A model for cation adsorption in closed systems: Application to calcium binding to phospholipid vesicles

A Model for Cation Adsorption in Closed Systems: Application to Calcium Binding to Phospholipid Vesicles SHLOMO NIR Seagram Centre for Soil and Water Sciences, Faculty of Agriculture, Hebrew University of Jerusalem, Rehovot 76100, Israel Received December 1, 1983; accepted April 25, 1984 A model for cation adsorption to negatively charged surfaces is developed and applied to the analysis of recent results on Ca 2÷ binding to phosphadidylserine (PS), phosphadylcholine (PC)/PS, and phosphatidylethanolamine (PE)/PS vesicles. The model combines electrostatics with specific binding and considers explicitly cation depletion from solution due to adsorption. Effect o f concentration of surface sites on cation adsorption is elucidated for various combinations o f cation concentrations and binding coefficients. Binding of Ca 2+ to PS in pure and mixed vesicles which are intact and disperse is well accounted for by employing intrinsic binding coefficients whose values are Kca = 30 M -r, KN~ = 0,8 M -r, and KMg = 20 M -~. Vesicles undergoing aggregation and fusion exhibit enhanced affinity of binding of Ca 2+ to PS. It is proposed that the enhanced binding is associated with vesicle fusion, © 1984 AcademicPress,Inc. INTRODUCTION

The coupling of specific binding with the electrostatic equations is a well established procedure in studies on the effects of cations on membranes. Studies on model membranes have demonstrated that consistent results are obtained by the application of several experimental procedures such as measurement of radioisotopes and atomic absorption of adsorbed cations (I-4), electron paramagnetic resonance (5), nuclear magnetic resonance (6-10), electrokinetic measurements (9, 11, 12), and surface potential measurements (7, 13). A consideration of cation binding was crucial for the explanation and prediction of panicle aggregation (14-20). Recently equations have been derived for cation binding to surfaces of various geometries (21, 22), but for the purpose of studying cation adsorption to vesicles in media containing cations at a concentration of at least 0.1 M the planar geometry suffices (21). This article introduces a new element in providing a procedure for an explicit calculation of cation depletion from the solution

due to adsorption. Consequently it becomes possible to elucidate the effect of the concentration of surface sites on the degree of adsorption. This extension is essential for the study of cation adsorption in certain cellular systems where solution concentrations of several cations, such as Ca or trace elements, are in the micromolar range. Here we apply our model to the analysis of recent results on Ca binding to vesicles composed of phosphatidylserine (PS) and its mixtures (23). The above study employed potentiometric titrations which give a measure of the concentration of Ca2+ in solution. In contrast to dialysis experiments, the solution concentrations in the above study vary with cation adsorption. Our analysis demonstrates that below a certain threshold of divalent cation concentrations, the intrinsic binding constants, or binding coefficients, are independent of the concentrations of the cations in the medium and the amounts of Ca2÷ adsorbed are adequately predicted by the calculations. Ekerdt and Papahadjopoulos (23) demonstrated that whenever aggregation and fusion of the vesicles occurred, there was

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0021-9797/84 $3.00 Copyright© 1984by AcademicPress,lnc. All rightsof reproductionin any form reserved.

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SHLOMO NIR

enhanced binding of Ca2+ to PS in PS or PS/ PE vesicles. Our analysis shows that the enhanced binding of Ca/PS amounts to an increase in the intrinsic binding coefficient of Ca to PS by one to two orders of magnitude.

= ¢o, and g = 272(80 × 298/eT) m. For a solution containing univalent and divalent cations and univalent anions, Eq. [4] can be written in the form

MATHEMATICAL DESCRIPTION OF ADSORPTION MODEL

where the coefficients A to D are given in the Appendix. The amount of adsorbed cations consists of two contributions, cations residing in enhanced concentrations in the double layer region and cations which chemically bind to the surface. The tightness of the chemical binding is expressed by the magnitude of the intrinsic binding constants, Ki. The binding of cations to singly charged negative sites P - on the surface, is given by

Modified equations of the Gouy-Chapman theory for the planar geometry are employed. The discrete charges are replaced by a homogeneously smeared surface charge. The electrical potential ~b satisfies the equation d2ff dx 2 -

4~'p ill

~ '

in which p(x) is the charge density in solution and E is the bulk dielectric constant. The variation of e with concentration of the solution, and with distance from the surface is neglected. Let N(x) and N be the numbers of ions of type i per unit volume at points x and oc, respectively. The charge density is given by

p(x) = ~ eZiNi(x) = ~ eZiNiy z', i

[2]

where e is the absolute magnitude of an electronic charge, Z i is the valence of an ion, being positive for a cation and negative for an anion, and

y(x) = exp - e~b(x)/kT,

[3]

where k is Boltzmann's factor and T is the absolute temperature. Let Ci be the total concentration of a given ion in the system in units of moles/liter, and Si will denote the concentration of ions i in solution, far away from the surface. Let a be the surface charge density in units of e/A 2. The solution of Eqs. [1] to [3] is (24, 25) 1

= - {Z

g

s l ( y ( O ) z' -

l)} '/2,

[4]

in which y(0) = Yo denotes the value of y at x = 0, where the surface potential is ~k(0) Journal of Colloid and Interface Science. Vol. 102, No. 2, December 1984

a z = ~ (ay~ + Byo + C + D/yo),

[51

P - + X• ~ PXi

[6]

[PX,] Ki = - [P-][x~ ]

[7]

i.e.,

In Eq. [7], the concentrations are given in units of moles/liter, or M, and the unit of Ki is M -1. The concentration of the cation, AT,., is to be taken at the surface, i.e., X;(0),

Xi(O) = SiyZo~= Si exp -- e~boZi/kT

[8]

For ¢0 = -25 mV, Y0 --- exp(l) at room temperature. Equations [7] and [8] give

[PXi] = K~[P-]Sty z'.

[9]

Divalent cations can form a 1-1 complex which will be denoted [PXf] since it is positively charged and a 2-1 complex which is neutral and will be denoted [PXj]. The corresponding binding constants will be denoted Kjl and Kj2. The surface charge density ao denotes the charge per area of a free site. In Fxl. [10] below both 1-1 and 2-1 complexes are considered. [P-] - Z [PX~-]

~r

-- =

eo

[e-] +

[lO]

[exi] mono

+ 2 Z [PXjl + ~ [PX]] di

di

CATION ADSORPTION IN CLOSED SYSTEMS

315

the excess concentrations of monovalent and divalent cations in the double layer region. 1 - yo Z K , S ? These quantities are proportional to the ao I + yo Z K, S+ + y2 ~ KjS2 +, [111 quantities given in Ref. (2), Eqs. (28) and (27), respectively. where Kj = Kj, + Kj2. If the quantities Si, Sj and Ki, Kj are COMPUTATIONAL PROCEDURE known, then Eqs. [4] and [11 ] suffice to find The calculations employ an iterative prothe unknowns a and y(0). However, in most cedure which consists of the following stages: cases the independent quantities are the total (i) in the first step, a guess is made for the concentrations C~, Cj, whereas the quantities values of S °), S ~2), y(0), and [P-]. (ii) The S/, Sj are modified due to adsorption. In the quantities St, Sj are calculated by using Eqs. next section we describe an iterative proce[15] and [16]. (iii) The coefficients of the dure for the determination of y(0), the polynomial equation for y(0) are obtained as amount of surface sites bound by each cation, shown in the Appendix. Then a positive root and the solution concentrations Si, Sj. It will greater than unity is found for y(0). The be assumed that the quantities Ci, Cj are polynomial equation is of the order seven if known. divalent cations are present and of the order The total concentration of cation i consists four if only monovalent cations are present. of three terms, (i) solution concentration, Si, For a 2:1 stoichiometry of divalent cation (ii) concentration of bound cations, PXg, and binding Bentz and Nir (27) have shown that (iii) cations residing in the double layer region, there is only one root which satisfies y(0) in excess of solution concentration. Analytical expressions for the latter quan- > 1. (iv) The concentrations of the complexed tity were derived (2) by integrating the solu- sites are calculated by using Eq. [9]. (v) [P-] tion (26) for the case of a surface in contact is calculated from the relation with a solution containing both monovalent [P-] = [PT] - ( ~ [PXi] and divalent cations. We define + 2 Z [PXj] + Z [PXf]), [17] S °) = Z S, [121 in which [PT] is the total concentration of 1220110 sites. (vi) The quantities Ql and Q2 are S ~2) = Z Sj. [131 calculated. di Then the solution concentrations are recalculated, and another cycle of calculations For monovalent cations of y(0) begins. Steps (ii) to (vi) are iterated Ci = St + [PXi] + Q i S d S °) [14] and the criterion for convergence is the agreement of the values calculated in the and the use of Eq. [9] gives previous cycle with those obtained in the current one. The requirement of a relative ct S { = 1 + [P-]yoK~ + Qj/S °)" [15] error of 10-3 resulted in a few up to a few tens of iterations. An independent test is the Similarly, for divalent cations mass conservation for each ion. By making use of Eq. [9]

S~+=

C~+ 1 + [e-]y2(Kjt + Kj2/2) + Q2/S (2)"

EFFECT OF CONCENTRATION OF SURFACE SITES

The binding of cations to surface sites and ~ their accumulation in the double layer region The quantities Q~ and Q: are, respectively, results in the depletion of their solution [16]

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concentrations, which in turn results in a lation increase with particle concentration, smaller degree of binding per site. Hence, and that this effect is more pronounced with while the absolute amount of cations bound cations of higher valencies. On the other increases with the concentrations of sites, the hand, Bentz and Nir (18, 19) showed that binding per site decreases. This effect is illus- the average aggregate size and aggregation trated in Table I, which gives the amount of kinetics increase with particle concentration. a divalent cation bound and adsorbed as a Our view is that with increase in surface site function of its total concentrations, C(2), and concentration there may be a significant reits binding constant, K(2). The medium also duction in the solution concentration of catincludes a monovalent cation whose binding ions. The current treatment enables calculaconstant and concentration are 1 M -~ and tions of both charge neutralization and cation 0.1 M, respectively. depletion from solution as a function of I n case I where PT, the concentration of surface site concentration, and to obtain essurface sites, equals 2 × I0 -3 M the fraction timates of the rates and extents of aggregation of surface charge neutralized by the divalent (15, 18, 19). Clearly, cation depletion is more cation (i.e., twice its binding ratio) is propor- pronounced with higher cation valencies, but tional to values of C(2) in the range of 10-4 specificity of binding plays an important role. to 10-6 M. In comparison, in case 2(PT ANALYSIS OF BINDING RESULTS = 10-5 M) corresponding reduction in bindOF Ca ~+ TO PS ing and in solution concentration is more The experimental results in Tables II, III, moderate as C(2) decreases from I0 -4 to 10 -6 M. Despite the two orders of magnitude and IV are taken fi'om Ekerdt and Papahaddifference in binding affinities of the divalent jopoulos (23). For brevity these tables include cation between cases I (K = 1000 M -I) and only a sample of the whole range of the 3 (K = 10 M-I), its bound amounts differ results. Table II provides a comparison beby a factor of 2 only, for C(2) values in the tween experimental and calculated values of range of 10-a to 10-6 M. This puzzling result Ca bound per PS. This comparison demoncan be rationalized by comparing the corre- strates that the use of a single binding constant sponding solution concentrations. Due to in Eqs. [9], [15], and [16] can explain the strong binding and cation depletion in Case whole set of binding data for intact and 1, the values of S(2) are smaller than C(2) dispersed vesicles. The corresponding range values by more than two orders of magnitude. of cation concentrations spans two orders of In contrast, when the concentration of surface magnitude in the final solution concentration sites is relatively small, the bound amounts of Ca ~+. The binding constants used are gNa are more sensitive to the binding affinities, = 0.8 M -l and Kca = 30 M -l, and the as can be seen by comparing Cases 2 and 4. calculations employed a 2:1 stoichiometry of In passing it may be noted that the results in binding of Ca to PS. These values of the Table I indicate that trace cations whose binding constants are close to previously concentrations are in the micromolar range reported values, which are Kca = 35 M -~ (2, exhibit a simple linear adsorption pattern. 6); Kca = 30 M -~ (7); KNa = 0.8 M -~ (2); Both the bound amounts and solution con- (0.8 +-- 0.2) (6); KNa = 0.6 M -~ (7, 12). centrations of trace elements are proportional McLaughlin et at. (9) employed a value of 12 M -~ for a 1:1 binding constant of Ca to to their total concentration in the system. The explicit consideration of surface site PS. Bentz et al. (29) pointed out that the concentration may be also applied to the different procedures quoted above yield simproblem of particle aggregation. Overbeek ilar calculated values of Ca/PS. Similarly, an (28) pointed out that in certain systems the inspection of Table III indicates that calcuconcentrations of cations needed for coagu- lations which employ the above binding conJournal of Colloid and Interface Science. Vol. 102, No. 2, December 1984

317

CATION ADSORPTION IN CLOSED SYSTEMS TABLE I Effect of Concentration, Binding Constant, and Site Concentration on Binding, Adsorption, and Solution Concentration of a Divalent Cation C(2) Total concentration of divalent cation (M)

S(2) Solution concentration of divalent cation (M)

Case 1 K(2) = 1000 M -~ PT = 0.002 M

0.01 0.001 10-4 10-5 10-6

0.009 2.2 X 10-4 9.3 X 10-7 7.6 X 10-8 7.5 X 10-9

0.95 0.78 0.10 0.01 0.001

0.96 0.78 0.10 0.01 0.001

l1 40 73 75 75

Case 2 K(2) = 1000 M -~ PT = 10-SM

0.01 0.001 104 10-5 10-6

0.01 0.001 9.6 X 10-5 8.1 X 10 -6 6.5 X 10-7

0.95 0.87 0.71 0.37 0.07

0.96 0.88 0.71 0.37 0.07

11 28 47 65 73

Case 3 K(2) = 10 M -~ PT = 0.002 M

0.01 0.001 10-4 10-5 10-6

0.009 6.5 × 10-4 4.5 X 10-S 4.2 X 10 -6 4.2 X 10 -7

0.66 0.33 0.05 0.006 0.0006

0.73 0.35 0.05 0.006 0.0006

44 65 74 75 75

Case 4 K(2) = 10 M -l PT = 10-5 M

0.01 0.001 10-4 10-~ 10-6

0.01 0.001 10-4 9.9 X 10 -6 9.9 × 10 -7

0.67 0.39 0.10 0.012 0.0012

0.74 0.42 0.10 0.013 0.0013

44 63 72 74 75

Fraction of surface charge neutralized by divalent cation

Total adsorption of divalent cation (fraction of sites adsorbed)

Surface potential -~(0) (mV)

Note. The area per molecule is 70 ~k2. The medium contains a monovalent cation at a concentration of 0. I M and a divalent cation at total concentrations as specified; K(l) = 1 M -t. Room temperature is assumed. The stoichiometry of binding of the divalent cation is 2: I. Fraction of surface charge neutralized indicates the fraction of the molecules of the membrane in complexes with cations. By fraction of sites adsorbed we designate the total concentration of cations adsorbed divided by the concentration of molecules, or sites, of which the surfaces are composed.

stants c a n p r o v i d e g o o d p r e d i c t i o n s for t h e b i n d i n g r a t i o s o f C a / P S in P S / P E a n d P S / P C 1:1 vesicles, as l o n g as vesicle a g g r e g a t i o n and fusion do not occur. The current binding calculations do not d i s t i n g u i s h b e t w e e n P S / P C o r P S / P E vesicles, a n d i n d e e d , t h e r e is n o d e t e c t a b l e d i f f e r e n c e in t h e e x p e r i m e n t a l values. I n P S / P C v e s i c l e s t h e s i m p l e b e h a v i o r still h o l d s in t h e p r e s e n c e o f l a r g e r C a 2+ c o n c e n t r a t i o n s , e.g., 2 r a M . A s e x p l a i n e d b e f o r e (4, 15) t h e b i n d i n g ratios C a / P S are s m a l l e r in m i x e d vesicles b e c a u s e o f t h e r e d u c e d s u r f a c e c h a r g e d e n s i t y d u e to

t h e d i l u t i o n o f PS in m e m b r a n e s o f m i x e d vesicles. T h e o t h e r results in T a b l e s II a n d III i n d i c a t e t h a t Kca s h o u l d b e i n c r e a s e d in o r d e r to e x p l a i n t h e b i n d i n g o f C a to PS in f u s e d vesicles. T h e v a l u e o f Kca = 75 M - l has b e e n u s e d b e f o r e (15, 16), b u t t h e c u r r e n t results i n d i c a t e t h a t a c o m p l e t e fit o f c a l c u l a t e d v a l u e s to t h e e x p e r i m e n t a l results requires ultimately more than two orders of m a g n i t u d e i n c r e a s e i n Kca. I n c o n f o r m i t y to t h e s u g g e s t i o n s o f P o r t i s et al. (30) a n d E k e r d t a n d P a p a h a d j o p o u l o s (23) it c a n be Journal of Colloid and Interface Science, Vol. 102, No. 2, December 1984

SHLOMO NIR

318

TABLE II Variation of Binding Affinity of Ca to PS with Ca Concentration. PS Vesicles in a Medium Containing Ca 2* + Na + a [Ca:+] free Exp. (M)

7.6 1.0 3.3 9.3 1.2 2.5 5.7 9.33 9.26

X x × × × × × × ×

Total Ca2+ conc. (M)

8.0 9.6 1.6 3.1 3.8 5.7 9.8 1.6 1.8

10 -6

10-5 10-5 I0 -5 10 -4

10-4 10-4 10-4 10-4

3.05 × 10 -3

X X × × X × × × ×

Experimental adsorption ratio Ca/PS

10-5 10-5 10-4 10-4 10-4 10-4 10-4 10-3 I0 -3

0.04 0.04 0.06 0.11 0.13 0.16 0.22 0.34 b 0.42 b

4.05 × 10-3

0.50 b

Calculated Ca2+/PS

Intrinsic binding coefficient of Ca to PS (M"-I)

0.03 0.04 0.06 0.11 0.13 0.18 0.24 0.34 0.35 0.36 0.42 0.40 0.45 0.48

30 30 30 30 30 30 30 75 75 100 500 75 500 5000

° The experimental results are at room temperature and are taken from Ekerdt and Papahadjopoulos (23). The calculations assume a 2:1 stoichiometry for the binding of Ca/PS, Kc, = 30 M -l. The medium also includes a monovalent cation (mainly sodium) at a concentration of 0.11 M and with a binding constant of 0.8 M -t. The employment of a 1:1 binding constant Kc, = 12 M -t and KN, = 0.6 M -j (9) gave similar results. The surface area per PS molecule is 70 ,~2. Lipid concentration is 2 nO//. b In these cases vesicle fusion occurs.

f o r m u l a t e d that the affinity o f b i n d i n g o f C a

Table IV describes the time course of the

to PS increases w i t h vesicle aggregation a n d

i n c r e a s e o f C a b i n d i n g t o P S w h i c h is i n i t i a t e d

fusion, reflecting structural c h a n g e s in the

upon the addition of 6 mM

vesicles a n d c o n f o r m a t i o n a l c h a n g e s o f t h e

t h e a d d i t i o n o f M g 2+ t h e v e s i c l e s d o n o t f u s e

M g 2+. P r i o r t o

PS molecules.

or aggregate for long periods of incubation

TABLE IIl Variation of Binding Affinity of Ca/PS with Ca Concentration. PS/PC and PS/PE (1:1) Vesicles Vesicle composition

[Ca2+] free Exp. (M)

PS/PC

10-5

or

10 -4

PS/PE PS/PC PS/PE PS/PE PS/PE

3.2 X 10 -4 2 × 10-3 7.6 × 10-4 10-3 3.2 × 10-3

Total Ca2+ cone. (M)

5 3 6.5 2.5 1.3 1.8 4.8

× × X × × X ×

10-5 10-4 10 -4

10-3

Experimental Ca2+/PS

0.02 a 0.1 a 0.17 a 0.25 c

10 -3

0.25 b

10-3 10-3

0.42 b 0.5 b

Calculated Ca2÷/PS

Intrinsic binding coefficient of Ca2+ to PS (M-j)

0.02 0.08 0.15 0.27 0.26 0.39 0.48

30 30 30 30 75 500 5000

a Conditions are as in Table II. The experimental binding results are almost the same in PS/PC or PS/PE vesicles in these media. b In these cases vesicle fusion occurs. c In this case vesicle aggregation occurs without fusion. Journal of Colloid and Interface Science, Vol. 102, No. 2, December 1984

319

CATION ADSORPTION IN CLOSED SYSTEMS TABLE IV Variation of Binding Affinity of Ca2*/ps with Time after Addition of 6 mM Mg2÷ to PS Vesicles Time (min)

[Ca2+] free Exp. (M)

Total Ca~÷ c o n c . (M)

Experimental Ca2+/PS

Calculated

Intrinsic binding coefficient of Ca2+ to PS

Ca2+/PS

(M-~)

Before addition of Mg2÷ 0 1 2

3.8 × 10-4 7.4 X 10-4 3.2 X 10-4 10-4

8 X 10-4 8 × 10-4 8 X 10-4 8 X 10-4

0.21 0.05 0.23 0.35

3

3.2 × 10-5

8 X 10-4

0.38

30

1.1 × 10~

8 × 10-4

0.39

0.22 0.06 0.23 0.35 (0.34) 0.37 (0.37) 0.37

30 30 500 5000 (500) 10,000 (1,000) 10,000

Note. Conditions are as specified in Table 11 KMg = 20 M-~. Values in parenthesis were calculated by setting KNa = KM~= 0.

and using a binding constant Kca = 30 M can account for the experimental values. This binding affinity is retained in the presence of 2 m M Mg 2+ and at the m o m e n t of addition of 6 m M Mg 2+, i.e., at t = 0 in Table 4. At this m o m e n t the binding ratio Ca/PS is rapidly diminished by a factor of 4. This initial drop in Ca binding arises because of competition of Ca with Mg for the binding sites and because of the reduction in surface potential in the presence of several millimoles Mg 2+. A few seconds after the addition of Mg 2+ there is a rise in the binding ratio Ca/ PS, which is described in Table IV in terms of an increase in Kca. During this period the vesicles are known to aggregate and fuse. Wilschut et al. (3 l) showed that these vesicles (of a diameter ~ 1000/~) aggregate but do not fuse in the presence of Mg 2+ alone. Ekerdt and Papahadjopoulos (23) proposed that the presence of Mg z+ promotes the aggregation of these vesicles and then an interm e m b r a n e complex between Ca and PS can form. The enhanced binding of Ca/PS occurs in the presence of 6 m M Mg 2+ at a small Ca 2+ concentration, and at a binding ratio of Ca/PS m u c h below 0.23, i.e., under such conditions that the vesicles are definitely stable in the presence of Ca 2+ and N a +. It has to be mentioned that the calculations assumed that the intrinsic binding constants,

gNa and

KMg, remained unchanged. The decrease of KN~ and /(Ms would require a smaller increase in Kc~, and vice versa. However, the calculations indicate that even in the extreme and unlikely case that KN~ = KMg = 0, there is still a very significant increase in Kca, up to 1000 M -~ (see values in parenthesis in Table IV). DISCUSSION The results of Ekerdt and Papahadjopoulos (23) demonstrate a dramatic increase in the binding of Ca to PS in vesicles undergoing aggregation and fusion. The significance of this finding is emphasized by expressing the increase in binding ratios of Ca/PS in terms of intrinsic binding coefficients which increase by two orders of magnitude. It should be noted that dramatic changes in binding ratios of Ca/PS do not necessarily reflect changes in binding affinities. For instance, the initial decrease by a factor of four in Ca/PS binding ratio upon the addition of Mg 2÷ can be predicted by assuming constant intrinsic binding coefficients. The calculations in this work consider isolated surfaces. Electrostatic calculations (22) indicate that vesicle aggregation should yield enhanced binding, even when the binding affinity remains constant. It is possible to Journal of Colloid and Interface Science, Vol. 102, No. 2, December 1984

320

SHLOMO

extend the present model to certain geometries of closely packed vesicles. However, it has to be noted that in a system containing Ca 2÷ and Mg2+ both cations would show a similar degree of enhanced binding due to increase in surface potentials upon vesicle aggregation. In contrast, the results demonstrate a very significant enhancement of Ca2+ binding at the expense of Mg2÷, at the onset of fusion of PS vesicles. It is proposed that the observed increase in binding is not merely due to vesicle aggregation. Ekerdt and Papahadjopoulos (23) noted than when large PS/PC I: 1 vesicles are aggregated but are not fused, there is no enhanced binding of Ca to PS (See Table III). It could be argued that this aggregation of large vesicles occurs in a secondary minimum (14, 20), where the separation between apposed surfaces does not enable the formation of intermembrane complexes. This point can be tested by measuring Ca binding in small PS and mixed vesicles, which are induced to aggregate without fusion in the presence of Ca2+ and large Na ÷ concentrations. There is some evidence that small sonicated PS vesicles remain stable (for several minutes at least) in an aggregated state in a primary minimum, provided that the binding ratio Ca/PS is below a certain threshold (29, 32). More kinetic studies of binding and fusion are required in order to determine whether the enhanced binding of Ca to PS occurs with the onset of fusion or following its progress in time. CRITIQUE

The actual values of the increased binding constant of Ca to PS may turn out to be below 1000 M -j, because of an increase in the surface potential of aggregating vesicles. Hence, the enhanced Ki values reflect the values used by employing equations for isolated vesicles. However, aggregation of nonfusing vesicles does not result in enhanced binding of Ca/PS. Furthermore, an increase in surface potential of closely apposed vesicles should have resulted in a similar increase in Journal of Colloid and Interface Science, Vol. 102, No. 2, December 1984

NIR

binding to PS of both Ca2+ and Mg2+, whereas the results clearly indicate enhanced Ca binding at the expense of Mg binding, implying the occurrence of molecular changes at the onset of vesicle fusion. The constancy of Kg values should be viewed as an approximation, even in the case of isolated vesicles. Cohen and Cohen (33) presented a statistical mechanical treatment which anticipates changes in binding due to changes in the numbers of accessible nonbound molecular pairs. It may be worthwhile to emphasize, that from the thermodynamic point of view, the constant quantities are the thermodynamic equilibrium constants, KEg (34), which are given by gel =

a(eXi)

a(P-)a(X+)

[17]

where ai are the activities. The equilibrium constant Kei is related to the binding constant by KEi =

f (PXt)55.5 f(p_)f(X+) Kg

[18]

in which the quantities f are the activity coefficients. Now, the activity coefficients are expected to vary with ionic strength. However, a similar change in the binding constants of two competing cations can result in a compensation of the effect. While the current results as well as those of previous studies (1, 2, 4, 6-9) lend support to the adequacy of the employment of constant binding coefficients, this question may be further pursued under extreme variations of ionic strengths. The current results on Ca and Mg binding to fusing PS vesicles demonstrate a different situation where the actual equilibrium constants do change, reflecting changes at the molecular level. APPENDIX

The coefficients A to D in Eq. [5] are given by

A=ZS? B= C = -(3A + 2B) O = 2A + B

[I1]

CATION ADSORPTION IN CLOSED SYSTEMS

By taking the square of Eq. [1 1] and dividing it into F_xl. [51, we obtain a polynomial equation of the form 7

b~v8 = 0

[12]

n=0

b7 = A E 2 b6 = B E 2 + 2 A E G b5 = A G 2 + 2 A E + 2 B E G + C E 2 - ( F H : ) b4 = 2 A G + 2 B E + B G 2 + D E 2 + 2 C E G b3 = A + 2 B G + 2 C E + C G 2 + 2DEG +

(2FH)

b2 = B + 2CG + 2 D E + D G 2 b1= C + 2DG + F

bo = D in which

[I31 G = ~

g

+

iSi

e = Z

u=

£ s]+rj,

+

[14]

and F = o'02g2

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