Shape Instabilities in Charged Lipid Domains
A.Cebers*
,²
and P. A.Janmey*
,‡
Institute of Physics, UniVersity of LatVia, Salaspils-1, LV-2169, LatVia, and
2
Institute for Medicine and
Engineering, Departments of Physiology and Physics, UniVersity of PennsylVania, Philadelphia, PA
ReceiVed: July 24, 2002
A theoretical model describing the formation and shapes of domains in amphiphilic monolayers driven by
screened electrostatic interactions is presented. The model is related to previous studies of pattern formation
in magnetic systems in Hele-Shaw cells. Using quantitative estimates of line tension, charge density, and
ionic strength appropriate for biological systems, this model accounts for the appearance of circular and
irregular domains in mixed lipid monolayer systems containing the anionic lipid phosphatidylinositol
monophosphate. The results may have implications for pattern formation in both synthetic materials and
biological membranes.
Introduction
Domain formation in lipid bilayers and other thin films
formed by amphiphilic molecules is a common feature of many
materials including the cell membrane, where lateral demixing
of specific lipids into specialized domains is related to several
aspects of signal transduction and other cellular functions. The
forces governing the formation of such domains are the subject
of many studies, and a number of reports have adressed shape
instabilities of dipolar domains in amphiphile monolayers.
1-7
These calculations are based on the introduction of a cutoff
length representing the distance of closest approach between
perpendicularly oriented dipoles. A somewhat different approach
has been considered regarding the shape instabilities of magnetic
fluid droplets in Hele-Shaw cells,
8
where a natural cutoff
lengthsthe thickness of the Hele-Shaw cellsappears. It has been
shown that the two models are equivalent.
9
An illustration of
the utility of this approach is the description of the undulation
instability of a foam in an amphiphile monolayer,
10
which is in
good agreement with experimental data and is based on the
relation obtained for the analogous instability of a magnetic
liquid foam in the limiting case of vanishing thickness in a
Hele- Shaw cell. In contrast to the large bibliography of works
concerning shape instabilities in amphiphile monolayers gov-
erned by steric or dipole-dipole interactions, a model for the
shape instability of domains of charged lipids interacting by
screened electrostatic interactions in the amphiphile monolayer
is not yet developed. This problem has great interest due to the
role that charged lipids play in the regulation of cell structure
and function
11
and in various biotechnological applications.
12
Model and Instability of the Circular Domain
The free energy of a charged lipid domain in the monolayer
under the assumption that the Debye-Hu ¨ckel theory is valid is
expressed as
13
where σ is the charge density and ψ is the electrostatic potential.
This free energy accounts for electrostatic interactions as well
as entropy of the counterion distribution. Since the dielectric
permeability of water is much higher than that of the surrounding
air, the one-sided model
14
is assumed in which the electrostatic
potential of the planar domain is expressed as follows:
where G(|r b |) ) e
-|r b |
/|r b | is the fundamental solution of the
Helmholtz equation,
2
) (4π∑z
i
2
e
2
n
0i
)/(ǫk
B
T) is the screening
parameter of the surrounding electrolyte solution, and ǫ is the
dielectric permeability of water. The line energy of the domain
boundary will be expressed as
where γ is the line tension.
Let us consider the energy of the domain E ) E
e
+ E
S
with
the deviation of its shape from circular with radius R to new
shapes given in polar coordinates as r ) R + (). By simple
transformations similar to those used before,
8
the electrostatic
part of the free energy E
e
up to second-order terms in can be
written as follows:
where G
1
( - ′) ) G(R 2(1-cos(-′))), and E
e
0
is the
electrostatic free energy of the circular domain with radius R.
Representing
and expressing a
0
from the condition of the domain area
²
E-mail: aceb@sal.lv.
‡
E-mail: janmey@mail.med.upenn.edu.
E
e
)
1
2
∫
σψ dS (1)
ψ( r b) )
2σ
ǫ
∫
G(| r b- r b′|)dS′ (2)
E
S
) γ
∫
dl (3)
∆Ee ) E
e
- E
e
0
(R) )
1
2π
∫
0
2π
dE
e
0
dR
()d +
1
2π
∫
0
2π 1
2
d
2
E
e
0
dR
2
2
()d -
σ
2
2ǫ
∫
0
2π
d
∫
0
2π
d′ R
2
G
1
( -
′)((′) - ())
2
(4)
() )
∑
n)0
∞
a
n
cos n
BATCH: jp12a34 USER: jld69 DIV: @xyv04/data1/CLS_pj/GRP_jp/JOB_i48/DIV_jp026598+ DATE: October 17, 2002
10.1021/jp026598+ CCC: $22.00 © xxxx American Chemical Society
PAGE EST: 2.5 Published on Web 00/00/0000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67