Transport of charged macromolecules across a biological charged membrane

Computer Programs in Biomedicine 19 (1985) 107-117

107

Elsevier CPB 00676

Transport of charged macromolecules across a biological charged membrane M. V a n D a m m e a n d M . P r e v o s t Queen Elisabeth Medical Foundation, Avenue J.J. Crocq 1, 1020 Brussels, Belgium

The glomerular capillary wall of the kidney behaves as an electronegatively charged structure consisting of three layers, the lamina densa and the two laminae rarae, which are differently charged. Thus, a three layer model is proposed to analyse the transport of charged macromolecules across this wall. A modified Nernst-Planck equation describes the macromolecule flux across the wall and a Donnan equilibrium is assumed at each interface. For a given value of the fixed charge concentration in each layer, the local sieving coefficient of the macromolecule, i.e. the ratio between the concentrations in the filtrate and in the plasma, is calculated. A sieving curve which relates the sieving coefficient to the Einstein-Stokes radius of the macrosolute is obtained. The fixed charge concentrations in each layer are iteratively modified until simultaneous adjustment is achieved between calculated and experimental curves, for positively and negatively charged tracers and their neutral equivalent. Biophysical models Basementmembrane Permeability of charged membranes Macromolecules Transport of macromolecules

1. I N T R O D U C T I O N The transport of polyanions across the glomerular capillary wall of the kidney is restricted c o m pared with that of neutral macromolecules of the same molecular size [1-3]. Conversely, the sieving of polycations is facilitated [4]. This p h e n o m e n o n is attributed to the presence of fixed negative charges in the glomerular capillary wall, especially the glomerular basement m e m b r a n e (GBM) [5-10]. In a previous theoretical work, Deen et al. idealized the glomerular wall as a porous and uniformly charged m e m b r a n e [11,12]. Assuming the existence of a D o n n a n equilibrium at the interface between the m e m b r a n e and the plasma or the filtrate, they estimated the value of the fixed charge concentration in the membrane. The 'electrostatic partitioning' of the charged solutes due to the fixed charge concentration represented the only factor responsible for charge selectivity, since the

electrical potential gradient developed within the m e m b r a n e was shown to be negligible. In our model [13], the morphological structure of the glomerular capillary wall is better taken into account. Three layers in series, characterized by different pore sizes and different concentrations of fixed negative charges, form the membrane. The p r o g r a m described in this paper calculates the concentration of fixed negative charges within each layer of the model. This study is a generalization of a p r o g r a m describing the transport of uncharged macromolecules across the glomerular capillary wall [14-17].

2. T H E M A T H E M A T I C A L M O D E L The glomerular capillary network is idealized as a single tube of u n k n o w n length and radius such that its surface area (Am) is equal to the sum of

0010-468X/85/$03.30 © 1985 Elsevier Science Publishers B.V. (Biomedical Division)

108

/

\

I t t t t - qpa-y

I I

I I

I i

i

i

,

~o

¢~i

Oti + / X O(

i I I

~n

Fig. 1. Idealized representation of the glomerular capillary network. Symbols in this figure are as defined in the text and Appendix.

CapiItary [umen I

LRI 2

lib 6m LR

mb LD 3

AmLO

IV@

IVb LRE 4

5m LR Va

Vb Urinary space 5

Fig. 2. Schematic representation of the three layers of the glomerular basement membrane, a and fl denote position in the tube and the capillary wall respectively. A m LR and A m LD are the L R and the L D thicknesses. Levels II, III, IV and V represent the inlet of compartment 2, 3, 4 and 5 respectively. At interface I, the subscript a represents the outlet of compartment i - 1 and b represents the inlet of compartment i (i ~ 1).

the filtering surfaces of all the capillaries present in the network (Fig. 1) [14,16,18]. The glomerular capillary wall is modelled as a three-layer membrane with different charge concentrations and different pore sizes. The anatomical equivalents of these layers are the lamina rara interna (LRI), the lamina densa (LD) and the lamina rara externa (LRE) of the GBM. The mechanical resistance is assumed lower in the LR than in the LD, which implies that the pores are larger in the LR. This assumption is based on the fact that large dextran molecules penetrate into the LRI but not into the LD [19]. In each layer, the pores are straight and uniformly distributed cylinders of equal length and radius. Separation of the GBM into three layers with different charge concentrations is justified by the presence of different charged chemical components in the LR (heparan sulphate-like components) [7-10] and in the LD (sialoglycoproteins) [5]. Moreover, the binding of electropositive molecules to the anionic sites in the LR suggests that the LR play a predominant role in the constitution of the electrical barrier. The plasma and the glomerular filtrate are approximated as solutions containing Na and monovalent anions (An). Plasma also contains a certain amount of proteins, which have a net negative charge, while the glomerular filtrate is almost protein free. As illustrated in Fig. 2, as the macromolecule travels from the capillary lumen to the urinary space, it passes through five media (i) and crosses four interfaces (l).

2.1. The hydrodynamic model The macromolecules under study are used at tracer concentrations, so that water flow is not influenced by the presence of solute but depends only on the local effective ultrafiltration pressure (Put) and on the hydraulic permeability of the membrane (Lp). The hydrostatic pressure is assumed to vary linearly along the tube.

p¢(a) = pc.(1 - e - a )

(1)

where Pc. is the afferent hydrostatic pressure, a is the normalized distance from the afferent end, e is

109 the pressure drop equal to (Pc,-Pco)/P~., and Pco is the efferent hydrostatic pressure. The local effective ultrafiltration pressure is given by Starling's law:

The hydrostatic pressure in the urinary space (Pus) is constant. Plasma oncotic pressure (%) is related to plasma protein concentration (%) by:

%(a) =

%(a) 2 (3)

where CPR01, CPR02 and CPR03 are numerical factors respectively equal to 1.630, 0.294 and 0.0 for rats [20] and 2.283, 0.0 and 0.022 for dogs [21]. The local values of plasma flow are given by: dqp(a)

da

KfP°f(a)

(4)

where the ultrafiltration coefficient (Kf) is equal to Lp x Am. The cumulative filtration rate along the tube, from the inlet 0 to point a is: y(O,a)=K,,

foap.f(a)da

where %, and %, are plasma flow and plasma protein concentration, respectively, at the entry of the tube.

2.2. Transport of solute We shall call X, (i = 2 to 4) the fixed negative charge concentration in compartment i. The electrical potential difference (~klb- ~l,) at the interface l between the compartments i and (i + 1) may be calculated applying the Donnan equations

(7)

CNa(i+ 1)

(8) (9)

CNa(l ) "~- CAn(1) "{-¢p

where R is the gas constant, F is the Faraday, T is the absolute temperature, CNa(1) and CAnO) are the sodium and small anion plasma concentrations, respectively, and la and I b represent both sides of interface 1. The solute molecules are simulated by rigid spheres of Einstein-Stokes radius a s. The net transport of these macromolecules results from the diffusion, ion migration and convection processes. In a stationary state, the solute flow (Js) and the solvent flow (Jr) per unit of membrane area are both independent of the position (fl) across the capillary wall. At each point of coordinates (a, fl), Js is given by the local modified Nernst-Planck equation:

ys(,O=

(6)

CNaO)

) + 4 CNa(1)CAn(l)] 1/2 /(i+1) + [ X(i+, 2 2

CNa(i+ 1) =

(5)

As filtration proceeds, plasma protein concentration increases and, at point a along the tube, it is expressed by

qP, Cp(a)=Cpaqp(a )

~kl, - ~l~ = - ~ l n

(2)

po,, (,,,) = pc (,,,) - P.s - .,,-,. (,,,)

CPRO1. Cp(a) + CPR02. 1 - CPR03. Cp(a)

[11]:

K2

X

&.yv(

B) -/m SD

[ des(a, f l ) Z ~ ' F c - a ,

dfl

+ ~

s(

fl)

d~d~ ] (io)

The first term accounts for convective transport and depends on jr(a) and on the local solute concentration, cs(a, fl). The second term represents the diffusion and is proportional to the concentration gradient across the membrane and to Ds, the free diffusion coefficient of the solute calculated by Einstein's law. fm is the ratio of pore area and membrane area. The third term takes into account the effects of electrical charge: it reduces to zero when the solute under study is uncharged or if the electrical potential does not vary along the fl axis. Z s is the charge number of

110 the solute. S D and S F a r e steric hindrance factors for diffusion and filtration, and are derived as described previously [16]. 1 / K 1 and K 2 / K a are the wall frictional factors for diffusion and filtration, calculated by Haberman and Sayre for rigid spheres in axial position [22,23]. These last four parameters depend only on the ratio a J r , where r is the pore radius. The macromolecule concentration in the filtrate is the ratio between the solute flow and the solvent flow:

= js( )

(11)

Jv( )

q ( a ) is the solute concentration in the plasma and ~ ( a ) is the sieving coefficient of the macromolecule, i.e. the ratio between the concentration of the macromolecule in the filtrate and that in the plasma. After transformation, the Nernst-Planck equation (eq. 10) is written:

equilibrium relation is obtained between the macromolecule concentration on both sides of each interface: Cs(O/, l b ) = C s ( a , la) exp

Zs

_

(13)

The solvent flow per membrane surface area is calculated by application of Poiseuille's law:

~rr4N j v ( a ) = 1333.22 8*/.Am .Am puf(a)

(14)

N is the total number of pores, A m is pore length, ~/the viscosity of the filtrate and pue(a) the pressure drop across the pore. The factor 1333.22 takes into account the different units on both sides of eq. 14. To characterize the flow across the three layers of the membrane, the pressure drop in the laminae rarae (LR) and in the lamina densa (LD) are determined as follows: puL?(a) = ~¢puf(O~)

(15)

pLD(a) = (1 -- 2 y ) p o f ( a )

(16)

d[c,(a, fl)/Cl(a)] dB

Jv( )

L,-Ds[

cs( , ,)

K,

<(,,)

sD

cs( , #) zs.r c,(a)

RT

dfl

]

(12)

Equation 12 is a local equation and must be integrated across the three layers of the membrane. The variation of G/ca along the fl axis and the sieving coefficient ~ are calculated at each point along the a axis, taking into account the variation of c 1 along the tube. 2.2.1. Calculation of cJc~ along the fl axis In each layer of the membrane, eq. 12 describes the transport of the macromolecule. As shown by Deen et al., the electrical potential gradient within the layers is expected to be quite small, so that the migration term in eq. 12 is neglected [11]. Assuming that the resistance to transport is much lower at interfaces than within the layers, an

Two arbitrarily chosen values are given to ~, (0.01 and 0.1) which are supposed to describe limit situations. From eqs. 14-16 and assuming the number of pores to be identical in the three layers, the ratio of the pore radius in the LR and LD is given by:

=

[1-2 ~

Am LD

(17)

where the ratio A m L R / A m LD is derived from morphological estimations. For the molecular coils under study (dextran sulphate (DS) and diethylaminoethyl dextran (DEAE.D)), the charge number is derived from electrophoretic measurements and is related to the Einstein-Stokes radius by: Z s = CH1 .a s + CH2

(18)

where a s is given in .& and CH1 and CH2 are numerical factors. From electrophoretic mobihty

111

measurements of DS, we derived CH1 and CH2 equal to -0.32 and 0.756 [21]. For DEAE.D, Deen et al. found CH1 and CH2 equal to + 0.090 and 0.0 [11].

on which all values are considered to be constant, the following relationships are applicable:

2.2.2. Calculation of cI along the a axis c 1 is modified along the tube under the combined influence of water loss and solute transport across the membrane. At each point along the tube, c 1 is given by:

qp(a + A a ) = q p ( a ) - y ( a ,

c,(a) = q(O) qp~ - z(O, a) q p . - y ( O , a)

(19)

where z(0, a) is the cumulative plasma clearance of the solute from the afferent end up to the considered point, the clearance being the solute flow rate across the membrane divided by its arterial concentration, ca(0 ) . z(0, a) =

gf

a

f ¢ ( a ) . p u f ( a ) . c,(a) da go

(20)

The mean calculated sieving coefficient (!~cal) is defined as the ratio of the solute plasma clearance ( Z ) and the total filtration (Y): Z • ¢,,I = -~

(21)

The sum of weighted quadratic errors on ~ , l is calculated over all molecular sizes, as previously described [17]. )-"~E = Y'~a~ [ ~ex(as) - q~cal(as)] 2 ~ex(as)

(22)

where q~e~ is the experimental value of the sieving coefficient.

3. COMPUTATIONAL METHOD

(23) a + Aa)

y ( a , a + Aa) = KrAa[ pc(a) - Pus - ~rp(a)]

(25)

Calculation is made step-by-step along the tube from the afferent extremity (a0) to the efferent end (a,). The same method is used to solve the differential equation for z. If the value of Kf is not imposed, the program adjusts the value of Kf by the Newton method [26], in order to equalize the calculated and experimental values of Y (relative error on Y < 10-5). The mean integrated ultrafiltration pressure fiuf is calculated as Y / K f .

3.2. Integration of the transport equations From the capillary lumen to the urinary space, levels II, III, IV and V represent the inlet of compartments 2, 3, 4 and 5 respectively (Fig. 2). At interface l, the subscript a represents the outlet of compartment i - 1 and b represents the inlet of compartment i (i = l).

3.2.1. Calculation of %/c I along the fl axis We shall call c*(a, 1) the ratio between cs(a, 1) and cs(a, IIa), where c~(a, II~) is the solute concentration at level II, and is identical to ca(a), the solute concentration in the plasma, cs(a, l) is the solute concentration at level l ( l = II, to Vb). C~(a, Vb), the solute concentration at level V b, is identical to cs(et ), the solute concentration in the glomerular filtrate. The first order differential equation (eq. 12) may be integrated analytically since the migration term is neglected. At interface 1-2

3.1. Water flow across the membrane The first order differential equations for qp and y are solved by the Euler method [24,25]. If Aa is defined as a nondimensionalized length element

(24)

[ ZsF"

c*(a, IIb) = c*(a, IIa) exp[-~-~-( qql ° -

+n~)] (26)

112 At interface 4-5

Within 2, i.e. within the LRI c*(a, IIIa)

K~-~(,~)

[ ZsF" 6vb)] (32) c*(a, Vb)= C*(a, Va) exp[-~-~-('kva-

K~-S~

and + [c*(a, Itb) -- K~i~(~) ]

c~*(., vb) = c s ( . ) / c , ( . ) = 0 ( ~ ) [jv(a)'K~" S~. Am LR ×exp[

"f~ 7-~2:~-~

](27)

f" is the ratio between the pore surface area and the membrane surface area in the laminae rarae. The superscript prime indicates that the values are calculated in the laminae rarae. At interface 2-3

(33)

@(a) is obtained by successive iterations using the secant method [26]. The initial value given to ~(a) is the experimental sieving coefficient ~ex" The iterative process is stopped when identity is obtained between the initial value of g,(a) and the calculated value, i.e. c*(a, Vb) (relative error on ¢(a) < 10-2).

3.2.2. Calculation of cI along the a axis [ ZsF,

c*(a, I l l b ) = c*(a, Ilia) exp[-~-~-{ ~bin" --~lllb)

]

As explained above, the first order differential equation for Z is solved by the Euler method [24]. The following relationship then writes:

(28)

z(a, a + a a ) = ¢ ( a ) ~ K , . A a . p , , ( a )

Within 3, i.e. within the LD

(34)

Combining eq. 34 with eqs. 19 and 25, we obtain:

c*(a, IVa) = K,. ¢ ( a ) K2•SF

+ [ c*(a'IIIb)

z(a, a + Aa) = ,#(a) qp %- ~--, z(0, a) a).y(a, a + Aa)

K,. q~(a) K~-~F ]

(35)

[jv(a):_K2- SF_:~toLD ×exp[

fro" Ds" SD

]

(29)

3.3. Minimization of EE At interface 3-4

[( ~ZsF c*(~, IVO = c*(~, ivy) exp/-k-Y 'v't

_ ~biVb)] (30)

Within 4, i.e. within the LRE

,

For a given value of X3, X2 and X4 are modified until the minimum value of EE is obtained. The minimization procedure is that proposed by Rosenbrock [27]. The calculation stops when EE decreases by less than 10-3 between two consecutive full iterations. The process is repeated for different values of X3 until a minimal value of EE is obtained.

, + c*(a, IVb)

K2•SF

[j~(~)-/q. s~. ,amL~]

Xexp[

"~£. "D--/T.~-~-

]

4. THE PROGRAM (31)

The program structure is presented in Fig. 3, with details in Figs. 4-6.

113

Initiatise parameters a* the entry of the tube p~o).qpl~ol, cp(~ o)

]

]Catco~ate nel=ol I Fig. 3. Program structure.

I MOELEC (main program): Reads data, performs successive iterations until minimization of ~ E , print results (Fig. 4). SUBRI: For a given Kf, calculates pressure, flow and filtration along the tube (Fig. 5). SUBR2: Calculates the sieving coefficient ~bca~ and the sum of quadratic errors on ~cal(EE) (Fig. 6). SUBR3: Calculates the local values of cs/c 1 along the fl axis. SUBR10: Calculates the potential differences using the values of the fixed charge concentrations. SUBR12: Calculates S D, S F, K~ and K 2.

1

[ Catculate PuFt=il

I [

I Catcutate ylo',,~,.d] Calcutate qploci. 1 ),pc(OCi,1}.Cp(~i.l). '/1p (o¢i.1 )

~

Yes

[__

(atculate Y, I~UF PUFIC~n)' J

Fig. 5. Flow diagram of SUBR1.

~ Read

input

data

on

Tape 1

[alculate So,SF K~,K2 for the given vatues of as and rISUBR12) Yes

l AdJust K t in order to min,mize the error on Y

I Compute f i t t r a t i o n curve for a given conductance (SUBR1)

1 Adjust X2 and X~,for a given value of X3,in order to minimize ~E Write results

1

on Tape 2

Fig. 4. Flow diagram of MOELEC.

5. SAMPLE RUN

5.1. Input parameters Pc., e, Pus, qp,, Cp, and Yex. L K F and initial value or imposed value of Kf. AmES~AmEn: ratio between LR and LD thickness. rLD: pore radius in the LD. ~,: ratio between the ultrafiltration pressures LR Put //Put"

NMOL: total number of macromolecules. a s (radius) and ~ex (experimental sieving coefficient) for each macromolecule. CH1, CH2: coefficients for calculation of the charge number Z s. CPRO1, CPR02, CPR03: coefficients for calculation of ~rp. Initial values of X 2, X3 and X4. CNaO), CAn(/): Na and anion concentrations. The logical variable LKF is set to 'false' for an

114 SUBROUTINE

SUER2

UATA AfFERENY PRESSURE

51.35 0.00 20.00 157.80 s.90

INlRIC.NYD.PRES~RV)E(nlHE~ DROP(Z)

EXTRAC.NYB.PRESSURE(KNH6) LFFERENT PLASKA fLOUoIL/KIN) LFFERENT PROT.CONC.~GRZ) EXP.FILTRATION RllE(KL/KIK) INITIAL KFoIL/KIN*KKH6~ CALCUL.blEP NUWER lHICKllESS RR110 LR/LD PORE RhBIUS IN LDtA) PRESSURE RATIO LR/LD ROLECULES NUKBER X2-(tlEO/L) X3-(HER/L) X4-(KEQIL)

45.10 3.50 20 *SO 51.11 .Ol 9 450.00 120.00 s10.00 lSO.00 134.00

EN&(l)-(KEQ/L) CAR(l,-(KEQ/L, RESULTS X2 tKEQ/L) 4so.00 449.29 450.71 4s1.41 452.12 452.83 453.54 4S4.24 454.95 455.66 456.36 457.07 457.76 4s0.49 459.19 459.90

Fig. 6. Flow diagram of SUBR2.

imposed K, and to ‘true’ for a calculated K,. In the second case, K, is adjusted to minimize the . . error on ftltratton. CNa(i) is equal to 150 mEq/l and CAn(lJ to 134 mEq/l. 5.2. Output parameters

460.61 461.31 462.02 462.73 413.44 462.51 461.01 463.22 462.66 462.40 462.04 462.67 461.60 463.65 ((2.60

x3 tKEO/L) 120.00 120.00

5.3. Example Real data obtained from experiments in the dog [13] and illustrations of output are given in Fig. 7.

s10.00 SOP.29 510.71 511.41 512.12 512.83 513.54 514.24 514.9s 515.66 516.36 517.07 517.78 518.49 519.19 s19.90 520.61 521.31

120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00

522.02 522.73 523.44 522.51 523.22 521.81 522.36 521.38 523.35 522.39 522.57 522.21 522.39

120.00 120.00 120.00 120.00 120.00 120.00 120.00 120.00

ERROR

.?14SE-01 .717?E-01 .711bE-01 .?089E-01 .7063E-01 .?040E-01 .7OlBE-01 .699?E-01 .69?9E-01 .6962E-01 .694?E-01 .6934E-01 .6922E-01 .b912E-01 .6904E-01 .689?E-01 .6892E-01 .6889E-01 .6887E-01 .608bE-01 .6818E-01 .68866-01 .6887E-01 .bOwE-01 .&ME-01 .I)BDE-Cl .688?E-01 .688bE-01 .6087E-01 .6887E-01 .68EbE-01

EFFERENT IKTRRC.NYO.PRESSURE(NKH6) KF(tlL/flIN*RtlN6)

51.35 3.64 45.10 8.31

CALCUL.FILTRATIOR RATEoIL/KIW) EFFERENT ULlR6F.PRESSURE~lllltl!3) KERN INTE6.ULTRAf.PRESSURE1KMNG)

+ P,, K,, Y our, and p,,,. Values of X,, X,, X, and CE for each iteration. a,, @_, and Qe,, for each macromolecule and for the final values of the fixed charge concentrations.

X4 IKEQILt

CAL.

.4202 15.00 .21?2 17.00 .OPB9 19.00 .0422 21.00 .01?6 23.00 .00?3 25.00 .0031 27.00 .0014 29.00 .OOOb 31.00 EOI ENCOUNTERED. /

H

12.39 EXP.

.3200 .2100 .1229 .Ob?O .0360 .OlBS .0090 .0046 .0022

Fig. 7. An example of sample run with real data and illustra tions of output.

115

6. H A R D W A R E CATIONS

AND

S O F T W A R E SPECIFI-

T h e p r o g r a m is c o d e d in F O R T R A N I V ext e n d e d for t h e C Y B E R 1 7 0 / 1 5 0 c o m p u t e r at the ' U n i v e r s i t 6 L i b r e de B r u x e l l e s ' c o m p u t e r centre. S t o r a g e r e q u i r e m e n t s are 21500 w o r d s .

7. M O D E

OF AVAILABILITY

A c o p y o f this p r o g r a m m a y b e o b t a i n e d f r o m the authors.

ACKNOWLEDGEMENTS W e a c k n o w l e d g e the h e l p o f the ' U n i v e r s i t 6 L i b r e de B r u x e l l e s ' c o m p u t e r centre. T h i s w o r k was s u p p o r t e d b y the N a t i o n a l F o u n d a t i o n for M e d i c a l R e s e a r c h ( B e l g i u m ) , g r a n t s 3.4581.75 a n d 3.4532.82. W e s i n c e r e l y t h a n k P r o f e s s o r P.P. L a m b e r t a n d D r . P. B e r g m a n n for t h e i r a d v i c e d u r i n g the e l a b o r a t i o n o f this study.

REFERENCES [1] B.M. Brenner, M.P. Bohrer, C. Baylis and W.M. Deen, Determinants of glomerular permselectivity (Editorial review), Kidney Int. 12 (1977) 229-237. [2] R.L.S. Chang, W.M. Deen, C.R. Robertson and B.M. Brenner, Permselectivity of the glomerular capillary wall. III. Restricted transport of polyanions, Kidney Int. 8 (1975) 212-218. [3] Y. Vanrenterghem, R. Vanholder, M. Lammens-Verslijpe and P.P. Lambert, Sieving studies in urea-induced nephropathy in the dog, Clin. Sci. 58 (1980) 65-75. [4] M.P. Bohrer, C. Baylis, H.D. Humes, R.J. Glassock, C.R. Robertson and B.M. Brenner, Permselectivity of the glomerular capillary wall: facilitated filtration of circulating polycations, J. Clin. Invest. 61 (1978) 72-78. [5] B.K. Nicholes, C.A. Krakower and S.A. Greenspon, The chemically isolated lamina densa of the renal glomerulus, Proc. Soc. Exp. Biol. Med. 142 (1972) 1316-1321. [6] N.A. Kefalides, Current status of chemistry and structure of basement membranes, in: Biology and Chemistry of Basement Membranes, ed. N.A. Kefalides, pp. 215-228 (Academic Press, New York, 1978). [7] Y.S. Kanwar and M.G. Farquhar, Anionic sites in the glomerular basement membrane, in vivo and in vitro localization to the laminae rarae by cationic probes, J. Cell Biol. 81 (1979) 137-153.

[8] Y.S. Kanwar and M.G. Farquhar, Presence of heparan sulfate in the glomerular basement membrane, Proc. Natl. Acad. Sci. U.S.A. 76 (1979) 1303-1307. [9] Y.S. Kanwar and M.G. Farquhar, Isolation of glycosaminoglycans (heparan sulphate) from glomerular basement membrane, Proc. Natl. Acad. Sci. U.S.A. 76 (1979) 4493-4497. [10] M.P. Cohen, Glycosaminoglycans are integral constituents of renal glomerular basement membrane, Biochem. Biophys. Res. Commun. 92 (1980) 343-348. [11] W.M. Deen, B. Satvat and J.J. Jamieson, Theoretical model for glomerular filtration of charged solutes, Am. J. Physiol. 238 (1980) F126-F139. [12] W.M. Deen and B. Satvat, Determinants of the glomerular filtration of proteins, Am. J. Physiol. 241 (1981) F162-F170. [13] M. Van Damme, M. Pr6vost and P.P. Lambert, Passive transport of charged and neutral macromolecules across a biological charged membrane, in: Electrical and magnetic separation and filtration technology, vol. 6, eds. R. Van Brabant and R. Weiler, pp. 61-66 (Antwerp, 23-25 May 1984). [14] R.L.S. Chang, C.R. Robertson, W.M. Deen and B.M. Brenner, Permselectivity of the glomerular capillary wall to macromolecules. I. Theoretical considerations, Biophys. J. 15 (1975) 861-886. [15] P. Decoodt, R. Du Bois, J.P. Gass6e, A. Vemiory and P.P. Lambert, A model for sieving of macromolecules by the glomerular membrane of the kidney, Comput. Programs Biomed. 4 (1975) 180-188. [16] R. Du Bois, P. Decoodt, J.P. Gass6e, A. Verniory and P.P. Lambert, Determination of glomerular intracapillary and transcapillary pressure gradients from sieving data. I. A mathematical model, Pfliigers Arch. 356 (1975) 299-316. [17] P.P. Lambert, R. Du Bois, P. Decoodt, J.P. Gass6e and A. Verniory, Determination of glomerular intracapillary and transcapillary pressure gradients from sieving data. II. A physiological study in the normal dog, Pfliagers Arch. 359 (1975) 1-22. [18] W.M. Deen, C.R. Robertson and B.M. Brenner, A model of glomerular ultrafiltration in the rat, Am. J. Physiol. 223 (1972) 1178-1183. [19] J.P. Caulfield and M.G. Farquhar, The permeability of glomerular capillaries to graded dextrans, J. Cell Biol. 63 (1974) 883-903. [20] C. Baylis, W.M. Deen, B.D. Meyers and B.M. Brenner, Effects of some vasodilator drugs on transcapillary fluid exchange in renal cortex, Am. J. Physiol. 230 (1976) 1148-1158. [21] P.P. Lambert, Personal communication. [22] W.L. Haberman and R.M. Sayre, Motion of rigid and fluid spheres in stationary and moving liquids inside cylindrical tubes (David Taylor Model Basin, report 1143, Department of the Navy, Washington D.C., 1958). [23] A. Verniory, R. Du Bois, P. Decoodt, J.P. Gass6e and P.P. Lambert, Measurement of the permeability of biological membranes, application to the glomerular wall, J. Gen. Physiol. 62 (1973) 489-507.

116 [24] E. Stiefel, Introduction h la Math6matique num6rique (Dunod, Paris, 1967). [25] M. Van Damme, S. Pegoff and P.P. Lambert, The pressure and flow distribution within a filtering capillary network, Comput. Programs Biomed. 13 (1981) 239-250. [26] B. Demidovitch and I. Maron, El6ments de calcul num6rique (MIR, Moscow, 1979). [27] H.H. Rosenbrock, An automatic method for finding the greatest or least value of a function, Comput. J. 3 (1960) 175-184.

APPENDIX: LIST OF SYMBOLS

Subscripts a and e indicate afferent and efferent extremities of the tube. i for i = 1 to 5 indicates the quantity evaluated in compartment i. la and I b for 1 = II to V represent both sides of any interface 1. Lower case letters are used to indicate the local value of the parameter under consideration. Am: Membrane surface area (cm2). as: Einstein-Stokes radius of the solute molecule (10- s cm). CAn: Small anions concentration (mEq/l). c / F o r i = 1 to 5, solute concentration in compartment i (M/ml). Crqa: Sodium concentration (mEq/1). %: Plasma protein concentration (g/100 ml). Cs: Local solute concentration (M/ml). c*: Ratio between local solute concentration and solute concentration in compartment 1. CH1, CH2: Numerical factors for calculation of the molecular charge of the solute under study ((10 - 8 cm)- 1, dimensionless) CPR01, CPR02, CPR03: Numerical factors for calculation of plasma oncotic pressure (mmHg/ (g/100 ml), mmHg/(g/100 ml) and 1 / ( g / 1 0 0 ml)). D~: Free diffusion coefficient of the solute (cm2/s). DEAE.D: Polycation diethylaminoethyl dextran. DS: Polyanion dextran sulphate. EE: Sum of errors on the calculated sieving coefficients (see eq. 22). fro, f ' : Ratio of pore surface area and membrane surface area in the lamina densa and lamina rata, respectively.

F: Faraday's constant (96520.1 C/M). GBM: Glomerular basement membrane. Js: Solute flow per unit of membrane area (M/(cm 2. min)). Jr: Water volume flow per unit of membrane area (ml/(cm2. min)). Kf: An ultrafiltration coefficient equal to Lp × A m (ml/(min • mmHg)). K 1, K2: Wall frictional factors in the lamina densa. K(, K~: Wall frictional factors in the lamina rara. Lp: Water permeability of the glomerular wall (ml/(min. mmHg) per cm2 of A m). LD: Lamina densa. LKF: Logical variable, 'true' if calculated Kf and 'false' if imposed Kf. LR: Laminae rarae. LRE: Lamina rara externa. LRI: Lamina rara interna. Am: Pore length (10 -8 cm). AmLD: Thickness of the lamina densa, i.e. lamina densa pore length (10-8 cm). AmLR: Thickness of the lamina rara, i.e. lamina rara pore length (10-8 cm). N: Total number of pores. NMOL: Total number of macromolecules. Pc: Local intracapillary hydrostatic pressure (mmHg). Puf: Local effective glomerular ultrafiltration pressure (mmHg). LD. puf. Local effective glomerular ultrafiltration pressure in the lamina densa (mmHg). LR. Local effective glomerular ultrafiltration Puf. pressure in the lamina rara (mmHg). Puf: Mean integrated ultrafiltration pressure (mmHg). Pus: Extracapillary hydrostatic pressure (mmHg). qp: Local plasma flow (ml/min). r: Pore radius (10 -8 cm). rLD: Pore radius in the lamina densa (10 -8 cm). rLR: Pore radius in the lamina rara (10 -8 cm). R: Gas constant (8.31662 J / ( M - K)). SD, SF: Steric hindrance factors for diffusion and convection in the lamina densa. S~, S~: Steric hindrance factors for diffusion and convection in the lamina rara. T: Absolute temperature (K). X~: For i = 2 to 4, fixed negative charge concentration in compartment i (mEq/1).

117

y: Cumulative filtration rate along the tube (ml/min). Y: Calculated glomerular filtration rate (ml/min). Yex: Experimental glomerular filtration rate (ml/min). z: Cumulative plasma clearance of the solute (ml/min). Z: Total plasma clearance of the solute (ml/min). Zs: Charge number of the solute under study. a: Normalized distance along the tube. Aa: Calculation step along the tube. fl: Denotes position within the capillary wall (10-8 cm).

y: Ratio of the ultrafiltration pressure in the LR and the total ultrafiltration pressure across the membrane. e: Pressure drop from the afferent end to the efferent extremity of the tube (Pea - Pco) as a percentage of Pea" 71: Filtrate viscosity (poise, i.e. dyne-s/cm2). %: Plasma oncotic pressure (mmHg). ~,: Local sieving coefficient. ~al: Calculated sieving coefficient. ~e~: Experimental sieving coefficient. ~: Electrical potential (mV).