Research article

Viability Kernel Algorithm for Shapes Equilibrium

  • Viability is a very important feature of dynamic systems under state constraints whose initial value problem does not ensure uniqueness of solutions. In this paper, we introduce an hybrid automaton to address the question of viability of a cellular tissue. This hybrid automaton couples two dynamical models: differential equations manage the energy of the system and morphological equations govern the growth of the tissue. The cells can proliferate when they have enough access to oxygen and nutrient to produce the energy, remain quiescent when this energy is between two levels, or die when this energy is too low. The constraint we choose is to maintain the number of cells of the tissue during a certain time horizon. We have shown that for all the 1029 2D-tissues of 16 cells with an associate genotype, only 5 are viable for this constraint in a long time horizon. Moreover, for all these tissues, they renew there cells periodically. These periodic shapes are like periodic limit cycles in the state space of shapes.

    Citation: Alexandra Fronville, Abdoulaye Sarr, Vincent Rodin. Viability Kernel Algorithm for Shapes Equilibrium[J]. AIMS Cell and Tissue Engineering, 2017, 1(2): 118-139. doi: 10.3934/celltissue.2017.2.118

    Related Papers:

  • Viability is a very important feature of dynamic systems under state constraints whose initial value problem does not ensure uniqueness of solutions. In this paper, we introduce an hybrid automaton to address the question of viability of a cellular tissue. This hybrid automaton couples two dynamical models: differential equations manage the energy of the system and morphological equations govern the growth of the tissue. The cells can proliferate when they have enough access to oxygen and nutrient to produce the energy, remain quiescent when this energy is between two levels, or die when this energy is too low. The constraint we choose is to maintain the number of cells of the tissue during a certain time horizon. We have shown that for all the 1029 2D-tissues of 16 cells with an associate genotype, only 5 are viable for this constraint in a long time horizon. Moreover, for all these tissues, they renew there cells periodically. These periodic shapes are like periodic limit cycles in the state space of shapes.


    加载中
    [1] Fronville A, Harrouet F, Desilles A, et al. (2010) Simulation tool for morphological analysis. In ESM 2010, pages 127-132.
    [2] Fernández JD, Vico F, Doursat R (2010) Complex and diverse morphologies can develop from a minimal genomic model. In Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation, GECCO '12, pages 553-560, New York, NY, USA, ACM.
    [3] Chavoya A, Duthen Y (2007) An artificial development model for cell pattern generation. In Wiles J Randall M, Abbass H, editor, Progress in artificial life, Lecture notes in computer science, pages 61-71. Springer International Publishing.
    [4] Müller G, Newman S (2003) Origination of organismal form : beyond the gene in developmental and evolutionary biology. MIT Press.
    [5] Turing AM (1952) The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Ser B, Biol Sci 237: 37-72. doi: 10.1098/rstb.1952.0012
    [6] Mercier F, Kitasako JT, Hatton GI (2002) Anatomy of the brain neurogenic zones revisited: fractones and the fibroblast/macrophage network. J Comp Neurol 451: 170-188. doi: 10.1002/cne.10342
    [7] Chyba M, Mercier F, Rader J, et al. (2011) Dynamic mathematical modeling of cell-fractone interactions. J Math-For-Industry 3: 79-88.
    [8] Chyba M, Tamura-Sato A (2017) Morphogenesis modelization of a fracture-based model. Discrete Cont Dyn-B 22: 29-58. doi: 10.3934/dcdsb.2017149
    [9] Melani C, Campana M, Lombardot B, et al. (2007) Cells tracking in the live zebrafish embryo. In Conf. Proc. IEEE Eng Med Biol Soc 1: 1631-1634.
    [10] Aubin J-P (1991) Viability theory. Birkhauser.
    [11] Aubin J-P (2000) Mutational and morphological analysis: tools for shape regulation and morphogenesis. Birkhauser.
    [12] Lorenz T (2010) Mutational Analysis A Joint Framework for Cauchy Problems In and Beyond Vector Spaces. Springer.
    [13] Fronville A, Sarr A, Rodin V (2017) Modelling multi-cellular growth using morphological analysis. Discrete Cont Dyn-B 22: 83-99.
    [14] Lorenz T (2008) Shape evolutions under state constraints: A viability theorem. J Math Anal Appl 340: 1204-1225. doi: 10.1016/j.jmaa.2007.08.030
    [15] Sarr A, Fronville A, Rodin V (2016) Emerging Trends in Applications and Infrastructures for Computational Biology, Bioinformatics and Systems Biology, chapter 2 - A directional cellular dynamic under the control of a diffusing energy for tissue morphogenesis: phenotype and genotype. pages 17-35. Elsevier.
    [16] Katsufumi D, Kang S, Mitani S, et al. (2016) Syndecan defines precise spindle orientation by modulating wnt signaling in c. elegans. Development 141: 4354-4365.
    [17] Olivier N, Luengo-Oroz MA, Duloquin L, et al. (2010) Cell lineage reconstruction of early zebrafish embryos using label-free nonlinear microscopy. Science 329: 967-971. doi: 10.1126/science.1189428
    [18] Gorre A (1997) Evolutions of tubes under operability constraints. J Math Anal Appl 216: 1-22. doi: 10.1006/jmaa.1997.5476
    [19] Saint-Pierre P (1994) Approximation of the viability kernel. Appl Math Optim 29: 187-209. doi: 10.1007/BF01204182
    [20] Deffuant G, Chapel L, Martin S (2007) Approximating viability kernels with support vector machines. IEEE Trans Autom Control 52: 933-937. doi: 10.1109/TAC.2007.895881
    [21] Coquelin P-A, Martin S, Munos R (2007) A dynamic programming approach to viability problems. In IEEE ADPRL, Proceedings of the 2007 IEEE Symposium on Approximate Dynamic Programming and Reinforcement Learning (ADPRL 2007), pages 178-184.
    [22] Sarr A, Miglierini P, Fronville A, et al. (2016) Directional cellular dynamics for tissue morphogenesis and tumor characterization by aggressive cancer cells identification. In Proceedings of the 9th International Joint Conference on Biomedical Engineering Systems and Technologies. Rome, pages 290-295.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2937) PDF downloads(893) Cited by(0)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog