Towards Sub-cellular Modeling with Delaunay Triangulation

G. Grise; M. Meyer-Hermann

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 1, page 224-238
  • ISSN: 0973-5348

Abstract

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In this article a novel model framework to simulate cells and their internal structure is described. The model is agent-based and suitable to simulate single cells with a detailed internal structure as well as multi-cellular compounds. Cells are simulated as a set of many interacting particles, with neighborhood relations defined via a Delaunay triangulation. The interacting sub-particles of a cell can assume specific roles – i.e., membrane sub-particle, internal sub-particle, organelles, etc –, distinguished by specific interaction potentials and, eventually, also by the use of modified interaction criteria. For example, membrane sub-particles may interact only on a two-dimensional surface embedded on three-dimensional space, described via a restricted Delaunay triangulation. The model can be used not only to study cell shape and movement, but also has the potential to investigate the coupling between internal space-resolved movement of molecules and determined cell behaviors.

How to cite

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Grise, G., and Meyer-Hermann, M.. "Towards Sub-cellular Modeling with Delaunay Triangulation." Mathematical Modelling of Natural Phenomena 5.1 (2010): 224-238. <http://eudml.org/doc/197664>.

@article{Grise2010,
abstract = {In this article a novel model framework to simulate cells and their internal structure is described. The model is agent-based and suitable to simulate single cells with a detailed internal structure as well as multi-cellular compounds. Cells are simulated as a set of many interacting particles, with neighborhood relations defined via a Delaunay triangulation. The interacting sub-particles of a cell can assume specific roles – i.e., membrane sub-particle, internal sub-particle, organelles, etc –, distinguished by specific interaction potentials and, eventually, also by the use of modified interaction criteria. For example, membrane sub-particles may interact only on a two-dimensional surface embedded on three-dimensional space, described via a restricted Delaunay triangulation. The model can be used not only to study cell shape and movement, but also has the potential to investigate the coupling between internal space-resolved movement of molecules and determined cell behaviors.},
author = {Grise, G., Meyer-Hermann, M.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {cell shape; cell movement; sub-cellular model; delaunay triangulation; voronoi tessellation; surface reconstruction; Delaunay triangulation; Voronoi tessellation},
language = {eng},
month = {2},
number = {1},
pages = {224-238},
publisher = {EDP Sciences},
title = {Towards Sub-cellular Modeling with Delaunay Triangulation},
url = {http://eudml.org/doc/197664},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Grise, G.
AU - Meyer-Hermann, M.
TI - Towards Sub-cellular Modeling with Delaunay Triangulation
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/2//
PB - EDP Sciences
VL - 5
IS - 1
SP - 224
EP - 238
AB - In this article a novel model framework to simulate cells and their internal structure is described. The model is agent-based and suitable to simulate single cells with a detailed internal structure as well as multi-cellular compounds. Cells are simulated as a set of many interacting particles, with neighborhood relations defined via a Delaunay triangulation. The interacting sub-particles of a cell can assume specific roles – i.e., membrane sub-particle, internal sub-particle, organelles, etc –, distinguished by specific interaction potentials and, eventually, also by the use of modified interaction criteria. For example, membrane sub-particles may interact only on a two-dimensional surface embedded on three-dimensional space, described via a restricted Delaunay triangulation. The model can be used not only to study cell shape and movement, but also has the potential to investigate the coupling between internal space-resolved movement of molecules and determined cell behaviors.
LA - eng
KW - cell shape; cell movement; sub-cellular model; delaunay triangulation; voronoi tessellation; surface reconstruction; Delaunay triangulation; Voronoi tessellation
UR - http://eudml.org/doc/197664
ER -

References

top
  1. M. J. Miller, S. H. Wei, I. Parker M. D. Cahalan. Two photon imaging of lymphocyte motility and antigen response in intact lymph node. Science296 (2002), 1869–1873 
  2. S. Stoll, J. Delon, T. M. Brotz R. N. Germain. Dynamic Imaging of T Cell-Dendritic Cell Interactions in Lymph Nodes. Science296 (2002), 1873–1876 
  3. U. H. von Andrian. T cell activation in six dimensions. Science296 (2002), 1815–1817 
  4. R. F. Murphy. Putting proteins on the map. Nat. Biotechnol.24 (2006), 1223–1224 
  5. W. Schubertet al.Analyzing proteome topology and function by automated multidimensional fluorescence microscopy. Nat. Biotechnol.24 (2006), 1270–1278 
  6. W. Alt R. T. Tranquillo. Basic morphogenetic system modeling shape changes of migrating cells: how to explain fluctuating lamellipodial dynamics. Journal of Biol. Systems3 (1995), No. 4905–916 
  7. M. T. Figge, A. Garin, M. Gunzer, M. Kosco-Vilbois, K.-M. Toellner M. Meyer-Hermann. Deriving a germinal center lymphocyte migration model from two-photon data. Journal of Exp. Med.205 (2008), No. 13, 3019–3029 
  8. M. Meyer-Hermann, M. T. Figge K.-M. Toellner. Germinal centres seen through the mathematical eye: B cell models on the catwalk. Trends in Immunology30 (2009), No. 4, 157–164 
  9. F. Graner J. A. Glazier. Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys. Rev. Lett.69 (1992), No. 13, 2013–2016 
  10. M. E. Meyer-Hermann P. K. Maini. Interpreting two-photon imaging data of lymphocyte motility. Phys. Review E71 (2005), No. 6, 061912–061923 
  11. F. A. Meineke, C. S. Potten M. Loeffler. Cell migration and organization in the intestinal crypt using a lattice-free model. Cell Prolif.34 (2001), No. 4, 253–266 
  12. T. Beyer M. Meyer-Hermann. Mechanisms of organogenesis of primary lymphoid follicles. Int. Immunol.20 (2008), No. 4, 615–623 
  13. M. Bock, A. K. Tyagi, J.-U. Kreft, W. Alt. Generalized Voronoi tessellation as a model of two-dimensional cell tissue dynamics. arXiv:0901.4469v2 [physics.bio-ph].  Zbl1202.92008
  14. J. Galle, M. Hoffmann G. Aust. From single cells to tissue architecture – a bottom-up approach to modeling the spatio-temporal organisation of complex multi-cellular systems. J. Math. Biol.58 (2009), 261–283 Zbl1161.92021
  15. T. J. Newman. Modeling multicellular systems using subcellular elements. Mathematical Biosciences and Engineering2 (2005), No. 3, 611–622 Zbl1079.92025
  16. S. A. S, ersius T. J. Newman. Modeling cell rheology with the Subcellular Element Model. Phys. Biol.5 (2008), No. 1Cell migration, 015002–015014 
  17. D. E. Ingber. Cellular tensegrity I. Cell structure and hierarchical systems biology. J. Cell Sci.116 (2003), 1157–1173 
  18. D. E. Ingber. Tensegrity II. How structural networks inuence cellular information-processing networks. J. Cell Sci.116 (2003), 1397–1408 
  19. G. Schaller, M. Meyer-Hermann. Kinetic and dynamic Delaunay tetrahedralizations in three dimensions. Comput. Phys., Commun.162 (2004), No. 1Cell migration, 9–23.  Zbl1196.65039
  20. T. Beyer, G. Schaller, A. Deutsch M. Meyer-Hermann. Parallel dynamic and kinetic regular triangulation in three dimensions. Comput. Phys. Commun.172 (2005), No. 2, 86–108 
  21. A. Okabe, B. Boots, K. Sugihara, S. N. Chiu. Spatial tessellations: concepts and applications of Voronoi diagrams. Probability and Statistics. John Wiley & Sons, Inc., New York, 1992.  Zbl0946.68144
  22. E. Mücke. A robust implementation for three-dimensional Delaunay triangulations. Internat. J. Comput. Geom. Appl.2 (1998), No. 8, 255–276 Zbl1035.68539
  23. F. Cazals e J. Giesen. Delaunay triangulation based surface reconstruction: ideas and algorithms. Institut National De Recherche En Informatic et en AutomatiqueRapport de recherche No. 5393 (2004).  Zbl1116.65022
  24. M. Meyer-Hermann. Delaunay-Object-Dynamics: cell mechanics with a 3D kinetic and dynamic weighted Delaunay-triangulation. Curr. Top. Dev. Biol.81 (2008), 373–399.  
  25. T. Beyer M. Meyer-Hermann. The treatment of non-flippable configurations in three dimensional regular triangulations. WSEAS Trans. Syst.5 (2006), No. 5, 1100–1107 
  26. G. Schaller M. Meyer-Hermann. Multicellular tumor spheroid in an off-lattice Voronoi/Delaunay cell model. Phys. Rev. E71 (2005), No. 5, 051910–051925 
  27. G. V. Reddy, M. G. Heisler, D. W. Ehrhardt E. M. Meyerowitz. Real-time lineage analysis reveals oriented cell divisions associated with morphogenesis at the shoot apex of Arabidopsis thaliana. Development131 (2004), No. 17, 4225–4237 
  28. N. Amenta, M. Bern, M. Kamvysselis. A new Voronoi-based surface reconstruction algorithm. SIGGRAPH ’98: Proceedings of the 25th annual conference on computer graphics and interactive techniques, ACM, New York, 1998.  
  29. N. Amenta M. Bern. Surface reconstruction by Voronoi filtering. Discrete and Computational Geometry22 (1999), No. 4, 481–504 Zbl0939.68138
  30. G. Grise, M. Meyer-Hermann. Surface reconstruction using Delaunay triangulation for applications in life sciences. Submitted (2009).  Zbl1221.65046
  31. L. Verlet. Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev.159 (1967), No. 1Cell migration, 98–103 
  32. L. Verlet. Computer experiments on classical fluids. II. Equilibrium correlation functions. Phys. Rev.165 (1968), No. 1Cell migration, 201–214 
  33. Wu-Yi Hsiang. On the sphere packing problem and the proof of Kepler’s conjecture. Internat. J. Math.4 (1993), No. 5, 739–831 Zbl0844.52017

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