route for water has never been
satisfactorily resolved. The paracellular route, in particular, has offered a
substantial challenge because the structural correlate for the differently
sized pores or their frequency and cross-sectional geometry are still unknown.
Preisig and Berry (27)
concluded that paracellular water permeability cannot be >2% of
transepithelial water permeability. They measured the permeabilities of
mannitol and sucrose, which are believed to traverse the epithelium only via
the paracellular pathway, and then used the single-pore/slit theory (Renkin
equation) to predict the dimensions of the pores/slits, which satisfied the
permeabilities for both solutes. TJ water permeability was then predicted
using these pore/slit dimensions. Weinstein
(35) argued that paracellular
water permeability should be comparable to that of the transcellular pathway
to accommodate the low transepithelial NaCl reflection coefficient. In his
compartmental model (35),
water permeability for the TJ has a value that is one and one-half times that
of the measured transepithelial water permeability. The additional hydraulic
resistance is associated with the lateral interspace and solute polarization
by the basement membrane.
The pore/slit theoretical approach was questioned by Fraser and Baines
(8) because they noted that the
pore/slit theory underestimated the water permeability of man-made gel
membranes compared with the fiber matrix model developed by Curry and Michel
(5). They
(8) introduced a fiber matrix
model based on the theory of Curry and Michel
(5) to estimate TJ water and
solute permeability. In their model, the TJ is modeled as a homogeneous fiber
matrix gel with polymers of several-nanometer radius that fill the space
between TJ strands. The model provides a consistent picture for rabbit
proximal tubule, but when applied to the rat proximal tubule it predicted
small ion permeabilities that were an order of magnitude smaller than those
measured. This model treated the TJ complex as a uniform structure without
discontinuities. Therefore, it did not allow for the possibility of
low-resistance, large-pore/slit pathways. In addition, the model was applied
to a matrix that filled the space between the strands and not to the strands
themselves. In the present study, it is the strands themselves that account
for most of the paracellular resistance for solute transport.
In this paper, we propose an ultrastructural model for TJ strands that
consists of infrequent large "slit breaks" and numerous small
circular pores. We also ask that this model be consistent with the parameter
selection in the compartmental model in Weinstein
(35). In the next section, we
reconsider single-pore/slit analysis, as it applies to NaCl permeability, as
well as to the passage of mannitol and sucrose. We then introduce a
dual-pathway model, and its additional parameters (pore/slit dimensions and
frequency) are used to represent TJ attributes, which had been previously
estimated (35,
36). It will be argued that
the pore/slit attributes are morphologically realistic.
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SINGLE-PORE/SLIT MODEL
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Solute Permeabilities
We first examine the single-pore/slit theory for compatibility with
transepithelial water permeability, TJ solute permeabilities, and the NaCl
reflection coefficient for the entire epithelium. This means estimating the
dimensions of the pores or slits in the TJ strands that are required to
satisfy the measured permeabilities of both small ions and nonelectrolytes.
For a pore, the critical parameters are the pore radius,
Rpore, and the total pore area per unit surface area/pore
depth, Apore/
. For a slit, the corresponding
parameters are the slit height, W, and the total slit area per unit
surface area/slit depth, Aslit/
. We modify the approach in
Preisig and Berry (27), who
used the TJ permeabilities of sucrose and mannitol to determine the dimensions
of the paracellular pathway. In their approach, they apply the Renkin equation
to two solutes, mannitol and sucrose, whose radii are close in size, 3.6 and
4.6 Å, respectively. Alternatively, it should provide better
discrimination in pore/slit dimensions to use permeabilities of solutes with
large variation in their radii, such as salt and either mannitol or sucrose.
NaCl permeability data have been obtained by many investigators, and the
radius of NaCl differs significantly from those of both sucrose and mannitol.
Thus we can use TJ permeability for NaCl together with that for either sucrose
or mannitol to determine the dimensions of a pore/slit paracellular
pathway.
In single circular pore theory, the water and solute permeabilities of the
TJ strands, LTJ(pore) and HTJ(pore),
respectively, are given by
 | (1) |
 | (2) |
Here,
is the pore depth, Rpore is the pore radius,
Apore is the total pore area per unit surface area, and
µ is the viscosity of water, whose assumed value is 0.0007 Pa s.
Dpore is the diffusion coefficient for a solute in a
circular pore. An empirical expression, the Renkin equation
(27), is used to relate
Dpore to the free diffusion coefficient,
Dfree, and a, the solute radius
 | (3) |
There are two multiplicative factors in Eq. 3. The first factor,
(1a/Rpore)2, is the partition
coefficient, representing the steric exclusion from the pore. The second
factor describes the hydrodynamic interaction of the solute with the pore
walls. From Eq. 2
 | (4) |
Using measured permeabilities for two distinct solute species, Eqs. 3
and 4 provide a means of calculating Rpore and
Apore/
for a single-pore pathway. The lefthand side
of Eq. 4 is a function of HTJ, solute radius
a, and Rpore. If two solutes share the same
transport pathway, then Rpore and
Apore/
will be the same for that pathway. Thus the
right-hand side of Eq. 4 will have the same value for these two
solutes, and the left-hand side of Eq. 4, when plotted as a function
of Rpore, will yield a compatible solution for
Rpore, provided the two curves for
HTJ/Dpore intersect.
Preisig and Berry (27)
measured the permeabilities of sucrose and mannitol, which are believed to
traverse the epithelium only via the paracellular route. These measured
permeabilities are HTJ(mannitol) = 0.87 x
105 cm/s and HTJ(sucrose) =
0.43 x 105 cm/s. The estimated TJ
permeability for NaCl is HTJ(NaCl) = 13 x
105 cm/s
(35). Thus we can plot three
curves for the left-hand term of Eq. 4 for NaCl, mannitol, and
sucrose as a function of Rpore
(Fig. 1A). The
intersection of any two curves provides a compatible Rpore
that satisfies the Renkin equation for those two solutes. In this calculation,
the Stokes-Einstein radii for NaCl, mannitol, and sucrose are 1.47, 3.6, and
4.6 Å, respectively. Their corresponding free diffusion coefficients
(Dfree; x105
cm2/s) are 2.21, 0.90, and 0.70.

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Fig. 1. A: plot of Eq. 4 for Apore/
(or HTJ/Dpore) for NaCl, mannitol, and
sucrose as a function of pore radius, where Apore is total
pore area per unit surface area, is the pore depth,
HTJ is tight junctional (TJ) permeability, and
Dpore is diffusion coefficient for a solute in a circular
pore. The compatible solutions for the mannitol/sucrose pair are 1.41 nm, the
NaCl/mannitol pair, 0.80 nm, and the NaCl/sucrose pair, 0.95 nm. B:
plot of Eq. 8 for Aslit/ (or
HTJ/Dslit) for NaCl, mannitol, and
sucrose as a function of half-slit height, where is the depth of the
slit, Aslit is the total slit area per unit surface area
of epithelium, and Dslit is the solute diffusion
coefficient for an infinite slit. The compatible solution for the
mannitol/sucrose pair is 0.77 nm, the NaCl/mannitol pair, 0.46 nm, and the
NaCl/sucrose pair, 0.55 nm.
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The solutions for Rpore obtained from the intersections
of the curves in Fig.
1A are summarized in
Table 1.
Apore/
can then be found using Eq. 4 and
LTJ calculated using Eq. 1. These results are
also given in Table 1. Our
results for Rpore and water permeability, which satisfy
mannitol and sucrose permeabilities, are the same as the results given
previously by Preisig and Berry
(27). This water permeability
is <2% of the measured total transepithelial water permeability,
0.120.15 cm/s1
(27). Although the single-pore
model predicts slightly larger TJ water permeabilities, 0.0023 and 0.0028
cm/s, when computed using NaCl/mannitol and NaCl/sucrose pairs rather than a
mannitol/sucrose pair, 0.0018 cm/s, the values are still <3% of the
transepithelial water permeability.
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Table 1. Compatible pore radius or half-slit height for solute pairs and their
corresponding water permeability based on a single-pore model or a single-slit
model
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Similarly, a single-slit model can be used to estimate the slit height and
the total area of open slit per unit surface area/slit depth. Again, we assume
all solutes share the same transport pathway. In the slit model, the water and
solute permeability, LTJ(slit) and
HTJ(slit), respectively, are given by
 | (5) |
 | (6) |
Here,
is the depth of the slit, Aslit is the total
slit area per unit surface area of epithelium, and W is the height of
the slit. Dslit, the solute diffusion coefficient for an
infinite slit, is given by the Renkin equation
(27)
 | (7) |
The first factor in Eq. 7, 12
/W, describes the
steric exclusion and the second the increased hydrodynamic resistance of the
slit walls. From Eq. 6
 | (8) |
Following the same argument as in circular pore theory, we have plotted in
Fig. 1B the left-hand
side of Eq. 8 vs. the slit half height, W/2, for three
solutes, i.e., NaCl, mannitol, and sucrose. The intersections of the curves
provide the solutions to Eq. 8 for each solute pair. In these
calculations, the solute permeabilities, HTJ, are the same
as used previously for the pore calculations. The solutions for W/2
obtained from the intersections of the curves in
Fig. 1B are summarized
in Table 1.
Aslit/
can then be found using Eq. 8 and
LTJ calculated using Eq. 5. The results
summarized in Table 1 are
similar to those for a circular pore. LTJ(slit) for the
mannitol/sucrose pair is
1.5% of the transepithelial water permeability,
Lp, as previously predicted in Preisig and Berry
(27). Although
LTJ(slit) for the NaCl/mannitol pair or NaCl/sucrose pair,
0.0018 and 0.0023 cm/s, is a little larger than that for the sucrose/mannitol
pair, 0.0013 cm/s, it is still <2% of Lp. Thus
neither a pore model nor a slit model predicts values for
LTJ that are a significant fraction of
Lp.
Salt Reflection Coefficient
Instead of using solute permeability pairs to determine pore or slit
dimensions, one can use Lp, TJ salt permeability,
HTJ(NaCl), and the transepithelial reflection coefficient
for NaCl,
(NaCl), for the entire epithelium to determine the
dimensions of the paracellular pathway. Experiments show that the rat proximal
tubule epithelium has a
(NaCl) that is close to 0.7
(32). Accordingly, we shall
attempt to satisfy the measured values of Lp,
, and TJ NaCl permeability but relax the constraints on the nearly
impermeant solutes, sucrose and mannitol. For a single-pore/slit model for the
TJ, one assumes that water and NaCl will traverse the TJ, sharing the same
pore or slit pathway. This approach leads to pores or slits that are much
larger and less frequent than the single-pore/slit model just considered for
paired solutes, but one finds the permeabilities for sucrose and mannitol are
far too large, as we show next.
There is no directly measured value for LTJ. However, a
compartment model has been used to relate LTJ to
Lp, the measured transepithelial water
permeability (35). In
compartment models, the properties of the entire epithelium are determined by
the properties of its components: the cell barrier, the TJ barrier, and the
basement membrane barrier (Fig.
2). Conversely, the overall epithelial permeabilities will serve
as constraints for determining the component parameters, and these have been
displayed in Table 2. The
values for Lp,
, and H used in
the model of Weinstein (35)
were all taken from those compiled by Ullrich
(32). Preisig and Berry
(27) subsequently determined
an overall Lp about one-half that found by
Ullrich (32), and a lower
value is used in the present model. The reflection coefficient for the cell
membrane is 1.0 (26,
33) and that for the basement
membrane is 0.0 (38). The rate
of active osmolar transport across the basolateral membrane, N, was
taken to be approximately twice the rate of net epithelial sodium transport
(32). For the diffusive salt
permeability of TJ, HTJ, the value selected (if applied to
both Na and Cl) yields a realistic estimate for TJ electrical resistance
(9). Isotonicity of proximal
tubule volume transport is embodied in the parameter C*, which is the
decrement in luminal osmolality required to yield a reabsorbate osmolality
equal to that of the lumen. Experimental determinations of luminal osmolality
indicate that this value is no greater than 23% of blood osmolality,
but a more precise definition has not been possible. Its exact value may vary
with peritubular protein concentration and luminal anion composition, but
model calculations indicate that C* depends largely on the overall rate of
sodium reabsorption relative to cell membrane water permeability
(35).

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Fig. 2. Compartment model for rat proximal tubule epithelium. The cell and the TJ
are in parallel and form a composite barrier. The cell barrier has the ability
to actively transport sodium. This composite barrier is in series with the
basement membrane. In our model, the reflection coefficient of the basement
membrane for NaCl is zero, and the water and solute permeability of the
basement membrane are much larger than that of the composite luminal barrier.
N, active transport flux across the basolateral cell membrane due to
sodium-potassium pump. JVTJ and JSTJ,
tight junctional volume flux and solute flux, respectively;
JVC and JSC, transcellular volume flux
and solute flux, respectively; JVB and
JSB, basement membrane volume flux and solute flux,
respectively.
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For this initial calculation, consider the cell barrier and the TJ barrier
in parallel and omit for simplicity the resistance of the highly permeable
basement membrane barrier. For this simplified composite pathway model, the
transepithelial water permeability and the reflection coefficient for NaCl,
Lp and
, respectively, are given by
(36)
 | (9) |
 | (10) |
Here, LC is the water permeability of the cell barrier,
LTJ is the water permeability of the TJ,
C is the NaCl reflection coefficient of the cell barrier,
and
TJ is the NaCl reflection coefficient of the TJ barrier.
Reasonable values for
and
C for NaCl are
=
0.7 and
C = 1.0, as stated above. From Eq. 10, we
can see that if LTJ/Lp
<< 1,
is close to 1 rather than 0.7.
Combining Eqs. 9 and 10, one has
 | (11) |
According to pore theory, the reflection coefficient can be written as
(23)
 | (12) |
Here,
is the partition coefficient, which for a circular pore is given
by (23)
 | (13) |
Combining Eqs. 11, 12, and 13, we find that
 | (14) |
From Eqs. 1 and 2, we have two independent relationships for
Apore/
 | (15a) |
 | (15b) |
After we substitute Eq. 14 into Eq. 15a, the only unknown
variable on the right-hand side of Eq. 15a is
Rpore. Similarly, the only unknown on the right-hand side
of Eq. 15b is Rpore if we know the solute
permeability HTJ and the solute radius a (Eq.
3). If water and solute share the same transport pathway,
Apore/
must be the same for that pathway. Thus, if
we plot the right-hand sides of Eqs. 15a and 15b vs.
Rpore, the intersection of two curves provides the
compatible Rpore (Fig.
3A). This compatible solution for
Lp = 0.15 cm/s, HTJ(NaCl) =
13 x 105 cm/s,
= 0.7, and
a = 0.147 nm, is Rpore = 5.2 nm. Once
Rpore is determined, we can use either Eq. 15a or
Eq. 15b to obtain Apore/
, 6.64
cm1. Because the predicted
Apore/
now is nearly one-half the predicted values
for NaCl/mannitol and NaCl/sucrose pair in
Table 1 and the predicted
Rpore here is at least five times greater than the values
predicted in Table 1, there are
many fewer pores in the TJ strands when we try to satisfy the measurements for
NaCl and water permeability. The permeability of any solute can now be
calculated using Eq. 2. The corresponding permeabilities of sucrose
and mannitol are HTJ(mannitol) = 4.42 x
105 cm/s and HTJ(sucrose) =
3.16 x 105 cm/s. These permeabilities are
5.0 (mannitol) to 7.4 (sucrose) times greater than the experimental values in
Preisig and Berry (27). The
predicted Rpore is much greater than the sodium radius.
Thus, from Eqs. 12 and 13, the TJ reflection coefficient for
NaCl,
TJ, is close to zero. From Eq. 11,
LTJ is nearly 30% of Lp.
A similar analysis can be performed for the single-slit model, and the slit
dimensions for the TJ can be determined using the same values for
Lp, HTJ(NaCl), and
as
for the circular pore. To simplify our calculation, we assume
TJ is zero because we anticipate that the slit height
W >> 2a and
TJ
0. Thus from
Eq. 11, LTJ
0.3 Lp.
From Eqs. 5 and 6
 | (16a) |
 | (16b) |
The right-hand sides of Eqs. 16a and 16b are plotted vs.
W/2 in Fig.
3B for the same values of Lp
and HTJ as for the circular pore. One finds that the
compatible slit half height, W/2, is 3.2 nm and
Aslit/
, from Eq. 16a or Eq. 16b,
is 6.5 cm1. This value of W/2 is at
least five times greater than the values in
Table 1. The predicted slit
half height W/2 = 3.2 nm is much larger than the sodium radius. Thus
our assumption, that
TJ is close to zero, is valid. Once
W and Aslit/
are determined, the
corresponding permeabilities of sucrose and mannitol can be determined using
Eq. 6. They are HTJ(mannitol) = 4.6 x
105 cm/s and HTJ(sucrose) =
3.4 x 105 cm/s. These permeabilities are
again 5.3 (mannitol) to 7.8 (sucrose) times larger than the experimentally
measured values in Preisig and Berry
(27).
These model calculations indicate that a single-pore/slit model cannot
satisfy the well-documented experimental measurements for
Lp, TJ solute permeability, and the overall
reflection coefficient for small ions for rat proximal tubule. The
calculations above in Solute Permeabilities suggest that the
dimensions of the single pore/slit based on TJ solute permeability alone are
rather small. This small pore/slit will offer a great resistance for water
transport and account for <3% of the measured
Lp. Thus LTJ contributes
insignificantly to Lp. The calculations in this
section, which are based on Lp and
for
small ions for the entire epithelium, suggest that pores or slits whose
dimensions are at least a factor of five larger are required to accommodate
Lp and
. However, these larger pores/slits
predict a much larger solute permeability for sucrose and mannitol than the
experimental values. Thus a single-pore/slit model is unable to reconcile all
the experimental data.
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DUAL-PORE/SLIT MODEL
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TJ Barrier in a Compartment Model of Rat Proximal Tubule
Epithelium
These contradictions lead to consideration of a dual-pathway
ultrastructural model to reconcile the junctional permeabilities of water,
ions, and small nonelectrolytes. Our proposed model for the TJ strands
contains two parallel transport pathways: infrequent large slit breaks formed
by junction strand discontinuities and numerous small circular pores in the
claudin-occludin TJ complexes. The large slit allows for a significant passage
of water. Most importantly, these junctional strand breaks, which allow for
flow through a double-strand complex, are very few in number. This transport
pathway will also allow small ions to pass, but it is not the dominant route
for ions because of the very low probability that an open pathway will be
formed by breaks in a TJ complex of two or more strands. Numerous small
circular pores are the primary pathway for small ions. This small-pore pathway
allows for a solute flux for molecules <1.0-nm diameter but offers large
resistance for the passage of water. The key idea in the model is the
distinction between volume (water) and solute transport pathways. One cannot
use the solute transport pathway to estimate water permeability nor the
small-pore pathway to evaluate nonelectrolyte permeability and water
permeability. The heterogeneity in ultrastructure also provides an alternative
view of the fiber matrix model of Fraser and Baines
(8).
Experimental data from rat proximal tubule are for the transepithelial
permeabilities of water and salt and for the transepithelial NaCl reflection
coefficient. Therefore, a compartment model will be used first to estimate
LTJ and
TJ from the whole epithelial
coefficients. Of note, the cell in this model is treated as a barrier in
parallel with the junctional pathway. Compartment models for rat proximal
tubule epithelium were introduced to explore the potential significance of a
permeable TJ (37). The
compartment model was later extended to include the compliance of the lateral
intercellular space (35) and
the impact of TJ convection in the epithelial transport equations
(36). In this study, we shall
apply the 1984 compartment model to provide an estimate of the properties of
the TJ barrier (35).
In the compartment model of Weinstein
(35), the cells and the TJ are
in parallel and form a composite barrier, which are both in series with a
lateral interspace basement membrane (Fig.
2). In this model, the cell itself is a barrier, not a
compartment. In the Weinstein model
(35),
Lp, the transepithelial NaCl permeability
(H) and the NaCl reflection coefficient (
) for the entire
epithelium are given by
 | (17) |
 | (18) |
 | (19) |
where LMB is defined as
 | (20) |
Here, R is the gas constant, T is absolute temperature, and
C0 is a reference osmolality. Following Weinstein
(35), we replace the mean
membrane osmolality with the reference osmolality C0 (290
mosmol/kgH2O) to avoid nonlinearities and keep accuracy.
HM,
M, and LM are
the NaCl permeability, the NaCl reflection coefficient, and the water
permeability of the composite barrier formed by the cells and TJ complex.
HB and LB are the NaCl permeability
and the water permeability of the basement membrane. As in Weinstein
(35), we have assumed that the
reflection coefficient of the basement membrane is zero. In our model, we
assume the basement membrane has a higher permeability to water and solutes
than the composite barrier formed by the cells and the TJ complex.
From Eqs. 18 and 19, HM can be expressed in
terms of
M
 | (21) |
Using Eq. 18, HB can be written as
 | (22) |
Equation 17 can be written so that LMB appears
explicitly.
 | (23) |
If Eq. 20 is rewritten as
 | (24) |
LM can be determined if LB is
prescribed and LMB is evaluated using Eq. 23. All
the parameters appearing in Eqs. 1719 for the composite
barrier, except LM, can be determined if
M can be evaluated and Lp,
, and H are measured. However, it is argued in Weinstein
(35) that
LB >> LM and, thus
LM
LMB. Thus we need to obtain
only one additional independent relationship for
M.
Water reabsorption in the proximal tubule is driven by active transport and
the osmotic pressure differences that are established by this active
transport. Weinstein (35)
defines a measure of transport isotonicity which is given by
 | (25) |
Here N is the active transport flux across the basolateral cell
membrane due to the sodium-potassium pump,
M is the mucosal
(luminal) oncotic pressure, and
S is the serosal (peritubular)
oncotic pressure. Equation 25 defines the luminal osmolality
difference when the transported fluid has the same osmolality as the reference
osmolality C0. We will focus on the first term and thus require
that transport be isotonic even in the absence of peritubular protein. The
value of this term defines a constraint between
Lp and
M because
HM, HB, and LMB
are all functions of
M and LMB is
related to Lp through Eq. 23. Thus
M can be determined if we know the transepithelial values
for H, Lp, and
along with an estimate of
C*. After
M is determined, HM,
HB, and LMB can be evaluated using
Eqs. 21, 22, and 23 as described previously.
Once LM,
M, and
HM are determined, one next evaluates their TJ components,
LTJ and
TJ. These predicted values of
LTJ and
TJ are then used to assess the
detailed TJ structure. The properties of the composite barrier consisting of
the cell barrier and the TJ barrier can be expressed in terms of their
individual parameters. Let LC and LTJ
denote the water permeabilities of the cell and the TJ complex,
HC and HTJ be their NaCl
permeabilities, and
C and
TJ be their NaCl
reflection coefficients. Then
 | (26) |
 | (27) |
 | (28) |
The last term on the right-hand-side in Eq. 28 describes the
solute-solvent interaction for a heteroporous parallel pathway with different
reflection coefficients
(36).
Equations 26, 27, and 28 can be manipulated to provide a
constraint between LTJ and
TJ. From
Eqs. 26 and 27, the fractional water permeability of the
cell barrier, LC/LM, is related to
TJ by
 | (29) |
The fractional water permeability of the TJ is
 | (30) |
From Eq. 30, LTJ/LM cannot be less
than
C
M. Equation 28 can
be rewritten using Eqs. 26, 29, and 30 as
 | (31) |
 | (32) |
Equation 32 provides the required constraint between
TJ and LTJ. This assumes that all three
permeabilities on the left-hand-side of Eq. 32 are known,
C = 1, and
M has been related to
Lp using Eq. 25. HM has been
already determined by the compartment model in terms of H and
M (Eq. 21). HC is very small
(35).
HTJ is independently estimated from the expression for
transepithelial electrical resistance
 | (33) |
Here,
is transepithelial electrical resistance, z is the
valence for NaCl (z = 1), F is Faraday's constant, and
is the mean ion concentration (the same
reference osmolality C0 as in Eq. 17 is used). Because the
basement membrane and the composite barrier are in series in the compartment
model and the conductance of the basement membrane is much larger than that of
the composite barrier,
is approximated by the resistance of the TJ.
The NaCl permeability H varies from 13.7 to 19.1 x
105 cm/s (the corresponding transepithelial
resistance varies from 57
· cm2). In this
model, we have selected a value for HTJ that is at the
lower limit for H, 13 x 105
cm/s.
There are two unknowns,
TJ and LTJ,
in Eq. 32. A simple way to solve for
TJ and
LTJ is to replace LM by
LMB in Eq. 31, because LB
>> LM in Eq. 24. Equation 31 can then be
approximated by
 | (34) |
From Eq. 34,
TJ can be expressed explicitly as
 | (35) |
Once
TJ is determined, LTJ can be
calculated from Eq. 32
 | (36) |
Heteroporous Model for TJ Strands
As discussed above, we propose that TJ strands contain numerous small
circular pores and infrequent large slit breaks, the former associated with
junctional particle pairs and the latter associated with junctional strand
discontinuities, as sketched in Fig.
4. The model predictions for the sizes of the pores and the slits
strongly suggest this structure. A heteroporous model that includes
solute-solvent interaction must be used because the reflection coefficients
and the water permeabilities differ greatly for each pathway. Let 1 and 2
denote the two pathways, 1 for large slit breaks and 2 for small circular
pores. Based on the theory in Weinstein
(36), the composite values for
the TJ, LTJ, HTJ, and
TJ are
 | (37) |
 | (38) |
 | (39) |
Here, C0 is a reference osmolality for each solute. Equation
39 is applied separately for NaCl, mannitol, and sucrose. The last term
in Eq. 39 again represents the solute-solvent interaction as in
Eq. 28. For NaCl, the reference osmolality is 290
mosmol/kgH2O used in Eq. 17. A rough calculation indicates
that the value for the interaction term for NaCl does contribute to
HTJ and will be retained in the calculation for NaCl. In
contrast, for mannitol and sucrose, this term is small by virtue of small
C0 for these solutes. Thus for mannitol and sucrose, the
interaction term in Eq. 39 is dropped in the calculation. The
magnitude of this neglected term can be estimated after the TJ ultrastructure
is determined.

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|
Fig. 4. Two possible ultrastructural models for the TJ strand based on the present
predictions of the dual-pathway model. There are infrequent large slit breaks
and numerous small circular pores associated with particle pairs in the TJ
strand. The circular pore is either in the middle (A) or between 2
neighboring particle pairs (B). In the dual-pathway model, there is 1
pore every 20 nm in the TJ strands. In this figure, the particle spacing is
assumed to be 20 nm.
|
|
The water permeability and solute permeability due to the infrequent large
slit breaks in the TJ strands can be expressed by
 | (40) |
 | (41) |
Here,
1 is the effective depth of the large slit breaks,
A1 is the total area of open slits per unit surface area,
and W1 is the slit height. Equation 40, like
Eq. 5, is based on infinite slit theory. Dslit,
the solute diffusion coefficient in the large slit breaks, is given by Eq.
7.
The water permeability and solute permeability due to the small circular
pores in the TJ strands can be expressed by
 | (42) |
 | (43) |
Here,
2 is the effective depth of the small circular pores,
A2 is the total area of open pores per unit surface area,
and R2 is the pore radius. Dpore, the
solute diffusion coefficient in the circular pores, is given by Eq.
3.
The expressions for the reflection coefficients for large slit break and
small circular pore pathways differ. For both cases, the reflection
coefficient is defined in terms of the partition coefficient
(23)
 | (44) |
For large slit breaks (23)
 | (45a) |
For small circular pores (23)
 | (45b) |
Four unknowns describe the geometry of the large slit break and small
circular pore pathways, W1,
A1/
1, R2, and
A2/
2. Four constraints are needed to
determine this dual-pore/slit geometry. These four constraints are
LTJ,
TJ, and TJ NaCl and sucrose
permeabilities. We relax the constraint of TJ mannitol permeability. For
sucrose, we use the measured permeability values in Preisig and Berry
(27). TJ NaCl permeability is
determined from the transepithelial electrical resistance in Eq. 33,
as described earlier. The estimated value for the TJ NaCl permeability, 13
x 105 cm/s, in Weinstein
(35) is used. There are no
measured values for LTJ and
TJ. However,
an estimate of
TJ and LTJ can be
provided from the analysis of the compartment model, Eqs. 35 and
36, as described in the previous section. After the dimensions of
both large slit breaks and small circular pores are determined, TJ mannitol
permeability will be evaluated and compared with its measured value.
To further explore the dual-pathway model, the fraction of the total TJ
length occupied by the large slit breaks and the average spacing of small
circular pores in rat proximal tubule are examined. The fraction of the total
TJ length occupied by the large slit breaks in rat proximal tubule,
f1, can be expressed as
 | (46) |
Here, lTJ is the total TJ length in the selected segment
of the rat proximal tubule, and S the total surface area excluding
the brush border of the same segment of proximal tubule.
SA1 is the total area of the large slit breaks in the same
segment, and SA1/W1 is the total
length of large slit breaks in the same segment. To calculate f1,
one must specify
1 to find A1 after
A1/
1 is determined.
The average spacing of the small circular pores in the rat proximal tubule,
2, can be expressed as
 | (47) |
Here, SA2 is the total area of small pores in the selected
segment of the rat proximal tubule and
is the number of small
pores in the same segment. Equation 47 provides an estimate of the
average distance between pores in the TJ strand. Again, we assume the pore
depth
2 is specified after A2/
2
is determined.
 |
PARAMETER VALUES
|
---|
The parameter values used in the compartment model are summarized in
Table 2. The reference
osmolality C0 = 290 mosmol/kgH2O, T =
310.15°K, and C* = 5.94 mosmol/kgH2O. The active transport flux
N = 18.5 nmol · s1 ·
cm2 epithelium. The sodium permeability of the
cell barrier HC is very small, and the value used in
Weinstein (35), 3.1 x
1010 cm/s, is adopted. The reflection coefficient
of the basement membrane
B is zero.
Lp of proximal tubule has been measured in
several species using different techniques
(16,
27,
32). Early measurements and
methods before 1983 are summarized in Berry
(3). These and more recent
experiments reveal a significant variation in Lp
for rat proximal tubule. Berry reported values that varied from 0.20.3
cm/s (1.872.80 x 107 cm ·
1 · mmHg1).
Lp measured by Preisig and Berry
(27) is 0.120.15 cm/s
(1.121.40 x 107 cm ·
s1 · mmHg1),
depending on whether the NaCl reflection coefficient is assumed to be 1.0 or
0.7. The microperfusion measurements in Green and Giebisch
(16) provided a value for
Lp of 0.10 cm/s (0.94 x
107 cm · s1
· mmHg1).
The measured values for
vary from 0.59
(16) to 0.7
(32). In work by Van de Goot
et al. (33), the NaCl and KCl
reflection coefficients are measured and found to be close to unity for both
plasma and intracellular membrane vesicles. In our model,
C
= 1 and transepithelial
for NaCl = 0.68. This transepithelial
for NaCl is the same as the value used in Weinstein
(35).
The measured mean values for NaCl permeability of rat proximal tubule vary
between 13.3 (16) and 24.7
x 105 cm/s
(32). The value for H
in this model is the same as the value in Weinstein
(35), i.e., H = 22.0
x 105 cm/s. In this study, we assume that
the electro-diffusive NaCl flux passes nearly exclusively through the TJ and
that the barrier associated with HB offers little
resistance. Thus HTJ is estimated from Eq. 33.
The selected value, 13 x 105 cm/s, is the
same as that used in Weinstein
(35). The corresponding
transepithelial electrical resistance is 7.35
·
cm2.
The parameters for the dual-pathway model are summarized in
Table 3. The viscosity µ =
0.0007 Pa s. In this calculation, the Stokes-Einstein radii for NaCl,
mannitol, and sucrose are 1.47, 3.6, and 4.6 Å, respectively. Their
corresponding free diffusion coefficients are 2.21, 0.90, and 0.70 x
105 cm2/s. The nonelectrolyte
permeability of the TJ is at least one order of magnitude smaller than the
small-ion permeability. The measured permeability values for mannitol and
sucrose in rat proximal tubule are 0.87 and 0.43 x
105 cm/s
(27). These values are adopted
in our calculation.
The measured luminal epithelial surface excluding microvilli and the TJ
length in the S2 segment of rat proximal tubule are 96 x 103
µm2/mm tubule and 68.8 mm/mm tubule
(21). We shall see that these
data suggest a very torturous cell boundary. The effective depth
(apical-to-basal direction) of large slit breaks is 100 nm. This is typically
the spacing between the strands in the depth direction of the cleft. In
proximal tubule, there is usually a two-strand structure that is divided into
small compartments by cross-bridging segments between the longitudinal
strands. The slit break occurs when the breaks in each of the TJ strands
coincide, providing a pathway through the TJ from lumen to lateral space.
In this study, the effective small circular pore depth is 10 nm. We assume
that the space between the lateral membranes of neighboring cells will offer
little resistance compared with the small pores in the TJ. This 10-nm pore
depth assumes that there are 5-nm-long circular pores in each strand of the
two-strand structure in the TJ complex.
 |
RESULTS
|
---|
We first examine the model data used by Weinstein
(35). When
Lp is 2.4 x
107 cm · s1
· mmHg1,
M from Eq.
25 has the value 0.84 for H = 22 x
105 cm/s,
= 0.68, and C* = 5.94
mosmol/kgH2O. In this case, our model predicts that
TJ = 0.62 and LTJ = 3.02 x
107 cm · s1
· mmHg1. Both values are slightly smaller
than the values in Weinstein
(35). In our model,
LM was replaced by LMB. Because
LMB is always less than LM, a smaller
TJ is needed to balance both sides of Eq. 34. A
smaller
TJ results in a smaller LTJ (see
Eq. 36 and Tables 2
and 3).
We next consider the results for the compartment model with a reduced
Lp. When Lp = 1.59
x 107 cm ·
s1 · mmHg1,
M from Eq. 25 has the value 0.94 for H =
22 x 105cm/s,
= 0.68, and C* = 5.94
mosmol/kgH2O. The NaCl permeability of the composite barrier,
HM, is 30.5 x 105 cm/s,
and the water permeability, LMB, is 5.66 x
107 cm · s1
· mmHg1. The value of
HB from this calculation is 79.3 x
105cm/s, or six times greater than
HTJ, close to the value used previously. There is some
security to this value, in the sense that HB is the key
parameter in determining the osmotic gradient against which the proximal
tubule can transport water. Model predictions of the magnitude of this
gradient have been found to be coherent with experimental determinations
(17). After
LM is replaced with LMB,
TJ = 0.0079 and LTJ = 0.34 x
107 cm · s1
· mmHg1 (see Tables
2 and
4). Equation 25
introduces uncertainty in the model because C* is not known precisely. In
Fig. 5 we have plotted the
relationship among
M, Lp, and
C* for three values of C*. Increasing Lp results
in a decreasing
M when C* is kept constant, while increasing
C* results in a nearly uniform downward shift of
M for all
Lp. Improper combinations of
Lp and C* will result in a value of
M that exceeds unity and is physically impossible. When
Lp is 2.4 x
107 cm · s1
· mmHg1 and C* = 5.94
mosmol/kgH2O,
M = 0.84; the value used
in Weinstein (35) is
recovered.
TJ Can be estimated from Eq. 35 if one replaces
LM with LMB in Eq. 31. In
Fig. 5 we plot the relationship
among
M, Lp, and
TJ for two values of
TJ, 0.0 and 0.05.
TJ << 1 Because this is required for any pore or slit
that admits a substantial water flow. As shown in
Fig. 5, a compatible value for
Lp to satisfy both C* = 5.94
mosmol/kgH2O and 0 <
TJ < 0.05 is
1.6
x 107 cm ·
s1 · mmHg1. A
sensitivity analysis, which will be described later in this section, has been
performed to show how the dual-pore/slit geometry varies as a function of C*
and
TJ. For each value of C*, there is a family of solutions
in a narrow range of
TJ near zero that enable one to satisfy
the LTJ and
TJ predicted by the
compartment model and the TJ permeability for NaCl and sucrose. We shall also
show that the dual-pore/slit geometry is insensitive to C* for a specified
value of
TJ. When C* = 5.94 mosmol/kgH2O, one
finds that this family of solutions will also independently satisfy the
measured permeability for mannitol if
TJ = 0.0079 and
LTJ = 0.336 x 107
cm·s1·mmHg1.
This solution is defined as a best fit, and the results for this case are
summarized in Table 4.
The theoretically estimated values for
TJ and
LTJ and the TJ solute permeabilities for NaCl and sucrose
are used to predict the four unknowns describing the geometry of the
dual-pathway model. The predicted results are listed in
Table 5. The predicted gap
height of the large slit breaks is 19.6 nm.
A1/
1 for these breaks is 0.525
cm1. The predicted small-pore radius is 0.668 nm,
and A2/
2 is 15.8
cm1.
1 For the large slit
breaks is very close to zero, 2.26 x 104,
whereas
2 for the small circular pores is 0.153. The
predicted TJ mannitol permeability is 0.89 x
107cm/s. Thus this pore/slit geometry provides
excellent agreement for the measured permeability of mannitol.
The reported values for the rat proximal tubule area and the total TJ
length (21) are used to
provide the estimation of the fraction of the total TJ length occupied by the
large slit breaks and the average spacing of the small circular pores. We
first assume that the effective depth of the large slit pathway is 100 nm.
This value for
1 assumes that the gap height of the pathway
through the strands is nearly uniform, as observed in endothelial junctions
(2). However, because the
average length of the breaks observed in individual strands is typically 100
nm, coincident breaks in a dual-strand structure are rare (see
DISCUSSION). Then, if S = 96 x 103
µm2/mm tubule and lTJ = 68.8 mm/mm tubule,
f1 = 3.75 x 104. This implies
that only 0.0375% of lTJ is occupied by aligned large slit
breaks. For small circular pores, if we assume the pore depth is 10 nm, then
f2 = 20.2 nm. This implies that on average there is a small pore
every 20.2 nm.
An important prediction of the dual-pathway model is that 95.0% of
LTJ is accommodated by the infrequent large slit breaks,
whereas only 5.0% is accounted for by the far more numerous small circular
pores. In contrast to LTJ, nearly 91.2% of
HTJ for NaCl is accounted for by these numerous small
circular pores. Only 8.65% of HTJ is accounted for by the
large slit breaks. The solute-solvent coupling term in Eq. 39
accounts for the remaining 0.16%. The model predicts that only 21.7% of the
sucrose transport is through the small circular pores and 78.3% through the
large slit breaks. The contribution of the large slit breaks to the predicted
TJ permeability for mannitol is 49.2%. The model thus predicts that nearly
one-half of mannitol transport is through the large slit breaks.
Figure 5 provides the
essential link between the compartment and the dual-pore/slit models. In the
compartment model, one has the freedom to choose large values of
TJ, such as 0.65 in Weinstein
(35). These larger values are
not compatible with the dual-pathway model because most of the water passes
through the large slit breaks and
for this pathway is close to zero.
Thus even if the
for small pores is close to unity,
TJ in Eq. 38 would still be small because little
water passes through the small-pore pathway. We shall show that the largest
realizable
TJ is limited to roughly 0.03.
Four unknowns are required to define the dual pathway in the TJ strands,
W1, A1/
1,
R2, and A2/
2.
However, the measured values for TJ salt, sucrose, and mannitol permeability
and the compartment model predictions for
TJ and
LTJ provide five constraints for predicting the dimensions
of the dual-pathway geometry. Therefore, we need to relax one of the
constraints. The logical choice is to relax either mannitol or sucrose
permeability because the radii of both of these solutes are close in size and
thus do not provide strong independent constraints, as already emphasized in
the single-pathway model. Thus we chose TJ water, salt, and sucrose
permeability values but relaxed the constraint on mannitol permeability. This
choice has the advantage that it satisfies the constraints on
TJ and LTJ required by both the
compartment and pore/slit models and thus unifies the two approaches.
In Table 6 we have listed
the predicted dimensions of the dual pathway for several different
combinations of Lp and C*. In the first section
of the table, we vary C* from 4 to 8 mosmol/kgH2O while maintaining
TJ nearly constant. Although Lp
varies significantly with C*, there are only minor changes in
LTJ from 0.33 to 0.34
x107 cm ·
s1 · mmHg1.
This can be explained using Eq. 36. Because
TJ
<< 1 and
M varies from 0.92 to 0.96, Eq. 36
can be approximately rewritten using Eq. 21 as
 | (48) |
where HC is very small and has been neglected. Because
M changes little, LTJ undergoes minor
changes. Thus the dual-pathway geometry is insensitive to C* if both
LTJ and
TJ are nearly constant. We then
conclude that keeping
TJ constant and varying C* does not
significantly alter pore/slit geometry, although
Lp changes significantly.
Lp is determined primarily by the transcellular
pathway, and the changes in C* are associated with the water permeability of
the cell membranes. LTJ << LM,
and most of the water enters through the transcellular pathway.
In the second section of Table
6, we predict the dimensions of the dual pathway by keeping C* =
5.94 mosmol/kgH2O and letting
TJ increase from
0.00666 to 0.0304. Equation 48 predicts that the changes in
LTJ are very small and the changes in
Lp even smaller because LC is
maintained constant and LTJ <<
LC. When
TJ = 0.0304 and
LTJ = 0.352 x 107 cm
· s1 ·
mmHg1, the pore spacing
2 is
only 0.17 nm larger than the pore diameter, 0.63 nm. When
TJ
= 0.00666 and LTJ = 0.3350 x
107 cm · s1
· mmHg1, the small-pore spacing is 40.2 nm
and the large slit gap height is 29.5 nm. Thus the upper bound of the
physiological range for
TJ is slightly larger than 0.03;
otherwise, the small pores would form a continuous narrow slit, which is not
compatible with recent views of the claudin-occludin structure of the TJ
strand (7,
15). The realizable lower
bound is 0.006; otherwise, the large slit height will be >30 nm, a value
significantly greater than the typical 20-nm gap height observed for the large
slit breaks in endothelial TJs
(2). These results for
small-pore spacing and large slit height are plotted in
Fig. 6.
Because the physically realizable range of
TJ is from
0.0067 to 0.03, one expects that small changes in
TJ can
produce large changes in pore/slit geometry. In fact, one expects there to be
an important transition in behavior as the permeability to NaCl of the small
pore increases. Because
2 >>
1,
TJ will be dominated by the second term in Eq. 38
when the small pore is small enough for
2 to significantly
exceed zero. Although
2 will decrease as the small pore
increases in size, the first term for large slit breaks will always be <3%
of the second term. The large increase in gap height as
TJ
approaches 0.006 is due to the increase in salt permeability through the
small-pore pathway from 86.6 to 96.1% as the small-pore radius increases. From
Table 6, the salt permeability
through the large slit pathway decreases at the same time from 11.1 to 3.8%
due to the threefold decrease in slit area
(A1/
1), whereas there is only a small
change in water permeability through the same pathway. The large increase in
gap height as
TJ approaches 0.006 is needed to maintain the
nearly constant value of LTJ required by the compartment
model.
Varying
TJ along a constant C* curve and varying C* along
a constant
TJ curve in
Fig. 5 have very different
effects. The former produces large changes in small-pore radius and spacing,
modest changes in large slit height and area, minor changes in
LTJ, and negligible changes in
Lp. In contrast, varying C* while holding
TJ and LTJ nearly constant has little
effect on pore/slit geometry but a substantial effect on
Lp.
The foregoing sensitivity analysis is summarized in
Fig. 6, in which we have
plotted the predicted results for large slit breaks (A) and small
circular pores (B) and the evaluated mannitol permeability
(C) from the dual-pathway model. In
Fig. 6A, we plot the
large slit height and spacing vs.
TJ. The large slit spacing
is defined as the average length of TJ strand between two large slit breaks if
their average length was 200 nm. If f1 is the fraction of the total
length occupied by the large slit breaks, T is the average length of
a large slit break, the large slit break spacing
is given by
 | (49) |
One observes that the large slit height and spacing are nearly constant when
TJ > 0.015. When
TJ > 0.015, the
small-pore radius is less than the sucrose radius. Thus 100% of sucrose
permeability is associated with the large slit break pathway. Because the
sucrose radius is much less than the large slit height, the transport area
available for sucrose transport is nearly constant (see Eq. 6). The
water permeability through the large slit pathway changes little; thus the
large slit height and spacing are nearly constant.
In contrast, the small-pore radius and small-pore spacing continue to
decrease when
TJ increases from 0.015 to 0.03. In Eq.
27,
TJ is mainly determined by the second term
(small-pore pathway), and the contribution of the first term (large slit
breaks) can be neglected. One can increase
TJ by either
increasing
2, increasing L2, or both.
However, the small-pore pathway has little capacity to allow for a large water
permeability, as shown in Preisig and Berry
(27). We also show in
Table 6 that the fractional TJ
water permeability through the small-pore pathway never exceeds 6.6% of the
LTJ. Thus the more likely way for
TJ to
increase is to increase
2 by decreasing the
Rpore. However, this also greatly increases the steric
exclusion and the hydrodynamic resistance for salt transport (see Eq.
3). One has to increase the total pore area to maintain the measured TJ
salt permeability. Thus the small pores will decrease in size but be more
frequent, and their spacing will greatly decrease. When
TJ
> 0.03, the small pores nearly overlap and form a continuous slit in
contradiction to the observed ultrastructure of the TJ strands. From the
standpoint of steric exclusion,
TJ could approach
L2/LTJ, but as noted above, the
realistic upper limit is
0.03.
The evaluated mannitol permeability in
Fig. 6C is nearly
constant and less than the experimental measurements when
TJ
varies from 0.015 to 0.03. In this range of
TJ, the
predicted small-pore radius is close to the mannitol radius, and the steric
exclusion and the hydrodynamic resistance greatly limit mannitol permeability
through the small-pore pathway. Most of the TJ mannitol permeability is due to
the large slit breaks. Because the large slit height and spacing change little
in this range of
TJ, the mannitol permeability does not
change significantly. At the lower limit,
TJ cannot be
<0.006. In this limit, the small-pore radius increases rapidly and allows
for a large increase in sucrose and mannitol permeability that exceeds the
experimental measurements in Preisig and Berry
(27). Again, we find that the
realizable range of
TJ is from 0.006 to 0.03. If we use the
measured mannitol permeability as an independent constraint on
TJ, one finds that the measured value, 0.87 x
105 cm/s in Preisig and Berry, can be achieved
with high precision when
TJ = 0.0079. At this value of
TJ, the large slit spacing is 533 µm and the small-pore
spacing is 20.2 nm. This is the best-fit solution, whose results were
discussed earlier in Table
4.
 |
DISCUSSION
|
---|
In this paper, we have proposed a new ultrastructural model for the TJ
strands in rat proximal tubule epithelium that attempts to satisfy the
measured permeabilities for water, NaCl, and nonelectrolytes. To achieve this,
we have developed a dual-pathway model that combines infrequent large slit
breaks and numerous small circular pores in the TJ strand. The
LTJ and reflection coefficient
TJ are
used together with TJ NaCl and sucrose permeability to provide insight into
the structure and function of the TJ complex. Although dual-pathway models
have been proposed in the past, nearly all of these models have been based on
heterogeneous circular pore theory, which is not an adequate description of
the large slit breaks observed in the TJ strands. The present model is
intended to provide a more realistic description of the actual TJ strand
ultrastructure, one that includes our latest understanding of its molecular
composition.
Single-Pore/Slit Models
The single-pore/slit pathway model in Preisig and Berry
(27) was developed to satisfy
only the measured permeabilities of mannitol and sucrose. The radii of
mannitol and sucrose differed by only 1 Å, and this limited an accurate
determination of the pore radius or slit height in the TJ strands. The model
further assumed that all the mannitol and sucrose molecules traverse the TJ
via the same pore/slit ultrastructure. Our model allows that this may not be
the case. In our proposed dual-pathway model, the small pores account for
nearly one-half of the mannitol flux, but, 78.3% of the sucrose follows a
second pathway, namely, large slit breaks in the TJ strand. The
single-pore/slit theory is unable to accommodate any substantial water
permeability.
We also examined the capacity of the single-pore/slit theory to satisfy TJ
NaCl permeability and either mannitol or sucrose permeability, because these
solutes differ significantly in size. This approach leads to the prediction
that the pore radius/slit height is smaller than predicted in Preisig and
Berry, but the available transport area is three times larger
(Table 1). Thus the predicted
water permeability of the TJ is a little larger than the predicted water
permeability of the paracellular pathway in Preisig and Berry. However, this
predicted TJ water permeability is still <3% of the entire transepithelial
water permeability.
In addition, we tried to jointly satisfy the TJ water and NaCl permeability
using a single-pore/slit model while relaxing the constraints on mannitol and
sucrose permeabilities. This approach leads to significantly larger
pores/slits in the TJ strand. However, it predicted a mannitol and sucrose
permeability that was approximately five times larger than the measured
values. In summary, we confirm that a single-pore/slit model cannot
simultaneously satisfy the measured values for transepithelial water
permeability, the transepithelial NaCl reflection coefficient, and
paracellular mannitol and sucrose permeabilities. The greater flexibility of a
dual-pathway model is needed to reconcile these discrepancies.
Relationship of Dual-Pathway Model for TJ Ultrastructure to
Compartment Model of Proximal Tubule
The effort to determine the dimensions of the dual-pathway pore/slit
structure using TJ water, NaCl, and sucrose permeabilities and the TJ
reflection coefficient is limited by the fact that there are no measured
values for TJ water permeability, LTJ, and the TJ
reflection coefficient,
TJ. However, estimated values for TJ
parameters are available from a compartment model of rat proximal tubule
epithelium (35). In the
dual-pathway model, the small circular pore and the large slit break pathways
are in parallel. The water permeability of the circular pores (5.0%) is small,
and the solute reflection coefficient of these small pores is close to 0.153
for NaCl. In contrast, the reflection coefficient for the large slit breaks
will approach zero, whereas its contribution to LTJ will
be large (95.0%). Thus the composite reflection coefficient of the TJ,
TJ, will be much less than unity (0.0079). This prediction
from the dual-pathway model contradicts the estimated value for the TJ
reflection coefficient in Weinstein
(35),
TJ =
0.65. We had to find a new set of parameter values to be used in the
dual-pathway model but one that would be consistent with the compartment
model. The compartment model remains necessary to provide a relationship
between the transepithelial reflection coefficient for NaCl,
= 0.68,
and the reflection coefficient for the TJ.
Measured Lp, NaCl permeability (H),
and the transepithelial NaCl reflection coefficient,
, along with the
constraint of isotonic transport C*, are first used to predict the composite
luminal membrane NaCl permeability, HM, NaCl reflection
coefficient,
M, and water permeability,
LM. This model can also provide a constraint between
LTJ and
TJ (see Eq. 32).
TJ Is first determined with Eq. 35 by assuming
LM
LMB. LTJ
is then determined using the constraint (Eq. 32). The predicted value
for LTJ is 21.2% of the transepithelial water
permeability. This estimate of
TJ and
LTJ is then applied in the dual-pathway model of the TJ to
determine the dimensions of the pores and slits in the TJ strand.
With respect to the model prediction of the magnitude of TJ water flux, one
may note the observations of Schnermann et al.
(29), who found that mice
genetically defective for the proximal tubule cell membrane water channel
aquaporin-1 had a reduction in proximal tubule epithelial water permeability
of
80% compared with control mice. That finding has been used by some to
conclude that 20% is an upper limit on TJ water flow in proximal tubule.
Although this is compatible with the present work [but not with Weinstein
(35)], it must be acknowledged
that there are no measurements of solute reflection coefficients in any strain
of mouse, so constraints on the size and locus of the water pathways are
unknown.
Large Slit Breaks
Our combined dual-slit/pore model predicts that there will be infrequent
large slit breaks in the TJ strands. These large slit discontinuities in the
TJ strands are responsible for the large increase in TJ water permeability
above that predicted in Preisig and Berry
(27). The predicted length of
the large slit breaks is only a small fraction (
3.75 x
104) of the entire length of the TJ strands in
the rat proximal tubule, but they account for 95.0% of TJ water permeability,
78.3% of TJ sucrose permeability, and nearly one-half of TJ mannitol
permeability. However, these large slit breaks account for only 8.7% of TJ
NaCl permeability. Thus they form a secondary route for the passage of small
solutes.
The total TJ length has been reported in Maunsbach and Christensen
(21) as 68.8 mm/mm tubule.
There are
300 cells in an S2 segment of 1-mm length in rat proximal
tubule. Thus the average length of the TJ surrounding one cell is 2 x
68.8 mm/300 = 459 µm, where the factor of 2 reflects the sharing of the TJ
between neighboring cells. Because the predicted fractional length of a large
slit pathway is 3.75 x 104, then the length
of a large slit in one cell is 459 µm x 3.75 x
104 = 172 nm. If the length of a typical large
slit break in an individual TJ strand of a two-strand junctional complex is
200 nm, as observed in Orci et al.
(25), then our model predicts
that one such slit can be found on average in 200/172 or every 1.2 cells.
The above estimate of the open fractional length is based on an examination
of the TJ complex and its compartment structure, as observed in Figs. 13 and
17 in Orci et al. (25). One
notes that the TJ in rat proximal tubule is typically a two-strand structure
with polygonal compartments that are roughly 100 nm on a side with traverse
segments interspersed between the basic longitudinal strands. Occasionally,
more than one transverse compartment can separate the two longitudinal
strands. Large slit breaks that allow for water passage are created when a
break in one strand happens to be aligned with a break in the second strand.
Only when this occurs is there an open water pathway across the two-strand
complex. This is a rare event due to the interspersed compartmental structure.
If this compartmental organization were absent, water could enter at a break
at any location in the first strand, travel in the channel between strands,
and eventually leave through a distant break in the second strand. However,
with compartments that are roughly of the same length as the breaks, the
probability of finding a through pathway is the product of finding overlapping
breaks in each strand. Thus if the probability of finding a 100-nm break in
the first strand is 0.01, the likelihood of finding two overlapping breaks in
two strands in series is 104. Because the
predicted probability of finding an open pathway through a dual-strand
structure is 3.75 x 104, the likelihood of
finding a break in each strand is 1.94%.
Multiple TJ strands can be found in both epithelium and endothelium.
However, in most continuous capillaries, endothelial TJs do not form small
polygonal compartments. The TJ ultrastructure in capillary endothelium has
been best quantified in frog mesentery capillary, where there are, on average,
1.4 strands/cross section, but only one nearly continuous strand
(2). The average length of the
large slit breaks in frog mesentery capillary is 150 nm. This is on the same
order as the 200-nm TJ discontinuities observed in proximal tubule
(25). The slit height in frog
mesentery capillary, 20 nm, is very close to the predicted gap height, 19.6
nm, in the present model. The frequency of the large slit breaks in frog
mesentery, one open slit of 150-nm length in 4,320 nm
(19), or a probability
150/4,320 = 0.035, is about twofold greater than the result predicted herein,
0.0194 for finding a 172-nm break in either strand of a two-strand
complex.
The important insight that the TJ strands of rat proximal tubule might have
discontinuities was deduced from the paper of Adamson and Michel
(2), in which it was
demonstrated by both serial sectioning and the tracer wakes of lanthinum that
penetrated the TJ of frog mesentery that discontinuities of significant length
could exist in the particle strands comprising the TJ. This conclusion cannot
be definitively deduced from the particle patterns observed in freeze fracture
because one does not have double replicas in which particle gaps in the E-face
can be matched in a mirror image with a gap in the P-face. Because the water
permeability coefficient of frog mesentery and rat proximal tubule are on the
same order, this provides a clue that such breaks might also be present in the
proximal tubule, although the comparable ultrastructural studies have not yet
been performed. Finally, the large slit breaks should be viewed as dynamic
rather than static structures, because the TJ strands may break and reform, in
response to either regulatory signals or pharmacological agents. Adamson et
al. (1) have observed that both
the average number of TJ strands and the water permeability can be modulated
by cAMP in frog mesentery capillaries, but the situation in epithelia is less
certain.
Small Circular Pores
In addition to the infrequent large slit breaks in the TJ strand, our model
also predicts that there are numerous small circular pores in the TJ strands.
These numerous small circular pores are the primary pathways for small
solutes. Our model predicts that 91.2% NaCl flux across the TJ is accommodated
via this pathway. One-half of the mannitol transport can traverse via this
pathway, whereas nearly one-fifth of the sucrose flux goes by this route. Our
model also suggests that there is one circular pore every 20.2 nm in the TJ
strands. It must be acknowledged that the calculations of this paper utilize
an equivalent nonelectrolyte reflection coefficient for NaCl, despite the fact
that the fluxes are ionic in nature and could be influenced by pore charge.
More specifically, one may ask whether the pore size determined using this
equivalent nonelectrolyte reflection coefficient is meaningful, given the
possible charge effects. The larger issue of relating overall salt
coefficients to the component ionic coefficients has been addressed
(20). However, the reliability
of the equivalent pore radius obtained from the neutral salt has never been
investigated. Despite this uncertainty, the calculations in this paper suggest
that the equivalent small pore is actually smaller than that estimated by
Preisig and Berry (27), and
thus it remains a poor candidate for the water pathway. The structural
correlates for the small pores are sketched in
Fig. 4. On average, there is
roughly one circular pore associated with each particle pair in the TJ strand,
assuming that junction particles are spaced every 20 nm along a TJ strand, an
average value in Figs. 13 and 17 in Orci et al.
(25). The pore could be formed
by particle pairs in apposing membranes, as shown in
Fig. 4A, or by the
interstices of adjacent particles, as shown in
Fig. 4B.
Recently, a dual-pathway model for the TJ has been demonstrated for
intestinal cell monolayers in vitro
(34). Polyethylene glycols
(PEGs) of increasing radius are used as paracellular probes to detect the
paracellular pathway in Caco-2 and T84 cell lines by measuring their
permeability. A restrictive pore (radius 0.430.45 nm) and a
nonrestrictive pore responsible for permeability of large molecules are found
in both cell lines. A mathematical model was developed to analyze the
permeability of different size PEGs. In that model, however, pore size was
determined by considering only steric exclusion, and the hydrodynamic
resistance due to the pore walls was neglected.
The TJ strands may be viewed as chains of particles, with typical spacing
of these particles, as seen in freeze-fracture electron micrographs, being on
the order of 20 nm (25). These
particles are thought to be the integral proteins of the claudin family and
occludin (7,
11,
13). Occludin is believed to
be a functional component of the TJ
(22) and a possible
determinant of TJ permeability in endothelial cells
(18). Claudin-1 and claudin-2
were the first members of the claudin family to be identified and could
reconstitute TJ strands (11,
14). When claudin-2 is
introduced into the Madin-Darby canine kidney I (MDCK-I) cells, a conversion
from a very "tight" junction to a leaky junction is observed
(12). Claudin-1 and claudin-4
are abundant in MDCK-I cells, which have very tight junctions, whereas
claudin-2 expression is found in MDCK-II cells, which have a much leakier TJ
than do MDCK-I cells, although the number of strands in these two cell types
is similar (12). This suggests
that claudin-2 could be responsible for the leakiness of the MDCK-II cells and
the formation of small pores between apposing TJ strands. Indeed, when the
claudins expressed in MDCK cells are selectively modified, the paracellular
conductance of small electrolytes can be modulated, including the anion/cation
selectivity preference (4).
Claudin-2 exists throughout the proximal tubule and in a contiguous early
segment of the thin descending limb of long-looped nephrons in mouse kidney
(6). Thus claudin-2 may be a
key component of the paracellular pores in the TJ of the mouse proximal tubule
and the integral protein responsible for its leaky permeability properties for
small ions. Homotypic interactions between claudin-2 in apposing TJ strands or
heterotypic interactions between claudin-2 and claudin-1 are possible
candidates for the small circular pores in the present model
(15).
Relationship to Prior Theory
The picture of proximal tubule water flow provided by this analysis is
different in several important ways from the view derived from the compartment
model of Weinstein (35). These
differences are featured in Tables
2 and
4. In this model, the TJ water
permeability is only 6.3% or (0.336/5.314) that of the cellular pathway,
although LTJ is 21.2% of Lp,
and
TJ is near zero. This means that even with a hypertonic
lateral interspace, there will be little transjunctional flux of water. In
Weinstein (35), the water
permeabilities of cell and junction were nearly equal, and small increases in
interspace salt concentration could drive large transjunctional water flows.
Thus in the present model, the composite luminal
M is
substantially higher than that used previously. This occurs despite the fact
that the overall salt reflection coefficients,
= 0.68, are the same
for both models. Here, the constraint on the overall reflection coefficient is
accommodated by virtue of the smaller basement membrane solute permeability,
and thus more solute polarization within the lateral interspace. The departure
from previous parameters is mandated by the assumed pore structure, and the
obligation that a large nondiscriminatory water pore has a high solute
permeability. In the present model, we have fashioned what seems to be the
largest TJ water flow possible, and this still yields a high composite luminal
reflection coefficient.
One difficulty with the present dual-pore/slit formulation, however, is
that there is no apparent way to accommodate the finding of substantial
differences among the ionic reflection coefficients. The careful experiments
of Fromter et al. (10)
provided values of 0.7, 0.5, and 1.0 for the overall reflection coefficients
of Na+, Cl, and
. These differences in reflection
coefficients were predicted to yield a force for proximal tubule water
reabsorption when luminal
concentrations are less than and Cl concentrations are
greater than their concentration in peritubular fluid. These predictions were
confirmed experimentally (28).
Although this is a nonelectrolyte pore/slit model, and the observations relate
to ions, it is difficult to see how charge effects could turn the large pore
into a discriminatory pathway for which small ions will have a non-zero
reflection coefficient. Indeed, one prediction from this model is that
claudins impact only the small-pore properties (ionic conductance). To our
knowledge, there have been no measurements of water permeability or reflection
coefficients in cultured epithelia in which claudins have been modified. Our
model suggests that modification of claudins should have little effect on the
water transport pathway. In sum, the very small
TJ is a
major difference with the earlier work
(35) but was necessary to
bring this model into compatibility with pore theory. Weinstein's choice of
reflection coefficient did attempt to satisfy compatibility with measured
ionic reflection coefficients. Unfortunately, the means to reconcile these two
constraints are not apparent. Despite this limitation, the added flexibility
of the present model provides an approach that fits conceptually into recent
views of the molecular structure of the TJ strands and the junction particle
patterns observed in freeze-fracture electron micrograph studies of the TJ
complex.
 |
DISCLOSURES
|
---|
This work was supported by National Institute of Diabetes and Digestive and
Kidney Diseases Grant 1-R01-DK-29857 (to A. M. Weinstein).
 |
ACKNOWLEDGMENTS
|
---|
P. Guo thanks Dr. Bingmei Fu for financial support.
This research was performed in partial fulfillment of the requirements for
the PhD degree from the City University of New York (P. Guo).
 |
FOOTNOTES
|
---|
Address for reprint requests and other correspondence: A. M. Weinstein, Dept.
of Physiology and Biophysics, Weill Medical College of Cornell Univ., 1300
York Ave., New York, NY 10021 (E-mail:
alan{at}nephron.med.cornell.edu).
The costs of publication of this article were defrayed in part by the
payment of page charges. The article must therefore be hereby marked
``advertisement'' in accordance with 18 U.S.C. Section 1734 solely to
indicate this fact.
1 Two sets of units, cm/s and cm · s1
· mmHg1, are used in this paper to
describe water permeability. The relationship between them is
Here, Pf is the water permeability (in cm/s) and
Lp is the water permeability (in cm ·
s1 · mmHg1).
VW is the molar volume of water, VW = 18
cm3/mol, R = 8.3145 J ·
mol1 · K1,
and T = 310.15 K. For water permeability,
Lp of 1 x 107
cm · s1 ·
mmHg1, the corresponding Pf =
0.107 cm/s. 
 |
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