Lecture #3 Notes

Lecture#3 Notes

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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 Gauss-Bonnet 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 10-100kB 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 two-layer 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 Canham-Helfrich 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.

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