A gel-membrane model of glomerular charge and size selectivity in series

Maria Ohlson1, Jenny Sörensson1, and Börje Haraldsson1,2

Departments of 1 Physiology and 2 Nephrology, Göteborg University, Göteborg 405 30, Sweden


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INTRODUCTION
THEORETICAL DEVELOPMENT
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We have analyzed glomerular sieving data from humans, rats in vivo, and from isolated perfused rat kidneys (IPK) and present a unifying hypothesis that seems to resolve most of the conflicting results that exist in the literature. Particularly important are the data obtained in the cooled IPK, because they allow a variety of experimental conditions for careful analysis of the glomerular barrier; conditions that never can be obtained in vivo. The data strongly support the classic concept of a negative charge barrier, but separate components seem to be responsible for charge and size selectivity. The new model is composed of a dynamic gel and a more static membrane layer. First, the charged gel structure close to the blood compartment has a charge density of 35-45 meq/l, reducing the concentration of albumin to 5-10% of that in plasma, due to ion-ion interactions. Second, the size-selective structure has numerous functional small pores (radius 45-50 Å) and far less frequent large pores (radius 75-115 Å), the latter accounting for 1% of the total hydraulic conductance. Both structures are required for the maintenance of an intact glomerular barrier.

capillary permeability; macromolecular transport; two-pore model


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ONE OF NATURE'S WELL KEPT SECRETS is the intricate mechanism underlying the permselectivity of the glomerular barrier. Normally, 180 liters of primary urine per day is produced in healthy humans with minimal losses of proteins. In fact, the causes for proteinuria, which is a hallmark of renal disease, are most often completely unknown. In what are now classic studies, Brenner and associates (2, 5, 11) used dextran to reveal the size-, charge-, and shape-selective properties of the glomerular membrane. However, the established theories of glomerular function have recently been questioned because the use of dextrans may have certain technical drawbacks (6).

Many researchers suggest the basement membrane to be the most significant component of the barrier, whereas others suggest it to be the podocytes. Indeed, there are genetic disorders that affect components in these regions. For example, the newly discovered nephrin has been found to be missing in hereditary nephrosis of the Finnish type (27). However, if the principal barrier were to be anywhere distal to the endothelial wall, this would require active mechanisms for the return of huge amounts of protein. Indeed, such a mechanism of "reuptake" of intact albumin from the proximal tubuli has been proposed (38), but there is in fact no experimental evidence to support it. In contrast, in a classic study, Maunsbach (32) showed that practically all albumin molecules taken up by the proximal tubules are degraded, a finding still considered to be valid.

Most investigators have considered the glomerular wall to behave like a membrane with functional pores of various dimensions, whereas others have used the fiber matrix concept developed by Curry and Michel (8). Alternatively, the barrier can be treated like a gel having both size- and charge-selective properties (54). Such a gel concept can be regarded as an extension of the fiber matrix model, and it has several interesting dynamic features not present in a static membrane. The combined effects of charge and size in a single structure result in highly complex equations. The first steps in such an analysis were recently published for a fiber matrix (23), but present equations for the effects on convection must be improved (26). Indeed, the predictions of the model do not seem to fit biological data (47).

In the present study the experimentally determined transglomerular passage of neutral and charged solutes were analyzed by using various theoretical models. The analysis led to the development of a new integrative gel-membrane model that can predict transport of molecules depending on their charge and size.


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INTRODUCTION
THEORETICAL DEVELOPMENT
EXPERIMENTAL METHODS
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This section begins with a description of the gel-membrane model and how its parameters are determined. The equations for a gel-membrane model of the glomerular capillary wall are then summarized.

The present gel-membrane model is composed of two separate barriers: a negatively charged gel, which acts like an ion-exchange chromatography column, and a second, size-selective membrane with two functional pore pathways. The model assumes that there is no discrimination of molecular size in the charged gel and no discrimination of charge in the porous membrane. In addition, the model requires that the molecules have a rather spherical shape. Under these conditions, the properties of the two components of the glomerular barrier are readily calculated from the fractional clearances (theta ) for neutral solutes with a range of molecular sizes (e.g., Ficoll) and one or more anionic molecules (e.g., albumin).

The following procedure is used for determining the gel-membrane model parameters from clearance data of albumin and Ficoll. First, the experimentally determined theta -ratio for albumin and the size-matched neutral Ficoll are inserted into Eqs. 1-5 below to give the concentration of fixed charges within the gel compartment. Second, calculated values from a two-pore model are fitted to experimentally obtained urine over plasma (U/P) sieving data for Ficoll molecules of different sizes [Stoke-Einstein radius (aSE)12-72 Å] by using a nonlinear regression analysis. There are four unknown parameters in the two-pore model, namely, the small- and large-pore radii (rs and rL), the large-pore fraction of the hydraulic conductance (fL), and the unrestricted pore area over diffusion distance (A0/Delta x). The subsequent equations (7-21) are used in the two-pore model to estimate the theta  as a function of molecular size. Finally, once the parameters of the gel-membrane model have been determined, the fractional clearance for any solute can be predicted from its size and charge.

Estimation of the Charge Density in the Gel

Assume a gel by the glomerular membrane with a homogenous distribution of fixed charges of a certain density, omega . The concentration of an anionic solute will be lower in the gel than in plasma due to ion-ion interactions (for details, consult Refs. 14, 53). Let us, for the sake of simplicity, assume that there are only two ions present, sodium and chloride, leaving the small contribution of other polyanions out of the equations. Electroneutrality requires plasma sodium (NaP) concentration to equal plasma chloride concentration (Clp) plus the charges given by albumin (Alb; zA), where p denotes the concentration in plasma
Na<SUB>p</SUB><IT>=</IT>Cl<SUB>p</SUB><IT>+</IT>−<IT>z</IT><SUB>A</SUB><IT>·</IT>Alb<SUB>p</SUB> (1)
According to the Donnan equation, the intramembraneous distribution of sodium (Namem) and chloride (Clmem) will be
&Dgr;E=<FR><NU>RT</NU><DE>z<SUB>Na</SUB><IT>F</IT></DE></FR><IT>·</IT>ln <FR><NU>Na<SUB>p</SUB></NU><DE>Na<SUB>mem</SUB></DE></FR><IT>=</IT><FR><NU><IT>RT</IT></NU><DE><IT>z</IT><SUB>Cl</SUB><IT>F</IT></DE></FR><IT>·</IT>ln <FR><NU>Cl<SUB>p</SUB></NU><DE><IT>Cl<SUB>mem</SUB></IT></DE></FR> (2A)
where Delta E is the electrical potential difference between plasma and the gel (mem); z is the valence of sodium and chloride, zNa = +1 and zCl = -1; R is the gas constant; and T is the absolute temperature. Equation 2A can thus be reduced to
Na<SUB>p</SUB>Cl<SUB>p</SUB><IT>=</IT>Na<SUB>mem</SUB>Cl<SUB>mem</SUB> (2B)
In a membrane with a charge density of omega , the chloride and sodium concentrations must be in balance
Cl<SUB>mem</SUB><IT>=</IT>Na<SUB>mem</SUB><IT>−&ohgr;</IT> (3)
Moreover, the gel-plasma partitioning of a charged solute, X, with a net charge of zX is directly related to the gel-plasma partitioning of chloride according to
<FR><NU><IT>X</IT><SUB>mem</SUB></NU><DE><IT>X</IT><SUB>P</SUB></DE></FR><IT>=</IT><FENCE><FR><NU>Cl<SUB>mem</SUB></NU><DE>Cl<SUB>P</SUB></DE></FR></FENCE><SUP><IT>−z</IT><SUB>X</SUB></SUP> (4)
Thus, if the chloride concentration ratio is 0.9 and X has a net charge of -20, then Xmem/Xp equals 0.920; i.e., the gel concentration of X is 12% of that in plasma. Hence the theta  of two differently charged solutes of similar size, e.g., albumin and Ficoll36Å (F), is related to the gel-plasma partitioning of chloride according to
&thgr;<SUB>ratio</SUB><IT>=</IT><FR><NU><IT>&thgr;</IT><SUB>Alb</SUB></NU><DE><IT>&thgr;</IT><SUB>F</SUB></DE></FR><IT>=</IT><FR><NU><FENCE><FR><NU>Cl<SUB>mem</SUB></NU><DE>Cl<SUB>p</SUB></DE></FR></FENCE><SUP><IT>−z</IT><SUB>Alb</SUB></SUP></NU><DE><FENCE><FR><NU>Cl<SUB>mem</SUB></NU><DE>Cl<SUB>p</SUB></DE></FR></FENCE><SUP><IT>−z<SUB>F</SUB></IT></SUP></DE></FR> (5)
To calculate the properties of the gel compartment, we must know NaP, ClP, and Albp, the experimentally determined fractional clearance ratio of albumin and Ficoll (theta ratio), and the net charge of these solutes (zAlb and zF). For albumin, the net charge, zAlb, is -23 (17), whereas it is 0 for Ficoll (zF = 0) or -1 for FITC-Ficoll (3). There are three unknown parameters: the omega  of the membrane, Namem, and Clmem. Thus a unique solution can be found by solving Eqs. 2B, 3, and 5.

The analysis shows that there is an imbalance between the concentrations of free ions in the gel and those in plasma. This imbalance will give rise to an electroosmotic pressure (pi el) (53) that amounts to
&pgr;<SUB>el</SUB><IT>=R·T·&Dgr;</IT>C<SUB>Na<IT>+</IT>Cl</SUB><IT>≈19.33</IT> (6A)

<IT>·</IT>[(Na<SUB>mem</SUB><IT>+</IT>Cl<SUB>mem</SUB>)<IT>−</IT>(Na<SUB>p</SUB><IT>+</IT>Cl<SUB>p</SUB>)]
where pi el is given in millimeters mercury for ion concentrations (in mM). As the gel omega  increases, pi el increases dramatically. At a omega  of 30 meq/l, the pi El is 29 mmHg compared with 690 mmHg at omega  of 150 meq/l. The following empirical function describes closely the relationship between pi el (in mmHg) and omega  (in meq/l) in our perfused kidneys
&pgr;<SUB>el</SUB>(<IT>&ohgr;</IT>)<IT>=0.0299&ohgr;<SUP>2</SUP>+0.119&ohgr;</IT> (6B)
Once the omega  has been determined, the gel-plasma partitioning can be predicted for any solute on the basis of its net charge by using Eq. 4. Thus the concentration of a negatively charged solute will be lower in the gel than in plasma. The size-selective porous membrane will further reduce its concentration before the molecule reaches Bowman's space. Hence, the effect of negative molecular charge is to reduce the concentration of the solute before it reaches the pore entrance.

Estimation of the Size Selectivity

There are several theoretical models that describe the effects of size on the passage of molecules across a membrane. In general, the solutes are assumed to be spherical and uncharged. In this context, aSE is used to describe solute size, because it is derived from the free diffusion constant. The experimental data were analyzed with the following size-selective models: one- or two-pore analysis, normal or lognormal pore distribution models with or without shunts, and a fiber matrix model with or without a shunt pathway. The various models gave qualitatively similar results as long as they were heteroporous, whereas the homoporous equations were unable to simulate the biological data. Therefore, the least complex heteroporous model, i.e., the two-pore model (42), was chosen for further analysis.

For two-pore analysis, experimental sieving data for Ficoll molecules of different radii (12-72 Å, ~100-250 data pairs) are compared with the modeled data by using a set of physiological equations. As mentioned above, the four unknown parameters (rs, rL, fL, and A0/Delta x) are determined by iterative calculations using nonlinear regression analysis. The procedure requires reasonable start values for the four parameters, which will be modified by the nonlinear regression analysis to give the best fit between experimental and modeled data. The best fit was quantitated by using the residual square sum or using Powell's method (see Ref. 36 for details).

First, the fluxes of fluid and solutes are calculated separately for each pore pathway in heteroporous membrane. Thus the total hydraulic conductance, LpS, equals that for the small (s) and the large (L) pores
Lp<IT>S=</IT>Lp<IT>S</IT><SUB>s</SUB><IT>+</IT>Lp<IT>S</IT><SUB>L</SUB><IT>=</IT>(<IT>1−</IT>f<SUB>L</SUB>)<IT>·L</IT>p<IT>S+</IT>f<SUB>L</SUB><IT>·L</IT>p<IT>S</IT> (7)
where fL is the fraction of LpS accounted for by the large-pore pathway. Because there only are two pore pathways, fs + fL = 1. Moreover, the reflection coefficient, sigma , gives important information about the membrane properties. It approaches unity for an ideal semipermeable membrane and is close to zero for extremely large pores when there is no restriction at all. For a heteroporous membrane, the average reflection coefficient, <A><AC>&sfgr;</AC><AC>&cjs1171;</AC></A>, is given by
<A><AC>&sfgr;</AC><AC>&cjs1171;</AC></A>=f<SUB>s</SUB><IT>&sfgr;</IT><SUB>s</SUB><IT>+</IT>f<SUB>L</SUB><IT>&sfgr;</IT><SUB>L</SUB> (8)
where the individual pore sigma  values are calculated by using Eq. 14 below.

LpS can be estimated from
Lp<IT>S=</IT><FR><NU>GFR</NU><DE><IT>&Dgr;</IT>P<IT>−<A><AC>&sfgr;</AC><AC>&cjs1171;</AC></A>·&pgr;</IT><SUB>p</SUB></DE></FR> (9)
where Delta P is the hydrostatic pressure difference across the glomerular barrier, <A><AC>&sfgr;</AC><AC>&cjs1171;</AC></A> is the average reflection coefficient for proteins, and pi p is the colloid osmotic pressure of the plasma proteins, assuming the oncotic pressure in Bowman's space to be zero. Delta P can be experimentally determined, but its absolute value has little impact on the parameters in the two-pore model (Ohlson M, Sörensson J, Lindström K, Blom A, Fries E, and Haraldsson B, unpublished observations).

The fluid flux, JV, can thus be estimated for each pore pathway as
J<SUB>v<SUB>s</SUB></SUB><IT>=</IT>(<IT>1−</IT>f<SUB>L</SUB>)<IT>·L</IT><SUB>p</SUB><IT>S·</IT>[<IT>&Dgr;</IT>P<IT>−<A><AC>&sfgr;</AC><AC>&cjs1171;</AC></A></IT><SUB>s</SUB><IT>·&pgr;</IT><SUB>p</SUB>] (10)

J<SUB>v<SUB>L</SUB></SUB><IT>=</IT>f<SUB>L</SUB><IT>·L</IT><SUB>p</SUB><IT>S·</IT>[<IT>&Dgr;</IT>P<IT>−<A><AC>&sfgr;</AC><AC>&cjs1171;</AC></A></IT><SUB>L</SUB><IT>·&pgr;</IT><SUB>p</SUB>] (11)
Second, the clearance for a solute (Cl) is calculated separately for each pore pathway by using the following nonlinear flux equation (42)
Cl<IT>=</IT><FR><NU><IT>J</IT><SUB>V</SUB><IT>·</IT>(<IT>1−&sfgr;</IT>)</NU><DE><IT>1−&sfgr;·e</IT><SUP><IT>−</IT>Pe</SUP></DE></FR> (12)
Where Pe is the Peclet number that describes the relative contribution of diffusion and convection
Pe<IT>=</IT><FR><NU><IT>J</IT><SUB>V</SUB><IT>·</IT>(<IT>1−&sfgr;</IT>)</NU><DE><IT>PS</IT></DE></FR> (13)
For each pore pathway, sigma  and PS must be determined. This is readily done for spherical molecules (with given aSE) in cylindrical pores, because the restriction factors are described by the following equations using the ratio of solute radius to pore radius (lambda ) (4, 12). Other often used equations (see Ref. 15, for instance) give similar results.

Thus sigma  over a wide range of lambda  values is given by
&sfgr;=1−W(&lgr;) (14)
where the restriction for convective solute transport, W(lambda ), is
W(&lgr;):= <FR><NU><IT>K</IT><SUB>S</SUB>(<IT>&lgr;</IT>)<IT>·&phgr;</IT>(<IT>&lgr;</IT>)<IT>·</IT>[<IT>2−&phgr;</IT>(<IT>&lgr;</IT>)]</NU><DE><IT>2·K</IT><SUB>t</SUB>(<IT>&lgr;</IT>)</DE></FR> (15)
Thus the hindrance factor for convective solute transport, W (as well as the restriction for diffusion, H; see Eq. 20), approaches unity as lambda  right-arrow 0 and approaches zero as lambda  right-arrow 1 (complete exclusion from the pores). The partition coefficient, phi (lambda ), in Eq. 15 (and Eq. 20) describes the effects of steric exclusion from the membrane pores and is expressed as
&phgr;(&lgr;):= (<IT>1−&lgr;</IT>)<SUP><IT>2</IT></SUP> (16)
KS(lambda ) and Kt(lambda ) have empirically (4) been described as
K<SUB>t</SUB>(<IT>&lgr;</IT>):= <FR><NU><IT>9</IT></NU><DE><IT>4</IT></DE></FR><IT>·&pgr;<SUP>2</SUP>·</IT><RAD><RCD><IT>2</IT></RCD></RAD><IT>·</IT>(<IT>1−&lgr;</IT>)<SUP><IT>−5/2</IT></SUP><IT> · </IT><FENCE><IT>1−</IT><FR><NU><IT>73</IT></NU><DE><IT>60</IT></DE></FR><IT>·</IT>(<IT>1−&lgr;</IT>)<IT>+</IT><FR><NU><IT>77,293</IT></NU><DE><IT>50,400</IT></DE></FR><IT>·</IT>(<IT>1−&lgr;</IT>)<SUP><IT>2</IT></SUP></FENCE> (17)

<IT>−22.5083−5.6117·&lgr;−0.3363·&lgr;<SUP>2</SUP>−1.216 · &lgr;<SUP>3</SUP>+1.647·&lgr;<SUP>4</SUP></IT>

K<SUB>S</SUB>(<IT>&lgr;</IT>):= <FR><NU><IT>9</IT></NU><DE><IT>4</IT></DE></FR><IT>·&pgr;<SUP>2</SUP>·</IT><RAD><RCD><IT>2</IT></RCD></RAD><IT>·</IT>(<IT>1−&lgr;</IT>)<SUP><IT>−5/2</IT></SUP><IT>·</IT><FENCE><IT>1+</IT><FR><NU><IT>7</IT></NU><DE><IT>60</IT></DE></FR><IT>·</IT>(<IT>1−&lgr;</IT>)<IT>−</IT><FR><NU><IT>2,227</IT></NU><DE><IT>50,400</IT></DE></FR><IT>·</IT>(<IT>1−&lgr;</IT>)<SUP><IT>2</IT></SUP></FENCE> (18)

<IT>+4.0180−3.9788·&lgr;−1.9215·&lgr;<SUP>2</SUP>+4.392·&lgr;<SUP>3</SUP>+5.006·&lgr;<SUP>4</SUP></IT>
PS (ml/min), required for the calculations of clearance (Eq. 13), is given by
PS(&lgr;)=H(&lgr;)·D<SUB>a<SUB>SE</SUB></SUB><IT>·</IT><FR><NU><IT>A<SUB>0</SUB></IT></NU><DE><IT>&Dgr;x</IT></DE></FR> (19)
where A0/Delta x is the unrestricted pore area over diffusion distance, H(lambda ) is a diffusive restriction factor, and DaSE is the free diffusion constant for the solute. Thus the diffusion of a solute is hindered in the cylindrical pore by the function H(lambda ), which in turn is calculated as
H(&lgr;):= <FR><NU><IT>6·&pgr;·&phgr;</IT>(<IT>&lgr;</IT>)</NU><DE><IT>K</IT><SUB>t</SUB>(<IT>&lgr;</IT>)</DE></FR> (20)
where Kt(lambda ) and phi (lambda ) are defined above. The free diffusion coefficient at 37°C is related to aSE of the solute
D=<FR><NU>R·T</NU><DE>6·&pgr;·&eegr;·N·a<SUB>SE</SUB></DE></FR> (21)
where eta  is viscosity, and N is the Avogadros number. At 37°C, D (in cm2/s) is ~3.28 × 10-6/aSE, with aSE expressed in nanometers.

The total clearance is the sum of the small- and large-pore clearances (Cls + ClL), the calculations of which require separate determinations of Jv, PS, and sigma . Finally, the sieving coefficient, theta , for a neutral solute is obtained by dividing the total clearance by the glomerular filtration rate (GFR). The latter requires that the tracer molecule is neither taken up, secreted, bound, or degraded in the tubular system. These conditions are met in the isolated kidneys perfused at low temperature and in vivo for a few solutes, e.g., Ficoll.

How to Estimate the Combined Effects of Glomerular Charge and Size Selectivity

When the functional parameters of the gel-membrane model have been determined, the fractional clearance of any (spherical) molecule X can be predicted on the basis of its size and charge. If the solute is charged, its gel-to-plasma concentration ratio, Xmem/XP, is calculated by using Eq. 4. The effect of molecular size is then calculated by using Eqs. 7-21 to estimate the primary U/P for a neutral solute. Finally, the combined effects of the charge and size barriers in series on the sieving of a charged molecule give a predicted fractional clearance of Xmem/XP times U/P. Thus in this model anionic molecules will have a lower concentration on the "blood" side of the pores, resulting in a further decrease in their urine concentration.


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In a previous paper from our group (34), theta  values for Ficoll molecules with aSE of 12-72 Å and albumin (aSE = 36 Å) were estimated in isolated perfused rat kidneys (for details, see Ref. 34). Ficoll was chosen due to its spherical nonflexible shape and absence of net molecular charge.

In the present paper three experimental groups were analyzed: isolated perfused rat kidneys (IPK) at 8°C with 18 g/l of albumin in Tyrode solution; IPK with similar solutions at 37°C; and IPK at 37°C with 50 g/l albumin in the perfusate. Multiple samples were analyzed in each experiment as previously described in detail (34). The experiments were conducted under controlled conditions, and the results were analyzed according to the gel-membrane model as described in THEORETICAL DEVELOPMENT. In the nonlinear regression analysis of the IPK data, a minimal value of 10,000 cm was used for A0/Delta x.

For comparison, similar two-pore analysis was performed by using previously published Ficoll data (see Table 1) on healthy and nephrotic humans (1) and on intact rats (36, 37). Tables showing the exact distribution of Ficoll molecular radii and U/P-ratios can be found in the aforementioned papers.

                              
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Table 1.   Studies in the literature of the glomerular sieving of Ficoll


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Biological Data

Figure 1 shows the sieving coefficient for Ficoll as a function of solute radius in IPK perfused with albumin solutions. The three experimental groups differed in temperature (8 or 37°C) and albumin concentration (18 or 50 g/l) but were otherwise maintained under identical conditions. As can be seen from Fig. 1, the urine-plasma concentration curves mainly differed for molecules >35 Å. Thus there was an increased flux of larger Ficoll molecules with increasing temperature and albumin concentration.


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Fig. 1.   The experimentally obtained sieving curves for Ficoll during 3 control situations in the IPK. Bottom curve: data from cooled isolated perfused kidney (cIPK) perfused with 18 g/l of albumin solution at 8°C (n = 7, black-lozenge ). Middle curve: data from 8 IPKs perfused with 18 g/l of albumin solution at 37°C (gray squares). Top curve: data from IPKs (n = 3, open circle ) perfused with 50 g/l of albumin at 37°C. The 3 curves represent the best fit for the calculated urine-plasma curve data from the 2-pore analysis for A, B, and C, respectively.

Charge Selectivity

The theories of ion-ion interactions presented in THEORETICAL DEVELOPMENT were applied to experimental data (34). Thus an albumin concentration of 18 g/l (0.269 mM) accounts for approx 6 meq/l of charge. For the sake of simplicity, the ionic composition is assumed to be sodium chloride, leaving the small contribution of other ions out of the equations. To obtain electroneutrality in the perfusate, the Nap of 147 mM requires the Clp to be 147 - 6 = 141 mM. Fractional clearance ratios were calculated from the theta  for albumin and for Ficoll of similar size (36 Å). Statistical analysis was performed on the logarithmic values due to the uneven distribution of data. Thus at 8°C the geometric mean of the theta ratio was 18.7 (-6.9, +11.0, n = 7, which corresponds to an average omega  of 43.1 meq/l). At 37°C the geometric mean of the theta ratio was 7.6 (-2.3, +3.3, n = 8). The average omega  at 37°C was 31.2 meq/l. Please note that the density of fixed negative charges dramatically affects the distribution of charged solutes within the charged membrane (or gel), as illustrated in Fig. 2. The estimated pi el in a gel structure with a omega  of 43 meq/l is close to 60 mmHg (see Eqs. 6A and 6B).


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Fig. 2.   The predicted effect of solute charge in a hydrated gel with a charge density (omega ) of 43 (solid line) or 100 meq/l (hatched line). The vertical dotted lines represent 2 solutes with net charges of -23 (e.g., albumin) and 0 (e.g., Ficoll), respectively.

The Two-Pore Model

The two-pore parameters for the three experimental situations are shown in Fig. 3. For the cooled IPK (cIPK), the estimated rs was 45.8 ± 0.4 Å, rL was 86.7 ± 2.1 Å, fL was 0.82 ± 0.1%, and A0/Delta x was 10,000 cm. The residual square sum (SS) was 0.226, and Powell's chi 2 was 0.095. For the IPK at 37°C, rs was 46.8 ± 0.3 Å, rL was 80.4 ± 0.7 Å, and fL was 1.36 ± 0.1%. The SS was 0.139, and Powell's chi 2 was 0.075. For the IPK at 37°C with 50 g/l of albumin in the perfusate, rs was 46.4 ± 0.4 Å, rL was 79.2 ± 0.7 Å, and fL was 3.50 ± 0.3%. The SS was 0.124, and Powell's chi 2 was 0.072. As shown in Fig. 3, rs and rL were rather similar in the three groups, whereas the number of large pores (reflected by fL) increased with temperature and albumin concentration. The best fit of the calculated U/P ratios from the two-pore analysis is shown with the experimentally obtained U/P ratios in Fig. 1.


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Fig. 3.   Two-pore model parameters in IPKs perfused with 18 g/l of albumin solution at 8°C (hatched bars), 18 g/l of albumin solution at 37°C (crosshatched bars), and 50 g/l albumin solution at 37°C (open bars). The analysis is based on the sieving of Ficoll in the molecular radius range 12-72 Å. Previously determined filtration pressures (25) were used in the analysis.

The stability of the two-pore model can be demonstrated by analyzing the Ficoll-sieving data in Fig. 1, assuming various Delta P. Figure 4 shows the two-pore parameters plotted against effective filtration pressure. All parameters except fL were rather insensitive to the assumed value of Delta P. Delta P has previously been determined in this experimental model (25). Thus, in IPKs perfused with 18 g/l of albumin solution at 8°C with a GFR of 0.2 ml · min-1 · g wet wt-1, Delta P is close to 18 mmHg, which for a colloid osmotic pressure of 6 mmHg would give a filtration pressure of 12 mmHg (see Ref. 25). In IPKs perfused with 18 g/l of albumin solution at 37°C with a GFR of 0.3 ml · min-1 · g wet wt-1, Delta P will be close to 14 mmHg (filtration pressure 8 mmHg). Finally, in IPKs perfused with 50 g/l of albumin solution at 37°C with a GFR of 0.3 ml · min-1 · g wet wt-1, Delta P will be close to 28 mmHg (filtration pressure 8 mmHg).


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Fig. 4.   Two-pore model parameters plotted against assumed filtration pressure. The analysis is based on the sieving of Ficoll in IPK perfused with 18 g/l of albumin solution at 8°C (black-lozenge ), 18 g/l of albumin solution at 37°C (gray sqaures), and 50 g/l albumin solution at 37°C (open circle ). Top and middle: small- and large-pore radius, respectively. According to the previously published pressure-flow relationship (25), the filtration pressure in the 3 groups would be close to 12, 8, and 8 mmHg, respectively. Bottom: the large-pore fraction of the hydraulic conductance, fL, increased with filtration pressure, temperature, and albumin concentration. The other pore parameters were, however, rather constant.

Table 2 shows the results of applying the two-pore model on glomerular sieving data for Ficoll from four different studies (1, 34, 36, 37). The effective filtration pressures presented in the cited studies were used in the analysis. Note that the average pore values obtained are rather similar in humans, rats in vivo, and in isolated perfused rat kidneys. Indeed, the glomerular small-pore Peclet numbers are almost identical in the different species in vivo and in the isolated rat kidneys both at 8 and 37°C (Ohlson M, Sörensson J, Lindström K, Blom A, Fries E, and Haraldsson B, unpublished observations). The analysis of Peclet numbers shows that diffusion is the dominating transport force (Pe <1) for molecules <30 Å in hydrodynamic radius, whereas convection dominates (Pe >1) for larger solutes.

                              
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Table 2.   The results of applying a 2-pore model on the Ficoll data from four studies

The Integrative Gel-Membrane Hypothesis

The charge and size selectivity are easily brought together in an integrated model with a negatively charged gel and a heteroporous membrane in series. The consequences of such a model in terms of albumin concentrations and fluxes are shown in Table 3 and illustrated in the drawing denoted as Fig. 5. According to the model, there is actually not a true charge barrier. Instead, significant charge selectivity is obtained because of the ion-exchange properties of the negatively charged gel maintaining lower concentrations of ions in the gel than in plasma. The gel-membrane model allows for predictions of transglomerular passage based on molecular size and charge. Figure 6 illustrates the sieving coefficients (or theta ) for the glomerular barrier for solutes with different molecular radii (16-72 Å) and z (+5 to -25); thus a solute with a aSE of 36 Å and a z of -23 (e.g., albumin) is predicted to have a theta  of 0.002. 

                              
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Table 3.   Predicted fluxes of albumin and fluid across the human glomerular barrier



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Fig. 5.   A schematic drawing of the model of glomerular permeability with charge and size selectivity in series. Jv, the fluid flux across the membrane; arrows, albumin molecules are indicated by the symbols. There is a Donnan equilibrium between the blood and gel compartments. The gel contains ~40 meq/l of fixed negative charges. Hereby, the effective albumin concentration at the "plasma side" of the membrane (M) will be 1/10 of the value predicted from a plasma sample. Restriction due to molecular size in the membrane causes a further reduction of the albumin concentration. See Table 3.



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Fig. 6.   The glomerular sieving coefficients for spherical solutes with molecular radii of 16-72 Å and net charges between +5 and -25 meq/l, as predicted from the integrative charge- and size-selective gel-membrane model.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
THEORETICAL DEVELOPMENT
EXPERIMENTAL METHODS
RESULTS
DISCUSSION
REFERENCES

To treat renal disorders such as nephrotic syndromes, we must understand the intricate glomerular barrier. The present paper is one of the first attempts to describe both the charge and size selectivity in one unifying model. We have found that size and charge effects must be exerted by different components of the glomerular wall to explain experimental data. Similar results were found when the gel-membrane model was applied to glomerular sieving data from humans, intact rats, IPK perfused at 37°C, and IPK perfused at low temperatures (8°C). The latter model is particularly interesting from a physiological point of view, because the experimental conditions can be modified in a manner that never can be done in vivo. Thus unique information about the glomerular barrier has been obtained by modification of perfusate ionic composition (48), osmolarity (30), temperature (34), and orosomucoid content (21, 24). In addition, the effects on the barrier of changing the perfusion pressure and/or the GFR can be studied in detail. Below we will discuss each of the functional components of the glomerular barrier separately and in combination. Finally, we will address some pathophysiological and clinical consequences of the analysis.

Our findings strongly support the classic concept of a highly permselective glomerular barrier with considerable charge selectivity. Thus, when the fractional clearances for albumin and Ficoll, with similar aSE (36 Å), were analyzed by using the equations given in THEORETICAL DEVELOPMENT, the gel omega  was 43 meq/l. This value is much lower than the 120-170 meq/l (13) that can be calculated from dextran sulfate data. Similar high values were obtained (omega  = 100-130 meq/l) when the omega  was estimated from the clearance rates for horseradish peroxidase (HRP) from Rennke and co-workers (40). However, as indicated at the beginning of this study, the charge barrier was probably overestimated by the use of sulfated dextrans. Similarly, the enzymatic determination of HRP (40) probably overestimated omega  (38). Note that for albumin, this difference in omega  implies two orders of magnitude difference in fractional clearance (see Fig. 2). Also, a omega  of 150 meq/l would give an pi el, and hence balancing, hydrostatic pressure of close to 700 mmHg (see Eq. 6B). In contrast, a omega  of 43 meq/l gives rise to an pi el of 60 mmHg, which is in the physiological range of pressure.

Thus the presently suggested gel omega  of 35-45 meq/l is similar to that found by using native and charge-modified myoglobin (52), lactate dehydrogenase isoenzymes (LDH) (31), as well as native and charge-modified HRP (48). Interestingly, Huxley and co-workers (22) estimated the omega  in single mesenteric capillaries perfused with plasma and reported a value of 34 meq/l. Thus glomerular and peripheral (20) capillaries may have similar charge-selective properties but differ in their rs and rL. In contrast to the studies cited above, low omega  (7, 38, 55) or absence of a glomerular charge barrier (39) has been reported. The apparent conflict regarding the charge barrier can, however, be resolved if one assumes the charged gel to be sensitive to hypoxia, as will be further discussed below.

We have applied several different size-discriminating models to glomerular sieving data of Ficoll, which is a neutral and nonflexible sphere. The heteroporous models (2-pore model, lognormal + shunt, fiber matrix + shunt) gave qualitatively similar results, whereas the homoporous equations were unable to simulate the biological data. This is in accordance with previous studies on glomerular permeability (1, 12, 19, 36, 37, 46) and, in fact, seems to apply to almost all microvascular beds (41, 42). Because of its less complex nature, the two-pore model was chosen for further analysis.

The theta  for larger Ficoll molecules increased during perfusion at 37°C and with high albumin concentration (see Fig. 1). This can be due to an increased rL, an increased number of large pores, and/or an increased convective flow through the large-pore pathway. Analysis with the two-pore model revealed that the pore radii were rather stable, but the number of large pores increased with increasing temperature and albumin clearance (see Fig. 3). The two-pore model describes the biological data just as well, or even better than, any other theory. This is in agreement with Deen et al. (12), who found that the two-pore model was superior to all other models in describing their dextran data. They found, however, an inverse correlation between two of the three unknown parameters in the two-pore model, a phenomenon not observed in the present study.

The studies using Ficoll, reported in Table 2, gave similar results in terms of rs and rL, with one exception (36). Thus the rs is 45-50 Å, and the rL is 75-90 Å [61 Å in the study with a narrow range of Ficoll and hence less sensitivity in the large-pore range (1)]. Moreover, the large-pore pathway seems to contribute to 1% of the LpS. A similar rL (110-115 Å) was recently reported from analysis of protein in intact rats with inhibited tubular uptake (50). These results are qualitatively similar to a large number of studies with dextran (12, 28, 33, 46). However, the flexible nature of the dextran molecule probably explains the numerical differences in pore radii. The rL ~80 Å is only one-third of that reported for other capillaries (42, 49). The glomerular large pore is therefore far from a shunt and possesses both size- and charge-selective properties (31).

In the present gel-membrane model, there are two distinct layers; the gel discriminates due to molecular charge, and the membrane is neutral and size selective. The reason for using two separate components is the observation that charge and size selectivity may change independently of each other, for example, when the ionic composition of the perfusate is altered (48). The charged fiber matrix of Johnson and Deen (23) presently represents the most extensive analysis of the effects of molecular charge and size. However, it cannot predict the effects on glomerular charge selectivity of reducing perfusate ionic strength (47).

From a logistical point of view, the glomerular charge barrier should ideally be placed at the endothelial level close to the plasma compartment (29). Indeed, experimental evidence for this was presented three decades ago, in a seldom cited but relevant paper by Ryan and Karnovsky (44). Most investigators have instead focused on the basement membrane as the morphological counterpart to the charge barrier. Observations on isolated basement membrane have revealed that these structures are rather nonselective (3, 7, 10, 16, 18). However, all such studies assume that the basement membranes do not alter their properties during the preparation procedure (3). Recently, the endothelial glycocalyx was suggested to be much thicker than previously thought, i.e., almost twice as thick as the basement membrane. Thus electron micrographs taken after fixation with glutaraldehyde in fluorocarbons seem to preserve the glycocalyx better than glutaraldehyde-based fixative (43).

Placing the charge barrier in the endothelial glycocalyx has several advantages. First, there is the logistical argument above. Second, it could explain how charge selectivity can be affected by the plasma composition, such as the beneficial effect of orosomucoid (21). Third, the glomerular omega  can reversibly be altered by changes in the perfusate ionic strength (48), but not by alterations in osmolality (30). Fourth, it can explain why some investigators fail to demonstrate charge selectivity, because the endothelial glycocalyx has been shown to be sensitive to hypoxia-reperfusion in other vascular beds (9, 51).

In agreement with these observations, IPKs perfused at 37°C had slightly less fixed negative charges (31 meq/l) compared with the omega  at 8°C (43 meq/l, n.s.). Erythrocyte-free perfusion is known to give hypoxic tubular cell damage at 37°C (45). With other experimental approaches, involving reperfusion, the glycocalyx is thus likely to be damaged, giving a reduced charge barrier, which then could be almost abolished by the administrations of various toxins (39). The apparently conflicting views of the glomerular barrier could therefore be due to differences in experimental approach affecting the delicate endothelial glycocalyx to various degrees.

Another virtue of a charged gel is that one would expect a Donnan equilibrium to maintain a reduced concentration of negatively charged solutes in the gel compared with that of plasma. However, such a "self-rinsing" ability would most likely be impaired due to concentration gradients and stagnant layers if there were to be significant size discrimination in the gel.

The size-selective barrier is probably situated at the podocyte slit membrane, having a large number of functional pores with 45 to 50 Å radii and a few pores with radii from 75 to 115 Å. Genetic or acquired defects in the basement membrane may disturb endothelial function or affect the three-dimensional glomerular structure, hence causing proteinuria. However, we suggest that most nephrotic syndromes are secondary to changes in the podocytes and/or in the endothelial glycocalyx.

To conclude, there is substantial evidence for the glomerular barrier to be heteroporous with charge-selective properties in qualitative agreement with the classic view. We extend the analysis and propose a gel-membrane model with charge and size selectivity in series. The charge selectivity is probably confined to a delicate endothelial glycocalyx acting as an ion-exchange gel sensitive to perfusate flow, pressure, and plasma composition. We have previously shown that glomerular charge and size selectivity can be changed independently of each other, which suggests that a different component is responsible for the size barrier, most probably the podocyte slit membrane. The glycocalyx has been shown to be highly sensitive to hypoxia-reperfusion in other organs, and that may explain why some investigators have found lower glomerular omega . Indeed, we would suspect certain nephrotic syndromes to be caused by changes in the hitherto completely neglected and poorly understood glomerular endothelial glycocalyx.


    ACKNOWLEDGEMENTS

This study was supported by the Swedish Medical Research Council Grants 9898 and 2855, the Knut and Alice Wallenberg Research Foundation, the Ingabritt and Arne Lundberg Research Foundation, the National Association for Kidney Diseases, the Göteborg Medical Society, and Sahlgrenska University Hospital Grant LUA-B31303.


    FOOTNOTES

Address for reprint requests and other correspondence: B. Haraldsson, Dept. of Physiology, Göteborg Univ., Box 432, SE-405 30 Göteborg, Sweden (E-mail: bh{at}kidney.med.gu.se).

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 9 May 2000; accepted in final form 10 October 2000.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
THEORETICAL DEVELOPMENT
EXPERIMENTAL METHODS
RESULTS
DISCUSSION
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