Budding of crystalline domains in fluid membranes
T. Kohyama,
1,
* D. M. Kroll,
2
and G. Gompper
1
1
Institut fu ¨r Festko ¨rperforschung, Forschungszentrum Ju ¨lich, D-52425 Ju ¨lich, Germany
2
Supercomputing Institute, University of Minnesota, 599 Walter Library, 117 Pleasant Street S.E., Minneapolis, Minnesota 55455, USA
Received 25 July 2003; published 17 December 2003
Crystalline domains embedded in fluid membrane vesicles are studied by Monte Carlo simulations of
dynamically triangulated surfaces and by scaling arguments. A budding transition from a caplike state to a
budded shape is observed for increasing spontaneous curvature C
0
of the crystalline domain as well as
increasing line tension . The location of the budding transition is determined as a function of C
0
, , and the
radius R
A
of the crystalline domain. In contrast to previous theoretical predictions, it is found that budding
occurs at a value of the spontaneous curvature C
0
, that is always a decreasing function of the domain size R
A
.
Several characteristic scaling regimes are predicted. The distribution of five- and sevenfold disclinations as the
budding transition is approached is determined, and the dynamics of the generation of defects is studied.
DOI: 10.1103/PhysRevE.68.061905 PACS numbers: 87.16.Dg, 64.70.Dv, 82.70.-y
I. INTRODUCTION
The primary new feature in two-component—compared
to single-component—fluids is the possibility of phase sepa-
ration. Canonically, mixtures have a lower miscibility gap,
that is, the system is homogeneously mixed at high tempera-
tures, but demixes at low temperatures into two coexisting
phases that are enriched in one of the two components. How-
ever, upper miscibility gaps and closed coexistence loops
also exist, typically in systems in which the hydrophobic
effect is important. The inverted phase behavior of these sys-
tems is due to the orientational degrees of freedom of the
water molecules, which are distributed isotropically at high
temperatures, but have a preferred orientation in the neigh-
borhood of polar solutes.
It is therefore natural to expect phase separation in two-
component amphiphilic membranes. Indeed, phase separa-
tion in Langmuir monolayers at the water-air interface has
been well documented for many years, and has been inves-
tigated in considerable detail 1,2. However, in bilayer
membranes, phase separation turns out to be much more dif-
ficult to observe. Initial evidence showed gel-fluid coexist-
ence in some systems 3, while fluid-fluid coexistence re-
mained elusive for a long time. Only very recently have
experiments using three-component membranes revealed
very clear and convincing evidence for both gel-fluid 4,5
and fluid-fluid 6,7 coexistence.
The coupling of phase separation and membrane shape in
flexible bilayer membranes opens the possibility for the bud-
ding of domains 8,9. The physical mechanism of this phe-
nomenon is the competition between the line tension energy
of the phase boundary and the curvature energy of the mem-
brane. Since the curvature energy is scale invariant, so that
the curvature energy of a spherical vesicle is independent of
the vesicle radius, and the line tension energy is proportional
to the domain perimeter, i.e., to the domain radius, it is im-
mediately clear that a budding transition occurs when the
domain radius R is on the order of /, where is the bend-
ing rigidity and is the line tension. Similarly, a membrane
patch with spontaneous curvature C
0
has a budding transi-
tion at R 1/C
0
.
The coexistence of two phases in biological membranes
has also received considerable attention recently. The exis-
tence of ‘‘lipid rafts’’ 10 may indeed play an important role
in the control of the activity of membrane proteins. Another
kind of two-phase coexistence in biological membranes oc-
curs when domains of adsorbed proteins form spontaneously.
A famous, and biologically very important, example is the
adsorption of clathrin molecules on the plasma membrane
11. Clathrin molecules assemble to form a regular hexago-
nal network on the membrane surface 12–14. By forming
first a coated pit and then a complete bud see Fig. 1, these
clathrin coats control endo- and exocytosis, i.e., the forma-
tion and detachment of small transport vesicles from the cell
membrane. The formation of clathrin cages is therefore an
example of the budding of crystalline membrane patches em-
bedded in a fluid lipid membrane.
*Permanent and present address: Department of Physics, Faculty
of Education, Shiga University, Hiratsu 2-5-1, Otsu, Shiga 520-
0862, Japan.
FIG. 1. Rounded clathrin-coated pits in normal chick fibroblasts
a – d and coated pits on membrane fragments derived from cells
that have been broken open and left in p H 7 buffer for 10 min at
25 °C before fixation and freeze drying e – f. The width of the
field of view of the individual pictures is 0.4 m. Reproduced from
Ref. 14 by copyright permission of The Rockefeller University
Press.
PHYSICAL REVIEW E 68, 061905 2003
1063-651X/2003/686/06190515/$20.00 ©2003 The American Physical Society 68 061905-1