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Dynamics of Biomembranes: Effect of the Bulk Fluid

Published online by Cambridge University Press:  10 August 2011

A. Bonito ,
R.H. Nochetto  and
M.S. Pauletti
A. Bonito*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas, USA
R.H. Nochetto
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland, USA
M.S. Pauletti
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas, USA
*
⋆ Corresponding author. E-mail: bonito@math.tamu.edu
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Abstract

We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. The membrane is characterized by its Canham-Helfrich energy (Willmore energy with area constraint) and acts as a boundary force on the Navier-Stokes system modeling an incompressible fluid. We give a concise description of the model and of the associated numerical scheme. We provide numerical simulations with emphasis on the comparisons between different types of flow: the geometric model which does not take into account the bulk fluid and the biomembrane model for two different regimes of parameters.

Keywords

biomembranevesiclered blood cellhelfrichwillmorebending energyparametric finite elementLagrange multipliers
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 25 - 43
Copyright
© EDP Sciences, 2011

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References

R.A. Adams, J.J.F. Fournier. Sobolev spaces. Pure and Applied Mathematics, second edition, Amsterdam, 2003.
Bänsch, E.. Finite element discretization of the Navier-Stokes equations with a free capillary surface. Numer. Math., 88 (2001), No. 2, 203–235. CrossRefGoogle Scholar
Bänsch, E., Morin, P., Nochetto, R.H.. A finite element method for surface diffusion: the parametric case. J. Comput. Phys., 203 (2005), No. 1, 321–343. CrossRefGoogle Scholar
Barrett, J.W., Garcke, H., Nürnberg, R.. Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput., 31 (2008), No. 1, 225–253. CrossRefGoogle Scholar
Bonito, A., Nochetto, R. H., Pauletti, M. S.. Geometrically consistent mesh modification. SIAM J. Numer. Anal., 48 (2010), No. 5, 1877–1899. CrossRefGoogle Scholar
Bonito, A., Nochetto, R.H., Pauletti, M.S.. Parametric fem for geometric biomembranes. J. Comput. Phys., 229 (2010), No. 9, 3171–3188. CrossRefGoogle Scholar
Canham, P.B.. The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. Journal of Theoretical Biology, 26 (1970), No. 1, 61–81. CrossRefGoogle ScholarPubMed
P.G. Ciarlet, J.-L. Lions, editors. Finite element methods. Part 1. Handbook of numerical analysis, 2, North-Holland, Amsterdam, 1991.
Clarenz, U., Diewald, U., Dziuk, G., Rumpf, M., Rusu, R.. A finite element method for surface restoration with smooth boundary conditions. Comput. Aided Geom. Design, 21 (2004), No. 5, 427–445. CrossRefGoogle Scholar
Davis, T.A.. Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software, 30 (2004), No. 2, 196–199. CrossRefGoogle Scholar
Deckelnick, K., Dziuk, G.. Error analysis of a finite element method for the Willmore flow of graphs. Interfaces Free Bound., 8 (2006), No. 1, 21–46. CrossRefGoogle Scholar
Deckelnick, K., Dziuk, G., Elliott, C.M.. Computation of geometric partial differential equations and mean curvature flow. Acta Numer., 14 (2005), 139–232. CrossRefGoogle Scholar
M.C. Delfour, J.-P. Zolésio. Shapes and geometries. Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.
Demlow, A.. Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal., 47 (2009), No. 2, 805–827. CrossRefGoogle Scholar
Doǧan, G., Morin, P., Nochetto, R.H., Verani, M.. Discrete gradient flows for shape optimization and applications. Comput. Methods Appl. Mech. Engrg., 196 (2007), No. 37-40, 3898–3914. CrossRefGoogle Scholar
Droske, M., Rumpf, M.. A level set formulation for Willmore flow. Interfaces Free Bound., 6 (2004), No. 3, 361–378. CrossRefGoogle Scholar
Du, Q., Liu, C., Wang, X.. A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys., 198 (2004), No. 2, 450–468. CrossRefGoogle Scholar
Du, Q., Liu, C., Wang, X.. Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys., 212 (2006), No. 2, 757–777. CrossRefGoogle Scholar
Du, Q., Liu, Ch., Ryham, R., Wang, X.. Energetic variational approaches in modeling vesicle and fluid interactions. Physica D, 238 (2009), No. 9-10, 923–930. CrossRefGoogle Scholar
Dziuk, G.. An algorithm for evolutionary surfaces. Numer. Math., 58 (1991), No. 6, 603–611. CrossRefGoogle Scholar
Dziuk, G.. Computational parametric willmore flow. Numer. Math., 111 (2008), No. 1, 55–80. CrossRefGoogle Scholar
Esedoglu, S., Ruuth, S.J., Tsai, R.. Threshold dynamics for high order geometric motions. Interfaces Free Bound., 10 (2008), No. 3, 263–282. CrossRefGoogle Scholar
Evans, E.A., Skalak, R.. Mechanics and thermodynamics of biomembranes .2. CRC Critical reviews in bioengineering, 3 (1979), No. 4, 331–418. Google ScholarPubMed
M. Giaquinta, S. Hildebrandt. Calculus of variations. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 310, Springer-Verlag, Berlin, 1996.
Helfrich, W.. Elastic properties of lipid bilayers - theory and possible experiments. Zeitschrift Fur Naturforschung C-A Journal Of Biosciences, 28 (1973), 693. CrossRefGoogle ScholarPubMed
D. Hu, P. Zhang, W. E. Continuum theory of a moving membrane. Phys. Rev. E (3), 75 (2007), No. 4, 11.
Jenkins, J.T.. The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math., 32 (1977), No. 4, 755–764. CrossRefGoogle Scholar
Köster, D., Kriessl, O., Siebert, K.G.. Design of finite element tools for coupled surface and volume meshes. Numer. Math. Theor. Meth. Appl., 1 (2008), No. 3, 245–274. Google Scholar
W. Losert. Personal communication, 2009.
M.S. Pauletti. Parametric AFEM for geometric evolution equation and coupled fluid-membrane interaction. Thesis (Ph.D.)–University of Maryland, College Park, 2008.
Rusu, R.E.. An algorithm for the elastic flow of surfaces. Interfaces Free Bound., 7 (2005), No. 3, 229–239. CrossRefGoogle Scholar
A. Schmidt, K.G. Siebert. Design of adaptive finite element software: The finite element toolbox ALBERTA, Lecture Notes in Computational Science and Engineering, 42, Springer-Verlag, Berlin, 2005.
Seifert, U.. Configurations of fluid membranes and vesicles. Advances in Physics, 46 (1997), No. 1, 13–137. CrossRefGoogle Scholar
Steigmann, D.J.. Fluid films with curvature elasticity. Arch. Ration. Mech. Anal., 150 (1999), No. 2, 127–152. CrossRefGoogle Scholar
Steigmann, D.J., Baesu, E., Rudd, R.E., Belak, J., McElfresh, M.. On the variational theory of cell-membrane equilibria. Interfaces Free Bound., 5 (2003), No. 4, 357–366. CrossRefGoogle Scholar
R. Temam. Navier-Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam, 1977.