INTRODUCTION

TOP ABSTRACT INTRODUCTION STRUCTURE AND COMPOSITION FILTRATION OF WATER FILTRATION OF MACROMOLECULES REFERENCES

THE CONCEPT OF THE GLOMERULUS as a highly refined ultrafiltration device, capable of filtering large volumes of plasma while efficiently retaining proteins within the circulation, has long been one of the cornerstones of renal physiology. Although that basic view of glomerular function is due to earlier generations of researchers, efforts to achieve a quantitative understanding of glomerular filtration received a distinct stimulus some 30 years ago. Beginning about 1970, new animal models (e.g., the Munich-Wistar rat) and advances in micropuncture pressure measurement techniques permitted a much more direct examination of glomerular forces and flows in mammals than had been possible previously. Those and other developments have stimulated a large number of investigations into the dynamics of water filtration and the selective retention of macromolecules by the glomerulus in health and disease. A comprehensive review is available of results published through about 1990 (66).

Among the more recent lines of research are efforts begun in the early 1990s to relate the functional properties of the glomerular capillary wall to its unique structural features, on the cellular and even macromolecular level. This represents a significant departure from earlier analyses of glomerular barrier function by us and others, which mainly sought to express the available micropuncture and clearance measurements in terms of hydraulic permeabilities and effective pore sizes. In other words, in the 1970s and 1980s the glomerular capillary wall was regarded largely as a black box with certain measurable properties, whereas more recent biophysical analyses have sought to explain its permeability properties in terms of specific structures. This has been done by combining morphometric results, in vitro data using isolated glomeruli, and detailed hydrodynamic models of the capillary wall. Such work is the focus of this review.

As background, we begin with a brief overview of the structure and composition of the glomerular capillary wall. The permeability properties will be affected by features spanning a wide range of length scales, from the dimensions of cells to the dimensions of the macromolecules that form the basement membrane and junctional complexes (slit diaphragms). Those in the ~100- to 1,000-nm and ~0.1- to 10-nm ranges are conveniently labeled as "microstructural" and "nanostructural," respectively. Following the structural description is a section on water permeability, in which the main elements of the structure-based hydrodynamic models are discussed and their predictions compared with experimental findings in vivo. Emphasized there are insights into the reduced filtration capacity for water in several forms of glomerular disease. The last section concerns the selectivity of the glomerular barrier to macromolecules. Efforts to understand glomerular size selectivity in terms of structural models are reviewed, and various unresolved issues are discussed. One of the most controversial issues is the extent to which charge selectivity is important for glomerular barrier function. Also included is a discussion of recent findings concerning the effects of serum albumin on the sieving of macromolecular tracers.

	STRUCTURE AND COMPOSITION

TOP ABSTRACT INTRODUCTION STRUCTURE AND COMPOSITION FILTRATION OF WATER FILTRATION OF MACROMOLECULES REFERENCES

Microstructural Idealizations

View larger version (18K): [in this window] [in a new window]   Fig. 1.   Idealized structural unit of the glomerular capillary wall, corresponding to one filtration slit. The figure is modified from Ref. 35. W, width of the structural unit; L, the thickness of the glomerular basement membrane (GBM); w, the width of the filtration slit; x and z, coordinates.

Data for healthy humans suggest a slit width similar to that in rats, w = 43 nm (37) but a significantly larger subunit width and GBM thickness, W = 500 nm and L = 400 nm, respectively (58, 97). A morphometric index used to describe slit spacing is the filtration slit frequency (FSF), which is related to the subunit width by W = (2/)(1/FSF); the factor 2/ accounts for the random angle of sectioning (33).

Slit Diaphragm

View larger version (13K): [in this window] [in a new window]   Fig. 2.   Representations of the epithelial slit diaphragm (SD). A: view perpendicular to the flow direction, as in Fig. 1. B: view parallel to the flow direction, showing the "zipper" configuration. C: another view parallel to the flow direction, showing the "ladder" configuration. Figure is modified from Ref. 36.

Recent efforts to elucidate the structure of the slit diaphragm have centered on its component molecules, particularly the newly identified protein nephrin. Nephrin has a molecular mass of ~150 kDa and has been shown to be expressed exclusively by glomerular podocytes in the slit diaphragm region (44, 89). Lack of proper expression of the nephrin gene has been shown by Tryggvason and co-workers (63, 102) to be linked to the congenital nephrotic syndrome of the Finnish type, a glomerular disorder that results in severe proteinuria and that is associated with normal GBM and the loss of foot processes and slit diaphragms. Genetic analysis of the coding region of the nephrin gene has demonstrated that it is a single-pass, membrane-spanning protein with eight Ig motifs and a type III fibronectin domain (102). It has been hypothesized that nephrin molecules extending out from adjacent podocytes might interact in a homophilic manner to form the zipper structure (102). Such proposals remain speculative, as the interaction of nephrin with other protein components of the slit diaphragm is not yet known. It has been demonstrated that cultured podocytes form linking structures that are similar to filtration slits in vivo and that these intercellular linking structures contain the proteins zonula occludens-1, P-cadherin, and -, -, and -catenin (82).

GBM

Endothelial Glycocalyx

	FILTRATION OF WATER

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Structure-Based Model

(1)

Finite-element solutions of Stokes' equation (the low-Reynolds-number form of the Navier-Stokes equation) were used to characterize flow in the epithelial filtration slits (30). The results indicated that the slit diaphragm is the dominant resistance to water flow between the foot processes, implying that the slit length is not an important parameter for water filtration. With the use of the zipper structure, with all dimensions as given in Rodewald and Karnovsky (87), the permeability of the slit diaphragm (in SI units) was estimated as k_s = 7.9 × 10⁸ m · s¹ · Pa¹. Because what is desired is a filtrate velocity (or volume flux) averaged over an entire structural unit, and because the slits only occupy a fraction _s of the surface area, the epithelial permeability is k_ep = _s k_s. With the use of the representative dimensions for the rat given above, _s = 0.11 and k_ep = 8.6 × 10⁹ m · s¹ · Pa¹. It was shown that the resistances to water flow of the zipper and ladder structures are similar, provided they are assumed to have the same ratio of wetted cylinder area to cross-sectional area (30).

Finite-element solutions of Stokes' equation were used also to characterize the hydraulic resistance of a water-filled fenestra (31). By using the dimensions given in Lea et al. (62), the permeability of a single fenestra was estimated as k_f = 1.0 × 10⁶ m · s¹ · Pa¹. With the fenestrae occupying 20% of the filtering surface (_f = 0.20), it was found that k_en = _f k_f = 2.0 × 10⁷ m · s¹ · Pa¹. Comparing this with the epithelial result, it is found that k_en/k_ep  20. This suggests that the dominant cellular contribution to k is that of the slit diaphragms and that the water flow resistance of the fenestrae is negligible. This assumes, however, that the flow resistance of the glycocalyx is unimportant (see below).

Water flow through the GBM was described by Drummond and Deen (31) using Darcy's law

(2)

where v is the local fluid velocity vector,  is the Darcy permeability, µ is the fluid viscosity, and P is the local pressure gradient. This relation is commonly used to model flow through porous or fibrous materials in situations where the pore spacings or interfiber spacings are much smaller than the dimensions of the sample. Microstructural details such as fiber concentration and fiber size are ignored, except as they influence the value of  (units of m²). This approach is suitable when the underlying structure is complex, but pressure-flow data are available from which  can be evaluated. Such data are provided by studies of filters made by consolidating isolated GBM, an approach used by Robinson and co-workers (86, 106) and by Daniels and her associates (9, 27, 34). Typical results are  = 1-3 nm².

Equation 2 was combined with that which describes local conservation of mass ( · v = 0) and solved for the idealized GBM geometry shown in Fig. 1 (31). Although the actual fenestral openings are circular, a comparison of three-dimensional finite-element solutions for circular openings with two-dimensional analytical solutions for slitlike openings showed that equivalent results were obtained if the value of _f was the same. Moreover, for the relative dimensions typical of the GBM, it was found that the infinite-series expression obtained from the analytical solution was well approximated by

(3)

where n_f is the number of fenestral "slits" per structural unit. The left-hand side is the hydraulic resistance of the GBM (1/k_bm) relative to the resistance it would exhibit if its surface were not partially blocked by cells. The resistance of "bare" GBM is µL/, as obtained by applying Eq. 2 to one-dimensional flow across a simple barrier of thickness L. All of the terms following the "1" on the right-hand side of Eq. 3 describe the increased GBM resistance due to the fact that only parts of its upstream and downstream surfaces are accessible to filtrate. That is, the channeling of fluid flow caused by the cellular coverage has the effect of increasing the flow resistance in the GBM. In this sense, the percentages of the overall flow resistance ascribed to the cells and to the GBM are somewhat arbitrary. Although we favor simply comparing the three terms in Eq. 1, one could argue that doing so understates the cellular contribution, because both cell layers reduce k_bm.

The main trends predicted by Eq. 3 are illustrated in Fig. 3, which shows the relative GBM resistance to water flow for various combinations of _s, _f, and n_f. The parameter values used are those for the normal rat (Table 1). The GBM resistance in vivo is predicted to be 2.3 times that of bare GBM. Decreases in _s, _f, and n_f all exaggerate the channeling phenomenon, thereby increasing the water flow resistance.

View larger version (14K): [in this window] [in a new window]   Fig. 3.   Dimensionless resistance to water flow in the GBM, as a function of the fractional areas of filtration slits (s) and fenestra (f) and the number of fenestra per structural unit (nf). The hydraulic resistance of the GBM (1/kbm) is expressed relative to that in the absence of cells (µL/), such that a value of 1 on the ordinate corresponds to "bare" GBM.

Setting  = 2.7 nm² and using the dimensions for the rat, found that k_bm = 8.3 × 10⁹ m · s¹ · Pa¹ Drummond and Deen (31). Because k_bm  k_ep k_en, it was concluded that the GBM and epithelial resistances to water filtration in the normal rat are about equal and that the resistance of the endothelium is negligible. From Eq. 1, the overall hydraulic permeability was predicted to be k = 4.1 × 10⁹ m · s¹ · Pa¹. This is well within the range of values estimated from micropuncture measurements, which is roughly from 3 × 10⁹ to 5 × 10⁹ m · s¹ · Pa¹ (31).

The hydraulic resistance of the GBM is proportional to 1/ (Eq. 3), and the  value used above is larger than more recent estimates, including  = 1.5 nm² (34) and  = 1.2 nm² (9). Thus the GBM may actually account for somewhat more of the overall resistance than indicated. If one uses  = 1.2 nm² instead of  = 2.7 nm², the contribution of the GBM increases from 50 to 69% of the total resistance. Although the overall hydraulic permeability is then reduced by 38% to k = 2.5 × 10⁹ m · s¹ · Pa¹, the predicted value is still in reasonable agreement with the experimental range.

There are uncertainties also in the cellular contributions to the hydraulic permeability. The value of k_en quoted above was computed by assuming that a fenestra is a short, water-filled channel of varying radius. An alternative model is that it is a gel-filled channel, due to the endothelial glycocalyx. When that possibility was explored by solving Brinkman's equation (related to Darcy's law) in a fenestra, with  = 2.7 nm² as for the GBM, k_en was decreased to 1.3 × 10⁸ m · s¹ · Pa¹ (31). That change alone decreases the overall hydraulic permeability from 4.1 × 10⁹ to 3.2 × 10⁹ m · s¹ · Pa¹, with the endothelium now accounting for 24% (instead of just 2%) of the total resistance. The main obstacle to refining the estimate of k_en is the unknown  of the glycocalyx.

Whereas the hydraulic resistance of the endothelium may have been underestimated, depending on the actual properties of the glycocalyx, that of the epithelium may have been overestimated. As already mentioned, the zipper structure is far too "tight" a barrier to be consistent with the relatively large test macromolecules that appear in normal glomerular filtrate. Larger openings in the slit diaphragm would also tend to increase the value of k_ep. To refine models either for water flow or for macromolecule movement through the filtration slits, an improved representation of the slit diaphragm geometry is needed.

Uncertainties in the individual contributions notwithstanding, the success of the water flow model in predicting the overall hydraulic permeability suggests that the overall balance between the GBM and cellular resistances is approximately correct. Indeed, the tendency to underestimate the endothelial contribution may well have canceled a tendency to overestimate the epithelial contribution. In most of the applications to pathophysiological situations described below, the fenestral and slit diaphragm permeabilities are each assumed to be constant, and the main factor considered is the calculated change in k_bm. Under those conditions, precisely apportioning the cellular resistance between the two layers is much less important than describing the effects of the cells on k_bm.

Applications of Water Flow Model to Glomerular Disease

The model has been applied also to human glomerular disease. In each of the four diseases studied to date, impairment of k appears to be the predominant cause of glomerular filtration rate (GFR) depression early in the course of the disorder. The conditions examined include minimal change, membranous, and diabetic nephropathies, and preeclamptic toxemia (33, 58, 79, 97). In each instance depression of GFR by 30-50% was associated with alterations in glomerular hemodynamics that should not have reduced the net ultrafiltration pressure and hence the GFR. By exclusion, we infer that GFR depression must have been due to a decline in the ultrafiltration coefficient (K_f). K_f is the product of glomerular hydraulic permeability and filtration surface area (K_f = kS), expressed either on a single-nephron or whole-kidney basis; single-nephron values are employed here.

In the human studies to be discussed, glomeruli obtained by biopsy were subjected to morphometric analysis to determine L, FSF (allowing calculation of W), filtration surface area per glomerulus (S), and certain other quantities. The value of S was computed from the product of filtration surface density and glomerular volume (33, 97). Except where indicated, the values of parameters employed in the water flow model (k_en, _f, n_f, , k_s, w) other than L and W were assumed to be the same as the original set used for normal rats (31), as given above. Control values of L and W were provided by groups of subjects with normal glomeruli (living kidney transplant donors). In the controls and in three forms of glomerular injury (diabetic, minimal change, and membranous nephropathy), transmission electron micrographs showed large and numerous endothelial fenestrae, and the endothelial resistance to water flow was neglected. In membranous (33, 97), minimal change (33), and diabetic nephropathy (79), the main contribution to the reduction in k was found to be the increase in W. In preeclamptic toxemia, an observed reduction in the size and number of fenestrae made the calculated endothelial contribution important (58).

An example is provided by findings in a group of 15 patients with membranous nephropathy. Each had a severe glomerular injury characterized by persistent nephrosis and a progressive decline in GFR over a 2- to 5-yr period of observation ( 97). Glomerular structure and ultrafiltration capacity were examined on two occasions, at the time of presentation and diagnostic biopsy (baseline) and again after 2-5 yr. At baseline, glomerular volume was larger than control, and it was estimated that S increased by some 40%. Membranous nephropathy at this time was accompanied by an approximate doubling of L and a roughly fourfold increase in W, reflecting a marked widening of both the GBM and the epithelial foot processes. Using Eqs. 1 and 3, it was found that there was a marked depression of k, 0.79 ± 0.09 × 10⁹ m · s¹ · Pa¹ in membranous nephropathy vs. 2.8 ± 0.09 × 10⁹ m · s¹ · Pa¹ in controls. The corresponding values of K_f predicted by the model were 3.4 ± 0.7 nl · min¹ · mmHg¹ in membranous nephropathy and 7.1 ± 0.6 nl · min¹ · mmHg¹ in controls. This estimated 52% reduction in K_f was sufficient to account for the observed reduction in GFR (56 ± 8 vs. 102 ± 2 ml/min in controls).

The later analysis (2-5 yr beyond baseline) revealed no further changes in FSF (or W), but there were increases in L to roughly four times that of control and reductions in S to ~30% below control values. The persistent nephrosis was associated with an additional, significant decline in GFR in each individual. Because k at this later time was computed to be not significantly different from that at baseline, it was concluded that the further reduction in GFR was attributable entirely to the reduced S. To summarize, the serial observations permit the conclusion that progressive hypofiltration in membranous nephropathy is a consequence of a biphasic loss of glomerular filtration capacity, consisting of an initial reduction in k that is later exacerbated by a loss of S (97).

Given that the GBM is a significant contributor to the overall water flow resistance, one might expect that the doubling of GBM width between biopsies in the membranous nephropathy patients would have lowered k even further below that of controls. However, with a very low FSF, as was the case in that disorder, much of the flow within the GBM is parallel to its surfaces, rather than directly across. With the path length for filtrate thereby determined largely by W, there is relatively little sensitivity of k to L. Thus FSF becomes the principal determinant of k when FSF is small enough. A similar observation was made in a comparison of results for membranous nephropathy and minimal change nephropathy (33). Similar values of FSF in the two groups led to similar predictions of k, despite approximately twofold larger values of L in membranous nephropathy. Because the measured values of S and of the hemodynamic determinants of GFR did not differ greatly, this explained the similar values of GFR in the two groups.

A group of glomerular diseases that fit loosely into the category of "thrombotic microangiopathy" or "hemolytic uremic syndrome" can lower GFR while having no discernable effect on the GBM or epithelial foot processes. Rather, this group of glomerulopathies is associated with substantial injury to glomerular endothelial cells. In subjects with preeclamptic toxemia, which is an example of a thrombotic microangiopathy, GFR was found to be depressed by 39% relative to healthy gravid controls (58). Reductions in filtration surface density due to mesangial interposition were partially offset by glomerular hypertrophy, resulting in values of S that tended to be slightly lower than in controls. Neither GBM thickness nor FSF was altered, but there were extensive, dense, subendothelial deposits of fibrinoid material that substantially lengthened the filtration pathway (from fenestral interface to slit diaphragm). The circumferential rim of endothelial cytoplasm was characterized by swollen segments that were devoid of fenestrae. A morphometric analysis of "en face" sections of endothelium by scanning electron microscopy revealed that _f was drastically reduced, from 0.16 in controls to 0.014-0.087 in the subjects with preeclamptic toxemia. The fenestrae were also smaller, as evidenced by a reduction in their area-to-perimeter ratio to one-half that of controls. From this structural information, it was estimated that k was reduced by ~30% in preeclamptic toxemia. Taken together with the trend toward lower S, it was calculated that K_f was likely to have been depressed by ~40% in preeclamptic toxemia, similar to the reduction in GFR.

GBM Nanostructure and

(4)

With   0.1, as has been reported for GBM (21, 85), realistic values for  (in the range of 1-2 nm²) are obtained from any of the theoretical expressions if the fiber radius is assumed to be ~1 nm (8). However, if r_f = 3-4 nm is employed, corresponding to the radii of fibers visible in electron microscopic images, the predicted value of  is an order of magnitude too large. This led to the suggestion that GBM be modeled as a mixture of coarse and fine fibers, the former corresponding roughly to collagen IV fibrils and the latter to glycosaminoglycan chains (8, 34). Underlying this suggestion is the presumption that the fine fibers would not have been resolved in the electron micrographs. With coarse and fine fiber radii of 3.5 and 0.5 nm, respectively, and roughly a 1:1 mixture (by volume) of the two fiber types, it was possible to reconcile the measured values of  and  with the electron microscopic appearance of GBM. Parameter values for this two-fiber model of the GBM, which should be viewed as quite tentative, are summarized in Table 2.

Additional quantitative information on the composition and the spatial arrangement of proteins and proteoglycans would be invaluable in efforts to reach more definite conclusions about the nanostructural basis for  in the GBM. Analogous information is needed to estimate  in the endothelial glycocalyx and thereby better define the endothelial resistance to water flow.

	FILTRATION OF MACROMOLECULES

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General Relationships

The relationship between the overall sieving coefficient at any position along a capillary () and those of the individual layers can be approximated as

(5)

For example, _bm is the concentration at the downstream edge of the GBM divided by that at the upstream edge, with both concentrations evaluated just inside the GBM. To the extent that _i  1 for layer i, that layer will not contribute to the observed selectivity of the barrier. It is important to note, though, that the product in Eq. 5 implies that a 10% change in any individual _i will affect the overall  by the same 10%, whether layer i is highly selective (e.g., _i = 0.001) or not (e.g., _i = 0.9). This contrasts with the situation for water flow, where the additive series-resistance relationship (Eq. 1) implies that if layer i contributes a negligible fraction of the overall resistance (i.e., if 1/k_i 1/k), then a 10% change in k_i will have no noticeable effect on k. Thus the layers combine to influence macromolecule selectivity in a fundamentally different way than they combine to influence water filtration. To obtain a more precise relationship between the overall  and those of the individual layers, additional factors must be included in Eq. 5 to account for the effects of soluble proteins (e.g., albumin) on the equilibrium partitioning of macromolecules (61).

Another important distinction between water filtration and macromolecule sieving is that the individual _i values affect one another, whereas the individual k_i values could be computed independently. Moreover, the _i values depend in general on the filtrate velocity, whereas the k_i values could be approximated as constants. (Constancy of k assumes, of course, that the applied pressures are not so large as to alter the structure of the capillary wall). The interdependence of the layer sieving coefficients and the effects of filtrate velocity are illustrated next by a somewhat simplified model for transport in the GBM. As discussed later, an extension of that approach is a central feature of a structure-based model that has been proposed to describe glomerular size selectivity.

As in the application of Darcy's law (Eq. 2), the GBM will be regarded as an isotropic medium, such as an array of randomly oriented fibers. In such a material the local flux (N) of an uncharged macromolecule may be expressed as

(6)

where D is the solute diffusivity in free solution, v is the local fluid velocity vector, C is the solute concentration, and K_d and K_c are hindrance factors for diffusion and convection, respectively. The local solute concentration is based here on total volume (water plus solids), as is usually done in describing equilibrium partitioning or transport in gels. Just as Eq. 2 relates the local fluid velocity to the pressure gradient, Eq. 6 relates the local solute flux to the concentration gradient and the fluid velocity.

The diffusivity and hindrance factors in Eq. 6 all depend on molecular size. The standard measure of molecular size is the Stokes-Einstein radius (r_s), because knowing it is equivalent to knowing D. For a spherical molecule of radius r_s in water at 37°C, the relationship is D = (3.28 × 10⁵ cm²/s)/r_s (where r_s is in Å). In general, steric and hydrodynamic interactions between a macromolecular solute and the fixed polymeric fibers of a membrane or gel will cause K_d and K_c to be less than unity, with both decreasing as r_s increases. The experimental estimation of these hindrance factors in GBM is discussed later. Another property of a fibrous membrane or gel that influences transport and depends on r_s is the equilibrium partition coefficient (). The partition coefficient is a thermodynamic quantity that describes the tendency of steric and/or electrostatic interactions to exclude macromolecules from the material. As with the hindrance factors, it is typically less than unity and decreases with increasing r_s. As defined here, if the GBM were in equilibrium with plasma, then C = C_p, where C_p is the plasma concentration. Steric exclusion from the GBM is important, but it appears that electrostatic interactions are not (9). Although the partition coefficient does not appear in Eq. 6, it enters the analysis when concentrations within the GBM are related to those in plasma or the other structures.

Assume for the moment that the GBM extends from z = 0 to z = L, that the solute concentration depends only on z, and that the solute flux and fluid velocity (magnitudes N and v, respectively) are each constant. This "one-dimensional" model, involving just z, corresponds to a hypothetical GBM with fully accessible surfaces (i.e., _f = _s = 1). As will be seen later, only a slight modification of the results is needed to describe the more realistic situation where the surfaces are largely blocked by cells. In the one-dimensional model, the solute concentration profile in the GBM can be derived analytically for any specified values of _en and _ep. This allows the sieving coefficient in the GBM to be evaluated. The result is

(7)

where Pe is the Péclet number

(8)

Notice in Eq. 7 that the sieving coefficient in the GBM depends on that for the epithelium (filtration slits). Notice also the effect of v, which is in the numerator of Pe (Eq. 8). The physical significance of the Péclet number is that it measures the importance of convection relative to diffusion; convection tends to dominate for large Pe and diffusion for small Pe. Equation 8 has been written with the common factor  in the numerator and denominator to emphasize that, because only the products K_c and K_d appear there and in Eq. 7, those two lumped quantities are sufficient to describe the intrinsic size selectivity of a membrane such as the GBM. That is, , K_c, and K_d need not be known separately. Although the simplified model employed here assumes that  for a tracer (e.g., Ficoll) has the same value at both sides of the GBM, a more detailed theory indicates that it depends on the local concentration of albumin and other abundant proteins (61). Accordingly, it is expected to differ at the two sides of the GBM, as discussed later.

The dependence of _bm on _ep predicted by Eq. 7 is illustrated by the curve labeled "1-D model" in Fig. 4. In these calculations Pe and K_c were held constant at values representative of a macromolecule with r_s = 35 Å in rat GBM. It is seen that _bm is predicted to range from values above unity for a highly selective filtration slit (_ep  0) to values below unity for a nonselective one (_ep = 1). The behavior for highly selective slits reflects concentration polarization within the GBM, as noted in Edwards et al. (35). That is, a concentration increase in the direction of flow arises to provide a diffusional driving force in the other direction. The opposing contributions of diffusion and convection in the GBM reduce N to what can be accommodated by the slit, thereby maintaining the steady state. Inspection of Eq. 7 reveals that the upper limit of the polarization effect in the GBM is _bm  exp(Pe) for _ep  0. It is also seen that GBM polarization disappears exactly (i.e., _bm = 1) if _ep = K_c, for any Pe. Only for _ep > K_c is the slit permeable enough to allow the basement membrane to enhance the overall selectivity (i.e., _bm < 1), rather than degrade it. A final noteworthy aspect of Eq. 7 is that it shows that _bm  1 as Pe  0, for any positive values of _ep and K_c. This is an example of a well-known phenomenon in ultrafiltration, which is the tendency for filtrate and retentate concentrations to equilibrate as diffusion becomes more important. In this instance, the equilibration is just across the GBM.

View larger version (12K): [in this window] [in a new window]   Fig. 4.   Dependence of the GBM sieving coefficient (bm) on that in the epithelial filtration slit (ep) for a molecule with a Stokes-Einstein radius (rs) = 35 Å. The predictions are based on Eq. 7, with a Péclet no. (Pe) = 0.065 for the 1-dimensional (1-D) model and Pe replaced by Pe* = 0.14 for the 2-D model.

The simplified, one-dimensional analysis just discussed illustrates an important, general point, which is that the individual sieving coefficients depend on one another and on the relevant Péclet number(s). Although the Péclet number discussed was that for the GBM, analogous Péclet numbers for the fenestrae and filtration slits can be expected to influence _en and _ep, respectively. Such effects have been discussed in models of the slit diaphragm (32, 35). A consequence of the dependence of the sieving coefficient on the Péclet numbers is that great care must be taken in extrapolating results from one experimental situation to another. For example, one cannot expect a sieving coefficient measured for GBM in vitro to equal that in vivo, even if the isolated GBM preparation is perfect. The thickness of a filter made by consolidating GBM fragments will greatly exceed that of a single layer of GBM and the filtrate velocity is unlikely to equal that in vivo; both of these differences will affect Pe (Eq. 8). Moreover, the modifying effect of the epithelial sieving coefficient will be absent.

Experimental Assessment of GBM and Cellular Contributions

View larger version (20K): [in this window] [in a new window]   Fig. 5.   Sieving coefficient of Ficoll (F) as a function of rs for isolated rat GBM. The symbols with error bars represent the data of Bolton et al. (9). Theoretical curves are shown for a solution without BSA, for a BSA solution with osmotic effects only, and for the complete theory with osmotic and partitioning effects. The figure is from Ref. 61.

The sieving results for Ficoll and Ficoll sulfate in protein-free solutions were analyzed by Lazzara and Deen (61) to estimate values of K_d and K_c for GBM. The data were fitted using a sieving relationship similar to Eq. 7 (but with _ep = 1) and assumed expressions of the form

(9)

(10)

The values of the empirical constants A and B were very similar for Ficoll and Ficoll sulfate, with averages of A = 0.130 Å¹ and B = 0.072 Å¹ for the two sets of data. Equations 9 and 10 have no theoretical basis, except for the expectation that both quantities should be near unity for small r_s and should decline to zero for very large molecules. Nonetheless, as shown by the lower curve in Fig. 5, excellent fits to the data for 20  r_s  50 Å were obtained with just the two adjustable parameters. Empirical expressions similar to Eqs. 9 and 10 were also employed previously (9, 35).

The use of Eqs. 9 and 10 to make inferences about the glomerular capillary wall assumes, of course, that the isolated GBM was not functionally different from that in vivo. The possibility that GBM is altered during the isolation process has been examined using a variety of methods. Immunofluorescent microscopy of consolidated GBM filters demonstrated the presence of type IV collagen, laminin, and the core protein of heparan sulfate proteoglycan (27), the main components of GBM. The sulfated side chains of GBM proteoglycans are also present in GBM isolated using N-lauryl sarcosine to lyse cells (25), the procedure employed to obtain the data from which Eqs. 9 and 10 were derived (9). The permeability of GBM filters was not changed when a milder detergent, Triton X-100, which has been shown to preserve heparan sulfate proteoglycan, was used to lyse glomerular cells (25). That isolated GBM is relatively intact is suggested also by electron microscopy studies: the spatial distribution of cationic ferritin has been found to be similar to that in vivo (55).

A technical advance due to Daniels and co-workers (26, 36) that has permitted the measurement of diffusional permeabilities for macromolecules is the use of confocal microscopy to monitor the movement of fluorescent tracers across segments of isolated glomerular capillaries. Experiments have been performed with intact glomeruli, freshly isolated from rats, and with glomeruli in which the cells have been removed by detergent lysis, leaving only GBM. Thus it has been possible to compare the diffusional permeability of intact capillary walls (p) with that of bare GBM (p_bm). Diffusional permeabilities of series barriers obey a resistance formula like Eq. 1, so that

(11)

The two cellular contributions, which cannot be distinguished using this approach, have been lumped together in the second equality as p_cell. Edwards et al. (36) measured p and p_bm for four narrow fractions of Ficoll (r_s = 30-62 Å) and found that p_bm for each molecular size was an order of magnitude larger than p. It was calculated that the GBM contributes only 13-26% of the diffusional resistance of the intact capillary wall (depending on r_s). The finding that p_cell p_bm for Ficoll is qualitatively similar to earlier results for dextran (26).

The experimental estimates of the GBM hindrance factors for Ficoll are plotted in Fig. 6. The results for K_d and K_c derived from sieving data (Eqs. 9 and 10) are compared with values of K_d calculated from p_bm. The relationship between the diffusional permeability and diffusional hindrance factor is p_bm = K_dD/L, where L (the GBM thickness) was taken to be 200 nm. The agreement between the two independent estimates of K_d is remarkably good, given the different experimental preparations and the several assumptions required in making this comparison. The finding that K_c K_d for Ficoll is qualitatively consistent with data for globular proteins and Ficoll in agarose gels (49, 52, 53).

View larger version (18K): [in this window] [in a new window]   Fig. 6.   Diffusive (Kd) and convective (Kc) hindrance factors for Ficoll in GBM as a function of rs. , Values of Kd calculated from the confocal microscopy data of Edwards et al. (36); solid lines, the estimates from sieving data in isolated GBM without BSA (Eqs. 9 and 10); dashed curves include the predicted effect of BSA on ; increases in  due to a BSA concentration of 6.2 g/dl at the upstream side of the membrane were computed as in Ref. 61 and Eqs. 9 and 10 modified accordingly.

Using v = 4 µm/s as a typical average filtrate velocity for the rat (corresponding roughly to single-nephron GFR = 40 nl/min), Pe calculated from Eqs. 8-10 ranges from 0.016 at r_s = 20 Å to 0.22 at r_s = 50 Å. These small values of Pe indicate that diffusion within the GBM is relatively rapid in vivo (compared with convection), even for relatively large molecules. A consequence of this is that concentration polarization within the GBM will tend to be minimal, even if the filtration slits are highly selective barriers. This tends to mitigate objections that are sometimes made to a glomerular capillary "design" in which the limiting barrier is the one farthest downstream. Although diffusion in the GBM is rapid relative to convection, it is still much slower than diffusion in water. This is indicated by the small values of K_d in Fig. 6. For example, K_d = 0.01 (the value for r_s = 35 Å) means that the diffusional permeability of the GBM is only 1% of that of a film of water of equivalent thickness.

Not considered in Fig. 6 are the possible effects of GBM compressibility on macromolecule partition coefficients and diffusive or convective hindrance factors. In particular, the sieving data used were obtained at an applied pressure of P = 60 mmHg (9), whereas the diffusion experiments (36) corresponded to P = 0. The hydraulic permeabilities and/or  values of filters made from isolated GBM have been found to decrease with increases in applied pressure (P) (27, 34, 86, 106). Because f() in Eq. 4 decreases with increasing , one would expect  to decrease if compression of the GBM forces out water and thereby increases the volume fraction of solids. On the basis of theories for fiber matrices, increases in  are expected to also result in decreases in  (60, 71) and K_d (49, 81). Experimental results for proteins and Ficoll in agarose suggest that K_c would decrease as well (52, 53). Attempts have been made to model the effects of pressure on K_d and K_c (34, 35), but these efforts are confounded by the lack of an adequate theory for K_c in fibrous materials and by the probable effects of BSA on the values of  for Ficoll (61). The effects of BSA are an issue because BSA has been present in some sieving experiments with isolated GBM, but not others.

The interpretation of p_cell depends, of course, on the relative contributions of the endothelium and epithelium to the diffusional resistance of the intact capillary wall. Assuming that the cellular resistance resides in the slit diaphragm, and modeling that structure as a row of parallel cylinders (as in the "ladder" of Fig. 2), Edwards et al (36) found that the diffusion results could be explained by a cylinder spacing that followed a lognormal distribution, with small areas (~0.2%) devoid of cylinders. That representation of the cellular barrier was incorporated into later simulations of macromolecule filtration in vivo (35). The one significant difference was that in healthy subjects, at least, there was no evidence for "shunts" created by small areas of the slit diaphragm devoid of cylinders.

As already stated, it was found that sieving curves measured in isolated GBM for Ficoll and its anionic derivative, Ficoll sulfate, were indistinguishable. Only when the ionic strength of the solutions was reduced below physiological levels, thereby amplifying the effects of electrostatic interactions, was _bm for Ficoll sulfate less than that of neutral Ficoll (9). This finding of little or no charge selectivity is generally consistent with other studies of isolated GBM. That is, Bray and Robinson (11) found only small differences in sieving curves for dextran and dextran sulfate (DS), and Bertolatus and Klinzman (5) noted only small differences in the filtration rates of native (anionic) and cationized BSA. Procedures used in those laboratories to neutralize GBM charge, including methylation of carboxyl groups (5) and reductions in pH from 7.4 to 5.7 (the isolectric point of GBM) (85) had little effect on the sieving of BSA. Similarly, Daniels (25) found that treat