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Well-posedness and Long-time Behavior of a Bulk-surface Coupled Cahn-Hilliard-diffusion System with Singular Potential for Lipid Raft Formation

Hao Wu, Shengqin Xu

Abstract

We study a bulk-surface coupled system that describes the processes of lipid-phase separation and lipid-cholesterol interaction on cell membranes, in which cholesterol exchange between cytosol and cell membrane is also incorporated. The PDE system consists of a surface Cahn-Hilliard equation for the relative concentration of saturated/unsaturated lipids and a surface diffusion-reaction equation for the cholesterol concentration on the membrane, together with a diffusion equation for the cytosolic cholesterol concentration in the bulk. The detailed coupling between bulk and surface evolutions is characterized by a mass exchange term $q$. For the system with a physically relevant singular potential, we first prove the existence, uniqueness and regularity of global weak solutions to the full bulk-surface coupled system under suitable assumptions on the initial data and the mass exchange term $q$. Next, we investigate the large cytosolic diffusion limit that gives a reduction of the full bulk-surface coupled system to a system of surface equations with non-local contributions. Afterwards, we study the long-time behavior of global solutions in two categories, i.e., the equilibrium and non-equilibrium models according to different choices of the mass exchange term $q$. For the full bulk-surface coupled system with a decreasing total free energy, we prove that every global weak solution converges to a single equilibrium as $t\to +\infty$. For the reduced surface system with a mass exchange term of reaction type, we establish the existence of a global attractor.

Well-posedness and Long-time Behavior of a Bulk-surface Coupled Cahn-Hilliard-diffusion System with Singular Potential for Lipid Raft Formation

Abstract

We study a bulk-surface coupled system that describes the processes of lipid-phase separation and lipid-cholesterol interaction on cell membranes, in which cholesterol exchange between cytosol and cell membrane is also incorporated. The PDE system consists of a surface Cahn-Hilliard equation for the relative concentration of saturated/unsaturated lipids and a surface diffusion-reaction equation for the cholesterol concentration on the membrane, together with a diffusion equation for the cytosolic cholesterol concentration in the bulk. The detailed coupling between bulk and surface evolutions is characterized by a mass exchange term . For the system with a physically relevant singular potential, we first prove the existence, uniqueness and regularity of global weak solutions to the full bulk-surface coupled system under suitable assumptions on the initial data and the mass exchange term . Next, we investigate the large cytosolic diffusion limit that gives a reduction of the full bulk-surface coupled system to a system of surface equations with non-local contributions. Afterwards, we study the long-time behavior of global solutions in two categories, i.e., the equilibrium and non-equilibrium models according to different choices of the mass exchange term . For the full bulk-surface coupled system with a decreasing total free energy, we prove that every global weak solution converges to a single equilibrium as . For the reduced surface system with a mass exchange term of reaction type, we establish the existence of a global attractor.
Paper Structure (32 sections, 16 theorems, 299 equations)

This paper contains 32 sections, 16 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Statement of results
  5. The full bulk-surface coupled system \ref{['1.a']}--\ref{['1.f']}
  6. The reduced system \ref{['r1.a']}--\ref{['r1.f']}
  7. Well-posedness of the Full Bulk-Surface Coupled System
  8. Existence and uniqueness
  9. A regularized system
  10. The Galerkin approximation
  11. Uniform estimates w.r.t $n$
  12. Extension of approximate solutions to $[0,T]$
  13. Passage to the limit as $n\to +\infty$
  14. Passage to the limit as $\kappa\to 0$
  15. Mass conservation and energy identity
  16. ...and 17 more sections

Key Result

Lemma 2.1

Suppose that $\Gamma$ is a two-dimensional smooth, compact and connected hypersurface without boundary and $F$ satisfies the assumption (H1).

Theorems & Definitions (34)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • Remark 2.3
  • Lemma 2.2
  • Remark 2.4
  • Definition 2.1: Weak solutions
  • Remark 2.5
  • Theorem 2.1: Existence and uniqueness
  • Remark 2.6
  • ...and 24 more