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Free energy of the electrostatic interaction of cells with adjacent charged glycoprotein layer: A theoretical approach
J. theor. Biol. (1982) 98, 269-282
Free Energy of the Electrostatic Interaction of Cells with Adjacent Charged Glycoprotein Layer: A Theoretical Approach ANDREAS
VOIGT
Forschungsabteilung Biomaterialien der Universitiitsklinik Innere Medizin “Theodor Brugsch” der Humboldt-Universitiit Berlin, G.D.R. EDWIN
DONATH
AND REINHART
fiir zu
HEINRICH
Bereich Biophpsik der Sektion Biologie der Humboldt- Universitiit zu Berlin, G.D.R. (Received 21 October 1981, and in revised form 29 March 1982) The distance-dependentfree energy of the electrostatic interaction is calculated for thick dielectric membranesof low dielectric constant in electrolyte solutionswith adjacentcharge-carryingglycoprotein layer (glycocalix). The chargesare assumedto be continuouslydistributed. For the description of the spacedependenceof the charge density on the space coordinate perpendicularto the membranesurfacesvarious functions are used. The free energy correspondingto the spacechargearrangementis determined by a stepwisecharging processwhere the mobile ions are in thermodynamicequilibrium.For various chargearrangementsof the interacting glycocalicesthe electric potential profile between the cells and the electrostaticcontribution to the free energy are calculated.The modelcan be appliedwithin the rangeof validity of the linearizedPoisson-Boltzmann equation. 1. Introduction The search for a unified physical concept to describe the energy of cell-cell
interactions in suspensions and tissues and of cells with non-biogenic substrate surfaces has been of increasing success within the last few years. The extension and application of the DLVO theory (Derjaguin & Landau, 194 1; Verwey & Overbeek, 1948) originally developed for colloidal systems to complicated geometrical structures and arrangements of interacting partners has proved also a key for understanding initial events in cell-cell and cell-substrate contact (Gingell, 1971; Parsegian & Gingell, 1972; Parsegian & Weiss, 1972; Gingell & Parsegian, 1972; Gingell & Form%, 1976; Parsegian & Gingell, 1980). 269
0022-5193/82/180269+14$03.00/0
@ 1982 Academic Press Inc. (London) Ltd.
270
A.
VOIGT
ET
AL.
In this connection “steric” contributions of macromolecular adsorption layers (Napper, 1970; Hesselink, Vrij & Overbeek, 1971), various shortrange molecular bonding energies (Bell, 1978) and changes of interface energy (Dolowy & Holly, 1978) were recently considered. For the interpretation of electrophoretic measurements Pastushenko & Donath (1976~) introduced and considered a finitely thick glycocalix with space charge density function perpendicular to the membrane surface. Gingell (1968) calculated characteristic electric potentials between two thick interacting membranes with non-adjacent plane charge layers as a function of membrane distance. Lerche (1980) tried to apply the equation for the force balance in overlapping Gouy-Chapman double layers to the case of spatially extended fixed charge arrangements. In the present paper a model is developed which allows the calculation of the profile of electric potential as well as the free energy of electrostatic interaction of plane thick membranes with adjacent space charge layers as functions of membrane distance. It is supposed that the charge arrangement can be described as a function of space charge density depending only on the normal distance from the membrane surface. Additional plane charge distributions (including membrane surface charges) are also admitted at any distance from the membrane surface. 2. Theory
Figure 1 gives a schematic representation of the model geometry. pr(x) and P&X) are profiles of charge density normal to the membrane surface for the two glycocalices, respectively. Denoting by d the distance of the two membranes and by 6i and S2 the thicknesses of charged surface layers, we may rewrite -d/25x s-d/2+6,, x >--d/2+6,,
(1)
d/2-&5x x
(2)
The whole fixed charge density profile in the interstice is then given by P(X)
=m(x)+p2b).
(3)
In accordance with Verwey & Overbeek (1948) the following relationship is used to determine the electrostatic free energy density E of the system: E=
(4)
ELECTROSTATIC
INTERACTION
OF
271
CELLS
Bulk phase
,
Membrane , /
,i /
,’ ,
/
;
/
i
I -L--L
d
-- 2
-d+8, 2
0 Dlstonce,
-x
2”
62
d 2
FIG. 1. Geometrical and physical parameters of the model. S1 and & are the thicknesses of the first and second glycocalices; pi(x) and p*(x) are the respective space charge densities and E, and E, are the relative dielectric constants of the membrane and the bulk phase, respectively.
The integrand represents the energy to increase the fixed charge by dfi within an infinitesimal volume element at a given potential I++due to the actual distribution of fixed charges p’. It should be stressed that the stepwise isothermal charging process which was considered at the derivation of equation (4) proceeds within a medium of mobile co- and counterions (electrolyte). Therefore I,&) stands for the profile of the electric potential in equilibrium, that is, after redistribution of mobile ions in accordance with the charging state already reached. The following substitution is introduced: P’=hp.
(5)
dfi =p dh.
(6)
Thus we obtain Here, A ranges from 0 to 1. If the potential is related by a linear differential equation to the fixed charge, the integration process can be considerably facilitated using equation (5). In the linear case we may write ita
= W(P)
(7)
and equation (4) simplifies to d/2 +1 2 _ cL(X)p(x)dx. I d/2
(8)
272
A.
ET
VOIGT
AL.
For the calculation of the electric potential profile, therefore, the linearized Poisson-Boltzmann equation is used: d2+ 7dx with the Debye-Hiickel
K2$L-L Es&o
parameter
F, R and T are the Faraday constant,
gas constant and absolute temperature, respectively. es is the relative and e. the absolute dielectric constants and Zj and cj are the valence and concentration of the jth ionic species, respectively. The following boundary conditions complete equation (9): d4 dx
=-4-m
W dxI
x-d/2-0
(114
3 Es&O
x+-d/2+0
_dm) ,
Ulb)
Es&O
where 4-d/2) and ~(42) are the surface charges at the two membranes. The boundary conditions [equations (lla, llb)] are justified by consideration of the thickness and the low dielectric constant of biological membranes (see also Heinrich, Gaestel &L Glaser, 1981). The integral over the total charge density (fixed and mobile charges), extending through the whole space between the two membrane surfaces from -d/2 to d/2, must disappear. The solution of equation (9) with boundary conditions (11) can be easily represented by means of the respective Green function G (x, x’) :
G(X) =J-y’;G(x,
x’)p (x’) dx’
(12)
with G(x, x’) = E~&OK
1 sinh (Kd)
x cash [K (x’+ d/2)] cash cash [K (x’ -d/2)] cash
[K
(X
[K
(x
-d/2)], d/2)],
+
x >x’, x
(13)
ELECTROSTATIC
INTERACTION
OF
CELLS
273
In equation (12) the charge density p(x’) may also include plane charge densities positioned at various points x’ = xi. An additional term C UiS(X’--Xi), with ai the plane charge density at x’ = xi, must be added to the fixed charge density p(x’); 8(x’-xi) is Dirac’s S-function. The integration has to be performed with the respective rules for the ‘i-function. In the case of surface charges at Xi = -d/2 or Xi = d/2 we define d/2
I
-d,2f(X’)6(X’-Xt)
dX’=f(Xi).
(14)
For d + COthe integration of equation (12) with equation (13) yields the potential profile obtained by Pastushenko & Donath (19766) at the outside of a single membrane with glycocalix. For a large class of functions p(x’) analytical solutions for the potential profiles and the free energies can be found. As a rule, long and intricate terms are obtained by solving equations (8) and (12). For this reason the potential profiles as well as the free energies of the charge arrangements were calculated numerically. However, for the practically relevant case of a cell approaching a smooth charged surface, an analytical expression is given below. The free energy of electrostatic interaction Eint as a function of the distance d of the two membranes may be expressed in the following way: Ei,,(d)=E(d)-(E,+E,), where E(d) stands for membrane distance d while El and E2 denote at d + 03 (no interaction
(15)
the free energy of charge arrangement at a given (with interaction of the two space charge layers) the free energies of the given charge arrangement of the two space charge layers). 3. Results (A)
POTENTIAL
PROFILES
Figure 2(a) shows three hypothetical charge arrangements within the glycocalix of the membranes. The curves 1-3 are characterized by the following functions p:(x), p?(x) and p:(x) written here for the left-hand glycocalix: p:(x)=p
=u/S1=const.,
(164
A.
0
-5
VOIGT
ET
0 x(nm)
AL.
5
ELECTROSTATIC
P:(x) =3x
d(x) = exp
INTERACTION
OF
+d2),
-
275 (16b)
PK& (K~I)
CELLS
1
eXp
[K
(X
+d/2)].
(164
The respective expressions for the right-hand glycocalix (d/2 - Sz I x I d/2) are derived by replacing d/2 +x by d/2 -x and 6i by SZ. The total charge for each glycocalix and each selected charge arrangement was considered to be the same. In all examples, both for thicknesses of glycocalix and for the respective total charges, realistic values were used from the relevant literature (Donath, 1979) for human erythrocytes (Si = Sz = 5.5 nm; (+ = 0.02 C/m’). The calculations were performed with the Debye-Hiickel parameter K = lo9 m-r. All examples shown in Fig. 2(a)-(c) are related to glycocalices with a distance of 0.5 nm from each other, i.e. between the two glycocalices there remains an interstice free from fixed charges. For these charge arrangements the integral in equation (12) for the determination of the potential profile can be solved analytically. However, long terms result which are difficult to handle. Figure 2(b) illustrates the calculated potential profiles for the spatial charge arrangements shown in Fig. 2(a). At linearly and exponentially increasing charge densities in the glycocalix [Fig. 2(a), curves 2 and 31, the electric field strength continuously changes its magnitude and direction inside each layer of fixed space charge [Fig. 2(b), curves 2 and 31, which can be seen from the location of the maxima of the potential profiles. Similar but discontinuous changes in magnitude and direction of the electric field strength can be obtained with a two-dimensional arrangement of the fixed charge. In this case the peaks on the potential profiles representing the discontinuities of the field strength coincide with the positions of the plane charges. Curves l-3 of Fig. 2(c) show the potential profiles for a successive subdivision of the total charge into one (curve 1: 0.02 C/m2 FIG. 2. (a) Charge density profiles in the two glycocalices. Constant (curve l), linearly (curve 2) and exponentially (curve 3) increasing charge densities; total fixed charge per glycocalix is 0.02 C/m2 and S1 = S2 = 5.5 nm. Explicit functions for the charge density profiles are given in the text. (b) Electric potential profiles between two interacting membranes as a function of various charge densities. Constant (curve l), linearly (curve 2) and exponentially (curve 3) increasing chargedensitieswithS~=S~=5~5nm,d-(S~+S~)=O~5nmand~=10gm~1. (c) Electric potential profiles between two interacting membranes as a function of various plane charge density arrangements. Total fixed charge is successively subdivided into one (curve l), two (curve 2) and four (curve 3) plane charge densities at different positions before the membranes (exact positions are given in the text). Sr = S2 = 5.5 nm, d - (6, +S,) = 0.5 nm and K = 10’ m-‘.
276
A.
VOIGT
ET
AL.
at x, = -d/2 + 81, d/2 - &), two (curve 2: O-01 C/m2 at xi = -d/2, -d/2 + 61, d/2-SZ, d/2) and four (curve 3: O*OOS C/m2 at Xi = -d/2, -d/2+ 2 nm, -d/2+4 nm, -d/2+S1,d/2-S2,d/2-4 nm,d/2-2 nm,d/2)planecharge densities for each glycocalix. (B)
FREE
ENERGIES
The free energy E, of a single glycocalix with a space charge layer of thickness St and with p(x) = p = const.
(17)
can be easily found by solving the integrals of equations (8) and (12). The following expression is obtained:
E,=
p2 --+2&
-
l/K
+ (l/K)
eXp
(--2K&)].
(1%
~&S&OK
At fixed total charge, in the limit 6r -0, one obtains from this equation the term for the free energy E, of the diffuse Gouy-Chapman double layer (Verwey & Overbeek, 1948):
(1% with ff = p&.
(20)
For the ratio of the two energies E,, and E, one gets E,/&
= $?/(K&)
+ 1/(K61)2(eXp
(-2K&)-
I)].
(21)
Figure 3 gives a graphical representation of the energy ratio E,/Ev as a function of 1/(~8,). Thus, with a given Debye-Hiickel parameter K and a given thickness S1 of the space charge layer, E, can be easily calculated from E, and vice versa. For low ionic strengths, i.e. for l/(~Sl)> 1 the free electrostatic energy of spatial arrangement does not differ very much from a two-dimensional one. This is understandable because, with greater Debye-Hiickel lengths, the spatial extension of charges becomes less effective. For arbitrary arrangements of the charges of the two space charge layers, the free energy of electrostatic interaction of these layers can be found, generally, by equation (15), after solving the following integrals, explicitly
ELECTROSTATIC
INTERACTION
OF
277
CELLS
I
5 Iu’
0
1
2
3
4
5
(K SJ’ FIG. 3. Ratio of the free energies of spatially distributed charge of constant density (I$,) and surface charge at the membrane (E,) as a function of the inverse product of Debye-Hickel parameter and thickness of glycocalix.
deduced from equation (8):
,
PZ+; a2$(x
-xi)
1dx*
(24)
Here, E(d) is again the free energy of the system. E1 and EZ are the free energies for d + co. The indices of potentials and charge densities indicate the type of charge (p corresponds to space charge and c to surface charge) and the number of the space charge layer (fir;: or second glycocalix). An important case in adhesion research is the mutual approach of a cell with a glycocalix and a smooth charged surface. Assuming a constant space charge density of fixed charges, the free interaction energy is expressed
278
A.
analytically
VOIGT
ET
AL.
as follows:
Eint=
’ %?&
2 sinh i sinh
p sinh
(K~)CT (Ed)
-
K
(-24-
-&Lexp
(~6)
sinh [K sinh (Kd)
11+&[coth
(d
6)] I
(Kd)
s
-
-
11.
(25)
Figures 4 and 5 illustrate the results for the free energies of the electrostatic interaction for different charge distributions as a function of membrane distance. Figure 4 shows a comparison of those cases where all the charge is located at the membrane surfaces (curve 4) or the space charge density is constant (curve l), linearly (curve 2) or exponentially (curve 3) increasing in each glycocalix [cf. Fig. 2(a)]. Figure 5 represents the results when subdividing the total charge into one, two or four plane charges per glycocalix, as was done before calculating the potential profiles of Fig. 2(c). It should be pointed out that the potential profiles given in Fig. 2(b) and (c) correspond to the same charge arrangements selected for Figs 4 and 5 at d = 11.5 nm. Figure 4 shows that the spatially distributed charge results in much greater interaction energy than surface charge, which is due to the diminished distance between the fixed charges. Of course, a similar high interaction energy in the case of surface charge distribution (curve 4) is only reached with remarkable glycocalix compression. L
-3. -4. d CT -5~
-61 2
23 \A
-7
, 4
-8 0
5
IO
15
ird
FIG. 4. Logarithm of the free energy of interaction of membranes with charge-carrying glycocalices as a function of the product of membrane distance and Debye-Hickel parameter. Curves 1-3 illustrate the effect of the charge density profiles given in Fig. 2(a), curve 4 represents charged membrane surfaces with 0.02 C/m . 6t = 8r = 5.5 nm and K = lo9 m-t.
ELECTROSTATIC
INTERACTION
OF
-71 0
5
IO
I5
279
CELLS
I
Kd FIG. 5. Logarithm of the free energy of interaction of membranes with plane charges at various positions in the glycocalices as a function of the product of membrane distance and Debye-Htickel parameter. Magnitude and position of the plane charge densities are given in the text. S1 = & = 5.5 nm and K = lo9 m-l.
For linearly and exponentialiy increasing space charge densities (Fig. 4, curves 2 and 3) and for two-dimensional charge arrangements at a considerable distance from the membrane surface (Fig. S), the dependence of the free energy of electrostatic interaction on the membrane distance exhibits a remarkable behaviour. At penetration of the glycocalices, there are pronounced local minima of free energy. The site and depth of the minima depend on charge distribution. Their existence is generally associated with a charge density increasing towards the outer edge of the glycocalix. One may conclude that such a stiff arrangement of charges can be stabilized by electrostatic effects under suitable conditions and after overcoming repulsive energy. Maxima on energy-distance curves result in changes of signs on forcedistance curves. In the interaction of spatial charge arrangements in Fig. 4 (curves l-3) the force vs. distance curve is continuous. If charge arrangement is two-dimensional, as in Figs 5 and 6, the force-distance function shows discontinuities whenever planes with plane charge densities of the two glycocalices exactly traverse each other. So the force abruptly changes its direction at these membrane distances, which is, of course, a mathematical idealization, but illustrates the qualitative difference between the two cases most clearly. Figure 6 shows, for two-dimensional charge arrangements with identical plane charge densities at the edges of the two glycocalices, the dependence of the free energy of electrostatic interaction on membrane distance and Debye-Hiickel parameter. With decreasing ionic strength (decreasing K)
280
A.
-717 0
5
VOIGT
ET
IO
AL.
15
20
d(m) FIG. 6. Logarithm of the free energy of interaction of membranes with a two-dimensional charge arrangement at the edges of the glycocalices as a function of distance and the Debye-Hiickel parameter. Plane charge densities are positioned at x = -d/2+6, and x = d/2 -6s with CT= 0.02 C/m*. Debye-Hiickel parameter K varies from lo9 m-’ (curve l), to 3 x 10’ m-r (curve 2) to 108m-’ (curve 3).
the energy minimum for overlapping glycocalices disappears, at a simultaneous global increase of the energy. Thus, due to the large Debye-Hiickel length, the screening of the fixed charge within the glycocalices is not sufficient and the system does not feel the distance between the charge planes. Whether such or similar charge arrangements investigated in this paper are of practical relevance for biological structures cannot be decided at present. To answer this question, the free energy minimum of a glycocalix structure will have to be determined under a number of further restrictive conditions, such as the given distribution of fixed charges along the macromolecular chains, primary and secondary intra- and intermolecular interactions, influence of interphase, etc. With flexible macromolecular chains stabilized by secondary bonds, the electrostatic contribution to the free energy becomes noticeable for determinihg the conformation of the molecules only if it lies in the order of secondary bonding energies, which, however, can as a rule be realized by suitable selection of milieu parameters, at least in vitro. Even at the present stage of this study an important conclusion may be drawn, that the usually assumed attraction constant (e.g. effective Hamaker constant) is overestimated. Indeed, a fit of experimental adhesion results in order to estimate the effective Hamaker constant, taking into account only surface charge densities at the membrane surfaces, fails due to overesti-
ELECTROSTATIC
INTERACTION
OF
CELLS
281
mated electrostatic repulsion. Preliminary calculations have shown that the effective Hamaker constant of cell interaction in aqueous solutions seems to be no higher than approximately 1O-22 J, this is an order of magnitude lower than the values often implicated (Nir & Andersen, 1977). This low value has also been predicted recently by van Oss er al. (1980), analysing the interface thermodynamics of blood cell interactions in serum. Nevertheless, for albumin adsorption at polystyrene latex Norde & Lyklema (1978) proposed a discontinuous distribution of charges inside the spatially extended adsorption layer. Two charge-carrying zones at the surface of polystyrene latex and at the edge of the adsorption layer are separated by a broad and free of fixed charges third zone. Only recently, a model for the structure of glycophorin A, the main charge-carrying glycoprotein on the human erythrocyte surface, has been proposed by Stibenz & Geyer (1980). It remains for further experimental and theoretical studies to determine the real distribution of fixed charges in the macromolecular layers on cell surfaces. A free electrostatic energy minimization approach is given by Donath & Voigt (1982). The concept presented in this paper can be helpful in the search for the energetically favoured arrangement of fixed charges and is applicable especially where, on the one hand, the effects of discrete charge distributions can be neglected and, on the other hand, the free electrostatic energy is not too great. This study was supported by the HFR “Kiinstlicher Organersatz und Biomaterialien” of the Ministry of Health of the G.D.R. The authors wish to express their gratitude to Professor Dr R. Glaser and Doz. Dr H. Wolf for their aid in writing this paper. REFERENCES BELL, G. DERJAGUIN, DOLOWY, DONATH, DONATH, GINGELL, GINGELL, GINGELL, GINGELL, HEINRICH, HESSELINK. LERCHE, NAPPER, NIR, S. &
I. (1978). Science, Wash. 200,618. 8. V. & LANDAU, L. D. (1941). Acta physicochim. U.S.S.R. 14,633. K. & HOLLY, F. J. (1978). j. theor. Biol. 75, 373. E. (1979). Dissertation, Humboldt-UniversitLt zu Berlin. E. & VOIGT, A. (1982). Submitted to J. theor. Biol. D. (1968). I theor. Biol. 19, 340. D. (1971). J. theor. Biol. 30, 121. D. & FORNBS, J. A. (1976). Biophys. J. 16, 1131. D. & PARSEGIAN, V. A. (1972). J. theor. Biol. 36,41. R., GAESTEL, M. & GLASER, R. (1981). Acta biol. med. germ. 40,765. F. TH.. VRLJ. A. & OVERBEEK. J. TH. G. (1971). J. ohvs. Chem. 75.2094. D: (198Oj. Dissdrtation, Humboldt:Universitlt’zu B&in. I D. H. (1970). J. Colloid Interface Sci. 32, 106. ANDERSEN, M. (1977). /. Membrane Biol. 31, 1.
282
A.VOIGT
ET AL.
NORDE, W. & LYKLEMA J.(1978).J. CoiloidZnterface VAN OSS,C.J.,NEUMANN, A. W.,GOOD, R.J.&
Sci.66,285.
ABSOLOM,D.R.(~~~O).
Ado.
Chem.
Ser. 188, 107.
PARSEGIAN, V. A. & WEISS, G. H. (1972). J. Colloid Interface Sci. 40, 35. PARSEGIAN,V.A.&GINGELL,D.(~~?~).B~~~~~S.J.~~,I~~~. PARSEGIAN,~. A.& GINGELL,D.(~~~O). J. CellSci.41,151. PASTUSHENKO,~.& DoNATH,E.(~~~~~). Stud.biophys.,Berl.56,7 and B2. PASTUSHENKO,~.& DONATH,E. (1976b). Stud.biophys.,Beri.56,9 andB8. STIBENZ,D. & GEYER, G. (1980). Folia haematol.,Leipzig 107,787. VERWEY, E. J. V. & OVERBEEK, J. TH. G. (1948). Theory of the Stability of Lyophobic Colloids. The Znteraction of Sol Particles Having an Electric Double Layer. Amsterdam: Elsevier.
Free Energy of the Electrostatic Interaction of Cells with Adjacent Charged Glycoprotein Layer: A Theoretical Approach ANDREAS
VOIGT
Forschungsabteilung Biomaterialien der Universitiitsklinik Innere Medizin “Theodor Brugsch” der Humboldt-Universitiit Berlin, G.D.R. EDWIN
DONATH
AND REINHART
fiir zu
HEINRICH
Bereich Biophpsik der Sektion Biologie der Humboldt- Universitiit zu Berlin, G.D.R. (Received 21 October 1981, and in revised form 29 March 1982) The distance-dependentfree energy of the electrostatic interaction is calculated for thick dielectric membranesof low dielectric constant in electrolyte solutionswith adjacentcharge-carryingglycoprotein layer (glycocalix). The chargesare assumedto be continuouslydistributed. For the description of the spacedependenceof the charge density on the space coordinate perpendicularto the membranesurfacesvarious functions are used. The free energy correspondingto the spacechargearrangementis determined by a stepwisecharging processwhere the mobile ions are in thermodynamicequilibrium.For various chargearrangementsof the interacting glycocalicesthe electric potential profile between the cells and the electrostaticcontribution to the free energy are calculated.The modelcan be appliedwithin the rangeof validity of the linearizedPoisson-Boltzmann equation. 1. Introduction The search for a unified physical concept to describe the energy of cell-cell
interactions in suspensions and tissues and of cells with non-biogenic substrate surfaces has been of increasing success within the last few years. The extension and application of the DLVO theory (Derjaguin & Landau, 194 1; Verwey & Overbeek, 1948) originally developed for colloidal systems to complicated geometrical structures and arrangements of interacting partners has proved also a key for understanding initial events in cell-cell and cell-substrate contact (Gingell, 1971; Parsegian & Gingell, 1972; Parsegian & Weiss, 1972; Gingell & Parsegian, 1972; Gingell & Form%, 1976; Parsegian & Gingell, 1980). 269
0022-5193/82/180269+14$03.00/0
@ 1982 Academic Press Inc. (London) Ltd.
270
A.
VOIGT
ET
AL.
In this connection “steric” contributions of macromolecular adsorption layers (Napper, 1970; Hesselink, Vrij & Overbeek, 1971), various shortrange molecular bonding energies (Bell, 1978) and changes of interface energy (Dolowy & Holly, 1978) were recently considered. For the interpretation of electrophoretic measurements Pastushenko & Donath (1976~) introduced and considered a finitely thick glycocalix with space charge density function perpendicular to the membrane surface. Gingell (1968) calculated characteristic electric potentials between two thick interacting membranes with non-adjacent plane charge layers as a function of membrane distance. Lerche (1980) tried to apply the equation for the force balance in overlapping Gouy-Chapman double layers to the case of spatially extended fixed charge arrangements. In the present paper a model is developed which allows the calculation of the profile of electric potential as well as the free energy of electrostatic interaction of plane thick membranes with adjacent space charge layers as functions of membrane distance. It is supposed that the charge arrangement can be described as a function of space charge density depending only on the normal distance from the membrane surface. Additional plane charge distributions (including membrane surface charges) are also admitted at any distance from the membrane surface. 2. Theory
Figure 1 gives a schematic representation of the model geometry. pr(x) and P&X) are profiles of charge density normal to the membrane surface for the two glycocalices, respectively. Denoting by d the distance of the two membranes and by 6i and S2 the thicknesses of charged surface layers, we may rewrite -d/25x s-d/2+6,, x >--d/2+6,,
(1)
d/2-&5x x
(2)
The whole fixed charge density profile in the interstice is then given by P(X)
=m(x)+p2b).
(3)
In accordance with Verwey & Overbeek (1948) the following relationship is used to determine the electrostatic free energy density E of the system: E=
(4)
ELECTROSTATIC
INTERACTION
OF
271
CELLS
Bulk phase
,
Membrane , /
,i /
,’ ,
/
;
/
i
I -L--L
d
-- 2
-d+8, 2
0 Dlstonce,
-x
2”
62
d 2
FIG. 1. Geometrical and physical parameters of the model. S1 and & are the thicknesses of the first and second glycocalices; pi(x) and p*(x) are the respective space charge densities and E, and E, are the relative dielectric constants of the membrane and the bulk phase, respectively.
The integrand represents the energy to increase the fixed charge by dfi within an infinitesimal volume element at a given potential I++due to the actual distribution of fixed charges p’. It should be stressed that the stepwise isothermal charging process which was considered at the derivation of equation (4) proceeds within a medium of mobile co- and counterions (electrolyte). Therefore I,&) stands for the profile of the electric potential in equilibrium, that is, after redistribution of mobile ions in accordance with the charging state already reached. The following substitution is introduced: P’=hp.
(5)
dfi =p dh.
(6)
Thus we obtain Here, A ranges from 0 to 1. If the potential is related by a linear differential equation to the fixed charge, the integration process can be considerably facilitated using equation (5). In the linear case we may write ita
= W(P)
(7)
and equation (4) simplifies to d/2 +1 2 _ cL(X)p(x)dx. I d/2
(8)
272
A.
ET
VOIGT
AL.
For the calculation of the electric potential profile, therefore, the linearized Poisson-Boltzmann equation is used: d2+ 7dx with the Debye-Hiickel
K2$L-L Es&o
parameter
F, R and T are the Faraday constant,
gas constant and absolute temperature, respectively. es is the relative and e. the absolute dielectric constants and Zj and cj are the valence and concentration of the jth ionic species, respectively. The following boundary conditions complete equation (9): d4 dx
=-4-m
W dxI
x-d/2-0
(114
3 Es&O
x+-d/2+0
_dm) ,
Ulb)
Es&O
where 4-d/2) and ~(42) are the surface charges at the two membranes. The boundary conditions [equations (lla, llb)] are justified by consideration of the thickness and the low dielectric constant of biological membranes (see also Heinrich, Gaestel &L Glaser, 1981). The integral over the total charge density (fixed and mobile charges), extending through the whole space between the two membrane surfaces from -d/2 to d/2, must disappear. The solution of equation (9) with boundary conditions (11) can be easily represented by means of the respective Green function G (x, x’) :
G(X) =J-y’;G(x,
x’)p (x’) dx’
(12)
with G(x, x’) = E~&OK
1 sinh (Kd)
x cash [K (x’+ d/2)] cash cash [K (x’ -d/2)] cash
[K
(X
[K
(x
-d/2)], d/2)],
+
x >x’, x
(13)
ELECTROSTATIC
INTERACTION
OF
CELLS
273
In equation (12) the charge density p(x’) may also include plane charge densities positioned at various points x’ = xi. An additional term C UiS(X’--Xi), with ai the plane charge density at x’ = xi, must be added to the fixed charge density p(x’); 8(x’-xi) is Dirac’s S-function. The integration has to be performed with the respective rules for the ‘i-function. In the case of surface charges at Xi = -d/2 or Xi = d/2 we define d/2
I
-d,2f(X’)6(X’-Xt)
dX’=f(Xi).
(14)
For d + COthe integration of equation (12) with equation (13) yields the potential profile obtained by Pastushenko & Donath (19766) at the outside of a single membrane with glycocalix. For a large class of functions p(x’) analytical solutions for the potential profiles and the free energies can be found. As a rule, long and intricate terms are obtained by solving equations (8) and (12). For this reason the potential profiles as well as the free energies of the charge arrangements were calculated numerically. However, for the practically relevant case of a cell approaching a smooth charged surface, an analytical expression is given below. The free energy of electrostatic interaction Eint as a function of the distance d of the two membranes may be expressed in the following way: Ei,,(d)=E(d)-(E,+E,), where E(d) stands for membrane distance d while El and E2 denote at d + 03 (no interaction
(15)
the free energy of charge arrangement at a given (with interaction of the two space charge layers) the free energies of the given charge arrangement of the two space charge layers). 3. Results (A)
POTENTIAL
PROFILES
Figure 2(a) shows three hypothetical charge arrangements within the glycocalix of the membranes. The curves 1-3 are characterized by the following functions p:(x), p?(x) and p:(x) written here for the left-hand glycocalix: p:(x)=p
=u/S1=const.,
(164
A.
0
-5
VOIGT
ET
0 x(nm)
AL.
5
ELECTROSTATIC
P:(x) =3x
d(x) = exp
INTERACTION
OF
+d2),
-
275 (16b)
PK& (K~I)
CELLS
1
eXp
[K
(X
+d/2)].
(164
The respective expressions for the right-hand glycocalix (d/2 - Sz I x I d/2) are derived by replacing d/2 +x by d/2 -x and 6i by SZ. The total charge for each glycocalix and each selected charge arrangement was considered to be the same. In all examples, both for thicknesses of glycocalix and for the respective total charges, realistic values were used from the relevant literature (Donath, 1979) for human erythrocytes (Si = Sz = 5.5 nm; (+ = 0.02 C/m’). The calculations were performed with the Debye-Hiickel parameter K = lo9 m-r. All examples shown in Fig. 2(a)-(c) are related to glycocalices with a distance of 0.5 nm from each other, i.e. between the two glycocalices there remains an interstice free from fixed charges. For these charge arrangements the integral in equation (12) for the determination of the potential profile can be solved analytically. However, long terms result which are difficult to handle. Figure 2(b) illustrates the calculated potential profiles for the spatial charge arrangements shown in Fig. 2(a). At linearly and exponentially increasing charge densities in the glycocalix [Fig. 2(a), curves 2 and 31, the electric field strength continuously changes its magnitude and direction inside each layer of fixed space charge [Fig. 2(b), curves 2 and 31, which can be seen from the location of the maxima of the potential profiles. Similar but discontinuous changes in magnitude and direction of the electric field strength can be obtained with a two-dimensional arrangement of the fixed charge. In this case the peaks on the potential profiles representing the discontinuities of the field strength coincide with the positions of the plane charges. Curves l-3 of Fig. 2(c) show the potential profiles for a successive subdivision of the total charge into one (curve 1: 0.02 C/m2 FIG. 2. (a) Charge density profiles in the two glycocalices. Constant (curve l), linearly (curve 2) and exponentially (curve 3) increasing charge densities; total fixed charge per glycocalix is 0.02 C/m2 and S1 = S2 = 5.5 nm. Explicit functions for the charge density profiles are given in the text. (b) Electric potential profiles between two interacting membranes as a function of various charge densities. Constant (curve l), linearly (curve 2) and exponentially (curve 3) increasing chargedensitieswithS~=S~=5~5nm,d-(S~+S~)=O~5nmand~=10gm~1. (c) Electric potential profiles between two interacting membranes as a function of various plane charge density arrangements. Total fixed charge is successively subdivided into one (curve l), two (curve 2) and four (curve 3) plane charge densities at different positions before the membranes (exact positions are given in the text). Sr = S2 = 5.5 nm, d - (6, +S,) = 0.5 nm and K = 10’ m-‘.
276
A.
VOIGT
ET
AL.
at x, = -d/2 + 81, d/2 - &), two (curve 2: O-01 C/m2 at xi = -d/2, -d/2 + 61, d/2-SZ, d/2) and four (curve 3: O*OOS C/m2 at Xi = -d/2, -d/2+ 2 nm, -d/2+4 nm, -d/2+S1,d/2-S2,d/2-4 nm,d/2-2 nm,d/2)planecharge densities for each glycocalix. (B)
FREE
ENERGIES
The free energy E, of a single glycocalix with a space charge layer of thickness St and with p(x) = p = const.
(17)
can be easily found by solving the integrals of equations (8) and (12). The following expression is obtained:
E,=
p2 --+2&
-
l/K
+ (l/K)
eXp
(--2K&)].
(1%
~&S&OK
At fixed total charge, in the limit 6r -0, one obtains from this equation the term for the free energy E, of the diffuse Gouy-Chapman double layer (Verwey & Overbeek, 1948):
(1% with ff = p&.
(20)
For the ratio of the two energies E,, and E, one gets E,/&
= $?/(K&)
+ 1/(K61)2(eXp
(-2K&)-
I)].
(21)
Figure 3 gives a graphical representation of the energy ratio E,/Ev as a function of 1/(~8,). Thus, with a given Debye-Hiickel parameter K and a given thickness S1 of the space charge layer, E, can be easily calculated from E, and vice versa. For low ionic strengths, i.e. for l/(~Sl)> 1 the free electrostatic energy of spatial arrangement does not differ very much from a two-dimensional one. This is understandable because, with greater Debye-Hiickel lengths, the spatial extension of charges becomes less effective. For arbitrary arrangements of the charges of the two space charge layers, the free energy of electrostatic interaction of these layers can be found, generally, by equation (15), after solving the following integrals, explicitly
ELECTROSTATIC
INTERACTION
OF
277
CELLS
I
5 Iu’
0
1
2
3
4
5
(K SJ’ FIG. 3. Ratio of the free energies of spatially distributed charge of constant density (I$,) and surface charge at the membrane (E,) as a function of the inverse product of Debye-Hickel parameter and thickness of glycocalix.
deduced from equation (8):
,
PZ+; a2$(x
-xi)
1dx*
(24)
Here, E(d) is again the free energy of the system. E1 and EZ are the free energies for d + co. The indices of potentials and charge densities indicate the type of charge (p corresponds to space charge and c to surface charge) and the number of the space charge layer (fir;: or second glycocalix). An important case in adhesion research is the mutual approach of a cell with a glycocalix and a smooth charged surface. Assuming a constant space charge density of fixed charges, the free interaction energy is expressed
278
A.
analytically
VOIGT
ET
AL.
as follows:
Eint=
’ %?&
2 sinh i sinh
p sinh
(K~)CT (Ed)
-
K
(-24-
-&Lexp
(~6)
sinh [K sinh (Kd)
11+&[coth
(d
6)] I
(Kd)
s
-
-
11.
(25)
Figures 4 and 5 illustrate the results for the free energies of the electrostatic interaction for different charge distributions as a function of membrane distance. Figure 4 shows a comparison of those cases where all the charge is located at the membrane surfaces (curve 4) or the space charge density is constant (curve l), linearly (curve 2) or exponentially (curve 3) increasing in each glycocalix [cf. Fig. 2(a)]. Figure 5 represents the results when subdividing the total charge into one, two or four plane charges per glycocalix, as was done before calculating the potential profiles of Fig. 2(c). It should be pointed out that the potential profiles given in Fig. 2(b) and (c) correspond to the same charge arrangements selected for Figs 4 and 5 at d = 11.5 nm. Figure 4 shows that the spatially distributed charge results in much greater interaction energy than surface charge, which is due to the diminished distance between the fixed charges. Of course, a similar high interaction energy in the case of surface charge distribution (curve 4) is only reached with remarkable glycocalix compression. L
-3. -4. d CT -5~
-61 2
23 \A
-7
, 4
-8 0
5
IO
15
ird
FIG. 4. Logarithm of the free energy of interaction of membranes with charge-carrying glycocalices as a function of the product of membrane distance and Debye-Hickel parameter. Curves 1-3 illustrate the effect of the charge density profiles given in Fig. 2(a), curve 4 represents charged membrane surfaces with 0.02 C/m . 6t = 8r = 5.5 nm and K = lo9 m-t.
ELECTROSTATIC
INTERACTION
OF
-71 0
5
IO
I5
279
CELLS
I
Kd FIG. 5. Logarithm of the free energy of interaction of membranes with plane charges at various positions in the glycocalices as a function of the product of membrane distance and Debye-Htickel parameter. Magnitude and position of the plane charge densities are given in the text. S1 = & = 5.5 nm and K = lo9 m-l.
For linearly and exponentialiy increasing space charge densities (Fig. 4, curves 2 and 3) and for two-dimensional charge arrangements at a considerable distance from the membrane surface (Fig. S), the dependence of the free energy of electrostatic interaction on the membrane distance exhibits a remarkable behaviour. At penetration of the glycocalices, there are pronounced local minima of free energy. The site and depth of the minima depend on charge distribution. Their existence is generally associated with a charge density increasing towards the outer edge of the glycocalix. One may conclude that such a stiff arrangement of charges can be stabilized by electrostatic effects under suitable conditions and after overcoming repulsive energy. Maxima on energy-distance curves result in changes of signs on forcedistance curves. In the interaction of spatial charge arrangements in Fig. 4 (curves l-3) the force vs. distance curve is continuous. If charge arrangement is two-dimensional, as in Figs 5 and 6, the force-distance function shows discontinuities whenever planes with plane charge densities of the two glycocalices exactly traverse each other. So the force abruptly changes its direction at these membrane distances, which is, of course, a mathematical idealization, but illustrates the qualitative difference between the two cases most clearly. Figure 6 shows, for two-dimensional charge arrangements with identical plane charge densities at the edges of the two glycocalices, the dependence of the free energy of electrostatic interaction on membrane distance and Debye-Hiickel parameter. With decreasing ionic strength (decreasing K)
280
A.
-717 0
5
VOIGT
ET
IO
AL.
15
20
d(m) FIG. 6. Logarithm of the free energy of interaction of membranes with a two-dimensional charge arrangement at the edges of the glycocalices as a function of distance and the Debye-Hiickel parameter. Plane charge densities are positioned at x = -d/2+6, and x = d/2 -6s with CT= 0.02 C/m*. Debye-Hiickel parameter K varies from lo9 m-’ (curve l), to 3 x 10’ m-r (curve 2) to 108m-’ (curve 3).
the energy minimum for overlapping glycocalices disappears, at a simultaneous global increase of the energy. Thus, due to the large Debye-Hiickel length, the screening of the fixed charge within the glycocalices is not sufficient and the system does not feel the distance between the charge planes. Whether such or similar charge arrangements investigated in this paper are of practical relevance for biological structures cannot be decided at present. To answer this question, the free energy minimum of a glycocalix structure will have to be determined under a number of further restrictive conditions, such as the given distribution of fixed charges along the macromolecular chains, primary and secondary intra- and intermolecular interactions, influence of interphase, etc. With flexible macromolecular chains stabilized by secondary bonds, the electrostatic contribution to the free energy becomes noticeable for determinihg the conformation of the molecules only if it lies in the order of secondary bonding energies, which, however, can as a rule be realized by suitable selection of milieu parameters, at least in vitro. Even at the present stage of this study an important conclusion may be drawn, that the usually assumed attraction constant (e.g. effective Hamaker constant) is overestimated. Indeed, a fit of experimental adhesion results in order to estimate the effective Hamaker constant, taking into account only surface charge densities at the membrane surfaces, fails due to overesti-
ELECTROSTATIC
INTERACTION
OF
CELLS
281
mated electrostatic repulsion. Preliminary calculations have shown that the effective Hamaker constant of cell interaction in aqueous solutions seems to be no higher than approximately 1O-22 J, this is an order of magnitude lower than the values often implicated (Nir & Andersen, 1977). This low value has also been predicted recently by van Oss er al. (1980), analysing the interface thermodynamics of blood cell interactions in serum. Nevertheless, for albumin adsorption at polystyrene latex Norde & Lyklema (1978) proposed a discontinuous distribution of charges inside the spatially extended adsorption layer. Two charge-carrying zones at the surface of polystyrene latex and at the edge of the adsorption layer are separated by a broad and free of fixed charges third zone. Only recently, a model for the structure of glycophorin A, the main charge-carrying glycoprotein on the human erythrocyte surface, has been proposed by Stibenz & Geyer (1980). It remains for further experimental and theoretical studies to determine the real distribution of fixed charges in the macromolecular layers on cell surfaces. A free electrostatic energy minimization approach is given by Donath & Voigt (1982). The concept presented in this paper can be helpful in the search for the energetically favoured arrangement of fixed charges and is applicable especially where, on the one hand, the effects of discrete charge distributions can be neglected and, on the other hand, the free electrostatic energy is not too great. This study was supported by the HFR “Kiinstlicher Organersatz und Biomaterialien” of the Ministry of Health of the G.D.R. The authors wish to express their gratitude to Professor Dr R. Glaser and Doz. Dr H. Wolf for their aid in writing this paper. REFERENCES BELL, G. DERJAGUIN, DOLOWY, DONATH, DONATH, GINGELL, GINGELL, GINGELL, GINGELL, HEINRICH, HESSELINK. LERCHE, NAPPER, NIR, S. &
I. (1978). Science, Wash. 200,618. 8. V. & LANDAU, L. D. (1941). Acta physicochim. U.S.S.R. 14,633. K. & HOLLY, F. J. (1978). j. theor. Biol. 75, 373. E. (1979). Dissertation, Humboldt-UniversitLt zu Berlin. E. & VOIGT, A. (1982). Submitted to J. theor. Biol. D. (1968). I theor. Biol. 19, 340. D. (1971). J. theor. Biol. 30, 121. D. & FORNBS, J. A. (1976). Biophys. J. 16, 1131. D. & PARSEGIAN, V. A. (1972). J. theor. Biol. 36,41. R., GAESTEL, M. & GLASER, R. (1981). Acta biol. med. germ. 40,765. F. TH.. VRLJ. A. & OVERBEEK. J. TH. G. (1971). J. ohvs. Chem. 75.2094. D: (198Oj. Dissdrtation, Humboldt:Universitlt’zu B&in. I D. H. (1970). J. Colloid Interface Sci. 32, 106. ANDERSEN, M. (1977). /. Membrane Biol. 31, 1.
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Sci.66,285.
ABSOLOM,D.R.(~~~O).
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PARSEGIAN, V. A. & WEISS, G. H. (1972). J. Colloid Interface Sci. 40, 35. PARSEGIAN,V.A.&GINGELL,D.(~~?~).B~~~~~S.J.~~,I~~~. PARSEGIAN,~. A.& GINGELL,D.(~~~O). J. CellSci.41,151. PASTUSHENKO,~.& DoNATH,E.(~~~~~). Stud.biophys.,Berl.56,7 and B2. PASTUSHENKO,~.& DONATH,E. (1976b). Stud.biophys.,Beri.56,9 andB8. STIBENZ,D. & GEYER, G. (1980). Folia haematol.,Leipzig 107,787. VERWEY, E. J. V. & OVERBEEK, J. TH. G. (1948). Theory of the Stability of Lyophobic Colloids. The Znteraction of Sol Particles Having an Electric Double Layer. Amsterdam: Elsevier.