Abstract
We start with a model for the actin–cytoskeleton in a symmetric lamellipodium (cp. Oelz et al. in Cell Adh Migr 2(2):117–126, 2008) which includes the description of the life-cycle of chemical bonds based on age-structured models. Based on the assumption that their average lifetime is actually small as compared to the time scale of the dynamics in which we are interested, we pass, after applying an appropriate scaling, to a limit where this average lifetime goes to zero. We obtain a gradient flow model and formulate a time step approximation scheme. We use it to construct solutions analytically, proving their local in time existence, and present a typical numerical solution based on this scheme.
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References
Ambrosio L., Gigli N., Savaré G.: Gradient flows in metric spaces and in the space of probability measures. In: Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 2005
Courant R., Friedrichs K., Lewy H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100(1), 32–74 (1928)
De Giorgi, E.: New problems on minimizing movements. Boundary Value Problems for Partial Differential Equations and Applications. In: RMA Research Notes in Applied Mathematics, vol. 29, pp. 81–98. Masson, Paris, 1993
De Giorgi E., Marino A., Tosques M.: Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68(3), 180–187 (1980)
Li F., Redick S.D., Erickson H.P., Moy V.T.: Force measurements of the [alpha]5[beta]1 integrin–fibronectin interaction. Biophys. J. 84(2), 1252–1262 (2003)
Oelz D., Schmeiser C.: How do cells move? Mathematical modelling of cytoskeleton dynamics and cell migration. In: Chauviere, A., Preziosi, L., Verdier, C. (eds) Cell Mechanics: from Single Scale-Based Models to Multiscale Modelling., Chapman and Hall, London (2010)
Oelz D., Schmeiser C., Small J.V.: Modelling of the actin–cytoskeleton in symmetric lamellipodial fragments. Cell Adh. Migr. 2(2), 117–126 (2008)
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Communicated by D. Kinderlehrer
This work has been supported by the Austrian Science Fund (FWF) through the Wissenschaftskolleg Differential Equations and by the WWTF-Project “How do cells move? Mathematical modelling of cytoskeletal dynamics and cell migration” of C. Schmeiser and V. Small.
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Oelz, D., Schmeiser, C. Derivation of a Model for Symmetric Lamellipodia with Instantaneous Cross-Link Turnover. Arch Rational Mech Anal 198, 963–980 (2010). https://doi.org/10.1007/s00205-010-0304-z
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DOI: https://doi.org/10.1007/s00205-010-0304-z