Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics
- 503 Downloads
Voronoi tessellations have been used to model the geometric arrangement of cells in morphogenetic or cancerous tissues, however, so far only with flat hyper-surfaces as cell-cell contact borders. In order to reproduce the experimentally observed piecewise spherical boundary shapes, we develop a consistent theoretical framework of multiplicatively weighted distance functions, defining generalized finite Voronoi neighborhoods around cell bodies of varying radius, which serve as heterogeneous generators of the resulting model tissue. The interactions between cells are represented by adhesive and repelling force densities on the cell contact borders. In addition, protrusive locomotion forces are implemented along the cell boundaries at the tissue margin, and stochastic perturbations allow for non-deterministic motility effects. Simulations of the emerging system of stochastic differential equations for position and velocity of cell centers show the feasibility of this Voronoi method generating realistic cell shapes. In the limiting case of a single cell pair in brief contact, the dynamical nonlinear Ornstein–Uhlenbeck process is analytically investigated. In general, topologically distinct tissue conformations are observed, exhibiting stability on different time scales, and tissue coherence is quantified by suitable characteristics. Finally, an argument is derived pointing to a tradeoff in natural tissues between cell size heterogeneity and the extension of cellular lamellae.
KeywordsCell tissue model Circular Voronoi power diagram
Unable to display preview. Download preview PDF.
- Alberts, B., Johnson, A., Lewis, J., Roberts, K., Walter, P. (Eds.), 2002. Molecular Biology of the Cell. 4th edn. Garland, New York. Chaps. 16 and 19. Google Scholar
- 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, p. 431. Springer, Berlin. Google Scholar
- Aurenhammer, F., Klein, R., Voronoi Diagrams. Technical Report 198, FernUniversität Hagen (1996). http://wwwpi6.fernuni-hagen.de/Publikationen/tr198.pdf
- Bernal, J., Bibliographic notes on Voronoi diagrams. Technical Report 5164, U.S. Dept. of Commerce. National Institute of Standards and Technology (1993). ftp://math.nist.gov/pub/bernal/or.ps.Z.
- Brillouin, L., 1930. Les électrons dans les métaux et le classement des ondes de de Broglie correspondantes. C. R. Hebd. Séances Acad. Sci. 191, 292. Google Scholar
- Dieterich, P., Seebach, J., Schnittler, H., 2004. Quantification of shear stress-induced cell migration in endothelial cultures. In: Deutsch, A., Falcke, M., Howard, J., Zimmermann, W. (Eds.), Function and Regulation of Cellular Systems: Experiments and Models, Mathematics and Biosciences in Interaction, p. 199. Birkhäuser, Basel Google Scholar
- Hegerfeldt, Y., Tusch, M., Bröcker, E.-B., Friedl, P., 2002. Collective cell movement in primary melanoma explants: Plasticity of cell-cell interaction, β1-integrin function and migration strategies. Cancer Res. 62, 2125. Google Scholar
- Janke, W. (Ed.), 2008. Rugged Free Energy Landscapes: Common Computational Approaches to Spin Glasses, Structural Glasses and Biological Macromolecules. Lecture Notes in Physics, vol. 736. Springer, Berlin. Google Scholar
- Marie, H., Pratt, S.J., Betson, M., Epple, H., Kittler, J.T., Meek, L., Moss, S.J., Troyanovsky, S., Attwell, D., Longmore, G.D., Braga, V.M., 2003. The LIM protein Ajuba is recruited to cadherin-dependent cell junctions through an association with alpha-catenin. J. Biol. Chem. 278, 1220. CrossRefGoogle Scholar
- Möhl, C., Modellierung von Adhäsions- und Cytoskelett-Dynamik in Lamellipodien migratorischer Zellen. Diploma thesis, Universität Bonn (2005) Google Scholar
- Schaller, G., On selected numerical approaches to cellular tissue. PhD thesis, Johann Wolfgang Goethe-Universität, Frankfurt am Main (2005) Google Scholar
- Shamos, M., Hoey, D., 1975. Closest point problems. In: Proceedings of the 16th Annual IEEE Symposium on Foundations of Computer Science (FOCS), p. 151. Google Scholar
- Thiessen, A.H., 1911. Precipitation averages for large areas. Mont. Weather Rev. 39, 1082. Google Scholar
- Weliky, M., Oster, G., 1990. The mechanical basis of cell rearrangement. I. Epithelial morphogenesis during Fundulus epiboly. Development 109, 373. Google Scholar
- Weliky, M., Minsuk, S., Keller, R., Oster, G., 1991. Notochord morphogenesis in Xenopus laevis: Simulation of cell behavior underlying tissue convergence and extension. Development 113, 1231. Google Scholar
- Young, B., Heath, J.W. (Eds.), 2000. Wheater’s Functional Histology: A Text and Colour Atlas. Churchill, London. Google Scholar