A coupled surface-Cahn-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes
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Date
2015-09
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Abstract
We propose and investigate a model for lipid raft formation and dynamics in biological
membranes. The model describes the lipid composition of the membrane and an interaction
with cholesterol. To account for cholesterol exchange between cytosol and cell membrane we couple
a bulk-diffusion to an evolution equation on the membrane. The latter describes a relaxation
dynamics for an energy taking lipid-phase separation and lipid-cholesterol interaction energy into
account. It takes the form of an (extended) Cahn{Hilliard equation. Different laws for the exchange
term represent equilibrium and non-equilibrium models. We present a thermodynamic
justification, analyze the respective qualitative behavior and derive asymptotic reductions of the
model. In particular we present a formal asymptotic expansion near the sharp interface limit,
where the membrane is separated into two pure phases of saturated and unsaturated lipids, respectively.
Finally we perform numerical simulations and investigate the long-time behavior of
the model and its parameter dependence. Both the mathematical analysis and the numerical
simulations show the emergence of raft-like structures in the non-equilibrium case whereas in the
equilibrium case only macrodomains survive in the long-time evolution.
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Keywords
partial differential equations on surfaces, phase separation, Cahn-Hilliard equation, Ohta-Kawasaki energy, reaction-diffusion systems, singular limit