Stability, convergence, and sensitivity analysis of the Filament Based Lamellipodium Model and the corresponding FEM

This paper focuses on the study of the Filament Based Lamellipodium Model (FBLM) and the corresponding Finite Element Method (FEM) from a numerical point of view. We study fundamental numerical properties of the FEM and justify the further use of the FBLM. We exhibit that the FEM satisfies a timestep stability condition that is consistent with the nature of the problem. We propose a particular strategy to automatically adapt the time step of the method. We show that the FEM convergences with respect to the (two-dimensional) space discretization in a series of characteristic and representative experiments. We embed and couple the FBLM with a complex extracellular environment comprised of chemical and haptic components and study their combined time evolution. Under this prism, we study the sensitivity of the FBLM on several of its controlling parameters and discuss their influence in the development of the model.

[1]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[2]  T. Mitchison,et al.  Actin-Based Cell Motility and Cell Locomotion , 1996, Cell.

[3]  M. Dembo,et al.  Cell movement is guided by the rigidity of the substrate. , 2000, Biophysical journal.

[4]  Nikolaos Sfakianakis,et al.  A Multiscale Approach to the Migration of Cancer Stem Cells: Mathematical Modelling and Simulations , 2016, Bulletin of mathematical biology.

[5]  Gary G. Borisy,et al.  Self-polarization and directional motility of cytoplasm , 1999, Current Biology.

[6]  D. Ambrosi,et al.  Mechanics and polarity in cell motility , 2016, 1605.09713.

[7]  Sally H. Zigmond,et al.  Leukocyte locomotion and chemotaxis. New methods for evaluation, and demonstration of a cell-derived chemotactic factor. , 1973 .

[8]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[9]  Gary G. Borisy,et al.  Analysis of the Actin–Myosin II System in Fish Epidermal Keratocytes: Mechanism of Cell Body Translocation , 1997, The Journal of cell biology.

[10]  H. Moses,et al.  Stimulation of the chemotactic migration of human fibroblasts by transforming growth factor beta , 1987, Journal of Experimental Medicine.

[11]  V. Torre,et al.  Cytoskeletal actin networks in motile cells are critically self-organized systems synchronized by mechanical interactions , 2011, Proceedings of the National Academy of Sciences.

[12]  C. Schmeiser,et al.  An extended Filament Based Lamellipodium Model produces various moving cell shapes in the presence of chemotactic signals. , 2015, Journal of theoretical biology.

[13]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[14]  Gaudenz Danuser,et al.  Actin–myosin network reorganization breaks symmetry at the cell rear to spontaneously initiate polarized cell motility , 2007, The Journal of cell biology.

[15]  Mariano Carrion-Vazquez,et al.  The mechanical hierarchies of fibronectin observed with single-molecule AFM. , 2002, Journal of molecular biology.

[16]  Klemens Rottner,et al.  The lamellipodium: where motility begins. , 2002, Trends in cell biology.

[17]  U. Schwarz,et al.  Modeling cytoskeletal flow over adhesion sites: competition between stochastic bond dynamics and intracellular relaxation , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[18]  Wolfgang Alt,et al.  Continuum model of cell adhesion and migration , 2009, Journal of mathematical biology.

[19]  C. Schmeiser,et al.  Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments , 2008, Cell adhesion & migration.

[20]  J. Small,et al.  Polarity of actin at the leading edge of cultured cells , 1978, Nature.

[21]  E. Elson,et al.  A mechanical function of myosin II in cell motility. , 1995, Journal of cell science.

[22]  J. Fuhrmann,et al.  A free boundary problem for cell motion , 2015, Differential and Integral Equations.

[23]  W Marth,et al.  A mechanism for cell motility by active polar gels , 2014, Journal of The Royal Society Interface.

[24]  Luigi Preziosi,et al.  Cell Mechanics. From single scale-based models to multiscale modeling , 2010 .

[25]  S. Zigmond,et al.  L E U K O C Y T E LOCOMOTION AND C H E M O T A X I S NEW METHODS FOR EVALUATION, AND DEMONSTRATION OF A CELL-DERIVED CHEMOTACTIC FACTOR* , 2003 .

[26]  Harold P. Erickson,et al.  Force Measurements of the α5β1 Integrin–Fibronectin Interaction , 2003 .

[27]  Pekka Lappalainen,et al.  Actin stress fibers – assembly, dynamics and biological roles , 2012, Journal of Cell Science.

[28]  H U Keller,et al.  Chemotactic reorientation of granulocytes stimulated with micropipettes containing fMet-Leu-Phe. , 1981, Journal of cell science.

[29]  Mark A J Chaplain,et al.  Computational modeling of single-cell migration: the leading role of extracellular matrix fibers. , 2012, Biophysical journal.

[30]  D. Lauffenburger,et al.  Cell Migration: A Physically Integrated Molecular Process , 1996, Cell.

[31]  P. Devreotes,et al.  Temporal and spatial regulation of chemotaxis. , 2002, Developmental cell.

[32]  N. Kirchgessner,et al.  Quantitative mapping of averaged focal adhesion dynamics in migrating cells by shape normalization , 2012, Journal of Cell Science.