2019/12/21 by Charles M. Elliott, Luke Hatcher
Biochemistry, Genetics and Molecular Biology · Chemistry · Mathematics · #Caveolin-1 and cellular processes #Classical mechanics #Coupling (piping) #Curvature #Geometry #Lipid Membrane Structure and Behavior #Materials science #Mathematical analysis #Mathematics #Perturbation (astronomy) #Physics #Raft #Statistical physics #Surface (topology) #Surfactants and Colloidal Systems #Uniqueness #math.AP #msc:35J35 #msc:65J10 #msc:65N30
paper · pdf · doi:10.1017/s0956792520000297
27 pages, 11 figures
arxiv created 2019/12/21 · openalex publication_date 2020/09/18 · arxiv updated 2020/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We derive and analyse an energy to model lipid raft formation on biological membranes involving a coupling between the local mean curvature and the local composition. We apply a perturbation method recently introduced by Fritz, Hobbs and the first author to describe the geometry of the surface as a graph over an undeformed Helfrich energy minimising surface. The result is a surface Cahn–Hilliard functional coupled with a small deformation energy. We show that suitable minimisers of this energy exist and consider a gradient flow with conserved Allen–Cahn dynamics, for which existence and uniqueness results are proven. Finally, numerical simulations show that for the long-time behaviour raft-like structures can emerge and stabilise, and their parameter dependence is further explored.