## Abstract

It was found previously that the sieving coefficients of Ficoll and Ficoll sulfate across isolated glomerular basement membrane (GBM) were greatly elevated when BSA was present at physiological levels, and it was suggested that most of this increase might have been the result of steric interactions between BSA and the tracers (5). To test this hypothesis, we extended the theory for the sieving of macromolecular tracers to account for the presence of a second, abundant solute. Increasing the concentration of an abundant solute is predicted to increase the equilibrium partition coefficient of a tracer in a porous or fibrous membrane, thereby increasing the sieving coefficient. The magnitude of this partitioning effect depends on solute size and membrane structure. The osmotic reduction in filtrate velocity caused by an abundant, mostly retained solute will also tend to elevate the tracer sieving coefficient. The osmotic effect alone explained only about one-third of the observed increase in the sieving coefficients of Ficoll and Ficoll sulfate, whereas the effect of BSA on tracer partitioning was sufficient to account for the remainder. At physiological concentrations, predictions for tracer sieving in the presence of BSA were found to be insensitive to the assumed shape of the protein (sphere or prolate spheroid). For protein mixtures, the theoretical effect of 6 g/dl BSA on the partitioning of spherical tracers was indistinguishable from that of 3 g/dl BSA and 3 g/dl IgG. This suggests that for partitioning and sieving studies in vitro, a good experimental model for plasma is a BSA solution with a mass concentration matching that of total plasma protein. The effect of plasma proteins on tracer partitioning is expected to influence sieving not only in isolated GBM but also in intact glomerular capillaries in vivo.

- Ficoll
- sieving coefficient
- equilibrium partition coefficient
- fiber matrix theory

in a study designed to test the effects of molecular charge on the barrier properties of glomerular basement membrane (GBM), the sieving of polydisperse Ficoll and Ficoll sulfate was examined in vitro using filters prepared from isolated rat GBM (5). Sieving coefficients (Θ; the ratio of filtrate to retentate concentration) were determined for Stokes-Einstein molecular radii (*r*
_{s}) ranging from 20 to 50 Å. The principal finding was that the values of Θ for any given size of Ficoll and Ficoll sulfate were indistinguishable when buffer solutions of physiological ionic strength were employed. This indicates that the GBM is only a size-selective barrier and does not exhibit charge selectivity. Although the experiments failed to detect an effect of molecular charge, there was a very pronounced upward shift in the sieving curves (plots of Θ vs. *r*
_{s}) of either tracer when BSA was present in the retentate at a concentration of 4 g/dl. Because the hydraulic permeability of the GBM filters was unaffected by BSA, the shift in the sieving curves apparently was not due to an alteration of the intrinsic properties of the GBM (i.e., a result of binding of BSA to the membrane). One alternative explanation for the increase in Θ is the reduction in filtrate velocity (or volume flux) caused by the osmotic pressure of BSA. A well-known finding in ultrafiltration is that small filtrate velocities promote diffusional equilibration between the filtrate and retentate, causing Θ to approach unity even for large solutes if the velocity is small enough. Thus slow rates of filtration diminish the apparent size selectivity. However, calculations based on the measured filtrate velocities with and without BSA revealed that this could explain only about one-third of the increase in Θ. It was suggested that most of the effect of BSA might be due to a second physical phenomenon, namely, a tendency of steric interactions with BSA to facilitate entry of the tracers into the membrane. It is this second phenomenon, known for some time in the membrane science literature but not widely recognized in microvascular physiology, which was examined in more detail in the present work.

Several theoretical and experimental investigations have shown that the equilibrium partitioning of a macromolecule between a bulk solution and a porous or fibrous material is dependent on its concentration. In essence, steric interactions between molecules in a concentrated solution cause entry into the porous or fibrous material to be more favorable thermodynamically than if the solution were dilute. The net effect is that the partition coefficient (Φ; the concentration in the membrane divided by that in the external solution, at equilibrium) increases with the external concentration. For uniform pores of various shapes and for solutions containing a single type of rigid, spherical solute, this effect was predicted by Anderson and Brannon (2) and by Glandt (10) using statistical mechanical arguments. Fanti and Glandt (9) used density functional theory to obtain similar results for spheres partitioning in randomly oriented arrays of fibers. More recently, Lazzara et al. (17) used an excluded volume formulation to extend the results for rigid solutes to arbitrary mixtures of spheres or spheroids, and White and Deen (35) used Monte Carlo methods to predict the partitioning of concentrated solutions of flexible polymer chains. Experimentally, increases in Φ with increasing solute concentration have been demonstrated, for example, by Brannon and Anderson (6) for both dextran and BSA in controlled pore glass and by White and Deen (35) for dextran in agarose gels. Additionally, the Θ of Ficoll and BSA in synthetic membranes were found to increase with increasing solute concentration, consistent with theoretical predictions for porous media (24).

Most of the work just cited involved concentrated solutions of single solutes, whereas what is of primary interest here is the effect of an abundant solute (e.g., BSA) on the partitioning of a dissimilar tracer (e.g., Ficoll). The partitioning of a spherical tracer molecule between a solution and a fiber matrix is depicted in Fig.1. When only the tracer is present, as in Fig.1
*A*, solute-solute interactions are negligible and steric exclusion of the tracer by the fibers causes Φ to be less than unity. This is the situation considered in the classic analysis by Ogston (25). The balance is altered when a second solute is added at high concentration, as in Fig. 1
*B*. When very little of the abundant solute is able to enter the membrane, it will tend to exclude the tracer from the solution, partially canceling the effects of the fibers. Accordingly, although Φ for the tracer is still less than unity, it is larger than for a very dilute solution.

Bolton et al. (5) were unable to satisfactorily model the effect of BSA on Ficoll or Ficoll sulfate partitioning or sieving because the theories for concentrated solutions that were available at that time were limited to single solutes. The recent results of Lazzara et al. (17), which may be applied to any number of spheroidal solutes, make it possible to further analyze the data and test whether the predicted effect of BSA on partitioning is sufficient to explain its influence on Ficoll and Ficoll sulfate sieving in GBM. That was the objective of the work reported here.

The paper is organized as follows. First, there is a discussion of the relationship between the Θ and Φ, including the effects of filtrate velocity and the novel behavior caused by the presence of an abundant solute. The key partitioning relationships from Lazzara et al. (17) are then summarized, to complete the description of the theory. After some general results are presented to illustrate the effects of solute concentration and filtrate velocity on Θ, a comparison is made between the theoretical predictions and the data for the GBM. We conclude with predictions of the effects of mixed solutions of proteins (e.g., serum albumin and globulins) and with a discussion of the physiological significance of this phenomenon. As will be explained, abundant, poorly filtered proteins such as albumin are likely to influence the sieving behavior of test macromolecules in vivo in much the same way that they influence their sieving in isolated GBM.

## MODEL DEVELOPMENT

### Relationship Between Sieving and Partitioning

In an isotropic medium, such as an array of randomly oriented fibers, the flux (**N**) of a macromolecular solute may be expressed as
Equation 1where *D*
_{∞} is the solute diffusivity in free solution, **v** is the fluid velocity vector, C is the solute concentration, and *K*
_{d} and*K*
_{c} are hindrance factors for diffusion and convection, respectively. 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, although*K*
_{c} may exceed unity for small solutes. This has been demonstrated, for example, for Ficoll and globular proteins in agarose gels (14, 16). Consider a membrane extending from*x* = 0 to *x* = *L* that is in contact with solutions of concentration C_{0} and C_{L}, respectively. For steady transport in the*x* direction, integration of *Eq. 1
* reveals that the solute flux is related to the external concentrations and filtrate velocity by
Equation 2where Φ_{0} and Φ_{L} are the equilibrium partition coefficients at the upstream and downstream surfaces, respectively, and Pe is the membrane Péclet number. The Φ is the concentration just inside the GBM, divided by that in the adjacent external solution. Pe is
Equation 3An implicit assumption in *Eq. 2
* is that there is an approximate thermodynamic equilibrium between the membrane and the external solutions at *x* = 0 and *x* =*L*. In ultrafiltration, the filtrate concentration is determined by the ratio of the solute and volume fluxes (i.e., C_{L} = N/*v*), and the membrane Θ is defined as Θ = C_{L}/C_{0}. These substitutions allow *Eq. 2
* to be rearranged as
Equation 4Similar expressions for Θ have been employed in many studies of ultrafiltration across synthetic or biological membranes. The one novel feature of *Eq. 4
* is the distinction between the upstream and downstream partition coefficients. Whereas with dilute solutions Φ_{0} = Φ_{L}
*,* an abundant solute in the retentate will tend to make Φ_{0} > Φ_{L} for the tracer.

The effects of filtrate velocity are described by the term in *Eq.4
* that contains Pe. With high filtrate velocities and/or thick membranes, such that Pe ≫ 1, we obtain
Equation 5In this limit, Θ depends on the upstream Φ and convective hindrance factor and is insensitive to filtrate velocity. This standard result is often expressed in terms of a reflection coefficient (ς), where Φ_{0}
*K*
_{c} = 1 − ς. Because Φ_{0}
*K*
_{c} < 1, we expect that Θ < 1 for any macromolecular tracer if the filtrate velocity is large enough. The limit for low filtrate velocities and/or thin membranes is
Equation 6In contrast to the usual result of Θ → 1 for Pe → 0, Θ is determined now by the ratio of the partition coefficients. Because a large, abundant solute will tend to make Φ_{0} > Φ_{L} , the Θ of an uncharged tracer could exceed unity. Although perhaps counterintuitive, this prediction has a firm physical basis. In general, *Eqs. 4
*-*
6
*indicate that an abundant solute will increase Θ of a tracer (Θ_{T}) at all values of Pe by increasing Φ_{0}. The extent of the increase will depend also on*K*
_{c} and Pe.

### Effects of Concentration on Partitioning

The effects of solute concentrations on partition coefficients were modeled, using the excluded volume theory of Lazzara et al. (17). In that theory, partition coefficients are calculated by summing the volumes excluded to a solute in the membrane and bulk phases due to the fixed structures of the membrane and to other solute molecules that may be present. Long-range intermolecular forces are ignored, limiting this method to media where steric considerations dominate and the effects of electric charge are negligible. As indicated earlier, this appears to be a valid approximation for GBM. The model generates a coupled set of nonlinear algebraic equations for the partition coefficients, one for each solute present. The most complicated situation to be considered here is a three-solute system of Ficoll, serum albumin, and IgG partitioning into a fibrous membrane composed of two distinct types of fibers. We will treat Ficoll and IgG as spherical molecules and BSA as a prolate spheroid. In the equations that follow, the solute *indexes 1*, *2*, and *3* refer to Ficoll, serum albumin, and IgG, respectively; there are also *indexes 1* and*2* for the two types of fibers. If we use the notation of Lazzara et al. (17), the expressions for the partition coefficients in such a system are
Equation 7a
Equation 7b
Equation 7c
where φ_{i} denotes the volume fraction of fibers of type *i* in the membrane, and χ_{j} denotes the volume fraction of solutes of type *j* in the bulk solution. The quantities*α _{ij}(x,y)* are dimensionless geometric parameters that are used to describe the interaction of a test solute

*i*of shape

*x*with a set of objects

*j*of shape

*y*(s = sphere, p = prolate spheroid, f = fiber). For example,

*α*

_{11}

*(*s,s

*)*describes the steric interaction between two spheres of

*type 1*. For spheres of radius

*r*and

_{i}*r*, the excluded volume parameter is Equation 8Expressions for the remaining

_{j}*α*,

_{ij}(x*y)*parameters, some of which are quite lengthy, can be found in Lazzara et al. (17). Once those parameters were specified,

*Eqs.7a*–

*7c*were solved, using Newton-Raphson iteration, with the dilute solution values for the partition coefficients as initial guesses.

All other systems considered here may be viewed as special cases of*Eqs. 7a
*–*
7c
*, obtained by setting certain terms equal to zero. Thus the dilute solution values of the partition coefficients were found by setting χ_{j} = 0 for all *j*. Results for just two solutes (Ficoll and albumin) were computed by setting χ_{3} = 0 and dropping *Eq. 7c
*. Calculations for membranes with just one type of fiber were done by setting φ_{2} = 0. Among the results that may be recovered in this manner is the partitioning expression of Ogston (25) for dilute solutions of spheres in random arrays of a single type of fiber; it corresponds to *Eq.7a
* with φ_{2} = 0 and all χ_{j} = 0.

## RESULTS AND DISCUSSION

### General Trends

Examples of the theoretical increase in the Φ of a spherical tracer due to BSA are shown in Fig. 2. The tracer Φ is denoted as Φ_{T}, and the φ of BSA in bulk solution is χ_{BSA}. These results were computed for a hypothetical fiber array with a volume fraction of φ = 0.2 and a fiber radius of *R*
_{f} = 10 Å. Those parameter values were selected so that BSA would be largely excluded from the membrane (Φ_{BSA} ≅ 0.01 for dilute solutions), as is true for GBM; otherwise, the choices are arbitrary. As discussed previously (17), BSA was represented as a prolate spheroid with an axial ratio of *η* = 3.3 (major and minor semiaxes of 70 and 21 Å, respectively). With this assumed shape, the mass concentration that corresponds to χ_{BSA} = 0.1 is 8.6 g/dl. Results are shown for tracers with *r*
_{s} = 20, 30, 40, and 50 Å. It is seen that Φ_{T} increases with increasing χ_{BSA} in each case. The greatest percentage variations in Φ_{T} were obtained for the largest molecule, where the dilute-solution partition coefficient (Φ_{T} for χ_{BSA} = 0) was smallest. These results demonstrate that the effect of an abundant solute on Φ_{T} can be quite large.

Figure 3 shows the predicted effects of BSA on the Θ of a spherical tracer with*r*
_{s} = 30 Å, for the same fiber matrix as in Fig. 2. As shown in *Eq. 4
*, which was applied here to both the tracer and BSA, Θ depends on *K*
_{c} and Pe, as well as Φ. Because there is not yet a reliable theory for predicting the *K*
_{c} in a random fiber matrix (16), *K*
_{c} = 0.75 was used as a representative value, both for the tracer and for BSA.^{1} Results were computed for a wide range of Pe values and BSA concentrations in the retentate. As seen in Fig. 3, Θ_{T} is predicted to be elevated as the BSA concentration is increased, for any fixed value of Pe > 0. Note that increasing the BSA concentration would also tend to decrease Pe, because of the osmotic pressure opposing filtration. Thus the usual effect of adding BSA would be to move toward the higher curves in Fig.3, making Θ_{T} even more sensitive to the BSA concentration. In contrast to the situation discussed in connection with *Eq. 6
*, Θ_{T} in Fig. 3 does not exceed unity even for Pe = 0 and large concentrations of BSA. The reason is that, with identical assumed values of *K*
_{c} for the tracer and BSA, and with their roughly comparable molecular sizes, Θ_{BSA}→ 1 as Pe → 0, much as Θ_{T} → 1. As the BSA concentration in the filtrate approaches that in the retentate, the tracer partition coefficients at the two membrane surfaces become equal. With Φ_{L} → Φ_{0}, *Eq.6
* indicates that Θ_{T} → 1, consistent with the behavior in Fig. 3 for Pe → 0. For Θ_{T} to exceed unity at small Pe, BSA (or another abundant protein) would have to be excluded from the membrane much more efficiently than the tracer.

### Sieving in Isolated GBM

To predict the effects of BSA on Ficoll and Ficoll sulfate sieving in isolated GBM, values were needed for the convective and diffusive hindrance factors for the range of molecular sizes studied by Bolton et al. (5). Because it was not possible to measure*K*
_{c} and *K*
_{d} independently under those experimental conditions, and because there is not yet a reliable theory for predicting hindrance factors in a material as complex as GBM, we elected to estimate the necessary quantities by fitting the Ficoll and Ficoll sulfate sieving curves measured in the absence of BSA. With protein-free solutions, the partition coefficients at the two sides of the membrane are equal; that is, Φ_{0} = Φ_{L} = Φ.*Equations 3
* and *
4
* indicate that knowledge of the products Φ*K*
_{c} and Φ*K*
_{d}is sufficient to find the Pe and Θ. Both of these products are expected to decline from values of near unity for very small molecules to nearly zero for large molecules. Accordingly, the empirical forms chosen for the fitting were^{2}
Equation 9a
Equation 9bThe constants *a* and *b* were evaluated by using Powell's method to minimize the norm of the error between the data and Θ predicted using *Eq. 4
*. The resulting values were *a* = 0.126 Å^{−1} and *b*= 0.075 Å^{−1} for Ficoll and *a* = 0.134 Å^{−1} and *b* = 0.069 Å^{−1} for Ficoll sulfate. The nearly identical values of *a* and*b* computed for Ficoll and Ficoll sulfate reflect the fact that the sieving curves of these neutral and anionic tracers in GBM were indistinguishable. The values of *a* and *b*given here for Ficoll differ slightly from those reported by Bolton et al. (5). The reason is that, in the present work, an effort was made to correct for nonselective “shunts” or “leaks” in the filters made by consolidating cell-free glomeruli. This was done by subtracting from each Φ the value measured for the largest Ficoll or Ficoll sulfate studied, where *r*
_{s} = 80 Å. Although this had only a modest effect on the results to be shown for 20 ≤ *r _{s}
* ≤ 50 Å, the “corrected” Θ are the ones plotted.

The central element of the theory used to predict the effects of BSA on Ficoll and Ficoll sulfate sieving was the partitioning model. It has been argued recently that representing GBM as a randomly oriented array of uniform fibers fails to account for its electron microscopic appearance, its measured volume fraction of solids, and its measured hydraulic (or Darcy) permeability (4). However, the assumption that it consists of a mixture of coarse and fine fibers, which correspond roughly to collagen IV and glycosaminoglycan chains, leads to behavior consistent with all of those properties. Accordingly, we adopted a two-fiber model with parameter values as suggested in Bolton and Deen (4): the radii of the coarse and fine fibers were taken to be 35 and 5 Å, respectively; the corresponding volume fractions were 0.046 and 0.054, for a total solid fraction of 0.10. Also needed for the partitioning calculations are the concentrations of BSA at the upstream and downstream surfaces of the GBM layer studied in vitro. Correcting the retentate value for concentration polarization and using the measured sieving coefficient for BSA (Θ_{BSA} = 0.085) (5), the upstream and downstream concentrations were found to be 6.2 and 0.53 g/dl, respectively. With BSA represented as a prolate spheroid, as described above, the corresponding volume fractions are χ_{BSA} = 0.072 and 0.0061. Determining its concentrations in this manner from experimental data, it was not necessary to specify *K*
_{c} and*K*
_{d} for BSA.

Theoretical sieving curves are compared with the GBM data for Ficoll and Ficoll sulfate in Figs. 4 and5, respectively. As shown by the*bottom* curves in each plot, the simple expressions adopted for the hindrance factors (*Eqs. 9a
* and *
9b
*) yielded excellent fits to the sieving data obtained in the absence of BSA. Also shown in Figs. 4 and 5 are the respective sieving curves measured in the presence of BSA and two predictions for that case. One prediction includes only the osmotic effect of BSA. In those calculations, BSA was assumed to reduce Pe (due to the lower filtrate velocity) without affecting Φ_{T}. As shown in both figures, and as noted in Bolton et al. (5), this purely osmotic effect of BSA accounts for only about one-third of the upward shift in the sieving curves. The remaining curves in each plot are based on the complete theory, including both partitioning and osmotic effects. It is seen that the predicted effect of BSA is more than sufficient to account for the upward shifts in the Ficoll and Ficoll sulfate sieving curves. The tendency of the theory to overestimate the effect of BSA, especially for the largest solutes, might be the result of limitations in the representation of the GBM as an array of randomly oriented fibers. Indeed, although glycosaminoglycan chains (and possibly other components) may be relatively disordered, there is evidence from electron microscopy that collagen IV fibers assemble into a branching polygonal network in at least some basement membranes (36). Thus it might be more accurate to model GBM as a partially ordered fibrous structure filled with smaller, randomly oriented fibers. Predictions for such mixed structures, however, are beyond the capabilities of current partitioning theories.

### Effects of Protein Size and Shape

Although our calculations have focused on BSA, any abundant protein should influence the partition and sieving coefficients of tracer macromolecules. This leads to the question of whether protein size and/or shape are important factors. This was examined in two ways: first, to see whether modeling BSA as a sphere would alter the predictions in Figs. 4 and 5 and, second, to see whether a mixture of albumin and globulins would behave differently than an albumin solution.

In the preceding calculations BSA was treated as a prolate spheroid with an axial ratio of 3.3 and major and minor semiaxes of 70 and 21 Å. This model appears to be most consistent with its partial specific volume (1), intrinsic viscosity (32), and*r*
_{s} (13, 17). However, a much simpler representation is a sphere of radius*r*
_{s} = 36 Å (*r*
_{s} of BSA). If the spherical model is adopted, then χ_{BSA} = 0.1 corresponds to a mass concentration of 5.8 g/dl. Repeating the calculations in Figs. 4 and 5 for a spherical BSA molecule resulted in curves that were virtually indistinguishable from those for the prolate spheroid. Thus the shape of BSA does not appear to be an important determinant of its effect on the partitioning of tracers in GBM, for the protein concentrations considered here. This is not a general finding, in that molecular shape has been shown to influence the effects of solute concentration on partitioning in other hypothetical situations (17).

To examine the effects of a protein mixture, we simulated partitioning into GBM from a BSA solution or a “plasma” represented as a 1:1 mixture (by mass) of BSA and IgG. Once again, the two-fiber GBM model was employed, and BSA was treated as a prolate spheroid. For simplicity, we did not attempt to model the “Y” shape of IgG, representing IgG instead as a sphere of 52-Å radius (29). The results are shown in Fig. 6 as plots of Φ_{T} vs. tracer size for various protein solutions. The presence of BSA at 6 g/dl is predicted to roughly double Φ_{T} of intermediate size. For the smaller molecules the percent changes are lower than for the larger molecules. Interestingly, if the total protein consists of 3 g/dl albumin and 3 g/dl IgG, the predicted partition coefficients are barely distinguishable from those for 6 g/dl albumin. This suggests that, from a partitioning viewpoint, a good experimental model for plasma is a BSA solution with a mass concentration that matches that of total plasma protein. (Such a solution is less accurate from an osmotic viewpoint, in that the osmotic pressure of BSA exceeds that of mixed plasma proteins, for a given mass concentration.) If the total protein content is reduced to 3 g/dl (either BSA or an albumin-IgG mixture), the augmentation of the Φ is very nearly one-half that for 6 g/dl. Thus the effects of abundant proteins on partitioning in the GBM are predicted to be nearly linear in the protein concentration.

### Application to Intact Capillaries

Attempts to extrapolate these findings to glomerular filtration in vivo are complicated by the fact that the barrier properties of the capillary wall are determined only partly by the GBM. The various factors to be considered will be identified first, and then some conclusions will be reached concerning filtration in intact capillaries. The overall sieving coefficient at any point along a glomerular capillary (i.e., Θ; the concentration in Bowman's space relative to plasma) depends on two kinds of quantities. First, there are the individual sieving coefficients for each of the three layers of the capillary wall: Θ_{en} for the endothelial fenestrae, Θ_{bm} for the GBM, and Θ_{ep} for the epithelial filtration slits. As used here, Θ_{i} is the concentration at the downstream edge of layer *i* divided by that at the upstream edge. These are “internal” Θ in the sense that the upstream and downstream concentrations are evaluated just inside the layer under consideration. As exemplified by *Eq.4
*, these sieving coefficients are dynamic quantities that depend on filtrate velocity, as well as on the respective diffusive and convective hindrance factors and thicknesses of the layers. Second, there are equilibrium partition coefficients that describe the step changes in concentration that occur at the phase boundaries. At the boundary between layer *i* and layer *j*, we denote the concentration in *i* divided by that in *j* as Φ_{i/j}. Of importance, Φ_{i/j} depends not just on the structural characteristics of layers *i* and *j*, such as their pore sizes or fiber spacings, but also on the concentration of albumin (or other abundant proteins) in the vicinity of the boundary. With these definitions, the overall sieving coefficient is given by
Equation 10where subscripts p and b denote plasma and Bowman's space, respectively. Thus seven quantities are needed to describe the concentration changes that occur across the three layers and at the four boundaries, as one moves from plasma to Bowman's space. If the partition coefficients were not affected by the local protein concentrations within the glomerular capillary wall, then their concentration ratios would cancel^{3} and *Eq. 10
*would simplify to
Equation 11as used previously (8). Thus it is the steric effect of proteins on tracer partitioning that requires the four additional terms in the more general expression.

Among the many possibilities that can be imagined, in which proteins might affect any or all of the four partition coefficients in *Eq.10
*, we focus now on two of the more likely scenarios. Both are motivated by the finding of Ryan and Karnovsky (31) that albumin is almost completely excluded from the GBM. Thus the common aspect of the two scenarios is the assumption that almost no protein reaches the downstream side of the GBM and the filtration slits, from which it follows that Φ_{ep/bm}Φ_{b/ep} = Φ_{b/bm} = 1/Φ_{bm/b}, where Φ_{bm/b} is the partition coefficient that would apply if the GBM were in direct contact with Bowman's space (or simply water). Suppose, now, that albumin passes freely through the endothelial fenestrae and that the limiting step for it is entry into the GBM. In other words, assume that the fenestrae act only as wide, water-filled channels. This assumption corresponds to Φ_{en/p} = Θ_{en} = 1 and Φ_{bm/en} = Φ_{bm/p}. Thus for water-filled fenestrae, *Eq. 10
*reduces to
Equation 12The steric effect of albumin (and other retained proteins) would be to make Φ_{bm/p}/Φ_{bm/b} > 1. Accordingly, in this scenario the effect of albumin on the overall Θ will closely resemble its effect on isolated GBM, as already described.

Alternatively, one could assume that the endothelial glycocalyx is the limiting barrier and that only the upstream sides of the fenestrae are exposed to protein. For this situation, algebraic manipulations like those above reduce *Eq. 10
* to
Equation 13The first term is similar to *Eq. 12
*, except that the partition coefficients are now those for the fenestral glycocalyx. Because the steric effect of abundant proteins on tracer partitioning will be directionally similar for any porous or fibrous material, we expect that Φ_{en/p}/Φ_{en/b} > 1. Thus for either of the situations represented by *Eqs. 12
* and *
13
*, the effects of abundant proteins on partitioning will be to increase the overall sieving coefficient of a tracer macromolecule.

As *Eqs. 12
* and *
13 *were obtained, it was assumed that the limiting barrier for albumin and other abundant proteins was upstream of the GBM. However, qualitatively similar trends are predicted if the limiting barrier is at the level of the slit diaphragm. In other words, no matter what the limiting barrier is for the protein, there will be a tendency for an abundant, poorly filtered protein to augment Θ_{T}. Although the location of the protein barrier does not influence the direction of the effect, it will determine its magnitude. If the effect is mediated by partitioning in the GBM, it can be estimated from our analysis of sieving data for isolated GBM. If it is mediated by partitioning elsewhere (e.g., between plasma and glycocalyx), then the paucity of information on material properties makes its magnitude more uncertain.

The likelihood that steric interactions with plasma proteins elevate tracer sieving coefficients has an interesting implication for studies of human disease. That is, it suggests that the low plasma protein concentrations characteristic of the nephrotic syndrome will tend to mask some of the glomerular injury revealed by fractional clearance measurements with tracer molecules such as Ficoll. Although the Ficoll (or dextran) sieving coefficients in nephrotic subjects might still be much higher than in healthy individuals, they will not be as high as if plasma protein levels were normal. In this sense, the true extent of the injury will be partly concealed. Similarly, variations in perfusate protein concentration in studies using the isolated perfused kidney (26-28) complicate efforts to assess the intrinsic size selectivity of the barrier. The steric effects we have described would cause apparent (calculated) pore radii to increase with increasing protein concentration, even without any structural change in the capillary wall.

### Other Effects of Proteins

This paper has focused mainly on the idea that steric interactions with plasma proteins tend to elevate the glomerular Θ_{T}. Such steric effects are entirely physical and nonspecific and will be present to varying degrees with any globular protein and any ultrafiltration membrane. Several other effects of proteins on microvascular permeability have been reported, some of them quite specific. The glycoprotein orosomucoid has been shown to influence the permeability of both glomerular and peripheral capillaries by maintaining charge selectivity (7, 11, 12, 15). Studies using frog mesenteric capillaries have revealed effects of albumin itself: omitting albumin from perfusates increased the hydraulic permeability and decreased the reflection coefficients for Ficoll (19, 22, 23). Specificity was demonstrated by showing that the effect was abolished by chemical modification of arginine residues of albumin (23). It was hypothesized that albumin (and also ferritin) might influence capillary permeability by ordering the fibers of the glycocalyx (21). Lowered protein concentrations have been shown to increase the permeability of capillaries in a variety of other vascular beds (18, 20, 30,34). In contrast, micropuncture studies in rats have shown that decreases in plasma protein concentration reduce the glomerular ultrafiltration coefficient, the product of hydraulic permeability and surface area for filtration (3, 33). The underlying mechanism for this remains unknown, but the observation that BSA did not affect the hydraulic permeability of isolated GBM (5) suggests involvement of endothelial cells and/or epithelial foot processes, rather than the GBM.

### Conclusions

The theory presented here suggests that BSA (or other abundant proteins) can markedly increase the sieving coefficients of tracer macromolecules in the GBM, largely as a consequence of steric interactions that favor tracer partitioning into the membrane. The predicted effect of these steric interactions, combined with the osmotic effect of BSA, is large enough to account for the marked elevation of Ficoll sieving coefficients in isolated GBM when BSA is present, reported previously (5). The magnitude of this protein effect is predicted to be less dependent on protein size and shape than it is on the total concentration of protein. It is a factor that should be taken into account in efforts to characterize the intrinsic barrier properties of the glomerular capillary wall.

## Acknowledgments

This work was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-20368.

## Footnotes

↵1 For simplicity, the values of

*K*_{c}and Pe for BSA in Fig. 3 were chosen to be equal to those of the 30-Å tracer. Because*r*_{s}= 36 Å for BSA, its*K*_{c}in the same fiber matrix would probably be smaller than that of a 30-Å sphere. Additionally, one would expect that Pe for BSA would be larger than that of a 30-Å sphere, in part because of the reduced value of*D*_{∞}and perhaps also because of larger values of the ratio*K*_{c}/*K*_{d}.*Equation4*indicates that using too large a*K*_{c}and/or too small a Pe for BSA would increase its predicted Θ. In other words, the filtrate concentration of BSA was probably overestimated. Because anything that tends to reduce the transmembrane concentration difference for BSA also tends to minimize its effect on tracer sieving, the effects shown in Fig. 3 should be viewed as conservative estimates.↵2 The known limiting behavior of Φ

*K*_{d}and Φ*K*_{c}for point-sized solutes in random arrays of fibers can be incorporated into*Eqs. 9a*and*9b*by changing the preexponential coefficients from unity to [1 − (5/3)φ] and [1 − φ], respectively. Using these modified expressions had virtually no effect on the ability to fit the sieving data without BSA or on the predicted sieving curves with BSA. Thus, although the modified forms are more exact for*r*_{s}→ 0, that had little consequence for the range of molecular sizes studied here.↵3 In the absence of protein effects, the partition coefficients obey relationships of the form Φ

_{i}_{/j}Φ_{j}_{/k}= Φ_{i}_{/k}. The cancellation of terms in*Eq. 10*follows from that and the fact that, without proteins, Φ_{b/p}= 1.Address for reprint requests and other correspondence: W. M. Deen, Dept. of Chemical Engineering, 66–572, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139 (E-mail: wmdeen{at}mit.edu).

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