Ion binding to charged lipid monolayers: The role of double layer and ion binding models

Ion Binding to Charged Lipid Monolayers: The Role of Double Layer and Ion Binding Models I A N S. G R A H A M , *'~'2 J O E L A. C O H E N , t

AND M A R T I N

J. Z U C K E R M A N N : ~

*Department of Physics, McGill University, 3600 University Street, Montreal, Quebec, Canada H3A 2T8; ?Department of Physiology, University of the Pacific, San Francisco, California 94115; and ~;Department of Physics, McGill University, 3600 University Street, Montreal, Quebec, Canada H3A 2T8 Received November 14, 1988; accepted May 15, 1989 We present a new approach to the analysis of monolayer surface potential experiments, based on a lattice model for binding of ions to the monolayer and two models for the electrical double layer: GouyChapman-Debye-Hfickel (GCDH) theory and a recently developed model that takes into account the finite polarizability of solvent dipoles and finite ion sizes. We demonstrate this approach by reanalyzing two recent monolayer surface potential experiments. For highly charged monolayers (Lakhadar-Ghazal, F., Tichadou, J. L., and Tocanne, J. F., Eur. J. Biochem. 134, 531 (1985)) we find that the new double layer model yields a description of surface properties which is distinctly different from that of GCDH theory, while at low surface charge densities the two models are effectively the same. The results of S. Ohki and R. Kurland (Biochim. Biophys. Acta 645, 170 ( 1981 )), fall in the latter regime, so our analysis of these data serves to contrast different binding models. Our results suggest that interactions between bound ions, or between lipid molecules in different ionization states, may be important in understanding surface properties when there is a high degree of ion binding. © 1990AcademicPress,Inc.

INTRODUCTION I o n b i n d i n g is k n o w n to p l a y an i m p o r t a n t role in lipid bilayer phase behavior ( 1 - 3 ) , lipid lateral phase s e p a r a t i o n ( 4 - 6 ) , a n d b i l a y e r vesicle fusion ( 7 - 9 ) . However, the m e c h a n i s m b y w h i c h i o n b i n d i n g induces such p h e n o m e n a is p o o r l y understood. This situation largely reflects a lack o f q u a n t i t a t i v e e x p e r i m e n t a l t e c h n i q u e s c a p a b l e o f directly p r o b i n g the int e r a c t i o n b e t w e e n the a d s o r b e d ions a n d the lipid m o n o l a y e r . Instead, the available experi m e n t s m e a s u r e m a c r o s c o p i c p r o p e r t i e s o f the e l e c t r o l y t e - m o n o l a y e r / b i l a y e r system, such as the e l e c t r o p h o r e t i c m o b i l i t y o f lipid bilayer vesicles, the lateral pressure o f lipid m o n o l a y ers, or the lipid m o n o l a y e r interfacial p o t e n tial. T h e m i c r o s c o p i c m e c h a n i s m m u s t t h e n Present address: National Research Council, Industrial Materials Research Institute, 75 de Mortagne Boulevard, Boucherville, Quebec, Canada, J4B 6Y4. 2 To whom all correspondence should be addressed.

be inferred t h r o u g h the use o f a p p r o p r i a t e models. O n e o f the simplest such e x p e r i m e n t s is m e a s u r e m e n t o f the p o t e n t i a l across a lipid m o n o l a y e r s p r e a d at an a i r - w a t e r interface. In these studies the c o n c e n t r a t i o n a n d ionic c o m p o s i t i o n o f the s u s p e n d i n g electrolyte are varied, giving rise to variation in the m e a s u r e d interfacial potential. These m e a s u r e m e n t s can be used to infer surface p r o p e r t i e s b y use o f a m o d e l for the electrostatic p o t e n t i a l o f the electrical d o u b l e layer plus a m o d e l for ion b i n d i n g to the m o n o l a y e r surface. T o date Gouy-Chapman-Debye-Hfickel (GCDH) t h e o r y has b e e n used as the s t a n d a r d m o d e l for the d o u b l e layer, while various i s o t h e r m s have b e e n p r o p o s e d to describe the b i n d i n g o f ions to the surface ( 10, 1 1 ). In this p a p e r we present a n e w theoretical a p p r o a c h to the p r o b l e m o f ion b i n d i n g to lipid m o n o l a y e r s a n d bilayers. W e use a recent m o d e l for the electrical d o u b l e layer b y G r a -

335 0021-9797/90 $3.00 Journal of Colloid and Interface Science, Vol. I35, No. 2, March 15, 1990

Copyright © 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

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GRAHAM, COHEN, AND ZUCKERMANN

ham et aL (12), which is a generalization of G C D H theory that takes into account the finite polarizability of solvent dipoles and finite ion sizes. In this paper we briefly summarize the results of that work and introduce a new lattice model for ion binding to lipid monolayers or bilayers. This model is a generalization of Langmuir adsorption which allows for non 1-1 binding stoichiometries. This paper is organized as follows. In the next section we describe the monolayer surface potential experiments and discuss the approximations commonly used in the analysis of such data. We also present a brief description of our new double layer model and give a new derivation of the lattice model for ion binding to the lipid monolayer. The third section contains the results of our reanalysis of some recent monolayer experiments (i.e., LakhdarGhazal et al. ( 13 ); Ohki and Kurland (14)), while the final section summarizes these results and suggests further avenues of investigation. THEORY

The Interracial Potential In the experiments we wish to consider, the interracial potential diV of a lipid monolayer suspended a t an air-water interface is measured as a function of the concentration and ionic composition of the electrolyte. This potential 2xV can be considered to arise from three sources. First is a contribution due to the reference air-water interface, which we denote by di Vo. Second is a large contribution from the average dipole moment perpendicular to the bilayer plane, due to the lipid molecules and the polarized lipid-water interface, which we denote by diVp. Finally there is the electrostatic contribution due to the electrical double layer, diVEs. The total interracial potential is then given by the expression diV = diV0 + dive + 2XVEs.

other c o m m o n approximation is to assume that the surface polarization contribution, 2xVp, is also independent of the ionic nature of the electrolyte. In this case changes in the interfacial potential, dxdiV, arise solely through changes in the double layer surface electrostatic potential, didi VES, and we can write didiV = didiVEs.

[21

This approximation was used by Ohki and Kurland (14) and will also be used in our experimental analysis. We note, however, that Lakhdar-Ghazal et al. (13) have analyzed their results using a model in which di Vp varies as a function of ion binding. Recent experiments indicate that surface polarization probably does vary with the degree of ion binding, and we discuss this aspect of the problem under Results and Discussion.

The Double Layer Model In a previous paper (12) we developed a new model for the electrical double layer. Here we review this model and describe its relationship to G C D H theory. We treat the solvent molecules as point dipoles and the dissolved ions as point charges. The same specific volume is assigned to both ions and solvent molecules, along with the additional requirement that the total particle concentration (ions plus solvent molecules) remain constant throughout the electrolyte, with a value equal to the inverse specific volume Co. In the previous paper we derived the equations governing the double layer by minimizing the appropriate free energy functional. This derivation, however, was limited to the case of monovalent electrolytes. The result can easily be extended to multicomponent electrolytes by taking the spatial-mixing entropy density (Eq. [2.4] o f ( 1 2 ) ) to be

Ill

In general we assume that the term di V0 is independent of the ionic strength and composition of the electrolyte; in any event it is small relative to the other contributions (13). AnYournal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990

., ,, / c " ( x ) l ] + c tx'ml-7{-o ) I.

[3]

CHARGED

LIPID

Here k is the Boltzmann constant, T the temperature in degrees Kelvin, and ci(x) are the local densities of ionic species i at distance x from the planar charged surface, while c6 are the bulk values of these concentrations, ca(x) is the local dipole density, given by

cd(x) = CO -- ~ ci(x). i

337

MONOLAYERS

dV + FP, dx

/~ = E + F P -

where F is a constant related to dipole-dipole coupling and E ( x ) is the macroscopic electric field. The potentials and fields are related by the equations of classical electrostatics:

[4]

D ( x ) = cooE(x) + 4~rP(x) dD

Minimization of the free energy now yields the following differential equations (in cgs units):

dx

[ 11 ]

- 47rp(x) = 4~rq ~ zici(x) i D ( x = 0) = 4~xr.

d2V c~ dx 2 -

-

-

4~r

_

-

dP dx -

4~rq

~

ziCio e-ziqV/kT

[5 ]

ZT(V, J~) i and P(x) =pce(x) zv(V,E)

[~J'

[61

[71

Zr(V, E) = 1 Z c6e-ziqv(x)/kr Co i

[12] [13]

Equation [ 13 ] is the boundary condition at the interface between the electrolyte and the lipid monolayer/bilayer, where a is the charge density at the lipid surface. In the limit of small fields and potentials we have Z ~_ ZT --~ 1 and we may linearize the Langevin function of Eq. [6 ]. In this case Eqs. [ 5 ] and [ 6 ] may be combined to give the standard GCDH description of the double layer, dzV co ~dx -

where the functions Z(/~) and ZT(V, E) are given by Z(E) = sinh(pE/kT) pJ~/kT

[10]

4~r ~ zic~e -z~qv/kT,

[14]

i where c0 is the bulk dielectric constant of the electrolyte, provided we make the following identification: 4~-p[c0 - Z c~] i

{

+ 1--

1 ~ Co

}

c6 z(t?).

c0 =

[8]

Here z~ is the valence of ionic species i, q is the magnitude of the elementary charge, and p is the solvent dipole moment, while V(x) is the electrostatic potential and P ( x ) is the polarization field due to the solvent dipoles. The parameter e~oallows for linear polarization contributions to the electric displacement that do not arise from the dipoles (12). J~(x) is the Langevin function, given by ~(x)=cothx

--.

1

Coo +

[9]

X

/~ is the microscopic field felt by the dipoles, defined by

3kT

{ ×

P2[CO-- ~CiO] 1-

3kT i

} -1 P

[15]

We impose this requirement so that our model reduces to GCDH theory in the limit of small fields. We chose the following values for these parameters (12): T = experimental value (-~300 K) Co = experimental value at temperature T (we chose e0 = 78) c~ = 4 (experimental dielectric constant above first dispersion: see (12)) p = 2.35 × 10 -18 esu-cm Co = (1/3.1)3A -3 I" = value calculated from Eq. [ 15 ] for a given

JournalofColloidandInterfaceScience,Vo.

135, No. 2, March 15, 1990

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GRAHAM,

COHEN, AND ZUCKERMANN

set of bulk salt concentrations and at the given temperature. The properties of the new double layer model are discussed extensively in our previous paper (12). At low surface charge densities (~<1 unit charge/200 A 2) our model is essentially equivalent to G C D H theory. At higher surface charge densities our model produces a depression of the dielectric constant close to the interface and surface potentials much higher than those arising from G C D H theory, for the same value of surface charge density. There are, of course, many other models for the double layer region. We contrast our model with other treatments at the conclusion of this paper. Ion Binding to the Surface

The simultaneous solution of Eqs. [5] and [ 6 ] (or solution of the G C D H Eq. [ 14 ] ) gives rise to a family of solutions V0(~), where V0 is the electrostatic potential at the surface and o- is the surface charge density. A second relation linking Vo and ~ is provided by the binding isotherm, which describes the electrochemical equilibrium between ions in the electrolyte and ions bound to the surface. This equilibrium condition can be written as # s _ #Li = -ziqVo,

[16]

where u s is the chemical potential of an ion of species i bound on the surface, ~ is the chemical potential of the same ion in the bulk electrolyte, zi is the valence of species i, q is the magnitude of the unit charge, and V0 is the surface potential, since we are assuming that ions bind at the surface-charge plane. The chemical potential g s is a function of the number of ions bound and as such must be derived from the free energy of the model chosen to describe ion binding to the surface. We choose to model the adsorbing surface as a lattice and adsorption as binding to the lattice sites. In principle an ion can bind to any number of adjacent sites on the lattice, and we shall refer to an ion bound with a given stoichiometry as an "object" on the lattice. We also Journal of ColloM and Interface Science, Vol. 135, No. 2, March 15, 1990

assume that lattice objects do not interact with each other and that each lattice site can be part of at most one object. We then use the Guggenheim combinatorial approximation (15) to derive an expression for the lattice free energy. We next calculate the chemical potential of a bound ion by the appropriate thermodynamic derivative of this free energy. The resulting expression is then substituted into Eq. [ 16] to yield the binding isotherm. Let the adsorbing surface be a lattice having N s sites and coordination number s. We let Ni be the number of objects of type i lying on the lattice, and let the integer ri denote the number of connected sites occupied by each of these objects. The subscript i thus labels a particular binding stoichiometry for each ionic species binding to the surface. By convention, the subscript i = 0 will denote vacant lattice sites (holes), with r0 = 1. Ns, Ni, and ri are linked by the relationship Ns = ~ Niri.

[17]

i=0

We d e f i n e r , the fraction of lattice sites occupied by objects of species i, by fi -

Niri

Ns

[181

Following Guggenheim ( 15 ) we define the parameter qi via qis = r i s - 2ri + 2.

[19]

The quantity qis is the number of nearestneighbor lattice sites adjacent to an isolated object of size ri, if the object is a simple extended chain. More generally, qis is the sum of the numbers of sites that are nearest-neighbors to each of the r~ sites of an isolated object, excluding, for each of the r~ sites, only those nearest-neighbors that are both occupied by the object and directly linked to that site by a single object "bond." Thus, depending on the geometry of the object, some nearest-neighbor sites may be counted more than once, and some may be occupied by the object itself. This definition includes objects comprising simple

CHARGED

or branched chains that may bend back on themselves, but it excludes objects having closed rings. We next define the quantity Nq by

Nq = ~ qiNi.

[20]

i=0

Since the number of bonded nearest-neighbor pairs within an object of size ri is r~ - 1, the total number of such pairs is ~ Ni(ri - 1 ). Since the total number of nearest-neighbor pairs on the lattice is ½sNs, it follows from Eqs. [ 17], [ 19], and [20] that ½sNqisthetotal number of nonbonded nearest-neighbor pairs on the lattice. Following Costas and Sanctuary (16), we now define the parameter q~ by

where pi denotes the number of distinct configurations of an object of type i lying on the lattice, m is hence a measure of the internal degrees of freedom for objects of type i. We are now ready to write down the free energy of the surface. The expression is

F ( T , Ns, {Ni}) = - k T l n ~2 + ~, NiEi, i=0

[251 where Ei is the binding energy for an object of type i. We have extended the sum to the index i = 0 since Ei = 0. Substitution of Eq. [24] into Eq. [25] gives the free energy per site:

F ( T , Ns, ~i = s

-

( r i - qi)Ni

--

{Ni})

Ns

Thus, using Eqs. [ 18 ], [ 19 ], and [ 21 ], ~if

339

LIPID MONOLAYERS

Ns

kTs l ~ ( - ~ ~ f ) I n ( l - Z Oil) i

,

- kT ~ f so that 4~ is the ratio of the fraction of nearestneighbor pairs on the lattice that are bonded in objects of type i, to the fraction of lattice sites occupied by such objects. It follows from Eqs. [17], [18], [20], and [22] that 1-

~ O~f - Nq .

i

i

[22]

[231

ln(ripi) + l n f + ~-~ . [261

i=0 ri

In the case of l - l adsorption of a single species Eq. [26] simplifies to

F - (1 - f ) l n ( 1 kTNs

Ns

The expression in Eq. [23] is the fraction of nearest-neighbor pairs on the lattice that are not bonded within any objects. Thus it is a measure of the external degrees of freedom of the objects on the lattice (16). Using the definitions in Eqs. [ 18 ] through [23] we rewrite the Guggenheim combinatorial expression for the number of configurations available to a system specified by Ns sites and { Ni } bound ions as follows:

-f) E + f ln f + f ~ ,

[271

which corresponds to Langmuir adsorption. Our model is therefore a generalization of Langmuir adsorption which allows for non 1-1 binding stoichiometries. We now evaluate the chemical potentials of the adsorbed species. These are obtained from the free energy:

{llNi

~2({Ni}) = 1F-[(r~p~)Ni \ f ] i=0

× (1 - ~ chif) sNs(l Zmf~)/2, [24]

ri

OF

i

Journal of Colloid and Interface Science, VoL 135, No. 2. March 15. 1990

340

GRAHAM,

COHEN,

Straightforward algebra yields an expression for u s which can be substituted into Eq. [16], giving the result

AND

ZUCKERMANN

Finally we calculate the surface charge density for a given set { f } ,

= ~o + q- ~ zi_f_f,

f f-~ (1 - ~ 4)if) sri*i/2 i=0

= ripie(UP-Ei)/kre-ziqV°/kr.

[29]

We now need to know the chemical potential of an ion in solution. Using standard notation we write the chemical potential of the ion in terms of its value in an arbitrary reference state, ~0, as

#L = t~Oi+ k T l n c~ + k T l n 7i({c~}), [301 where 7i is the chemical activity, which is a function of the concentrations of all the dissolved species. There is no general expression for the function 7i( { c J } )- The Debye-Htickel expression for the chemical potential of a bulk electrolyte is

#L = 0 + k T l n C~o+ f ( I ) ,

[311

where f is a function of I, the ionic strength of the electrolyte. However, this approximation is valid only at low ionic strengths (less than a few m M ) . We instead can assume that k T l n 7i is constant or that it is small compared to the difference between the bulk ideal chemical potential at standard concentration (~t0) and the binding energy E~. In the case of a dilute electrolyte the former condition is true provided that the ionic strength is kept constant. In either case the term k T In 7~ can be absorbed into a binding "constant," defined by K i = ripi d(u°i+kTln~'i-Ei)/kT.

[32]

Equation [ 32 ] allows us to write the equations describing the equilibrium state (Eq. [ 29 ] ) as f-~ (1 - ~

~ i f i ) sri~Si/2 =

[34]

a i= 1 F i

Kicre -z'qv°/kr. [33]

i=0

In the Appendix we show that for competitive binding of monovalent and divalent ions our model is equivalent to the Bethe approximation used by Cohen and Cohen (10). Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990

where a is the surface area per lattice site and ~o is the charge density of the monolayer in the absence of associated ions. Equations [ 33 ] and [34] give the desired second relationship between V0 and o-. RESULTS

AND

DISCUSSION

The purpose of this section is to examine experimental results for lipid monolayers at an air-water interface in light of the theory presented above. We have chosen two sets of experimental results appropriate for the following tasks: (i) Analysis for the case where the ion binding is as simple as possible ( 1-1 ) and the monolayer surface charge density is sufficiently high that the new electrolyte model differs significantly from GCDH theory. This situation allows us to compare the two electrolyte models independently of the binding model. (ii) Analysis for the case where binding stoichiometry may be important but where the surface charge is sufficiently low that the new electrolyte model is indistinguishable from GCDH theory. This situation allows us to compare binding models independently of the electrolyte model. The experimental data consist of measurements of the change in surface potential, ~x2xV, as a function of changes in the concentration and ionic composition of the electrolyte. Our analysis of these data is as follows. We choose a model for the electrolyte (GCDH theory or our electrolyte model) along with a model for competitive ion binding. We analyze the surface potential data by finding the intrinsic binding constants Ki which give the best fit to the experimental data. The different models (i.e., GCDH plus 1-1 binding, new electrolyte model plus 1-2 binding, etc.) can be contrasted by comparing values of x 2 per degree

341

C H A R G E D LIPID M O N O L A Y E R S

of freedom, the lower value corresponding to the better fit.

High Surface Charge Density We begin with the experimental data of Lakhdar-Ghazal et al. (13), which were kindly made available to us by Professor J.-F. Tocanne. These authors studied the surface potential o f dilauroylphosphatidylglycerol ( D L P G ) monolayers at an air-water interface as a function of pH and of monovalent salt concentration (NaC1, LiC1, and CsC1). The experiments were performed at a temperature of 20°C and at a fixed area per lipid molecule of 75 ~2. Each data point had a standard error of_+ 5 mV. In these experiments the interfacial potential was measured as a function of pH for electrolytes containing either 0.01 or 0.1 M monovalent salt. Additional experiments were performed at a fixed pH of 5.6 for monovalent salt concentrations varying between 10-6 and 0.1 M. The fitted results for the 0.01 M data, assuming competitive 1-1 binding, are shown in Table I. Here we give the results for inde-

pendent fits to the three sets of data and also the result for a fit to all three sets constrained to have the same proton binding constant KI~. The fits for the case of the c o m m o n proton binding constant are shown in Figs. l and 2. We now compare the fits obtained using our electrolyte model with those obtained using G C D H theory. The new electrolyte model provides fits to these data which are qualitatively (Figs. 1A and 2 ) and quantitatively (Table I) somewhat better than the corresponding G C D H fits. In all cases the X2/df resulting from the new electrolyte model are lower than those obtained using G C D H theory. In addition we note that the best G C D H fits often gave negative monovalent-cation binding constants, which are clearly unphysical. Therefore the G C D H binding constants were, when necessary, restricted to be greater than an arbitrarily small value, here 1 X 10-s M-~. The principal interest of these fits is in the predicted properties at the monolayer surface. A comparison of the new electrolyte model with G C D H theory for the case of NaC1 is given in Figs. 1B through 1D. (The results for

TABLEI FiaedParameters ~rthe0.01MMonovalentSaltDataofLakhdar-Ghazal~ Salt

No. of points

Kn (M -z)

KM (M -~)

x2/df

a/.(13) Constraints

(A) Fits using the new electrolyte model NaCI LiC1 CsCI

12 14 16

0.25 0.28 0.30

Alls~ts

42

0.29

0.005 0.005 0.004 KNa = 0.005 KLi = 0.006 Kc~ = 0.004

0.42 6.6 2.3 2.9

(B) Fits using G C D H theory NaC1 LiC1 CsC1

12 14 16

1.75 2.48 2.29

1 X 10-5 0.013 1 × 10 -s

0.73 7.9 3.4

All salts

42

2.25

KN. = 0.005 KLi = 0.010 K c s ~ 1 X 10 -5

3.7

KNa/> 1 X 10 -5 M -~ Kcs >/ 1 X 10 -s M -1

K¢~>~ 1 X 10- s M -l

Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990

342

GRAHAM, COHEN, AND ZUCKERMANN

200

i

,

,

(A) > E ,E

/..

O.01M NaCt

150

GcDNeWHElectr°lyt~ Model

/

/

10o

<~

/~-"_ M

///*

50 ^_. j •

0 1.0

i

I

Q

I

N \

0.8

//

///

0.6

b x

I

(B)

-.

fH

/ "'.x(

0.4

0.2

__f._° .......... 0.0

9

6

3

O

pH FIG. 1. Fits to the 0.01 M NaCI data of Lakhdar-Ghazal et al. ( 13 ). The solid and dashed lines give the results using the new electrolyte model described in the text and GCDH theory, respectively. The binding constants from Table I are K~a = 0.005 M ~and KH = 0.29 M -~ for our new electrolyte model and 0.005 M -~ and 2.25 M -~, respectively, for the GCDH case. (A) Predicted surface potential AAVas a function of pH. (B) Variation in fNa and fm the bound ion fractions, and in a, the fraction of unattached surface sites, as a function of pH. (C) Predicted variation in the electrostatic surface potential A Vzs as a function of pH. (D) Prediction, using the new electrolyte model, for the dependence of the surface dielectric constant (from Eq. [35]) on pH.

LiC1 a n d CsC1 are n o t displayed, as they are qualitatively the s a m e as for NaC1.) Figure 1B shows the fraction o f sites b o u n d b y a p r o t o n , fH, the fraction b o u n d b y the m o n o v a l e n t salt, fNa, a n d the fraction o f u n a t t a c h e d sites, c~, all as a f u n c t i o n o f p H . ~ is also the a r e a / s u r f a c e charge o f a fully dissociated surface ( h e r e 75 ~ 2 ) d i v i d e d b y the a r e a / s u r f a c e charge o f the i o n - b o u n d surface. W e see t h a t a for o u r new electrolyte m o d e l varies f r o m ~ 0 . 4 to ~ 0.8, which c o r r e s p o n d s to a n a r e a / s u r f a c e charge b e t w e e n 190 A2 a n d 95 A2. Since o u r m o d e l gives a steeper d e p e n dence o f surface p o t e n t i a l o n surface charge density t h a n does G C D H t h e o r y ( 1 2 ) , we exJournal of Colloid and Interface Science, Vol. 135,No. 2, March 15, 1990

pect the G C D H result to require a considerably greater change in surface charge density t h a n does o u r m o d e l for a similar d e p e n d e n c e o f surface p o t e n t i a l on p H , as seen in Fig. 1A. E x a m i n a t i o n o f Fig. 1B c o n f i r m s this expectation, with the G C D H fits predicting a r e a s / surface charge b e t w e e n 400 a n d 80 ~2. Therefore, a l t h o u g h o u r t h e o r y a n d G C D H t h e o r y can fit similar curves o f surface p o t e n t i a l as a f u n c t i o n o f p H , the p r e d i c t e d b e h a v i o r at the surface is m a r k e d l y different, with the two m o d e l s requiring c o n s i d e r a b l y different degrees o f surface binding. This result is also reflected in the very different fitted values o f the p r o t o n b i n d i n g c o n s t a n t s seen in Table I.

CHARGED

LIPID

343

MONOLAYERS

i

t

1

///

(C )

-40

//

> -80 E

///

"~-120 > /

-160 -200 -240 I

I

I

(D)

~_ 6o 0 U U

_J W Q W U

I

50

4o 30

~ c~ 2o 09 I 3

I 6

q0

I O

pH FIG. 1.--Continued

The difference between the two models is further reflected in Fig. 1C, which shows that our model predicts ~ 3 6 mV larger contribution to the magnitude of the interfacial potential due to the double layer than does GCDH theory. Our model consequently predicts a larger surface polarization contribution than does GCDH theory, since the double layer potential contributions are negative, whereas the net monolayer surface potential (including the polarization contribution) is positive (13). In the new double layer model the dielectric constant at the monolayer surface is a function of the surface charge density (12). It is calculated via D(O) 40)

-

E(0)

'

[351

where D(0) and E(0) are, respectively, the macroscopic electric displacement and electric

field at the monolayer surface. For the PG monolayer experiments our model predicts a significant reduction of the dielectric constant at the surface relative to its bulk value of 78. This effect is shown in Fig. 1D, which indicates a surface dielectric constant varying between 20 and 50 over the pH range studied. This result is in good agreement with recent experimental studies which indicate a value of approximately 30 for the dielectric constant at the membrane surface ( 17-19 ). We have used the above fitted binding constants to predict the behavior of the remaining data of Lakhdar-Ghazal et al. (13) and have tabulated the resulting x 2 / d f values in Table II. On the whole these parameter-free fits are rather poor. We believe this failure results from our assumption that the polarization potential, 2x Ve of Eq. [ 1], is constant and independent of salt concentration. Several recent experiments indicate that ion binding can perturb Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990

344

GRAHAM, COHEN, AND ZUCKERMANN

200

, (A)

:>

0.01M L]CL New Electrolyte Model

150

E c-

7//~ o /'/

> 100

...... GCDH O

~ / zz ~"

•

i

50

O

9

r~- -

6

3

0

t

pH 200

i

(B) > E

/ O.

150

>100 <] 50 0

o I

9

0

I

o ~

~

6

L

L

3

0

pH

FIG. 2. Fits to the 0.01 M LiC1 (A) and CsC1 ( B ) data of Lakhdar-Ghazal et al. ( 13) showing AA V as a function of pH. The solid and dashed lines give the results using the new electrolyte model described in the text and GCDH theory, respectively. The binding constants (from Table I) used with the new electrolyte model are Ku = 0.006 M -1, Kcs = 0.004 M -1, and Kn = 0.29 M -1. For the GCDH case these constants are 0.01 M -l, 1 × 10 5 M i and 2.25 M -~, respectively. A proton binding constant common to the three monovalent salt experiments is assumed in these fits (cf. Table I).

the polar region o f the lipid head group (1, 20) and that lithium, in particular, can strongly affect the phase behavior and polar head c o n f o r m a t i o n o f anionic bilayers (21, 22). We note that the poorest fit in Table II is for the lithium binding data. Lakhdar-Ghazal et al. (13 ) have analyzed their results allowing for a polarization potential that varies linearly with the b o u n d fraction o f the different species. However, we note that (i) surface p o l a r i z a t i o n / h y d r a t i o n is almost certainly a cooperative p h e n o m e n o n ; and (ii) the presence ( a n d probable disruption) o f a hydration layer in the neighborhood o f the Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990

polar heads implies the existence o f a large free energy contribution which is not properly treated, either by the approach o f LakhdarGhazal et al. (13) or by the use o f simple binding constants in our own binding model. We consequently believe that the good fit obtained by L a k h d a r - G h a z a l et al. ( 13 ) is largely due to extra fitting parameters and does not necessarily reflect the appropriateness o f their approximation. Unfortunately there as yet exists no binding model which includes surface "polarization" or " h y d r a t i o n " effects, and we therefore have not included a variable polarization potential in our analysis.

CHARGED LIPID MONOLAYERS

345

or tetramethylammonium chloride, Me4NC1) as a function o f p H , and (b) at fixed p H (7.4) × 2 per Degree of Freedom for the Remaining Data and monovalent salt concentration (0.1 M ) of Lakhdar-Ghazal et al. (13) as a function of divalent salt concentration No. of (CaC12 or MgC12). The temperature was 24 data Salt points Model x z/dr _+ 2°C and the average area/lipid molecule was 65 + 2 A2. The standard errors were +1 0.1 M NaCl 10 New electrolyte model 1.4 m V for the NaC1 data and +2 m V for the GCDH 2.6 Me4NC1 data. NaC1 (pH 5.6) 7 New electrolyte model 3.8 In these experiments the degree of surface GCDH 3.1 binding is high so that the net surface charge LiCI (pH 5.6) 6 New electrolyte model 65 density is low. In this regime the new double GCDH 67 layer model and G C D H theory are practically equivalent. Consequently we only present reCsC1 (pH 5.6) 6 New electrolyte model 12 GCDH 14 sults using the new electrolyte model. Our interest here is in contrasting the predictive beNote. We have used the binding constants from Table havior of different binding models. I assuming a common proton binding constant for the Phosphatidylserine has three possible bindthree different salt experiments. ing sites (phosphate, carboxyl, and amino groups) which in principle renders the binding analysis more difficult than is the case for PG. Last, we look at the G C D H fits and compare A recent study has reported the apparent pKa our fitted binding constants with others in the values of these sites, in 0.1 MNaC1, to be 0.5, literature. In particular we find KNa ~ 0 M -1, 5.5, and 11.5, respectively ( 1 ). Thus only the and KI~ ~ 2 M - 1, which is consistent with the carboxyl group should undergo significant devalues KNa = 0 to 0.005 M -~ and KH = 1.2 to protonation over the interfacial p H range 2 to 1.6 M -~ suggested by the monolayer study of 9, and we can therefore use our binding model Toko and Yamafuji (23). However, electroto analyze experiments in which the p H is phoretic mobility and N M R studies of bilayer varied over this range. Similarly the carboxyl vesicles both suggest a sodium binding congroup is expected to be the only site signifistant in the range 0.6 to 0.8 M 1 (20, 24, 25). cantly affected by changes in ionic composiThis inconsistency m a y be due to the different tion and strength at a constant p H of 7.4, so natures of the experiments (the latter being we can treat divalent-ion binding experiments studies of ion binding to egg PG bilayers, the using the same model. former studies of proton binding to monolayWe have analyzed these data as follows. We ers of homogeneous acyl chain P G ) , or it m a y first determined the monovalent-salt binding reflect a fundamental difference between PG constants by fitting the monovalent salt data, monolayers and bilayers. assuming competitive 1-1 adsorption of the monovalent species. The results are s u m m a Low Surface Charge Density rized in Table III. Taking the monovalent-salt We now consider the results of Ohki and binding constants as fixed, we then fit the diKurland (14), who measured changes iin the valent salt data, assuming competitive adsurface potential of bovine brain phosphati- sorption of the monovalent and divalent spedylserine (PS) monolayers at an air/w~ ter in- cies and either a 1-1 or a 1-2 divalent-salt terface as a function of electrolyte corn :entra- binding stoichiometry. The 1-2 binding isotion and ionic composition. These ,,xperi- therm requires the specification of the lattice ments were performed (a) at ccnstant coordination n u m b e r s, i.e., the n u m b e r of monovalent salt concentration (0.1 3~ NaC1 nearest-neighbor sites surrounding any site on TABLE II

Journal of Colloid and lnterface Science, Vol. 135, No. 2, March 15, 1990

346

GRAHAM, COHEN, AND ZUCKERMANN TABLE III

Fitted Parameters for the 0.1 M Monovalent Salt Data of Ohki and Kurland (14) as a Function of pH, Using the New Electrolyte Model

Salt

No. of points

Kn (M -x)

KM(M -l)

x2/df

16 17 33

1770 3900 2325

1.08 0.52 KNa = 1.33 KM~4N= 0.334

3.0 4.4 3.8

NaC1 Me4NC1 NaC1 and Me4NC1

the lattice. W e have chosen s = 6, which reflects the h e x a g o n a l p a c k i n g o f PS h y d r o c a r b o n chains in p h o s p h o l i p i d bilayers ( 2 6 ) . T h e X 2 values for the different fits to the d i v a l e n t - i o n b i n d i n g d a t a are s u m m a r i z e d in T a b l e IV. Characteristic fits, c o r r e s p o n d i n g to the case where the different m o n o v a l e n t - s a l t results were c o n s t r a i n e d to have the s a m e di-

valent-ion binding constant, are shown in Figs. 3 a n d 4. W e first c o n s i d e r the c a l c i u m results. T h e X 2 values o f T a b l e IV suggest t h a t c a l c i u m b i n d s to PS m o n o l a y e r s with a 1-1 stoichio m e t r y . W e note, however, that the different m o n o v a l e n t - s a l t e x p e r i m e n t s yield rather different p r e d i c t i o n s for the c a l c i u m b i n d i n g constants a n d that the best fit to the c o m b i n e d data, shown in Fig. 3, yields a m u c h better fit to the NaC1 curve t h a n to the Me4NC1 curve. In the case o f m a g n e s i u m a clear distinction b e t w e e n 1-1 a n d 1-2 b i n d i n g is n o t a p p a r e n t . Quantitatively, the X 2 values c o r r e s p o n d i n g to separate fits to each m o n o v a l e n t - s a l t exp e r i m e n t seem to s u p p o r t a 1-1 b i n d i n g stoic h i o m e t r y . H o w e v e r the fit using a single m a g n e s i u m b i n d i n g c o n s t a n t c o m m o n to the two e x p e r i m e n t s supports the 1-2 b i n d i n g scheme. Curves c o r r e s p o n d i n g to the latter case are shown in Fig. 4.

TABLE IV Fits to Calcium and Magnesium Binding Data of Ohki and Kurland (14) Salt

No. of points

Kca(M-1)

K M (M 1)

xZ/df

Divalent binding stoichiometry

(A) Calcium binding

CaC12/NaC1

9 9

13.73 100

1.33 1.33

1.3 18.3

1-1 1-2

CaCIz/Me4NC1

9 9

22.0 128

0.334 0.334

1.5 4.5

1- 1 1-2

CaCl2/both monovalent salts

18 18

15.34 106

gNa, KMe4N

as above

3.1 10.7

1-1 1-2

KM (M 1)

xZ/df

Divalent binding stoichiometry

(B) Magnesium binding Salt

No. of points

KM, (M -j)

MgCI2/NaC1

8 8

5.5 33.4

1.33 1.33

1.3 2.2

1-1 1-2

MgCIz/Me4NC1

9 9

11.2 53.1

0.334 0.334

1.2 2.3

1-1 1-2

17 17

6.72 38.7

KNa, KMe4N as above

4.9 2.9

1-1 1-2

MgClz/both monovalent salts

Note. The monovalent salt binding constants are those values from Table III, using a common proton binding constant for the two monovalent salts. The electrolyte is treated by the new double layer model. Journal of Colloid and Interface Science, Vol. 135, No. 2, March 15, 1990

CHARGED LIPID MONOLAYERS 8 0

i

70

i

(A)

i

-

<8

i

0.1M Me 4 N C [

60 c 50

347

.S/'" # ./27~" / / - f/ , -

1-1 Binding

-

........ 1-2 Binding

30

~/

. -'0""

20

~-d'"

. "" ~

"

0 1.0

I

I

I

I

I

(B) 0.8

_--fMe4N

(U 0 . 6

~~

,,

fCa

.:'~: \\

0.4

/

0.2

-.

_

// I

-6

-5

-4

-3

-2

I.Oglo [Ca *+] ( M )

FIG. 3. Fits to the calcium ion data of Ohki and Kurland (14) using the new electrolyte model. From Table IV the monovalent-ion binding constants are KNa = 1.33 M t and KMe4N= 0.334 M -1. The divalention binding constants are Kca = 15.34 M -~ for 1-1 binding and Kc~ - 106 M -I for 1-2 binding. Divalention binding constants common to the two monovalent salt experiments are assumed in these fits. (A) Predicted surface potential 2xAV as a function of calcium concentration, assuming either a 1- 1 or 1-2 CaPS binding stoichiometry. (B) Fractional surface charge (a) and the fraction of sites bound by calcium (fc,) and by the ion Me4N + (fMe4N)for each calcium binding stoichiometry.

W e suspect that i n t e r a c t i o n s at the m o n o layer surface are responsible in part for these difficulties. T h e M e 4 N + i o n is large on the scale o f the lipid h e a d g r o u p a n d it is therefore possible t h a t a b o u n d M e 4 N + i o n interacts with a d j a c e n t lipid molecules. Such interactions c o u l d b e very different f r o m those d u e to b o u n d s o d i u m ions, the differences being greatest w h e n m o s t o f the lipids possess b o u n d c o u n t e r i o n s , as is the case in these experiments. O u r b i n d i n g m o d e l does n o t i n c l u d e surface interactions; therefore it m a y be ina p p r o p r i a t e to fit s i m u l t a n e o u s l y the s a m e div a l e n t - i o n b i n d i n g c o n s t a n t to the d a t a for the two m o n o v a l e n t salts. In this case we c a n bet-

ter trust the i n d i v i d u a l fits in T a b l e IV, a n d these results favor a 1-1 m a g n e s i u m b i n d i n g stoichiometry. C o n t r a s t i n g the fitted m o n o v a l e n t - s a l t b i n d ing constants for P G with those for PS provides tentative s u p p o r t for the existence o f such interactions at high surface coverage. T a b l e III shows that the individual fits to the PS b i n d i n g e x p e r i m e n t s c o n d u c t e d at varying p H yield quite different values for the p r o t o n b i n d i n g c o n s t a n t d e p e n d i n g o n the m o n o v a l e n t salt used. This is quite different f r o m the results o f the P G e x p e r i m e n t s o f L a k h d a r - G h a z a l et al. ( 1 3 ) ( T a b l e I ) , which show r e m a r k a b l y consistent values o f KH for the three different Jot~rnal of Colloid and lnterface Science, Vol. 135, No. 2, March 15, 1990

348

GRAHAM, COHEN, AND ZUCKERMANN 80 70

i

i

i

(A)

i

i

0.1M M e 4 NCL

n ~

60 2>

E

7 /

50

r'-" o-- 40 1> <:1 30

1-1

Binding

~...~""

. . . . . 1-2 Bindlng

/~/¢~

20

d

j.~o

~

¢'/J NaCL

10 0 1,0

I

I

I

1

I

(B) 0o8

s

fMe4N ~)

x

~ ~

~ ~

0.6

0.4

~

~" ~

----=

.....

fl,4g

eS_

0,2

-6

-5

-4

-3

-2

IOglo [Mg ++] (M) FIG. 4. Fits to the magnesium ion data of Ohki and Kurland (14) using the new electrolyte model. The binding constants are from Table IV: the monovalent-ion binding constants are as in Fig. 3, while the divalent-ion binding constants are KMg = 6.72 M -1 for 1-1 binding and KMg = 38.7 M -1 for 1-2 binding. Divalent-ion binding constants common to the two monovalent salt experiments are assumed in these fits. (A) Predicted surface potential AAV as a function of magnesium concentration, assuming either a 1-1 or 1-2 Mg-PS binding stoichiometry. (B) Fractional surface charge (cQ and the fraction of sites bound by magnesium (fMg) and by the ion MegN + (fMe+N)for each magnesium binding stoichiometry.

salts NaC1, CsC1, a n d LiCI. This is the behavior we w o u l d expect if short-range surface interactions are i m p o r t a n t at high lattice coverage, since the PS monolayers, at 0.1 M m o n o v a l e n t salt, have a m u c h higher density o f b o u n d ions t h a n do the P G m o n o l a y e r s , at 0.01 M m o n o v a l e n t salt, s t u d i e d b y L a k h d a r - G h a z a l et al. ( 1 3 ) . W e therefore suggest t h a t i n t e r a c t i o n b e t w e e n lipid m o l e c u l e s in their different ionization states m a y be i m p o r t a n t w h e n the surface is heavily b o u n d with counterions. Since o u r fits are largely i n d e p e n d e n t o f the d o u b l e layer m o d e l used, we can realistically c o m p a r e o u r b i n d i n g c o n s t a n t s with those in the literature. These m e a s u r e m e n t s , based Journal of Colloid and lnterface Science, Vol. 135, No. 2, March 15, 1990

p r i m a r i l y on e l e c t r o p h o r e t i c m o b i l i t y studies o f bilayer vesicles, give the following values for these b i n d i n g constants: KNa = 0.6 to 1.0 M - l

( M c L a u g h l i n et al.

( 2 7 ) , N i r a n d Bentz ( 2 8 ) , Portis et al. ( 2 9 ) ) KMe4N ~< 0.02 M - l /(Ca ~

12 M -1

(Eisenberg et al. ( 2 4 ) )

( 1-1 s t o i c h i o m e t r y ) ( M c L a u g h l i n et al. ( 2 7 ) )

KMg ~ 8 M -1

(1-1 stoichiometry) ( M c L a u g h l i n et al. ( 2 7 ) ) .

CHARGED LIPID MONOLAYERS Except for Me4NC1, our binding constants (Tables III and IV) are consistent with these values. However, we note that Eisenberg et al. (24) report anomalous behavior for Me4NC1 compared to the alkaline salts. CONCLUSIONS We have presented a new approach to the analysis of monolayer surface potential experiments, using a simple lattice model for binding of ions to the monolayer and two models for the electrical double layer: GCDH theory and the recently developed model of Graham et al. (12). At moderate to high surface charge densities the new electrolyte model predicts monolayer surface properties which are significantly different from those of GCDH theory. Table I contrasts the difference in the predicted binding constants, while Fig. 1 shows the very different predictions for different degrees of ion binding (Fig. 1B), different absolute contributions to the electrostatic surface potential (Fig. 1C), and a suppression of the surface dielectric constant from its bulk value of approximately 80 (Fig. 1D). Recent experiments have indicated that the dielectric constant is depressed close to the bilayer surface (19), in agreement with our predictions. We note, however, that other models which deal specifically with the interracial region also predict depression of the dielectric constant close to the lipid interface. These other approaches attempt to model the effect of hydration or polarization of the interface (30-32) or nonelectrostatic effects due to ion binding (33). These aspects of the interface are absent from our model. It should therefore be kept in mind that our electrolyte model is still a poor approximation to the double layer, albeit hopefully less so than GCDH theory. We treat the electrolyte as a system of hard-sphere ions and dipoles within a mean field approximation, and we consider the charged bilayer surface to be flat and uniformly charged. Our approach therefore ignores such important effects as hydrogen

349

bonding (both at the surface and between the water molecules) and the finite width and molecular nature of the interfacial region. Given these approximations our model predicts, for high surface charge densities, surface potentials which are significantly higher than those predicted by GCDH theory. This trend is consistent with other, more sophisticated approaches, such as the mean spherical approximation analysis of Blum and Henderson (34) or the mean electrostatic potential treatment of Outhwaite (35). Other treatments can be found in the work of Carnie and Chan (36) and Dong et al. (37). Unfortunately there are no simulation (Monte Carlo) results to test these theories, which are for the most part limited to dilute electrolytes and low surface charge densities. Double layer theory is an area of ongoing interest, with the hard ion and dipole model being one of the more complicated approaches. A good review of double layer theory is given by Carnie and Torrie (38). Some interesting recent work involving hard ions with hard dipoles and quadrupoles is found in the papers of Torrie et al. (39). There are also recent treatments (Gruen and Marrelja ( 32 ), Luzar et al. (30), Gur et al. (31)) in which the hydrated, or polarized, surface is included within a description of the double layer. We note that for a given surface charge density all three of these models predict higher surface potentials than does GCDH theory. Our intent therefore is not to emphasize the appropriateness of our model. Rather it is to stress the thct that different double layer models can give rise, under appropriate conditions (e.g., high surface charge density), to distinctly different descriptions of the interface. Such differences can be seen in macroscopic averages such as the surface potential and surface charge density, which are relevant to surface binding models. Correct knowledge of these quantities is essential if one is to extend the binding analysis beyond the noninteracting Langmuir-like models used to date. With regard to divalent-ion binding to PS membranes, where the results are independent Journal of Colloid and lnterface Science, Vol. 135, No. 2, March 15, 1990

350

GRAHAM, COHEN, AND ZUCKERMANN

APPENDIX of the double layer model used, our analysis at low surface charge densities suggests a 1-1 In this Appendix we consider ion binding binding stoichiometry for the binding of calto a lipid surface where each binding site (lipid cium and magnesium to PS monolayers. molecule) carries an intrinsic unit negative While this result is certainly reasonable it needs charge. We first consider the case of a single to be supported by direct experimental evimonovalent species (bulk concentration Cl, dence, for example, 2H-NMR as used by Macbinding constant/£1, with r~ = 1, z~ = +1, q~l donald and Seelig (20) to investigate calcium = O, f~ = N 1 / N s ) binding to a negatively binding to mixed phosphatidylethanolaminecharged lipid monolayer. The isotherm (Eq. phosphatidylserine membranes. [33]) then reduces to a single equation More significantly, our analysis suggests that surface interactions may be important, f _l-fo whereas our binding model, which lumps all - -- K 1 c l e - q v ° / k T , [A1] the chemistry of binding into a single parame t e r - t h e binding constant--does not allow since ~ f = f0 + f~ = 1. Rearrangement gives for interactions between sites on the monolayer surface and also ignores the quantum1 mechanical nature of the binding mechanism fo = 1 + h'11cle-qV°/kz' [A2] (40). Given these caveats the quasi-chemical approximation allows for straightforward ex- which is the standard Langmuir isotherm. If tension of the lattice model to include nearest- the intrinsic surface charge density of the neighbor interactions. However, it must be (negatively charged) monolayer is o-0 = - q / a , noted that this analysis introduces many new where a is the area per lattice site and q is the parameters whose use can only be justified by magnitude of the elementary charge, we can appropriate experimental data. From our write the monolayer charge density (Eq. analysis it seems clear that monolayer studies [34]) as cannot provide the needed results. Similar problems are likely to exist with electrophoo" = ~ 0 - flo'0 = foa0. [A3] retic mobility studies, owing to the difficulty of determining the exact position of the plane The generalization to competitive bindof shear (41) and to possible nonelectrostatic ing among multiple monovalent species is contributions to the electrophoretic mobility straightforward, since the resulting equations (30). A good summary of the successes (and (from Eq. [33]) are all linear. some of the limitations) of the GCDH analysis We now consider competitive binding beas applied to electrophoretic mobility experi- tween a monovalent species (parameters given ments is found in (41) and (42). above) and a divalent species (bulk concenTherefore, although GCDH theory allows tration c2, z2 = +2) which binds to the sura good qualitative description of ion binding face described above with a l-1 stoichiomto lipid bilayer membranes, it is unlikely to etry (binding constant K2, r2 = 1, ~2 = 0, provide the quantitative basis for a more so- f2 = N z / N s ) . In this case the isotherm is given phisticated binding analysis. Such analysis will by the two equations require either a more detailed statistical-mechanical treatment of the bilayer-electrolyte f l = K 1 cle -~v°/kT [A4 ] system or experiments which can more easily f0 separate the behavior of the surface from that of the double layer. Unfortunately the requisite f 2 __ K z c z e - Z q V o / k T , [A5] fo experiments have not been done to date. Journal of Colloidand InterfaceScience, VoL 135,No, 2, March 15, 1990

351

CHARGED LIPID MONOLAYERS

which are coupled by the requirement that Z f = f o + f l + f 2 = 1. These linear equations can be rearranged to give

f0 = 1 1 + K1¢1 e-qV°/kT q- K2c2 e-zqv°/kT'

f3(1 --f3/s)

K3C3e-2qV°/kT.

[AS]

Since ~ f = f0 + f3 = 1 we can write this as (As+

1)f~+(s-2)fo+(1-s)=O,

[A9] where we have defined A = 1£3 c3e-2qV°/kT.This quadratic equation can be solved for f0. The surface charge density is then given by ~r -- ~r0(1 --f3) = foa0.

[A10]

Our last example illustrates competitive binding between a monovalent species (bulk concentration Cl, binding constant K1 ) and a divalent species which binds with a 1-2 stoichiometry (bulk concentration c3, binding constant/£3). The binding isotherm is now given by the two equations

fo

_ KlCle_qVo/kT -~ A

B

[A13]

(1 + A ) 2'

a = a0(1 - f l - f 3 ) =f0a0.

[A14]

Comparison of Eq. [ A 13 ] with Eqs. [ 46a ] and [43b] of Cohen and Cohen (10) shows that, for this case, our isotherm is equivalent to the monomer-dimer adsorption isotherm calculated in the Bethe approximation. There is, however, a subtle notational difference between the two results, which arises from different definitions of the binding constants. In our analysis the binding constant /£3 (and hence the constant B) contains the term r3p3, where p3 counts the number of distinct configurations of a dirner on the lattice (see the definition of the binding constant in Eq. [ 32 ]). For dimers with indistinguishable ends P3 = s / 2 , where the factor 2, the symmetry number of the dimer, takes account of the fact that we can flip the ends of the dimer without changing the configuration. Cohen and Cohen (10) have defined their binding constants to exclude the term r303 = s (see their Eqs. [15] and [20]), retaining it instead within the combinatorial factor g u ( N D ) (see their Eqs. [ 24 ] and [ A 1] ). Consequently, their isotherm contains, compared to our result, an extra factor of 1/ s. As a result the two models give rise to equivalent isotherms, but with different numerical values for the dimer binding constant KD. The relationship linking these dimer binding constants is K(this model)

D

fl _ 1 - fo-f3

_

which is quadratic in J3. The surface charge density, for the surface described above, is

[A71

The generalization to electrolytes of more complicated composition is again straightforward, provided we retain 1-1 binding stoichiometries. Our third example is that of a divalent species (bulk concentration c3, z3 = +2) which binds to the lipid monolayer with a 1-2 stoichiometry (binding constant/£3, r3 = 2, q~3 = 1 / s , f 3 = 2 N 3 / N s ) . In this case the isotherm is given by the single equation f3(1---~)=

(1 --f3) 2

[A6]

where the surface charge density is now given by o- = ~o(1 - f , - 2f2) = ~ro(f0 - f 2 ) .

where we have used the fact that Z f = fo + fl + f 3 = 1. Substitution of the expression for f0 from Eq. [A11] into Eq. [A12] gives the following equation for f3:

T1(Cohenand Cohen)

= S~D

[A15]

[All]

fo REFERENCES f3(l--~)=

fo

g3c3e-2qV°/kT~B,

[A12]

1. Cevc,G.,.Watts,A., and Marsh,D., Biochemistry 20, 4955 (1981). Journal of Colloid and Interface Science, Vol. 135,No. 2, March 15, 1990

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2. Vogel, H., and Stockburger, M., Chem. Phys. Lipids 32, 91 (1983). 3. Miner, V. W., and Prestegard, J. H., Biochim. Biophys. Acta 774, 227 (1984). 4. van Dijek, P. W. M., de Kruijff, B., Verkleij, A. J., van Deenen, L. L. M., and de Gier, J., Biochim. Biophys. Acta 512, 84 (1978). 5. Hui, S. W., Boni, L. T., Stewart, T. P., and Isac, T., Biochemistry 22, 3511 (1983). 6. Graham, I. S., Gagn~, J., and Silvius, J. R., Biochemistry 24, 7123 (1985). 7. Wilschut, J., Diizgiines, N., Fraley, R., and Papahadjopoulos, D., Biochemistry 19, 6011 (1980). 8. Sundler, R., and Papahadjopoulos, D., Biochim. Biophys. Acta 649, 743 ( 1981 ). 9. Silvius, J. R., and Gagn6, J., Biochemistry 23, 3232 (1984). 10. Cohen, J. A., and Cohen, M., Biophys. J. 36, 623 ( 1981 ). 11. Cohen, J. A., and Cohen, M., Biophys. J. 46, 487 (1984). 12. Graham, I. S., Georgallas, A., and Zuckermann, M. J., J. Chem. Phys. 85, 6010 (1986). 13. Lakhdar-Ghazal, F., Tichadou, J.-L., and Tocanne, J.-F., Eur. J. Biochem. 134, 531 (1983). 14. Ohki, S., and Kurland, R., Biochim. Biophys. Acta 645, 170 (1981). 15. Guggenheim, E. A., Proc. R. Soc. London Ser. A 183, 203 (1944). 16. Costas, M., and Sanctuary, B., J. Phys. Chem. 85, 3153 (1981). 17. Overath, P., and Trauble, H., Biochemistry 12, 2625 (1973), 18. Fernandez, P., and Fromhertz, J., J. Phys. Chem. 81, 1755 (1977). 19. Fragata, M., and Bellemare, F., J. Colloid Interface Sci. 107, 553 (1985). 20. Macdonald, P. M., and Seelig, J., Biochemistry 26, 1231 (1987). 21. Ceve, G., Seddon, J. M., and Marsh, D., Biochim. Biophys. Acta 814, 141 (1985). 22. Hauser, H., and Shipley, G., J. Biol. Chem. 256, 11377 ( 1981 ).

JournalofColloidandInterfaceScience,Vol.135,No.2, March15, 1990

23. Toko, K., and Yamafuji, K., Chem. Phys. Lipids 26, 79 (1980). 24. Eisenberg, M., Gresalfi, T., Riccio, T., and MeLaughlin, S., Biochemistry 18, 5213 (1979). 25. Lau, A., McLaughlin, A., and MeLaughlin, S., Biochim. Biophys. Acta 645, 279 ( 1981 ). 26. Hauser, H., Paltauf, F., and Shipley, G. G., Biochemistry 21, 1061 (1982). 27. McLaughlin, S., Mulrine, N., Gresalfi, T., Vaio, G., and McLaughlin, A., J. Gen. Physiol. 77, 445 (1981). 28. Nir, S., and Bentz, J., J. Colloid. Interface Sci. 65, 399 (1978). 29. Portis, A., Newton, C., Pangborn, W., and Papahadjopoulos, D., Biochemistry 18, 780 (1979). 30. Luzar; A., Svetina, S., and Zeks, B., Bioelectrochem. Bioenerg. 13, 473 (1984). 31. Gur, Y., Ravine, I., and Babchin, A. J., J. Colloid. Interface Sci. 64, 333 (1978). 32. Gruen, D. W. R., and Mar6elja, S., J. Chem. Soc. Faraday Trans. 2 79, 211 (1983). 33. Cevc, G., and Marsh, D., J. Phys. Chem. 87, 376 (1983). 34. Blum, L., and Henderson, D., J. Chem. Phys. 74, 1902 (1981). 35. Outhwaite, C. W., Mol. Phys. 48, 599 (1983). 36. Carnie, S. L., and Chan, D. Y. C., J. Chem. Soc. Faraday Trans. 2 78, 695 (1982). 37. Dong, W., Rosinberg, M. L., Perera, A., and Patey, G. N., J. Chem. Phys. 89, 4994 (1988). 38. Carnie, S. L., and Torrie, G. M., Adv. Chem. Phys. 56, 141 (1984). 39. Torrie, G. M., Kusalik, P. G., and Patey, G. N., J. Chem. Phys. 89, 3285 (1988); see also Torrie, G. M., Kusalik, P. G., and Patey, G. N., J. Chem. Phys. 88, 7826 (1988). 40. Gresh, N., Biochim. Biophys. Acta 597, 345 (1980). 41. McLaughlin, A., Eng, W.-K., Vaio, G., Wilson, T., and McLaughlin, S., J. Membr. Biol. 76, 183 (1973). 42. Winiski, A. P., McLaughlin, A. C., MeDaniel, R. V., Eisenberg, M., and McLaughlin, S., Biochemistry 25, 8206 (1986).