Recent Res. Devel. Electrochem., 9 (2013): 81-126 ISBN ... A3.pdf · processes occurring in...

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Review Article

Recent Res. Devel. Electrochem., 9 (2013): 81-126 ISBN: 978-81-7895-594-0

5. Electrochemical impedance spectroscopy

as a method for characterization of the

equilibria in bilayer lipid membranes and

between bilayers‟ components and ions

present in the aqueous solution

Monika Naumowicz Institute of Chemistry, University of Bialystok, Al. J. Pilsudskiego 11/4, 15-443 Bialystok, Poland

Abstract. The quantitative descriptions of various equilibria that

take place in lipid membranes, both natural and artificial, are

extremely important and essential for the interpretation of

processes occurring in biological membranes.

Physicochemical and electrochemical researches of lipid

membrane properties are carried out by various research groups

from many years. However, knowledge concerning the electrical

properties of bilayers and of influence of various factors on them

(e.g., the membrane composition and the electrolyte solution

composition) is still not enough in order to give complete

description of all functions of lipid membranes. Electrochemical

studies are particularly promising here.

Therefore, this chapter regards quantitative analysis of

definite models of biological systems. The presented researches of

biolipid interaction are concentrated on description of equilibria that

Correspondence/Reprint request: Dr. Monika Naumowicz, Institute of Chemistry, University of Bialystok

Al. J. Pilsudskiego 11/4, 15-443 Bialystok, Poland. E-mail: [email protected]

Monika Naumowicz 82

take part within the bilayer as well as on its surface. Assumed models of interaction

between amphiphilic molecules and the equilibria that take place there as well as

acid-base equilibria are described by mathematical equations for the systems studied.

These theoretical models are verified experimentally using electrochemical

impedance spectroscopy technique.

1. Introduction

The inspiration for lipid bilayer research, without question, comes from

the biological world. Because of the structural diversity of cell membranes, it

is extremely difficult to examine their properties. That is why scientific

experiments are carried out on simplified models usually composed of one or

more compounds. Although the first report on self-assembled bilayer lipid

membranes in vitro was reported in 1961 [1], experimental scientists

including surface, colloid, and bioscientists have been dealing with these

interfacial phenomena since Robert Hooke‟s time (1672).

The most popular cell membrane models are planar bilayer lipid

membranes (BLMs) which have many advantages. Primarily, they allow

studying the properties of a singular component of membranes. Secondly,

the structures of BLMs are well understood and theoretically described.

Furthermore, a composition of artificial lipid bilayers can be easily modified

during the experiment. Finally, the BLM system allows the exquisite

investigation of electrical properties (membrane potentials, electrical

resistance, current-voltage curves and electrical capacitance). Unlike most

other model membranes, BLMs are dynamic, ultrathin and liquid-crystalline.

Indeed, studies of the BLMs facilitate the initial testing of working

hypotheses, which have generated guidelines for a better choice of

reconstituted membrane experiments and have led to potential applications

[2]. Bilayer lipid membranes have been used in a number of applications

ranging from basic membrane biophysics studies to the conversion of solar

energy via water photolysis, and to biosensor development that uses

supported bilayer lipid membranes [3].

Cell membranes and their models are primary composed of amphiphiles,

a special class of surface-active molecules, which are characterized by

having a hydrophilic and a hydrophobic group in the same molecule.

Usually, zwitterionic or nonionic lipids are used as the basic lipids for the

preparation of bilayers. These lipids can be categorized into three principal

types: phospholipids, sphingolipids, and sterols. They each play different

roles in the membranes.

Phosphatidylcholine (PC) is the most abundant phospholipid in

mammalian cell membranes, constituting 40–50 % of total phospholipid [4].

Impedance of bilayer lipid membranes 83

The second most abundant mammalian membrane phospholipid is

phosphatidylethanolamine (PE), which constitutes 20–50 % of total

phospholipids [4]. Phosphatidylcholine and phosphatidylethanolamine differ

considerably in their effective sizes, hydration properties, and interhydrogen

bonding capability of their polar headgroups [5] as well as their intrinsic

curvatures [6] pertinent to the effective structure and intermolecular

interactions of lipid molecules in membranes. Although phosphatidylserine

(PS) is distributed widely among animals, plants, and microorganisms,

it is usually less than 10 % of the total phospholipids, with the greatest

concentration being in myelin from brain tissue. However, it may comprise

10 to 20 % of the total phospholipid in the cell membrane bilayer [7], where

it exerts important functions. Sphingolipids differ from phospholipids in

being based on a lipophilic amino alkohol (sphingosine) rather than glycerol.

Sphingomyelin (SM) is a major sphingolipid species (sphingolipids can

comprise 10-20 % mol of the plasma membrane). Ceramide (Cer) is the

biological building blocks of more complex sphingolipids. The cellular

concentration of ceramides can be quite high in reaching levels of

1–10 mol % of the total phospholipid concentration [8]. Cholesterol (Ch),

the major sterol present in plasma membranes of higher eukaryotes, can

comprise up to 50 mol % of the total plasma membrane lipids [9]. Ergosterol

(Erg) is the major component present in lower eukaryotes such as certain

protozoa, yeast, and other fungi, and in insects such as Drosophila [10].

The second dominating group contained in cell membranes are proteins

embedded in membranes; they are macromolecules that are responsible for

many functions essential for living, i.e. transporting of molecules across the

lipid membrane, enzymatic properties and interactions with the other cell

membranes.

In addition proteins and lipids, biological membranes also contain:

carbohydrates, vitamins, free fatty acids, carotenoids, lysolipids and many

other compounds that constitute extraordinarily important elements in the

biomembranes functioning. For example, -tocopherol ( -T) is the form of

vitamin E that is preferentially absorbed and accumulated in humans [11]. It

is believed to have two primary functions in cells: a major role is thought to

be as an antioxidant acting to prevent free radical damage to the tissues [12]

and a subsidiary function is to stabilize the structure of membranes [13].

Fatty acids and amines contain the same functional groups that are present in

natural membranes and are suitable for introduction into model membranes.

Their simple structures make them invaluable sources of information that

may be extrapolated to more complex biological membranes.

Most of the lipid bilayer studies concentrate on the surface potential [14]

or surface pressure measurements [15], spectroscopy [16] and microscopic

Monika Naumowicz 84

visualization of lateral domains [17]. The possibility to form lipid bilayers

between two aqueous solutions allows placing the electrodes in both aqueous

solutions being in contact with the membrane, and thus permits on utilization

of electrochemical techniques to study the properties of the lipid bilayers.

Thus, as is well known, biological membranes and black lipid membranes

used as model membranes can also be very well characterized by

electrochemical impedance spectroscopy [18-19]. Beginning in the 1970s,

research electrochemists and materials scientists began to discover the power

of electrochemical impedance as a tool for studying difficult and

complicated systems. Even today, it often provides the only non-invasive

method for detailed structural-functional studies of these systems [18]. This

is especially so of systems in which important processes occur at the

molecular level, such as those processes associated with biological and

synthetic membranes and interfaces that form between solutions and various

solids, e.g., metals and colloid particles [18].

Electrochemical impedance spectroscopy (EIS) may be used in the

characterization of membrane capacitance and resistance. Both these

electrical parameters are affected by such factors as medium pH and the

presence of certain substances, such as cholesterol, other lipids, fatty acids or

amines incorporated in the lipid bilayer. In this paper, the effect of various

agents on impedance parameters values of bilayer lipid membranes is

discussed.

2. Electrical capacitance and electrical resistance of single-

component bilayer lipid membranes

Capacitance is a characteristic and well-reproducible parameter of

BLMs. It is considered to be the best tool for probing the stability and formal

goodness of the lipid bilayer [20]. For this reason it is measured for every

bilayer, even when other properties are the main focus of the researches.

Membrane capacitance measurements are made for various purposes

including, for example, membrane thickness determinations [21], studies of

membrane stability and of its formation process [22], studies of surface

potentials [23], toxicological studies [24], investigation of photo-effects [25]

and studies of transport phenomena through the membrane [26].

Measurements of membrane capacitance can be performed by various

methods, differing in the way of capacitance determination and the type of a

measurement signal applied to the membrane: measurements with a sinusoidal

signal [27] or a triangular signal [28], pulse methods [29], compensation

methods [30], bridge methods [31], conversion of capacitance to frequency

[32] and measurements of the minimum capacitance potential [33].


Resistance (reciprocal of conductance) as an electrical property of

nonpermeabilized lipid bilayer can be measured only during application of

voltage or current signal [20]. Resistance may vary by at least one order of

magnitude, possibly because of impurities of the bilayer, border leakage at

the membrane support, the appearance of lipid “crystals” at the periphery of

the bilayer or way of introducing the lipid solution (if the forming solution is

introduced with a microsyringe, instead of a brush, the irreproducibility of

bilayer can be minimized). The resistance of a single membrane, however, is

usually constant until a short time before the membrane ruptures. Therefore,

any changes in resistance due to addition of ions, proteins, drugs, etc., can be

determined with a relatively high degree of accuracy [34]. Galluci et al. [35]

have developed the system for measuring conductance and capacitance

simultaneously and continuously as a function of time. This method allows

measurement of electrical properties of nonpermeabilized planar bilayers as

well as during the process of defect formation and electroporation

(electroporation is a significant increase in the electrical conductivity

and permeability of the membrane caused by an externally applied

electrical field).

Capacitance and resistance values of pure bilayer lipid membranes

formed from various components, obtained in our laboratory, are

summarized in Table 1. The EIS technique was used to characterize

the membrane features as this method has been shown to measure the

electrical parameters of BLMs accurately. The capacitance value for

phosphatidylcholine is equal to 0.62 ± 0.02 F cm-2

. This value has been

reported in the subject literature [1,20,36].

From the table above it can be concluded that values of resistance often definitely show greater scattering than values of capacity, and therefore

effects of measurements of the conductance are most often treated as supplementary data. The conductance is burdened with random errors caused by the presence of the solvent and ions in a bilayer. The presence of the solvent and ions always carries a meaningful, easily noticeable error as well as scattering to results. This effect is not occurring in such a visually perceptible manner in the capacity. There is no reliable literature data on capacitance and resistance values

for membranes formed from pure fatty acids, amines, -tocopherol and

ergosterol because these components do not form single-component

membranes. Therefore, the hypothetical values of the impedance parameters

for bilayers built from the above components were determined (values

presented in Table 1). In the case of decanoic acid and decylamine, the

capacitance hypothetical values for membranes built from these components

were determined adjusting the experimental curve with the polynomial of the

Monika Naumowicz 86

Table 1. Impedance parameters values of single-component bilayer lipid membranes.

Electrolyte, 0.1 M KCl.

other mark extrapolating the x2 = 1 value. The hypothetical capacitance value for membrane built from stearic acid was estimated by fitting the experimental curve with a straight line for four different molar fractions of fatty acid and extrapolating x2 = 1 value. The hypothetical values of the resistance for bilayers built from pure fatty acids and amines is not provided here because of the spread in the resistance measurements and the irreproducibility of the results. In the case of -tocopherol and ergosterol, the hypothetical values of the impedance parameters were evaluated from plateau values appearing at the end of the experimental curves.

3. Equilibria of 1:1 complex formation in bilayer lipid

membranes

In spite of a wide variety of experimental methods for the study of lipid bilayers, some long-lasting problems remain. One of them is the complex formation between two kinds of lipids in bilayers. Moreover, there is still little of the quantitative description of the lipid-lipid systems. It is required for a better understanding of the processes that take place in biological membranes with the aim of forming the artificial membranes that would very closely resemble the properties of the natural membranes. Therefore, the knowledge of molecular structure and organization of biolipids is necessary.


3.1. Theoretical considerations

One, two, or several membrane components can be contained in a forming solution used to obtain artificial lipid bilayers. These components may or may not form another compound in the membranes. The model, which has been presented in full detail previously [38,42,43] assumes that in the case, where the membrane components do not form chemical compounds, any two-component system, regardless of whether it forms a two-dimensional monolayer or a three-dimensional bilayer, can be described by the equations expressing additivity of the capacitance and the conductance:

1 1 1 2 2 2

s s

mC C c S C c S (1a)

and

-1 -1 -1

1 1 1 2 2 2

s s

mR R c S R c S (1b)

here:

11

1 2

s

s s

cx

c c (2)

1 2 1x x (3)

where:

mC [ F cm-2

] – the measured capacitance of the membrane referred to the

unit area of the membrane;

1 2C C, [ F cm-2

] – the capacitance of the membrane built by components 1

and 2, respectively, referred to the unit area of the membrane; 1-

mR [

-1 cm

-2] – the measured conductance of the membrane referred to the

unit area of the membrane; 1 1

1 2R ,R [

-1 cm

-2] – the conductance of the membrane built by components

1 and 2, respectively, referred to the unit area of the membrane;

1 2,s sc c [mol m-2

] – the surface concentration of components 1 and 2,

respectively, in the membrane;

Monika Naumowicz 88

1 2S S, [m2 mol

-1] – the surface area, occupied by one mole of components 1

and 2, respectively;

1 2x x, – the molar fraction of components 1 and 2, respectively.

After solution of the equations system (1)-(3), dependences of linear

type are derived:

21 1 2 2

1

- - -m m

SC C x C C x

S

(4a)

-1 -1 -1 -121 1 2 2

1

( - ) - ( - )m m

SR R x R R x

S

(4b)

Lipid membranes may also be assembled from two components capable

of forming a complex (compound 3). The stoichiometry of the complex may

vary, but because the first stability constant in complexes, as the most

essential one, is usually the largest and should be taken into consideration

[46,47], mainly the existence of 1:1 complex in the analyzed system was

assumed. The equilibrium between the individual components and the complex is presented by:

A + B AB

(Component 1) (Component 2) (Compound 3)

Then, the set of equations (1)-(3) is modified because the impedance

parameters (electric capacity and electric conductance) are the sum of the

contributions from each compound [38,42,43]:

1 1 1 2 2 2 3 3 3

s s s

mC C c S C c S C c S (5a)

1 1 1 1

1 1 1 2 2 2 3 3 3

- - s - s - s

mR R c S R c S R c S

(5b)

here:

3

1 2

s

R s s

cK

c c


1 31

1 2 32

s s

s s s

c cx

c c c (7)

1 1 3

s s s

tc c c (8)

2 2 3

s s s

tc c c (9)

1 2 1x x (10)

where:

C3 [ F cm

-2] – the capacitance of the membrane built by compound 3,

referred to the unit area of the membrane; 1

3R [

-1 cm

-2] – the conductance of the membrane built by compound 3,

referred to the unit area of the membrane;

3

sc [mol m-2

] – the surface concentration of compound 3 in the membrane;

1 2,s s

t tc c [mol m-2

] – the total surface concentration of components 1 and 2,

respectively, in the membrane;

3S [m2 mol

-1] – the surface area, occupied by one mole of compound 3;

RK [m2 mol

-1] – the stability constant of compound 3.

After solution of the equations system (5)-(10), the following basic

equations are obtained:

1 2 1 2 1 2 3 1 2 1 3 2 1 2 1 2 1 2- - - - - -m mC C B x C C B x C C B x C C B x C C x x

1

3 1 2 1 2 1 3 1 2 2 1 2 3 2 1

-

R m m m mK S B B C -C x - x C -C B x C -C x - x (C -C )B x (11a)

and

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1

1 2 1 2 1 2 3 1 2 1 3 2 1 2 1 2 1 2- - - - - -m mR R B x R R B x R R B x R R B x R R x x

1 1 1 1 1 1 1 1 1

3 1 2 1 2 1 3 1 2 2 1 2 3 2 1

- - - - - - - - -

R m m m mK S B B R - R x - x R - R B x R - R x - x R - R B x (11b)

in which:

Monika Naumowicz 90

31

1

SB =

S and 3

2

2

.S

B = S

Eqs. (11) are equations of second degree with respect to mC and -1mR , to

the complex composition as well as with respect to the constants: -1 -1 -1

1 2 3 1 2 3 1 2, , , , , , ,C C C R R R B B . Opening of parentheses results in a great

complexity of the equations, and it is troublesome when directly applied to

the determination of constants. The capacitance and resistance values of the

membranes formed from pure components 1 and 2 can be measured directly

(or 2C and -12R can be determined in such a way as described in Section 2). The

remaining constants can be determined in individual cases using simplified

forms of these equations.

Eqs. (11) may be simplified taking into account the sufficiently high value

of the stability constant of the complex (in limit → ∞) and approximating x2 to

low or high values. The criterion of rightness of the accepted assumption is the

agreement between theoretical and experimental values.

Compound 1 and the complex are contained in the two-component lipid

membrane for low x2 values (x2 0). These compounds determine in this

case the values of function of capacitance and conductance. Therefore,

Eqs. (11) may be presented as the following straight lines:

1 2

1 1 3 1

2

-- -m m

x xC C B C B C

x

(12a)

-1 -1 -1 -11 21 1 3 1

2

-- -m m

x xR R B R B R

x (12b)

Compound 2 and the complex are contained in the two-component lipid

membrane for high x2 values (x2 1). In this case only these two

compounds determine the values of capacitance and conductance functions.

This is why Eqs. (11) may be written as other straight lines:

2 12 2 3 2

1

-- -m m

x xC C B C B C

x (13a)

-1 -1 -1 -12 12 2 3 2

1

-- -m m

x xR R B R B R

x (13b)


When calculating the stability constant for the complex, Eqs. (11) can be

simplified to 1 2x x :

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

2 1 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 3- - -m mC S C S C S S C S C S C S C S C S S S S C

2-1 2 -1 2

1 2 3 3( ) ( ) -R mK S S S C C (14a)

and

-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 12 1 1 2 1 2 2 1 1 2 2 1 1 2 1 2 1 2 3- - -m mR S R S R S S R S R S R S R S R S S S S R

2-1 2 -1 2 -1 -11 2 3 3( ) ( ) -mRK S S S R R (14b)

The parameters describing the complex ( -13 3, ,RK C R , and

3S ), determined

using Eqs. (12)–(14), may be used to calculate theoretical points using the

equation presented below (agreement between theoretical and experimental

values implies that the system is well described by the above equations):

(15a)

where:

and

(15b)

in which:

-1 -1 -1 -1 -1 -1 -1 -1 -1

4 3 1 2 2 1 1 3 1 2 2 3 2 1- - - -a S R R x x R R x S R R x S

Monika Naumowicz 92

For bilayer membranes assembled from two components, 1:1 complex

formation may be assumed to be the explanation for deviation from the

additivity rule. Model curves can be constructed using calculated parameters

such as equilibrium constants, molecular areas of the complexes, electrical

capacitance and electrical conductance of molecules and complexes. The

accuracy of the models is then verified by comparison to experimental

results.

3.2. Lipid-lipid systems

The phospholipid-cholesterol interactions have been widely studied

since 1925 [48] because of their importance from a biological point of view.

Cholesterol is a significant factor in controlling physical properties of cell

membranes and their functions [49]. As a result of a growing interest in

biological membranes, many studies are directed at the cholesterol-

phospholipid system [50]. Cholesterol increases both the order of the

hydrocarbon chains (an ordering effect) and the surface density of the membrane

(condensing effect) in a model and in biological membranes [51,52]. Ordering

and condensing effects are not limited to specific phospholipids, they are

shown by phosphatidylcholines, phosphatidylethanolamines and also

phosphatidic acid, sphingomyelin, phosphatidylglycerol or other derivatives of

phospholipids [53].

There have been a number of proposals for existence of the complexes at

different stoichiometries formed from two kinds of lipids. However, there is

no agreement concerning what stoichiometries of the formed complexes

have. The 1:1, 1:2, 2:1, 1:3 and 3:1 complexes are mostly claimed to be in

bilayers [54,55]. On the other hand, subsequent investigators have suggested

that specific associates [56], phase separation [57], domains [58] or lattice-

based structures [59] could be formed in the PC–Ch membranes.

In this paragraph, a 1:1 complex has been proposed to exist in bilayer

composed of two kinds of lipids. It was verified experimentally using

electrochemical impedance spectroscopy technique. Area, capacitance,

conductance, and stability constant values for the complexes were calculated

based on the model assuming 1:1 stoichiometry, and the model was

validated by comparison of these values to experimental results. Description

of the complex formation equilibria is presented below on the example of

phosphatidylcholine bilayers modified with ceramide.

Typical impedance spectra of the PC, PC-Cer (1:1 molar ratio) and Cer

membranes are presented in Fig. 1. Very simple impedance diagrams were

obtained for all the examined membranes. They have the form of semicircles

in the entire analyzed frequency range. Their centers lie on the real axis,


provided that the lipid bilayers are considered as dielectric layers with

leakage. The pure PC bilayers have different impedance than the PC-Cer

membranes, confirming that ceramide has been successfully incorporated

into the lipid bilayers and has a contrast effect on the electrical properties of

the membranes. It caused both capacitance of the membrane Cm and resistance

of the membrane Rm to decrease. The complex nonlinear least squares fits

(according to the equivalent circuit illustrated in Fig. 2) are represented by

solid lines and are in good agreement with the data obtained.

0.0E+00

4.0E+04

8.0E+04

1.2E+05

0.0E+00 4.0E+04 8.0E+04 1.2E+05 1.6E+05 2.0E+05 2.4E+05

Z' / cm2

-Z"

/ c

m2

phosphatidylcholine phosphatidylcholine-ceramide (1:1 molar ratio) ceramide

0.E+00

3.E+03

0.E+00 6.E+03

Figure 1. Dependence of an imaginary part −Z” on the real part Z‟ for a membrane

made of phosphatidylcholine, phosphatidylcholine-ceramide and ceramide. The solid

lines represent the results of the fitting procedure.

R0

Rm

Cm

Figure 2. The equivalent circuit used for impedance data analysis: R0 represents the

resistance of the electrolyte, Cm is the capacitance of the membrane and Rm is the

resistance of the membrane.

Monika Naumowicz 94

The validity of the results obtained by electrochemical impedance

spectroscopy was verified using electrical equivalent circuit model which is

presented in Fig. 2. This equivalent circuit consists of a parallel arrangement

of capacitance mC and resistance mR , attributed to the electrical properties

of the bilayer, completed by a serial resistance 0R for the conductivity of the

bulk. The possibility of misinterpretation of the recorded data is reduced by

the simplicity of the circuit. This electric circuit is characteristic for an

artificial lipid membrane only when ionophore systems, specific channels-

pores and adsorption are absent [60]. The electrochemical parameters of the

circuit were evaluated by employing the ZsimpWin software. A very high

correlation was observed between experimental results and the results

calculated with the best fitting electrical equivalent circuit model, where χ2

was minimized below 10-3

. An examination of the data obtained for analyzed

systems indicate that the proposed equivalent circuit can be used to describe

the experimental results (Fig. 1).

Dependences of the capacitance and the conductance of phosphatidylcholine

(component 1) - ceramide (component 2) membranes are illustrated in Figs. 3a

and 3b as functions of the molar fraction of ceramide. The resulting dependences

deviate from linearity indicating that specific interactions between membrane

components are presented in the membrane. The capacitance and the

conductance values obtained for a pure PC bilayer are equal to 0.62 0.02

F cm-2

and (4.35 0.91) 10-6 -1

cm-2

, respectively. The capacitance and the

conductance values for a pure Cer bilayer are equal to 0.17 0.01 F cm-2

and

(3.29 1.35) 10-4

-1

cm-2

, respectively. Figs. 3a and 3b also show the

theoretical values, marked by the curve, obtained from Eqs. (15a) and (15b)

(describing the complex formation process). These theoretical values were

calculated using values whose determination will be described in further

parts of the paper. It can be seen from these figures that the theoretical

values agree well with the experimental data in the whole analyzed

concentration range which suggests the existence of PC-Cer complex in the

examined membranes.

Figs. 4a and 4b present graphs (Cm -C1)x1 versus -(Cm –C2)x2 and -1 -1

1 1( - )mR R x versus -1 -1

2 2-( - )mR R x described by Eqs. (4a) and (4b),

respectively. The dependences are expressed in the coordinate systems in

which the plots should yield straight lines when the membrane components

do not interact. Their actual shapes prove that they do not correspond to

Eqs. (4a) and (4b), suggesting that there are specific interactions in the

analyzed bilayer. Therefore, the formation of a complex in this system was

assumed. Because the existence of a 1:1 complex is a typical case [46], the

formation of a 1:1 PC-Cer complex was accepted.


a)

0.00

0.20

0.40

0.60

0.80

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x 2

Cm

/F

cm

-2

experimental points theoretical curve

b)

1.0E-06

1.0E-05

1.0E-04

1.0E-03

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x 2

Rm

-1/

-1 cm

-2


Figure 3. Dependences of capacitance Cm (a) and conductance Rm–1 (b) of the

phosphatidylcholine-ceramide bilayers on the molar fraction of ceramide x2. Error

bars indicate the experimental scatter. The experimental values are denoted by points

and the theoretical ones, calculated according to Eqs. (15), by curves.

Consequently, Eqs. (5a) and (5b), and the stability constant KR,

describing a complex formed in this system, complete the theoretical

description. After simple modifications of Eqs. (5a) and (5b), one can obtain

information of great interest from our point of view, presented by Eqs. (11a)

and (11b). Assuming that the value of the stability constant of the PC-Cer

complex was sufficient with respect to the simplified Eqs. (11) to Eqs. (12)

and (13), the constants B1, B2, C3, R3-1

were obtained.

Monika Naumowicz 96

a) -0.08

-0.06

-0.04

-0.02

0.00

-0.30-0.20-0.100.00

-(C m -C 2)x 2/ F cm-2

(Cm

-C1)x

1/

F c

m-2

x2

x2

b)

0.0E+00

1.0E-05

2.0E-05

3.0E-05

4.0E-05

5.0E-05

-3.0E+04-2.0E+04-1.0E+040.0E+00

-(R m-1

-R 2-1

)x 2/-1

cm-2

(Rm

-1-R

1-1

)x1/

-1 c

m-2

x2

Figure 4. Plots representing Eqs. (4): Cm, Rm–1 - capacitance and resistance of the

membrane, respectively; C1, R1–1 - capacitance and resistance of the pure

phosphatidylcholine membrane, respectively; C2, R2–1 - capacitance and resistance of

the pure ceramide membrane, respectively; x1, x2 - molar fraction of the

phosphatidylcholine and the ceramide, respectively. The arrows denote the direction

of the increasing x2 values and the dashed lines indicate the order of points.

Figs. 5a, 5b, 6a and 6b present the plots of functions (12a), (13a), (12b)

and (13b), respectively. The presented dependences are transformed into

straight lines when KR is high and the values x2 are low (Figs. 5a and 6a) or

the values x2 are high (Figs. 5b and 6b). If at least three successive points fall

on the same straight line, it is reasonable to assume that the conditions for

simplification of Eqs. (11) have been met and straight line passing through

these three points may be described by Eqs. (12) and (13).


a)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.0 0.2 0.4 0.6 0.8

C m F cm-2

(C1-C

m)x

1-x

2/x

2/

F c

m-2

0.58 0.60 0.62

b)

-1.0

-0.7

-0.4

-0.1

0.2

0.5

0.0 0.2 0.4 0.6 0.8

C m / F cm-2

(C2-C

m)x

2-x

1/x

1/

F c

m-2

Figure 5. Plots illustrating the dependences for phosphatidylcholine-ceramide

complex described with Eq. (12a) (a) and Eq. (13a) (b). Straight lines join the points,

from which parameters B1, B2, S3 and capacitance C3 can be determined.

Points fulfilling both the aforementioned limitations of x2 values and

forming straight lines are joined together in Figs. 5 and 6. From the B1 and

B2 constants, which were determined on the basis of these equations, it was

possible to calculate the capacitance value of the complex C3 and the

conductance value of the complex R3-1

.

The values of B1 and B2 were determined from the slopes of the lines. The

intersections of the straight lines with the ordinate provide -B1C3 and -B2C3

and -B1R3-1

and -B2R3-1

, which can be used to determine C3 and R3-1

. The mean

Monika Naumowicz 98

a)

-8.0E-05

4.0E-05

1.6E-04

2.8E-04

0.0E+00 1.1E-04 2.2E-04 3.3E-04

R m-1

/-1

cm-2

(R1

-1-R

m-1

)x1-x

2/x

2/

1 c

m-2

5.0E-06 1.0E-05 1.5E-05

b)

-4.0E-04

5.0E-05

5.0E-04

0.0E+00 1.0E-04 2.0E-04 3.0E-04

R m-1

/-1

cm-2

(R2

-1-R

m-1

)x2-x

1/x

1/

-1 c

m-2

Figure 6. Plots illustrating the dependences for phosphatidylcholine-ceramide

complex described with Eq. (12b) (a) and Eq. (13b) (b). Straight lines join the points,

from which parameters B1, B2, S3 and conductance R3-1 can be determined.

values obtained from these points equal to 0.58 ± 0.01 F cm-2

and

(1.30 ± 0.13) 10-5

-1

cm-2

, respectively.

Eqs. (12) and (13) could also be applied to calculate the surface area per

a single complex molecule S3. The values of the surface area, occupied by one

mole of components 1 and 2, are necessary for this calculation. The surface

area occupied by the PC molecule depends on the way the phospholipid is

prepared, because this affects the length, conformation and degree of

unsaturation of the fatty acids chains. Therefore, the values in the literature

range between 54 and 99 Å2 [61,62]. In our case, we chose S1 value,

determined in our laboratory as 85 Å2 [63]. The surface area occupied by Cer

molecule, reported in the literature, is equal to 50 Å2 [64]. The resulting S3

value amounts to 75 Å2 molecule

-1 and is lower than the sum of areas

occupied by each component of the complex (135 Å2).


The stability constant of the complex could be determined from

Eqs. (14a) and (14b) by setting 1 2 0.5x x . The resulting mean value of

the stability constant is (8.76 ± 0.42) 107 m

2 mol

-1.

The parameters determined on the basis of Eqs. (11) and (14) were

applied to represent the agreement of the data, evaluated from Eqs. (11)

(solid lines) with the experimental data (points) in Figs. 3a and 3b using

Eqs. (15). Both of Eqs. (15) can yield two solutions, as they are of a

second order. The values ensuring better agreement of the experimental

points with the predicted ones by the equations describing the complex

formation between membrane lipid components were chosen. It can be seen

from the Figs. 3a and 3b that the agreement between experimental and

theoretical points is good. This agreement means that our theoretical model

(presented in Section 3.1) is sufficient to describe the analyzed system. This

agreement also justifies the statement that other complexes do not represent

a significant component of this system. Still, the fact remains that we cannot

detect other complexes based on our experimental data.

Table 2 lists several physicochemical parameters for 1:1 complexes

existing in BLMs. These parameters were determined in our laboratory using

electrochemical impedance spectroscopy.

Table 2. Selected physicochemical parameters for lipid-lipid complexes (1:1) in

bilayer lipid membranes. Electrolyte, 0.1 M KCl.

Monika Naumowicz 100

Analysis of the results presented in the table leads to the following

conclusions:

1. The values of stability constant of all analyzed complexes are

relatively high, providing additional support for the prevalence of

1:1 complexes in mixed bilayers. These values also confirm that the

assumptions used to simplify Eqs. (11) were correct.

2. The stability constant of the Ch-containing complexes are higher

(besides SM-Ch complex) than of complexes without cholesterol in

their structures indicating that these complexes are more stable.

3. The experimentally obtained values for the surface area occupied

by the PC-Ch and PE-Ch complexes 3

eS are lower than values of the

surface area 3

tS calculated for both molecules; 3

eS and 3

tS values in

the case of SM-Ch complex are the same. These result confirm that

addition of cholesterol to membranes formed from phospholipids

resulted in increased stability and reproducibility. Cholesterol

condenses some membrane components, making the membrane

structures more rigid. Ch also improves the packing of membrane

lipids as, unlike sphingolipid molecules, it preferentially occupies

voids in the hydrophobic layer of the membrane rather than spaces

in the polar region. 4. The 3

eS values for PC-Cer and PS-Cer complexes are also lower than values of 3

tS calculated for these complexes. It is interpreted as condensing effect of Cer, which is in agreement with the subject literature [65-67].

5. Good agreement between the experimental and the theoretical points verifies the assumption of existence of 1:1 complex in the lipid bilayer. The lack of variation between theoretical and experimental points indicates that our theoretical model is sufficient to describe the interaction in analyzed systems. The agreement between the experimental results and the model predictions for the PC-Ch, PE-Ch, SM-Ch, PC-PE, PC-Cer and PS-Cer membranes justifies the statement that other complexes do not represent a significant component of these systems.

6. The mathematically derived and experimentally confirmed results

presented here are of great importance for the interpretation of

phenomena occurring in lipid bilayers. These results can help lead

to a better understanding of the physical properties of biological

membranes. The simple and very interesting methods proposed in

this Section may be used with success to determine the equilibrium

constant values of 1:1 complexes in any lipid–lipid systems.


3.3. Lipid-fatty acid and lipid-amine systems

Amphiphiles such as fatty acids or amines are known to strongly modify

the structures of biological membranes and their functions such as transport

[68] or enzyme adsorption [69]. These effects may be investigated by

examining the molecular interaction of fatty acids or amines with

phospholipids in the bulk [70,71] or in monolayers [72]. Calorimetric studies

[70,71] of phospholipid–fatty acid mixtures in the bulk phase have shown

that saturated fatty acids partition preferentially into the solid-like domains of

the lipids; a 1:1 or 1:2 complex formation was also suggested.

The dependence of lipid membrane capacitance on composition for

PC-decanoic acid (DA), PC-decylamine (DE), and PC-stearic acid (SA) systems

were studied over a possible concentration range. The results are depicted in Fig. 7.

These dependences were executed in the function of the composition to 0.52

molar fraction of the decanoic acid, to 0.88 molar fraction of the decylamine

and to 0.80 molar fraction of the stearic acid, because only to such contents

of component 2 (decanoic acid, decylamine, stearic acid) the bilayer

membrane formation was possible.

The conductance values were treated as supplementary data and were

not used for equilibrium calculations.

The values of C3 for the PC-DA (0.33 F cm-2

), PC-DE (0.94 F cm-2

),

and PC-SA (1.52 F cm-2

) complexes were calculated using Eqs. (12a) and

(13a). Equations (12a) and (13a) could also be used to calculate the areas

occupied by PC-DA, PC-DE, and PC-SA complexes (the surface area

occupied by decanoic acid, decylamine, and stearic acid molecule is 22 [72],

20 [72], and 19 Å2

[73], respectively), which are 239, 152, and 122 Å2,

respectively. Obtained values of the surface areas are much larger than the

sum of the surface area occupied by each component of the complexes. It is

probably connected with the arrangement of PC molecules in the complex

and also connected with the structural construction of such complexes.

The stability constants of the analyzed complexes were determined

using Eq. (14a). The KR of PC-DA, PC-DE, and PC-SA complexes are

2.82 105, 2.06 10

7 and 2.87 10

7 m

2 mol

-1, respectively. The KR values

obtained for PC-DE and PC-SA complexes are relatively high, giving

evidence for the prevailing of the 1:1 complex in bilayers composed from

phosphatidylcholine and decylamine or stearic acid. These values also confirm

that the assumptions used to simplify Eq. (11a) were correct.

The verification of the assumption about the formation of 1:1 complexes in

the PC-DA, PC-DE, and PC-SA bilayers is shown in Fig. 7. The experimental

values are marked by points and compared with the values calculated from

Eq. (15a). These calculated („„theoretical‟‟) results are presented as the solid


a)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6

x 2

Cm

/F

cm

-2

b)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

x 2

Cm

/F

cm

-2

c)

0.0

0.4

0.8

1.2

1.6

0.0 0.2 0.4 0.6 0.8 1.0

x 2

Cm

/F

cm

-2

Figure 7. Dependence of capacitance Cm of the phosphatidylcholine membranes

modified with decanoic acid (a), decylamine (b) and stearic acid (c) on the molar

fraction of decanoic acid, decylamine or stearic acid x2. Error bars indicate the

experimental scatter. The solid line represents the theoretical values calculated from

Eq. (15a).


lines. It can be seen that, in the case of PC-DE and PC-SA membranes, the agreement between experimental and calculated points is good. This agreement means that our theoretical model (presented in Section 3.1) is sufficient to describe the interaction in these systems. Obtained results indicate that a 1:1 complex with high stability constant forms in mixed membranes of PC and DE or SA, similar to the complex between PC and unsaturated fatty acids containing a trans or cis double bond in the acyl chain [74]. The results also verify our choice of C2 value for component of the membrane. The small variances between the experimental and the theoretical capacitance values indicate that complexes of different stoichiometries or associates are also possible in the PC-DE and PC-SA bilayers.

On the contrary, the lack of agreement between the experimental results and values calculated from Eq. (15a) for PC-DA bilayers indicates that complexes of stoichiometry other than 1:1 are present in these membranes. Presumably, the existence of a 1:2 complex PC-DA is the most probable explanation, as in the case of the mixtures of PC with longer-chain (C12–C20)

fatty acids [71,75]. During the course of presented investigations, the formation of PC-DA, PC-DE, and PC-SA complexes with 1:1 stoichiometry was assumed. These complexes arise by producing a connection between the −N

(+)(CH3)3 group from PC molecule and –COO

(−) groups from DA

(or SA), in the case of the complex PC-decanoic acid (or PC-stearic acid), and between the –PO

(−) group from phosphatidylcholine and −N

(+)H3 group

from decylamine. The dissociation constants of the −N(+)

(CH3)3 group from PC and –COO

(−) groups from the decanoic acid are equal 10

−5.7 [63] and

about 10−5

[76], respectively. It should be noted that the dissociation constants of –PO

(−) group from PC and −N

(+)H3 group from DA are equal

10−2.6

[63] and about 10−10

[76], respectively. Therefore the connection between PC and decanoic acid will be stronger and it is possible to expect

that the stability constant of the PC-DA should be higher than the stability constant of the PC–DE complex. Thus, in order to correctly characterize the interactions between phosphatidylcholine and decanoic acid in the bilayers, complexes with stoichiometry other than 1:1 should be taken into consideration during derivation of mathematical formulas. Next, suitable modifications should be applied in equations.

4. Equilibria of domain formation in bilayer lipid membranes

Lipid lateral organization is an important issue in membrane biology.

There is increasing evidence that functional lipid domains of micrometer

sizes exist in biological membranes under physiological conditions [77,78],

but relatively less is known about lipid organization at the molecular level

inside and outside those domains.


4.1. Theoretical considerations During formation of a two-component lipid membrane, the spatial

regionalization of components can occur in various lengthscales from

nanometer to micrometer scale. It is due to specific interactions between

membrane components resulting in the appearance of membrane regions of

diverse chemical character, structure, and functions. Such specialized

structures of various sizes have been called domains [79]. The equilibrium of

domain formation can be described in terms of such physicochemical

parameters as electric capacity and electric conductance. Let us assume that,

in the domain (compound 3) formation process in a two-component lipid

membrane, every molecule of component 2 is surrounded by certain,

possibly to determine, quantity of component 1. The equilibrium state of the

discussed system is described by a previously published set of equations

[44,45,80]:

1 1 1 3 3 3

s s

mC C c S C c S (16a)

and

-1 -1 -1

1 1 1 3 3 3

s s

mR R c S R c S (16b)

in which:

11

1 3

s

s s

cx =

c c (17)

1 1 3 3 1s sc S c S (18)

where -1

3 3 3, sC ,R c and

3S symbols were described in Section 3.1.

Knowing that 1 3 1x x , where x3 is the molar fraction of compound 3,

and eliminating 1

sc and 3

sc , the above system of equations can be presented as:

1 1 3 3 1 1 3

1 3 1 3

( )

( )m

C S C S C S xC

S S S x (19a)


-1 -1 -1-1 1 1 3 3 1 1 3

1 3 1 3

( )

( )m

R S R S R S xR

S S S x (19b)

The systems of Eqs. (1)-(4) and (16)-(19) are mathematically equivalent,

although the parameters for component 2 and compound 3 are different. However,

Eqs. (1)-(4) refer to the case where no domains exist and Eqs. (16)-(19) refer to the

case where domains are created. Therefore both cases are considered separately.

In consequence, the final Eqs. (4a)-(4b) and (19a)-(19b) are presented in the form

suited for a discussion of the results obtained.

Eqs. (19) are quotients of polynomials. Dividing the numerator of each

quotient by its denominator yields a series of increasing exponents of the

power of molar fraction, x3. Further, taking into account two first terms of

each series results in linear expressions, which are correct at low molar

fractions (for 03x ):

-1 -1 -1

3 1 3 3 1 1 3( )mC x C x C C S S (20a)

-1 -1 -1 -1 -1 -1 -1

3 1 3 3 1 1 3( )mR x R x R R S S (20b)

4.2. Phospholipid- -tocopherol and phospholipid-ergosterol

domains Based on derived mathematical equations, a simple method for

calculation of the surface area of the domain between phospholipid and

alpha-tocopherol or ergosterol was proposed. This method can be used with

success for the quantitative determination of area and describe the

stoichiometry of domains formed in any two-component system.

The dependences of impedance parameters on composition for

PC-ergosterol (Erg), PC-alpha-tocopherol ( -T), and PE- -T membranes

were studied over a possible concentration range. These dependences were

executed in the function of the composition to 0.11 molar fraction of the Erg

for PC-Erg membrane, to 0.33 molar fraction of the -T for PC- -T

membrane and to 0.27 molar fraction of the -T for PE- -T membrane,

because only to such contents of component 2 (ergosterol or -tocopherol)

the bilayer membranes formation was possible. The new method developed

by us is presented below on example of PC-Erg domains.

Fig. 8 shows typical impedance plots of chosen phosphatidylcholine

membranes, pure and containing different amounts of ergosterol. For the sake

of clarity, spectra for some molar fractions have been omitted (otherwise the


0.0E+00

5.0E+05

1.0E+06

0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06

Z'/ cm2

-Z"

/ c

m2

0.00 0.06 0.07 0.08 0.09 0.11

Figure 8. Complex plane impedance diagrams of phosphatidylcholine bilayers

modified with ergosterol. A different content of ergosterol (expressed as a molar

fraction) is illustrated by the different point‟s shapes of the impedance spectra. The

solid lines represent the results of the fitting procedure.

figure would be illegible by superimposed spectra caused by too little

differences in the impedance parameters values). The equivalent circuit used

for data analysis of the system in which domain formation is assumed is the

same as the equivalent circuit used for the system where 1:1 complex

formation appears (presented in Fig. 2).

Dependences of the capacitance and the conductance of the PC-Erg

bilayer on the molar fraction of ergosterol are presented in Figs. 9a and 9b,

respectively. The resulting curves deviate from linearity indicating that some

bonds are formed in the membrane.

In Figs. 9, points denote experimental values and solid lines are

calculated on the basis of Eqs. (19a) and (19b) (describing the domain

formation process) using values whose determination will be presented in

further parts of this paper. The theoretical values agree well with the

experimental data in the range 0.00-0.06 molar fraction of ergosterol

suggests the existence of PC-Erg domains with a defined stoichiometry and a

constant area in this region. Erg had significant effect on the capacitance and

the conductance membranes up to 0.09 molar fraction, Cm and 1-

mR reached a

plateau value when the ergosterol molar fraction was over 0.09. At a molar

fraction of ergosterol in the range 0.06-0.09, the lipid composition and

physicochemical properties of PC-Erg domains change in comparison with

the domains formed in the range 0.00-0.06. The C1 and 1

1R values obtained

for a pure bilayer of phosphatidylcholine are equal to 0.62 F cm-2

and


4.35 10-6

-1

cm-2

, respectively. The C2 and 1

2R values for a pure bilayer of

Erg (evaluated from plateau values) are equal to 0.37 F cm-2

and

5.12 10-7

-1

cm-2

, respectively.

Fig. 10 presents the dependences illustrating the Eqs. (20) in the whole

analyzed molar fraction of the Erg range. Provided that a domain is formed, the plots of Eqs. (20) show straight lines. The six points obtained for the

lowest concentrations of ergosterol lie on ideal straight lines. These points

correspond to the ergosterol content up to 0.06 molar fraction and confirm,

that in this molar fraction range there are created domains with a defined

stoichiometry and a relatively constant area.

a)

0.30

0.40

0.50

0.60

0.70

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11

x 2

Cm

/F

cm

-2

experimental points teoretical curve

b)

1.0E-07

1.0E-06

1.0E-05

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11

x 2

Rm

-1/

-1 c

m-2


Figure 9. Dependence of capacitance Cm (a) and conductance 1-

mR (b) of the

phosphatidylcholine-ergosterol membrane on the molar fraction of ergosterol x2. The

solid line represents the theoretical values calculated from Eqs. (19).


0.0E+00

2.0E+01

4.0E+01

6.0E+01

8.0E+01

0.0E+00 3.0E+01 6.0E+01 9.0E+01 1.2E+02

x 3-1

Cm

x3

-1/

F c

m-2

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

Rm

-1 x

3-1

/-1

cm

-2

experimental points theoretical curves

Figure 10. A plot illustrating Eqs. (20), from which the surface area of

phosphatidylcholine-ergosterol domain can be determined Cm represents capacitance

of the membrane, -1

mR conductance of the membrane and x3 molar fraction of the

domain. Solid lines represent the theoretical lines calculated according to Eqs. (20).

The Eqs. (20) can be written in the form: y = ax + b. The a and b

coefficients were determined for six of the least (smallest) molar fractions of

Erg using a linear regression. Next, these coefficients were applied to present

the agreement of the Eqs. (20) data (solid lines) with the experimental data

(points) in Fig. 10. The slope values of the straight lines are equal to

C1 and -1

1R , respectively, and are in agreement (in deviation limits) with

experimental values obtained for a pure PC membrane. The intersections of

the straight lines with y-axes yield the -

3( )C C S S13 1 1 and -1 -1 -1

3 1 1 3( )R R S S ,

respectively, which allow one to determine an area occupied by one domain

(denoted by S3). Knowing the S1 (85 Å2), C1 (0.62 F cm

-2) and C3

(0.37 F cm-2

), as well as -1

1R (4.35 10-6

-1

cm-2

) and -1

3R (5.12 10-7 -1

cm-2

),

the area occupied by one PC-Erg domain could be determined. The resulting

S3 values were 243 Å2 from capacitance measurements and 281 Å

2 from

conductance measurements, which gave the mean value amounting to 262 Å2.

Taking into account the surface area occupied by one Erg molecule

(equal to 38.5 Å2 [81]), the surface area occupied by one PC molecule

(mentioned above), the mean experimental surface area occupied by one

PC-Erg domain determined by us and the possibility of existence of

condensation and ordering effects, we suggest that the stoichiometry of

PC-Erg domain is equal to 3:1. The mean experimental S3 value is higher

than the sum of areas per two PC and one Erg molecule (208.5 Å2) but lower

than the sum of areas per three PC and one Erg molecule (293.5 Å2).


Table 3 contains the values of surface area of both components of the

domains formed in analyzed two-component systems. These values are taken

together with values of area and stoichiometry of the domains.

Deviation of the surface area of all analyzed domains calculated on the

basis of experiments from the theoretical values show a negative value, which

means that a reduction in the surface area occurs as a result of component

mixing. This is thus a condensation effect observed between the

phospholipids and alpha-tocopherol or ergosterol, which indicates that

-T and Erg can work as a reinforcer for phosphatidylcholine and

phosphatidylethanolamine bilayers. The condensation can be attributed to

area changes within the PC and PE when these lipids are mixed with -T or

-T and Erg are rigid molecules whose area does not significantly

change when PC- -T and PE- -T bilayers are formed. This is in agreement

with the model of intermolecular interactions, which assumes that at a

constant surface pressure the area per rigid molecule (e.g., sterol) is also

constant. The condensation effect of -T and Erg is related to the decrease of

the area per phospholipid molecule. The surface area occupied by one PE

molecule within the PE- -T domain amounts to about 65 Å2. Alpha-

tocopherol causes similar condensation effect on the phosphatidylcholine

bilayer; it reduces the molecular area of the PC from 85 to 77 Å2. The surface

Table 3. Selected physicochemical parameters of domains occurring in examined

bilayer lipid membranes. Electrolyte, 0.1 M KCl.


area occupied by one PC molecule within the PC-Erg domain amounts to 75

Å2, whereas the presence of cholesterol in the PC bilayer can reduce the

molecular area of the phosphatidylcholine to 56 Å2 [34].

5. Acid-base equilibria between lipid membranes and

electrolyte solution

The studies on acid-base equilibria between the BLMs and solutions

around membranes are of great importance in understanding the phenomena

which take place in living organisms. The pH inside the human body varies

from about 1 to 8 [82-84]. The pH solution impacts many membrane

parameters, such as dipole and zeta potential [85], bending stiffness [85] or

surface tension [86]. It is well-known that many membrane-mediated

processes are affected by pH changes, e.g. phase transition between gel and

liquid-crystal [87], acid induced membrane fusion [88] or drug – membrane

interaction [89]. Although literature is full of examples on how pH affects

model and biological membranes there are still only few publications which

describe the impact of pH on electrical properties of lipid bilayers [90,91].

5.1. Theoretical considerations

A phosphatidylcholine molecule contains groups that are positively and

negatively charged, with net electrostatic charge of zero. Representative

negatively charged group of phosphatidylcholine includes the phosphate

group, which is linked to the glycerol backbone. The positively charged

group is the choline moiety, which constitutes the free end of the headgroup.

Thus, the PC bilayer observed from the aqueous solution side has uniformly

distributed –PO(-)

and –N(+)

(CH3)3 groups. The PC membrane surface can be

modeled in two ways. In model I, the membrane surface is continuous with

uniformly distributed functional groups being the active centers of

adsorption of the hydrogen and hydroxide ions. In model II, the bilayer

surface is composed of nonbonded PC molecules and of molecules with

bonded H+ and OH

- ions [86,92].

Model I

The electrical capacitance of the lipid membrane results from acid-base

equilibria existing between uniformly distributed active centers at the

membrane surface and solution ions. Assuming that the H+ and OH

- ions are

adsorbed at the PC surface, the adsorption equilibria can be presented

in the form:


A H AH (21)

B OH BOH (22)

where:

A- is –PO

(-) group of phosphatidylcholine and B

+ is –N

(+)(CH3)3 group of

phosphatidylcholine.

Consequently, the four groups A-, AH, B

+ and BOH are present at the

bilayer surface.

Acid-base constants can be calculated from surface concentrations of the

bilayer lipid components and volume concentrations of the ions present in

the solution according to the equations:

-

AHA

A H

aK

a a (23)

BOHB

B OH

aK

a a (24)

Taking into account the acid-base equilibria – Eqs. (21) and (22), the surface

concentration of the phosphatidylcholine s can be expressed as:

- AHAa a s (25)

BOHBa a s (26)

where:

BOHBAHAaaaa ,,, [mol m

-2] – the concentrations on the membrane

surface of the membrane components, respectively.

Assuming additivity of the contributions of the individual forms of the

phosphatidylcholine molecule to the electrical capacitance of membrane Cm,

the following equation can be written:

-m AH BOHA BC C C C C (27)


in which:

-

- -

0 A

A A

aC C

s (28)

0 AHAH AH

aC C

s (29)

0 B

B B

aC C

s (30)

0 BOHBOH BOH

aC C

s (31)

where:

-

0 0 0 0, , ,AH BOHA BC C C C [ F cm

-2] – the specific capacitances of the membrane

components, respectively.

Elimination of BOHBAHAaaaa ,,, from the equations system (21)–(31)

yields the equation:

-

0 0 0 01 1

1 1 1 1

BA OHHm AH BOHA B

A A B BH H OH OH

K aK aC C C C C

K a K a K a K a

(32)

Eq. (32) describes the dependence of the electrical capacitance of the lipid

membrane on the pH of the electrolyte solution.

Model II

Since the PC molecule possesses a zwitterionic character, it can

participate in acid-base equilibria with both H+ and OH

-:

PC H PCH (33)

PC OH PCOH (34)

PC HOH PCHOH (35)


Consequently, Eqs. (33)-(35) can be considered as the description of an

adsorption process. As a result of adsorption of hydrogen and hydroxide ions

on the surface of phosphatidylcholine layer, the PC molecule can exist in

four different forms: PCH+ with H

+ adsorbed, PCOH

- with OH

- adsorbed,

PCHOH with both H+ and OH

- ions adsorbed on the surface and a free

phosphatidylcholine molecule PC i.e. with no ions adsorbed (Fig. 11). A

phosphatidylcholine bilayer is assumed to consist of these four forms. The

relative contributions of above forms are dependent on pH, according to

Eqs. (33)-(35).

Assuming that the electrical capacitance of membrane is the sum of the

contributions of the individual forms of the phosphatidylcholine molecule,

the following equation can be presented:

m PCHOH PCPCH PCOHC C C C C (36)

The surface concentrations of the groups considered in Model I can be

written depending on the forms of the PC molecule postulated in Model II:

- -PCA PCOHa a a (37)

AH PCHOHPCHa a a (38)

O N(CH3)3

R2

O

O

R1

O

O

O

O-

PO

H (-) (+)

(+)

OH

PCHOH

(-) O N(CH3)3

R2

O

O

R1

O

O

O

O-

PO

H (-) (+)

(+)

PCH+

O N(CH3)3

R2

O

O

R1

O

O

O

O-

PO

(-)

(+)

OH

PCOH-

(-) O N(CH3)3

R2

O

O

R1

O

O

O

O-

PO

(-)

(+)

PC

Figure 11. Four forms of phosphatidylcholine occurring as a result of adsorption of

H+ and OH- ions on the bilayer surface.


PCB PCHa a a (39)

BOH PCHOHPCOHa a a (40)

where:

PC PCHOHPCOH PCHa ,a ,a ,a [mol m

-2] – the concentrations on the membrane

surface of the membrane components, respectively.

Then, the association acid constant of the groups PC and PCOH- with

the H+ ions (Eq. 23) may be expressed as:

-

PCHOHPCHA

PCH PCOH

a aK

a a a

(41)

and the association base constant of the groups PC and PCH+ with the

OH- ions (Eq. 24) may be presented as:

PCHOHPCOHB

PCOH PCH

a aK

a a a

(42)

On the basis of Eqs. (37)-(40), the surface concentration of the lipid can

be written as:

PCHOH PCPCH PCOHa a a a s (43)

The relationship between the surface concentrations of the membrane

components, the surface concentration of the lipid and the electrical

capacitance values takes a form:

0 0 0 0PCHOH PCH PCOH PCm PCHOH PCPCH PCOH

a aa aC C C C C

s s s s (44)

in which:

0 0 0 0, , ,PCHOH PCPCH PCOH

C C C C [ F cm-2

] – the specific capacitances of the

membrane components, respectively.


The equations connecting surface concentrations of the lipid forms

PC PCHOHPCOH PCHa ,a ,a ,a with surface concentrations of the groups

BOHBAHAaaaa ,,, can be written:

PCHOH BOHAHa aa

s s s (45)

1PCH BOHAHa aa

s s s (46)

1PCOH BOH AHa a a

s s s (47)

1 1PC BOHAHa aa

s s s (48)

The below equations can be obtained on the basis of Eqs. (41)-(43):

1

A HAH PCHOHPCH

A H

K a sa a a

K a (49)

1

B OHBOH PCHOHPCOH

B OH

K a sa a a

K a (50)

Substituting Eqs. (49)-(50) to Eqs. (45)-(48) yields the dependence:

0 0 1

1 1 1 1

BA AOHH Hm PCHOH PCH

A B A BH OH H OH

K aK a K aC C C

K a K a K a K a

0 01 1 1

1 1 1 1

B OHPCPCOH

A B A BH OH H OH

K aC C

K a K a K a K a

(51)

Eq. (51) represents the dependence of the electrical capacitance of the lipid

bilayer on the pH of the electrolyte solution.


5.2. Phosphatidylcholine membranes

The electrochemical impedance spectroscopy measurements were

carried out in pH range from 3.32 to 7.27. Attempts to form bilayer stable

enough in the solution of lower or higher pH were unsuccessful. Presumably,

the large electrostatic forces due to the dissociation of polar groups at the

interface may prevent the formation of a stable bilayer. The minimal Cm

value obtained at pH 4.13 is equal to 0.552 F cm-2

. A quantitative

description of acid-base equilibria based on Rm is not carried out because of

the spread in the measurements of this electrical parameter and the

uncertainty of the results.

Typical impedance diagrams of chosen PC membranes registered in

different pH are presented in Fig. 12. These diagrams have the form of

semicircles or part of semicircles in the entire analyzed frequency range (the

point distribution is logarithmic). They confirm that pH of the electrolyte

solution has considerable effect on the electrical properties of the

membranes. An equivalent circuit models the electric properties of the

analyzed lipid membranes appeared in Fig. 2.

Electrical capacitance of the PC bilayer is depicted in Fig. 13 vs. the pH

of the electrolyte solution. The experimental values are marked by points,

the total values obtained on the basis of Eq. (32) are presented by solid line

and the capacitance of the individual bilayer components, i.e. A-, B

+, AH and

0.0E+00

1.0E+07

2.0E+07

3.0E+07

0.0E+00 1.0E+07 2.0E+07 3.0E+07 4.0E+07 5.0E+07 6.0E+07

Z '/ cm2

-Z"

/ c

m2

pH = 3.03

pH = 3.77

pH = 4.64

pH = 6.02

pH = 7.27

Figure 12. Complex plane impedance diagrams of phosphatidylcholine bilayers

recorded at different pH values. The solid lines represent the results of the fitting

procedure. The equivalent circuit used for impedance data analysis is shown in Fig. 2.


-4.0E-03

0.0E+00

4.0E-03

8.0E-03

1.2E-02

0 2 4 6 8 10 12

pH of electrolyte solution

Cm

/F

m-2

experimental values A B AH BOH A+AH+B+BOH

Figure 13. The participation of the A and B groups, calculated from the Model I, in

associated and dissociated forms in the electrical capacitance of the

phosphatidylcholine bilayer, as a function of pH of the electrolyte solution.

BOH are marked with broken lines. This figure refers to the earlier described

structural Model I of a phosphatidylcholine membrane surface in which the

functional groups have been assumed to be uniformly distributed on its

surface from the aqueous solution side. Fig. 13 shows that the capacitance

has a minimum value around pH 4.13, and increases almost symmetrically

toward low pH and high pH (similar variation in Cm with pH was reported in

[93]). Disturbances of the symmetry may be caused by difference in the sizes

of the positive and negative ionized groups. The minimal electrical

capacitance of the analyzed bilayers appears at the isoelectric point of the

phosphatidylcholine, which corresponds to the pH at which surface formed

from PC has no net electrical charge, or where the negative and positive

charges are equal [94].

As it is seen in Eq. (32), the total capacitance value of the PC bilayer is

the sum of the capacitance values of its components. In order to calculate the

values of the specific capacitance of these components, the equilibrium

constants of adsorption processes of hydrogen and hydroxide ions on

phosphatidylcholine have to be known. The acid-base equilibrium constants

for PC bilayer were reported in the paper [63], their values are: KA = 102.581

and KB = 105.687

. The association acid constant is assigned to the –PO(-)

group

and the association base constant is attributed to the –N(+)

(CH3)3 group.

From comparison of the association constants it appears that the OH- ion is

more strongly adsorbed than the H+ ion. Introduction of KA and KB values to


Eq. (32) results in specific capacitance values of the individual forms of the

phosphatidylcholine bilayer. The -

0 0 0 0, , ,AH BOHA BC C C C values calculated using

the linear regression method are -0.40 F cm-2

, 0 F cm-2

, 0.94 F cm-2

and

1.12 F cm-2

, respectively. When the specific capacitances of the membrane

components have zero or negative values it is possible to suppose that no

bilayer formed from these forms do exist.

The degree of coverage of the PC membrane surface by associated and

dissociated forms of the groups present at the membrane surface as function

of pH of the electrolyte solution is plotted in Fig. 14. It is seen that at

extreme acidic pH‟s the degree of coverage of the membrane by the H+ ions

is over 0.9, e.g. in this pH range the bilayer is covered by hydrogen ions. As

the pH changes to less acidic values, the concentration of the adsorbed

protons decreases e.g. AH groups begin to lose partially their proton

(AH A-). Around the isoelectric point of PC (pH equal to 4.13), the

surface is almost not covered by the H+ ions. In the proximity of isoelectric

point of PC, there is also almost no coverage of the membrane by OH- ions.

As the pH of the electrolyte solution increases up to basic values, the

concentration of hydroxide ions in solution grows and therefore such ions

begin to bind with B+ groups (B

+ BOH). The coverage with the OH

- ions

is favored at basic pH‟s, it is almost one at pH 7.

The pH dependence of the electrical capacitance of the bilayer

membrane in the electrolyte solution is also plotted in Fig. 15. Points present

the experimental values, the total values of Cm calculated from Eq. (51) are

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12


deg

ree

of

cove

rag

e

AH BOH A B

Figure 14. The degree of coverage of the phosphatidylcholine bilayer surface,

calculated from the Model I, with associated and dissociated forms of the A and B

groups as a function of pH of the electrolyte solution.


0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

0 2 4 6 8 10 12


Cm

/F m

-2

experimental values PCH

PCOH PC

PCHOH PCHOH+PCH+PCOH+PC

Figure 15. The participation of the individual forms of the phosphatidylcholine

molecules, calculated from the Model II, in the electrical capacitance of the

phosphatidylcholine bilayer, as a function of pH of the electrolyte solution.

presented by a continuous line and the capacitance values of PC membrane

components are marked with broken lines. Fig. 15 refers to the above

presented Model II, in which the PC bilayer surface is composed of

nonbonded molecules and of molecules with bonded H+ and OH

- ions. Based

on Eq. (51), the total membrane capacitance is a sum of capacitances of its

components, i.e. PC, PCH+, PCOH

- and PCHOH. Specific capacitance

values of individual components of the PC membrane were determined by

linear regression method. The obtained values of 0 0 0 0, , ,PC PCHOHPCH PCOHC C C C

amount to 0.53 F cm-2

, 0.94 F cm-2

, 0.72 F cm-2

and 0 F cm-2

,

respectively. When the specific capacitance of the membrane component has

zero value it is possible to suppose that the bilayer formed from this form

does not exist.

The degree of coverage of the phosphatidylcholine membrane surface by

the H+ and OH

- ions as function of pH of the electrolyte solution is shown in

Fig. 16. It is seen that the PC form predominates in the membrane surface

when the pH is near the isoelectric point (like in Fig. 14); the membrane

surface is almost not covered by the hydrogen and hydroxide ions there. In

both descriptions, the degree of coverage of the PC membrane surface by the

H+ and OH

- ions remains unchanged in the ranges below 1.5 and above 7.5.

A head group of amphoteric PC contains two separated oppositely charged

moieties. Then, there is a possibility of strong electrostatic attraction

between hydrophobic parts and appropriate local charge. The negative charge


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12


deg

ree

of

cove

rage

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03PCH PCOH PC PCHOH

Figure 16. The degree of coverage of the phosphatidylcholine bilayer surface,

calculated from the Model II, with the individual forms of the molecules as a function

of pH of the electrolyte solution.

of the phosphate group is distributed among four oxygen atoms, while the

positive charge of ammonium group is concentrated on a single nitrogen

atom, which is favorable for electrostatic interaction with anions [95].

The effect of pH on membrane mechanics and electrostatics has not

been systematically studied in membrane bilayers. It was found that the

electrical capacitance of the bilayer depended upon pH. The Cm had a

minimum value around pH 4 for PC membrane and its value increased as the

pH became lower or higher [93]. From the investigation of the

electrophoretic mobility of bilayers, it has been inferred that the isoelectric

point of PC is around pH 4 and in this pH zeta potential is equal zero. Zeta

potential is positive below pH 4 and this is due to the high concentration of

H+

ions in the solution which bind to PC membrane generating a positive

charge on the bilayer surface. Above pH 4 zeta potential is negative and this

is caused by high concentration of OH- ions which are attached to the

membrane inducing a negative charge on the surface [96]. The effect of pH

on PC membrane interfacial tension has been described in [63,86]. The

interfacial tension of PC lipids was observed to increase near pH 4. It was

shown that the description, in which the assumption of the additivity of the

contributions of each individual forms of the PC molecule to the overall

interfacial tension of the membrane was made, resulted in significantly better

description of the dependence on pH than the description based on the Gibbs

isotherm [63].


In this paper, two models describing the surface of the membrane build

from phosphatidylcholine are presented. Model I assumes the equilibria

between -PO(-)

and -N(+)

(CH3)3 groups uniformly distributed on membrane

surface and hydrogen and hydroxide ions. Model II supposes the existence

of four different forms of PC as a result of adsorption of H+ and OH

- ions on

the bilayer surface. Figs. 13-16 show that Model II gives more reliable

results than Model I, in which the negative value of the specific capacitance

of -PO(-)

form shows that this model is not likely from the physicochemical

point of view.

6. Materials and experimental details

6.1. Chemicals and preparation of the forming solutions

All substances used for membrane formation were dissolved in

chloroform and mixed in appropriate proportions to achieve the desired

molar fractions. The solvent was evaporated under a stream of argon.

The dried residues were dissolved in appropriate solvents mixture (e.g.

n-hexadecane: n-butanol 10:1 by volume) to produce a concentration of

20 mg ml-1

. During membrane formation, the solvent mixture was removed,

resulting in a membrane with the same composition as the solution. Samples

were stored at 4 C for less than a week. The preparation and storage

methods provided reproducible electrochemical properties when samples

prepared at different times were examined using impedance spectroscopy.

0.1M potassium chloride solution was used as electrolyte for the

experiment. Potassium chloride was heated before use at 400 C for 4 h to

remove traces of organic material.

The acetate buffer was used as the electrolyte in the pH range of 3.32 to

7.27 in the experiment, in which acid-base equilibria at interface separating

electrolyte solution and phosphatidylcholine bilayer were assumed.

The experiments were performed at a temperature of about 293 1 K.

6.2. Preparation of the bilayer membranes

Bilayer membranes were obtained as bubbles at the Teflon cap

comprising a portion of the measuring vessel. The use of n-hexadecane as a

solvent made it possible to obtain membranes with thickness and capacity

values similar to those of monolayer membranes [97]. The small quantity of

n-butanol had a negligible effect on the impedance parameters of the

bilayers, yet it considerably accelerated membrane formation.


Thinning of the membranes was monitored using reflected light

microscopy with a high-brightness yellow LED source and by measuring the

membrane capacitance.

6.3. Impedance analysis and modeling

Electrochemical impedance spectroscopy was performed with an a.c.

impedance system that included a personal computer, a two-phase lock-in

amplifier and a potentiostat/galvanostat. The electrochemical cell was

connected with a potentiostat via a self-constructed four-electrode

preamplifier with high-impedance inputs; the measuring cell was described

exactly in [27,43]. The use of the four-electrode system in the studies of

electric phenomena occurring in membranes makes it possible to

considerably reduce the errors caused by electrode and electrolyte

impedance [98]. A 4-mV amplitude sine-wave signal perturbation was

applied in the 0.1-10 000 Hz frequency range.

Impedance data were analyzed by using the nonlinear least-squares

fitting to a model represented by an equivalent electrical circuit. The

modeling process was iterative, using the chi-square (χ2) value for the entire

model and the percent error values for each circuit component to determine

the fit of a given model to the experimental data. Components were chosen

basing on theories from electrochemical cell studies and using the Boukamp

suggestion that each component addition should reduce the χ2 value by one-

order of magnitude. According to the literature [99], a χ2 of on the order of

1 10-3

or below was acceptable for a given model.

The membrane images were captured with a color CCD camera. The

bilayer areas were calculated from the photographs, taking into consideration

the spherical nature of the surface and using the equations provided in [100].

The area of the bilayer membranes were between 4 10-2

-8 10-2

cm2

(the

values were given for the bilayer area with subtracted margin).

Each value of the measured parameters was calculated as an arithmetic

mean and standard deviation for at least six membranes. The experimental

impedance values are related to the bilayer surface-area unit.

Conclusion

Investigations presented in this paper have provided valuable

information necessary for the quantitative description of bilayer lipid

membranes. Further studies will be certainly aiming at examinations of

natural membranes. However, preparation of natural membranes so that

possible was access to them from both sides is extremely difficult. Efforts


in this direction can already be seen in the subject literature e.q. patch-clamp

technique.

Acknowledgement

Impedance spectroscopic measurements of the following systems

were supported by a grant from the Polish Committee of Scientific Research

No 1 T09A 070 30: phosphatidylcholine-phosphatidylethanolamine,

phosphatidylethanolamine-cholesterol, sphingomyelin-cholesterol, and

phosphatidylethanolamine- -tocopherol.

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