Stability of MultiComponent Biological Membranes

Sefi Givli, Ha Giang, Kaushik Bhattacharya
2012 SIAM Journal on Applied Mathematics  
Equilibrium equations and stability conditions are derived for a general class of multicomponent biological membranes. The analysis is based on a generalized Helfrich energy that accounts for geometry through the stretch and curvature, the composition, and the interaction between geometry and composition. The use of nonclassical differential operators and related integral theorems in conjunction with appropriate composition and mass conserving variations simplify the derivations. We show that
more » ... ons. We show that instabilities of multicomponent membranes are significantly different from those in single component membranes, as well as those in systems undergoing spinodal decomposition in flat spaces. This is due to the intricate coupling between composition and shape as well as the nonuniform tension in the membrane. Specifically, critical modes have high frequencies unlike single component vesicles and stability depends on system size unlike in systems undergoing spinodal decomposition in flat space. An important implication is that small perturbations may nucleate localized but very large deformations. We show that the predictions of the analysis are in qualitative agreement with experimental observations. 1. Introduction. Biological membranes (BMs) are fundamental building blocks of cell walls, mitochondria, and other organelles. They protect the cell by providing a barrier, and control almost all interaction with the surroundings, including transport, signaling, transduction, and adhesion. A key to this diverse functionality is the coupling between mechanical signals carried by the BM and biochemical events in the cell [13, 24, 31, 37, 47] and the rich phenomena that this coupling creates; see, e.g., [3, 16, 34, 28, 36] . For example, gated mechano-sensitive ion channels open to form a large conductance pore in response to membrane stretching [14, 17, 18, 33, 38, 25] . BMs are primarily made of a lipid bilayer, but also contain proteins, rigid cholesterol molecules, and other functional molecules [9] . Moreover, for the same lipid, various phases may be found, such as gels, liquid disordered phases, and liquid ordered phases. These phases differ in their mechanical properties, which makes the BM a heterogeneous mechanical structure. Moreover, BMs are dynamic structures whose molecular arrangements can change with conditions. Depending on the type of lipids and the functional molecules involved, as well as the external conditions like osmotic pressure and temperature, the BM can remain homogeneous or segregate into different phases/domains. The latter changes the stress distribution in the BM and either absorbs or releases energy. Therefore, just like other heterogeneous materials, deformation of the BM is dictated by composition. However, unlike standard mechanical structures, composition is modulated by the shape of the BM [26]. * The inherent coupling between the shape and composition of the BM is an important avenue through which cells sense their environment, and is a key mechanism for mechano-transduction in cells and other organelles. For example, proteins that act on the membrane like wedges lead to areas with high curvature. In addition, certain types of functional proteins concentrate in domains of curvature that they prefer, leading to the formation of functionalized domains [30, 29] . The formation of such domains controls membrane transport as well as cellular sensing and signaling. While the mechanics of BMs has been studied theoretically extensively [27, 46, 10] since the pioneering work of Helfrich [12] , much of it focuses on single component (homogeneous) membranes. Work on multicomponent BMs either relies on advanced numerical methods such as nonlinear finite elements and phase field methods [21, 7, 20, 11, 8] , or uses models with various simplifying assumptions such as axi-symmetry, small deformations, spherical caps landscape, and complete separation of the phases [2, 15, 35, 5] . Nevertheless, the literature still lacks a systematic derivation of equilibrium equations along with stability conditions for the general class of heterogeneous BMs. In this work we systematically derive the equilibrium equations and (linear) stability conditions for a general class of multicomponent BMs motivated by the following facts: (i) stable configurations are the observable in most experiments; (ii) chemomechanical instabilities in cell membranes often relate to critical changes in biochemical processes, cell behavior, or fate. Examples are formation of focal adhesions, initiation of filopodia, and opening of ion channels; (iii) knowledge of the stability conditions can be used to measure, indirectly, mechanical and chemical properties of lipids, protein aggregates, and other functional components of the membrane. We consider a closed BM composed of two phases. These can represent two different lipid phases (e.g., liquid ordered and liquid disordered phases), two different types of lipid molecules, or mobile membrane proteins embedded in a lipid phase. Equilibrium equations and stability conditions are obtained by calculating the first and second variations of a generalized Helfrich energy functional. We assume that overall composition, i.e., the total number of molecules of each phase, does not change in the course of the experiment. In calculating the variations of the energy functional we take advantage of nonclassical differential operators and related integral theorems developed by Yin and collaborators [42, 40, 41, 44, 39, 43] . Further, we introduce density and composition conserving variations, so that the use of Lagrange multipliers is avoided. In addition we account for the spatial nonuniform stretching of the membrane. This feature, which is commonly ignored by assuming a constant membrane area, is especially important in multicomponent membrane applications, and can have important implications in processes such as the activity of gated ion channels. The manuscript is arranged as follows. Our model is introduced in section 2. Section 3 provides some mathematical preliminaries including the operators and identities that enable this calculation and the form of the perturbations. Calculation of the first variation and corresponding equilibrium equations are detailed in section 4. Stability conditions are derived in section 5. Section 6 specializes to a uniform spherical membrane and provides a detailed analysis of the stability. Main conclusions are discussed in section 7. A model of a multicomponent vesicle. 2.1. The energy functional. We consider a closed lipid membrane composed of two components, which we shall refer to as type I and type II. These can represent two different lipid phases (e.g., liquid ordered and liquid disordered phases), two different
doi:10.1137/110831301 fatcat:gligmxgbunfxll54cdrb5f27uy