Electrochemical properties of spherical polyelectrolytes

Electrochemical Properties of Spherical Polyelectrolytes II. Hollow Sphere Model for Membranous Vesicles 1 MICHEL MILLE2 University oj Wisconsin-Madison, School oj Pharmacy, 425 North Charter Street, Madison, Wisconsin 53706 AND

GARRET VANDERKOOI Chemistry Department, NorthernIllinois University, DcKalb, Illinois 60115

Received March 29, 1976; accepted December 6,1976 The nonlinearized Poisson-Boltzmann equation has been numerically solved to obtain the electrostatic potential inside and outside of hollow spherical shells suspended in an aqueous medium. The geometrical characteristics of the shells were chosen to correspond to those of the small (l00 to 600 A radius) vesicles typically formed by sonication of phospholipids or membranes. The boundary conditions were specified using the following assumptions: (i) The shells are permeable to ions, permitting an electrochemical equilibrium to exist between the inner and outer solutions. (ii) Both inner and outer surfaces of the shell bear ionizable groups, but only the total (in plus out) degree of dissociation is given. (iii) The cell model for finite concentrations of vesicles was employed. The additional assumption of electroneutrality within the vesicles was used in most of the reported calculations. It is shown that for vesicles having radii in the range studied here, the computed results obtained with and without the electroneutrality constraint are nearly the same. The electrostatic potential was studied as a function of salt concentration, vesicle concentration, shell thickness, radius of shell, and surface area per ionizable group. The ionic distribution, activity coefficients, Ll.pK, and osmotic coefficient were obtained from the electrostatic potential. The degree of dissociation under all circumstances was found to be less on the inside than the outside. The average potential of the inside solution is larger in magnitude than that outside the shell at low salt and polyelectrolyte concentrations. At high salt concentration the potentials are similar inside and outside, whereas at high polyelectrolyte concentration, hut low salt, the magnitude of the potential is larger outside than inside. INTROD UCTIO'\

The purpose of this paper is to explore the electrochemical properties of membranous vesicles through the use of the nonlinearized Poisson-Boltzmann equation. Phospholipids 1 A report on some of this work was given in Minneapolis at the Biochemistry/Biophysics 1974 Meeting, June 2-7,1974, and in Puerto Rico at the International Conference on Colloids and Surfaces, June 21-25,1976. 2 Work done in partial fulfillment of the requirements for the degree of Ph.D. in Biophysics at the University of Wisconsin. Current address: Center for Theoretical Studies, University of Miami, P.O. Box 249055, Coral Gahles, Florida 33124.

readily form closed spherical shells or vesicles when dispersed in water by ultrasonic oscillation. The shell consists of lipid in the bilayer arrangement, with the polar groups of the lipid molecules forming both the inner and outer surfaces. An aqueous cavity is enclosed by the shell. If lipids with ionizable polar groups are employed, the possibility exists for both the inner and outer surfaces of the vesicles to bear a net charge. The major objective of the present work was to compute the electrostatic potential as a function of radial position, both within and outside of charged vesicles of this type, and to derive from it the

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Journal of Colloid and l nterfuce Science. Vol. 61. No.3, October I, 1977 ISSN 0021-9797

456

MILLE AND VAJ.'W ERKOOI

"'--+--t----~

R

FIG. 1. Hollow sphere model for the solution of the Poisson- Boltzmann equation. Th e radii a and bare, respectively, the inner and outer radii of the polyelectrolyte particle, and R is the cell radius of the solution volume associated with the particle. The spherical shell defined by a < r < b is the lipid bilayer j the space inside and out side of the bilayer is aqueous.

concentration) and as a function of the surface charge density and salt concentration. Using the potentials so obtained, we have derived the ..1pK, the salt distribution inside and outside the bilayer, the interior to exterior ratio of groups ionized, and the osmotic coefficient. The Poisson-Boltzmann equation was numerically solved for the general case of finite salt and finite particle density, but results are also given for the simpler cases of either zero salt or infinite particle dilution ; the latter results are also applicable for the case of high salt concentration because of the electrostatic shielding between particles afforded by the salt. Results obtained with the linearized Poisson-B oltzmann equation are given for comparison ; we found that the linearized equation greatly overestimat es the surface potential even in the presence of 0.1 M salt. The math ematical methods for treating hard spheres at finite salt and polyelectrolyt e concentration were presented in Paper I of this series (2). It was shown in that paper that experimental data on spherical detergent micelles (especially osmotic coefficient data) could be satisfactorily interpreted by the theoretical results. The mathemat ical meth ods employed for the present more-complicated system were developed as an extension of those used in the hard sphere work. Thus , only those aspects of the methods which differ from those already described in Pap er I will be given here. The glossary of symbols given in Paper I is also applicable for the present paper." The published theoretical work on charged vesicles is very sparse. Israelachvili (3) has studied the vesicle problem for the purpose of determining the equilibrium distribution of charged groups between the inner and outer surfaces of lipid vesicles. He gave an approximate expression for determining the ratios of inner to outer surface charge and surface charge density in vesicles. We will show that the Israelachvili method and the linearized solu-

expected equilibrium distribution of ions between the inside and outside of the vesicles. A uniform distribution of ions is precluded by virtue of the net surface charge on the vesicles. Furthermore, the difference in curvature of the two surfaces prevents an a priori assumption that the degree of dissociation of the ionizable groups on the two surfaces will be the same. The computations were done specifically for the case of vesicles consisting only of phosphatidic acid. The phosphate group of this lipid has two dissociable hydrogens, the first of which ionizes at very low pH , and the second near neutral pH. Paper III in this series (1) gives the results of pH titration studies of phosphatidic acid vesicles; that paper also compares the theoretical results with the experiment. The results obtained here are by no means restricted to phosphatidic acid vesicles, however, but should be applicable to any lipid or lipoprotein membrane vesicle system which bears charge on both surfaces. We have computed the electrostatic po3 Except , in this paper, 1 + and L are defined by the tential as a function of geometrical param eters quantities in brackets in Eqs. [6J and [ 7J of the (particle size, bilayer thickness, and particle present paper. J ournal of Colloid and I nterfuce Science,Vol. 61, No.3, Oct ober 1, 1977

MEMBRANOUS VESICLES

tion of the Poisson-Boltzmann equa tion overestimate this ratio as compared to the result given by the nonlinearized solution, but the agreement improves at large valu es of «a. PHYSICAL MODEL

The cell model was used to calculate the electrostatic potential if;for the condition of finite vesicle concentration. In this approach, the volum e of the solution is divided by the number of polyelectrolyte particles to give the average volume of solution per polyelectr olyte particle. This allotted volume is assumed to be spherical, and the polyelectrolyte particle, taken as a spherical shell, is situated in the center of this volume (Fig. 1). The assumptions and approximations underlying the use of the cell model are well known (4, 5) since it has been widely used for problems of spherical or cylindrical symmetry. Many results are also given in this paper for the case of infinite dilution , in which the cell model app roximations are not needed. The total surface char ge per polyelectrolyte particle Qt is equal to the sum of the inner Qi and outer Qo surface charges, each of which is assumed to be smeared uniformly over the respecti ve surface of the polyelectrolyte vesicle. Th e mobile ions (counterions a nd coions) are treated as point charges. We hav e assumed that the individual phospholipid molecules occupy the same surface area on both the inner and outer surfaces of the vesicles; the number of molecules facing each surface is different, however, because of the different radius of curvature of the two surfaces. The number on each surface is given by the area of the spherical surface divided by the assumed area per molecule. We have also assumed that the total degree of dissociation a t is known, bu t that the inner and outer degrees of dissociati on a i and llo are unkn own and may differ from each other and from a t. ll i and a o are relate d to a t by the equat ion a t= -

Vi

-

-

+ Vo

[ 1J

457

where Vi and Vo are the number of monomers on the inner and outer surfa ces, respectively. The se values are in turn specified by the assumed geomet rical parameters of the particles, i.e., a (inner radius), b (outer radius), and 5 ' (surface area per molecule), as follows: 2 2 Vi = 41ra / S' and Vo = 41rb /S'. (Note that 5' is area per monomer whereas an unprimed S has been used in Paper I to denote the surface area per unit charge.) Th ere is a maximum of two charges per molecule of phosphatidic acid. The total volume of solution associated with one vesicle is given by 41rR3/3 = (Vi 1'0) X 108/ (CpN) ern', where N is Avogadro's number, C p is the phospholipid concentration in mole/liter, and R is the radius of the sphere of solution , or cell, associated on the average with one vesicle. Th e region within the vesicle shell, a < r < b, was assumed to be devoid of mobile ions, due to its hydrocarbon-like chara cter and consequent low dielectric constant. The shell is considered to be permeable to ions, however, so that an electrochemical equilibrium must exist bet ween the interior and exterior spaces. We assume tha t the tot al region, 0 ~ r ~ R , is electrically neutral. In the calculati ons report ed here, it was assumed that the centers of the mobile ions could approach directly to the rad ius of the smeared charge, since unpublished trial calculations showed that no advantage could be gained in explaining experimental dat a by assuming a nonzero closest approach of ions. The assumptions listed so far yield a sufficient num ber of boundary conditions to completely specify the problem. We used one additional assumption in most of the calculations, however, in order to simplify the solution : This was that the interior of the vesicle (0 ~ r ~ a) is also electricall v neutral. Comparison of calculations made with and without this assumption shows that it is a good approximation except at low surface charge and/ or vesicle radius. Three physical parameters which must be specified before proceeding with the calcula-

+

J ournal of Colloid and I nterf ace Sc ience,Vol. 61. No. 3. Octo ber I. 1977

458

MILLE AND VANDERKOOI

tions are the surface area per molecule, the were for pure phosphatidic acid vesicles, howbilayer thickness, and the vesicle radius. Since ever. Our electron micrographic examination these values are not known with certainty for of negatively stained phosphatidic acid vesicles acidic lipids (e.g., phosphatidic acid), we have (admittedly an imprecise method ) showed simply selected values to be used routinely in vesicles having a radius in the 200-300 A most calculations, but then carried out other range, with some being considerably larger. calculations in which these parameters were Johnson (9) gave experimental evidence that varied independently in order to see how the average vesicle size increases as the surface strongly the derived quantities depended upon charge density increases. The concentration of phosphatidic acid emthese parameters. In particular, we have used 60 A2 as the surface area per molecule, 45 A as ployed in pH titration experiments (Paper III) the thickness of the bilayer, and 300 A as the was in the range of 1.9 to 2.8 mM. A value of 2 mM was therefore chosen for most calculaparticle radius. The bilayer thickness is, for our present tions; but again, the effect of variable phospurposes, defined as the distance between the phatidic acid concentration on derived values two spherical shells which bear the negative was also investigated. charge. Since the negative charge is distributed POISSON-BOLTz:.vrANN EQUATION mainly over two of the oxygen atoms of the phosphate group, the distance in question is We introduce the simplifying notation that between the average positions of the [2J x = r/R phosphate oxygens on opposite sides of the membrane. Levine and Wilkins (6) have given [3J cP = El/;/kT a value of about 35 A as the glycerol to glycerol distance across an egg lecithin bilayer, where € is the charge on a proton, k is Boltzon the basis of X-ray diffraction. Assuming mann's constant, T is the temperature in dethat the phosphate group is extended away grees Kelvin, and cP is the reduced potential. We assume the presence of a simple salt from the membrane plane in the acidic lipids, with only two ionic species, the counterion of a value of 45 A is a reasonable estimate for the polyelectrolyte being identical with one of the P0 4 to P0 4 distance (7). these species. The valences of these mobile The value of 60 A2 for the surface area per ions are represented as z+ and a.. With this phospholipid is typical of the values which notation, the Poisson-Boltzmann equation for have been reported for phospholipid dispersions and hydrated lamellae in the liquid the hollow sphere is crystalline state. This is approximately the cP value obtained for egg lecithin lamellae (6, 8), ~~ (X2d ) = 0, a < r < b, [4J dx for mixed phosphatidic acid-lecithin disper- x2 dx sions (9), and for phosphatidyl serine lamellae (10). The larger surface areas per phospholipid also used in this paper may be thought of as applying to phosphatidic acid diluted with the Os r S a, appropriate amount of neutral or zwitterionic b S r SR. [5J lipid, rather than for pure phosphatidic acid. The vesicle radius was taken routinely as The dielectric constant D was taken as that 300 A, but results are also reported for values for water at the appropriate temperature. as low as 100 A. Values of 100 to 250 A for the 1t+(O) and 1L(0) are the number concentraouter radius of charged vesicles have appeared tions of positive and negative ions at a place in the literature (9-11). None of these values where cP = o. The general solution to Eq. [4J J ou fllol 01 (:9110i
459

yIEMBRANOUS VESICLES

+

is cP = Alx A', where A and A' are constant in the region a ~ r ~ b, and its derivative is zero. These physical consideraconstants. If we make the assumption that the region tions and assumptions yield the following boundary conditions for Eqs. [4J and [5]: ~ r ~ a, is electrically neutral, then by Gauss' law, dcPldr = within the membranous det>1 shell (a::; r::; b). et> is then constant within -- = 0, at x = 0, [10J the shell, and the potentials at the two surdx faces (at r = a and r = b) are equal to this -aiVi€ kT det>1 constant and to each other. D----=-- , at x = af R, [llJ The normalization conditions for the hollow €R dx a2 sphere are cPl = cP2 = constant, at x = af R, [12J

°

°

aov o € kT dcP3 D---=- , €R

dx

et>2 =

b2

et>~,

det>3

-- = 0, dx

at x = bfR,

[13J

at x = blR,

[14J

at x = 1.

[15J

The subscripts 1, 2, and 3 on et> refer to regions o r ::; a, a < r < b, and b::; r ::; R, respectively. Equations [llJ and [13J are written specifically for a negative surface charge. Because cP2 = constant, there are only five unknown integration constants; these, together with ai and aD, make seven unknowns to solve for. The six boundary conditions plus Eq. [1J make seven equations, so that there is a unique solution (plus or minus a constant) to Eqs. [4J and [5]. If the assumption is not made that the vesicle interior is electroneutral, then the boundary conditions given by Eqs. [llJ, [12J, and [13J must be replaced by Eqs. [llaJ, [12aJ, and [13aJ, but Eqs. [lOJ, [14J, and [15J still apply.

<

where V = t1l"(R3 - b3 + a3) . The quantities in brackets on the right-hand side of Eqs. [6J and [7J are denoted by 1+ and L, respectively. The conservation of charge equation is l1+V = l1_V

+ alV,

[8J

where v = Vi + v.; The total salt concentration C in moles per liter, is converted to the number concentration n, in number per ern' by correcting for the shell volume:

eN n,

R3

= -.

1000 R3 - b3 + a:J

[9J

+

Boundary Conditions

The electric field, dif;I dr = (11R)det>1dx, must be zero at the center of the sphere in order for the potential to be finite there. Because the total volume 0 ::; x ::; 1 is electrically neutral, the electrical field is zero at x = 1, by Gauss' law. We first give the set of boundary conditions which apply if the region r ::; a is assumed to be electrically neutral. et> is then

°::;

kT det>1 kT det>2 - (ai l - 1)Vi€ D---- - D 2 - - - - = - - - - - €R dx €R dx a2

at x

=

at R,

[11aJ

at x = afR, [12aJ kT det>3 kT #2 D - - --- - D 2 - - -

€R dx

€R dx

(a o + l - l)v o €

= -----b2

at x = biR. [13aJ

Journal of Colloid and l ntcrfcce Science. Vol. 61, No.3, October 1, 1977

460

MILLE AND VANDERKOOI

D 2 is the dielectric constant within the membraneous shell (a ~ r ~ b) ; l is an integer indicating the proton dissociation in question (l = 1 or 2 for phosphatidic acid, which has two dissociable protons); and ai and a o vary from 0 to 1. Free energy minimization considerations, of the type given by Marcus (12), yield another boundary condition which relates the inner and outer surface charge densities and degrees of ionization:

Linearized Poisson-BoltzmannEquation

We also present the linearized form of Eq. [18J for the inner and outer regions of the hollow sphere, valid for cP « 1:

ao

ai

cP(a) - In - 1 - ai

The boundary conditions are identical to those for the finite cell model, except that dr replaces dx or Rdx where they appear. For the infinite dilution case, cP = cP'.

=

cP (h) -In - - . 1 - ao

[16J

[19J where

This relationship applies provided that the dissociation of the primary hydrogens on the phosphate group is completed before dissociation of the secondary hydrogens begins. It is also implicitly assumed that the intrinsic dissociation constants are the same on the inner and outer surfaces. There are now eight equations , and eight unknowns to be solved for. As in the case for the impermeable sphere model, cP can be related to a simple salt phase in osmotic equilibrium with the polyelectrolyte solution. We use cP' to denote the potential with respect to the equilibrium salt solution, while 4> will denote the potential relative to cP = 0 at x = 1. Infinite Dilution

[20J

METHODS OF SOLUTION

The method of solution used for the hollow sphere model at finite polyelectrolyte concentration is quite analogous to that already described for the impermeable sphere (2), but is more involved on account of the larger number of boundary conditions and equations. The essential features of the method wiII be summarized in what follows; greater detail can be found in (13). Trial values of I+ and L (as defined in Eqs. [ 6J and [7J) were used to solve the system of equations

For the case of infinite dilution of polyelectrolyte particles, the cell model is no longer relevant. Mathematically, this involves letting R go to infinity, with the zero of potential being taken at infinity, and ii+L = iLL there. The reduced length parameter becomes mean- dw ingless when R ~ ao , so the Poisson-Boltzmann -dx equation is written in terms of r rather than x :

2~(r2dcP) =0 2 r dr

dr

[21J

=-

X241rf2R2 - --[n+(O)z+y-z+

DkT

-IL(O)Lyz-J, [22J for a

< r < b,

[17J

along with

dI+ldx

= rr

z

+,

dLldx = x2yZ-,

for 0 ~ r ~ a and b ~ r ~ R. [18J

[23J [24J

where y = e'" and w = ;\:2 (dcPl dx). The boundary conditions [1OJ-[15J were also expressed in terms of y and w.

J our nal of Colloid and I nterf ace S cience,Vol. 61, No . 3, October 1, 1977

MEMBRA~O US

Integrations using the trial values of [+ and L were begun at x = 0 with yeO) = 1 and w(O), [ +(0), and L (O) all initially set equal to zero. Th e values obtained for y and w at x = a/ R, along with the boundary conditions and Eq. [lJ were used to determine the initial values to use at :r = bf R. Analy tical expressions can be given for y (b) and web) if the assumption of internal electroneutrality is employed, as follows:

VESICLES

461

I nfinite Dilution of Polyel ectrolyte

The high salt or low polyelectrolyte concentration cases gave computational difficulties with the meth od described above for finite polyelectr olyt e concentration, but these cases could readily be solved by using the limiting case of infinite dilution of polyelectrolyte model. (See Brenner and Roberts (14) for a variational technique by which the PoissonBoltzmann equation can be solved for spherical particle s at infinite dilution. ) For this model the absolute value of the y(b) = Yea), [25J potential decreases from the outer surface at web) = w(a) + (f2/DkTR)at(vi + vo) . [26J r = bof the polyelectrolyte and goes to zero at infinity. In the region where Ic/J I is small the Solutions of Eqs. [21J~[24J were then carried linearized form of the Poisson~Boltz~ann through to x = 1. This yielded a set of calcuequation should be a good approximation ; lated values for w(l), 1+, and L. The intertherefore, the nonlinearized form was solved relationships between the trial and calculated only out to a point where Ic/J(r) I ~ 0.01 (for a values of these parameters, and the "ac tual" uni-univalent salt ), for r » b, and the linearor "correct" value for the parameters was ized form can be used beyond that. For high developed in (13). The use of these relati onsalt concentr ations or a large inner radius of ships showed the direction in which the trial the hollow sphere, the potential near r = 0 is values had to be incremented in order to conalso close to zero; here we used the linearized verge on the self-consistent solution to the form of the Poisson-Boltzmann equation for simultaneous equations. values of r for which Ic/J (r) I ~ 10-". As writte n, Eq. [ 21J is undefined at x = 0, Trial values of yeO) = e9 (O) were used to since w(O) = O. However, since w/ x2 = d4J/ dx, solve the following system of equations: by Eq , [10J, (d4J/ dx)x=o = 0, and since y = 1 dy yUJ at x = 0, dy/dx = at x = O. So dy/dx = [27J was used at x = O. dr r2 The estimated propagated error was ob411"f2r2 tained at x = a by halving the step sizes. A dw [n (O)z y-z+ similar estimate of the propagated error from DkT + + dr x = b to x = 1 was not useful, since We al e (1) "'" O. However, based on potential calcula-n_(O)LyZ-J, [28J tions with different trial values of 1+, L near 2(d the "best" ones, and the deviation between where w = r 4J/ dr) . In the region r »b the calculated values for the right -hand side where Iep (r) I ~ 0.01, the values obtained for of Eqs. [6J and [7J and the known values for y(r) and w(r) using Eqs. [ 27J and [28J the left-hand side, we conclude that the pre- should sati sfy a solution to the linearized forms cision of most of the computed values is on the ?f these equati ons if the trial value of :v(O) is III fact the actual value. The following condiorder of 0.1%. tion should hold for the linearized solution: Th e case for no added salt was solved as a separate pr oblem, since there is one less undc/J/dr = - c/J (l /r + K) specified value. Trial values for f + only, and or [ 29J not 1_, were required. w = - (1 + Kr)r In y.

°

°

J ourn al oj Colloid and I nterf ace Sci ence,Vol. 61. No, 3, Octo ber 1. 1971

462

MILLE AND VANDERKOOI

Whether the computed w(r) for r» b is less than or greater than that given by Eq. [29J indicates in which direction the trial value of yeO) must be changed to obtain the correct value for yeO).

Linearized Poisson-BoltzmannEquation Equation [19J can be solved analytically if the assumption of electroneutral vesicle interior is employed:

E

l/J = - - - - - - - - - - - - - - kT Da(sinh Ka) [K coth (Ka) E

EO!t (Vi

~ +:: (K +

r

o ~ r ~ a,

[30J

< r < i,

[31J

+ va)

l/J = - - - - - - - - - - - - -kT Da2[Kcoth (Ka)

DJ

sinh «r

-~+ ::(K+DJ

a

[32J

One of the unknowns must be solved for numerically if the vesicle interior is not assumed to be electroneutral.

distinguishable from the curve obtained for zero added salt (not shown); the two curves differ by not more than 0.2% at any point. Figure 3 gives the reduced potential, ¢ vs r, CQ1IPUTATIONAL RESULTS for the case of infinite dilution of polyelectrosalt concentrations in the range 10- 1 We give here the computational results' ob- lyte, for 4 tained using the auxiliary assumption of elec- to 10- M. Comparison of Figs. 2 and 3 shows 4 troneutral vesicle interiors. Under the final that when 10- M salt is present, the potentials subheading in this section, these results are calculated using the cell model and by the compared in selected cases with those obtained infinite dilution method are essentially equivawhen this auxiliary assumption is not em- lent. Thus, if only the potential is of interest, 4 ployed. It is sliown that the results of the at salt concentrations above 10- M there is simpler method are sufficiently accurate to be no advantage in using the cell model for 2 mM phospholipid, and indeed, the numerical soluusable for most applications. tion of the cell model becomes very difficult at higher salt concentrations. Electrostatic Potential It can be seen in Figs. 2 and 3 that the Curves of q/, the reduced potential , for potential in the center of the vesicle is signififinite particle concentration expressed relative cantly higher than that at the cell boundary to the potential of an equilibrium salt solution for all salt concentrations of 10-3 M or lower. are plotted as a function of the reduced length The potential drops effectively to zero, both parameter :r in Fig. 2. Curves are presented inside and outside of the vesicles, when 10- 1 M for a phospholipid concentration of 2 mM. salt is present. The curve of ¢ (relative to x = 1) for the The surface potential as calculated with the 10-7 M salt shown in this figure is nearly inlinearized Poisson-Boltzmann equation (Eqs. [30J-[32J) is far greater than that obtained 4 Extensive tables are found in (13). J our nal of Colloid and In ter/ace Science, Vol. 61, No . 3, October I. 1977

j\IEMRR:\NOl~S

463

VESICLES

r(A) 0

200

600

400

800

1000

120 0

1400

1600

1800

-7

16 14 12 10

-¢' 8 6 4 2 00

.2

.6

.4

.8

10

X

FIG. 2. Plot of the reduced potential 1>' vs x and r for several uni-univalent salt concentrations, computed for finite polyelectrolyte concentration using the cell model. Th e numbers on the curves a re th e logs of the molarity of uni-univalent salt present. The parameter s used were T = 298°K, D = 78.54, at = 1.0, S' = 60 ,\ 2, b = 300 Ai (b - a) = 45 A, and Cp (concentra tion of phospholipid) = 2.0 mM. For these par ticular conditions, R = 1860 ..\., and the r values are as given on the upper abscissa.

by the nume rical solution for infinite polyelectrolyte dilution even in the presence of 0.1 M salt. This is noted in Fig. 3. The dis-

16

crepanc y is magnified as the salt concentration is decreased or the surface charge density increased.

Log C

n

I I

! 12

- 1 (Iln.]

I

I

I

-4

-tfJ 8

0 1-~+~'-4--'::::=;~--'::;:::;;==r==~

o

400

r (A )

800

1200

FIG. 3. Plot of the reduced potential 1> vs r for several uni-univalent salt concentra tions using the infinite dilution of polyelectrolyt e model. The num bers on the curves are the logs of the molarity of uni-univalent salt present. The parameter s used were T = 298 OK, D = 78.54, b = 300 A; (b - a) = 45A ; S ' = 60 A2, and (l, t = 1.0. Th e dashed curve gives the potential obta ined using the linearized approximat ion to the Poisson-Boltzmann equation , for 10- 1 ,l[ salt. J ournal of Colloid and lnt erface Scie nce, Vol. 61, :sio. 3. Oc tober I , 19 i7

464

MILLE AND VANDERKOOI 10....----.,..-----,--..,....---__-,.-----.---....----.,..-----,

8

6

_¢' 4

2

~ 200

600

400 r(4)

800

FIG. 4. Plot of the reduced potential ,p' vs r for several phospholipid concentrations, computed with the cell model. The numbers on the curves are the molar concentrations of phospholipid, Cpo The fixed parameters used were b = 300 A, (b - a) = 45 A, S' = 60 A2, at = 1.0, C = 10-3 M, T = 298°K, D = 78.54, and z+ = z.: = 1. The values of R corresponding to each value of Cp are given in Table VI.

Figure 4 shows the reduced potential rp' vs r as a function of phospholipid concentration. At 0.02 M the curve is still quite similar to the 2 mM curve (see Fig. 3); however, as the concentration of phospholipid increases the absolute value of the potential at the center decreases, while it increases at the cell boundary. Eventually for high concentrations, 0.2 M

in Fig. 4, the potential at the center is less than at the cell boundary. This is a consequence of the decrease in volume on the outside as the concentration of phospholipid increases. Ionic Distribution

The localconcentrations ofpositive and negative ions can be obtained from the electrostatic

2.--------------------.,

-:j\,

~----:--::----:..------"-"-"-----~

-4

..,'"

~

~,., ...

E -6 ---............

..2'

\

-8

I

f

r

_10~!r -12

~

1

-14

o

.2

.4

x

.6

.8

1.0

FIG. 5. Computed values of the local concentrations of mobile ions, n+ and «., as a function of position. Concentrations are expressed as log (moles of ions per liter). The physical parameters are the same as in Fig. 2. The solid line is for C = 10- 7 M, and the dashed line for C = 10-4 M. n+ (inside) is identical for these two salt concentrations. Journal of Colloid and Interface Science,Vol. 61, No.3, October I, 1977

~\'fEMBRANOUS

-2

n,( O )

...... -4

n, ( R)

465

VESICLES

n;•......_...«'···~······

III

c:

~ -6 Cl

o

- -8 -10

n-
- 00

-7

-5

-6

-1

-4 -3 log (C)

FIG. 6. Ionic concentrations at the vesicle center [nt CO) and n_CO)] the cell boundary [nt CR) and n_CR)], and in an equilibrium salt solut ion CnO±), as a function of tota l average salt concentration C. The parameters employed were the same as for Figs. 2 or 3. The cell model was employed for salt concentrations from 0 to 10-- 4 M, but the infinite dilution meth od was used for higher salt concent rations. The computed points are indicated by th e symbols on the lines. Filled symbols denote negative ions and open symbols denote positive ions. Th e circles are for concentrations inside the vesicles and the triangles are for the concentrations outside of the vesicles.

potential using the Boltzmann distrib ution: 1t+(x )

=

It- (X)

= nO_ez- ' = nR_ez- .

nO+e- Zt ¢'

=

nR+e- Zt ,

[33J

These functions are plott ed in Fig. 5 for two salt concentrations, at low but finite polyelectrolyte concentration, using the same parameters as for Fig. 2. Within the vesicle, the counterion concentration is higher, and the coion concentration lower, than is found a comparab le distance away from the vesicle on the exterior. This is as we would expect on

0 -2

the basis of the electrostatic potential curves in Fig. 2. It is interest ing that the interior counterion concentrations are found to be nearly constant between 10-4 and 10- 7 M added salt, so that only a single line appears in Fig. 5 for this quant ity. Th e interior coion concentrations, on the other hand, differ in proportion to the total salt concentrat ion. The individual ion concentrations are plotted in Fig. 6 as a function of added salt concentration for the vesicle center and for the cell boundary. The concentration of the equilib-

ii,(in)

n

n..( our )

~ -4

.2

Cl

.2 -6 n_(out)

-8 -10 -00

iU in )

-7

-6

-5

-4

-3

-2

-1

log(C)

FIG. 7. A verage ionic concentrations, nt and ii.; inside and outside of the vesicle, as a function of salt concentration. The para meter s and symbols are the same as for Fig. 6. J ournal of Colloid and I nterfa ce S cience,Vol. 61, No. 3. October I , 19 77

466

MILLE AN D VANDER KOOI

-2

-3

Log Cp

o

-1

'in t:

g-2 Cl

o

... -4

.002

.01 .02

.1

.2.3

Cp FIG. 8. Average ionic concentrations, ii+ and ii_ inside and outside the vesicle, as a function of phosphatidic acid concentration CpoThe param eters and symbols are the same as in Fig. 6, with C = 10- 3 M.

rium salt solution, nO±, for which z+no+ = z-no_, is included for comparison. In Fig. 7, the aeerage individual ion concentrations are plotted for the interi or and exterior of the vesicles as a function of added salt concentration, for 2 mM phosphatidic acid. In this case, the ion concentrations were averaged over the int erior vesicle space, or over the space of th e cell exterior to the vesicle. Th e results shown here are qualitatively similar to the point values plotted in Fig. 6, in that the average concentra tions of counterions is also higher inside the vesicle than outside. The difference between the inside and outside concentrati ons decreases as salt is added, as is shown in both Figs. 6 and 7. The latter . 8,...--~----,--~---.---,---.,..---.......,

result could have been deduced from the earlier observation that the potential difference between the vesicle center and the cell boundary goes to zero in 0.1 M salt. At the high vesicle concentrations which cause the average value of e- ';' inside the vesicles to be less than that outside, the average counterion concentr ation will also be lower inside the vesicles than on the exterior. This is illustrated in Fig. 8. While the foregoing discussion ostensibly dealt with salt ions, the comments about counterion distribution across the vesicle apply equally well to hydronium ions. A hydronium ion concentrati on difference will exist between the int erior and exterior of the vesicle under the same low salt conditions which give a salt concentration difference.

.6

Activity Coqtficients

.2 -----~=:::::::::~.--------

Poo

-7

-6

-5

-4 -3 log (C)

-2

-,

FIG. 9. Compu ted values of the mean inner and outer activity coefficien ts, "Y± in and "Y± out, plotted as a function of the log of the salt concentration C. The physical parameters are the same as in Figs. 2 and 3. Up to and including C = 10-3 M the cell model was used, but at higher concentra tions the infinite dilution method was used.

In general the products of the average ion concentrations interior and exterior are not equal: (ii+i n)(iLin) ~ (ii+ out ) (jL out ) ' However, as can readily be seen from Eq. [33J , the produ ct of the point ion concentrations, ll+(X) lL(X) , is constant throughout the entire system: inside, out side, and in an equilibrium simple salt solution if present. To relate this product to the average inner and outer ion concentrations we can therefore define inner and outer mean ion activity coefficients similar to those derived by Ma rcus for the total cell

J ournal of Colloid and I nterf ace S cience,V ol. 6 1, No . 3, Oct ober I , 1977

~l

model (12) :

[34J

1/out

[35J

th en

Eq. [36J gives the ratio of outer to inner charged phospholipids for the condition of complete ionization. For the particular case of only one type of phospholipid present which is capable of dissociating, Eq . [37J is equivalent to a o/ai if the area per phospholipid is the same on both surfaces. The linearized Poisson-Boltzmann solution can also be used to obtain an approximate expression for Ct.o/Ct.i . The inner and outer surface charges, expressed in terms of ai and a o , can be obtained from the boundary conditions (- DaJ/tjar)a = 41rCTi and (DaJ/tjar)b = 4 7l"(To . ' usmg Eqs . [30J and [32]. The result is

'Y 2±i n(n+ in) (n-i n) = 'Y2± out Ui+ out)(iL out).

Values of 'Y±in and 'Y±out for various salt concentrations are shown in Fig. 9. 'Y ± in is larger than 'Y±out for low salt concentra tions, but 'Y± out begins to increase at a lower salt concentration than 'Y± io.

Degree of Dissociation The fractional degree of dissociat ion on the inner and outer surfaces will not in general be the same, due to the differing radii of curvature. The degree of dissociation on the inside surface, a i, is less than that on the outside surface, a o, for all the conditions we have employed. (T hese quantities are unknowns in the PoissonBoltzmann equation, and are obta ined in the course of solution of the equation .) Israelachvili (3) has derived an approximate expression for the ratio of the outer to inner surface charge and surface charge densities. These are Eqs. [ 6J and [ 8J of (3), which in our notation read as follows:

b(Kb

Q o

-

+ 1)

Qi

a(Ka - 1)

lI o

a(Kb + 1)

IIi

,

b(Ka - 1) ,

467

E:'IIBRANOUS VESI CLE S

[36J

a(Kb + 1)

ai

b(Ka coth (KG ) - 1)

[38J

For Ka » 1, Eq . [3 8J approaches Eq . [37]. Table I (A) gives the a o/ai ratio as a function of salt concentration, as computed with the nonlinearized Poisson-Boltzmann equation, the linearized Poisson-Boltzmann equation, and the Israelachvili expression. In Table I (B), the values of a o/Ct.i are given as a function of particle radiu s, at constant salt concentration (10- 3 M ). The results obtained by the three meth ods converge in the limit of high salt or large radius. At low salt even the qualitative behavior differs. Using the nonlinearized Poisson-Boltzmann equation, a u/Ct.i is constant at low salt, whereas the ratio continu es to increase by the other two methods as the salt concentration app roaches zero. Table II shows the behavior of a o and ai as a function of at, as computed with the nonlinearized Poisson-Boltzmann equation. The ratio a o / Ct.i approaches 1 as at increases, but the differences a o - at and Ct.t - ai change very little over a wide range of a t.

Dissociation Constant

[37J

where the subscript 0 refers to outer and i to inner surfaces, Q is the surface charge, and II is the surface charge densit v. For the case of mixed zwitterionic and acidic phospholipids,

Ct. o

The apparent pK of ionizable groups on the surface of an impermeable polyelectrolyte particle is related to pIC, the intrinsic pK , in the following manner (4): ~pK

=

pKappareo t - pK;

<:::

- tjls/2.303. [39J

J ourn al oj Colloid awl I nterf aceScience, Vol. 6 1, No . 3, Oc to be r I, 19/j

468

MILLE AND VANDERKOOI TABLE I Ratio of a % l j for Various Salt Concentrations and Vesicle Radiia o / ai

Nonlinea rized Po issonBolt zmann equ ati on

Israelachvili

Linearized PoissonBolt zma nn eq uati on

( R ef. 3)

A. Salt concentration C (mole/liter) H/4 10-3 H/2 H/l

0.838 2.65 8.38 26.5

1.019 1.019 1.019 1.017

7.541 2.087 1.251 1.073

- 1Q.43 2.121 1.251 1.073

0.572 1.09 2.65 5.77

1.076 1.043 1.019 1.009

10.52 4.859 2.087 1.404

- 2.618 19.62 2.121 1.404

B. Vesicle radius" b (1) 100 150 3oo 6oo a

Physical parameters are as in Fig. 3 for infinite dilution of polyelectrolyte. 10--3 M .

be =

cjJs is the electrostatic potential at the surface. This equation must be modified for the case of hollow spherical vesiclesin which the surface potential may differ on the two surfaces. Following Marcus (12), we have obtained Eq. [ 40J , which is a general expression for LlpK in vesicular systems, based on minimization of the free energy with respect to the total degree of dissociation. This equation is consistent with the boundary condition given in Eq. [16J:

L1pK= log { [av(e (a)+ e (bl) -Vie ( c ) -voe (bl

+ {[ v\e (c) +voe (b)-av (e (a)+e (b) )J2 +4( 1-a)av2e(al+( bl)

used to relate cjJ(a) and LlpK.It is shown in a subsequent section that essentially the same result would have been obtained had this assumption not been made, since for the vesicle parameters employed, cjJ(a) and cjJ(b) are found to be nearly equal. LlpK is a function of the surface charge density and is th us related to the fracti onal degree of ionization at, which varies between o and 2 for phosphatidic acid since two dissociable groups are present. LlpK is plotted vs at in Fig. 10 for several salt concentrations between 0 and 0.1 M . It can be seen that whereas the actua l value of L1pK is strongly dependent upon the salt concentration, the

!J

X (2ave (a l+( b)) -

I) .

TABLE II

[ 40J

Eq uation 40 readily reduces to [39J if cjJs = cjJ(a) =cjJ(b), which is the case if the assumption of intern al electroneutrality applies. If cjJ(a) and cjJ(b) differ by only a small amount, on the other hand , it can be shown that Eq. [39J is still a good approximation if cjJs is set equal to the mean of cjJ(a) and cjJ(b). The calculations reported in this section were carried out using the assumption of electroneutra l interiors, and Eq . [39J was

Ra tio of a o/a; at Various atO

a

at

a o/a i

au

0.0375 0.3 0.5 0.75 1.0 1.5

1.442 1.064 1.038 1.026 1.019 1.013

0.00553 0.00773 0.00789 0.00797 0.00801 0.00805

C

=

Fig. 3.

J ourn al of Colloid and Int erface S cience,Vol. 6 1, No. 3, Octo ber I, 19i7

-a~

at -

a (

0.00765 0.01070 0.01092 0.01103 0.01109 0.01115

10- 3 M j other physical parameters are as in

469

MEMBRANOUS VESICLES 7 6

-4

5

-3

.4

.8

Oft

1.2

1.6

2.0

FIG. 10. Ll.pK versus at at various concentrations of salt. Other parameters are as in Figs. 2 or 3. The curves at 10- 4 M and higher salt were obtained using the infinite dilution method (as in Fig. 3); the curve for 0 salt was computed using the cell model (as in Fig. 2).

slope .1 (.1pK)j .1at is nearly independent of salt concentration above at """ 0.5. Below at """ 0.5 the slope changes rapidly since .1pK must equal zero when the surface charge IS zero. Tables III to V show the dependency of .1pK on the geometrical parameters of the particles. The results reported are for infinite dilution of polyelectrolyte, so as to eliminate interparticle effects. The variation with respect to particle radius is given in Table III. It can be seen that for the range of particle radii shown, only a variation in the third and fourth significant figures is observed. Thus the fact that experimentally obtained phospholipid vesicles show a size distribution, the mean of which is not readily determinable with great accuracy, is

not a serious handicap in comparing experimental and theoretical pH titration curves in the presence of salt. Table IV shows that out to four significant figures there is no variation in .1pK with bilayer thickness for the small range presented. As in the previous table, these values are for infinite dilution of polyelectrolyte. The variation of .1pK is tabulated as a function of surface area per phospholipid in Table V, with at = 1.0, at several salt concentrations. While the results are expressed in terms of surface area per molecule with constant degree of dissociation, they could equally well be interpreted in terms of surface area per unit charge, since the infinite dilution of polyelectrolyte assumption was employed. This equivalence would not obtain if finite poly-

TABLE III TABLE IV

Ll.pKat Various Vesicle Radii"

Ll.pKat Various Bilayer Thicknesses"

"C

=

b (X)

IlpK

100 150 300 600

4.306 4.312 4.315 4.315

10- 3 M; other parameters are as in Fig. 3.

"C

=

(b - a) (A)

IlpK

35 45 55

4.315 4.315 4.315

10- 3 M; other parameters are as in Fig. 3.

Journal of Colloid and Interface Science, Vol, 61, No, 3, October I, 1977

M ILLE AN D VANDERKOOI

470

T ARLE VI

.4

Osmotic Coefficient at Various Phosp holipid Concen trat ions"

.3 9 .2

/

.1

c; (mole /l ite r)

R (A)

g

0.002 0.01 0.02 0.1 0.2 0.3

1860 1088 863 505 401 350

0.341 0.0985 0.0558 0.0212 0.0229 0.0429

• -. o ------.- t1IO

-7

-6

-5

-4

-3

-2

log (C) FIG. 11. Plot of t he osmotic coefficient g vs the log of the salt concentra tion. The physical pa rameters are the same as in Fig. 2. a

electrolyte concentration were used, since the cell size (R) is dependent upon the phospholipid concentration and is thus also related to the surface area per molecule.

Osmotic Coeficien: Ma rcus (12) showed quite generally tha t the ionic concentration at the cell boundary can be interpreted as the concentr ati on of the ions that are osmotically active. The osmotic coefficient can be expressed as

where the ideal osmotic pressure is assumed to be due to all the phospholipid molecules, as if they were all in monomeric form, plus any salt and counterions. Equation [41J also holds if no added salt is present, for which case n R _ TABLE V ~p K a t Various Surface Areas per Ph ospholipid

and Salt Concentrations-

s- ( A')

42.1 53.3 60

80 100 160 320 1600 a

at

C (mole / liter) 10-1

W '"

to -'

2.62 2.42 2.32 2.07 1.88 1.49 0.960 0.231

3.62 3.42 3.32 3.07 2.87 2.46 1.87 0.667

4.62 4.42 4.32 4.06 3.87 3.46 2.86 1.47

= 1.0; pa rameters not specified are as in Fig . 3.

C = 10- 3 ,lif ; other par ameters ar e as in Fig. 2.

and n; would be zero. At infinite dilution of polyelectrolyt e or high salt concentr ation, g = 1 according to Eq . [ 41J, since the terms in V predominate . The osmotic coefficient is plotted as a function of the added salt concentration in Fig. 11. Th is shows that g is close to zero at low salt levels, but begins to noticeably increase when C > 10- ;; M . Computa tional difficulties with the finite polyelectrolyt e concentration method prevented the calculation of g at salt concentr at ions higher than 10-3 M for the parameters employed. Ta ble VI gives the dependence of g upon the phospholipid concentra tion. Since the geometrical parameters (par ticle radius, bilayer thickness, and surface area per molecule) are held constant, the particle density is directly proportional to the concentrat ion, or conversely, the average inte rparticle distance ( ""2R) varies inversely as the cube root of the concent ration. As the concentr ation of phospholipid is increased, g is seen to decrease (indicating an increased electrostatically induced depa rture from ideality), up to a phospholipid concentrat ion of 0.1 M . Since the molecular weight of phosphatidi c acid derived from egg lecithin is about no, 0.1 M corresponds to a 7.2% solution by weight. At a concentration of 0.3 M; however, g is seen to increase again. This concentration would correspond to a 22% solution by weight, which would be a slurry. A similar increase in osmotic coefficient of micelles has been noted both

J ournal of Colloid and I nterf ace S cience,Vol. 6 1, No. 3, October I, 1977

471

:VIENIBRANOUS VESICLES

TABLE VII Comparison of Calculations Using Hard and Hollow Sphere Models" -",(bl

g

X 10'

---_._-----_

~-~---~----

Hollow

Hard

12.62 14.01 14.82

12.59 13.99 14.80

9.76 7.75

9.74 7.68

Hollow

'Y± out

'Y±

Hard

Hollow

Hard

0.5669 0.4268 0.3419

0.1880 0.1338 0.1094

0.1895 0.1343 0.1097

5.443 1.782

0.3202 0.2742

0.3218 0.2809

-

..._---_ ..

Zero salt case b <>t

=

0.5 1.0 1.5

Cp

=

0.02 M 0.2

=

10- M 10- 2 10- 1

0.5684 0.4279 0.3428

Cell model with salt' 5.575 2.287

Infinite dilution of polyelectrolyte" C

3

9.94 7.63 5.34 -

---~-----

9.91 7.62 5.35 - - _ ..

_~~.

Parameters used were b = 3001, a = 255 1,5' = 601 2 for the hollow sphere; b = 3001,5 = 60/<>t 1 2 for the hard sphere. T = 298°K, D = 78.54; and Z+ = L = 1 for all cases. The values for the hard sphere were obtained by the methods described in Paper I (2), except in the infinite dilution of polyelectrolyte case Stigter's(15) numerical expression for the tables of Loeb et al. (16) was used. b Cp = 0.002 M. C <>t = 1.0, C = 10- 3 M. d <>t = 1.0. a

experimentally and theoretically at high detergent concentrations (2). The computed osmotic coefficients and other parameters at high particle concentrations must be viewed with caution, since the approximations of the cell model become progressively poorer as the interparticle distance decreases. The computed values under these conditions should be taken as a qualitative rather than quantitative measure of the actual behavior. Comparison with Hard Sphere Calculations

For many purposes, only the electrostatic potential in the region exterior to the polyelectrolyte vesicles is needed. It would therefore be useful to know how well the potential in the exterior space is approximated by the simpler hard sphere calculations. The hard sphere and hollow sphere results for the surface potential, osmotic coefficient, and activity coefficient are compared in Table VII. For the hard sphere calculations the total surface charge was obtained from the values of LYt

and 5' employed in the comparable hollow sphere calculation. It may readily be seen from the table that, although the results are not identical, the differences are small, which is as one might have expected. [On first sight, one might think that the results should have been identical, barring computer roundoff errors, since an electro neutral vesicle interior has been assumed, but this is not true since LYo (vesicle) oF- (Xt (hard sphere), and also since all of the salt is in the exterior space for the hard sphere, whereas some of it is in the interior for the vesicles.] At large particle radii and/or high salt concentration (Kb -,1> 00 ), the hard sphere can be approximated by a flat plate (16), thus Table VII supports the use of the Gouy-Chapman equation for large phospholipid vesicles in a concentrated salt solution. Vesicle Interiors notAssumed Electroneutral

The results reported in the preceding sections were obtained with the use of the simplifying approximation that the vesicle

Journal of Colloid and l ntcrjacc Science, Vol. 61, No.3, October 1, 1977

472

MILLE AND VANDERKOOI TABLE VIII Comparison of Surface Potentials Calculated With and Without Assumption of Electroneutral Vesicle Interiors Nonelectroneutral Interior

Conditions-

Electroneutral Interior -(a) = -(b)

-(a)

-q,(b)

8.142 12.641 14.826 15.298

7.883 12.603 14.807 15.280

7.990 12.619 14.815 15.287

3.997 6.156 6.627

3.969 6.139 6.610

3.98 b 6.14b 6.617

8.613 4.040

8.473 3.929

Zero added salt, finite vesicle concentration.

b = 300 A,

c, =

2 mM

at

= 0.05 = 0.5 =1.5 = 1.9

Infinite dilution of vesicles, added salt.

b = 300 A, C = 0.1 M

b = 100 A, at = 0.5

at

= 0.5 = 1.5 = 1.9

C = 0.001 M = 0.1 M

Not available Not available

-Tn all cases, (b-a)=45 Aj 5'=60 A2 j D=78.54j D 2=2j T=298°K; and salt assumed to be uni-univalent, b Obtained by interpolation.

interior (0:::; r :::; a) is electroneutral. This assumption is not consistent with free energy minimization with respect to the degree of dissociation. The boundary condition given in Eq. [16J is consistent with free energy minimization (12), but since it was derived upon the assumption that each dissociable group dissociates independently from any other, it is undefined for at = 0 or 1. The results of calculations using Eq. [16J as a boundary condition are given in this section and compared with the results obtained for the same vesicle parameters but with the assumption of electroneutral interiors. The largest differences, which are themselves small, were found in the surface potentials on the inner and outer surfaces of the vesicles. The values of the inner and outer surface potentials for several selected sets of parameters, calculated with the use of Eq. [16J, are given in Table VIII, and compared with the results obtained using the electroneutral interior assumption. The potential within the shell (a:::; r :::; b) varies as l/x, according to solution of Eq. [4]. It can be seen that the result obtained with the use of the latter

assumption is very close to the mean of ¢(a) and ¢(b) computed using Eq. [16J as a boundary condition. For this reason, the use of Eqs. [39J and [40J will give nearly identical results in the computed ~pK values. Although the difference in potential across the shell is evidently too small to be of significance in the interpretation of pH titration curves of lipid vesicles or membranes, the fact that a potential difference exists at equilibrium may be important for understanding the equilibrium distribution of lipids and other charged molecules between the inner and outer halves of a bilayer vesicle. DISCUSSION

The computational methods involved in the electrostatics calculations have been discussed at length in Paper 1. The extension of these methods to the hollow sphere problem was straightforward, but since it was necessary to solve for the inner and outer regions simultaneously, the solution required a longer computer time and, in general, resulted in a less precise answer, as compared to the hard sphere calculations.

Journal of Colloid and Interface Science,Vol. 61, No.3, October 1,)977

473

MEMBRANOUS VESICLES

We chose to express our results for conditions of specified surface area per molecule (S') and fractional degree of ionization on the inner and outer surfaces of the vesicle ((Xi and (Xo). This was done to facilitate comparison of the computations with experimental data on phospholipid vesicles. For the assumption that the interior of the hollow sphere is electroneutral, the inner and outer surface charge densities are the fundamental quantities, and these can readily be obtained from the expressed quantities using the equations

[42J For a given set of geometric parameters (particle and cell dimensions), temperature, salt concentrations, and valences, the computed results are applicable to any system which has the same Uj and a., values. The principal results of qualitative interest obtained in this work concern the difference in electrostatic potential across the membrane of a closed vesicle, and the ionic effects which accompany this difference. We have shown that for fairly dilute solutions of charged lipid vesicles at low salt concentrations coions will be virtually excluded from the interiors of the particles because of the electrostatic field in that space. This effect is decreased if the particles are made larger or if the salt concentration is increased. At very high particle densities the opposite situation occurs: Under these conditions, the electrostatic potential exterior to the particles may be greater than that in the interior, and the concentration of coions on the exterior may actually be less than that in the interior. Thus, the equilibrium distribution of ions between the interior and exterior space is a function of particle density. In practical terms, this means that a change in the equilibrium distribution may be expected to occur upon concentration or dilution of charged vesicles, as in ultracentrifugation or ultrafiltration, or other physical technique in which the vesicle concentration is highly altered. We have shown that for lipid vesicles consisting of one ionizable lipid species, the frac-

tional degree of ionization of the outer surface will be greater than that on the inner surface. Furthermore, the dissociation of water soluble molecules would be expected to be different in the interior solution of the vesicle from that in the exterior solution, since in the general case the average potential in the interior differs from that in the exterior. The difference in these potentials should be reflected in the measurable degree of ionization of water soluble indicator dyes in the interior and exterior spaces. Membrane-bound indicator dyes may be used, on the other hand, to register the electrostatic potential at or in the vicinity of the vesicle surfaces. The qualitative observations given here for lipid vesicles should be applicable to other types of small charged vesicles as well, including biological membranes. A point of caution is in order when applying these results to biological systems, however. The results given here are for systems at equilibrium, and are therefore not directly applicable to actively metabolizing biological systems which are not at equilibrium, other than to show what would be the equilibrium state. The fact that the equilibrium distribution of salt across a vesicle does not in general correspond to equal salt concentrations may be relevant for the interpretation of work on excitable membranes. One would expect a similar unequal salt distribution at equilibrium between the interior and exterior of hollow tubes (e.g., nerves). The significance of these differences under physiological conditions may be slight, since our results have shown that a salt concentration of 0.1 M is sufficient to nearly obliterate the unequal distribution in particles with a radius of 300 A. ACKNOWLEDGMENTS This work was supported in part by Grant GM12847 from the National Institute for General Medical Sciences, by a computing grant from the University of Wisconsin Research Committee, and by Grant K\lS7514425 from the National Science Foundation. We are indebted to Dr. David E. Green, in whose laboratories the initial phase of this work was carried out. We are also appreciative of many helpful discussions with Dr.

Journal 0/ Colloid and Inter/ace Science, Vol. 61. No.3. October 1. 1977

474

MILLE AND VANDERKOOI

G. Zografi and Dr. P. Mukerjee. One of us (M.::YI.) was the recipient of an NIH predoctoral training grant. REFERENCES 1. MILLE, M., AND VANDERKOOI, G., J. Colloid Interface Sci. 61, 475 (1977) (Paper III). 2. MILLE, M. AND VANDERKOOI, G., J. Colloid Interface Sci. 59, 211 (1977) (Paper I). 3. ISRAELACHVILI, J. N., Biochim. Biophys. Acta 323, 659 (1973). 4. SCHERAGA, H. A., KATCHALSKY, A., AND ALTERMAN, Z., J. Amer. Chem. Soc. 91, 7242 (1969). 5. KATCHALSKY, A., ALEXANDROWICZ, Z., AND KEDEM, 0., in "Chemical Physics of Ionic Solutions" (B. E. Conway and R. G. Barradas, Eds.), p. 295. Wiley, New York, 1966. 6. LEVINE, Y. K., AND WILKINS, M. H. F., Nature New Bioi. 230, 69 (1971). 7. VANDERKOOI, G., in Conformation of Biological Molecules and Polymers, "The Jerusalem Sym-

8. 9. 10.

11. 12. 13. 14. 15. 16.

posia on Quantum Chemistry and Biochemistry V" (E. D. Bergman and B. Pullman, Eds.), pp.469-479 Jerusalem, 1973. SMALL, D. M., J. Lipid Res. 8, 551 (1967). JOHNSON, S. M., Biochim. Biophys. Acta 307, 27 (1973). ATKINSON, D., HAl:SER, M., SHIPLEY, G. G., AND STUBBS, J. M., Biochim. Biophys. Acta 339, 10 (1974). MICHAELSON, D. M., HORWITZ, A. F., AND KLEIN, M. P., Biochemistry 12, 2637 (1973). MARCUS, R. A., J. Chem. Phys. 23, 1057 (1955). MILLE, M., Ph.D. thesis, University of WisconsinMadison, 1976. BRENNER, S. L., AND ROBERTS, R. E., J. Phys. Chem. 77,2367 (1973). STIGTER, D., J. Electroanal. Chem. 37, 61 (1972). LOEB, A. L., OVERBEEK, J. T. G., AND WIERSEMA, P. H., "The Electrical Double Layer Around a Spherical Colloid Particle." MIT Press, Cambridge, Massachusetts, 1961.

Journal of Colloid and Interface Science. Vol. 61, No.3, October 1, 1977