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Unformatted text preview: Gaussian curvature. The are the only two
independent scalar deformations of an inﬁnitely thin interface (they are the trace and determinant of the curvature
tensor deﬁning the second fundamental form). The parameters κ(S) and κG (S) are microscopic parameters with units
of energy. If there is no change in topology of a surface the κG (S)K (S) is just a constant from the GaussBonnet
Theorem. Thus, it suﬃces to consider only the mean curvature term. The bending rigidity κ(S) can depend on
surface position and composition (larger for higher cholesterol content) and is typically 10100kB T or lipid bilayers.
This is huge and in many situations it suﬃces to consider the low temperature regime, where thermal ﬂuctuations
are negligible. Thus, simple energy minimization does a good job in predicting vesicle shapes for example. The total
bending energy of a surface is Hb [S]dS = Hb = κ(S)
(C (S) − C0 (S))2 .
2 (10) The bending rigidity is a function of the molecular details of the lipid bilayer. A simple model for κ considers the
bilayer as a continuum elastic plate. For a homogeneous medium, the bending rigidity is proportional to the cube of
the plate thickness, κ ∝ d3 . However, lipid bilayers can have neutral surfaces (surfaces where bending does not result
in stretching or compression) that are located anywhere from the outer edges (near the head groups) to the midplane.
When the lateral size of the headgroups is large compared to that of the acyl chains, the area per lipid molecule is
controlled by the headgroups and bending stresses are transduced along the two outer surfaces of the bilayer. In this
extreme limit, κ ∝ d2 . The two leaftlets usually slide against each other so it has been necessary to consider the
membrane as a twolayer system in order to qualitatively predict experimentally observed vesicle shapes. C Bilayer Couple Model The bilayer couple model treats each leaﬂet as independent plates that can slide over each other and that follow the
CanhamHelfrich energy (12). However, for a closed vesicle for example, the two leaﬂets will have diﬀerent numbers
of lipid molecules and hence a diﬀerent areas. And since inextensibility demands that each leaftlet have the same
area, there is an additional constraint on...
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This note was uploaded on 11/14/2013 for the course MATH 201 taught by Professor Tomchou during the Fall '12 term at UCLA.
 Fall '12
 TomChou
 Math

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