Robust Stability of Multicomponent Membranes: The Role of Glycolipids
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Robust Stability of Multicomponent Membranes: The Role of Glycolipids Yuan Chen, Arjen Doelman, Keith Promislow Veerman
& Frits
Communicated by I. Fonseca
Abstract We present the multicomponent functionalized free energies that characterize the low-energy packings of amphiphilic molecules within a membrane through a correspondence to connecting orbits within a reduced dynamical system. To each connecting orbit we associate a manifold of low energy membrane-type configurations parameterized by a large class of admissible interfaces. The normal coercivity of the manifolds is established through criteria depending solely on the structure of the associated connecting orbit. We present a class of examples that arise naturally from geometric singular perturbation techniques, focusing on a model that characterizes the stabilizing role of cholesterol-like glycolipids within phospholipid membranes.
1. Introduction Amphiphilic molecules play a fundamental role in the self-assembly of nanostructured membranes. These include phospholipids, the building blocks of cellular membranes, and synthetic polymers that are finding applications to drug delivery compounds and as active materials for separator membranes in energy conversion devises, [5,15,27]. The scalar functionalized Cahn-Hilliard free energy has been proposed [14,16] as a model of the interaction of a single species of amphiphilic molecule with a solvent, characterizing the density of the amphiphile through a phase function u ∈ H 2 () via the free energy 1 2 ε2 (ε u − W (u))2 − ε p η1 |∇u|2 + η2 W (u) dx. (1.1) FFCH [u] := 2 2 Here W is a double well potential with two unequal depth minima at b− < b+ satisfying W (b− ) = 0 > W (b+ ). The amphiphilic volume fraction is related to the density u − b− with the equilibrium state u = b− corresponding to pure
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solvent. The strength of the lower order functionalization terms are characterized by the value of p, generically selected as 1 or 2, and the values of η1 and η2 . These parameters encode the affinity of the charged elements of the amphiphilic molecule for the solvent (called the solvent quality) and the aspect ratio of the amphiphilic molecule, respectively, (see [2,5,10]). Experimental investigations show that when single-species amphiphilic materials are dispersed in solvent and then allowed to self-assemble, in a process called casting, they form a diverse array of molecular-width structures [12,21]. The associated bifurcation diagram depends subtly upon both the aspect ratio of the amphiphilic molecule and the solvent quality. Molecules with aspect ratio near unity form two-molecule thick bilayer membranes familiar from cellular biology. Larger aspect ratio molecules form higher codimensional structures such as filaments and micelles and complex networks with triple junctions and end-caps. Within the casting experiments the genesis of this structural diversity has been referred to as the onset of ‘morphological complexity’ [20]. Gradient flows of the scalar FCH free energy provid
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