Multi-component phase separation and small deformations of a spherical biomembrane
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911040104562688 |
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| author | Caetano, Diogo Elliott, Charles M. Grasselli, Maurizio Poiatti, Andrea |
| author_facet | Caetano, Diogo Elliott, Charles M. Grasselli, Maurizio Poiatti, Andrea |
| contents | We focus on the derivation and analysis of a model for multi-component phase separation occurring on biological membranes, inspired by observations of lipid raft formation. The model integrates local membrane composition with local membrane curvature, describing the membrane's geometry through a perturbation method represented as a graph over an undeformed Helfrich minimising surface, such as a sphere. The resulting energy consists of a small deformation functional coupled to a Cahn-Hilliard functional. By applying Onsager's variational principle, we obtain a multi-component Cahn-Hilliard equation for the vector $\boldsymbolφ$ of protein concentrations coupled to an evolution equation for the small deformation $u$ along the normal direction to the reference membrane. Then, in the case of a constant mobility matrix, we consider the Cauchy problem and we prove that it is (globally) well posed in a weak setting. We also demonstrate that any weak solution regularises in finite time and satisfies the so-called "strict separation property". This property allows usto show that any weak solution converges to a single stationary state by a suitable version of the Lojasiewicz-Simon inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi-component phase separation and small deformations of a spherical biomembrane Caetano, Diogo Elliott, Charles M. Grasselli, Maurizio Poiatti, Andrea Analysis of PDEs 35B36, 35Q92, 92C15 We focus on the derivation and analysis of a model for multi-component phase separation occurring on biological membranes, inspired by observations of lipid raft formation. The model integrates local membrane composition with local membrane curvature, describing the membrane's geometry through a perturbation method represented as a graph over an undeformed Helfrich minimising surface, such as a sphere. The resulting energy consists of a small deformation functional coupled to a Cahn-Hilliard functional. By applying Onsager's variational principle, we obtain a multi-component Cahn-Hilliard equation for the vector $\boldsymbolφ$ of protein concentrations coupled to an evolution equation for the small deformation $u$ along the normal direction to the reference membrane. Then, in the case of a constant mobility matrix, we consider the Cauchy problem and we prove that it is (globally) well posed in a weak setting. We also demonstrate that any weak solution regularises in finite time and satisfies the so-called "strict separation property". This property allows usto show that any weak solution converges to a single stationary state by a suitable version of the Lojasiewicz-Simon inequality. |
| title | Multi-component phase separation and small deformations of a spherical biomembrane |
| topic | Analysis of PDEs 35B36, 35Q92, 92C15 |
| url | https://arxiv.org/abs/2410.05492 |