## Abstract

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: F848–F862, 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: F124–F131, 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.

## SINGLE-PORE/SLIT MODEL

### 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, *R*_{pore}, and the total pore area per unit surface area/pore depth, *A*_{pore}/δ. For a slit, the corresponding parameters are the slit height, *W*, and the total slit area per unit surface area/slit depth, A_{slit}/δ. 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, *L*_{TJ}(pore) and *H*_{TJ}(pore), respectively, are given by (1) (2) Here, δ is the pore depth, *R*_{pore} is the pore radius, *A*_{pore} is the total pore area per unit surface area, and μ is the viscosity of water, whose assumed value is 0.0007 Pa s. *D*_{pore} is the diffusion coefficient for a solute in a circular pore. An empirical expression, the Renkin equation (27), is used to relate *D*_{pore} to the free diffusion coefficient, *D*_{free}, and *a*, the solute radius (3) There are two multiplicative factors in *Eq. 3*. The first factor, (1–*a*/*R*_{pore})^{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 *R*_{pore} and *A*_{pore}/δ for a single-pore pathway. The lefthand side of *Eq. 4* is a function of *H*_{TJ}, solute radius *a*, and *R*_{pore}. If two solutes share the same transport pathway, then *R*_{pore} and *A*_{pore}/δ 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 *R*_{pore}, will yield a compatible solution for *R*_{pore}, provided the two curves for *H*_{TJ}/*D*_{pore} 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 *H*_{TJ}(mannitol) = 0.87 × 10^{–}^{5} cm/s and *H*_{TJ}(sucrose) = 0.43 × 10^{–}^{5} cm/s. The estimated TJ permeability for NaCl is *H*_{TJ}(NaCl) = 13 × 10^{–}^{5} 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 *R*_{pore} (Fig. 1*A*). The intersection of any two curves provides a compatible *R*_{pore} 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 (*D*_{free}; ×10^{–}^{5} cm^{2}/s) are 2.21, 0.90, and 0.70.

The solutions for *R*_{pore} obtained from the intersections of the curves in Fig. 1*A* are summarized in Table 1. *A*_{pore}/δ can then be found using *Eq. 4* and *L*_{TJ} calculated using *Eq. 1*. These results are also given in Table 1. Our results for *R*_{pore} 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, 0.12–0.15 cm/s^{1} (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.

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, *L*_{TJ}(slit) and *H*_{TJ}(slit), respectively, are given by (5) (6) Here, δ is the depth of the slit, *A*_{slit} is the total slit area per unit surface area of epithelium, and *W* is the height of the slit. *D*_{slit}, the solute diffusion coefficient for an infinite slit, is given by the Renkin equation (27) (7) The first factor in *Eq. 7*, 1–2α/*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. 1*B* 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, *H*_{TJ}, are the same as used previously for the pore calculations. The solutions for *W*/2 obtained from the intersections of the curves in Fig. 1*B* are summarized in Table 1. *A*_{slit}/δ can then be found using *Eq. 8* and *L*_{TJ} calculated using *Eq. 5*. The results summarized in Table 1 are similar to those for a circular pore. *L*_{TJ}(slit) for the mannitol/sucrose pair is ∼1.5% of the transepithelial water permeability, *L*_{p}, as previously predicted in Preisig and Berry (27). Although *L*_{TJ}(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 *L*_{p}. Thus neither a pore model nor a slit model predicts values for *L*_{TJ} that are a significant fraction of *L*_{p}.

### Salt Reflection Coefficient

Instead of using solute permeability pairs to determine pore or slit dimensions, one can use *L*_{p}, TJ salt permeability, *H*_{TJ}(NaCl), and the transepithelial reflection coefficient σ for NaCl, σ(NaCl), for the entire epithelium to determine the dimensions of the paracellular pathway. Experiments show that the rat proximal tubule epithelium has a σ(NaCl) that is close to 0.7 (32). Accordingly, we shall attempt to satisfy the measured values of *L*_{p}, σ, 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 *L*_{TJ}. However, a compartment model has been used to relate *L*_{TJ} to *L*_{p}, 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 *L*_{p}, σ, 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 *L*_{p} 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, *H*_{TJ}, 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 2–3% 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).

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, *L*_{p} and σ, respectively, are given by (36) (9) (10) Here, *L*_{C} is the water permeability of the cell barrier, *L*_{TJ} is the water permeability of the TJ, σ_{C} is the NaCl reflection coefficient of the cell barrier, and σ_{TJ} is the NaCl reflection coefficient of the TJ barrier. Reasonable values for σ and σ_{C} for NaCl are σ = 0.7 and σ_{C} = 1.0, as stated above. From *Eq. 10*, we can see that if *L*_{TJ}/*L*_{p} << 1, σ 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, ϕ 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 *A*_{pore}/δ (15a) (15b) After we substitute *Eq. 14* into *Eq. 15a*, the only unknown variable on the right-hand side of *Eq. 15a* is *R*_{pore}. Similarly, the only unknown on the right-hand side of *Eq. 15b* is *R*_{pore} if we know the solute permeability *H*_{TJ} and the solute radius *a* (*Eq. 3*). If water and solute share the same transport pathway, *A*_{pore}/δ must be the same for that pathway. Thus, if we plot the right-hand sides of *Eqs. 15a* and *15b* vs. *R*_{pore}, the intersection of two curves provides the compatible *R*_{pore} (Fig. 3*A*). This compatible solution for *L*_{p} = 0.15 cm/s, *H*_{TJ}(NaCl) = 13 × 10^{–}^{5} cm/s, σ = 0.7, and *a* = 0.147 nm, is *R*_{pore} = 5.2 nm. Once *R*_{pore} is determined, we can use either *Eq. 15a* or *Eq. 15b* to obtain *A*_{pore}/δ, 6.64 cm^{–}^{1}. Because the predicted *A*_{pore}/δ now is nearly one-half the predicted values for NaCl/mannitol and NaCl/sucrose pair in Table 1 and the predicted *R*_{pore} 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 *H*_{TJ}(mannitol) = 4.42 × 10^{–}^{5} cm/s and *H*_{TJ}(sucrose) = 3.16 × 10^{–}^{5} 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 *R*_{pore} is much greater than the sodium radius. Thus, from *Eqs. 12* and *13*, the TJ reflection coefficient for NaCl, σ_{TJ}, is close to zero. From *Eq. 11, L*_{TJ} is nearly 30% of *L*_{p}.

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 *L*_{p}, *H*_{TJ}(NaCl), and σ as for the circular pore. To simplify our calculation, we assume σ_{TJ} is zero because we anticipate that the slit height *W* >> 2*a* and σ_{TJ} ∼ 0. Thus from *Eq. 11, L*_{TJ} ∼ 0.3 *L*_{p}.

From *Eqs. 5* and *6* (16a) (16b) The right-hand sides of *Eqs. 16a* and *16b* are plotted vs. *W*/2 in Fig. 3*B* for the same values of *L*_{p} and *H*_{TJ} as for the circular pore. One finds that the compatible slit half height, *W*/2, is 3.2 nm and *A*_{slit}/δ, from *Eq. 16a* or *Eq. 16b*, is 6.5 cm^{–}^{1}. 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 σ_{TJ} is close to zero, is valid. Once *W* and *A*_{slit}/δ are determined, the corresponding permeabilities of sucrose and mannitol can be determined using *Eq. 6*. They are *H*_{TJ}(mannitol) = 4.6 × 10^{–}^{5} cm/s and *H*_{TJ}(sucrose) = 3.4 × 10^{–}^{5} 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 *L*_{p}, 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 *L*_{p}. Thus *L*_{TJ} contributes insignificantly to *L*_{p}. The calculations in this section, which are based on *L*_{p} and σ 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 *L*_{p} and σ. 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.

## DUAL-PORE/SLIT MODEL

### 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 *L*_{TJ} and σ_{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), *L*_{p}, the transepithelial NaCl permeability (*H*) and the NaCl reflection coefficient (σ) for the entire epithelium are given by (17) (18) (19) where *L*_{MB} is defined as (20) Here, *R* is the gas constant, *T* is absolute temperature, and C_{0} is a reference osmolality. Following Weinstein (35), we replace the mean membrane osmolality with the reference osmolality C_{0} (290 mosmol/kgH_{2}O) to avoid nonlinearities and keep accuracy. *H*_{M}, σ_{M}, and *L*_{M} are the NaCl permeability, the NaCl reflection coefficient, and the water permeability of the composite barrier formed by the cells and TJ complex. *H*_{B} and *L*_{B} 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, H*_{M} can be expressed in terms of σ_{M} (21) Using *Eq. 18, H*_{B} can be written as (22) *Equation 17* can be written so that *L*_{MB} appears explicitly. (23) If *Eq. 20* is rewritten as (24) *L*_{M} can be determined if *L*_{B} is prescribed and *L*_{MB} is evaluated using *Eq. 23*. All the parameters appearing in *Eqs. 17–19* for the composite barrier, except *L*_{M}, can be determined if σ_{M} can be evaluated and *L*_{p}, σ, and *H* are measured. However, it is argued in Weinstein (35) that *L*_{B} >> *L*_{M} and, thus *L*_{M} ∼ *L*_{MB}. Thus we need to obtain only one additional independent relationship for σ_{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, π_{M} is the mucosal (luminal) oncotic pressure, and π_{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 C_{0}. 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 *L*_{p} and σ_{M} because *H*_{M}, *H*_{B}, and *L*_{MB} are all functions of σ_{M} and *L*_{MB} is related to *L*_{p} through *Eq. 23*. Thus σ_{M} can be determined if we know the transepithelial values for *H, L*_{p}, and σ along with an estimate of C*. After σ_{M} is determined, *H*_{M}, *H*_{B}, and *L*_{MB} can be evaluated using *Eqs. 21, 22*, and *23* as described previously.

Once *L*_{M}, σ_{M}, and *H*_{M} are determined, one next evaluates their TJ components, *L*_{TJ} and σ_{TJ}. These predicted values of *L*_{TJ} and σ_{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 *L*_{C} and *L*_{TJ} denote the water permeabilities of the cell and the TJ complex, *H*_{C} and *H*_{TJ} be their NaCl permeabilities, and σ_{C} and σ_{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 *L*_{TJ} and σ_{TJ}. From *Eqs. 26* and *27*, the fractional water permeability of the cell barrier, *L*_{C}/*L*_{M}, is related to σ_{TJ} by (29) The fractional water permeability of the TJ is (30) From *Eq. 30, L*_{TJ}/*L*_{M} cannot be less than σ_{C}–σ_{M}. *Equation 28* can be rewritten using *Eqs. 26, 29*, and *30* as (31) (32) *Equation 32* provides the required constraint between σ_{TJ} and *L*_{TJ}. This assumes that all three permeabilities on the left-hand-side of *Eq. 32* are known, σ_{C} = 1, and σ_{M} has been related to *L*_{p} using *Eq. 25. H*_{M} has been already determined by the compartment model in terms of *H* and σ_{M} (*Eq. 21*). *H*_{C} is very small (35).

*H*_{TJ} is independently estimated from the expression for transepithelial electrical resistance (33) Here, Ω is transepithelial electrical resistance, *z* is the valence for NaCl (*z* = 1), *F* is Faraday's constant, and C̄ is the mean ion concentration (the same reference osmolality C_{0} 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, Ω is approximated by the resistance of the TJ. The NaCl permeability *H* varies from 13.7 to 19.1 × 10^{–}^{5} cm/s (the corresponding transepithelial resistance varies from 5–7 Ω · cm^{2}). In this model, we have selected a value for *H*_{TJ} that is at the lower limit for *H*, 13 × 10^{–}^{5} cm/s.

There are two unknowns, σ_{TJ} and *L*_{TJ}, in *Eq. 32*. A simple way to solve for σ_{TJ} and *L*_{TJ} is to replace *L*_{M} by *L*_{MB} in *Eq. 31*, because *L*_{B} >> *L*_{M} in *Eq. 24. Equation 31* can then be approximated by (34) From *Eq. 34*, σ_{TJ} can be expressed explicitly as (35) Once σ_{TJ} is determined, *L*_{TJ} 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, *L*_{TJ}, *H*_{TJ}, and σ_{TJ} are (37) (38) (39) Here, C_{0} 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/kgH_{2}O used in *Eq. 17*. A rough calculation indicates that the value for the interaction term for NaCl does contribute to *H*_{TJ} and will be retained in the calculation for NaCl. In contrast, for mannitol and sucrose, this term is small by virtue of small C_{0} 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.

The water permeability and solute permeability due to the infrequent large slit breaks in the TJ strands can be expressed by (40) (41) Here, δ_{1} is the effective depth of the large slit breaks, *A*_{1} is the total area of open slits per unit surface area, and *W*_{1} is the slit height. *Equation 40*, like *Eq. 5*, is based on infinite slit theory. *D*_{slit}, 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, δ_{2} is the effective depth of the small circular pores, *A*_{2} is the total area of open pores per unit surface area, and *R*_{2} is the pore radius. *D*_{pore}, 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 ϕ (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, *W*_{1}, *A*_{1}/δ_{1}, *R*_{2}, and *A*_{2}/δ_{2}. Four constraints are needed to determine this dual-pore/slit geometry. These four constraints are *L*_{TJ}, σ_{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 × 10^{–}^{5} cm/s, in Weinstein (35) is used. There are no measured values for *L*_{TJ} and σ_{TJ}. However, an estimate of σ_{TJ} and *L*_{TJ} 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, f_{1}, can be expressed as (46) Here, *l*_{TJ} 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. *SA*_{1} is the total area of the large slit breaks in the same segment, and *SA*_{1}/*W*_{1} is the total length of large slit breaks in the same segment. To calculate f_{1}, one must specify δ_{1} to find *A*_{1} after *A*_{1}/δ_{1} is determined.

The average spacing of the small circular pores in the rat proximal tubule, λ_{2}, can be expressed as (47) Here, *SA*_{2} 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 δ_{2} is specified after A_{2}/δ_{2} is determined.

## PARAMETER VALUES

The parameter values used in the compartment model are summarized in Table 2. The reference osmolality C_{0} = 290 mosmol/kgH_{2}O, *T* = 310.15°K, and C* = 5.94 mosmol/kgH_{2}O. The active transport flux *N* = 18.5 nmol · s^{–}^{1} · cm^{–}^{2} epithelium. The sodium permeability of the cell barrier *H*_{C} is very small, and the value used in Weinstein (35), 3.1 × 10^{–}^{10} cm/s, is adopted. The reflection coefficient of the basement membrane σ_{B} is zero. *L*_{p} 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 *L*_{p} for rat proximal tubule. Berry reported values that varied from 0.2–0.3 cm/s (1.87–2.80 × 10^{–}^{7} cm · σ–1 · mmHg^{–}^{1}). *L*_{p} measured by Preisig and Berry (27) is 0.12–0.15 cm/s (1.12–1.40 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}), 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 *L*_{p} of 0.10 cm/s (0.94 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}).

The measured values for σ 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, σ_{C} = 1 and transepithelial σ for NaCl = 0.68. This transepithelial σ 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 × 10^{–}^{5} cm/s (32). The value for *H* in this model is the same as the value in Weinstein (35), i.e., *H* = 22.0 × 10^{–}^{5} 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 *H*_{B} offers little resistance. Thus *H*_{TJ} is estimated from *Eq. 33*. The selected value, 13 × 10^{–}^{5} cm/s, is the same as that used in Weinstein (35). The corresponding transepithelial electrical resistance is 7.35 Ω · cm^{2}.

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 × 10^{–}^{5} cm^{2}/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 × 10^{–}^{5} 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 × 10^{3} μm^{2}/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 *L*_{p} is 2.4 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}, σ_{M} from *Eq. 25* has the value 0.84 for *H* = 22 × 10^{–}^{5} cm/s, σ = 0.68, and C* = 5.94 mosmol/kgH_{2}O. In this case, our model predicts that σ_{TJ} = 0.62 and *L*_{TJ} = 3.02 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}. Both values are slightly smaller than the values in Weinstein (35). In our model, *L*_{M} was replaced by *L*_{MB}. Because *L*_{MB} is always less than *L*_{M}, a smaller σ_{TJ} is needed to balance both sides of *Eq. 34*. A smaller σ_{TJ} results in a smaller *L*_{TJ} (see *Eq.* 36 and Tables 2 and 3).

We next consider the results for the compartment model with a reduced *L*_{p}. When *L*_{p} = 1.59 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}, σ_{M} from *Eq. 25* has the value 0.94 for *H* = 22 × 10^{–}^{5}cm/s, σ = 0.68, and C* = 5.94 mosmol/kgH_{2}O. The NaCl permeability of the composite barrier, *H*_{M}, is 30.5 × 10^{–}^{5} cm/s, and the water permeability, *L*_{MB}, is 5.66 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}. The value of *H*_{B} from this calculation is 79.3 × 10^{–}^{5}cm/s, or six times greater than *H*_{TJ}, close to the value used previously. There is some security to this value, in the sense that *H*_{B} 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 *L*_{M} is replaced with *L*_{MB}, σ_{TJ} = 0.0079 and L_{TJ} = 0.34 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1} (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 σ_{M}, *L*_{p}, and C* for three values of C*. Increasing *L*_{p} results in a decreasing σ_{M} when C* is kept constant, while increasing C* results in a nearly uniform downward shift of σ_{M} for all *L*_{p}. Improper combinations of *L*_{p} and C* will result in a value of σ_{M} that exceeds unity and is physically impossible. When *L*_{p} is 2.4 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1} and C* = 5.94 mosmol/kgH_{2}O, σ_{M} = 0.84; the value used in Weinstein (35) is recovered.

σ_{TJ} Can be estimated from *Eq. 35* if one replaces *L*_{M} with *L*_{MB} in *Eq. 31*. In Fig. 5 we plot the relationship among σ_{M}, *L*_{p}, and σ_{TJ} for two values of σ_{TJ}, 0.0 and 0.05. σ_{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 *L*_{p} to satisfy both C* = 5.94 mosmol/kgH_{2}O and 0 < σ_{TJ} < 0.05 is ∼1.6 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}. 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 σ_{TJ}. For each value of C*, there is a family of solutions in a narrow range of σ_{TJ} near zero that enable one to satisfy the *L*_{TJ} and σ_{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 σ_{TJ}. When C* = 5.94 mosmol/kgH_{2}O, one finds that this family of solutions will also independently satisfy the measured permeability for mannitol if σ_{TJ} = 0.0079 and *L*_{TJ} = 0.336 × 10^{–}^{7} cm·s^{–}^{1}·mmHg^{–}^{1}. This solution is defined as a best fit, and the results for this case are summarized in Table 4.

The theoretically estimated values for σ_{TJ} and *L*_{TJ} 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. *A*_{1}/δ_{1} for these breaks is 0.525 cm^{–}^{1}. The predicted small-pore radius is 0.668 nm, and *A*_{2}/δ_{2} is 15.8 cm^{–}^{1}. σ_{1} For the large slit breaks is very close to zero, 2.26 × 10^{–}^{4}, whereas σ_{2} for the small circular pores is 0.153. The predicted TJ mannitol permeability is 0.89 × 10^{–}^{7}cm/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 δ_{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 × 10^{3} μm^{2}/mm tubule and *l*_{TJ} = 68.8 mm/mm tubule, f_{1} = 3.75 × 10^{–}^{4}. This implies that only 0.0375% of *l*_{TJ} is occupied by aligned large slit breaks. For small circular pores, if we assume the pore depth is 10 nm, then f_{2} = 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 *L*_{TJ} 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 *L*_{TJ}, nearly 91.2% of *H*_{TJ} for NaCl is accounted for by these numerous small circular pores. Only 8.65% of *H*_{TJ} 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 σ_{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 σ for this pathway is close to zero. Thus even if the σ for small pores is close to unity, σ_{TJ} in *Eq. 38* would still be small because little water passes through the small-pore pathway. We shall show that the largest realizable σ_{TJ} is limited to roughly 0.03.

Four unknowns are required to define the dual pathway in the TJ strands, *W*_{1}, *A*_{1}/δ_{1}, *R*_{2}, and *A*_{2}/δ_{2}. However, the measured values for TJ salt, sucrose, and mannitol permeability and the compartment model predictions for σ_{TJ} and *L*_{TJ} 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 σ_{TJ} and *L*_{TJ} 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 *L*_{p} and C*. In the first section of the table, we vary C* from 4 to 8 mosmol/kgH_{2}O while maintaining σ_{TJ} nearly constant. Although *L*_{p} varies significantly with C*, there are only minor changes in *L*_{TJ} from 0.33 to 0.34 ×10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}. This can be explained using *Eq. 36*. Because σ_{TJ} << 1 and σ_{M} varies from 0.92 to 0.96, *Eq. 36* can be approximately rewritten using *Eq. 21* as (48) where *H*_{C} is very small and has been neglected. Because σ_{M} changes little, *L*_{TJ} undergoes minor changes. Thus the dual-pathway geometry is insensitive to C* if both *L*_{TJ} and σ_{TJ} are nearly constant. We then conclude that keeping σ_{TJ} constant and varying C* does not significantly alter pore/slit geometry, although *L*_{p} changes significantly. *L*_{p} is determined primarily by the transcellular pathway, and the changes in C* are associated with the water permeability of the cell membranes. *L*_{TJ} << *L*_{M}, 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/kgH_{2}O and letting σ_{TJ} increase from 0.00666 to 0.0304. *Equation 48* predicts that the changes in *L*_{TJ} are very small and the changes in *L*_{p} even smaller because *L*_{C} is maintained constant and *L*_{TJ} << *L*_{C}. When σ_{TJ} = 0.0304 and *L*_{TJ} = 0.352 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}, the pore spacing λ_{2} is only 0.17 nm larger than the pore diameter, 0.63 nm. When σ_{TJ} = 0.00666 and *L*_{TJ} = 0.3350 × 10^{–}^{7} cm · s^{–}^{1} · mmHg^{–}^{1}, 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 σ_{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 σ_{TJ} is from 0.0067 to 0.03, one expects that small changes in σ_{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 σ_{2} >> σ_{1}, σ_{TJ} will be dominated by the second term in *Eq. 38* when the small pore is small enough for σ_{2} to significantly exceed zero. Although σ_{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 σ_{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 (*A*_{1}/δ_{1}), whereas there is only a small change in water permeability through the same pathway. The large increase in gap height as σ_{TJ} approaches 0.006 is needed to maintain the nearly constant value of *L*_{TJ} required by the compartment model.

Varying σ_{TJ} along a constant C* curve and varying C* along a constant σ_{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 *L*_{TJ}, and negligible changes in *L*_{p}. In contrast, varying C* while holding σ_{TJ} and *L*_{TJ} nearly constant has little effect on pore/slit geometry but a substantial effect on *L*_{p}.

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. 6*A*, we plot the large slit height and spacing vs. σ_{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 f_{1} 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 Δ is given by (49) One observes that the large slit height and spacing are nearly constant when σ_{TJ} > 0.015. When σ_{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 σ_{TJ} increases from 0.015 to 0.03. In *Eq. 27*, σ_{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 σ_{TJ} by either increasing σ_{2}, increasing *L*_{2}, 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 *L*_{TJ}. Thus the more likely way for σ_{TJ} to increase is to increase σ_{2} by decreasing the *R*_{pore}. 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 σ_{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, σ_{TJ} could approach *L*_{2}/*L*_{TJ}, but as noted above, the realistic upper limit is ∼0.03.

The evaluated mannitol permeability in Fig. 6*C* is nearly constant and less than the experimental measurements when σ_{TJ} varies from 0.015 to 0.03. In this range of σ_{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 σ_{TJ}, the mannitol permeability does not change significantly. At the lower limit, σ_{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 σ_{TJ} is from 0.006 to 0.03. If we use the measured mannitol permeability as an independent constraint on σ_{TJ}, one finds that the measured value, 0.87 × 10^{–}^{5} cm/s in Preisig and Berry, can be achieved with high precision when σ_{TJ} = 0.0079. At this value of σ_{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 *L*_{TJ} and reflection coefficient σ_{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, *L*_{TJ,} and the TJ reflection coefficient, σ_{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 *L*_{TJ} will be large (95.0%). Thus the composite reflection coefficient of the TJ, σ_{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), σ_{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, σ = 0.68, and the reflection coefficient for the TJ.

Measured *L*_{p}, NaCl permeability (*H*), and the transepithelial NaCl reflection coefficient, σ, along with the constraint of isotonic transport C*, are first used to predict the composite luminal membrane NaCl permeability, *H*_{M}, NaCl reflection coefficient, σ_{M}, and water permeability, *L*_{M}. This model can also provide a constraint between *L*_{TJ} and σ_{TJ} (see *Eq. 32*). σ_{TJ} Is first determined with *Eq. 35* by assuming *L*_{M} ≈ *L*_{MB}. *L*_{TJ} is then determined using the constraint (*Eq. 32*). The predicted value for *L*_{TJ} is 21.2% of the transepithelial water permeability. This estimate of σ_{TJ} and *L*_{TJ} 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 × 10^{–}^{4}) 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 × 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 × 10^{–}^{4}, then the length of a large slit in one cell is 459 μm × 3.75 × 10^{–}^{4} = 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 10^{–}^{4}. Because the predicted probability of finding an open pathway through a dual-strand structure is 3.75 × 10^{–}^{4}, 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. 4*A*, or by the interstices of adjacent particles, as shown in Fig. 4*B*.

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 0.43–0.45 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 *L*_{TJ} is 21.2% of *L*_{p}, and σ_{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 σ_{M} is substantially higher than that used previously. This occurs despite the fact that the overall salt reflection coefficients, σ = 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 σ_{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

↵1 Two sets of units, cm/s and cm · s

^{–}^{1}· mmHg^{–}^{1}, are used in this paper to describe water permeability. The relationship between them is Here,*P*_{f}is the water permeability (in cm/s) and*L*is the water permeability (in cm · s_{p}^{–}^{1}· mmHg^{–}^{1}). V_{W}is the molar volume of water, V_{W}= 18 cm^{3}/mol,*R*= 8.3145 J · mol^{–}^{1}· K^{–}^{1}, and*T*= 310.15 K. For water permeability,*L*_{p}of 1 × 10^{–}^{7}cm · s^{–}^{1}· mmHg^{–}^{1}, the corresponding*P*_{f}= 0.107 cm/s.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.

- Copyright © 2003 the American Physiological Society