A dual-pathway ultrastructural model for the tight junction of rat proximal tubu - 《American Journal of Physiology》美国生理学杂志干细胞之家



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发表于 2009-4-21 13:42 |只看该作者 |倒序浏览 |打印

作者：Peng Guo, Alan M. Weinstein, and Sheldon Weinbaum作者单位：1 CUNY Graduate School and New York Center forBiomedical Engineering, Department of Mechanical Engineering, The City Collegeof the City University of New York, New York 10031; and 2 Department of Physiology and Biophysics, WeillMedical College of Cornell University, New York, New York 10021
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      : x! Q- ?( u" [' M5 x
      4 k+ R' {, a/ V5 t( J0 ?9 ]
      . T& R3 w/ Q% n0 A$ D' O: t1 y
      【摘要】
   A dual-pathway model is proposed for transport across the tight junction(TJ) in rat proximal tubule: large slit breaks formed by infrequentdiscontinuities in the TJ complex and numerous small circular pores, withspacing similar to that of claudin-2. This dual-pathway model is developed inthe context of a proximal tubule model (Weinstein AM. Am J Physiol RenalFluid Electrolyte Physiol 247: F848-F862, 1984) to provide anultrastructural view of solute and water fluxes. Tubule model paramters (TJ reflection coefficient and water permeability), plus the measured epithelialNaCl and sucrose permeabilities, provide constraints for the dual-pathwaymodel, which yields the small-pore radius and spacing and large slit heightand area. For a small-pore spacing of 20.2 nm, comparable to the distancebetween 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, occupying0.04% of the total TJ length. This pore/slit geometry also satisfies themeasured permeability for mannitol. The numerous small circular pores accountfor 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 waterpermeability 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 thesmall 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 is21.2% of the entire transepithelial water permeability and thus an order ofmagnitude greater than that predicted by the single-pore/slit theory (PreisigPA and Berry CA. Am J Physiol Renal Fluid Electrolyte Physiol 249: F124-F131, 1985).
      【关键词】 paracellular pathway water transport compartment model reflection coefficient' E# J* b8 N* R! ?6 m( j0 h! |9 J3 D6 e
   ----------

WATER AND SOLUTES CAN TRAVERSE the proximal tubule epithelium ofmammalian kidney via both transcellular and paracellular routes. The tightjunction (TJ) complex forms the major barrier in the paracellular route, andits ability to seal the paracellular route is variable. In freeze-fractureelectron micrographs, the TJ appears to be a set of long, parallel, and linearfibrils 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 ofclaudins interspersed with occludin from one cell may copolymerize to form astrand in a side-by-side manner( 15 ). Strands from neighboringcells form a pair in a head-to-head homotypic or heterotypic interaction ( 15, 31 ). Freeze-fracture electronmicroscopic observations show that the TJ of rat proximal tubule consiststypically of a two-strand complex that is shallow ( 100 nm) in theapical-basal direction and that these strands exhibit discontinuities that canexceed 0.1 µm in length( 25 ).

Although there are two basic transport routes, transcellular andparacellular, the relative importance of each route for water has never beensatisfactorily resolved. The paracellular route, in particular, has offered asubstantial challenge because the structural correlate for the differentlysized pores or their frequency and cross-sectional geometry are still unknown. Preisig and Berry ( 27 )concluded that paracellular water permeability 2% oftransepithelial water permeability. They measured the permeabilities ofmannitol and sucrose, which are believed to traverse the epithelium only viathe paracellular pathway, and then used the single-pore/slit theory (Renkin equation) to predict the dimensions of the pores/slits, which satisfied thepermeabilities for both solutes. TJ water permeability was then predictedusing these pore/slit dimensions. Weinstein ( 35 ) argued that paracellularwater permeability should be comparable to that of the transcellular pathwayto 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 thatof the measured transepithelial water permeability. The additional hydraulic resistance is associated with the lateral interspace and solute polarizationby the basement membrane.

The pore/slit theoretical approach was questioned by Fraser and Baines( 8 ) because they noted that thepore/slit theory underestimated the water permeability of man-made gelmembranes compared with the fiber matrix model developed by Curry and Michel( 5 ). They( 8 ) introduced a fiber matrixmodel based on the theory of Curry and Michel( 5 ) to estimate TJ water andsolute permeability. In their model, the TJ is modeled as a homogeneous fibermatrix gel with polymers of several-nanometer radius that fill the spacebetween TJ strands. The model provides a consistent picture for rabbitproximal tubule, but when applied to the rat proximal tubule it predictedsmall ion permeabilities that were an order of magnitude smaller than thosemeasured. This model treated the TJ complex as a uniform structure without discontinuities. Therefore, it did not allow for the possibility oflow-resistance, large-pore/slit pathways. In addition, the model was appliedto a matrix that filled the space between the strands and not to the strandsthemselves. In the present study, it is the strands themselves that accountfor most of the paracellular resistance for solute transport.

In this paper, we propose an ultrastructural model for TJ strands thatconsists of infrequent large "slit breaks" and numerous smallcircular pores. We also ask that this model be consistent with the parameterselection in the compartmental model in Weinstein( 35 ). In the next section, wereconsider single-pore/slit analysis, as it applies to NaCl permeability, aswell as to the passage of mannitol and sucrose. We then introduce adual-pathway model, and its additional parameters (pore/slit dimensions and frequency) are used to represent TJ attributes, which had been previouslyestimated ( 35, 36 ). It will be argued thatthe pore/slit attributes are morphologically realistic.# d: |8 t  I1 y: C1 S+ L

SINGLE-PORE/SLIT MODEL

Solute Permeabilities: R; D: {& u) W3 A7 i

We first examine the single-pore/slit theory for compatibility withtransepithelial water permeability, TJ solute permeabilities, and the NaClreflection coefficient for the entire epithelium. This means estimating thedimensions of the pores or slits in the TJ strands that are required tosatisfy 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/poredepth, A pore /. For a slit, the correspondingparameters are the slit height, W, and the total slit area per unitsurface area/slit depth, A slit /. We modify the approach inPreisig and Berry ( 27 ), who used the TJ permeabilities of sucrose and mannitol to determine the dimensionsof the paracellular pathway. In their approach, they apply the Renkin equationto two solutes, mannitol and sucrose, whose radii are close in size, 3.6 and4.6 Å, respectively. Alternatively, it should provide betterdiscrimination 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 theradius 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.) X8 _; p2 m" c9 T* d
$ ^, O- T' m5 g# f! n, ?5 _* X3 n
In single circular pore theory, the water and solute permeabilities of theTJ strands, L TJ (pore) and H TJ (pore),respectively, are given by) C- U6 j) {) R+ h- |
! i& w$ y/ R  N
Here, is the pore depth, R pore is the pore radius, A pore is the total pore area per unit surface area, andµ is the viscosity of water, whose assumed value is 0.0007 Pa s. D pore is the diffusion coefficient for a solute in acircular 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

There are two multiplicative factors in Eq. 3. The first factor,(1- a / R pore ) 2, is the partitioncoefficient, representing the steric exclusion from the pore. The secondfactor describes the hydrodynamic interaction of the solute with the porewalls. From Eq. 28 {7 I' g6 e, e) i7 X

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 sideof Eq. 4 is a function of H TJ, solute radius a, and R pore. If two solutes share the sametransport pathway, then R pore and A pore / will be the same for that pathway. Thus theright-hand side of Eq. 4 will have the same value for these twosolutes, and the left-hand side of Eq. 4, when plotted as a functionof R pore, will yield a compatible solution for R pore, provided the two curves for H TJ / D pore intersect.2 h# y4 n0 n+ K# o6 R! c7 U
4 Y2 @( U- U* A: C9 L% ?& K1 T
Preisig and Berry ( 27 )measured the permeabilities of sucrose and mannitol, which are believed totraverse the epithelium only via the paracellular route. These measuredpermeabilities are H TJ (mannitol) = 0.87 x 10 - 5 cm/s and H TJ (sucrose) =0.43 x 10 - 5 cm/s. The estimated TJpermeability for NaCl is H TJ (NaCl) = 13 x 10 - 5 cm/s( 35 ). Thus we can plot threecurves for the left-hand term of Eq. 4 for NaCl, mannitol, andsucrose 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, and4.6 Å, respectively. Their corresponding free diffusion coefficients( D free; x 10 - 5 cm 2 /s) are 2.21, 0.90, and 0.70.8 @1 W% E  Z7 Q6 S

Fig. 1. A : plot of Eq. 4 for A pore / (or H TJ / D pore ) for NaCl, mannitol, andsucrose as a function of pore radius, where A pore is totalpore area per unit surface area, is the pore depth, H TJ is tight junctional (TJ) permeability, and D pore is diffusion coefficient for a solute in a circularpore. The compatible solutions for the mannitol/sucrose pair are 1.41 nm, theNaCl/mannitol pair, 0.80 nm, and the NaCl/sucrose pair, 0.95 nm. B :plot of Eq. 8 for A slit / (or H TJ / D slit ) for NaCl, mannitol, andsucrose as a function of half-slit height, where is the depth of theslit, A slit is the total slit area per unit surface areaof epithelium, and D slit is the solute diffusioncoefficient for an infinite slit. The compatible solution for themannitol/sucrose pair is 0.77 nm, the NaCl/mannitol pair, 0.46 nm, and theNaCl/sucrose pair, 0.55 nm.

The solutions for R pore obtained from the intersectionsof the curves in Fig.1 A are summarized in Table 1. A pore / can then be found using Eq. 4 and L TJ calculated using Eq. 1. These results arealso given in Table 1. Ourresults for R pore and water permeability, which satisfymannitol and sucrose permeabilities, are the same as the results givenpreviously by Preisig and Berry( 27 ). This water permeabilityis

Table 1. Compatible pore radius or half-slit height for solute pairs and theircorresponding water permeability based on a single-pore model or a single-slitmodel
. |( b+ b2 J6 m: [8 W% {
Similarly, a single-slit model can be used to estimate the slit height andthe total area of open slit per unit surface area/slit depth. Again, we assumeall solutes share the same transport pathway. In the slit model, the water andsolute permeability, L TJ (slit) and H TJ (slit), respectively, are given by

Here, is the depth of the slit, A slit is the totalslit area per unit surface area of epithelium, and W is the height ofthe slit. D slit, the solute diffusion coefficient for aninfinite slit, is given by the Renkin equation( 27 )* k/ B9 k, Y/ w" n3 u1 Q' B; r
* _+ j- A5 U( G' E" B2 Z' R8 B
The first factor in Eq. 7, 1-2 / W, describes thesteric exclusion and the second the increased hydrodynamic resistance of the slit walls. From Eq. 6* {; o/ I  S+ J( r

Following the same argument as in circular pore theory, we have plotted in Fig. 1 B the left-handside of Eq. 8 vs. the slit half height, W /2, for threesolutes, i.e., NaCl, mannitol, and sucrose. The intersections of the curvesprovide the solutions to Eq. 8 for each solute pair. In thesecalculations, the solute permeabilities, H TJ, are the sameas used previously for the pore calculations. The solutions for W /2obtained from the intersections of the curves in Fig. 1 B are summarizedin Table 1. A slit / can then be found using Eq. 8 and L TJ calculated using Eq. 5. The results summarized in Table 1 aresimilar to those for a circular pore. L TJ (slit) for themannitol/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/mannitolpair, 0.0013 cm/s, it is still
- Q3 s% U+ `; ]/ ~5 O
Salt Reflection Coefficient
: P0 I! p( |) }, G' E( d
Instead of using solute permeability pairs to determine pore or slitdimensions, 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 proximaltubule epithelium has a (NaCl) that is close to 0.7( 32 ). Accordingly, we shallattempt to satisfy the measured values of L p,, and TJ NaCl permeability but relax the constraints on the nearlyimpermeant solutes, sucrose and mannitol. For a single-pore/slit model for theTJ, 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 muchlarger and less frequent than the single-pore/slit model just considered forpaired solutes, but one finds the permeabilities for sucrose and mannitol arefar too large, as we show next.+ P! k; q$ j' O, X4 ^) z

There is no directly measured value for L TJ. However, acompartment model has been used to relate L TJ to L p, the measured transepithelial waterpermeability ( 35 ). Incompartment models, the properties of the entire epithelium are determined bythe properties of its components: the cell barrier, the TJ barrier, and thebasement membrane barrier ( Fig.2 ). Conversely, the overall epithelial permeabilities will serveas constraints for determining the component parameters, and these have beendisplayed in Table 2. Thevalues for L p,, and H used inthe model of Weinstein ( 35 )were all taken from those compiled by Ullrich( 32 ). Preisig and Berry( 27 ) subsequently determinedan overall L p about one-half that found byUllrich ( 32 ), and a lowervalue is used in the present model. The reflection coefficient for the cellmembrane is 1.0 ( 26, 33 ) and that for the basementmembrane is 0.0 ( 38 ). The rateof active osmolar transport across the basolateral membrane, N, wastaken to be approximately twice the rate of net epithelial sodium transport( 32 ). For the diffusive saltpermeability of TJ, H TJ, the value selected (if applied toboth Na and Cl) yields a realistic estimate for TJ electrical resistance( 9 ). Isotonicity of proximaltubule volume transport is embodied in the parameter C*, which is thedecrement in luminal osmolality required to yield a reabsorbate osmolality equal to that of the lumen. Experimental determinations of luminal osmolalityindicate 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 varywith peritubular protein concentration and luminal anion composition, butmodel calculations indicate that C* depends largely on the overall rate ofsodium reabsorption relative to cell membrane water permeability( 35 ).
4 ^* e$ x, ?: j: U
Fig. 2. Compartment model for rat proximal tubule epithelium. The cell and the TJare in parallel and form a composite barrier. The cell barrier has the abilityto actively transport sodium. This composite barrier is in series with thebasement membrane. In our model, the reflection coefficient of the basementmembrane for NaCl is zero, and the water and solute permeability of thebasement membrane are much larger than that of the composite luminal barrier. N, active transport flux across the basolateral cell membrane due tosodium-potassium pump. J VTJ and J STJ,tight junctional volume flux and solute flux, respectively; J VC and J SC, transcellular volume fluxand solute flux, respectively; J VB and J SB, basement membrane volume flux and solute flux,respectively.

Table 2. Parameter values used in the compartment model, \. ?. Z5 X3 }

For this initial calculation, consider the cell barrier and the TJ barrierin parallel and omit for simplicity the resistance of the highly permeablebasement membrane barrier. For this simplified composite pathway model, thetransepithelial water permeability and the reflection coefficient for NaCl, L p and, respectively, are given by( 36 )0 ^0 F8 ?# B# m' F' ]3 A9 W8 v8 ?; J
3 [# H# J* J4 p: O8 f; c
(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, wecan see that if L TJ / L p
! g9 O) ^6 n0 f# k
Combining Eqs. 9 and 10, one has

(11)5 k9 ]4 b! d. v

According to pore theory, the reflection coefficient can be written as( 23 )$ Y6 r& j9 w4 t' V3 ~# s/ g5 o$ J
1 U* l0 o! |# D( q( J/ f: Y
(12)# `* T" x' A, ^
$ y- h0 ^, k3 d3 Z2 W
Here, is the partition coefficient, which for a circular pore is givenby ( 23 )

(13)
; U# k, n2 T9 J( [$ c$ q, j" K
Combining Eqs. 11, 12, and 13, we find that4 L+ N. }4 v; |$ _5 C/ [

(14), V% Z/ J* f, o& M/ k7 I
3 H2 L6 r- A( ~" r' X5 |( p
From Eqs. 1 and 2, we have two independent relationships for A pore /
4 B- o! M; Y4 M" h5 o
(15a)$ U9 L% c+ Q* x
! ]" A- g- K3 u. s
(15b)
; ?7 N3 ~" }) N+ l
After we substitute Eq. 14 into Eq. 15a, the only unknownvariable on the right-hand side of Eq. 15a is R pore. Similarly, the only unknown on the right-hand sideof 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, ifwe plot the right-hand sides of Eqs. 15a and 15b vs. R pore, the intersection of two curves provides thecompatible R pore ( Fig.3 A ). 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 /, 6.64cm - 1. Because the predicted A pore / now is nearly one-half the predicted valuesfor NaCl/mannitol and NaCl/sucrose pair in Table 1 and the predicted R pore here is at least five times greater than the valuespredicted in Table 1, there aremany fewer pores in the TJ strands when we try to satisfy the measurements forNaCl and water permeability. The permeability of any solute can now becalculated using Eq. 2. The corresponding permeabilities of sucroseand 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 are5.0 (mannitol) to 7.4 (sucrose) times greater than the experimental values inPreisig and Berry ( 27 ). Thepredicted R pore is much greater than the sodium radius.Thus, from Eqs. 12 and 13, the TJ reflection coefficient forNaCl, TJ, is close to zero. From Eq. 11,L TJ is nearly 30% of L p.

Fig. 3. A : plot of Eqs. 15a and 15b as a function ofpore radius. The compatible pore radius is 5.17 nm. In calculation,transepithelial permeabilty ( L p ) = 0.15 cm/s,reflection coefficient ( ) = 0.70, and H TJ = 13 x 10 - 5 cm/s. B : plot of Eqs.16a and 16b as a function of half-slit height. The compatiblehalf-slit height is 3.2 nm. In calculation, L p =0.15 cm/s, = 0.70, and H TJ = 13 x 10 - 5 cm/s.
: e: K, t' {2 H2 t' {4 J
A similar analysis can be performed for the single-slit model, and the slitdimensions for the TJ can be determined using the same values for L p, H TJ (NaCl), and asfor the circular pore. To simplify our calculation, we assume TJ is zero because we anticipate that the slit height W 2 a and TJ 0. Thus from Eq. 11, L TJ 0.3 L p.8 y9 |# A" v/ L, m' d
. P/ t& a2 o& Y6 ^
From Eqs. 5 and 6( w" B( }. [  H1 ]* t

(16a)6 S6 U* l/ ?+ _3 z/ W

(16b)( O- w& v' ~# o( ~7 ]5 d/ Y
: d5 s0 H+ W/ b: I1 a
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 thecompatible 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 atleast five times greater than the values in Table 1. The predicted slithalf height W /2 = 3.2 nm is much larger than the sodium radius. Thusour assumption, that TJ is close to zero, is valid. Once W and A slit / are determined, thecorresponding 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 areagain 5.3 (mannitol) to 7.8 (sucrose) times larger than the experimentallymeasured values in Preisig and Berry( 27 ).. P4 o* _7 O* D9 p: Q

These model calculations indicate that a single-pore/slit model cannotsatisfy the well-documented experimental measurements for L p, TJ solute permeability, and the overallreflection coefficient for small ions for rat proximal tubule. Thecalculations above in Solute Permeabilities suggest that thedimensions of the single pore/slit based on TJ solute permeability alone arerather small. This small pore/slit will offer a great resistance for watertransport and account for 4 y% j9 T7 c" a& _! i" u
" Q5 b$ p8 F7 a0 `. M% s
DUAL-PORE/SLIT MODEL: }0 s3 F$ v: I* q: G' s) _

TJ Barrier in a Compartment Model of Rat Proximal TubuleEpithelium$ N! ~* Z: B' T3 I2 V+ }( q/ s

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 strandscontains two parallel transport pathways: infrequent large slit breaks formedby junction strand discontinuities and numerous small circular pores in theclaudin-occludin TJ complexes. The large slit allows for a significant passage of water. Most importantly, these junctional strand breaks, which allow forflow through a double-strand complex, are very few in number. This transportpathway will also allow small ions to pass, but it is not the dominant routefor ions because of the very low probability that an open pathway will beformed 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 pathwayallows for a solute flux for molecules % s8 ?- U* ?# R) s( _
/ ~# ?4 q  p/ S/ I6 C9 q0 k7 ~
Experimental data from rat proximal tubule are for the transepithelial permeabilities of water and salt and for the transepithelial NaCl reflectioncoefficient. 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 inparallel with the junctional pathway. Compartment models for rat proximaltubule epithelium were introduced to explore the potential significance of apermeable TJ ( 37 ). Thecompartment model was later extended to include the compliance of the lateralintercellular space ( 35 ) andthe impact of TJ convection in the epithelial transport equations( 36 ). In this study, we shallapply the 1984 compartment model to provide an estimate of the properties ofthe TJ barrier ( 35 ).
6 v! Z, X5 F6 i  |4 [+ q! a# R
In the compartment model of Weinstein( 35 ), the cells and the TJ arein parallel and form a composite barrier, which are both in series with alateral interspace basement membrane ( Fig.2 ). In this model, the cell itself is a barrier, not acompartment. In the Weinstein model( 35 ), L p, the transepithelial NaCl permeability( H ) and the NaCl reflection coefficient ( ) for the entireepithelium are given by

(17)

(18)
5 Q9 K! U. C$ [& f, E
(19)
2 q5 ^1 n) T6 \" C
where L MB is defined as1 a' ~' I' [, d& q& A1 D5 L

(20)5 d" B% S9 W1 p% Y: S
/ M( K5 ^  @( p5 _
Here, R is the gas constant, T is absolute temperature, andC 0 is a reference osmolality. Following Weinstein( 35 ), we replace the mean membrane osmolality with the reference osmolality C 0 (290mosmol/kgH 2 O) to avoid nonlinearities and keep accuracy. H M, M, and L M are the NaCl permeability, the NaCl reflection coefficient, and the waterpermeability 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 thereflection coefficient of the basement membrane is zero. In our model, weassume the basement membrane has a higher permeability to water and solutesthan the composite barrier formed by the cells and the TJ complex.8 J  z: u; T# s1 A% j

From Eqs. 18 and 19, H M can be expressed interms of M
6 s6 J7 H0 }6 r; x" e: ]
(21)

Using Eq. 18, H B can be written as

(22)) S- W: }4 e2 l' J9 U, d( g5 J

Equation 17 can be written so that L MB appears explicitly.
* z" Z3 l  g$ H  a, W" G& @& B8 k
(23)
6 r& C$ y/ u' I3 Z7 C2 T
If Eq. 20 is rewritten as, ^# x, {1 o; D

(24)4 P% j% K& X0 D, ~
% c2 p; k. |5 H5 H
L M can be determined if L B is prescribed and L MB is evaluated using Eq. 23. Allthe parameters appearing in Eqs. 17-19 for the compositebarrier, 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 obtainonly one additional independent relationship for M.

Water reabsorption in the proximal tubule is driven by active transport andthe osmotic pressure differences that are established by this activetransport. Weinstein ( 35 )defines a measure of transport isotonicity which is given by; v3 f% A) l/ k6 Y
) O+ K. R: V  W: `! ?' w1 p: F# u6 i5 c
(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 referenceosmolality C 0. We will focus on the first term and thus requirethat transport be isotonic even in the absence of peritubular protein. Thevalue 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 isrelated 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 ofC*. After M is determined, H M, H B, and L MB can be evaluated using Eqs. 21, 22, and 23 as described previously.  S  _( }, |7 @; R
1 H9 E' k" ]' P
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 thedetailed TJ structure. The properties of the composite barrier consisting ofthe cell barrier and the TJ barrier can be expressed in terms of theirindividual 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 NaClpermeabilities, and C and TJ be their NaClreflection coefficients. Then# c8 D1 A" Q6 m2 H( [

(26)+ |& j, y. k, S9 h+ F& R! U' q
; x8 ?; p6 z' ~- D/ `$ r
(27)" a) n5 _) z0 x+ y# T

(28)7 F4 M2 S' s  h% r/ C
( m( @2 C, h" d) f- |- ]7 V8 v
The last term on the right-hand-side in Eq. 28 describes the solute-solvent interaction for a heteroporous parallel pathway with differentreflection coefficients( 36 ).: m2 @0 r. Q: Z) h( o/ ?1 G

Equations 26, 27, and 28 can be manipulated to provide aconstraint between L TJ and TJ. From Eqs. 26 and 27, the fractional water permeability of thecell barrier, L C / L M, is related to TJ by
! T& n' r! \" d% n) ^- p0 Z
(29)6 E6 B9 E# U5 [+ L0 F( ^# J4 ~! Y4 I
/ D% w- q4 W; g, ?- N! E
The fractional water permeability of the TJ is
. ^" i; ]& J0 H7 {- g" d
(30)% c/ r0 S/ T+ N! W# C
' `! K( W6 `3 ^% P1 P( H4 l
From Eq. 30, L TJ / L M cannot be lessthan C - M. Equation 28 canbe rewritten using Eqs. 26, 29, and 30 as
7 F( b$ @2 ^( B4 n- S% o
(31)
( m( X. x, Z  T5 k* x, |5 g
(32)

Equation 32 provides the required constraint between TJ and L TJ. This assumes that all threepermeabilities 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 beenalready determined by the compartment model in terms of H and M ( Eq. 21 ). H C is very small( 35 ).
% k6 K* @  |+ U
H TJ is independently estimated from the expression fortransepithelial electrical resistance0 |9 N! D9 f& C' Q2 t" g( @
. k  H7 B2 P3 c+ Z
(33)2 F& R# n& ^, J3 W) g9 x. S1 k3 }

Here, is transepithelial electrical resistance, z is thevalence for NaCl ( z = 1), F is Faraday's constant, and is the mean ion concentration (the samereference osmolality C 0 as in Eq. 17 is used). Because thebasement membrane and the composite barrier are in series in the compartmentmodel and the conductance of the basement membrane is much larger than that ofthe 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 · cm 2 ). In thismodel, we have selected a value for H TJ that is at thelower 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
' n7 ~/ G/ \' n0 C( e6 y7 }$ J
(34): k% H; g0 b  a4 `( K

From Eq. 34, TJ can be expressed explicitly as

(35)- c6 u( U* ~. e3 S  M! g  X
' K- ~' q! D& h$ p" u+ w
Once TJ is determined, L TJ can becalculated from Eq. 320 C5 Q: ]. s- {7 G2 }% I/ n( W

(36)
9 v' s9 g# Y) Z7 f& h" G
Heteroporous Model for TJ Strands
; W2 \+ Y! p% u$ k& [4 g
As discussed above, we propose that TJ strands contain numerous smallcircular pores and infrequent large slit breaks, the former associated withjunctional particle pairs and the latter associated with junctional stranddiscontinuities, as sketched in Fig.4. The model predictions for the sizes of the pores and the slitsstrongly suggest this structure. A heteroporous model that includessolute-solvent interaction must be used because the reflection coefficientsand the water permeabilities differ greatly for each pathway. Let 1 and 2denote the two pathways, 1 for large slit breaks and 2 for small circular pores. Based on the theory in Weinstein( 36 ), the composite values forthe TJ, L TJ, H TJ, and TJ are9 E1 H. A, C$ c" z( {: r

(37)! C6 o6 I. \0 ~& x5 _. r

(38). h5 ]8 p0 x: x- P1 Z: \' r

(39)4 e: g" U* n: i. L1 H
8 y2 M* t+ k$ L& h; O7 ~
Here, C 0 is a reference osmolality for each solute. Equation39 is applied separately for NaCl, mannitol, and sucrose. The last termin Eq. 39 again represents the solute-solvent interaction as in Eq. 28. For NaCl, the reference osmolality is 290mosmol/kgH 2 O used in Eq. 17. A rough calculation indicatesthat the value for the interaction term for NaCl does contribute to H TJ and will be retained in the calculation for NaCl. Incontrast, for mannitol and sucrose, this term is small by virtue of small C 0 for these solutes. Thus for mannitol and sucrose, theinteraction term in Eq. 39 is dropped in the calculation. Themagnitude of this neglected term can be estimated after the TJ ultrastructure is determined.- X$ O5 C( K8 q; h, Y0 c" C

Fig. 4. Two possible ultrastructural models for the TJ strand based on the presentpredictions of the dual-pathway model. There are infrequent large slit breaksand numerous small circular pores associated with particle pairs in the TJstrand. The circular pore is either in the middle ( A ) or between 2neighboring particle pairs ( B ). In the dual-pathway model, there is 1pore every 20 nm in the TJ strands. In this figure, the particle spacing isassumed to be 20 nm.) R: T$ g* `! ^" v1 [- e

The water permeability and solute permeability due to the infrequent largeslit breaks in the TJ strands can be expressed by

(40)8 r- @) ^1 {, h7 [

(41)5 R  ?# R1 ^0 I$ M5 X& [$ A

Here, 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.+ h5 X: Z. y' l  b1 |' T% f  G& f
7 q8 n. @& E8 w
The water permeability and solute permeability due to the small circularpores in the TJ strands can be expressed by
3 |! w7 J1 w: A+ \
(42)
/ B5 d8 Q$ M- [" O" ]
(43)( P2 i' n" h; Y& \+ q0 j% ~% `% O

Here, 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, thesolute diffusion coefficient in the circular pores, is given by Eq.3.

The expressions for the reflection coefficients for large slit break andsmall circular pore pathways differ. For both cases, the reflectioncoefficient is defined in terms of the partition coefficient ( 23 )
  S- x  q; V/ g( ?  N. l# b+ x
(44)0 p" Z- l+ N4 R
6 j# H' E2 p4 s6 d4 V
For large slit breaks ( 23 )
* R- u4 w3 p+ x! o
(45a)6 y' L/ w, t# D) ?8 V

For small circular pores ( 23 )
+ K* n: m& P% [8 O& \% T# K' L
(45b)

Four unknowns describe the geometry of the large slit break and smallcircular pore pathways, W 1, A 1 / 1, R 2, and A 2 / 2. Four constraints are needed todetermine this dual-pore/slit geometry. These four constraints are L TJ, TJ, and TJ NaCl and sucrosepermeabilities. We relax the constraint of TJ mannitol permeability. Forsucrose, 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 nomeasured values for L TJ and TJ. However, an estimate of TJ and L TJ can beprovided from the analysis of the compartment model, Eqs. 35 and 36, as described in the previous section. After the dimensions ofboth large slit breaks and small circular pores are determined, TJ mannitolpermeability will be evaluated and compared with its measured value.* u. F" z; O  g8 W
- K9 M8 e* f- Z) d
To further explore the dual-pathway model, the fraction of the total TJlength occupied by the large slit breaks and the average spacing of smallcircular pores in rat proximal tubule are examined. The fraction of the totalTJ length occupied by the large slit breaks in rat proximal tubule,f 1, can be expressed as

(46)8 f2 T: j1 T1 c5 v  _! A' T; k
& V1 A* b6 D7 z* x
Here, l TJ is the total TJ length in the selected segmentof the rat proximal tubule, and S the total surface area excludingthe brush border of the same segment of proximal tubule. SA 1 is the total area of the large slit breaks in the samesegment, and SA 1 / W 1 is the totallength of large slit breaks in the same segment. To calculate f 1, one must specify 1 to find A 1 after A 1 / 1 is determined.# q- T" Y2 |3 F- M) e% Z1 N

The average spacing of the small circular pores in the rat proximal tubule, 2, can be expressed as

(47)) [3 @7 y  Q9 ], j7 @7 y
9 y0 V7 p! \0 q2 O/ a5 l. |
Here, SA 2 is the total area of small pores in the selected segment of the rat proximal tubule and is the number of smallpores in the same segment. Equation 47 provides an estimate of theaverage distance between pores in the TJ strand. Again, we assume the poredepth 2 is specified after A 2 / 2 is determined.8 d: {8 ]" ]' Z
5 x/ `5 e2 W- w2 `
PARAMETER VALUES. Z3 G. Q0 U* r4 J  u3 S! w
& T4 F8 o/ T$ W0 K1 ~. b( s/ c
The parameter values used in the compartment model are summarized in Table 2. The referenceosmolality 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 thecell barrier H C is very small, and the value used inWeinstein ( 35 ), 3.1 x 10 - 10 cm/s, is adopted. The reflection coefficientof the basement membrane B is zero. L p of proximal tubule has been measured inseveral species using different techniques ( 16, 27, 32 ). Early measurements andmethods before 1983 are summarized in Berry( 3 ). These and more recentexperiments reveal a significant variation in L p for rat proximal tubule. Berry reported values that varied from 0.2-0.3cm/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 or0.7. 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 Gootet al. ( 33 ), the NaCl and KClreflection coefficients are measured and found to be close to unity for bothplasma 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 ).4 o, U3 t! }5 \. g( P9 A
: Q/ c! Y# ~- C" P6 p$ e2 G
The measured mean values for NaCl permeability of rat proximal tubule varybetween 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 thatthe electro-diffusive NaCl flux passes nearly exclusively through the TJ andthat the barrier associated with H B offers littleresistance. Thus H TJ is estimated from Eq. 33.The selected value, 13 x 10 - 5 cm/s, is thesame as that used in Weinstein( 35 ). The correspondingtransepithelial electrical resistance is 7.35 ·cm 2., e+ F, C# U4 V6 p/ k  e  [! T% O  |
) j5 U" P4 }# j% K. i
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. Theircorresponding free diffusion coefficients are 2.21, 0.90, and 0.70 x 10 - 5 cm 2 /s. The nonelectrolytepermeability of the TJ is at least one order of magnitude smaller than thesmall-ion permeability. The measured permeability values for mannitol andsucrose in rat proximal tubule are 0.87 and 0.43 x 10 - 5 cm/s( 27 ). These values are adoptedin our calculation.
+ Y. K3 m! t5 }6 P; M" j8 S
Table 3. Parameter values used in the dual-pathway model for the ultrastructureof the tight junctional strands
6 B" h" M& N! `
The measured luminal epithelial surface excluding microvilli and the TJlength in the S2 segment of rat proximal tubule are 96 x 10 3 µm 2 /mm tubule and 68.8 mm/mm tubule( 21 ). We shall see that thesedata suggest a very torturous cell boundary. The effective depth(apical-to-basal direction) of large slit breaks is 100 nm. This is typicallythe spacing between the strands in the depth direction of the cleft. In proximal tubule, there is usually a two-strand structure that is divided intosmall compartments by cross-bridging segments between the longitudinalstrands. The slit break occurs when the breaks in each of the TJ strandscoincide, providing a pathway through the TJ from lumen to lateral space.! ?0 q: m% U& t( p$ h2 [
8 B! U8 X, a( Y
In this study, the effective small circular pore depth is 10 nm. We assumethat the space between the lateral membranes of neighboring cells will offerlittle resistance compared with the small pores in the TJ. This 10-nm poredepth assumes that there are 5-nm-long circular pores in each strand of the two-strand structure in the TJ complex.4 \& F1 d# a3 V- ^4 J) K

RESULTS
/ u! T- ^: C# n+ i
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* = 5.94mosmol/kgH 2 O. 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. Asmaller TJ results in a smaller L TJ (see Eq. 36 and Tables 2 and 3 ).
1 J! x7 }& Y) L; ]
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 - 5 cm/s, = 0.68, and C* = 5.94mosmol/kgH 2 O. 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 - 5 cm/s, or six times greater than H TJ, close to the value used previously. There is somesecurity to this value, in the sense that H B is the keyparameter in determining the osmotic gradient against which the proximaltubule can transport water. Model predictions of the magnitude of thisgradient 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 therelationship among M, L p, andC* 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* = 5.94mosmol/kgH 2 O, M = 0.84; the value usedin Weinstein ( 35 ) isrecovered.2 G7 F. M3 i  C3 M
# V+ ?, [! {1 ?! ^+ B! |! I
Table 4. Predicted values in the compartment model
" \! K+ Q* n8 `0 L. v7 {3 K
Fig. 5. Relationships between L p and membranereflection coefficient ( M ) for 3 values of decrement inluminal osmolality required to yield a reabsorbate osmolality equal to that ofthe lumen (C*) values, i.e., 4, 5.94, and 8 mosmol/kgH 2 O, and 2values for the TJ reflection coefficient ( TJ ) values, 0.0and 0.05. In this figure, the curves for 2 TJ valuesintersect at a point where L p = 1.8 x 10 - 7 cm · s - 1 · mmHg - 1 and M = 1.0.0 a3 z& p# {6 q; U: Z
2 V& T, T6 m" f6 c- D: X# |
TJ Can be estimated from Eq. 35 if one replaces L M with L MB in Eq. 31. In Fig. 5 we plot the relationshipamong M, L p, and TJ for two values of TJ, 0.0 and 0.05. TJ
- i: k  k9 Q0 o" \! Y+ p6 K) S; g
The theoretically estimated values for TJ and L TJ and the TJ solute permeabilities for NaCl and sucroseare used to predict the four unknowns describing the geometry of thedual-pathway model. The predicted results are listed in Table 5. The predicted gapheight of the large slit breaks is 19.6 nm. A 1 / 1 for these breaks is 0.525cm - 1. The predicted small-pore radius is 0.668 nm,and A 2 / 2 is 15.8cm - 1. 1 For the large slitbreaks is very close to zero, 2.26 x 10 - 4,whereas 2 for the small circular pores is 0.153. Thepredicted TJ mannitol permeability is 0.89 x 10 - 7 cm/s. Thus this pore/slit geometry providesexcellent agreement for the measured permeability of mannitol.
; J( v4 X0 d* z6 ~. y+ g$ \
Table 5. Predicted results in the dual-pathway model for the ultrastructure ofthe tight junctional strands- s2 Y3 H- i- n' P! s; l" o+ b% l
2 w. o2 }4 S$ J) m) \% Y
The reported values for the rat proximal tubule area and the total TJlength ( 21 ) are used toprovide the estimation of the fraction of the total TJ length occupied by thelarge slit breaks and the average spacing of the small circular pores. Wefirst assume that the effective depth of the large slit pathway is 100 nm.This value for 1 assumes that the gap height of the pathwaythrough the strands is nearly uniform, as observed in endothelial junctions( 2 ). However, because theaverage length of the breaks observed in individual strands is typically 100nm, coincident breaks in a dual-strand structure are rare (see DISCUSSION ). Then, if S = 96 x 10 3 µm 2 /mm tubule and l TJ = 68.8 mm/mm tubule,f 1 = 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, thenf 2 = 20.2 nm. This implies that on average there is a small poreevery 20.2 nm.
3 d3 j! k0 `" f# f0 d) S
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 circularpores. In contrast to L TJ, nearly 91.2% of H TJ for NaCl is accounted for by these numerous smallcircular pores. Only 8.65% of H TJ is accounted for by thelarge slit breaks. The solute-solvent coupling term in Eq. 39 accounts for the remaining 0.16%. The model predicts that only 21.7% of thesucrose transport is through the small circular pores and 78.3% through thelarge slit breaks. The contribution of the large slit breaks to the predictedTJ permeability for mannitol is 49.2%. The model thus predicts that nearlyone-half of mannitol transport is through the large slit breaks.
8 c% L* h$ C0 x, }! E: Y
Figure 5 provides theessential link between the compartment and the dual-pore/slit models. In thecompartment model, one has the freedom to choose large values of TJ, such as 0.65 in Weinstein( 35 ). These larger values arenot 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 littlewater passes through the small-pore pathway. We shall show that the largest realizable TJ is limited to roughly 0.03.
9 K: H9 D- P" H6 Y7 H% ~* T
Four unknowns are required to define the dual pathway in the TJ strands, W 1, A 1 / 1, R 2, and A 2 / 2.However, the measured values for TJ salt, sucrose, and mannitol permeabilityand the compartment model predictions for TJ and L TJ provide five constraints for predicting the dimensionsof the dual-pathway geometry. Therefore, we need to relax one of theconstraints. The logical choice is to relax either mannitol or sucrosepermeability because the radii of both of these solutes are close in size andthus do not provide strong independent constraints, as already emphasized inthe single-pathway model. Thus we chose TJ water, salt, and sucrosepermeability values but relaxed the constraint on mannitol permeability. Thischoice has the advantage that it satisfies the constraints on TJ and L TJ required by both thecompartment and pore/slit models and thus unifies the two approaches.

In Table 6 we have listedthe predicted dimensions of the dual pathway for several differentcombinations of L p and C*. In the first sectionof the table, we vary C* from 4 to 8 mosmol/kgH 2 O 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 x 10 - 7 cm ·s - 1 · mmHg - 1.This can be explained using Eq. 36. Because TJ
, C0 x$ c5 O, u% l* z
(48)
% p; o! w; p9 i, h
where H C is very small and has been neglected. Because M changes little, L TJ undergoes minorchanges. Thus the dual-pathway geometry is insensitive to C* if both L TJ and TJ are nearly constant. We thenconclude that keeping TJ constant and varying C* does notsignificantly 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 ofthe cell membranes. L TJ

Table 6. Impact of input parameters on predicted dual-pathwaydimensions
+ o: o- O% S! q# |. ]
In the second section of Table6, we predict the dimensions of the dual pathway by keeping C* =5.94 mosmol/kgH 2 O and letting TJ increase from0.00666 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 ismaintained constant and L TJ
/ T( X( y4 |; s: q6 D5 X6 ~
Fig. 6. A : large slit height and spacing vs. TJ. Thedefinition of large slit spacing is given in Eq. 49, where averagelength of a large slit break ( T ) = 200 nm. B : small-poreradius and spacing vs. TJ. C : predicted mannitolpermeability in the dual-pathway model vs. TJ./ Y/ o2 k" s2 M- }- t" C
" x; z& `/ P. h" O+ |
Because the physically realizable range of TJ is from0.0067 to 0.03, one expects that small changes in TJ canproduce large changes in pore/slit geometry. In fact, one expects there to bean important transition in behavior as the permeability to NaCl of the smallpore increases. Because 2 1, TJ will be dominated by the second term in Eq. 38 when the small pore is small enough for 2 to significantlyexceed zero. Although 2 will decrease as the small poreincreases in size, the first term for large slit breaks will always be
" n2 a( R# B- R, P5 ?, _
Varying TJ along a constant C* curve and varying C* alonga constant TJ curve in Fig. 5 have very differenteffects. 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 TJ and L TJ nearly constant has littleeffect on pore/slit geometry but a substantial effect on L p.

The foregoing sensitivity analysis is summarized in Fig. 6, in which we haveplotted the predicted results for large slit breaks ( A ) and smallcircular pores ( B ) and the evaluated mannitol permeability( C ) from the dual-pathway model. In Fig. 6 A, we plot thelarge slit height and spacing vs. TJ. The large slit spacingis defined as the average length of TJ strand between two large slit breaks iftheir average length was 200 nm. If f 1 is the fraction of the totallength occupied by the large slit breaks, T is the average length ofa large slit break, the large slit break spacing is given by9 z. V4 P- G. t
9 o* z* L5 e9 x$ N! ?% |' I
(49)

One observes that the large slit height and spacing are nearly constant when TJ 0.015. When TJ 0.015, thesmall-pore radius is less than the sucrose radius. Thus 100% of sucrosepermeability is associated with the large slit break pathway. Because thesucrose radius is much less than the large slit height, the transport areaavailable for sucrose transport is nearly constant (see Eq. 6 ). Thewater permeability through the large slit pathway changes little; thus thelarge slit height and spacing are nearly constant.

In contrast, the small-pore radius and small-pore spacing continue todecrease when TJ increases from 0.015 to 0.03. In Eq.27, TJ is mainly determined by the second term(small-pore pathway), and the contribution of the first term (large slitbreaks) can be neglected. One can increase TJ by eitherincreasing 2, increasing L 2, or both.However, the small-pore pathway has little capacity to allow for a large waterpermeability, as shown in Preisig and Berry( 27 ). We also show in Table 6 that the fractional TJwater permeability through the small-pore pathway never exceeds 6.6% of the L TJ. Thus the more likely way for TJ toincrease is to increase 2 by decreasing the R pore. However, this also greatly increases the stericexclusion and the hydrodynamic resistance for salt transport (see Eq. 3 ). One has to increase the total pore area to maintain the measured TJsalt permeability. Thus the small pores will decrease in size but be morefrequent, and their spacing will greatly decrease. When TJ 0.03, the small pores nearly overlap and form a continuous slit incontradiction to the observed ultrastructure of the TJ strands. From thestandpoint of steric exclusion, TJ could approach L 2 / L TJ, but as noted above, the realistic upper limit is 0.03., h; R) W  @) J2 T  }! H4 z3 f! w
- n+ ~+ F! r5 F  U  d; y2 g
The evaluated mannitol permeability in Fig. 6 C is nearlyconstant and less than the experimental measurements when TJ varies from 0.015 to 0.03. In this range of TJ, thepredicted small-pore radius is close to the mannitol radius, and the stericexclusion and the hydrodynamic resistance greatly limit mannitol permeability through the small-pore pathway. Most of the TJ mannitol permeability is due tothe large slit breaks. Because the large slit height and spacing change littlein this range of TJ, the mannitol permeability does notchange significantly. At the lower limit, TJ cannot be

DISCUSSION7 ]* O  P* D$ g# p& |1 {

In this paper, we have proposed a new ultrastructural model for the TJstrands in rat proximal tubule epithelium that attempts to satisfy themeasured permeabilities for water, NaCl, and nonelectrolytes. To achieve this,we have developed a dual-pathway model that combines infrequent large slitbreaks and numerous small circular pores in the TJ strand. The L TJ and reflection coefficient TJ areused together with TJ NaCl and sucrose permeability to provide insight intothe structure and function of the TJ complex. Although dual-pathway modelshave been proposed in the past, nearly all of these models have been based onheterogeneous circular pore theory, which is not an adequate description of the large slit breaks observed in the TJ strands. The present model isintended to provide a more realistic description of the actual TJ strandultrastructure, one that includes our latest understanding of its molecularcomposition.
3 L: i1 t6 A2 }9 c; w( z
Single-Pore/Slit Models

The single-pore/slit pathway model in Preisig and Berry( 27 ) was developed to satisfyonly the measured permeabilities of mannitol and sucrose. The radii ofmannitol and sucrose differed by only 1 Å, and this limited an accuratedetermination of the pore radius or slit height in the TJ strands. The model further assumed that all the mannitol and sucrose molecules traverse the TJvia the same pore/slit ultrastructure. Our model allows that this may not bethe case. In our proposed dual-pathway model, the small pores account fornearly one-half of the mannitol flux, but, 78.3% of the sucrose follows asecond pathway, namely, large slit breaks in the TJ strand. Thesingle-pore/slit theory is unable to accommodate any substantial waterpermeability.
/ [1 @2 m, t3 H) {& x, v. ]
We also examined the capacity of the single-pore/slit theory to satisfy TJNaCl permeability and either mannitol or sucrose permeability, because thesesolutes differ significantly in size. This approach leads to the predictionthat the pore radius/slit height is smaller than predicted in Preisig andBerry, but the available transport area is three times larger( Table 1 ). Thus the predictedwater permeability of the TJ is a little larger than the predicted waterpermeability of the paracellular pathway in Preisig and Berry. However, thispredicted TJ water permeability is still + ?# _/ |1 B6 X* V0 U

In addition, we tried to jointly satisfy the TJ water and NaCl permeabilityusing a single-pore/slit model while relaxing the constraints on mannitol andsucrose permeabilities. This approach leads to significantly largerpores/slits in the TJ strand. However, it predicted a mannitol and sucrosepermeability that was approximately five times larger than the measured values. In summary, we confirm that a single-pore/slit model cannotsimultaneously satisfy the measured values for transepithelial waterpermeability, the transepithelial NaCl reflection coefficient, andparacellular mannitol and sucrose permeabilities. The greater flexibility of adual-pathway model is needed to reconcile these discrepancies.9 K5 ?! {9 h1 ?) u+ v; G

Relationship of Dual-Pathway Model for TJ Ultrastructure toCompartment Model of Proximal Tubule8 d, T8 t8 _# z
  P7 C! s. o& @" V' w1 o" M/ V7 F
The effort to determine the dimensions of the dual-pathway pore/slit structure using TJ water, NaCl, and sucrose permeabilities and the TJreflection coefficient is limited by the fact that there are no measuredvalues for TJ water permeability, L TJ, and the TJreflection coefficient, TJ. However, estimated values for TJparameters are available from a compartment model of rat proximal tubuleepithelium ( 35 ). In thedual-pathway model, the small circular pore and the large slit break pathwaysare 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 0.153for NaCl. In contrast, the reflection coefficient for the large slit breakswill approach zero, whereas its contribution to L TJ willbe large (95.0%). Thus the composite reflection coefficient of the TJ, TJ, will be much less than unity (0.0079). This predictionfrom the dual-pathway model contradicts the estimated value for the TJreflection coefficient in Weinstein( 35 ), TJ =0.65. We had to find a new set of parameter values to be used in thedual-pathway model but one that would be consistent with the compartmentmodel. The compartment model remains necessary to provide a relationshipbetween the transepithelial reflection coefficient for NaCl, = 0.68,and the reflection coefficient for the TJ.
1 ~) E0 m( V  Z- h. {
Measured L p, NaCl permeability ( H ),and the transepithelial NaCl reflection coefficient,, along with theconstraint of isotonic transport C*, are first used to predict the composite luminal membrane NaCl permeability, H M, NaCl reflectioncoefficient, M, and water permeability, L M. This model can also provide a constraint between L TJ and TJ (see Eq. 32 ). 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 valuefor L TJ is 21.2% of the transepithelial waterpermeability. This estimate of TJ and L TJ is then applied in the dual-pathway model of the TJ todetermine the dimensions of the pores and slits in the TJ strand./ c2 h& v  z) D( r0 a$ G% Y; Z0 U

With respect to the model prediction of the magnitude of TJ water flux, onemay note the observations of Schnermann et al.( 29 ), who found that micegenetically defective for the proximal tubule cell membrane water channelaquaporin-1 had a reduction in proximal tubule epithelial water permeability of 80% compared with control mice. That finding has been used by some toconclude 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 strainof mouse, so constraints on the size and locus of the water pathways areunknown.# ~3 X  ], a5 x  ~' R8 Z5 ?

Large Slit Breaks( f- R3 [5 L8 c; s6 Y9 C

Our combined dual-slit/pore model predicts that there will be infrequentlarge slit breaks in the TJ strands. These large slit discontinuities in theTJ strands are responsible for the large increase in TJ water permeabilityabove that predicted in Preisig and Berry( 27 ). The predicted length ofthe large slit breaks is only a small fraction ( 3.75 x 10 - 4 ) of the entire length of the TJ strands inthe 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 TJNaCl permeability. Thus they form a secondary route for the passage of smallsolutes.+ I) G9 F  D9 u: z+ Y

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 proximaltubule. Thus the average length of the TJ surrounding one cell is 2 x 68.8 mm/300 = 459 µm, where the factor of 2 reflects the sharing of the TJbetween neighboring cells. Because the predicted fractional length of a largeslit pathway is 3.75 x 10 - 4, then the length of a large slit in one cell is 459 µm x 3.75 x 10 - 4 = 172 nm. If the length of a typical largeslit 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.
( r0 i8 }6 k2 T$ ?
The above estimate of the open fractional length is based on an examinationof the TJ complex and its compartment structure, as observed in Figs. 13 and17 in Orci et al. ( 25 ). Onenotes that the TJ in rat proximal tubule is typically a two-strand structurewith polygonal compartments that are roughly 100 nm on a side with traversesegments interspersed4 P9 H9 b/ V- S  Y; _  c6 Q3 M9 ]
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