## Abstract

To investigate the charge effect of the endothelial surface glycocalyx on microvessel permeability, we extended the three-dimensional model developed by Fu et al. (*J Biomech Eng* 116: 502–513, 1994) for the interendothelial cleft to include a negatively charged glycocalyx layer at the entrance of the cleft. Both electrostatic and steric exclusions on charged solutes were considered within the glycocalyx layer and at the interfaces. Four charge-density profiles were assumed for the glycocalyx layer. Our model indicates that the overall solute permeability across the microvessel wall including the surface glycocalyx layer and the cleft region is independent of the charge-density profiles as long as they have the same maximum value and the same total charge. On the basis of experimental data, this model predicts that the charge density would be 25–35 meq/l in the glycolcalyx of frog mesenteric capillaries. An intriguing prediction of this model is that when the concentrations of cations and anions are unequal in the lumen due to the presence of negatively charged proteins, the negatively charged glycocalyx would provide more resistance to positively charged solutes than to negatively charged ones.

- model for charge effect of endothelial surface glycocalyx
- microvessel permeability to charged molecules
- interendothelial cleft

the endothelial cell glycocalyx is an extracellular matrix that is expressed on the luminal surface of the endothelial cells forming the microvessel wall. This matrix is believed to be composed primarily of proteoglycans, glycoproteins, and glycosaminoglycans (14, 18, 19, 21). Because of its distinct location in the transvascular pathway, in conjunction with the intercellular junctions in the cleft between adjacent endothelial cells, the surface glycocalyx is of great importance in determining the microvessel permeability (*P*) to water and solutes.

Adamson et al. (2) showed that for similar size globular proteins, α-lactalbumin (molecular weight = 14,176) and ribonuclease (molecular weight = 13,683), the permeability of the frog mesenteric capillary to positively charged ribonuclease (net charge = +3, including the charge effect from fluorescent probe labeling; *P*
^{ribonuclease} = 4.3 × 10^{−6} cm/s) was twice that of negatively charged α-lactalbumin (net charge = −11, including the charge effect from fluorescent probe labeling;*P*
^{α-lactalbumin} = 2.1 × 10^{−6} cm/s). Their experiments suggested that the microvessel wall contains negative charges, which enhance the transport of positively charged molecules but retard that of negatively charged molecules. With the use of a Donnan-type model (see below) for electrostatic partitioning, they estimated that the charge density in the microvessel wall was ∼11.4 meq/l.

This charge effect of the microvessel wall has also been shown in other experiments. Curry et al. (6) measured*P*
^{ribonuclease} and*P*
^{α-lactalbumin} in microvessels perfused with orosomucoid in a Ringer-albumin perfusate. They found that*P*
^{ribonuclease} was six times that of*P*
^{α-lactalbumin} in the presence of orosomucoid. In the presence of orosomucoid, *P*
^{α-lactalbumin}was only about one-half of the value in the absence of orosomucoid. They suggested that these results could be accounted for if orosomucoid increased the net negative charge on microvessel walls in the frog mesentery from 11.2 to 28 meq/l. Huxley and Curry (16) showed that the diffusive solute permeability to α-lactalbumin was lower during exposure to plasma [(*P*
^{α-lactalbumin})^{plasma} = 1.0 × 10^{−6} cm/s] than that during exposure to bovine serum albumin (BSA)-Ringer solution [(*P*
^{α-lactalbumin})^{BSA} = 5.0 × 10^{−6} cm/s]. Huxley et al. (17) further showed that (*P*
^{α-lactalbumin})^{plasma}/(*P*
^{α-lactalbumin})^{BSA}= 0.31, whereas there was no change in hydraulic conductivity (*L*
_{p}). They concluded that the actions of plasma were to confer charge selectivity for anionic solutes and modify the porous pathways of the microvessel wall to a lesser extent. With the use of the same model as in Refs. 2 and 6, they predicted an increase in charge from 11.2 meq/l in the presence of albumin to 34 meq/l in the presence of plasma.

In another line of investigation into the mechanism of decreasing*P*
^{α-lactalbumin} by plasma protein, Adamson and Clough (1) tried to test the hypothesis that plasma protein may modulate surface glycocalyx structural properties. With the use of cationized ferritin staining, they found that the total glycocalyx thickness in the presence of plasma was twice the value of that with BSA-Ringer perfusion. Their interpretation for this was that the increase in the thickness of surface glycocalyx layer is the result of a change in the orientation of surface glycoproteins to which cationized ferritin binds.

Previously, a simple Donnan-type model was proposed to describe the charge effect on microvessel permeability (2, 6, 16, 17). It was based on a Donnan equilibrium distribution of ions, which exists as a result of retention of negative charges on the capillary membrane. It was suggested (8, 9) that the steric and electrostatic exclusions be described in terms of an effective partition coefficient (Φ_{eff})
Equation 1Here, Φ_{steric} is the steric partition coefficient describing the size selectivity of the membrane, Δ*E* is the effective Donnan electrical potential difference across the membrane,*Z* is the charge on the solute, *R* is the universal gas constant, *F* is Faraday's constant, and T is temperature. *R*T/*F* is 25.2 mV at 20°C. ΔΨ is the dimensionless electrical potential difference and is equal to Δ*EF*/*R*T. With the use of this model, the fixed negative charge in the transport pathway in the frog mesentery (*C*
_{m}) was estimated to be ∼11 meq/l (2,6, 17). Although this model described the steric and electrostatic partition to a charged solute at the interface between the membrane and the solution, it neglected the thickness of the membrane and thus neglected the steric and electrostatic interactions between the solutes and the membrane components within the membrane.

A more sophisticated model for steric and diffusion resistance to solute transport in the fiber matrix was proposed by Weinbaum et al. (22). With the use of this theory for the entrance fiber layer, Fu et al. (10-13) developed a three-dimensional model for the interendothelial cleft to describe solute exchange across the microvessel. Whereas this model could successfully explain the size-restricted transport of a solute through the surface glycocalyx and the interendothelial cleft, it did not consider the electrical charge factors of the glycocalyx layer and the solute. Therefore, it can only be applied to describe the transvascular transport of electroneutral molecules. Recently, an electrochemical model was proposed by Stace and Damiano (7, 20) for the transport of charged molecules through the capillary glycocalyx. However, this model did not consider transport through the cleft region.

In the current study, we attempted to develop a two-dimensional model incorporating both size and charge effects so that it will provide, for the first time, a quantitative analysis of various experimental results expected to be associated with negative charges in transvascular pathways. Compared with the model in Fu et al. (13), this model features two new characteristics: *1*) the surface glycocalyx contains a negative electric charge, and *2*) there is an interface between the surface glycocalyx layer and the cleft entrance (15). This model will help to better understand both the physical and electrochemical mechanisms of selectivity in the endothelial surface glycocalyx layer and therefore provide a new method for controlling transport rates of charged or uncharged molecules in drug delivery.

## MODEL DESCRIPTION

### Model Geometry

The schematic of the new model geometry for the interendothelial cleft is shown in Fig. 1. −*L*
_{f} < *x* < 0 is the surface glycocalyx layer represented by a periodic square array of cylindrical fibers, where *L*
_{f} is the thickness of the entrance fiber matrix layer and *x* is the abscissa with the origin at the cleft entrance (Fig. 1). The radius of the fiber is *a*, and the gap spacing between the fibers is Δ.*L*
_{jun} is the junction strand thickness.*L*
_{1} and *L*
_{3} are the depths between the junction strand and the luminal and abluminal fronts.*L* is the total length of the cleft. There are two types of pores in the junction strand, as proposed in Fu et al. (11-13), based on Adamson and Michel's observations (3). One is an infrequent large break of width 2*d* and height 2*B*. The distance between the adjacent large breaks is 2*D*; another is a continuous narrow slit of width 2*b*
_{s.} The effect of a narrow slit is neglected because the solute considered in this study, the diameter of which is 4.02 nm, cannot penetrate the slit of width 2*b*
_{s}, ∼2 nm. The electric charge is assumed to only exist in the surface glycocalyx layer, and the charge density*C*
_{m}(*x*) is assumed to have four distribution profiles, as shown in Fig.2: *1*)*C*
_{m}(*x*) = constant =*C*
_{m0}(−*L*
_{f} < *x* < 0),*2*) *C*
_{m}(*x*) =*C*
_{m0}tanh(1 +*x*/*L*
_{f})/tanh(1) (−*L*
_{f} < *x* < 0),*3*) *C*
_{m}(*x*) =*C*
_{m0}tanh(−*x*/*L*
_{f})/tanh(1) (−*L*
_{f} < *x* < 0), and*4*) *C*
_{m}(*x*) =*C*
_{m0}tanh(2 + 2*x*/*L*
_{f})/tanh(1) (−*L*
_{f} < *x* < −*L*
_{f}/2);*C*
_{m0}tanh(−2*x*/*L*
_{f})/tanh(1) (−*L*
_{f}/2 < *x* < 0). These four profiles are chosen so that when the concentrations of cations (C_{+}) and anions (C_{−}) are equal in the solution, *case 1* has an electrical partition at both interfaces between the glycocalyx layer and the lumen (*x* = −*L*
_{f})/cleft (*x* = 0), *case 2* has a partition only at*x* = 0, whereas *case 3* has a partition at*x* = −*L*
_{f}, and *case 4*has an electrical partition at none of the interfaces. Similar assumptions in Ref. 8 are used in the glycocalyx layer:*1*) all charged solutes [ribonuclease, α-lactalbumin, and univalent cations (Na^{+} and anions, mainly Cl^{−})] obey a modified Nernst-Planck flux expression;*2*) overall electroneutrality is satisfied everywhere; and*3*) Donnan equilibria exist at the interfaces of the fiber layer between the vessel lumen (*x* = −*L*
_{f}) and between the cleft entrance (*x* = 0).

### Mathematical Model

#### Entrance fiber matrix layer.

As shown in Fig. 1, the glycocalyx (fiber matrix) layer lies in front of the cleft and covers the entire endothelial surface. The solution in the vessel lumen consists of monovalent cations (C_{+}) and monovalent anions (C_{−}) as well as a small amount of protein (e.g., albumin). The volume flux and flux of *solute i* are denoted by *J*
_{v} and*J _{i}
*, respectively. The concentrations of

*solute i*and the electrical potential within the fiber matrix layer are denoted as C′

_{i}(

*x*) and

*E*′(

*x*). At the interface of the vessel lumen and the fiber layer, C

_{i}(−

*L*

_{f}) and

*E*(−

*L*

_{f}) represent the concentration and electrical potential at

*x*= −

*L*

_{f}from the lumen side and C′

_{i}(−

*L*

_{f}) and

*E*′(

*L*

_{f}) from the fiber side. At the interface of the fiber layer and the cleft entrance, C

_{i}(0) and

*E*(0) represent the concentration and electrical potential at

*x*= 0 from the cleft side and C′

_{i}(0) and

*E*′(0) from the fiber side.

With the assumptions described in the model geometry and steady-state conditions, the governing equation for solute transport in the fiber layer can be written as
Equation 2or
Equation 2awhere Ψ′ = *FE*′/*R*T and is the dimensionless electrical potential, *R* is the universal gas constant, *F* is Faraday's constant, T is temperature,*E*′ is the electrical potential, *A* is an arbitrary constant, and C′_{i} is the solute concentration within fiber matrix.*D _{i}
*

_{,f}is the effective diffusion coefficient of

*solute i*in the fiber matrix layer, which includes both steric hindrance and diffusive resistance of fibers.

*Z*is the molecular charge number of

^{i}*species i*, and

*K*

_{i}_{,f}is the hindrance factor or retardation coefficient of

*solute i*in convection transport.

*Equation 2a*is a modified form of the Nernst-Planck equation, with contributions to solute flux resulting from diffusion, ion migration, and convection.

We defined Pe and Pe_{charge} as Pe =*K _{i}
*

_{,f}

*J*

_{v}

*L*

_{f}/

*D*

_{i}_{,f}and Pe

_{charge}=

*Z*× dΨ′/d

^{i}*x*×

*L*

_{f}. The dimensionless parameter Pe is often called the Peclet number, which is a measure of relative importance of convection and diffusion to the transport of a solute. Analogously, the dimensionless parameter Pe

_{charge}is a measure of the relative importance of ion migration and diffusion to transport of a charged solute. Under the experimental conditions for frog mesenteric capillaries (2,10, 16, 17), the hydraulic conductivity (

*L*

_{p}) = 2.0 × 10

^{−7}cm · s

^{−1}· cmH

_{2}O

^{−1}, the effective filtration pressure across the microvessel wall (Δp) < 5 cmH

_{2}O, the total length of the cleft per unit surface area of the microvessel (

*L*

_{jt}) = 2,000 cm/cm

^{2}, the cleft width 2

*B*= 20 nm, and the entrance fiber layer thickness (

*L*

_{f}) = 100 nm; Pe =

*K*

_{i}_{,f}

*J*

_{v}

*L*

_{f}/

*D*

_{i}_{,f}= (

*K*

_{i}_{,f}

*L*

_{p}Δp/

*L*

_{jt}2

*B*) ×

*L*

_{f}/

*D*

_{i}_{,f}< 0.05 for a solute of radius 2.01 nm in the fiber layer (

*K*

_{i}_{,f}= 0.65 and

*D*

_{i}_{,f}= 0.025 × 10

^{−6}cm

^{2}/s; Refs. 10 and 22). If we neglect Pe,

*Eq. 2a*can be rewritten as Equation 3The boundary conditions are Equation 3a Equation 3bAt the interface of the fiber layer and the cleft entrance Equation 3cIn

*Eq. 3c*, C (

*x*,

*y*) is the solute concentration in

*region 1*of the cleft. It is assumed that the resistance to transport at the glycocalyx-solution interfaces (

*x*= −

*L*

_{f}and

*x*= 0) is much smaller than that offered by the glycocalyx itself. Therefore, as shown in Ref.8, there is a Donnan equilibrium relationship between the solute concentration in the fiber layer [C (

*x*)] and that at the lumen or the cleft side [C

_{i}(

*x*)] Equation 4where Ψ′(

*x*) and Ψ(

*x*) are the dimensionless electrical potentials inside and outside the fiber layer, respectively. At the vessel lumen, Ψ(−

*L*

_{f}) = 0, which is the reference potential.

By combining *Eqs. 3
*, *a* and *b*, and *4*, the solution of *Eq. 3
*, which satisfies corresponding boundary conditions, is
Equation 5
Here C_{i}
_{L} is the solute concentration in the lumen, which is a constant. C_{i}(0, *y*) is the solute concentration at the cleft entrance *x* = 0, which can be obtained by jointly solving the governing equation in the cleft region.*y* is the vertical coordinate with the origin at the center line of a periodic unit of the cleft (Fig. 1). At the interface of the fiber layer and the cleft entrance (*x* = 0), *Eq.3c
* becomes
Equation 6For neutral solutes (*Z ^{i}
* = 0),

*Eq.6*reduces to the expression used in previous models for uncharged molecules (11-13).

#### Cleft region.

Because there is no charge in cleft *regions 1* and*3*, and, in our case, Pe in the cleft [Pe =*K _{i}
*

_{,c}

*J*

_{v}

*L*/

*D*

_{i}_{,c}= (

*K*

_{i}_{,c}

*L*

_{p}Δp/

*L*

_{jt}2

*B*) ×

*L*/

*D*

_{i}_{,c}] is in the order of 10

^{−2}[the retardation coefficient (

*K*

_{i}_{,c}) = 0.99, the diffusion coefficient of the solute in the cleft (

*D*

_{i}_{,c}) = 0.68 × 10

^{−6}cm

^{2}/s, and the cleft depth (

*L*) = 400 nm; Refs. 10, 13, and 22] for the solute of radius 2.01 nm, the governing equation for solute transport in the cleft region can be approximated by a steady two-dimensional diffusion equation averaged over the cleft height (11-13) Equation 7C , where

*j*= 1, 3, is the concentration in

*regions 1*and

*3*of the cleft. Boundary conditions for

*Eq. 7*are Equation 7a Equation 7b Equation 7c Equation 7d

*Equation 6*is the interface boundary condition, which represents the conservation of mass from the fiber region to the cleft region. Boundary condition

*Eq. 7a*shows the continuity across the junction break, and

*Eq. 7b*indicates the impermeability of the rest part of the junction strand.

*Equation7c*indicates that concentration is a constant, C

_{i}

_{A}, at the tissue side of the cleft.

*Equation 7d*is the symmetric boundary condition. To obtain the solution of

*Eq. 7*with boundary conditions

*Eq. 7*,

*a–d*, and interface condition

*Eq. 6*, we first found Ψ(0) and ∫

*e*

^{Zi }

^{Ψ′(x)}d

*x*in

*Eq. 6*by solving modified Nernst-Planck equations for monovalent ion concentrations in the fiber layer. This process is shown in the . A numerical method similar to that in Hu and Weinbaum (15) was applied to solve for C (

*x*,

*y*) in cleft

*regions 1*and

*3*. Finally, averaged

_{i}(0) = ∫ C

_{i}(0,

*y*)/2

*D*d

*y*was substituted into

*Eq. 5*for C

_{i}(0,

*y*) to obtain C (

*x*) in the fiber region. For the case presented in Hu and Weinbaum (15), due to the high filtration pressure (43 cmH

_{2}O), the high plasma oncotic pressure (26 cmH

_{2}O), and larger molecule albumin (radius = 3.55 nm), Pe is highly nonuniform along the

*y*direction behind the surface fiber layer (

*x*= 0). It can be as high as the order of 1 over the break region (

*y*<

*d*). Therefore, the contribution from filtration cannot be neglected in their case. However, in our case, the filtration pressure is <10 cmH

_{2}O, the plasma oncotic pressure is ∼5 cmH

_{2}O (2, 10, 16, 17), and the largest Pe in the junction break is only ∼0.3 for a solute of radius 2.01 nm (11). Neglecting the convection in our case is reasonable. In another work, we will present the convection effect as well as the charge effect when high filtration pressure occurs.

The diffusive permeability (*P*) of the microvessel to a solute is defined as
Equation 8Here, C_{i}
_{L} and C_{i}
_{A} are concentrations in the lumen and in the tissue space, *L*
_{jt} is the total length of the cleft per unit surface area of the microvessel, and 2*D* is the distance between the adjacent junction breaks.*L*
_{jt}/2*D* is the total number of the breaks per unit surface area of the microvessel. Q
is the solute mass flow rate through one junction break period, which is
Equation 9

### Parameter Values

#### Cleft and fiber layer geometry.

Figure 1 shows the three-dimensional model for the interendothelial cleft and the charged surface fiber layer. Model parameters are determined according to experimental data for frog mesenteric capillaries (1, 3), which are the same as those in Fu et al. (13). The total cleft length *L* = 400 nm. The junction strand is in the middle of the cleft and its thickness, *L*
_{jun}, which is in the order of 10 nm, can be neglected compared with *L*. Therefore,*L*
_{1} = *L*
_{3} =*L*/2 = 200 nm. The cleft height 2*B* = 20 nm. The large junction break width 2*d* = 150 nm, and the average spacing between the adjacent breaks is 2*D* = 2,640 nm. Because all of the charged solutes in the current study have a diameter of 4.02 nm, which is larger than the width of the small slit 2*b*
_{s} ∼ 2 nm, the small slit is impermeable to these solutes. The total cleft length per unit area (*L*
_{jt}) = 2,000 cm/cm^{2}. In the entrance fiber matrix layer, both periodic and random fiber arrays were examined. We used fiber radius *a* = 0.6 nm and gap spacing Δ = 7 nm if periodic fiber arrays exist or volume fraction of fiber matrix S_{f} = 0.11 if random fiber arrays exist. The values of Δ or S_{f} lead to a diffusion coefficient of a solute with radius *r*
_{s} = 2.01 nm in the fiber matrix of*D _{i}
*

_{,f}= 0.025 × 10

^{−6}cm

^{2}/s (13, 22).

#### Properties of ions and charged solutes.

Two solutions were considered for ion concentration in the vessel lumen. The first was Ringer solution, whose composition was (in mM) 111 NaCl, 2.4 KCl, 1.0 MgSO_{4}, 1.1 CaCl_{2}, 0.195 NaHCO_{3}, 5.5 glucose, and 5.0 HEPES (pH = 7.4). Ringer solution with 10 mg/ml BSA (molecular weight = 69,000) was the perfusate in the experiment (2, 5, 10, 16). At pH 7.4, the molecular charge of the albumin is about −19 (8). The charge density of albumin is 0.144 mM × 19 = 2.7 meq/l, which is negligible compared with the ion charge density (in the order of 100 meq/l). Thus, in Ringer solution, the cation concentration is taken as the same as that of anions: C_{+} = C_{−} = 118 M.

Another solution was blood plasma. The concentration of plasma proteins in the vessel lumen was 1 mM, and the valency was assumed to be −17 (4). A simplified plasma has cations (155 mM Na^{+}) and anions (138 mM Cl^{−}) to satisfy electrical neutrality. Therefore, C_{+} = 155 mM and C_{−} = 138 mM in plasma.

On the tissue side of the fiber layer at the cleft entrance, we assumed a negligible concentration of proteins because the fiber layer behaves as the molecular filter to proteins (22). To satisfy the electrical neutrality, C_{+} should be identical to C_{−} on the tissue side of the fiber layer, whose values are dependent on C_{+} and C_{−} in the lumen as well as the charge density of the surface glycocalyx at the interface of the cleft entrance. These values are calculated using *Eq. EA3
* in the
at *x* = 0.

Because most of the cations are Na^{+} and most of the anions are Cl^{−}, we used the diffusion coefficients of Na^{+} and Cl^{−} as those for cations and anions, respectively: *D*
_{+} = 1.506 × 10^{−5} cm^{2}/s and *D*
_{−}= 1.999 × 10^{−5} cm^{2}/s. These values of*D*
_{+} and *D*
_{−} were calculated for T = 20°C using the values for T = 37°C given in Ref. 8. T of 20°C is the temperature in the experiments for measuring the permeability of frog mesenteric microvessels.

*D*
_{free} = 1.07 × 10^{−6}cm^{2}/s and is the free diffusion coefficient of a solute with radius *r*
_{s} = 2.01 nm (both ribonuclease and α-lactalbumin) at T = 20°C. The corresponding diffusion coefficients in the cleft and in the fiber matrix layer are*D _{i}
*

_{,c}= 0.68 ×10

^{−6}cm

^{2}/s and

*D*

_{i}_{,f}= 0.025 × 10

^{−6}cm

^{2}/s, respectively (10, 22).

## RESULTS

### Electrical Potential Profiles

Figure 3 shows the electrical potential profiles across the surface glycocalyx layer. When the perfusate in the vessel lumen is Ringer solution, in which the cation and anion concentrations C_{+} = C_{−} = 118 mM, the results are shown in Fig. 3
*A*. Figure3
*B* shows the results for the plasma as the perfusate in which C_{+} = 155 mM and C_{−} = 138 mM. The electrical potential is determined by the distribution of cations C_{+} and anions C_{−} and the charge density*C*
_{m} of the surface glycocalyx. In Fig.3
*A*, in both the lumen and cleft regions, C_{+}= C_{−} = 118 mM. At the interface of the fiber layer and the lumen (*x* = −*L*
_{f}), for*case 1*, when *C*
_{m} =*C*
_{m0} = 25 meq/l, there is a step decrease in electrical potential from its initial value of*E*(−*L*
_{f}) = 0 to*E*′(−*L*
_{f}) = −2.7 mV. Because*C*
_{m} is constant across the fiber layer and*J*
_{v} = 0, there is no change in *E*′ within the fiber layer. At the interface of the fiber layer and the cleft entrance, there is a step increase in electrical potential from*E*′(0) = −2.7 mV to*E*(0) = 0. For *case 2* of*C*
_{m}(*x*) =*C*
_{m0} tanh (1 +*x*/*L*
_{f})/ tanh (1), where*C*
_{m}(−*L*
_{f}) = 0, there is no jump in *E* at *x* = −*L*
_{f}, whereas *E*′(*x*) decreased from *x* = −*L*
_{f} to 0 due to changes in *C*
_{m}(*x*). At the interface between the fiber layer and cleft entrance, there is a step increase in electrical potential from *E*′(0) = −2.7 mV to *E*(0) = 0. For *case 3* of*C*
_{m}(*x*) =*C*
_{m0} tanh (−*x*/*L*
_{f}) tanh (1), the change in the electrical potential is complementary to that of*case 2*. In *case 3*, there is a step decrease at*x* = −*L*
_{f}, whereas*E*′(*x*) increases from *x* = −*L*
_{f} to 0. At the interface between the fiber layer and cleft entrance, *x* = 0, there is no change in*E*. In *case 4*, when*C*
_{m}(*x*) =*C*
_{m0} tanh (2 + 2*x*/*L*
_{f})/ tanh (1) (−*L*
_{f} < *x* < −*L*
_{f}/2),*C*
_{m}(*x*) =*C*
_{m0} tanh (−2*x*/*L*
_{f})/ tanh (1) (−*L*
_{f}/2 < *x* < 0); there are no changes in *E* at both interfaces between the fiber layer and lumen/cleft entrance due to*C*
_{m}(−*L*
_{f}) =*C*
_{m}(0) = 0 and C_{+} = C_{−} in the lumen and in the cleft.*E*′(*x*) decreases first from *x* = −*L*
_{f} to −*L*
_{f}/2 and then increases from *x* = −*L*
_{f}/2 to 0. At the interface of the fiber layer and the cleft, it returns to its initial value*E*(0) =*E*(−*L*
_{f}) = 0. The difference between the maximum and the minimum in *E*, 2.7 mV, is the same for all the cases.

In Fig. 3
*B*, the perfusate in the lumen is plasma, in which C_{+} = 155 mM and C_{−} = 138 mM. These unequal ion concentrations induce a large difference in electrical potential profiles, although the charge density distributions are the same as those shown in Fig. 3
*A*. For *case 1* of constant *C*
_{m} = 25 meq/l, *E*decreases suddenly from 0 to −0.7 mV at *x* = −*L*
_{f}, does not change across the entire fiber layer, and suddenly increases from −0.7 to 1.5 mV at *x*= 0. For *case 2*, in which*C*
_{m}(−*L*
_{f}) = 0, there is a sudden increase in *E* at *x* = −*L*
_{f} from 0 to 1.5 mV; *E* then decreases gradually across the fiber layer to −0.7 mV at the fiber layer exit *x* = 0^{−} and finally jumps to 1.5 mV at the entrance of the cleft *x* = 0^{+}. For*case 3*, in which*C*
_{m}(−*L*
_{f}) =*C*
_{m0}= 25 meq/l, there is a sudden decrease in *E* at *x* = −*L*
_{f} from 0 to −0.7 mV; *E* then gradually increases across the fiber layer to 1.5 mV at*x* = 0. For *case 4*, in which*C*
_{m}(−*L*
_{f}) = 0, there is a sudden increase in *E* from 0 to 1.5 mV at*x* = −*L*
_{f}; *E* then gradually decreases to −0.7 mV at *x* = −*L*
_{f}/2 and increases to 1.5 mV at*x* = 0. The difference between the maximum and minimum in *E* is the same again for all the cases due to the same*C*
_{m0}. However, the difference is 2.2 mV, when C_{+} = 155 mM and C_{−} = 138 mM, instead of 2.7 mV, when C_{+} = 118 mM and C_{−} = 118 mM.

As shown earlier, charges can affect molecular transport in two ways: *1*) through the ion migration term in solute flux expression (the second term in *Eq. 2
*), and *2*) through electrostatic partitioning of charged solutes at interfaces (*Eq. 4
*). For *case 1* of constant*C*
_{m}, d*E*′/d*x* = 0 (or dΨ′/d*x* = 0), the charge effect on the solute transport comes only from the electrostatic partition at two interfaces of the fiber layer (*x* = −*L*
_{f} and*x* = 0). This partition is the same at both interfaces when C_{+} = C_{−} = 118 mM, whereas the partition is larger at *x* = 0 when C_{+} = 155 mM and C_{−} = 138 mM in the lumen. For all cases of*C*
_{m} =*C*
_{m}(*x*), d*E*′/d*x*≠ 0 (or dΨ′/d*x* ≠ 0), the charge contributes to the solute migration term in *Eq. 2
*. Under the condition of C_{+} = C_{−} = 118 mM, there is no partition at both interfaces for *case 4*, whereas there is a partition at *x* = −*L*
_{f} for*case 3* and a partition for *case 2* at*x* = 0. Under the conditions of C_{+} = 155 mM and C_{−} = 138 mM, there is a partition at*x* = −*L*
_{f} for *cases 3* and *4* and partitions at two interfaces for *case 2*.

### Concentration Distributions of Solutes

The dimensionless concentration distributions of positively charged ribonuclease (+3), negatively charged α-lactalbumin (−11), and a neutral solute (0) of same size are shown in Fig.4. Figure 4
*A* shows the cases when C_{+} = C_{−} = 118 mM, and Fig.4
*B* shows the cases when C_{+} = 155 mM and C_{−} = 138 mM. For a neutral solute that is not affected by charge, its dimensionless concentration (dotted line) decreased gradually from 1 in the lumen to 0.65 at the exit of the fiber layer. This decrease is due to the size effect, e.g., the steric hindrance and diffusion resistance of the fibers to the solute, and is the same under various charge densities (*C*
_{m}) for the same size solute. Figure 4, *top*, shows the concentration distributions when *C*
_{m} = constant = 25 meq/l, in which case there is no charge effect within the fiber layer (d*E*′/d*x* = 0). Under *condition a*, the concentration of positively charged ribonuclease (solid line) abruptly increased by electrical partition at *x* = −*L*
_{f,} gradually decreased in the fiber layer due to the size effect, and abruptly decreased further at *x*= 0 due to electrical partition. For negatively charged α-lactalbumin (dashed line), the electrical partitions at interfaces provide an opposite effect from that for ribonuclease. Under *condition b*, due to unequal electrical potential differences at*x* = −*L*
_{f} and *x* = 0 (see Fig. 3
*B*), the electrical partitions are different from those in *condition a*. For example, the concentration of ribonuclease first increased from 1 to 1.08 at *x* = −*L*
_{f}, gradually decreased to 0.76 across the fiber layer, and had a step decrease from 0.76 to 0.59 at*x* = 0. This value of 0.59 is even lower than the concentration for the neutral solute at the same location, which was 0.65. In contrast, the concentration of negatively charged α-lactalbumin at *x* = 0 jumped from 0.32 to 0.83, which is higher than 0.65. This electrical partition [Ψ(0) − Ψ′(0) > 0; see Fig.3
*B*] favors the transport of negatively charged α-lactalbumin (*Eq. 4
*). This induces an interesting effect on total permeability of a microvessel to solutes with different charges, which will be shown in Fig.5
*B*.

Figure 4, *top middle*, *bottom middle*, and*bottom*, show the cases when *C*
_{m} =*C*
_{m}(*x*), corresponding to those in Fig.2. In these cases, d*E*′/d*x* no longer equals to zero, and there is a contribution from d*E*′/d*x* to the transport of the charged solutes within the fiber layer in addition to electrical partitions at the entrance and exit of the fiber layer. In *cases 2* and *4*, where*C*
_{m}(−*L*
_{f}) = 0, under*condition a*, when C_{+} = C_{−}= 118 mM, there was no electrical partition at *x* = −*L*
_{f}; the concentration is the same as that in the lumen for all solutes with and without charges. However, under*condition b*, when C_{+} = 155 mM and C_{−} = 138 mM, the electrical partition at*x* = −*L*
_{f} favors the transport of negatively charged α-lactalbumin instead of positively charged ribonuclease due to the positive jump in the electrical potential at*x* = −*L*
_{f} (Fig. 3
*B*). In *cases 3* and *4*, where*C*
_{m}(0) = 0, there was no electrical partition at *x* = 0. Although concentration profiles in the fiber layer are varied for various*C*
_{m}(*x*), the concentration at*x* = 0, the exit of the fiber layer, or the entrance of the cleft, is the same for all cases. Under *condition a*, the exit concentration for ribonuclease is 0.69; for α-lactalbumin, it is 0.49. Under *condition b*, the exit concentration is 0.57 for ribonuclease but 0.96 for α-lactalbumin. The higher the exit concentration, the lower the resistance of the fiber layer to a solute. Because the resistance of the cleft region is the same for ribonuclease, α-lactalbumin, and a neutral solute with the same size, the total resistance or permeability of the microvessel wall to a solute is the same for all *C*
_{m}(*x*) shown in Fig. 2.

### Charge Effect of Surface Glycocalyx Layer on Permeability

Figure 5 shows the ratio of charged solute permeability to neutral solute permeability of the same size as a function of the maximum value *C*
_{m0} in charge density*C*
_{m} shown in Fig. 2. Solid lines with circles are the results for ribonuclease and the dashed lines with triangles are the results for α-lactalbumin when *C*
_{m} = constant = *C*
_{m0} (*case 1* in Fig. 2). The dotted lines with squares are the results for ribonuclease and the dashed-dotted lines with diamonds are the results for α-lactalbumin when *C*
_{m} is*C*
_{m}(*x*) (*cases 2–4* in Fig. 2). Figure 5
*A* shows the condition of C_{+} = C_{−} = 118 mM, and Fig.5
*B* shows the conditions of C_{+} = 155 mM and C_{−} = 138 mM. As discussed above, the permeability of the microvessel wall to a charged solute is identical for the different charge-density distributions *C*
_{m}(*x*) shown in Fig. 2. However, the permeability of the microvessel with varied *C*
_{m} =*C*
_{m}(*x*) is different from that with constant *C*
_{m}, although the maximum value*C*
_{m0} is the same. In general, the permeability under the constant *C*
_{m} case is larger than that under varied *C*
_{m} for positively charged ribonuclease but smaller for negatively charged α-lactalbumin. The reason for this is that the total charge of the surface glycocalyx for constant *C*
_{m} is larger than that for varied *C*
_{m}. The effect of varied maximum value *C*
_{m0} but fixed total charge is shown in Fig. 6.

### Comparison of Previous and Current Models

Figure 6 shows predictions of current and previous models for the ratio of ribonuclease permeability*P*
^{ribonuclease} to α-lactalbumin permeability *P*
^{α-lactalbumin}. Figure6
*A* shows C_{+} = C_{−} = 118 mM, and Fig. 6
*B* shows C_{+} = 155 mM and C_{−} = 138 mM. The dashed lines with squares in Fig. 6,*A* and *B*, are the results from the previous Donnan-type model (*Eq. 1
*), which was used in Refs.2, 6, 16, and 17. This model neglected the thickness of the fiber layer and therefore neglected the steric and electrostatic interactions between solutes and the glycocalyx within the layer. The solid lines with circles, dashed-dotted lines with diamonds, and dotted lines with triangles are the current model results, when *C*
_{m} = constant and *C*
_{m} =*C*
_{m}(*x*), correspondingly. The difference between the solid lines and dashed-dotted lines is that*C*
_{m0} in the dashed-dotted lines is one-half of that in the solid lines, but fiber layer thickness*L*
_{f} is doubled to keep the same total charge. We can see from Fig. 6 that with the same total charge and charge distribution, the lower the maximum value*C*
_{m0} and the larger the charge effect. Figure 6 also shows that the previous model may overestimate the charge effect of the surface glycocalyx layer by ignoring the layer thickness.

### Combined Effect of Charge and Thickness of the Glycocalyx Layer

The ratio of varied α-lactalbumin permeability *P* to its permeability under the condition of the zero-charge glycocalyx layer with thickness *L*
_{f} = 100 nm is shown in Fig. 7, as a function of charge density *C*
_{m0} and fiber layer thickness *L*
_{f}. Figure 7
*A* shows C_{+} = C_{−} = 118 mM, and Fig.7
*B* shows C_{+} = 155 mM and C_{−} = 138 mM. Obviously, α-lactalbumin permeability*P* decreases with increasing fiber layer thickness*L*
_{f}. When *L*
_{f} is increased from 100 to 400 nm, the reduction in*P*
^{α-lactalbumin} is always ∼50–70% no matter how much *C*
_{m0} is in both*conditions a* and *b*. With increasing*C*
_{m0}, the relative decreasing effect of *L*
_{f} becomes larger.

## DISCUSSION

The combined junction-orifice-fiber matrix model in Ref.13 demonstrated previously that the endothelial surface glycocalyx and the junction strands in the interendothelial cleft provided significant size restriction to the diffusion of intermediate-sized solutes across the walls of continuous microvessels of the frog mesentery. The results of our current model demonstrate that both solute charge and charge carried by the surface glycocalyx also determine the selectivity of the microvessel wall. The model predictions conform to the hypothesis that under previous experimental conditions, the fixed negative charge within transcapillary pathways for intermediate-sized proteins restricts the transport of negatively charged solutes more, relative to positively charged solutes of the same size. However, this effect is counteracted to some extent if the solution on the luminal side contains negatively charged proteins that cannot enter the layer.

From Fig. 6
*A*, to account for the twofold difference in*P*
^{ribonuclease} and*P*
^{α-lactalbumin} observed in Ref. 2under Ringer-BSA perfusion, *C*
_{m0}would be ∼25 meq/l for the case of charge density*C*
_{m} = constant (*case 1* in Fig. 2) and *C*
_{m0} would be ∼35 meq/l for cases of varied *C*
_{m} =*C*
_{m}(*x*) (*cases 2–4* in Fig. 2). These values are significantly larger than 11 meq/l, the prediction from the previous electric partition-only model (2). Under the condition of orosomucoid perfusion, the increased ratio of *P*
^{ribonuclease} to*P*
^{α-lactalbumin} from two to six and the decrease in *P*
^{α-lactalbumin} to 0.47 of*P*
^{α-lactalbumin} under Ringer-BSA perfusion (6) could be explained if*C*
_{m0} increased from ∼25 to ∼55 meq/l for *case 1* and from ∼35 to ∼75 meq/l for*cases 2–4*, a roughly twofold increase. The prediction of the previous model was from 11 to 28 meq/l (6). The change in *P*
^{α-lactalbumin} by plasma, to 0.31 of*P*
^{α-lactalbumin} under Ringer-BSA perfusion, observed in Huxley et al. (17) could be accounted for if*C*
_{m0} increased from ∼25 to ∼60 meq/l for *case 1* and from ∼35 to ∼80 meq/l for*cases 2–4*. The previous model predicted that*C*
_{m0} was increased from 11 to 34 meq/l. Because of the thickness of the glycocalyx layer was ignored, the simple Donnan-type model used previously may overestimate the charge effect of the fiber matrix.

Figure 4 shows the dimensionless concentration distribution in the glycocalyx layer for ribonuclease with a net charge +3, α-lactalbumin with a net charge −11, and a neutral solute with the same size. When C_{+} = C_{−} = 118 mM, for *cases 1* and *3*, there is a charge partition at the entrance of the glycocalyx layer (*x* = −*L*
_{f}), which favors the passage of positively charged ribonuclease; there is no charge partition for *cases 2* and *4*. When C_{+} = 155 mM and C_{−} = 138 mM, the charge partition at the entrance of the glycocalyx layer still favors the transport of ribonuclease in *cases 1* and *3*; however, in *cases 2* and *4*, the partition favors the negatively charged α-lactalbumin. The reason for this is that the electrical potential *E* across the interface (*x* = −*L*
_{f} or *x* = 0), which determines the charge partition, not only depends on charge density *C*
_{m}(*x*) of the glycocalyx layer but also on the concentrations of cations C_{+} and anions C_{−} in the lumen and cleft (Fig. 3). This dependence of*E* on *C*
_{m}(*x*), as well as C_{+} and C_{−}, induces the dependence of the solute permeability P on the solute charge and the charge density*C*
_{m}(*x*) of the glycocalyx, as well as C_{+} and C_{−}, which brings us the interesting phenomenon shown in Fig. 5
*B*.

Despite the various concentration distributions (Fig. 4) due to different profiles of charge density*C*
_{m}(*x*) within the glycocalyx layer, the concentrations of charged solutes are the same at the entrance of the cleft behind the glycocalyx layer for *cases 2–4*, provided that these density profiles have the same maximum value*C*
_{m0} and the same total charge (Fig.2). For this sake, the overall solute permeability across the microvessel wall including the surface glycocalyx layer and the cleft region is identical regardless of*C*
_{m}(*x*) profiles. If we keep the same maximum value *C*
_{m0}, the larger the total charge of fibers, the larger the charge effect. In contrast, for the same total charge, the larger the*C*
_{m0}, the smaller the charge effect. These results are shown in Fig. 6.

Figure 5 shows the ratio of *P*
^{ribonuclease} to*P*
^{neutral} ^{solute} and the ratio of*P*
^{α-lactalbumin} to*P*
^{neutral} ^{solute} as a function of*C*
_{m0}
_{,} the maximum value of charge density in the glycocalyx layer. When proteins exist in the plasma, there are unequal concentrations of cations (Na^{+}) and anions (Cl^{−}) in the lumen, C_{+} = 155 mM and C_{−} = 138 mM (4). Figure5
*B* shows that when*C*
_{m0} < 35 meq/l for constant*C*
_{m}, and*C*
_{m0} < 55 meq/l for varied*C*
_{m}(*x*), the permeability *P*of negatively charged α-lactalbumin is higher than that of positively charged ribonuclease, although the surface glycocalyx carries negative charges. As discussed above, this phenomenon is due to the dependence of *P* not only on solute charge and charge density of the fiber matrix *C*
_{m}(*x*), but also on C_{+} and C_{−.} In particular, if*C*
_{m0} is small, the negative charge in the layer is not enough to overcome the favored electrical partition to negatively charged α-lactalbumin due to the presence of negatively charged proteins in the lumen. This prediction may be used in controlled drug delivery by locally modulating*C*
_{m}(*x*), C_{+}, and C_{−} for a certain drug with fixed charge.

In plasma perfusion compared with albumin perfusion, the thickness of the glycocalyx is greatly increased (1). We tested the combined effect of charge and thickness of the glycocalyx on*P*
^{α-lactalbumin} in Fig. 7. When charge density*C*
_{m0} is in the range of 25–35 meq/l, as predicted by our model during albumin perfusion, the twofold increase in fiber thickness *L*
_{f} from 100 nm (albumin perfusion) to 200 nm (plasma perfusion) (1) would only decrease *P*
^{α-lactalbumin} by ∼33%, not enough to account for the ∼70% decrease observed in Ref.15. However, if we additionally increase*C*
_{m0} by twofold, the combined effect of increasing both charge density*C*
_{m0} and the thickness*L*
_{f} would account for the 70% decrease in*P*
^{α-lactalbumin}.

In summary, we developed a two-dimensional model incorporating the charge effects of the endothelial surface glycocalyx so that it can provide a more detailed quantitative analysis of various experimental results expected to be associated with negative charge in transvascular pathways. This will also help understand the physical mechanisms of glycocalyx selectivity and provide a new method for controlling transport rates of charged solutes in drug delivery. In the current study, we only analyzed the case when the hydraulic conductivity*L*
_{p} of the microvessel and the pressure difference across the vessel wall Δp are normal; the convection effect due to enhanced *L*
_{p} and Δp will be discussed in our next study.

## Acknowledgments

This work was supported by National Cancer Institute Grant R15-CA-86847-01 and University of Nevada (Las Vegas, NV) New Investigator Awards and Applied Research Initiative grants.

## Appendix

This appendix shows the equations used to calculate the electrical potential profiles in the surface glycocalyx layer [Ψ′(*x*)] and to find Ψ′(0) and ∫
*e*
^{Zi
}Ψ′(*x*)d*x*in *Eq. 5
*. It was assumed that overall electroneutrality is satisfied in the glycocalyx layer
Equation A1Here, C′_{i} is concentration of positive (*i* = +) and negative (*i* = −) monovalent ions and charged macromolecules [*i* = test solute (TS)] in the glycocalyx layer and *Z ^{i}
* is the corresponding electrical valence.

*Equation EA1*indicates that the negative charge

*C*

_{m}(

*x*) carried by the fiber matrix must be balanced by an excess of mobile positive ions. Usually, the concentration of charged macromolecules is negligible compared with the concentrations of ions (see

*Parameter Values*), and

*Eq. EA1*reduces to the balance between monovalent cations (

*Z*

^{+}= +1) and the summation of monovalent anions (

*Z*

^{−}= −1) and negative charges of the fiber matrix Equation A2At the interface between the fiber layer and lumen (

*x*= −

*L*

_{f}in Fig. 1) and at that between the fiber layer and cleft entrance (

*x*= 0), the Donnan equilibrium is satisfied. This gives

*Eq. 4*in the main text, which is Combining

*Eqs. 4*and

*EA2*gives Equation A3 The condition for no electrical current flows across the glycocalyx layer is Equation A4Neglecting the current due to the macromolecules,

*Eq.EA4*reduces to Equation A5Under normal conditions in frog mesenteric capillaries,

*L*

_{p}= 2.0 × 10

^{−7}cm · s

^{−1}· cmH

_{2}O

^{−1}, Δp ∼ 5 cmH

_{2}O,

*L*

_{jt}= 2,000 cm/cm

^{2}, 2

*B*= 20 nm, and

*L*

_{f}= 100 nm, so that Pe =

*J*

_{v}

*L*

_{f}/

*D*

_{i}_{,f}= (

*L*

_{p}Δp/

*L*

_{jt}2

*B*) ×

*L*

_{f}/

*D*

_{i}_{,f}≈ 10

^{−3}for both monovalent cations and anions.

*D*

_{i}_{,f}≈

*D*

_{+}for cations and

*D*

_{i}_{,f}≈

*D*

_{−}for anions. The modified Nernst-Planck equations written for positive and negative ions are Equation A6 Equation A7Here, f is the void volume of the fiber matrix; f = 0.98 in our case (13, 22). The conditions of electroneutrality (

*Eq. EA2*) and zero current flow (

*Eq. EA5*) can be used to eliminate C′

_{−}and

*J*

_{−}from

*Eqs. EA6*and

*EA7*so that Equation A8 Equation A9

*Equation EA8*is solved for C (

*x*) by numerical integration, with initial values of C at

*x*= −

*L*

_{f}obtained from

*Eq. EA3*. Substituting obtained C (

*x*) into

*Eq.EA9*and Ψ′(

*x*) is solved by numerical integration with Ψ′(−

*L*

_{f}) from

*Eqs. 4*and

*EA3*. An iterative procedure was used, with values of

*J*

_{+}adjusted until the following relation was satisfied by where

## Footnotes

Address for reprint requests and other correspondence: B. M. Fu, Dept. of Mechanical Engineering and Cancer Institute, Univ. of Nevada, 4505 Maryland Pkwy., Box 454027, Las Vegas, NV 89154 (E-mail: bmfu{at}nscee.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.First published December 12, 2002;10.1152/ajpheart.00467.2002

- Copyright © 2003 the American Physiological Society