## Abstract

In the current study we explore the electrostatic interactions on the transport of anionic Ficoll (aFicoll) vs. neutral Ficoll (nFicoll) over the glomerular filtration barrier (GFB) modeled as a charged fiber matrix. We first analyze experimental sieving data for the rat glomerulus, and second, we explore some of the basic implications of a theoretical model for the electrostatic interactions between a charged solute and a charged fiber-matrix barrier. To explain the measured difference in glomerular transport between nFicoll and aFicoll (Axelsson J, Sverrisson K, Rippe A, Fissell W, Rippe B. *Am J Physiol* 301: F708–F712, 2011), the present simulations demonstrate that the surface charge density needed on a charged fiber matrix must lie between −0.005 C/m^{2} and −0.019 C/m^{2}, depending on the surface charge density of the solute. This is in good agreement with known surface charge densities for many proteins in the body. In conclusion, the current results suggest that electrical charge makes a moderate contribution to glomerular permeability, while molecular size and conformation seem to be more important. Yet, the weak electrical charge obtained in this study can be predicted to nearly totally exclude albumin from permeating through “high-selectivity” pathways in a charged-fiber matrix of the GFB.

- capillary permeability
- fiber matrix
- anionic Ficoll
- charge selectivity

the relative importance of electrical charge in the sieving of plasma proteins across the glomerular filtration barrier (GFB) has been a matter of debate over the last few decades. The seminal data of Brenner an colleagues (8, 10), using differently charged dextran thus suggested that the glomerular transport of negatively charged, sulfated dextran molecules be much lower than that of neutral dextran of the same size (8). However, several authors have questioned these results. Thus some fractions of sulfated dextran seem to bind to plasma proteins or glomerular cells (18, 33). In addition, it has been shown that polysaccharides (such as dextran and Ficoll) exhibit a flexible molecular conformation, making them hyperpermeable compared with more rigid solutes, such as proteins, at least for molecular radii approaching the pore radius (1, 12). Ficoll apparently shows glomerular sieving characteristics somewhere between those of dextran and proteins (34). Several findings also suggest that polysaccharides undergo significant conformational changes during charge modification, making them even more flexible. Hence, Asgeirsson et al. (2) conducted an experiment investigating the glomerular sieving of carboxymethylated (CM) anionic Ficoll (aFicoll) and “unmodified” neutral Ficoll (nFicoll) in rats. The glomerular permeability to negatively charged CM-Ficoll was markedly enhanced compared with that of its neutral counterpart. In addition, size separation using high-performance size exclusion chromatography showed that the aFicoll tested eluted earlier than nFicoll, indicating that the charge-modification process had indeed increased the molecular radius (and altered the conformation) of Ficoll (2).

Schaeffer and colleagues (16, 31) published experiments showing that the difference in permeability between anionic and neutral dextran was negligible. In contrast to these results, there are extensive experimental results from both synthetic membranes and the glomerulus that demonstrate that globular proteins are selected based both on their size and charge (19, 34). Using anionic 5.8- and 20-nm (pore radius) silicon nanopore membranes, Fissell and colleagues (17) showed a charge-dependent permselectivity for aFicoll. Using the same charge-modification technique for CM-Ficoll, our own experiments have demonstrated a reduced transport of aFicoll relative to nFicoll across the rat GFB (4). Bhalla and Deen (6) published an elegant theoretical model for the transport of charged solutes over a regular array of charged fibers, and their results showed that the osmotic reflection coefficient for BSA was much larger than that for an uncharged system.

It is well-recognized that some of the major structural components of the GFB (e.g., perlecan, agrin, entactin/nidogen, and proteoglycans of the glycocalyx) and important plasma proteins, such as albumin and orosomucoid, carry a net negative charge during physiological conditions. Numerous experiments using charged barriers have demonstrated large to moderate effects of electrical charge in the hindrance of charged solutes, with the difference being strongly dependent on the ionic strength of the solution. Pujar and Zydney (27) showed that the clearance of BSA through a 100,000 molecular weight cutoff membrane decreased by nearly two orders of magnitude as the ionic strength (salt concentration) was reduced from 150 to 1.5 mM. At physiologic (“high”) ionic strengths the electrical field of the solute will be condensed and the “screening distance” for charge at which significant charge effects occur (i.e., the so-called Debye length, being ∼8 Å at physiological ionic strength) will be reduced (9). Isolated glomerular basement membranes (GBM) have failed to show charge selectivity at physiological ionic strengths when probed with neutral and negatively charged Ficoll (7). At the ionic strengths of 1.0, 0.11, and 0.01 M, Johnson et al. (22) demonstrated that the hindrance due to charge of BSA, ovalbumin, and lactalbumin was moderate in a charged agarose gel. The partition coefficient for BSA with an estimated net charge of −37 was reduced from 0.67 to 0.47 when the ionic strength was lowered from 1.0 to 0.11 M (22).

In the current study, the model of Johnson and Deen (23) as modified by Jeansson and Haraldsson (21) is used to analyze the sieving data for 20–35 Å (radius) aFicoll and nFicoll based on experimental data from Axelsson et al*.* (4) to determine the charge density needed in a fibrous barrier to account for the measured results. To our knowledge, this is the first attempt to calculate the surface charge density of the GFB. It will be shown that, in a randomly oriented anionic fiber matrix fitting to the mentioned Ficoll data, if the difference between the sieving coefficient of nFicoll and that of negatively charged albumin would be due to charge only, this would require a very highly charged barrier. In such a case, surface potentials exceeding −200 mV would be needed, which is equivalent to that of a charged spherical 35.5 Å (radius) solute with a net charge of approximately −200. Still, the electrical charge of the GFB certainly has a distinct effect on the glomerular permeability to charged solutes. The importance of this charge effect relative to the size selectivity will be explored in this study.

## METHODS

#### Theory.

The fiber-matrix model of Ogston supposes a barrier consisting of a random array of fibers of thickness *r*_{f}. The partition coefficient was determined from
*g*(*h*) is the probability density function (PDF) for finding the closest fiber at a surface-to-surface distance *h* from the solute sphere with radius *r*_{s}. For Ogston's original model for random fibrous media, the PDF is
*e*^{−E(h)/kT} can be introduced describing the relative probabilities of various energy states in the solute-fiber system.
*E*(*h*) is the electrostatic free energy associated with moving a charged sphere to a certain distance *h* from the fiber. The free-energy was calculated from the expression
_{r} is the dielectric permittivity of the solvent and
_{s}, and σ_{f} are described in Table 1. The coefficients *A*_{i} are given by

where the coefficients are given by Table 1 in Johnson and Deen (23).

#### Charged fiber-matrix + large-pore model.

The data were analyzed using a modified version of the theoretical model by Johnson and Deen (23), based on the extended Ogston model, as developed by Jeansson and Haraldsson (21). In this model, the theoretical sieving coefficients (θ_{model}) were calculated from the nonlinear global convection/diffusion equation (*30*) according to
*f*_{fib} is the fractional volume flow (*J*_{v,fib}/*J*_{v}) through the fiber-matrix pathway and *f*_{L} = 1 − *f*_{fib} represents the fraction of volume transported via the large-pore system. The Peclet numbers are defined by

For the fiber-matrix pathway, the reflection coefficient was estimated from the partition-coefficient (Φ) using the simple relation

The permeability-surface area coefficient was calculated from

For the large-pore system, the hydrodynamic estimates recommended for porous membranes (*Eqs. 16* and *18*) by Dechadilok and Deen (11) were used. The definitions of constants and symbols are essentially the same as those used in the two-pore model according to Rippe and Haraldsson (30). The hydraulic conductance through the fibrous pathway was estimated using the Kozeny-Carman equation
*G* is the Kozeny constant. The fractional total cross-sectional area over the large pores can be calculated using [cf. also *Eq. 27* in Rippe and Haraldsson (30)]

Optimal values for the fractional fiber volume ϕ = (1 − ε), *A*_{0}/Δ*x*, *r*_{L}, and α_{L} were calculated with the nonlinear regression method of Levenberg and Marquardt using the well-known MINPACK software library (25) with standard settings. For the data analysis of the aFicoll, only the fiber-matrix pathway was considered so that f_{L} = 0 and the nonlinear regression scheme was limited to the parameters ϕ and *A*_{0}/Δ*x*. Due to the confounding effects of the other model parameters, only the barrier surface charge density (*q*_{f}) was allowed to vary between the charged and uncharged models. Taking into account that the oncotic pressure gradient is different over different pathways (cf. *Eqs. 3*–*11*; Ref. 29), the volume flux through a pathway *i* in a heteroselective barrier can be approximated from
_{o} = Σα_{i}σ_{i} is the osmotic reflection coefficient and a plasma oncotic pressure Δπ of 28 mmHg is assumed. The improper integral in *Eq. 3* was evaluated numerically using a 21-point Gauss-Kronrod quadrature. Numerical calculations were performed using the software package GNU Science Library (14).

#### Physical properties of aFicoll.

The net charge of human serum albumin during physiological conditions is −22 (26). By approximating its surface area to that of a sphere, using a Stokes-Einstein radius of 35.5 Å, a surface charge density of approximately −0.022 C/m^{2} can be calculated. The electric charge of the aFicoll used in the experiments has been quantified in terms of the zeta potential ζ (see Fig. 1), which is the electric potential at the slipping plane, with values of −40 mV and −45 mV for CM-Ficoll 70 and CM-Ficoll 400, respectively (17). The surface charge density can be approximated using the Grahame equation
_{r} is the dielectric permittivity of the solvent and κ^{−1} is the Debye length. If assuming a relative permittivity of 74.3 for the solvent at 310° K (13) and an ionic strength of 0.15 M, a surface charge density of −0.033 C/m^{2} for Ficoll 70 and −0.037 C/m^{2} for Ficoll 400 can be calculated. The general problem of estimating the net charge of a protein is not trivial (35), but the use of the Grahame equation provides a good estimate at high ionic strengths (9). Since the perfusate used in the experiments consisted of both Ficoll 70 and Ficoll 400, the surface charge density is estimated as the arithmetic mean of the above values, i.e., −0.035 C/m^{2}. This charge density is equivalent to that of a 36 Å spherical molecule with a net charge of approximately −37. As described in Axelsson et al. (4), aFicoll showed a slight increase in molecular diameter as measured by the difference in elution time. This effect was compensated for in the current analysis by subtracting the measured difference in Stokes-Einstein radius from the aFicoll data. In addition, the glomerular filtration rate differed in the experiments using nFicoll from that in the anionic group. To compensate for this, the same glomerular filtration rate for both Ficoll species was used in the theoretical model.

### Statistical Analysis

Parameter values are presented as means ± SE. Differences among the models were tested using a nonparametric Mann-Whitney test. A Pearson χ^{2}-test was used for testing the “goodness of fit” for the data fitted to the model. Significance levels were set at *P* < 0.05, *P* < 0.01, and *P* < 0.001. All statistical calculations were made using the computer software R version 2.14.2 for Linux.

## RESULTS

#### Theoretical analysis.

In Fig. 2, several different fiber surface charge densities are simulated for a solute having a charge density similar to that of albumin (−22 mC/m^{2}) utilizing as starting point the optimized parameters calculated for nFicoll (or aFicoll) in an uncharged fiber matrix with characteristics shown in Table 2 (4). The rightmost sieving curve is the best-fitting curve to experimental data for nFicoll according to the charged fiber-matrix + large-pore model. A very low value (0.077) of the Pearson χ^{2} indicates a good data fit to the model. The leftmost curve represents an extreme scenario, where the barrier has a surface charge of −200 mC/m^{2}. With this charge, the theoretical aFicoll_{35.5Å} sieving coefficient is on the order of 1·10^{−5} (cf. albumin). In Fig. 3, *A* and *B*, the importance of ionic strength in screening the surface potential on a charged solute is demonstrated by plotting the partition coefficient vs. solute radii for different solute charge densities (*q*_{s}) at low (Fig. 3*A*) and physiologic (Fig. 3*B*) ionic strengths. As expected, the importance of charge on the partition coefficient is moderate at physiologic (“high”) ionic strengths. In these plots, it can also be seen that there is a slight difference between the original Ogston model and the charged fiber-matrix model, even for an uncharged molecule. However, this difference becomes negligible at physiologic ionic strengths.

#### Data analysis.

The sieving curves for the model best-fit versus the experimental data are shown in Fig. 4. The optimized model parameters for the uncharged random fiber-matrix model are shown in Table 3, whereas the parameters for the charged fiber-matrix model are shown in Table 4. Since the Grahame equation only gives a rough estimate of the actual surface charge density, two different solute surface charge densities were used in the models, −0.022 C/m^{2} (albumin) and −0.035 C/m^{2} (Ficoll approximation). Expectedly, a more anionic solute resulted in a lower charge density for the filtration barrier of about −5 mC/m^{2}. Interestingly, assuming the solute to have a charge similar to that of serum albumin gave values for fiber charge in the vicinity of known surface charge densities for many anionic plasma proteins. Only solute radii in the range 20–35 Å were analyzed. Two different fiber radii, 5 and 10 Å, were used in the modeling. For the neutral Ficoll data, the 5-Å fiber-radius model showed a better goodness of fit than the 10-Å model. It can also be seen that the fractional void volume is much lower for the 10-Å model, which is to be expected since the partition coefficient (*Eq. 1*) is an increasing function of *r*_{f,}, i.e., a more dense fiber array is needed for an array of large fibers to achieve the same partition coefficient as that of an array of thinner fibers. The hydraulic conductance was calculated assuming a Kozeny constant of 5 for the 5 Å-model, which yields a reasonable value for hydraulic conductance of the fiber array (LpS) of 0.13 ml·min^{−1}·mmHg, which is equivalent of a glomerular net pressure gradient of 5–7 mmHg. In the 10-Å model, a Kozeny constant of 1 was needed to achieve a similar gradient.

## DISCUSSION

The essential result of the present modeling is that the magnitude of negative charge present on the fibers in a charged fiber-matrix model separating aFicoll from nFicoll in the intact rat GFB (4) is predicted to be of the same order of magnitude as that obtained for a majority of anionic proteins in plasma, such as albumin or orosomucoid. At normal (physiologic) ionic strengths, electrical charge effects thus appear to contribute only moderately to glomerular permeability, whereas molecular size and conformation are more important. A factor of ∼2–3 difference in the sieving coefficient, as in the current study, is moderate compared with that of several other researchers, both historically (8) and recently, for example in Haraldsson et al. (19) where a factor of ∼20, for a solute charge density similar to that of albumin, was suggested. This does not deny the fact that in a more “tight” fiber matrix with a higher size selectivity, as that assessed for neutral proteins (corresponding to an equivalent small-pore radius of ∼37.4 Å), a negative electrical charge may critically exclude albumin from passing through “high-selectivity” pathways, redirecting albumin to rare “large-pore” pathways (24). In terms of Debye screening lengths (being ∼8 Å at physiologic ionic strength), the charge effect in this study was found to be only ∼20% of that predicted from the simple Debye-Hückel theory of ion-ion interactions. Hence, using pore theory and adding 1.5 Å (20% of 8 Å) to the molecules of radius of albumin (35.5 Å) and subtracting 1.5 Å from the small-pore radius (37.4 Å) implies that even the small charge effects obtained in the current study will totally screen out albumin from the “small-pore” pathway, thus, making the large-pore system the only pathway in the GFB for proteins of a similar or larger size than albumin under normal conditions. Leakage of albumin and large plasma solutes may, however, occur across the small pores in conditions where they are less selective, as has been observed in dialyzer membranes (5) or in the GFB after high doses of angiotensin II (3).

The mathematical model used in this study assumes a very simple structure of the GFB as a negatively charged barrier with random fibers (fiber matrix) and an idealized uniformly charged spherical “hard” solute (21). Conceivably, the former approximation should be more accurate for proteins than for polysaccharide molecules, such as dextran or Ficoll. For opposing surface charges, the single fiber-single sphere approach will tend to underestimate the restriction when there is significant interaction with several fibers, i.e., at a low void volume. Therefore, this approach should be most accurate at small Debye lengths (high ionic strengths) and/or at low fiber densities. Technically, the construction of a more complex model is always possible, but on the other hand, such models would have an increasing number of phenomenological parameters that would need to be approximated more or less arbitrarily, since the exact structure of the GFB is not known. In addition, a good model should be able to reproduce real experimental data. Despite the simplicity of the model used in this study, it shows a very good fit to the neutral Ficoll data for molecules of radius 20–35 Å. Thus, in a functional sense, the real and theoretical barriers are comparable. Only one model parameter, the surface charge density (*q*_{f}), was allowed to vary between the charged and uncharged barriers (due to the confounding effects of the other parameters). This may, in part, explain why the model fit for the aFicoll data was not as good as that for nFicoll since there are also small [cf. Axelsson et al. (4)] differences in the other parameters.

Given the lack of charge selectivity of isolated GBM (7), and the fact that the total abolition of the negatively charged GBM proteoglycans, agrin, and perlecan does not seem to affect the permeability of albumin across the GFB (15, 20), the endothelial glycocalyx may be implicated as the major charge barrier of the GFB. Actually, the best fit of nFicoll data to the fiber-matrix model was for a fiber radius of 5 Å and a fiber density of 11%, which would approximately fit the composition of a fiber matrix of glycoproteins, proteoglycans, and glucosaminoglycans, rather than a matrix of the much thicker collagen-IV and laminin molecules of the GBM. The glycocalyx, representing a cell surface coat of the composition mentioned above, is important for several basic cell functions, such as immunologic recognition of “self” and “nonself,” sensing of shear stress, and the presentation of receptors and adhesion molecules on the cell surface (32). The glycocalyx can be regarded as a dynamic network in which soluble plasma proteins and endothelial derived components are incorporated to form an even larger endothelial surface layer (ESL). Being in direct contact with the circulation, the ESL is continuously remodeled due to both enzymatic and shear-induced shedding (28). This means that the glycocalyx, and the ESL in toto, can hardly be viewed as a static structure. Still, the glycocalyx may be regarded as at least one of the structural candidates to the charged fiber-matrix concept in this study.

Examining glomerular permeability by studying the sieving of plasma proteins is complicated by the fact that there is extensive tubular processing of the glomerular ultrafiltrate with almost complete tubular reabsorption of filtered proteins. Furthermore, the sieving of a large number of proteins of discrete molecules of radii (*a*_{e}) has to be assessed to create a protein glomerular sieving curve. We have therefore preferred to use polysaccharides, uncharged and charged, as probe molecules, to estimate the sieving properties of the GFB. Polysaccharides are minimally processed by the renal tubule, and they allow the assessment of sieving of molecules of a wide range of *a*_{e}. Unfortunately, polysaccharides in general apparently exhibit a glomerular “hyperpermeability,” especially when *a*_{e} approaches the radius of the size-selective structures in the GFB. We have found, however, that Ficoll apparently behaves as a hard sphere (cf. proteins) for λ values (solute radius over pore radius) ≤0.65, implying that Ficoll data and protein data are similar for solutes up to a radius of 25 Å, i.e., across the “small-pore equivalent” of the GFB. Furthermore, Ficoll and proteins seem to be handled in a similar fashion for an *a*_{e} interval of 50–75 Å, i.e., in the equivalent “large-pore pathway” (pore radius ∼120 Å). Thus, as already discussed above, the size selectivity of the GFB measured using intermediately sized Ficoll molecules of radius of 25–50 Å might be in error. If instead the small-pore radius estimate determined for neutral protein permeation across the GFB (24), i.e., 37.4 Å, were correct, this implies that Ficoll molecules >37.4 Å would actually permeate the membrane in an anomalous fashion, conceivably by (increased) molecular deformability. In the present study we found that the sieving coefficients for aFicoll ≥35 Å in radius (for λ approaching or exceeding 1) significantly deviated from those predicted by the charged fiber-matrix model, in that sieving coefficients for aFicoll approached those for nFicoll. A similar phenomenon has been observed for aFicoll vs. nFicoll in artificial membranes (17). This may indicate that charge restriction effects tend to be markedly reduced in magnitude when the size of the charged molecules approaches the size of the transport limiting structures of the barrier or that charge selectivity is much more complex than predicted from a charged fiber-matrix model.

The present results suggest that electrical charge only makes a moderate contribution to glomerular permselectivity and that molecular size and conformation are far more important in this respect. In fact, in the present study it was shown that a supraphysiological surface charge density (*q*_{f}), being nearly −0.2 C/m^{2}, is needed to account for the difference in the sieving coefficients between nFicoll and (negatively) charged albumin, if charge-dependent restriction were the only factor affecting the glomerular sieving of albumin vs. Ficoll. For a more compact fiber matrix than that fitting to the sieving characteristics measured for nFicoll, i.e., a fiber matrix fitting to (neutral) protein data, the moderate negative charge determined in this study may still critically affect the permeation of albumin from blood to urine. In such a model, a weak electrical charge will more or less totally exclude albumin from a high-selectivity fiber matrix, to redirect it to some very rare low-selectivity glomerular transport pathways.

## GRANTS

This study was supported by grants from the Swedish Research Council Grant 08285, Heart and Lung Foundation, and Medical Faculty at Lund University (ALF Grant).

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: C.M. and B.R. conception and design of research; C.M. analyzed data; C.M. and B.R. interpreted results of experiments; C.M. prepared figures; C.M. and B.R. drafted manuscript; C.M. and B.R. edited and revised manuscript; C.M. and B.R. approved final version of manuscript.

## ACKNOWLEDGMENTS

We gratefully thank Josefin Axelsson for technical aid and Kerstin Wihlborg for expert secretarial assistance.

- Copyright © 2013 the American Physiological Society