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Soft Matter and Biological Physics

Eur. Phys. J. E 3, 323-335

Elastic energy of tilt and bending of fluid membranes

M. Hamm1 - M.M. Kozlov2

1 Department of Materials Science & Metallurgy, University of Cambridge, Cambridge CB2 3QZ, UK
2 Department of Physiology and Pharmacology - Sackler Faculty of Medicine, Tel-Aviv University, Tel Aviv 69978, Israel

mh264@cus.cam.ac.uk
misha@devil.tau.ac.il

Received 10 February 2000 and Received in final form 19 June 2000

Abstract
Tilt of hydrocarbon chains of lipid molecules with respect to membrane plane is commonly considered to characterize the internal structure of a membrane in the crystalline state. However, membranes in the liquid state can also exhibit tilt resulting from packing constraints imposed on the lipid molecules in diverse biologically relevant structures such as intermediates of membrane fusion, pores in lipid bilayers and others. We analyze the energetics of tilt in liquid membranes and its coupling with membrane bending. We consider three contributions to the elastic energy: constant tilt, variation of tilt along the membrane surface and membrane bending. The major assumption of the model is that the core of a liquid membrane has the common properties of an elastic continuum. We show that the variation of tilt and membrane bending are additive and that their energy contributions are determined by the same elastic coefficient: the Helfrich bending modulus, the modulus of Gaussian curvature and the spontaneous curvature known from previous studies of pure bending. The energy of a combined deformation of bending and varying tilt is determined by an effective tensor accounting for the two factors. In contrast, the deformation of constant tilt does not couple with bending and its contribution to the elastic energy is determined by an independent elastic constant. While accurate determination of this constant requires additional measurements, we estimate its value using a simplified approach. We discuss the relationships between the obtained elastic Hamiltonian of a membrane and the previous models of membrane elasticity.

PACS
68.10.-m Fluid surfaces and fluid-fluid interfaces - 68.10.Et Interface elasticity, viscosity, and viscoelasticity - 87.16.Dg Membranes, bilayers, and vesicles

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