Existence, bifurcation, and geometric evolution of quasi-bilayers in the multicomponent functionalized Cahn–Hilliard equation

• Published in: Journal of mathematical biology Vol. 75; no. 2; pp. 443 - 489
• Main Authors: Promislow, Keith, Wu, Qiliang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2017; Springer Nature B.V
• Subjects: Applications of Mathematics; Aspect ratio; Bifurcations; Cell Membrane; Cellular structure; Competition; Copolymers; Curvature; Drug delivery; Drug delivery systems; Evolution; Filaments; Free energy; Gradient flow; Interface stability; Interfaces; Lipid Bilayers - chemistry; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Membranes; Micelles; Models, Biological; 92B05; 35B36; Multi-component bilayer; Billiard limit; Canham–Helfrich energy; 35B40; 53A05; Functionalized Cahn–Hilliard energy
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-016-1089-y

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We present the multicomponent functionalized Cahn–Hilliard (mFCH) free energy as a model which allows competition between bilayers with distinct composition and between bilayers and higher codimensional structures, such as co-dimension two filaments and co-dimension three micelles. We construct symmetric and asymmetric homoclinic bilayer profiles via a billiard limit potential and show that co-dimensional bifurcation is driven by the experimentally observed layer-by-layer pearling mechanism. We investigate the stability and slow geometric evolution of multicomponent bilayer interfaces within the context of an H - 1 gradient flow of the mFCH, addressing the impact of aspect ratio of the amphiphile (lipid or copolymer unit) on the intrinsic curvature and the codimensional bifurcation. In particular we derive a Canham–Helfrich sharp interface energy whose intrinsic curvature arises through a Melnikov parameter associated to amphiphile aspect ratio.