Abstract
The morphological changes of lipid membranes upon interaction with an
amphiphilic polymer were studied experimentally and theoretically and
various shape instabilities were observed. My study focused mainly on the
pearling of hollow cylinders, the coiling of multilamellar tubes and
the budding and fingering of highly oblate vesicles. A single model,
based on the induction of local curvature by the polymer, is shown to
agree with all the observed phenomena.
The experiment
As a first step towards describing biological membranes, we study a simplified
model system composed of a lipid membrane interacting with an amphiphilic
polymer. The polymer (yellow) inserts its hydrophobic side-groups (red) into the
membrane (black).
Pearling
Snapshots of a multilamellar tubular vesicle of SOPC
undergoing a pearling instability: (a) 0, (b) 70 sec. and (c) 150 sec. after the
onset of pearling. (d) Image of an inhomogeneous pearled structure at late
stages of the instability, 900 sec. after the onset of pearling. The polymer
concentration is much larger than in (a-c). The scale bars represent 20
mm.
Budding
Buds form at the rim of highly oblate vesicles. This is probably due to the
fact that the rim has the greatest curvature. At low polymer concentrations, the
rim displays enhanced fluctuations (a)-(b). Nucleation of a bud suppresses rim
fluctuations (c). The bud grows into a tube (d-f). The scale bar represents 10
mm.
Alternatively, flushing the cell with polymer solution of large concentration, induces the simultaneous formation of buds all along the rim (b). Buds grow into tubes which subsequently pearl (c). At high polymer concentrations on the membrane, small structures of high curvature result (d). The scale bar represents 10 m.
Local injection leads to local budding
I. Tsafrir, Y. Caspi, T. Arzi, M.A. Guedeau-Boudeville, and J. Stavans, Budding and Tubulation
in Highly Oblate Vesicles by Anchored Polymers, Phys, Rev. Lett, 91,
138102 (2003)
Coiling
The cylindrical multilamellar tubes, called myelin figures, consist of a large
number of membrane bilayers reaching almost to the core. Due to the large number
of concentric layers, these structures cannot pearl, instead, when hydrated with
polymer they bend into coils. This system is unique in that coils form in the
absence of twist.
The model
The proposed model describing the system is based on the following assumptions:
Membrane curvature and polymer concentration are coupled. The polymer induces a
local spontaneous curvature H0. Polymers can diffuse on the membrane.
The membrane is modeled as two-dimensional lattice, where each site can be
occupied by at most one polymer molecule. Microscopic Hamiltonian:
Pearling
Analysis of the instability near threshold, revealed that the ratio between the
radius of the tube and the wavelength of the instability, 2pR0/P
is characteristic of spontaneous curvature.
When
the polymer is introduced locally, by using a micropipette, the instability
propagates from the region where the polymer is added. When the pipette is
removed, polymer continues to diffuse along the tube. The pearls open up
sequentially. Two theoretical mechanisms, ADE and SC, were shown to be
responsible. The timescale associated with ADE is roughly 50 times faster than
the one associated with SC.
Coiling
The equilibrium state of a tube is calculated as follows:
Pay energy for bending a tube, keeping polymer distribution homogeneous. Gain energy
from inhomogeneous polymer distribution. Pay for loss of mixing entropy. The energy gain
from an inhomogeneous polymer
distribution depends linearly on H0, while the bending energy and the
entropy do not . Thus for large enough values of H0 the free energy
of an inhomogeneous bent tube is larger than that of a homogenous straight one.
The curvature at every point is determined by the center line:
The free energy of a thick tube with constant centerline curvature, after integrating over the entire stack, is:
This energy predicts that below a critical polymer concentration the equilibrium shape is that of a straight tube, while above it the tube form maximally curved structures.
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Geometrical phase diagram for double helices, r0 is the outer radius, 2ph is the pitch of the central line, and r2 is the radius of the smaller central line. |
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Some comparisons between the theoretical models and actual observed coils:
