Remarks on the geometry of micelles, bilayers and cell membranes

Abstract

Starting from simple geometric properties of parallel surfaces, it is suggested that bilayers, and also monolayers, present two spontaneous principal curvatures gamma and gamma, so that a narrow disc of freely deformable bilayer might adopt either a 'saddle' shape, or a 'hat' shape, or a cylindrical shape. Besides the usually considered spontaneous splay c0 = gamma + gamma, there is also a spontaneous gaussian curvature g0= gammagamma, with noticeable effects in strongly curved bilayers. An excess of area of the median hydrophobic level with respect to the mean area occupied by the two hydrophilic layers creates a saddle shape, whereas a deficit leads to a hat shape, the equality corresponding to a cylindrical shape. The usual two layers theory of the spontaneous curvature seems to be improved by considering the role of a median layer. We have tried to illustrate this new point of view by many examples. Due to their asymmetry, monolayers and cell membranes give rise to micelles and vesicles of comparable geometries, but of very different sizes. At the considered scales, a term of order higher than quadratic, such as kt(cc- gammagamma)2, seems to be necessary in the expression of the elastic energy.