title: Free boundary problem for cell protrusion formations: theoretical and numerical aspects.
abstract: 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.
authors.affiliation: Team MONC, INRIA Bordeaux-Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251 and Université de Bordeaux, 351 cours de la Libération, 33405, Talence Cedex, France.
authors.forename: Olivier
authors.initials: O
authors.lastname: Gallinato
authors.affiliation: Department of Mathematics, Tokyo University of Science, Tokyo, Japan.
authors.forename: Masahito
authors.initials: M
authors.lastname: Ohta
authors.affiliation: Team MONC, INRIA Bordeaux-Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251 and Université de Bordeaux, 351 cours de la Libération, 33405, Talence Cedex, France. clair.poignard@inria.fr.
authors.forename: Clair
authors.initials: C
authors.lastname: Poignard
authors.affiliation: Division of Mathematical Science, Osaka University, Osaka, Japan.
authors.forename: Takashi
authors.initials: T
authors.lastname: Suzuki
mesh_terms: D002462:Cell Membrane / Q000737:chemistry / Q000502:physiology; D002465:Cell Movement; D004777:Environment; D008954:Models, Biological; D011554:Pseudopodia / Q000502:physiology*; D015398:Signal Transduction
keywords: Cell protrusion formation; Finite differences on cartesian grids; Free boundary problem; Mathematical biology