Effects of plasma proteins on sieving of tracer macromolecules
in glomerular basement membrane
Matthew J.
Lazzara1 and
William M.
Deen1,2
1 Department of Chemical Engineering and 2 Division
of Bioengineering and Environmental Health, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139
 |
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
 |
INTRODUCTION |
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 (
; the ratio
of filtrate to retentate concentration) were determined for
Stokes-Einstein molecular radii (rs) ranging from 20 to 50 Å. The principal finding was that the values of
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
vs. rs) 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
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
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
. 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 (
; 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
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
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.1A, solute-solute interactions are negligible and steric
exclusion of the tracer by the fibers causes
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. 1B. 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
for the tracer is still less
than unity, it is larger than for a very dilute solution.

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Fig. 1.
Schematic of the partitioning of a spherical tracer
( ) into a randomly oriented matrix of fibers. In
A, only tracer is present, and its partitioning is
determined by steric interactions with the fibers only. In
B, tracer interactions with the abundant solute
( ) tend to exclude tracer molecules preferentially from
the bulk solution and increase the partition coefficient of the
tracer.
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|
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
and
, 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
, 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
|
(1)
|
where D
is the solute diffusivity in
free solution, v is the fluid velocity vector, C is the
solute concentration, and Kd and
Kc 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 Kd and
Kc to be less than unity, although
Kc 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 C0 and
CL, 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
|
(2)
|
where
0 and
L are the
equilibrium partition coefficients at the upstream and downstream
surfaces, respectively, and Pe is the membrane Péclet number. The
is the concentration just inside the GBM, divided by that in the
adjacent external solution. Pe is
|
(3)
|
An 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.,
CL = N/v), and the membrane
is
defined as
= CL/C0. These
substitutions allow Eq. 2 to be rearranged as
|
(4)
|
Similar expressions for
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
0 =
L, an abundant solute
in the retentate will tend to make
0 >
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
|
(5)
|
In this limit,
depends on the upstream
and convective
hindrance factor and is insensitive to filtrate velocity. This standard
result is often expressed in terms of a reflection coefficient (
),
where
0Kc = 1
.
Because
0Kc < 1, we expect
that
< 1 for any macromolecular tracer if the filtrate
velocity is large enough. The limit for low filtrate velocities and/or
thin membranes is
|
(6)
|
In contrast to the usual result of
1 for Pe
0,
is
determined now by the ratio of the partition coefficients. Because a
large, abundant solute will tend to make
0 >
L , the
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
of a tracer
(
T) at all values of Pe by increasing
0.
The extent of the increase will depend also on
Kc 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
|
(7a)
|
|
(7b)
|
|
(7c)
|
where
i denotes the volume fraction of
fibers of type i in the membrane, and
j denotes the volume fraction of
solutes of type j in the bulk solution. The quantities
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,
11(s,s) describes the
steric interaction between two spheres of type 1. For
spheres of radius ri and
rj, the excluded volume parameter is
|
(8)
|
Expressions for the remaining
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
j = 0 for all j. Results for just two solutes (Ficoll and
albumin) were computed by setting
3 = 0 and
dropping Eq. 7c. Calculations for membranes with just one type of fiber were done by setting
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
2 = 0 and all
j = 0.
 |
RESULTS AND DISCUSSION |
General Trends
Examples of the theoretical increase in the
of a spherical
tracer due to BSA are shown in Fig. 2.
The tracer
is denoted as
T, and the
of BSA in
bulk solution is
BSA. These results were computed for a
hypothetical fiber array with a volume fraction of
= 0.2 and a
fiber radius of Rf = 10 Å. Those parameter
values were selected so that BSA would be largely excluded from the
membrane (
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
= 3.3 (major and minor semiaxes of 70 and
21 Å, respectively). With this assumed shape, the mass concentration
that corresponds to
BSA = 0.1 is 8.6 g/dl. Results
are shown for tracers with rs = 20, 30, 40, and 50 Å. It is seen that
T increases with increasing
BSA in each case. The greatest percentage variations in
T were obtained for the largest molecule, where the
dilute-solution partition coefficient (
T for
BSA = 0) was smallest. These results demonstrate that the effect of an abundant solute on
T can be quite
large.

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Fig. 2.
Partition coefficient of a spherical tracer
( T) as a function of BSA volume fraction
( BSA). Results are shown for various tracer
Stokes-Einstein radii (rs). The volume fraction
of fibers was = 0.2, the fiber radius was
Rf = 10 Å, and BSA was treated as a
prolate spheroid with axial ratio = 3.3.
|
|
Figure 3 shows the predicted effects of
BSA on the
of a spherical tracer with
rs = 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,
depends on Kc and Pe, as
well as
. Because there is not yet a reliable theory for predicting
the Kc in a random fiber matrix
(16), Kc = 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,
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
T even more sensitive to the BSA
concentration. In contrast to the situation discussed in connection
with Eq. 6,
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 Kc for
the tracer and BSA, and with their roughly comparable molecular sizes,
BSA
1 as Pe
0, much as
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
L
0, Eq. 6 indicates that
T
1, consistent with the
behavior in Fig. 3 for Pe
0. For
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.

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Fig. 3.
Sieving coefficient of a spherical tracer
( T) as a function of BSA and
membrane Péclet number (Pe). Results are shown for
rs = 30 Å and the hindrance factor for
convection (Kc) = 0.75, with other
conditions as in Fig. 2.
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|
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
Kc and Kd 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,
0 =
L =
.
Equations 3 and 4 indicate that knowledge of the
products
Kc and
Kd
is sufficient to find the Pe and
. 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 were2
|
(9a)
|
|
(9b)
|
The constants a and b were evaluated by
using Powell's method to minimize the norm of the error between the
data and
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
the value measured for the largest Ficoll
or Ficoll sulfate studied, where rs = 80 Å. Although this had only a modest effect on the results to be shown
for 20
rs
50 Å, the "corrected"
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 (
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
BSA = 0.072 and 0.0061. Determining its
concentrations in this manner from experimental data, it was not
necessary to specify Kc and
Kd for BSA.
Theoretical sieving curves are compared with the GBM data for Ficoll
and Ficoll sulfate in Figs. 4 and
5, 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
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.

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Fig. 4.
Sieving coefficient of Ficoll ( F) as a
function of Ficoll radius (rs) for isolated rat
glomerular basement membrance (GBM). The symbols with error bars
represent the data of Bolton et al. (5). 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.
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Fig. 5.
Sieving coefficient of Ficoll sulfate ( FS)
as a function of rs for isolated rat GBM. The
symbols with error bars represent the data of Bolton et al.
(5). 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.
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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
rs (13, 17). However, a much
simpler representation is a sphere of radius
rs = 36 Å (rs of
BSA). If the spherical model is adopted, then
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
T vs. tracer size for various protein solutions. The
presence of BSA at 6 g/dl is predicted to roughly double
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
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.

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Fig. 6.
T as a function of tracer radius
rs. Results are shown for a protein-free
solution and for solutions containing albumin and/or IgG.
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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.,
; 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:
en for the endothelial fenestrae,
bm for the GBM, and
ep for the epithelial
filtration slits. As used here,
i is the
concentration at the downstream edge of layer i divided by
that at the upstream edge. These are "internal"
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
i/j. Of importance,
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
|
(10)
|
where 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
cancel3 and Eq. 10
would simplify to
|
(11)
|
as 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
ep/bm
b/ep =
b/bm = 1/
bm/b, where
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
en/p =
en = 1 and
bm/en =
bm/p. Thus for water-filled fenestrae, Eq. 10
reduces to
|
(12)
|
The steric effect of albumin (and other retained proteins) would
be to make
bm/p/
bm/b > 1. Accordingly, in this scenario the effect of albumin on the overall
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
|
(13)
|
The 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
en/p/
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
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
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.
 |
ACKNOWLEDGEMENTS |
This work was supported by National Institute of Diabetes and
Digestive and Kidney Diseases Grant DK-20368.
 |
FOOTNOTES |
1
For simplicity, the values of
Kc and Pe for BSA in Fig. 3 were chosen to be
equal to those of the 30-Å tracer. Because
rs = 36 Å for BSA, its
Kc 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
Kc/Kd. Equation 4 indicates that using too large a Kc
and/or too small a Pe for BSA would increase its predicted
. 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
Kd and
Kc 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)
] and [1
], 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 rs
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
i/j
j/k =
i/k. The
cancellation of terms in Eq. 10 follows from that and the
fact that, without proteins,
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).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 29 March 2001; accepted in final form 11 June 2001.
 |
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