## Abstract

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.

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## References

Alt W, Kuusela E (2009) Continuum model of cell adhesion and migration. J Math Biol 58(1–2):135

Ambrosi D, Zanzottera A (2016) Mechanics and polarity in cell motility. Phys D 330:58–66

Blanchoin L, Boujemaa-Paterski R, Sykes C, Plastino J (2014) Actin dynamics, architecture, and mechanics in cell motility. Physiol Rev 94:235–263

Brunk A, Kolbe N, Sfakianakis N (2016) Chemotaxis and haptotaxis on a cellular level. In: Proceedings of XVI international conference on hyperbolic problems

Campos D, Mendez V, Llopis I (2010) Persistent random motion: uncovering cell migration dynamics. J Theor Biol 21:526–534

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 108:13978–13983

Chen WT (1981) Mechanism of retraction of the trailing edge during fibroblast movement. J Cell Biol 90(1):187–200

Freistühler H, Schmeiser C, Sfakianakis N (2012) Stable length distributions in co-localized polymerizing and depolymerizing protein filaments. SIAM J Appl Math 72:1428–1448

Fuhrmann J, Stevens A (2015) A free boundary problem for cell motion. Diff Integr Equ 28:695–732

Gerisch G, Keller HU (1981) Chemotactic reorientation of granulocytes stimulated with micropipettes containing fMet-Leu-Phe. J Cell Sci 52:1–10

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):923–34

Hestenes MR (1969) Multiplier and gradient methods. J Optim Theory Appl 4:303–320

Hundsdorfer W, Verwer JG (2003) Numerical solution of time-dependent advection–diffusion–reaction equations. Springer, Berlin

Iijima M, Huang YE, Devreotes J (2002) Temporal and spatial regulation of chemotaxis. Dev Cell 3(4):469–478

Jay PY, Pham PA, Wong SA, Elson EL (1995) A mechanical function of myosin II in cell motility. J Cell Sci 108(1):387–393

Jiang GS, Shu CW (1996) Efficient implementation of weighted eno schemes. J Comput Phys 126:202–228

Kennedy CA, Carpenter MH (2003) Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl Numer Math 1(44):139–181

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 273:353–376

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):491–539

Lauffenburger DA, Horwitz AF (1996) Cell migration: a physically integrated molecular process. Cell 84(3):359–69

LeVeque R (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge

Li F, Redick SD, Erickson HP, Moy VT (2003) Force measurements of the \(\alpha 5\beta 1\) integrin–fibronectin interaction. Biophys J 84(2):1252–1262

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:e2093

Lo CM, Wang HB, Dembo M, Wang YL (2000) Cell movement is guided by the rigidity of the substrate. J Biophys 79(1):144–152

Madzvamuse A, George UZ (2013) The moving grid finite element method applied to cell movement and deformation. Finite Element Anal Des 74:76–92

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 74:169–193

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 382:244–258

Manhart A, Oelz D, Schmeiser C, Sfakianakis N (2016) Numerical treatment of the filament based lamellipodium model (FBLM). In: Modelling cellular systems

Marth W, Praetorius S, Voigt A (2015) A mechanism for cell motility by active polar gels. J R Soc Interface 12:20150161

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):484–501

Mitchison TJ, Cramer LP (1996) Actin-based cell motility and cell locomotion. Cell 84(3):371–379

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 125:155–165

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:e1005862

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):433–47

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, London

Oelz D, Schmeiser C (2010) Derivation of a model for symmetric lamellipodia with instantaneous cross-link turnover. Arch Ration Mech Anal 198:963–980

Oelz D, Schmeiser C, Small JV (2008) Modeling of the actin–cytoskeleton in symmetric lamellipodial fragments. Cell Adhes Migr 2:117–126

Postlethwaite AE, Keski-Oja J (1987) Stimulation of the chemotactic migration of human fibroblasts by transforming growth factor beta. J Exp Med 165(1):251–256

Powell MJD (1969) Optimization. A method for nonlinear constraints in minimization problems. Academic Press, London, pp 283–298

Rubinstein B, Fournier MF, Jacobson K, Verkhovsky AB, Mogilner A (2009) Actin-myosin viscoelastic flow in the keratocyte lamellipod. Biophys J 97(7):1853–1863

Sabass B, Schwarz US (2010) Modeling cytoskeletal flow over adhesion sites: competition between stochastic bond dynamics and intracellular relaxation. J Phys Condens Matter 22:194112

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):1141–51

Schwarz US, Gardel ML (2012) United we stand—integrating the actin cytoskeleton and cell-matrix adhesions in cellular mechanotransduction. J Cell Sci 125:3051–3060

Scianna M, Preziosi L, Wolf K (2013) A cellular potts model simulating cell migration on and in matrix environments. Math Biosci Eng 10:235–261

Selmeczi D, Mosler S, Hagedorn PH, Larsen NB, Flyvbjerg H (2005) Cell motility as persistent random motion: theories from experiments. Biophys J 89:912–931

Sfakianakis N, Kolbe N, Hellmann N, Lukacova-Medvidova M (2017) A multiscale approach to the migration of cancer stem cells. Bull Math Biol 79:209–235

Shu CW (2009) High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev 51:82–126

Small JV, Isenberg G, Celis JE (1978) Polarity of actin at the leading edge of cultured cells. Nature 272:638–639

Small JV, Stradal T, Vignal E, Rottner K (2002) The lamellipodium: where motility begins. Trends Cell Biol 12(3):112–20

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):397–415

Tojkander S, Gateva G, Lappalainen P (2012) Actin stress fibers—assembly, dynamics and biological roles. J Cell Sci 125(8):1855–1864

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):631–644

Verkhovsky AB, Svitkina TM, Borisy GG (1999) Self-polarisation and directional motility of cytoplasm. Curr Biol 9(1):11–20

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):1207–1221

Zigmond SH, Hirsch JG (1973) Leukocyte locomotion and chemotaxis. J Exp Med 137:387–410

## Acknowledgements

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”.

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## Appendices

### Appendix A: The FEM for the FBLM

We numerically solve the FBLM (6) with a problem-specific FEM that was first presented in Manhart et al. (2016). Here we present some of its components.

The maximal filament length varies around the lamellipodium, and in effect the computational domain

is non-rectangular. For consistency and stability reasons we recover the orthogonality of the domain *B*(*t*), using the coordinate transformation

and replace it by

See also Fig. 17.

Accordingly, the weak formulation of (6), recasts into

with \(\mathbf {F}, \mathbf {G}\in H^1_{\alpha }\left( (0,2\pi );\,H^2_s(-\,1,0)\right) \). In a similar manner the modified material derivative and inextensibility conditions read

and

We decompose \(B_0\) into disjoined rectangular *computational cells* as follows:

for \(\alpha _i = (i-1)\Delta \alpha , \Delta \alpha = \frac{2\pi }{N_{\alpha }}, i = 1,\ldots ,N_{\alpha }+1\), and \(s_j = -\,1 + (j-1)\Delta s, \Delta s = \frac{1}{N_s - 1}, j = 1,\ldots ,N_s\). The resolution of the grid along the \(\alpha \) and *s* directions is denoted by \(\alpha _{\text {nodes}}, s_{\text {nodes}}\). The \(\alpha \)-periodicity assumption suggests that \(\alpha _{N_{\alpha }+1} = 2\pi \) is identified with \(\alpha _1 = 0\).

We follow Manhart et al. (2016) and set the conforming FE space

of continuous functions that are continuously differentiable with respect to *s*, and such that on each computational cell they coincide with a first-order polynomial in \(\alpha \), and a third-order polynomial in *s*.

In particular, we consider for \(i = 1,\ldots ,N_{\alpha }+1, i = j,\ldots ,N_s\), and \((\alpha ,s)\in C_{i,j}\), that

with

and that \(H_k^{i,j}(\alpha ,s)=0, k=1,\ldots ,8\), whenever \((\alpha ,s)\not \in C_{i,j}\). The basis functions are then defined as:

for \(i=1,\ldots ,N_{\alpha },\, j=1,\ldots ,N_s\), and the element \(\mathbf {F}\in {\mathcal {V}}\) can be represented in terms of the point values \(\mathbf {F}_{i,j}\) and the *s*-derivatives \(\partial _s \mathbf {F}_{i,j}\) at the discretization nodes, as:

The FE formulation of the lamellipodium problem on the time interval [0, *T*] is to find \(\mathbf {F}\in C^1\big ([0,T];\,{\mathcal {V}}\big )\), such that (42) holds for all \(\mathbf {G}\in C\big ([0,T];\,{\mathcal {V}}\big )\).

We only mention here that the enforcement of the inextensibility condition (3), and in effect the computation of \(\lambda _{\text {inext}}\) in (42) is done via and *augmented Lagrangian method* (Hestenes 1969; Powell 1969).

For the implementation details of the augmented Lagrangian and the rest of the terms of (42), we refer to Manhart et al. (2016).

### Appendix B: The FV Method the Environment

We solve the (27) using a FV method that was previously developed in Kolbe et al. (2016) and Sfakianakis et al. (2017) where we refer for details. Here we provide some information.

We consider the *advection–reaction–diffusion* (ARD) system

where \(\mathbf w\) represents the solution vector, and *A*, *R*, and *D* the *advection*, *reaction*, and *diffusion* operators, respectively.

We denote by \(\mathbf w_h(t)\) the corresponding (semi-)discrete numerical approximation, indexed by the maximal diameter of the spatial grid *h*, that satisfies the system of ODEs

where the numerical operators \({\mathcal {A}}, {\mathcal {R}}\), and \({\mathcal {D}}\) are *discrete approximations* of the operators *A*, *R*, and *D* in (49), respectively.

We *split* (50) in an *explicit* and an *implicit* part as

The details of the splitting depend on the particular problem in hand but in a typical case, the advection terms \({\mathcal {A}}\) are explicit in time, the diffusion terms \({\mathcal {D}}\) implicit, and the reaction terms \({\mathcal {R}}\) partly explicit and partly implicit, according to the reaction rates.

More precisely, we employ a diagonally implicit RK method for the implicit part and an explicit RK for the explicit part

where \(s=4\) are the stages of the IMEX method, \(\mathbf E_i=\mathcal E(\mathbf W_i), I_i={\mathcal {I}}(\mathbf W_i), i=1\ldots s, \{{\bar{b}},\, {\bar{A}}\}, \{b,\, A\}\) are, respectively, the coefficients for the explicit and the implicit part of the scheme, given in the Butcher Tableau in Table 5 (Kennedy and Carpenter 2003). The linear systems in (52) are solved using the *iterative biconjugate gradient stabilized Krylov subspace* method (Krylov 1931; Vorst 1992).

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Sfakianakis, N., Brunk, A. Stability, Convergence, and Sensitivity Analysis of the FBLM and the Corresponding FEM.
*Bull Math Biol* **80, **2789–2827 (2018). https://doi.org/10.1007/s11538-018-0460-0

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### Keywords

- Lamellipodium
- Actin filaments
- Cell motility
- Convergence
- Stability
- Sensitivity analysis