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Numerical investigations of the role of curvature in strong segregation problems on a given surface. (English) Zbl 1365.92012

Summary: Below the transition temperature in a multi-phase lipid vesicle membrane, phases separate into non-connecting domains that coarsen into larger areas. The free energy of phase properties determines the length of the boundaries separating the regions. Mathematical theory predicts that in a two-phase lipid vesicle, a small geodesic disk of the minority lipids forms at a point of the membrane where the Gauss curvature attains a maximum. The lipid bilayer enables crucial signaling and clustering activities in cell membranes and so it is critical to understand the interplay between membrane curvature and lipid separation. We use numerical simulation with finite elements to probe the connection between curvature and phase. Our numerical solutions affirm the assertion regarding patch formation of the minority lipid and suggest the analytical results are applicable to patches that are not necessarily small. To demonstrate these results, we focus on an ellipsoid-shaped vesicle and determine the phase distribution on this domain by directly minimizing a Landau-type free energy subject to a constraint that describes the proportion of each phase. We investigate the sensitivity of the solution process on the grid size \( h\) and the relation between \( h\), the diffusion coefficient \(\alpha\), the conservation constant \( m\) and the initial phase configuration.

MSC:

92C05 Biophysics
92C37 Cell biology
53A99 Classical differential geometry

Software:

DistMesh
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Full Text: DOI

References:

[1] Gillmor, S.; Lee, J.; Ren, X., The role of gauss curvature in a membrane phase separation problem, Physica D, 240, 1913-1927 (2011) · Zbl 1228.92019
[2] Van Meer, G.; Voelker, D.; Feigenson, G., Membrane lipids: where they are and how they behave, Nat. Rev. Mol. Cell Biol., 9, 2, 112-124 (2008)
[3] Baumgart, T.; Das, S.; Webb, W. W.; Jenkins, J. T., Membrane elasticity in giant vesicles with fluid phase coexistence, Biophys. J., 89, 1067-1080 (2005)
[4] Baumgart, T.; Hess, S. T.; Webb, W. W., Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension, Nature, 425, 821-824 (2003)
[5] Helfrich, W., Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforsch. C, 28, 693-703 (1973)
[6] Lipowsky, R., The conformation of membranes, Nature, 349, 475 (1991)
[7] Seifert, U.; Berndl, K.; Lipowsky, R., Shape transformations of vesicles: phase-diagram for spontaneous-curvature and bilayer-coupling models, Phys. Rev. A, 44, 1182-1202 (1991)
[8] Temam, R., Sur l’approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires II, Arch. Rat. Mech. Anal., 33, 377-385 (1969) · Zbl 0207.16904
[9] DeGiorgi, E., Sulla convergenza di alcune successioni d’integrali del tipo dell’area, Rend. Mat., 8, 6, 277-294 (1975) · Zbl 0316.35036
[10] Modica, L., The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal., 98, 2, 123-142 (1987) · Zbl 0616.76004
[11] Modica, L.; Mortola, S., Un esempio di \(\Gamma^-\)-convergenza, Boll. Un. Mat. Ital. B (5), 14, 1, 285-299 (1977) · Zbl 0356.49008
[12] do Carmo, M. P., Differential Geometry of Curves and Surfaces (1976), Prentice-Hall Inc.: Prentice-Hall Inc. Englewood Cliffs, NJ · Zbl 0326.53001
[14] Persson, P. O.; Strang, G., A simple mesh generator in MATLAB, SIAM Rev., 46, 2, 329-345 (2004) · Zbl 1061.65134
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