Journal of Mathematical Biology

, Volume 75, Issue 2, pp 263–307 | Cite as

Free boundary problem for cell protrusion formations: theoretical and numerical aspects

  • Olivier Gallinato
  • Masahito Ohta
  • Clair PoignardEmail author
  • Takashi Suzuki


In this paper, a free boundary problem for cell protrusion formation is studied theoretically and numerically. The cell membrane is precisely described thanks to a level set function, whose motion is due to specific signalling pathways. The aim is to model the chemical interactions between the cell and its environment, in the process of invadopodia or pseudopodia formation. The model consists of Laplace equation with Dirichlet condition inside the cell coupled to Laplace equation with Neumann condition in the outer domain. The actin polymerization is accounted for as the gradient of the inner signal, which drives the motion of the interface. We prove the well-posedness of our free boundary problem under a sign condition on the datum. This criterion ensures the consistency of the model, and provides conditions to focus on for any enrichment of the model. We then propose a new first order Cartesian finite-difference method to solve the problem. We eventually exhibit the main biological features that can be accounted for by the model: the formation of thin and elongated protrusions as for invadopodia, or larger protrusion as for pseudopodia, depending on the source term in the equation. The model provides the theoretical and numerical grounds for single cell migration modeling, whose formulation is valid in 2D and 3D. In particular, specific chemical reactions that occurred at the cell membrane could be precisely described in forthcoming works.


Mathematical biology Cell protrusion formation Free boundary problem Finite differences on cartesian grids 

Mathematics Subject Classification

65M06 65M12 92C37 



This study has been carried out with financial support from the JSPS Core to Core program Advanced Research Networks. O.G and C.P has been partly supported by the French National Research Agency (ANR) in the frame of the ”Investments for the future” Programme IdEx Bordeaux—CPU (ANR-10-IDEX-03-02). Numerical simulations presented in this paper were carried out using the PLAFRIM experimental tested, being developed under the Inria PlaFRIM development action with support from LABRI and IMB and other entities: Conseil Régional d’Aquitaine, FeDER, Université de Bordeaux and CNRS (see The authors would like to thank very warmly Professor Thierry Colin for his advices and suggestions that were helpful in the analytical and numerical study of the cell migration problem. They are also grateful to the reviewers that help in the improvement of the present paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Olivier Gallinato
    • 1
  • Masahito Ohta
    • 2
  • Clair Poignard
    • 1
    Email author
  • Takashi Suzuki
    • 3
  1. 1.Team MONC, INRIA Bordeaux-Sud-Ouest, Institut de Mathématiques de BordeauxCNRS UMR 5251 and Université de Bordeaux, 351 cours de la LibérationTalence CedexFrance
  2. 2.Department of MathematicsTokyo University of ScienceTokyoJapan
  3. 3.Division of Mathematical ScienceOsaka UniversityOsakaJapan

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