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Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches

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Abstract

Diffuse interface (phase field) models are developed for multi-component vesicle membranes with different lipid compositions and membranes with free boundary. These models are used to simulate the deformation of membranes under the elastic bending energy and the line tension energy with prescribed volume and surface area constraints. By comparing our numerical simulations with recent biological experiments, it is demonstrated that the diffuse interface models can effectively capture the rich phenomena associated with the multi-component vesicle transformation and thus offering great functionality in their simulation and modelling.

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References

  1. Anderson D., McFadden G., Wheeler A. (1998). Diffuse-interface methods in fluid mechanisms. Annu. Rev. Fluid Mech. 30: 139–165

    Article  MathSciNet  Google Scholar 

  2. Baumgart T., Das S., Webb W., Jenkins J. (2005). Membrane elasticity in giant vesicles with fluid phase coexistence. Biophys. J. 89: 1067–1080

    Article  Google Scholar 

  3. Baumgart T., Hess S., Webb W. (2003). Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature 425: 821–824

    Article  Google Scholar 

  4. Benvegnu D., McConnell M. (1992). Line tension between liquid domains in lipid monolayers. J. Phys. Chem. 96: 6820–6824

    Article  Google Scholar 

  5. Biben T., Kassner K., Misbah C. (2005). Phase-field approach to 3D vesicle dynamics. Phys. Rev. E 72: 041921

    Article  Google Scholar 

  6. Boettinger W., Warren J., Beckermann C., Karma A. (2002). Phase-field simulation of solidification. Ann. Rev. Mater. Res. 32: 163–194

    Article  Google Scholar 

  7. Burchard P., Cheng L.-T., Merriman B., Osher S. (2001). Motion of curves in three spatial dimensions using a level set approach. J. Comput. Phys. 170: 720–741

    Article  MATH  MathSciNet  Google Scholar 

  8. Caginalp, G., Chen, X.F.: Phase field equations in the singular limit of sharp interface problems. In: On the Evolution of Phase Boundaries (Minneapolis, MN, 1990–1991), pp. 1–27. Springer, New York (1992)

  9. Capovilla R., Guven J., Santiago J. (2002). Lipid membranes with an edge. Phys. Rev. E 66: 021607

    Article  Google Scholar 

  10. Carmo M. (2006). Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  11. Chen L.-Q. (2002). Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32: 113–140

    Article  Google Scholar 

  12. Ciarlet P.G. (1998). Introduction to Linear Shell Theory. Gauthier-Villars and Elsevier, Paris

    MATH  Google Scholar 

  13. Ciarlet, P.G.: Mathematical elasticity, III: Theory of shells. In: Studies in Mathematics and its Applications. North-Holland, Amsterdam (2000)

  14. Döbereiner, H., Käs, J., Noppl, D., Sprenger, I., Sackmann, E.: Budding and fission of vesicles. Biophys. J. 65, 1396C1403 (1993)

  15. Du Q., Li M., Liu C. (2007). Analysis of a phase field Navier–Stokes vesicle-fluid interaction model. Disc. Cont. Dyn. Sys. B. 8(3): 539–556

    MathSciNet  Google Scholar 

  16. Du Q., Liu C., Ryham R., Wang X. (2005). A phase field formulation of the Willmore problem. Nonlinearity 18: 1249–1267

    Article  MATH  MathSciNet  Google Scholar 

  17. Du Q., Liu C., Ryham R., Wang X. (2006). Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation. Commun. Pure Appl. Anal. 4: 537–548

    MathSciNet  Google Scholar 

  18. Du, Q., Liu, C., Ryham, R., Wang, X.: Modeling vesicle deformations in flow fields via energetic variational approaches (2006, preprint)

  19. Du Q., Liu C., Ryham R., Wang X. (2007). Diffuse interface energies capturing the euler number: relaxation and renormalization. Commun. Math. Sci. 5: 233–242

    MathSciNet  Google Scholar 

  20. Du Q., Liu C., Wang X. (2004). A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198: 450–468

    Article  MATH  MathSciNet  Google Scholar 

  21. Du Q., Liu C., Wang X. (2005). Retrieving topological information for phase field models. SIAM J. Appl. Math. 65: 1913–1932

    Article  MATH  MathSciNet  Google Scholar 

  22. Du Q., Liu C., Wang X. (2006). Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212: 757–777

    Article  MATH  MathSciNet  Google Scholar 

  23. Du Q., Wang X. (2007). Convergence of numerical approximations to a phase field bending elasticity model of membrane deformations. Inter. J. Numer. Anal Model. 4: 441–459

    MathSciNet  Google Scholar 

  24. Du, Q., Zhang, J.: Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations (2007, preprint)

  25. Gozdz W., Gompper G. (1999). Shapes and shape transformations of two-component membranes of complex topology. Phys. Rev. E 59: 4305–4316

    Article  Google Scholar 

  26. Jiang Y., Lookman T., Saxena A. (2000). Phase separation and shape deformation of two-phase membranes. Phys. Rev. E 6: R57–R60

    Article  Google Scholar 

  27. Juelicher F., Lipowsky R. (1996). Shape transformations of vesicles with intramembrane domains. Phys. Rev. E 53: 2670–2683

    Article  Google Scholar 

  28. Kumar P., Gompper G., Lipowsky R. (2001). Budding dynamics of multicomponent membranes. Phys. Rev. Lett. 86: 3911–3914

    Article  Google Scholar 

  29. Lipowsky R. (1992). Budding of membranes induced by intramembrane domains. Journal de Physique II, France 2: 1825–1840

    Article  Google Scholar 

  30. Lipowsky R. (1995). The morphology of lipid membranes. Curr. Opin. Struct. Biol. 5: 531–540

    Article  Google Scholar 

  31. Lipowsky R. (2002). Domains and rafts in membranes hidden dimensions of self-organization. J. Biol. Phys. 28: 195–210

    Article  Google Scholar 

  32. McWhirter J., Ayton G., Voth G. (2004). Coupling field theory with mesoscopic dynamical simulations of multicomponent lipid bilayers. Biophys. J. 87: 3242–3263

    Article  Google Scholar 

  33. Mukherjee S., Maxfield F. (2004). Membrane domains. Annu. Rev. Cell Dev. Biol. 20: 839–866

    Article  Google Scholar 

  34. Ou-Yang Z., Liu J., Xie Y. (1999). Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases. World Scientific, Singapore

    MATH  Google Scholar 

  35. Osher S., Fedkiw R. (2002). The Level Set Method and Dynamic Implicit Surfaces. Springer, Heidelberg

    Google Scholar 

  36. Saitoh A., Takiguchi K., Tanaka Y., Hotani H. (1998). Opening-up of liposomal membranes by talin. Proc. Natl. Acad. Sci. Biophys. 956: 1026–1031

    Article  Google Scholar 

  37. Seifert U. (1993). Curvature-induced lateral phase separation in two-component vesicles. Phys. Rev. Lett. 70: 1335–1338

    Article  MathSciNet  Google Scholar 

  38. Sethian J.A. (1999). Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science. Cambridge University Press, New York

    MATH  Google Scholar 

  39. Simons K., Vaz W. (2004). Model systems, lipid rafts and cell membranes. Annu. Rev. Biophys. Biomol. Struct. 33: 269–295

    Article  Google Scholar 

  40. Tu, Z., Ou-Yang, Z.: Lipid membranes with free edges. Phys. Rev. E 68, 061915 (1–7) (2003)

  41. Tu Z., Ou-Yang Z. (2004). A geometric theory on the elasticity of bio-membranes. J. Phys. A Math. Gen. 37: 11407–11429

    Article  MATH  MathSciNet  Google Scholar 

  42. Umeda, T., Suezaki, Y., Takiguchi, K., Hotani, H.: Theoretical analysis of opening-up vesicles with single and two holes. Phys. Rev. E 71, 011913 (1–8) (2005)

  43. Wang, X.: Phase Field Models and Simulations of Vesicle Bio-membranes. Ph.D thesis, Department of Mathematics, Penn State University (2005)

  44. Yin Y., Yin J., Ni D. (2005). General mathematical frame for open or closed biomembranes I: Equilibrium theory and geometrically constraint equation. J. Math. Biol. 51: 403–413

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Computational Sciences, Florida State University, Tallahassee, FL, 32306, USA

    Xiaoqiang Wang

  2. Department of Mathematics and Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA, 16802, USA

    Qiang Du

Authors
  1. Xiaoqiang Wang
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  2. Qiang Du
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Correspondence to Qiang Du.

Additional information

This research is supported in part by NSF-DMS 0409297 and NSF-ITR 0205232. Part of the work was performed while the first author was supported by the Institute for Mathematics and its Applications at the University of Minnesota.

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Wang, X., Du, Q. Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol. 56, 347–371 (2008). https://doi.org/10.1007/s00285-007-0118-2

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  • DOI: https://doi.org/10.1007/s00285-007-0118-2

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