Dynamics of Biomembranes: Effect of the Bulk Fluid

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 con- straint) 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.

[1]  Daniel,et al.  Design of Finite Element Tools for Coupled Surface and Volume Meshes , 2008 .

[2]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[3]  Eberhard Bänsch,et al.  Finite element discretization of the Navier–Stokes equations with a free capillary surface , 2001, Numerische Mathematik.

[4]  M. Giaquinta,et al.  Calculus of Variations I , 1995 .

[5]  Editors , 1986, Brain Research Bulletin.

[6]  Ricardo H. Nochetto,et al.  Discrete gradient flows for shape optimization and applications , 2007 .

[7]  Martin Rumpf,et al.  A finite element method for surface restoration with smooth boundary conditions , 2004, Comput. Aided Geom. Des..

[8]  Harald Garcke,et al.  Parametric Approximation of Willmore Flow and Related Geometric Evolution Equations , 2008, SIAM J. Sci. Comput..

[9]  C. M. Elliott,et al.  Computation of geometric partial differential equations and mean curvature flow , 2005, Acta Numerica.

[10]  Robert E. Rudd,et al.  On the variational theory of cell-membrane equilibria , 2003 .

[11]  W. Helfrich.  Elastic Properties of Lipid Bilayers: Theory and Possible Experiments , 1973, Zeitschrift fur Naturforschung. Teil C: Biochemie, Biophysik, Biologie, Virologie.

[12]  Philippe G. Ciarlet,et al.  Finite element methods , 1991 .

[13]  E. Christiansen,et al.  Handbook of Numerical Analysis , 1996 .

[14]  R. Skalak,et al.  Mechanics and thermodynamics of bio membranes 1 , 1979 .

[15]  Ricardo H. Nochetto,et al.  Parametric FEM for geometric biomembranes , 2010, J. Comput. Phys..

[16]  Gerhard Dziuk,et al.  Computational parametric Willmore flow , 2008, Numerische Mathematik.

[17]  G. Dziuk,et al.  An algorithm for evolutionary surfaces , 1990 .

[18]  James T. Jenkins,et al.  The Equations of Mechanical Equilibrium of a Model Membrane , 1977 .

[19]  M. Droske,et al.  A level set formulation for Willmore flow , 2004 .

[20]  Pingwen Zhang,et al.  Continuum theory of a moving membrane. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Q. Du,et al.  Energetic variational approaches in modeling vesicle and fluid interactions , 2009 .

[22]  Gerhard Dziuk,et al.  Error analysis of a finite element method for the Willmore flow of graphs , 2006 .

[23]  Q. Du,et al.  A phase field approach in the numerical study of the elastic bending energy for vesicle membranes , 2004 .

[24]  Ricardo H. Nochetto,et al.  Geometrically Consistent Mesh Modification , 2010, SIAM J. Numer. Anal..

[25]  Steven J. Ruuth,et al.  Threshold dynamics for high order geometric motions , 2006 .

[26]  Raluca E. Rusu.  An algorithm for the elastic flow of surfaces , 2005 .

[27]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.

[28]  P. Canham.  The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. , 1970, Journal of theoretical biology.

[29]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[30]  A. Schmidt,et al.  Design of Adaptive Finite Element Software , 2005 .

[31]  Matthew MacDonald,et al.  Shapes and Geometries , 1987 .

[32]  D. M,et al.  A level set formulation for Willmore flow , 2004 .

[33]  Ricardo H. Nochetto,et al.  A finite element method for surface diffusion: the parametric case , 2005 .

[34]  Qiang Du,et al.  Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions , 2006, J. Comput. Phys..

[35]  Stefan Hildebrandt,et al.  The Lagrangian formalism , 1996 .

[36]  Alan Demlow,et al.  Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces , 2009, SIAM J. Numer. Anal..

[37]  Udo Seifert,et al.  Configurations of fluid membranes and vesicles , 1997 .

[38]  Miguel Sebastian Pauletti,et al.  Parametric AFEM for geometric evolution equation and coupled fluid -membrane interaction , 2008 .

[39]  David J. Steigmann,et al.  Fluid Films with Curvature Elasticity , 1999 .

[40]  Kunibert G. Siebert,et al.  Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA , 2005, Lecture Notes in Computational Science and Engineering.

[41]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[42]  G. Burton.  Sobolev Spaces , 2013 .