Abstract
We describe in this work the numerical treatment of the Filament-Based Lamellipodium Model (FBLM). This model is a two-phase two-dimensional continuum model, describing the dynamics of two interacting families of locally parallel F-actin filaments. It includes, among others, the bending stiffness of the filaments, adhesion to the substrate, and the cross-links connecting the two families. The numerical method proposed is a Finite Element Method (FEM) developed specifically for the needs of this problem. It is comprised of composite Lagrange–Hermite two-dimensional elements defined over a two-dimensional space. We present some elements of the FEM and emphasize in the numerical treatment of the more complex terms. We also present novel numerical simulations and compare to in-vitro experiments of moving cells.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
W. Alt and M. Dembo. Cytoplasm dynamics and cell motion: Two-phase flow models. Math. Biosci., 156(1–2):207–228, 1999.
L. Blanchoin, R. Boujemaa-Paterski, C. Sykes, and J. Plastino. Actin dynamics, architecture, and mechanics in cell motility. Physiological Reviews.
D. Braess. Finite Elements. Theory, fast solvers, and applications to solid mechanics. Cambridge University Press, 2001.
G. Csucs, K. Quirin, and G. Danuser. Locomotion of fish epidermal keratocytes on spatially selective adhesion patterns. Cell Motility and the Cytoskeleton, 64(11):856–867, 2007.
F. Gittes, B. Mickey, J. Nettleton, and J. Howard. Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape. The Journal of Cell Biology, 120(4):923–34, 1993.
M.E. Gracheva and H.G. Othmer. A continuum model of motility in ameboid cells. Bull. Math. Biol., 66(1):167–193, 2004.
H.P. Grimm, A.B. Verkhovsky, A. Mogilner, and J.-J. Meister. Analysis of actin dynamics at the leading edge of crawling cells: implications for the shape of keratocytes. European Biophysics Journal, 32:563–577, 2003.
C.I. Lacayo, Z. Pincus, M.M. VanDuijn, C.A. Wilson, D.A. Fletcher, F.B. Gertler, A. Mogilner, and J.A. Theriot. Emergence of large-scale cell morphology and movement from local actin filament growth dynamics. 2007.
T. Lämmermann, M. Sixt, Mechanical modes of amoeboid cell migration. Current Opinion in Cell Biology 21(5), 636–644 (2009)
F. Li, S.D. Redick, H.P. Erickson, and V.T. Moy. Force measurements of the \(\alpha 5\beta 1\) integrin-fibronectin interaction. Biophysical Journal, 84(2):1252–1262, 2003.
I.V. Maly and G.G. Borisy. Self-organization of a propulsive actin network as an evolutionary process. Proc. Natl. Acad. Sci., 98:11324–11329, 2001.
A. Manhart, D. Oelz, C. Schmeiser, and N. Sfakianakis. An extended Filament Based Lamellipodium Model produces various moving cell shapes in the presence of chemotactic signals. J. Theor. Biol., 382:244–258, 2015.
A. F. M. Marée, A. Jilkine, A. Dawes, V. A. Grieneisen, and L. Edelstein-Keshet. Polarization and movement of keratocytes: a multiscale modelling approach. Bull. Math. Biol., 68(5):1169–1211, 2006.
R.W. Metzke, M.R.K. Mofrad, and W.A. Wall. Coupling atomistic simulation to a continuum based model to compute the mechanical properties of focal adhesions. Biophysical Journal, 96(3, Supplement 1):673a –, 2009.
S.J. Mousavi and M.H. Doweidar. Three-dimensional numerical model of cell morphology during migration in multi-signaling substrates. PLoS ONE, 10(3), 2015.
A.F. Oberhauser, C. Badilla-Fernandez, M. Carrion-Vazquez, and J.M. Fernandez. The mechanical hierarchies of fibronectin observed with single-molecule AFM. Journal of Molecular Biology, 319(2):433–47, 2002.
D. Oelz and C. Schmeiser. Cell mechanics: from single scale-based models to multiscale modeling., chapter How do cells move? Mathematical modeling of cytoskeleton dynamics and cell migration. Chapman and Hall, 2010.
D. Oelz and C. Schmeiser. Derivation of a model for symmetric lamellipodia with instantaneous cross-link turnover. Archive for Rational Mechanics and Analysis, 198:963–980, 2010.
D. Oelz and C. Schmeiser. Simulation of lamellipodial fragments. Journal of Mathematical Biology, 64:513–528, 2012.
D. Oelz, C. Schmeiser, and J.V. Small. Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments. Cell Adhesion and Migration, 2:117–126, 2008.
B. Rubinstein, K. Jacobson, and A. Mogilner. Multiscale two-dimensional modeling of a motile simple-shaped cell. Multiscale Model. Simul., 3(2):413–439, 2005.
B. Rubinstein, M.F. Fournier, K. Jacobson, A.B Verkhovsky, and A. Mogilner. Actin-myosin viscoelastic flow in the keratocyte lamellipod. Biophysical Journal, 97(7):1853–1863, 2009.
Y. Sakamoto, S. Prudhomme, and M.H. Zaman. Modeling of adhesion, protrusion, and contraction coordination for cell migration simulations. Journal of Mathematical Biology, 68(1–2):267–302, 2014.
T.E. Schaus, E. W. Taylor, and G. G. Borisy. Self-organization of actin filament orientation in the dendritic-nucleation/array-treadmilling model. Proc Natl Acad Sci USA, 104(17):7086–91, 2007.
J.V. Small, G. Isenberg, and J.E. Celis. Polarity of actin at the leading edge of cultured cells. Nature, bf 272:638–639, 1978.
J.V. Small, T. Stradal, E. Vignal, and K. Rottner. The lamellipodium: where motility begins. Trends in Cell Biology, 12(3):112–20, 2002.
A.B. Verkhovsky, T.M. Svitkina, and G.G. Borisy. Self-polarisation and directional motility of cytoplasm. Current Biology, 9(1):11–20, 1999.
M. Vinzenz, M. Nemethova, F. Schur, J. Mueller, A. Narita, E. Urban, C. Winkler, C. Schmeiser, S.A. Koestler, K. Rottner, G.P. Resch, Y. Maeda, and J.V. Small. Actin branching in the initiation and maintenance of lamellipodia. Journal of Cell Science, 125(11):2775–2785, 2012.
Acknowledgements
This work has been supported by the Austrian Science Fund through grant no. J-3463 and through the PhD program Dissipation and Dispersion in Nonlinear PDEs, grant no. W1245. The authors also acknowledge support by the Vienna Science and Technology Fund, grant no. LS13-029. N. Sfakianakis wishes to thank the Alexander von Humboldt Foundation and the Center of Computational Sciences (CSM) of Mainz for their support, and M. Lukacova for the fruitful discussions during the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Manhart, A., Oelz, D., Schmeiser, C., Sfakianakis, N. (2017). Numerical Treatment of the Filament-Based Lamellipodium Model (FBLM). In: Graw, F., Matthäus, F., Pahle, J. (eds) Modeling Cellular Systems. Contributions in Mathematical and Computational Sciences, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-45833-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-45833-5_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45831-1
Online ISBN: 978-3-319-45833-5
eBook Packages: EngineeringEngineering (R0)