Structure factor of a lamellar smectic phase with inclusions

Motivated by numerous X-ray scattering studies of lamellar phases with membrane proteins, amphiphilic peptides, polymers, or other inclusions, we have determined the modifications of the classical Caillé law for a smectic phase as a function of the nature and concentration of inclusions added to it. Besides a fundamental interest on the behavior of fluctuating systems with inclusions, a precise characterization of the action of a given protein on a lipid membrane (anchoring, swelling, stiffening ...) is of direct biological interest and could be probed by way of X-ray measurements. As a first step we consider three different couplings involving local pinching (or swelling), stiffening or curvature of the membrane. In the first two cases we predict that independent inclusions induce a simple renormalization of the bending and compression moduli of the smectic phase. The X-ray experiments may also be used to probe correlations between inclusions. Finally we show that asymmetric coupling (such as a local curvature of the membrane) results in a modification of the usual Caillé law. Received 10 March 2000 and Received in final form 30 August 2000