A poroelastic immersed boundary method with applications to cell biology

The immersed boundary method is a widely used mixed Eulerian/Lagrangian framework for simulating the motion of elastic structures immersed in viscous fluids. In the traditional immersed boundary method, the fluid and structure move with the same velocity field. In this work, a model based on the immersed boundary method is presented for simulating poroelastic media in which the fluid permeates a porous, elastic structure of small volume fraction that moves with its own velocity field. Two distinct methods for calculating elastic stresses are presented and compared. The methods are validated on a radially symmetric test problem by comparing with a finite difference solution of the classical equations of poroelasticity. Finally, two applications of the modeling framework to cell biology are provided: cellular blebbing and cell crawling. It is shown that in both examples, poroelastic effects are necessary to explain the relevant mechanics.

[1]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[2]  Robert Michael Kirby,et al.  Unconditionally stable discretizations of the immersed boundary equations , 2007, J. Comput. Phys..

[3]  T. Pollard,et al.  Cellular Motility Driven by Assembly and Disassembly of Actin Filaments , 2003, Cell.

[4]  Aaron L Fogelson,et al.  The Influence of Hindered Transport on the Development of Platelet Thrombi Under Flow , 2013, Bulletin of mathematical biology.

[5]  L. Mahadevan,et al.  Non-equilibration of hydrostatic pressure in blebbing cells , 2005, Nature.

[6]  James M. Drake,et al.  Biomechanics of the brain: A theoretical and numerical study of Biot's equations of consolidation theory with deformation-dependent permeability , 2005 .

[7]  Nancy Guillen,et al.  Dynamic instability of the intracellular pressure drives bleb-based motility , 2010, Journal of Cell Science.

[8]  Alba Diz-Muñoz,et al.  Control of Directed Cell Migration In Vivo by Membrane-to-Cortex Attachment , 2010, PLoS biology.

[9]  Dharshi Devendran,et al.  An immersed boundary energy-based method for incompressible viscoelasticity , 2012, J. Comput. Phys..

[10]  Andrzej GręObecki,et al.  Membrane and Cytoskeleton Flow in Motile Cells with Emphasis on the Contribution of Free-Living Amoebae , 1994 .

[11]  Héctor D. Ceniceros,et al.  A fast, robust, and non-stiff Immersed Boundary Method , 2011, J. Comput. Phys..

[12]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[13]  Peter Friedl,et al.  Compensation mechanism in tumor cell migration , 2003, The Journal of cell biology.

[14]  Jiannong Fang,et al.  A MLS-Based Lattice Spring Model for Simulating Elasticity of Materials , 2012 .

[15]  Charles S. Peskin,et al.  2-D Parachute Simulation by the Immersed Boundary Method , 2006, SIAM J. Sci. Comput..

[16]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[17]  C. Peskin,et al.  When vortices stick: an aerodynamic transition in tiny insect flight , 2004, Journal of Experimental Biology.

[18]  Bobby Philip,et al.  A Multigrid Method for a Model of the Implicit Immersed Boundary Equations , 2012 .

[19]  Arvind Santhanakrishnan,et al.  A numerical study of the effects of bell pulsation dynamics and oral arms on the exchange currents generated by the upside-down jellyfish Cassiopea xamachana , 2011, Journal of Experimental Biology.

[20]  Thomas Y. Hou,et al.  An efficient semi-implicit immersed boundary method for the Navier-Stokes equations , 2008, J. Comput. Phys..

[21]  D. Srolovitz,et al.  Brittle fracture in materials with random defects. , 1989, Physical review. B, Condensed matter.

[22]  P. Meakin.  A simple model for elastic fracture in thin films , 1987 .

[23]  Jessica A. Gorman.  Cell Biology of Physarum and Didymium. Volume 2: Differentiation, Metabolism, and Methodology. Henry C. Aldrich , John W. Daniel , 1983 .

[24]  Guillaume Charras,et al.  Blebs lead the way: how to migrate without lamellipodia , 2008, Nature Reviews Molecular Cell Biology.

[25]  S. Takagi,et al.  Locomotive mechanism of Physarum plasmodia based on spatiotemporal analysis of protoplasmic streaming. , 2008, Biophysical journal.

[26]  Martin Ostoja-Starzewski,et al.  Lattice models in micromechanics , 2002 .

[27]  Robert M. Kirby,et al.  A Comparison of Implicit Solvers for the Immersed Boundary Equations , 2008 .

[28]  Héctor D. Ceniceros,et al.  Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method , 2009, J. Comput. Phys..

[29]  N G Cogan,et al.  Multiphase flow models of biogels from crawling cells to bacterial biofilms , 2010, HFSP journal.

[30]  Y. Mukaigawa,et al.  Large Deviations Estimates for Some Non-local Equations I. Fast Decaying Kernels and Explicit Bounds , 2022 .

[31]  Ray W. Ogden,et al.  Nonlinear Elastic Deformations , 1985 .

[32]  Mohammad R. K. Mofrad,et al.  Rheology of the Cytoskeleton , 2009 .

[33]  Lucy T. Zhang.  Immersed finite element method for fluid-structure interactions , 2007 .

[34]  W Alt,et al.  Cytoplasm dynamics and cell motion: two-phase flow models. , 1999, Mathematical biosciences.

[35]  L. Heltai,et al.  On the hyper-elastic formulation of the immersed boundary method , 2008 .

[36]  B. Alberts,et al.  Molecular Biology of the Cell (Fifth Edition) , 2008 .

[37]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[38]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[39]  David Farrell,et al.  Immersed finite element method and its applications to biological systems. , 2006, Computer methods in applied mechanics and engineering.

[40]  Boyce E. Griffith,et al.  Immersed boundary method with finite element elasticity , 2010 .

[41]  William A. Curtin,et al.  Mechanics modeling using a spring network , 1990 .

[42]  J. Stockie.  Modelling and simulation of porous immersed boundaries , 2009 .

[43]  Paul Meakin,et al.  Formation of fractal cracks in a kinetic fracture model , 1986, Nature.

[44]  E. F. Haskins,et al.  Cell Biology of Physarum and Didymium , 1983 .

[45]  Wanda Strychalski,et al.  A computational model of bleb formation. , 2013, Mathematical medicine and biology : a journal of the IMA.

[46]  Jonathan Stricker,et al.  Mechanics of the F-actin cytoskeleton. , 2010, Journal of biomechanics.

[47]  B. Alberts,et al.  Molecular Biology of the Cell 4th edition , 2007 .

[48]  János Kertész,et al.  Fractal Shapes of Deterministic Cracks , 1989 .

[49]  L. Fauci,et al.  A computational model of aquatic animal locomotion , 1988 .

[50]  J. Tinevez,et al.  Role of cortical tension in bleb growth , 2009, Proceedings of the National Academy of Sciences.

[51]  Hans G. Othmer,et al.  A continuum model of motility in ameboid cells , 2004, Bulletin of mathematical biology.

[52]  C. Peskin,et al.  Implicit second-order immersed boundary methods with boundary mass , 2008 .

[53]  O. Fackler,et al.  Cell motility through plasma membrane blebbing , 2008, The Journal of cell biology.

[54]  F. MacKintosh,et al.  The mechanics and fluctuation spectrum of active gels. , 2009, The journal of physical chemistry. B.