Summary
We review two modelling approaches to obtain genuinely nonlinear systems of one hyperbolic transport equation (for density) accompanied by parabolic or elliptic equations (for mean velocity and, eventually, pressure), namely generalized Navier-Stokes or (pseudostationary) Stokes equations. Background and applications are related to models of interactive biological motion, namely for contractile polymer networks in intra-cellular motility, for cell movement and tissue formation during wound healing as well as for cohorts of migrating birds. One approach is to derive, after suitable scaling, a formal continuum limit of (stochastic) Hamiltonian equations for ‘visco-elastic’ multi-particle networks with specific interaction laws. The other consists in the derivation of (highly) viscous two-phase flow equations by minimization of a corresponding energy-loss functional. In both procedures there remain convergence or existence problems to be solved analytically. Some results and a few numerical simulations are shown, particularly for the 1-dimensional case. For further results, technical details and for comparison with other methods we give corresponding references.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
W. Alt, G. Hoffmann (eds.) (1990): Biological Motion. Lecture Notes in Biomath, vol. 89, Springer-Verlag, Berlin
W. Alt, O. Brosteanu, B. Hinz and H.W. Kaiser (1995): Patterns of spontaneous motility in videomicrographs of human epidermal keratinocytes (HEK). Biochemistry and Cell Biology, 73, 441–459
W. Alt (1996): Biomechanics of actymyosin mediated motility of keratinocytes. Biophysics, 41, 181–188
W. Alt, A. Deutsch, G. Dunn (eds.) (1997): Dynamics of Cell and Tissue Motion. Birkhäuser-Verlag, Basel, Boston, Berlin
W. Alt, M. Dembo (1999): Cytoplasm dynamics and cell motion: two-phase flow models. Math. Biosciences, 156. 207–228
W. Alt (2002): One-dimensional dynamics of stochastic skeins: condensation waves and continuum limits. Submitted to J. Math. Biol.
W. Alt, T. Bretschneider, R. Müller (2002): Interactive movement, aggregation and swarm dynamics. In: W. Alt, M. Chaplain, M. Griebel, J. Lenz (eds.) Polymer and Cell Dynamics — Multiscale Modelling and Numerical Simulation. To appear in Birkhäuser-Verlag, Basel
W. Alt, M. Chaplain, M. Griebel, J. Lenz (eds.) (2002): Polymer and Cell Dynamics — Multiscale Modelling and Numerical Simulation. To appear in Birkhäuser-Verlag, Basel
D.C. Bottino (1998): Modeling viscoelastic networks and cell deformation in the context of the immersed boundary method. J. Comp. Phys., 147. 86–113
D.C. Bottino (2001): Computer simulations of mechanochemical coupling in a deforming domain: Applications to cell motion. Preprint
T. Bretschneider (2002): Reinforcement of cytoskeleton-matrix bounds and tensiotaxis: a cell based model. In: A. Deutsch et al. (eds) Function and Regulation of Cellular Systems: Experiments and Models. To appear in Birkhäuser-Verlag, Basel
Z. Csahok, A. Czirok (1997): Hydrodynamics of bacterial motion. Physica A, 243, 304–318
A. Czirok, T. Vicsek (2000): Collective behaviour of interacting self-propelled particles. J. Phys. A, 281,17–29
M. Dembo, F.W. Harlow, W. Alt (1984): The biophysics of cell surface mobility. In: (A.D. Perelson, Ch. DelLisi, F.W. Wiegel eds.) Cell Surface Dynamics, Concepts and Models. Marcel Dekker, New York, Basel, 495–542
M. Dembo, F.W. Harlow (1986): Cell motion, contractile networks, and the physics of interpenetrating reactive flow. Biophys. J., 50, 109–121
M. Dembo (1986): The mechanics of motility in dissociated cytoplasm. Biophys. J., 50, 1165–1183
M. Dembo (1989): Field theories of the cytoplasm. Comments theor. Biol., 1, 159
M. Dembo (1994): On free boundary problems and amoeboid motion. In: N. Akkas (ed) Biomechanics of Active Movement and Division of Cells. NATO ASI Ser. H84. Springer, Berlin, p. 231
M. Dembo, Y. Wang (1999): Stresses at the cell-to-substrate interface during locomotion of fibroblasts. Biophys. J., 76, 2307–2316
D. Drasdo (2002): On selected individual-based approaches to the dynamics in multicellular systems. In: W. Alt, M. Chaplain, M. Griebel, J. Lenz (eds) Polymer and Cell Dynamics — Multiscale Modelling and Numerical Simulation. To appear in Birkhäuser-Verlag, Basel
D.A. Drew, L.A. Segel (1991): Averaged equations for two phase flow. Stud. Appl. Math., 50, 205–231
W. Ebeling, F Schweitzer (2001): Swarms of particle agents with harmonic interactions. Theory in Biosciences, 120. 207–224
X. He, M. Dembo (1997): On the mechanics of the first cleavage division of the sea urchin egg. Exper. Cell Res., pp. 233–252
F.H. Heppner, U. Grenander (1990):A stochastic nonlinear model for coordinated bird flocks. In: S. Krasner (ed) The Ubiquity of Chaos. Washington, DC: American Association for the Advancement of Science.
H. Holden, B. Oksendal, J. Uboe, T. Zhang (1996): Stochastic Partial Differential Equations. A modeling white noise functional approach, Birkhäuser, Basel
V. Lendowski, A. Mogilner (1997): Origin of actin-induced locomotion of Listeria. In: W. Alt, A. Deutsch, G. Dunn (eds) Dynamics of Cell and Tissue Motion. Birkhäuser-Verlag, Basel, Boston, Berlin, 93–99
S. R. Lubkin, T. Jackson (2001): Multiphase Mechanics of Capsule Formation in Tumors. J. Biomech Eng, in press
T. Oliver, M. Dembo, K. Jacobson (1999): Seperation of propulsive and adhesive traction stresses in locomoting keratocytes. J. Cell Biology, 145. 589–604
M. Mansour (2001): Singularity analysis of travelling wave solutions for degenerate diffusion and transport equations modelling cell motility. PhD thesis, Univ. Bonn
F.A. Meinicke, S.C. Porten, M. Loeffler (2001): Cell migration and organization in the intestinal crypt using a lattice-free model. Cell Prolif., 34, 253–266
R. Müller (2002): Dynamik in stochastischen Vielteilchensystemen zur Modellierung von Vogelschwärmen. Diploma thesis Univ. Bonn
K. Oehlschläger (1989): On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes. Th. Rel. Fields, 82, 565–589
K. Oehlschläger (1991) On the connection between Hamiltonian many-particle systems and the hydrodynamical equations. Arch. Rat. Mech. Anal., 115. 297–310
S. Olla, S.R.S. Varadhan, H.T. Yau (1993): Hydrodynamical limit for a Hamiltonian system with weak noise. Comm. Math. Phys., 155. 523–560
J. Toner, Y. Tu (1995): How birds fly together: Long range oder in a two-dimensional dynamical XY model. Phys. Rev. Lett. 75, 4326–4329
J. Toner, Y. Tu (1998): Rocks, herds and schools: A quantitative theory of flocking. Phys. Rev. E, 58, 4828–4858
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Alt, W. (2003). Nonlinear Hyperbolic Systems of Generalized Navier-Stokes Type for Interactive Motion in Biology. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-55627-2_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44051-2
Online ISBN: 978-3-642-55627-2
eBook Packages: Springer Book Archive