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. PACS. 87.22.Bt Membrane and subcellular physics and structure – 82.65.Dp Thermodynamics of surfaces and interfaces – 82.70.-y Disperse systems Lyotropic smectic phases (Lα phases) are liquid crystalline systems of parallel, regularly spaced lipid bilayers [1,2]. Those phases form a quasi-crystalline structure in one (the z) direction, while retaining their fluid properties in the two others directions (x-y)-plane. They have been studied extensively in recent years, as they form one of the most convenient lyotropic structures for the experimental study of fluctuating interfaces. They have been for instance particularly useful in demonstrating the importance of the so-called entropic (Helfrich) repulsion between fluctuating membranes [3]. Furthermore, the degree of dilution (the layer spacing) and other controlled parameters of the lyotropic Lα phase can be experimentally varied over a wide range, which allows for a precise study of the scattering intensity of the phase. This scattering intensity is expected to show several peaks corresponding to a stack of regularly spaced membrane. In these layered systems however, the thermal fluctuations destroy the longrange order at finite temperature, leading to weak —power law— singularities of the peaks. Inclusions in membrane have been actively studied for many years. Most of the earlier work was done by the biological community, for which it was a necessary step toward the comprehension of the complexity of biological (cell) membrane [4]. In the last twenty years or so, a more physical description of membrane-inclusion complexes, in terms of curvature energy and disruption of the a e-mail: [email protected] molecular order of the membrane, have led to interesting results on membrane-induced interactions between inclusions [5]. While this “physical” picture of the membraneinclusion interactions leaves aside many fundamental processes driven by specific interactions, it allows for a description of universal features of the complexes in terms of elastic couplings between membrane and inclusion. This description is applicable to a certain extent to membrane with adsorbed or grafted polymers or colloids [6,7], as well as to transmembrane inclusions resembling certain membrane proteins [5]. In recent years, numerous experimental studies have investigated the role of added membrane proteins in lyotropic smectic phase. This lipid phase seems to be of particular interest for the study in vitro of the bilayerprotein association [8–12], and may have some practical application for gene therapy [13]. From a physicist point of view, it is interesting to understand the effect of inclusions on the thermal fluctuations of flexible, fluid membranes. From a more “biological” point of view, a precise characterization of the physical action of a given protein onto a lipid membrane (anchoring, swelling, stiffening ...) is of direct biological interest, and could be reached via X-ray measurements. This work has some relevance in material science as well, since for practical purposes, a lamellar phase is very rarely composed of two components (solvent+surfactants) only. The addition of a third or fourth component, such as cosurfactant, polymers, proteins or colloids is of common practice to tune the properties 116 The European Physical Journal E of the phase [14,15]. The object of the present work is to determined the modifications of the smectic X-ray scattering due to the presence of inclusions in the lamellar phase. Ultimately, one could expect to map out different membrane-inclusion couplings by their effect on the Xray structure factor. We consider three model inclusions, leading to different modifications of the membrane elastic properties, and we show that the qualitative analysis of the diffraction spectrum is enough to discriminate between these inclusions. Several recent theoretical works deal with lamellar systems with flexible polymers [15] or other soluble inclusions (so-called doped solvent lamellar phases). These works are complementary, as we are interested in insoluble inclusions such as the membrane proteins used in [9], while [15] studied effects such as polymer depletion-induced interactions between lamellae, resulting from an equilibrium between the trapped inclusions and free inclusions in pure solvent. The standard description of Caillé [16] for pure Lα phases has been confirmed experimentally in numerous systems. It is based on the classical smectic Hamiltonian [17]: H0 = 1 2 ∫ dr ( B (∂zu) 2 +K (∇2⊥u)2) , (1) where u(r) is the (continuous) normal displacement of the layer at the point r, z is the coordinate normal to the layers, and ∇⊥ is the gradient along the layers. The elasticity of the smectic phase is characterized by the two coefficients K and B, which are related to the energy cost of bending and compressing the sample. Typically, K = κ/d, where κ is the bending modulus of a single membrane and d is the average layer spacing, and B is a function of the intermembrane interactions [1]. Those moduli define the characteristic smectic length λ = √ K/B [17]. The scattering intensity I(q) is related to the Fourier transform of the density correlation function Gn(r), itself related to the exponential of the layer displacement correlation function g(r) = q 0/2〈(u(r)− u(0))〉, where 〈...〉 represents a thermal average of the fluctuations, and q0 = 2π/d (see [2], Sect. 6.3.2). As usual, the Fourier decomposition of the layer displacement is used: u(r) = ∫ dq/(2π)uqe. Using equation (1):