Bulletin of Mathematical Biology

, Volume 80, Issue 11, pp 2789–2827 | Cite as

Stability, Convergence, and Sensitivity Analysis of the FBLM and the Corresponding FEM

  • N. SfakianakisEmail author
  • A. Brunk
Original Article


We study in this paper the filament-based lamellipodium model (FBLM) and the corresponding finite element method (FEM) used to solve it. We investigate fundamental numerical properties of the FEM and justify its further use with the FBLM. We show that the FEM satisfies a time step stability condition that is consistent with the nature of the problem and propose a particular strategy to automatically adapt the time step of the method. We show that the FEM converges with respect to the (two-dimensional) space discretization in a series of characteristic and representative chemotaxis and haptotaxis experiments. We embed and couple the FBLM with a complex and adaptive extracellular environment comprised of chemical and adhesion components that are described by their macroscopic density and study their combined time evolution. With this combination, we study the sensitivity of the FBLM on several of its controlling parameters and discuss their influence in the dynamics of the model and its future evolution. We finally perform a number of numerical experiments that reproduce biological cases and compare the results with the ones reported in the literature.


Lamellipodium Actin filaments Cell motility Convergence Stability Sensitivity analysis 



The authors would like to thank Christian Schmeiser, Anna Marciniak-Czochra, and Mark Chaplain for the fruitful discussions and suggestions during the preparation of this manuscript. NS acknowledges also the support of the SFB 873: “Maintenance and Differentiation of Stem Cells in Development and Disease”.

Supplementary material

11538_2018_460_MOESM1_ESM.mp4 (14 mb)
Supplementary material 1 (mp4 14352 KB)
11538_2018_460_MOESM2_ESM.mp4 (869 kb)
Supplementary material 2 (mp4 868 KB)
11538_2018_460_MOESM3_ESM.mp4 (27 mb)
Supplementary material 3 (mp4 27617 KB)

Supplementary material 4 (mp4 30581 KB)

11538_2018_460_MOESM5_ESM.mp4 (1.5 mb)
Supplementary material 5 (mp4 1582 KB)
11538_2018_460_MOESM6_ESM.mp4 (1.7 mb)
Supplementary material 6 (mp4 1745 KB)

Supplementary material 7 (mp4 17083 KB)


  1. Alt W, Kuusela E (2009) Continuum model of cell adhesion and migration. J Math Biol 58(1–2):135MathSciNetzbMATHGoogle Scholar
  2. Ambrosi D, Zanzottera A (2016) Mechanics and polarity in cell motility. Phys D 330:58–66MathSciNetCrossRefGoogle Scholar
  3. Blanchoin L, Boujemaa-Paterski R, Sykes C, Plastino J (2014) Actin dynamics, architecture, and mechanics in cell motility. Physiol Rev 94:235–263CrossRefGoogle Scholar
  4. Brunk A, Kolbe N, Sfakianakis N (2016) Chemotaxis and haptotaxis on a cellular level. In: Proceedings of XVI international conference on hyperbolic problemsGoogle Scholar
  5. Campos D, Mendez V, Llopis I (2010) Persistent random motion: uncovering cell migration dynamics. J Theor Biol 21:526–534MathSciNetCrossRefGoogle Scholar
  6. Cardamone L, Laio A, Torre V, Shahapure R, DeSimone A (2011) Cytoskeletal actin networks in motile cells are critically self-organized systems synchronized by mechanical interactions. PNAS 108:13978–13983CrossRefGoogle Scholar
  7. Chen WT (1981) Mechanism of retraction of the trailing edge during fibroblast movement. J Cell Biol 90(1):187–200CrossRefGoogle Scholar
  8. Freistühler H, Schmeiser C, Sfakianakis N (2012) Stable length distributions in co-localized polymerizing and depolymerizing protein filaments. SIAM J Appl Math 72:1428–1448MathSciNetCrossRefGoogle Scholar
  9. Fuhrmann J, Stevens A (2015) A free boundary problem for cell motion. Diff Integr Equ 28:695–732MathSciNetzbMATHGoogle Scholar
  10. Gerisch G, Keller HU (1981) Chemotactic reorientation of granulocytes stimulated with micropipettes containing fMet-Leu-Phe. J Cell Sci 52:1–10Google Scholar
  11. Gittes F, Mickey B, Nettleton J, Howard J (1993) Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape. J Cell Biol 120(4):923–34CrossRefGoogle Scholar
  12. Hestenes MR (1969) Multiplier and gradient methods. J Optim Theory Appl 4:303–320MathSciNetCrossRefGoogle Scholar
  13. Hundsdorfer W, Verwer JG (2003) Numerical solution of time-dependent advection–diffusion–reaction equations. Springer, BerlinCrossRefGoogle Scholar
  14. Iijima M, Huang YE, Devreotes J (2002) Temporal and spatial regulation of chemotaxis. Dev Cell 3(4):469–478CrossRefGoogle Scholar
  15. Jay PY, Pham PA, Wong SA, Elson EL (1995) A mechanical function of myosin II in cell motility. J Cell Sci 108(1):387–393Google Scholar
  16. Jiang GS, Shu CW (1996) Efficient implementation of weighted eno schemes. J Comput Phys 126:202–228MathSciNetCrossRefGoogle Scholar
  17. Kennedy CA, Carpenter MH (2003) Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl Numer Math 1(44):139–181MathSciNetCrossRefGoogle Scholar
  18. Kolbe N, Katuchova J, Sfakianakis N, Hellmann N, Lukacova-Medvidova M (2016) A study on time discretization and adaptive mesh refinement methods for the simulation of cancer invasion. Appl Math Comput 273:353–376MathSciNetGoogle Scholar
  19. Krylov AN (1931) On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined. Otdel Mat Estest Nauk 87(4):491–539Google Scholar
  20. Lauffenburger DA, Horwitz AF (1996) Cell migration: a physically integrated molecular process. Cell 84(3):359–69CrossRefGoogle Scholar
  21. LeVeque R (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  22. Li F, Redick SD, Erickson HP, Moy VT (2003) Force measurements of the \(\alpha 5\beta 1\) integrin–fibronectin interaction. Biophys J 84(2):1252–1262CrossRefGoogle Scholar
  23. Li L, Noerrelykke SF, Cox EC (2008) Persistent cell motion in the absence of external signals: a search strategy for eukaryotic cells. PLOS ONE 35:e2093CrossRefGoogle Scholar
  24. Lo CM, Wang HB, Dembo M, Wang YL (2000) Cell movement is guided by the rigidity of the substrate. J Biophys 79(1):144–152CrossRefGoogle Scholar
  25. Madzvamuse A, George UZ (2013) The moving grid finite element method applied to cell movement and deformation. Finite Element Anal Des 74:76–92MathSciNetCrossRefGoogle Scholar
  26. Manhart A, Schmeiser C (2017) Existence of and decay to equilibrium of the filament end density along the leading edge of the lamellipodium. J Math Biol 74:169–193MathSciNetCrossRefGoogle Scholar
  27. Manhart A, Oelz D, Schmeiser C, Sfakianakis N (2015) An extended filament based lamellipodium: model produces various moving cell shapes in the presence of chemotactic signals. J Theor Biol 382:244–258MathSciNetCrossRefGoogle Scholar
  28. Manhart A, Oelz D, Schmeiser C, Sfakianakis N (2016) Numerical treatment of the filament based lamellipodium model (FBLM). In: Modelling cellular systemsGoogle Scholar
  29. Marth W, Praetorius S, Voigt A (2015) A mechanism for cell motility by active polar gels. J R Soc Interface 12:20150161CrossRefGoogle Scholar
  30. Milišić V, Oelz D (2011) On the asymptotic regime of a model for friction mediated by transient elastic linkages. J Math Pures Appl 96(5):484–501MathSciNetCrossRefGoogle Scholar
  31. Mitchison TJ, Cramer LP (1996) Actin-based cell motility and cell locomotion. Cell 84(3):371–379CrossRefGoogle Scholar
  32. Möhl C, Kirchgessner N, Schäfer C, Hoffmann B, Merkel R (2012) Quantitative mapping of averaged focal adhesion dynamics in migrating cells by shape normalization. J Cell Sci 125:155–165CrossRefGoogle Scholar
  33. Nickaeen M, Novak IL, Pulford S, Rumack A, Brandon J, Slepchenko BM, Mogilner A (2017) A free-boundary model of a motile cell explains turning behavior. PLOS Comput Biol 13:e1005862CrossRefGoogle Scholar
  34. Oberhauser AF, Badilla-Fernandez C, Carrion-Vazquez M, Fernandez JM (2002) The mechanical hierarchies of fibronectin observed with single-molecule AFM. J Mol Biol 319(2):433–47CrossRefGoogle Scholar
  35. Oelz D, Schmeiser C (2010) Cell mechanics: from single scale-based models to multiscale modeling. How do cells move? Mathematical modeling of cytoskeleton dynamics and cell migration. Chapman and Hall, LondonCrossRefGoogle Scholar
  36. Oelz D, Schmeiser C (2010) Derivation of a model for symmetric lamellipodia with instantaneous cross-link turnover. Arch Ration Mech Anal 198:963–980MathSciNetCrossRefGoogle Scholar
  37. Oelz D, Schmeiser C, Small JV (2008) Modeling of the actin–cytoskeleton in symmetric lamellipodial fragments. Cell Adhes Migr 2:117–126CrossRefGoogle Scholar
  38. Postlethwaite AE, Keski-Oja J (1987) Stimulation of the chemotactic migration of human fibroblasts by transforming growth factor beta. J Exp Med 165(1):251–256CrossRefGoogle Scholar
  39. Powell MJD (1969) Optimization. A method for nonlinear constraints in minimization problems. Academic Press, London, pp 283–298Google Scholar
  40. Rubinstein B, Fournier MF, Jacobson K, Verkhovsky AB, Mogilner A (2009) Actin-myosin viscoelastic flow in the keratocyte lamellipod. Biophys J 97(7):1853–1863CrossRefGoogle Scholar
  41. Sabass B, Schwarz US (2010) Modeling cytoskeletal flow over adhesion sites: competition between stochastic bond dynamics and intracellular relaxation. J Phys Condens Matter 22:194112CrossRefGoogle Scholar
  42. Schlüter I, Ramis-Conde DK, Chaplain MA (2012) Computational modeling of single-cell migration: the leading role of extracellular matrix fibers. J Biophys 103(6):1141–51CrossRefGoogle Scholar
  43. Schwarz US, Gardel ML (2012) United we stand—integrating the actin cytoskeleton and cell-matrix adhesions in cellular mechanotransduction. J Cell Sci 125:3051–3060CrossRefGoogle Scholar
  44. Scianna M, Preziosi L, Wolf K (2013) A cellular potts model simulating cell migration on and in matrix environments. Math Biosci Eng 10:235–261MathSciNetCrossRefGoogle Scholar
  45. Selmeczi D, Mosler S, Hagedorn PH, Larsen NB, Flyvbjerg H (2005) Cell motility as persistent random motion: theories from experiments. Biophys J 89:912–931CrossRefGoogle Scholar
  46. Sfakianakis N, Kolbe N, Hellmann N, Lukacova-Medvidova M (2017) A multiscale approach to the migration of cancer stem cells. Bull Math Biol 79:209–235MathSciNetCrossRefGoogle Scholar
  47. Shu CW (2009) High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev 51:82–126MathSciNetCrossRefGoogle Scholar
  48. Small JV, Isenberg G, Celis JE (1978) Polarity of actin at the leading edge of cultured cells. Nature 272:638–639CrossRefGoogle Scholar
  49. Small JV, Stradal T, Vignal E, Rottner K (2002) The lamellipodium: where motility begins. Trends Cell Biol 12(3):112–20CrossRefGoogle Scholar
  50. Svitkina TM, Verkhovsky AB, McQuade KM, Borisy GG (1997) Analysis of the actin–myosin II system in fish epidermal keratocytes: mechanism of cell body translocation. J Cell Biol 139(2):397–415CrossRefGoogle Scholar
  51. Tojkander S, Gateva G, Lappalainen P (2012) Actin stress fibers—assembly, dynamics and biological roles. J Cell Sci 125(8):1855–1864CrossRefGoogle Scholar
  52. van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Sci Comput 13(2):631–644MathSciNetCrossRefGoogle Scholar
  53. Verkhovsky AB, Svitkina TM, Borisy GG (1999) Self-polarisation and directional motility of cytoplasm. Curr Biol 9(1):11–20CrossRefGoogle Scholar
  54. Yam PT, Wilson CA, Ji L, Herbert B, Barnhart EL, Dye NA, Wiseman PW, Danuser G, Theriot JA (2007) Actin–myosin network reorganisation breaks symmetry at the cell rear to sponaneously initiate polarized cell motility. J Cell Biol 178(7):1207–1221CrossRefGoogle Scholar
  55. Zigmond SH, Hirsch JG (1973) Leukocyte locomotion and chemotaxis. J Exp Med 137:387–410CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Institute of Applied MathematicsHeidelberg UniversityHeidelbergGermany
  2. 2.Institute of MathematicsJohannes Gutenberg-UniversityMainzGermany

Personalised recommendations