Abstract
We present a phase field model for vesicle growth or shrinkage induced by an osmotic pressure due to a chemical potential gradient. The model consists of an Allen–Cahn equation describing the evolution of the phase field parameter that describes the shape of the vesicle and a Cahn–Hilliard-type equation describing the evolution of the ionic fluid. We establish conditions for vesicle growth or shrinkage via a common tangent construction using free energy curves. During the membrane deformation, the model ensures total mass conservation of the ionic fluid, and we weakly enforce a surface area constraint of the vesicle. We develop a stable numerical scheme and an efficient nonlinear multigrid solver to evolve the phase and concentration fields, and we use this to evolve the fields to near equilibrium for 2D vesicles. Convergence tests confirm an \({\mathcal {O}}(t+h^2)\) accuracy for our scheme and near-optimal convergence for our multigrid solver. Numerical results reveal that the diffuse interface model captures the main features of cell shape dynamics: for a growing vesicle, there exist circle-like equilibrium shapes if the concentration difference across the membrane and the initial osmotic pressure are large enough; while for a shrinking vesicle, there exists a rich collection of finger-like equilibrium morphologies.
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Acknowledgements
SL acknowledges the support from the National Science Foundation (NSF), Division of Mathematical Sciences Grant DMS-1720420. SL was also partially supported by Grant ECCS-1307625. SMW acknowledges support from NSF Grant, DMS 2012634. JL acknowledges partial support from the NSF through Grants DMS-1714973, DMS-1719960, and DMS-1763272 and the Simons Foundation (594598QN) for a NSF-Simons Center for Multiscale Cell Fate Research. JL also thanks the National Institutes of Health for partial support through grants 1U54CA217378-01A1 for a National Center in Cancer Systems Biology at UC Irvine and P30CA062203 for the Chao Family Comprehensive Cancer Center at UC Irvine.
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Tang, X., Li, S., Lowengrub, J.S. et al. Phase field modeling and computation of vesicle growth or shrinkage. J. Math. Biol. 86, 97 (2023). https://doi.org/10.1007/s00285-023-01928-2
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DOI: https://doi.org/10.1007/s00285-023-01928-2
Keywords
- Allen–Cahn equation
- Cahn–Hilliard equation
- Osmosis
- Vesicle growth or shrinkage
- Nonlinear multigrid
- Convergence