16 Electrostatic InteractionBetween NonuniformlyCharged Membranes

16.1 INTRODUCTION

As a model for the electrostatic interaction between biological cells or their model

particles, we have developed a theory of the double-layer interaction between ion-

-penetrable charged membranes in Chapter. This theory, unlike the conventional

models for interactions of colloidal particles at constant surface potential or con-

stant surface charge density, assumes that the potential far inside the membranes

remains constant at the Donnan potential during interaction. We thus term this type

of interaction, the Donnan potential-regulated interaction. In Chapter 13, we con-

fined ourselves mainly to uniformly charged membranes. Since, however, in the

membranes of real cells the distribution of membrane-fixed charges may actually

be nonuniform rather than uniform, it is necessary to extend our theory to cover the

case of arbitrary distributions of membrane-fixed charges. In this chapter, we shall

consider the electrostatic interaction of ion-penetrable multilayered membranes as a

model for real cell membranes.

16.2 BASIC EQUATIONS

Consider two parallel planar ion-penetrable membranes 1 and 2, which may not

be identical, at separation h in a symmetrical electrolyte solution of valence z andbulk concentration n (Fig. 16.1). We take an x-axis perpendicular to the mem-

branes with its origin at the surface of membrane 1. The electric potential c(x) atposition x between the membranes (relative to the bulk solution phase, where

c(x) is set equal to zero) is assumed to be small so that the linearized Poisson–

Boltzmann equation can be employed. Membranes 1 and 2, respectively, consist

of N and M layers. All the layers are perpendicular to the x-axis. Let the thicknessand the density of membrane-fixed charges of the ith layer of membrane j ( j¼ 1,

2) be dð jÞi and rð jÞi . The linearized Poisson–Boltzmann equation for the ith layer

Biophysical Chemistry of Biointerfaces By Hiroyuki OhshimaCopyright# 2010 by John Wiley & Sons, Inc.

375

of membrane j is

d2cdx2

¼ k2c� rð1Þi

ereo; � d

ð1Þi�1 < x < �d

ð1Þi ; 1 � i � N ð16:1Þ

d2cdx2

¼ k2c� rð2Þi

ereo; hþ d

ð2Þi < x < hþ d

ð2Þi�1; 1 � i � M ð16:2Þ

and

d2cdx2

¼ k2c; 0 < x < h ð16:3Þ

with k being the Debye–Huckel parameter of the electrolyte solution, we have as-

sumed that the relative permittivity er takes the same value in the regions inside and

outside the membranes, and we have defined dð1ÞN ¼ d

ð2ÞM ¼ 0 and d

ð1Þ0 ¼ d

ð2Þ0 ¼ 1. It

is convenient to introduce the Donnan potential cð jÞDON;i of the respective layers. Since

the Donnan potential satisfies d2c/dx2¼ 0, we have from Eqs. (16.1) and (16.2)

cð jÞDON;i ¼

rð jÞi

ereok2ð16:4Þ

Now we assume that the distribution of mobile electrolyte ions is always at thermo-

dynamic equilibrium within the membranes as well as in the surrounding solution,

then the potential far inside the membranes remains constant during interaction. In

the present system the potentials far inside membranes 1 and 2, respectively, remain

constant at cð1ÞDON;1 and at c

ð2ÞDON;2.

16.3 INTERACTION FORCE

The interaction force P(h) driving the two membranes apart per unit area in the low

potential case is given by Eq. (8.25), namely,

PðhÞ ¼ 1

2ereok2 c2ð0Þ � dc

dx

����x¼0þ

� �2" #

ð16:5Þ

xh0

1 1… … … …

h+d1(2)h+di−1

(2)h+dM−1(1) h+di(2)−d1

(1) −di−1(1) −dN−1

(1)−di(1)

i iN M

Membrane 2Membrane 1

FIGURE 16.1 Interaction of two ion-penetrable membranes 1 and 2 consisting of N and

M layers, respectively.

376 ELECTROSTATIC INTERACTION BETWEEN NONUNIFORMLY CHARGED

The solution to Eqs. (16.1)–(16.3) subject to the boundary conditions that c and its

derivative are, respectively, continuous at the boundaries x ¼ �dð1Þi and x ¼ hþ d

ð2Þi

can easily be obtained. Since only the values of c and its derivative at x¼ 0 are

required to calculate P(h) via Eq. (16.5), we give explicitly the solution to

Eq. (16.3) below.

cðxÞ ¼ co1 e�kx þ co2 e

�kðh�xÞ ð16:6Þwhere co1 and co2 are, respectively, given by

cðxÞ ¼ 1

2cð1ÞDON;1 exp �kdð1Þ1

� �þ 1

2

XN�1

i¼2

cð1ÞDON;i

nexp �kdð1Þi

� �� exp �kdð1Þi�1

� �o

þ 1

2cð1ÞDON;N

n1� exp �kdð1ÞN�1

� �oð16:7Þ

and

co1 ¼ 1

2cð2ÞDON;1 exp �kdð2Þ1

� �þ 1

2

XM�1

i¼2

cð2ÞDON;i

nexp �kdð2Þi

� �� exp �kdð2Þi�1

� �o

þ 1

2cð2ÞDON;M

n1� exp �kdð2ÞM�1

� �oð16:8Þ

Evaluating c and its derivative at x¼ 0 via Eq. (16.6) and substituting the result into

Eq. (16.5), we obtain

PðhÞ ¼ 2ereok2co1co2 e�kh ð16:9Þ

It is interesting to note that co1 and co2, are, respectively, the unperturbed surface

potentials of membranes 1 and 2 (i.e., the surface potentials at h¼1) and that

Eq. (16.9) states that the interaction force is proportional to the product of the un-

perturbed surface potentials of the interacting membranes. This is generally true for

the Donnan potential-regulated interaction between two ion-penetrable membranes

in which the distribution of the membrane-fixed charges far inside the membranes

is uniform but may be arbitrary in the region near the membrane surfaces (see

Eq. (8.28)).

Our treatment is based upon the assumption that the Donnan potential of the

innermost layer of each membrane cð1ÞDON;1

�and cð2Þ

DON;1

�remains constant during

interaction. Let us examine the relative contributions from each layer of membrane

1 to the surface potential co1. From Eq. (16.7) it follows that the contribution from

the innermost layer, that is, a semi-infinite layer having the Donnan potential yð1ÞDON;1

occupying the region �1< x<�d1 is seen to be

1

2cð1ÞDON;1 exp �kdð1Þ1

� �ð16:10Þ

INTERACTION FORCE 377

The contribution of the ith layer, having a finite thickness dð1Þi � d

ð1Þi�1 ð2� i�

N� 1Þ and the Donnan potential cð1ÞDON;i, can thus be expressed as the difference in

contribution between two semi-infinite layers, both having the same Donnan poten-

tial cð1ÞDON;i, occupying the regions �1 < x < �d

ð1Þi and �1 < x < �d

ð1Þi�1, that is,

1

2cð1ÞDON;i exp �kdð1Þi

� �� exp �kdð1Þi�1

� �n oð16:11Þ

For the outermost layer (the Nth layer), we have

1

2cð1ÞDON;N 1� exp �kdð1ÞN�1

� �n oð16:12Þ

16.4 ISOELECTRIC POINTS WITH RESPECTTO ELECTROLYTE CONCENTRATION

Equations (16.10)–(16.12) show that owing to the factor exp �kdð1Þi

� �, the contri-

bution of the ith layer (1� i�N� 1) becomes negligible if kdð1Þi � 1. In other

words, the unperturbed surface potential co1 is determined mainly by the contribu-

tion from layers located within the depth of l/k from the membrane surface. This

implies that variation of the electrolyte concentration (and thus of l/k) changes therelative contributions to the unperturbed surface potentials from the respective lay-

ers of the interacting membranes so that the interaction force may also alter with

electrolyte concentration. It is of particular interest to note that if layers forming the

region within the depth of l/k are not of the same sign, the surface potential may

alter even its sign with changing electrolyte concentration. This implies the exis-

tence of isoelectric points with respect to variation of the electrolyte concentration.

Note, however, that the isoelectric point introduced here is the value of the electro-

lyte concentration at which the unperturbed surface potential becomes zero and

does not have the usual meaning that the total (net) amount of membrane-fixed

charges is zero.

It immediately follows from the existence of isoelectric points that if the two

interacting membranes have different isoelectric points, then the interaction force

alters its sign at these isoelectric points (note that the interaction force between

membranes with the same isoelectric point does not change its sign).

An example is given in Fig. 16.2, which shows the interaction force between two

membranes, each consisting of with two sublayers, in a monovalent symmetrical

electrolyte solution as a function of the electrolyte concentration. Here membranes

1 and 2 have different isoelectric points. It is seen that the interaction force is posi-

tive (i.e., repulsive) at concentrations less than 0.07M and higher than 0.18M and

negative (i.e., attractive) between 0.07 and 0.18M, corresponding to the isoelectric

points at 0.07M for membrane 1 and 0.18M for membrane 2. Membranes consist-

ing of more than two layers may exhibit more than one isoelectric point. The

378 ELECTROSTATIC INTERACTION BETWEEN NONUNIFORMLY CHARGED

interaction force between such membranes will show complicated behavior on vari-

ation of the electrolyte concentration. The present study has revealed that the sign

of the interaction force P(h) is determined not by the net or total amount of mem-

brane-fixed charges but, rather, depends strongly on the isoelectric points of the

interacting membranes. This behavior is not predicted by the conventional interac-

tion models assuming constant surface potential or constant surface charge density.

We would therefore suggest a possibility of selective aggregation in the system of

ion-penetrable membranes with different isoelectric points by using the difference

between their isoelectric points.

Equations (16.1) and (16.2) can be generalized to the case of arbitrary fixed-

-charge distributions in membranes 1 and 2. Let the membrane-charge distribution

be r(1)(x) in membrane 1 (�1< x< 0) and that in membrane 2 (h< x<þ1) be

r(2)(x), respectively. Then Eqs. (16.1) and (16.2) become

d2cdx2

¼ k2c� rð1ÞðxÞereo

; �1 < x < 0 ð16:13Þ

d2cdx2

¼ k2c� rð2ÞðxÞereo

; h < x < þ1 ð16:14Þ

FIGURE 16.2 Electrostatic force P between two ion-penetrable membranes, each consist-

ing of two layers, immersed in a monovalent symmetrical electrolyte solution as a function

of the electrolyte concentration n (M). The Donnan potentials at n¼ 0.1M in the respective

layers are as follows. cð1ÞDON;1 ¼ �20 mV, cð1Þ

DON;2 ¼ þ15 mV, cð2ÞDON;1 ¼ �10 mV, cð2Þ

DON;2 ¼þ10 mV, d

ð1Þ1 ¼ 0:1 nm, d

ð2Þ1 ¼ 0:5 nm, and h¼ 2.5 nm, er¼ 78.5, T¼ 298K. P> 0 corre-

sponds to repulsion and P< 0 to attraction. (From Ref. 1.)

ISOELECTRIC POINTS WITH RESPECT TO ELECTROLYTE CONCENTRATION 379

Equations (16.13), (16.14) and (16.3) under appropriate boundary conditions can be

solved to give

cðxÞ ¼ c1ðxÞ þ c2ðxÞ ð16:15Þ

where c1(x) and c2(x) are unperturbed potentials produced by membranes 1 and 2

in the absence of interaction and are given by

c1ðxÞ ¼1

2ereok

Z 0

x

e�kðt�xÞrð1ÞðtÞdt þZ x

�1e�kðx�tÞrð1ÞðtÞdt

� �; �1 < x < 0

ð16:16Þ

c1ðxÞ ¼ co1 e�kx; 0 < x < 1 ð16:17Þ

c2ðxÞ ¼1

2ereok

Z 1

x

e�kðt�xÞrð2ÞðtÞdt þZ x

h

e�kðx�tÞrð2ÞðtÞdt� �

; h < x < þ1

ð16:18Þ

c1ðxÞ ¼ co1 e�kðh�xÞ; �1 < x < h ð16:19Þ

with

co1 ¼1

2ereok

Z 0

�1ekxrð1ÞðxÞdx ð16:20Þ

co2 ¼1

2ereok

Z 1

h

e�kðx�hÞrð2ÞðxÞdx ð16:21Þ

where co1 and co2 are, respectively, the unperturbed surface potentials of mem-

branes 1 and 2. The interaction force P(h) per unit area between membranes 1 and

2 is given by the same equation as Eq. (16.9).

REFERENCE

1. R. Tokugawa, K. Makino, H. Ohshima, and T. Kondo, Biophys. Chem. 40 (1991) 77.

380 ELECTROSTATIC INTERACTION BETWEEN NONUNIFORMLY CHARGED