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A dual-pathway ultrastructural model for the tight junction of rat proximal tubule epithelium

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

	ABSTRACT

TOP ABSTRACT SINGLE-PORE/SLIT MODEL DUAL-PORE/SLIT MODEL PARAMETER VALUES RESULTS DISCUSSION REFERENCES

 
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

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

TOP ABSTRACT SINGLE-PORE/SLIT MODEL DUAL-PORE/SLIT MODEL PARAMETER VALUES RESULTS DISCUSSION REFERENCES

 
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/. For a slit, the corresponding parameters are the slit height, W, and the total slit area per unit surface area/slit depth, A_slit/. 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, 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/ 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/ 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 x 10^–5 cm/s and H_TJ(sucrose) = 0.43 x 10^–5 cm/s. The estimated TJ permeability for NaCl is H_TJ(NaCl) = 13 x 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. 1A). 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; x10^–5 cm²/s) are 2.21, 0.90, and 0.70.

View larger version (29K): [in this window] [in a new window]   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.

The solutions for R_pore obtained from the intersections of the curves in Fig. 1A are summarized in Table 1. A_pore/ 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/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.

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, 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, 1–2/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, 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. 1B are summarized in Table 1. A_slit/ 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 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 L_p, , 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, , 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).

View larger version (17K): [in this window] [in a new window]   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.

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 , 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, _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 L_TJ/L_p << 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 A_pore/

(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/ 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. 3A). This compatible solution for L_p = 0.15 cm/s, H_TJ(NaCl) = 13 x 10^–5 cm/s, = 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/, 6.64 cm^–1. Because the predicted A_pore/ 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 x 10^–5 cm/s and H_TJ(sucrose) = 3.16 x 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, _TJ, is close to zero. From Eq. 11, L_TJ is nearly 30% of L_p.

View larger version (26K): [in this window] [in a new window]   Fig. 3. A: plot of Eqs. 15a and 15b as a function of pore radius. The compatible pore radius is 5.17 nm. In calculation, transepithelial permeabilty (Lp) = 0.15 cm/s, reflection coefficient () = 0.70, and HTJ = 13 x 10–5 cm/s. B: plot of Eqs. 16a and 16b as a function of half-slit height. The compatible half-slit height is 3.2 nm. In calculation, Lp = 0.15 cm/s, = 0.70, and HTJ = 13 x 10–5 cm/s.

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 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, 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. 3B 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/, 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 _TJ is close to zero, is valid. Once W and A_slit/ are determined, the corresponding permeabilities of sucrose and mannitol can be determined using Eq. 6. They are H_TJ(mannitol) = 4.6 x 10^–5 cm/s and H_TJ(sucrose) = 3.4 x 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 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 . 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

TOP ABSTRACT SINGLE-PORE/SLIT MODEL DUAL-PORE/SLIT MODEL PARAMETER VALUES RESULTS DISCUSSION REFERENCES

 
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 _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 () 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 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. H_M, _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 _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 _M can be evaluated and L_p, , 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 _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 L_p and _M because H_M, H_B, and L_MB are all functions of _M and L_MB is related to L_p through Eq. 23. Thus _M can be determined if we know the transepithelial values for H, L_p, and along with an estimate of C*. After _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, _M, and H_M are determined, one next evaluates their TJ components, L_TJ and _TJ. These predicted values of L_TJ 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 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 _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 L_TJ and _TJ. From Eqs. 26 and 27, the fractional water permeability of the cell barrier, L_C/L_M, is related to _TJ by

(29)

The fractional water permeability of the TJ is

(30)

From Eq. 30, L_TJ/L_M 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 L_TJ. This assumes that all three permeabilities on the left-hand-side of Eq. 32 are known, _C = 1, and _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 _M (Eq. 21). H_C is very small (35).

H_TJ 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 10^–5 cm/s (the corresponding transepithelial resistance varies from 5–7 · cm²). In this model, we have selected a value for H_TJ that is at the lower limit for H, 13 x 10^–5 cm/s.

There are two unknowns, _TJ and L_TJ, in Eq. 32. A simple way to solve for _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, _TJ can be expressed explicitly as

(35)

Once _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 _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 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 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.

View larger version (38K): [in this window] [in a new window]   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. 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, 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. 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 (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 L_TJ, _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 10^–5 cm/s, in Weinstein (35) is used. There are no measured values for L_TJ and _TJ. However, an estimate of _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, f₁, 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. 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

TOP ABSTRACT SINGLE-PORE/SLIT MODEL DUAL-PORE/SLIT MODEL PARAMETER VALUES RESULTS DISCUSSION REFERENCES

 
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 · 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 x 10^–10 cm/s, is adopted. The reflection coefficient of the basement membrane _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 x 10^–7 cm · –1 · mmHg^–1). L_p measured by Preisig and Berry (27) is 0.12–0.15 cm/s (1.12–1.40 x 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 x 10^–7 cm · s^–1 · mmHg^–1).

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 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 x 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 x 10^–5 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 10^–5 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 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 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

TOP ABSTRACT SINGLE-PORE/SLIT MODEL DUAL-PORE/SLIT MODEL PARAMETER VALUES RESULTS DISCUSSION REFERENCES

 
We first examine the model data used by Weinstein (35). When L_p is 2.4 x 10^–7 cm · s^–1 · mmHg^–1, _M from Eq. 25 has the value 0.84 for H = 22 x 10^–5 cm/s, = 0.68, and C* = 5.94 mosmol/kgH₂O. In this case, our model predicts that _TJ = 0.62 and L_TJ = 3.02 x 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 _TJ is needed to balance both sides of Eq. 34. A smaller _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 x 10^–7 cm · s^–1 · mmHg^–1, _M from Eq. 25 has the value 0.94 for H = 22 x 10^–5cm/s, = 0.68, and C* = 5.94 mosmol/kgH₂O. The NaCl permeability of the composite barrier, H_M, is 30.5 x 10^–5 cm/s, and the water permeability, L_MB, is 5.66 x 10^–7 cm · s^–1 · mmHg^–1. The value of H_B from this calculation is 79.3 x 10^–5cm/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, _TJ = 0.0079 and L_TJ = 0.34 x 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 _M, L_p, and C* for three values of C*. Increasing L_p results in a decreasing _M when C* is kept constant, while increasing C* results in a nearly uniform downward shift of _M for all L_p. Improper combinations of L_p and C* will result in a value of _M that exceeds unity and is physically impossible. When L_p is 2.4 x 10^–7 cm · s^–1 · mmHg^–1 and C* = 5.94 mosmol/kgH₂O, _M = 0.84; the value used in Weinstein (35) is recovered.

View larger version (24K): [in this window] [in a new window]   Fig. 5. Relationships between Lp and membrane reflection coefficient (M) for 3 values of decrement in luminal osmolality required to yield a reabsorbate osmolality equal to that of the lumen (C*) values, i.e., 4, 5.94, and 8 mosmol/kgH2O, and 2 values for the TJ reflection coefficient (TJ) values, 0.0 and 0.05. In this figure, the curves for 2 TJ values intersect at a point where Lp = 1.8 x 10–7 cm · s–1 · mmHg–1 and M = 1.0.

_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 _M, L_p, 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 L_p to satisfy both C* = 5.94 mosmol/kgH₂O and 0 < _TJ < 0.05 is 1.6 x 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 _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 L_TJ 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 L_TJ = 0.336 x 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 _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. A₁/₁ for these breaks is 0.525 cm^–1. The predicted small-pore radius is 0.668 nm, and A₂/₂ is 15.8 cm^–1. ₁ For the large slit breaks is very close to zero, 2.26 x 10^–4, whereas ₂ for the small circular pores is 0.153. The predicted TJ mannitol permeability is 0.89 x 10^–7cm/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 l_TJ = 68.8 mm/mm tubule, f₁ = 3.75 x 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 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 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 _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 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 _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/kgH₂O while maintaining _TJ nearly constant. Although L_p varies significantly with C*, there are only minor changes in L_TJ from 0.33 to 0.34 x10^–7 cm · s^–1 · mmHg^–1. 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 H_C is very small and has been neglected. Because _M changes little, L_TJ undergoes minor changes. Thus the dual-pathway geometry is insensitive to C* if both L_TJ and _TJ are nearly constant. We then conclude that keeping _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/kgH₂O and letting _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 _TJ = 0.0304 and L_TJ = 0.352 x 10^–7 cm · s^–1 · mmHg^–1, the pore spacing ₂ is only 0.17 nm larger than the pore diameter, 0.63 nm. When _TJ = 0.00666 and L_TJ = 0.3350 x 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 _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.