Past exam of the quantum condensed matter course of the University of Cambridge / 2005 / Section A / A1 / b

A lipid bilayer self-assembles from amphiphiles: hydrophilic head groups face the surrounding water, while hydrocarbon tails form a hydrophobic interior. In a fluid membrane, molecules diffuse laterally within each membrane leaflet. Closing the bilayer into a vesicle avoids an exposed edge. The interior is a substantial barrier to isolated ions and many polar solutes; membrane proteins provide selective transport pathways in a cell membrane.

The Helfrich energy describes long-wavelength bending:Here $H=(c_1+c_2)/2$ and $K=c_1{c}_2$ are mean curvature and Gaussian curvature, $C_0$ is spontaneous curvature, ${\kappa }_b$ the bending modulus, $\bar{\kappa }$ the Gaussian-curvature coefficient and $\sigma$ the membrane tension. Bending is comparatively inexpensive, whereas stretching the molecular area strongly is costly. Leaflet asymmetry can favor a nonzero spontaneous curvature.

In micropipette aspiration, controlled suction draws part of a large vesicle into a narrow pipette. Pressure and geometry determine membrane tension, and the aspirated length measures apparent area change. At low tension, smoothing thermal undulations supplies much of that area; stronger tension probes actual stretching. Tracking vesicle shape fluctuations provides a complementary estimate of the bending modulus.