Morphogenesis modelization of a fractone-based model

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January 2017, 22(1): 29-58. doi: 10.3934/dcdsb.2017002

Morphogenesis modelization of a fractone-based model

Monique Chyba ^, and Aaron Tamura-Sato

Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall, Honolulu, HI, 96822, USA

* Corresponding author: chyba@hawaii.edu

Received September 2015 Revised May 2016 Published December 2016

Fund Project: This research received support from the National Science Foundation: NSF Award DGE-0841223.

It has been hypothesized that the generation of new neural cells (neurogenesis) in the developing and adult brain is guided by the extracellular matrix. The extracellular matrix of the neurogenic niches features specialized structures termed fractones, which are scattered in between stem/progenitor cells and bind and activate growth factors at the surface of stem/progenitor cells to influence their proliferation. We present a mathematical control model that considers the role of fractones as captors and activators of growth factors, controlling the rate of proliferation and directing the location of the newly generated neuroepithelial cells in the forming brain. The model is a hybrid control system that incorporates both continuous and discrete dynamics. The continuous dynamics of the model features the diffusion of multiple growth factor concentrations through the mass of cells, with fractones acting as sinks that absorb and hold growth factor. When a sufficient amount has been captured, growth is assumed to occur instantaneously in the discrete dynamics of the model, causing an immediate rearrangement of cells, and potentially altering the dynamics of the diffusion. The fractones in the model are represented by controls that allow for their dynamic placement in and removal from the evolving cell mass.

Keywords: Morphogenesis, fractones, hybrid automata, controllability.

Mathematics Subject Classification: 93C30, 92B05, 93A3.

Citation: Monique Chyba, Aaron Tamura-Sato. Morphogenesis modelization of a fractone-based model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 29-58. doi: 10.3934/dcdsb.2017002

References:

[1]	R. Araujo and D. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091. doi: 10.1016/j.bulm.2003.11.002. Google Scholar
[2]	A. Auley, Z. Werb and P. Mirkes, Characterization of the unusually rapid cell cycles during rat gastrulation, Development, 117 (1993), 873-883. Google Scholar
[3]	A. Bafico and S. Aaronson, Holland-Frei Cancer Medicine, 6th edition, BC Decker, 2003. Google Scholar
[4]	S. Banerjee, R. Cohn and M. Bernfield, Basal lamina of embryonic salivary epithelia. production by the epithelium and role in maintaining lobular morphology, The Journal of Cell Biology, 73 (1977), 445-463. Google Scholar
[5]	E. Bianconi, A. Piovesan, F. Facchin, A. Beraudi, R. Casadei, F. Frabetti, L. Vitale, M. Pelleri, S. Tassani, F. Piva, S. Perez-Amodio, P. Strippoli and S. Canaider, An estimation of the number of cells in the human body, Ann Hum Biol., 40 (2013), 463-471. doi: 10.3109/03014460.2013.807878. Google Scholar
[6]	G. Brodland, X. Chen, P. Lee and M. Marsden, From genes to neural tube defects (ntds): Insights from multiscale computational modeling, HFSP J, 4 (2010), 142-152. Google Scholar
[7]	R. Chaturvedi, C. Huang, B. Kazmierczak, T. Schneider, J. Izaguirre, T. Glimm, H. Hentschel, J. Glazier, S. Newman and M. Alber, On multiscale approaches to three-dimensional modelling of morphogenesis, J R Soc Interface, 2 (2005), 237-253. doi: 10.1098/rsif.2005.0033. Google Scholar
[8]	H. Chen and G. Brodland, Cell-level finite element studies of viscous cells in planar aggregates, J Biomech Eng, 122 (2000), 394-401. doi: 10.1115/1.1286563. Google Scholar
[9]	M. Chyba, M. Kobayashi, F. Mercier, J. Rader, A. Tamura-Sato, G. Telleschi and Y. Kwon, A new approach to modeling morphogenesis using control theory, Special volume of the São Paulo Journal of Mathematical Sciences in honor of Prof. Waldyr Oliva,, 5 (2011), 281-315. doi: 10.11606/issn.2316-9028.v5i2p281-315. Google Scholar
[10]	P. Ciarletta, M. Ben Amar and M. Labouesse, Continuum model of epithelial morphogenesis during caenorhabditis elegans embryonic elongation, Phil. Trans. R. Soc. A, 367 (2009), 3379-3400. doi: 10.1098/rsta.2009.0088. Google Scholar
[11]	D. Clausi and G. Brodland, Mechanical evaluation of theories of neurulation using computer simulations, Development, 118 (1993), 1013-1023. Google Scholar
[12]	I. Decimo, G. Fumagalli, V. Berton, M. Krampera and F. Bifari, Meninges: from protective membrane to stem cell niche, American Journal of Stem Cells, 1 (2012), 92-105. Google Scholar
[13]	V. Douet, E. Arikawa-Hirasawa and F. Mercier, Fractone-heparan sulfates mediate bmp-7 inhibition of cell proliferation in the adult subventricular zone, Neuroscience Letters, 528 (2012), 120-125. doi: 10.1016/j.neulet.2012.08.077. Google Scholar
[14]	M. Fitch and J. Silver, Glial cell extracellular matrix: Boundaries for axon growth in development and regeneration, Cell and Tissue Research, 290 (1997), 379-384. Google Scholar
[15]	H. Frieboes, X. Zheng, C. Sun, B. Tromberg, R. Gatenby and V. Cristini, An integrated computational/experimental model of tumor invasion, Cancer Res, 66 (2006), 1597-1604. doi: 10.1158/0008-5472.CAN-05-3166. Google Scholar
[16]	F. Gage, Neurogenesis in the adult brain, The Journal of Neuroscience, 22 (2002), 612-613. Google Scholar
[17]	T. Henzinger, The theory of hybrid automata, Logic in Computer Science. LICS '96. Proceedings. Eleventh Annual IEEE Symposium, (1996), 278-292. doi: 10.1109/LICS.1996.561342. Google Scholar
[18]	D. Ingber, J. Madri and J. Jamieson, Role of basal lamina in neoplastic disorganization of tissue architecture, Proc. Natl. Acad. Sci. USA, 78 (1981), 3901-3905. doi: 10.1073/pnas.78.6.3901. Google Scholar
[19]	A. Jacobson, G. Oster, G. Odell and L. Cheng, Neurulation and the cortical tractor model for epithelial folding, Journal of Embryology and Experimental Morphology, 96 (1986), 19-49. Google Scholar
[20]	K. Johansson, M. Egerstedt, J. Lygeros and S. Sastry, On the regularization of zeno hybrid automata, System & Control Letters, 38 (1999), 141-150. doi: 10.1016/S0167-6911(99)00059-6. Google Scholar
[21]	K. Johansson, J. Lygeros, S. Sastry and M. Egerstedt, Simulation of zeno hybrid automata, Decision and Control, 4 (1999), 3538-3543. doi: 10.1109/CDC.1999.827900. Google Scholar
[22]	G. Kempermann, L. Wiskott and F. Gage, Functional significance of adult neurogenesis, Current Opinion in Neurobiology, 14 (2004), 186-191. doi: 10.1016/j.conb.2004.03.001. Google Scholar
[23]	A. Kerever, J. Schnack, D. Vellinga, N. Ichikawa, C. Moon, E. Arikawa-Hirasawa, J. Efird and F. Mercier, Novel extracellular matrix structures in the neural stem cell niche capture the neurogenic factor fibroblast growth factor 2 from the extracellular milieu, Stem Cells, 25 (2007), 2146-2157. doi: 10.1634/stemcells.2007-0082. Google Scholar
[24]	Y. Kim, M. Stolarska and H. Othmer, A hybrid model for tumor spheroid growth in vitro i: Theoretical development and early results, Mathematical Models and Methods in Applied Sciences, 17 (2007), 1773-1798. doi: 10.1142/S0218202507002479. Google Scholar
[25]	L. Li, W. Dong, Y. Ji, Z. Zhang and L. Tong, Minimal-energy driving strategy for high-speed electric train with hybrid system model, IEEE Transactions on Intelligent Transportation Systems, 14 (2013), 1642-1653. Google Scholar
[26]	H. Lin and P. Antsaklis, Hybrid dynamical systems: An introduction to control and verification, Foundations and Trends in Systems and Control, 1 (2014), 1-172. Google Scholar
[27]	J. Lygeros, K. Johansson, S. Sastry and M. Egerstedt, On the existence of executions of hybrid automata, Decision and Control, 3 (1999), 2249-2254. doi: 10.1109/CDC.1999.831255. Google Scholar
[28]	J. Lygeros, K. Johansson, S. Simic, J. Zhang and S. Sastry, Continuity and invariance in hybrid automata, Decision and Control, 1 (2001), 340-345. doi: 10.1109/CDC.2001.980123. Google Scholar
[29]	J. Lygeros, K. Johansson, S. Simic, J. Zhang and S. Sastry, Dynamical properties of hybrid automata, Automatic Control, IEEE Transactions, 48 (2003), 2-17. doi: 10.1109/TAC.2002.806650. Google Scholar
[30]	J. Lygeros, S. Sastry and C. Tomlin, Hybrid Systems: Foundations, Advanced Topics, and Applications, Springer Verlag, 2012. Google Scholar
[31]	J. Meitzen, K. Pflepsen, C. Stern, R. Meisel and P. Mermelstein, Measurements of neuron soma size and density in rat dorsal striatum, nucleus accumbens core and nucleus accumbens shell: Differences between striatal region and brain hemisphere, but not sex, Neuroscience Letters, 487 (2011), 177-181. doi: 10.1016/j.neulet.2010.10.017. Google Scholar
[32]	F. Mercier and E. Arikawa-Hirakawa, Heparan sulfate niche for cell proliferation in the adult brain, Neuroscience Letters, 510 (2012), 67-72. doi: 10.1016/j.neulet.2011.12.046. Google Scholar
[33]	F. Mercier, J. Kitasako and G. Hatton, Anatomy of the brain neurogenic zones revisited: Fractones and the fibroblast/macrophage network, J Comp Neurol, 451 (2002), 170-188. doi: 10.1002/cne.10342. Google Scholar
[34]	F. Mercier, J. Kitasako and G. Hatton, Fractones and other basal laminae in the hypothalamus, J Comp Neurol, 455 (2003), 324-340. doi: 10.1002/cne.10496. Google Scholar
[35]	H. Minkowski, Space and time, in The Principle of Relativity, Calcutta University Press, 1920, 70-88, Trans. M Saha. Google Scholar
[36]	G. Oster, On the crawling of cells, Journal of Embryology and Experimental Morphology, 83 (1984), 329-364. Google Scholar
[37]	E. Palsson, Single-Cell-Based Models in Biology and Medicine, Mathematics and Biosciences in Interaction, 2007. doi: 10.1007/978-3-7643-8123-3. Google Scholar
[38]	J. Piovesan, C. Abdallah and H. Tanner, Preliminary results on interconnected hybrid systems, Control and Automation, (2008), 101-106. doi: 10.1109/MED.2008.4602157. Google Scholar
[39]	J. Piovesan, C. Abdallah and H. Tanner, Modeling multi-agent systems with hybrid interacting dynamics, American Control Conference, (2009), 3644-3649. doi: 10.1109/ACC.2009.5160419. Google Scholar
[40]	J. Piovesan, H. Tanner and C. Abdallah, Discrete asymptotic abstractions of hybrid systems, Decision and Control, (2006), 917-922. doi: 10.1109/CDC.2006.377733. Google Scholar
[41]	N. Poplawski, M. Swat, J. Gens and J. Glazier, Adhesion between cells, diffusion of growth factors, and elasticity of the aer produce the paddle shape of the chick limb, Physica A, 373 (2007), 521-532. doi: 10.1016/j.physa.2006.05.028. Google Scholar
[42]	J. Rader, Dynamic Modelling Of Neural Morphogenesis Using Mathematical Control Theory, Master's thesis, University of Hawai'i, 2011. Google Scholar
[43]	K. Rejniak and A. Anderson, Hybrid models of tumor growth, Wiley Interdiscip Rev Syst Biol Med, 3 (2011), 115-125. doi: 10.1002/wsbm.102. Google Scholar
[44]	C. Ronse and M. Tajine, Discretization in hausdorff space, Journal of Mathematical Imaging and Vision, 12 (2000), 219-242. doi: 10.1023/A:1008366032284. Google Scholar
[45]	J. Sanes, Extracellular matrix molecules that influence neural development, Ann. Rev. Neurosci, 12 (1989), 491-516. doi: 10.1146/annurev.ne.12.030189.002423. Google Scholar
[46]	M. Stolarska, Y. Kim and H. Othmer, Multi-scale models of cell and tissue dynamics, Phil. Trans. R. Soc. A, 367 (2009), 3525-3553. doi: 10.1098/rsta.2009.0095. Google Scholar
[47]	A. Teel, A. Subbaraman and A. Sferlazza, Stability analysis for stochastic hybrid systems: A survey, Automatica, 50 (2014), 2435-2456. doi: 10.1016/j.automatica.2014.08.006. Google Scholar
[48]	S. Tripakis and T. Dang, Modeling, verification and testing using timed and hybrid automata, in Model-Based Design of Embedded Systems (ed. M. P. Nicolescu G), CRC Press, 2009,383-436. doi: 10.1201/9781420067859-c13. Google Scholar
[49]	A. Voss-Bohme, Multi-scale modeling in morphogenesis: A critical analysis of the cellular potts model, PLoS ONE, 7 (2012), e42852. doi: 10.1371/journal.pone.0042852. Google Scholar
[50]	R. Vracko, Basal lamina scaffold-anatomy and significance for maintenance of orderly tissue structure, Am J Pathol, 77 (1974), 313-346. Google Scholar
[51]	D. Walker, S. Wood, J. Southgate, M. Holcombe and R. Smallwood, An integrated agent-mathematical model of the effect of intercellular signalling via the epidermal growth factor receptor on cell proliferation, J Theor Biol, 242 (2006), 774-789. doi: 10.1016/j.jtbi.2006.04.020. Google Scholar
[52]	J. Ward and J. King, Mathematical modelling of avascular-tumour growth, IMA J Math Appl Med Biol, 14 (1997), 39-69. doi: 10.1093/imammb/14.1.39. Google Scholar
[53]	M. Wyczalkowski, Z. Chen, B. Filas, V. Varner and L. Taber, Computational models for mechanics of morphogenesis, Birth Defects Res C Embryo Today, 96 (2012), 132-152. doi: 10.1002/bdrc.21013. Google Scholar
[54]	G. Xie and L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Transactions on Automatic Control, 49 (2004), 960-966. doi: 10.1109/TAC.2004.829656. Google Scholar
[55]	Z. Yang and M. Blanke, A unified approach to controllability analysis for hybrid control systems, Nonlinear Analysis: Hybrid Systems, 1 (2007), 212-222. doi: 10.1016/j.nahs.2006.08.002. Google Scholar
[56]	M. Zarandi, A. Bonakdar and I. Stiharu, Investigations on natural frequencies of individual spherical and ellipsoidal bakery yeast cells, in Proceedings of the COMSOL Conference 2010 Boston, 2010. Google Scholar

show all references

References:

[1]	R. Araujo and D. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091. doi: 10.1016/j.bulm.2003.11.002. Google Scholar
[2]	A. Auley, Z. Werb and P. Mirkes, Characterization of the unusually rapid cell cycles during rat gastrulation, Development, 117 (1993), 873-883. Google Scholar
[3]	A. Bafico and S. Aaronson, Holland-Frei Cancer Medicine, 6th edition, BC Decker, 2003. Google Scholar
[4]	S. Banerjee, R. Cohn and M. Bernfield, Basal lamina of embryonic salivary epithelia. production by the epithelium and role in maintaining lobular morphology, The Journal of Cell Biology, 73 (1977), 445-463. Google Scholar
[5]	E. Bianconi, A. Piovesan, F. Facchin, A. Beraudi, R. Casadei, F. Frabetti, L. Vitale, M. Pelleri, S. Tassani, F. Piva, S. Perez-Amodio, P. Strippoli and S. Canaider, An estimation of the number of cells in the human body, Ann Hum Biol., 40 (2013), 463-471. doi: 10.3109/03014460.2013.807878. Google Scholar
[6]	G. Brodland, X. Chen, P. Lee and M. Marsden, From genes to neural tube defects (ntds): Insights from multiscale computational modeling, HFSP J, 4 (2010), 142-152. Google Scholar
[7]	R. Chaturvedi, C. Huang, B. Kazmierczak, T. Schneider, J. Izaguirre, T. Glimm, H. Hentschel, J. Glazier, S. Newman and M. Alber, On multiscale approaches to three-dimensional modelling of morphogenesis, J R Soc Interface, 2 (2005), 237-253. doi: 10.1098/rsif.2005.0033. Google Scholar
[8]	H. Chen and G. Brodland, Cell-level finite element studies of viscous cells in planar aggregates, J Biomech Eng, 122 (2000), 394-401. doi: 10.1115/1.1286563. Google Scholar
[9]	M. Chyba, M. Kobayashi, F. Mercier, J. Rader, A. Tamura-Sato, G. Telleschi and Y. Kwon, A new approach to modeling morphogenesis using control theory, Special volume of the São Paulo Journal of Mathematical Sciences in honor of Prof. Waldyr Oliva,, 5 (2011), 281-315. doi: 10.11606/issn.2316-9028.v5i2p281-315. Google Scholar
[10]	P. Ciarletta, M. Ben Amar and M. Labouesse, Continuum model of epithelial morphogenesis during caenorhabditis elegans embryonic elongation, Phil. Trans. R. Soc. A, 367 (2009), 3379-3400. doi: 10.1098/rsta.2009.0088. Google Scholar
[11]	D. Clausi and G. Brodland, Mechanical evaluation of theories of neurulation using computer simulations, Development, 118 (1993), 1013-1023. Google Scholar
[12]	I. Decimo, G. Fumagalli, V. Berton, M. Krampera and F. Bifari, Meninges: from protective membrane to stem cell niche, American Journal of Stem Cells, 1 (2012), 92-105. Google Scholar
[13]	V. Douet, E. Arikawa-Hirasawa and F. Mercier, Fractone-heparan sulfates mediate bmp-7 inhibition of cell proliferation in the adult subventricular zone, Neuroscience Letters, 528 (2012), 120-125. doi: 10.1016/j.neulet.2012.08.077. Google Scholar
[14]	M. Fitch and J. Silver, Glial cell extracellular matrix: Boundaries for axon growth in development and regeneration, Cell and Tissue Research, 290 (1997), 379-384. Google Scholar
[15]	H. Frieboes, X. Zheng, C. Sun, B. Tromberg, R. Gatenby and V. Cristini, An integrated computational/experimental model of tumor invasion, Cancer Res, 66 (2006), 1597-1604. doi: 10.1158/0008-5472.CAN-05-3166. Google Scholar
[16]	F. Gage, Neurogenesis in the adult brain, The Journal of Neuroscience, 22 (2002), 612-613. Google Scholar
[17]	T. Henzinger, The theory of hybrid automata, Logic in Computer Science. LICS '96. Proceedings. Eleventh Annual IEEE Symposium, (1996), 278-292. doi: 10.1109/LICS.1996.561342. Google Scholar
[18]	D. Ingber, J. Madri and J. Jamieson, Role of basal lamina in neoplastic disorganization of tissue architecture, Proc. Natl. Acad. Sci. USA, 78 (1981), 3901-3905. doi: 10.1073/pnas.78.6.3901. Google Scholar
[19]	A. Jacobson, G. Oster, G. Odell and L. Cheng, Neurulation and the cortical tractor model for epithelial folding, Journal of Embryology and Experimental Morphology, 96 (1986), 19-49. Google Scholar
[20]	K. Johansson, M. Egerstedt, J. Lygeros and S. Sastry, On the regularization of zeno hybrid automata, System & Control Letters, 38 (1999), 141-150. doi: 10.1016/S0167-6911(99)00059-6. Google Scholar
[21]	K. Johansson, J. Lygeros, S. Sastry and M. Egerstedt, Simulation of zeno hybrid automata, Decision and Control, 4 (1999), 3538-3543. doi: 10.1109/CDC.1999.827900. Google Scholar
[22]	G. Kempermann, L. Wiskott and F. Gage, Functional significance of adult neurogenesis, Current Opinion in Neurobiology, 14 (2004), 186-191. doi: 10.1016/j.conb.2004.03.001. Google Scholar
[23]	A. Kerever, J. Schnack, D. Vellinga, N. Ichikawa, C. Moon, E. Arikawa-Hirasawa, J. Efird and F. Mercier, Novel extracellular matrix structures in the neural stem cell niche capture the neurogenic factor fibroblast growth factor 2 from the extracellular milieu, Stem Cells, 25 (2007), 2146-2157. doi: 10.1634/stemcells.2007-0082. Google Scholar
[24]	Y. Kim, M. Stolarska and H. Othmer, A hybrid model for tumor spheroid growth in vitro i: Theoretical development and early results, Mathematical Models and Methods in Applied Sciences, 17 (2007), 1773-1798. doi: 10.1142/S0218202507002479. Google Scholar
[25]	L. Li, W. Dong, Y. Ji, Z. Zhang and L. Tong, Minimal-energy driving strategy for high-speed electric train with hybrid system model, IEEE Transactions on Intelligent Transportation Systems, 14 (2013), 1642-1653. Google Scholar
[26]	H. Lin and P. Antsaklis, Hybrid dynamical systems: An introduction to control and verification, Foundations and Trends in Systems and Control, 1 (2014), 1-172. Google Scholar
[27]	J. Lygeros, K. Johansson, S. Sastry and M. Egerstedt, On the existence of executions of hybrid automata, Decision and Control, 3 (1999), 2249-2254. doi: 10.1109/CDC.1999.831255. Google Scholar
[28]	J. Lygeros, K. Johansson, S. Simic, J. Zhang and S. Sastry, Continuity and invariance in hybrid automata, Decision and Control, 1 (2001), 340-345. doi: 10.1109/CDC.2001.980123. Google Scholar
[29]	J. Lygeros, K. Johansson, S. Simic, J. Zhang and S. Sastry, Dynamical properties of hybrid automata, Automatic Control, IEEE Transactions, 48 (2003), 2-17. doi: 10.1109/TAC.2002.806650. Google Scholar
[30]	J. Lygeros, S. Sastry and C. Tomlin, Hybrid Systems: Foundations, Advanced Topics, and Applications, Springer Verlag, 2012. Google Scholar
[31]	J. Meitzen, K. Pflepsen, C. Stern, R. Meisel and P. Mermelstein, Measurements of neuron soma size and density in rat dorsal striatum, nucleus accumbens core and nucleus accumbens shell: Differences between striatal region and brain hemisphere, but not sex, Neuroscience Letters, 487 (2011), 177-181. doi: 10.1016/j.neulet.2010.10.017. Google Scholar
[32]	F. Mercier and E. Arikawa-Hirakawa, Heparan sulfate niche for cell proliferation in the adult brain, Neuroscience Letters, 510 (2012), 67-72. doi: 10.1016/j.neulet.2011.12.046. Google Scholar
[33]	F. Mercier, J. Kitasako and G. Hatton, Anatomy of the brain neurogenic zones revisited: Fractones and the fibroblast/macrophage network, J Comp Neurol, 451 (2002), 170-188. doi: 10.1002/cne.10342. Google Scholar
[34]	F. Mercier, J. Kitasako and G. Hatton, Fractones and other basal laminae in the hypothalamus, J Comp Neurol, 455 (2003), 324-340. doi: 10.1002/cne.10496. Google Scholar
[35]	H. Minkowski, Space and time, in The Principle of Relativity, Calcutta University Press, 1920, 70-88, Trans. M Saha. Google Scholar
[36]	G. Oster, On the crawling of cells, Journal of Embryology and Experimental Morphology, 83 (1984), 329-364. Google Scholar
[37]	E. Palsson, Single-Cell-Based Models in Biology and Medicine, Mathematics and Biosciences in Interaction, 2007. doi: 10.1007/978-3-7643-8123-3. Google Scholar
[38]	J. Piovesan, C. Abdallah and H. Tanner, Preliminary results on interconnected hybrid systems, Control and Automation, (2008), 101-106. doi: 10.1109/MED.2008.4602157. Google Scholar
[39]	J. Piovesan, C. Abdallah and H. Tanner, Modeling multi-agent systems with hybrid interacting dynamics, American Control Conference, (2009), 3644-3649. doi: 10.1109/ACC.2009.5160419. Google Scholar
[40]	J. Piovesan, H. Tanner and C. Abdallah, Discrete asymptotic abstractions of hybrid systems, Decision and Control, (2006), 917-922. doi: 10.1109/CDC.2006.377733. Google Scholar
[41]	N. Poplawski, M. Swat, J. Gens and J. Glazier, Adhesion between cells, diffusion of growth factors, and elasticity of the aer produce the paddle shape of the chick limb, Physica A, 373 (2007), 521-532. doi: 10.1016/j.physa.2006.05.028. Google Scholar
[42]	J. Rader, Dynamic Modelling Of Neural Morphogenesis Using Mathematical Control Theory, Master's thesis, University of Hawai'i, 2011. Google Scholar
[43]	K. Rejniak and A. Anderson, Hybrid models of tumor growth, Wiley Interdiscip Rev Syst Biol Med, 3 (2011), 115-125. doi: 10.1002/wsbm.102. Google Scholar
[44]	C. Ronse and M. Tajine, Discretization in hausdorff space, Journal of Mathematical Imaging and Vision, 12 (2000), 219-242. doi: 10.1023/A:1008366032284. Google Scholar
[45]	J. Sanes, Extracellular matrix molecules that influence neural development, Ann. Rev. Neurosci, 12 (1989), 491-516. doi: 10.1146/annurev.ne.12.030189.002423. Google Scholar
[46]	M. Stolarska, Y. Kim and H. Othmer, Multi-scale models of cell and tissue dynamics, Phil. Trans. R. Soc. A, 367 (2009), 3525-3553. doi: 10.1098/rsta.2009.0095. Google Scholar
[47]	A. Teel, A. Subbaraman and A. Sferlazza, Stability analysis for stochastic hybrid systems: A survey, Automatica, 50 (2014), 2435-2456. doi: 10.1016/j.automatica.2014.08.006. Google Scholar
[48]	S. Tripakis and T. Dang, Modeling, verification and testing using timed and hybrid automata, in Model-Based Design of Embedded Systems (ed. M. P. Nicolescu G), CRC Press, 2009,383-436. doi: 10.1201/9781420067859-c13. Google Scholar
[49]	A. Voss-Bohme, Multi-scale modeling in morphogenesis: A critical analysis of the cellular potts model, PLoS ONE, 7 (2012), e42852. doi: 10.1371/journal.pone.0042852. Google Scholar
[50]	R. Vracko, Basal lamina scaffold-anatomy and significance for maintenance of orderly tissue structure, Am J Pathol, 77 (1974), 313-346. Google Scholar
[51]	D. Walker, S. Wood, J. Southgate, M. Holcombe and R. Smallwood, An integrated agent-mathematical model of the effect of intercellular signalling via the epidermal growth factor receptor on cell proliferation, J Theor Biol, 242 (2006), 774-789. doi: 10.1016/j.jtbi.2006.04.020. Google Scholar
[52]	J. Ward and J. King, Mathematical modelling of avascular-tumour growth, IMA J Math Appl Med Biol, 14 (1997), 39-69. doi: 10.1093/imammb/14.1.39. Google Scholar
[53]	M. Wyczalkowski, Z. Chen, B. Filas, V. Varner and L. Taber, Computational models for mechanics of morphogenesis, Birth Defects Res C Embryo Today, 96 (2012), 132-152. doi: 10.1002/bdrc.21013. Google Scholar
[54]	G. Xie and L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Transactions on Automatic Control, 49 (2004), 960-966. doi: 10.1109/TAC.2004.829656. Google Scholar
[55]	Z. Yang and M. Blanke, A unified approach to controllability analysis for hybrid control systems, Nonlinear Analysis: Hybrid Systems, 1 (2007), 212-222. doi: 10.1016/j.nahs.2006.08.002. Google Scholar
[56]	M. Zarandi, A. Bonakdar and I. Stiharu, Investigations on natural frequencies of individual spherical and ellipsoidal bakery yeast cells, in Proceedings of the COMSOL Conference 2010 Boston, 2010. Google Scholar

Figure 1. Fractones in the fourth ventricle of a rat brain. The dot-like structures arrayed around the boundary of the 4th ventricle are the fractones.

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Figure 2. An individual fractone. Note the branching structure that lends it its name. Image taken by transmission electron microscope and used with permission from Dr. Frederic Mercier.

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Figure 3. Fractones in the third ventricle of a rat brain. The dot-like structures arrayed in the funnel-shaped ventricle (indicated by the arrows) are the fractones.

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Figure 4. Examples of an (a) admissible set of cell bodies and (b-c) two inadmissible sets of cell bodies. The dark spheres are the cell bodies, and the lighter surrounding ellipsoids to form the associated $\hat{B}$. The set in (b) is not admissible since

$\hat{B}$

is not connected. The set in (c) is not admissible since two of the cell bodies intersect.

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Figure 5. Examples of two overlapping cell spaces, one in black and one in white. Each cell has a radius of 4.5

$\mu$

m. The geometric distance between the cell spaces are (a) 4.5

$\mu$

m and (b) 9

$\mu$

m. Note that the existence of an outlier cell in (b) causes the much larger distance even though many of the other overlapping cells are close to one another.

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Figure 6. Example of two cells with the same radius and with centers 24

$\mu$

m apart. The cell-birth parameters are

$90$

minutes and

$120$

minutes. The age distance

$d_a$

(with

$kappa=4.5$

$\mu$

m/min) is therefore

$4.5|120-90|+24=159$

$\mu$

m.

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Figure 7. (a) A 2D figure of a biological structure and diffusion space and (b) a 3D figure of a biological structure with diffusion space. Cells are in dark grey, fractones in black, meninges in white, and diffusion space in light grey (although the diffusion space also includes the fractones).

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Figure 8. An example initial condition with a single cell (in black) and distribution of a single growth factor in the diffusion space surrounding it. The growth factor concentration is represented by the color intensity of the grey; close to the cell, the concentration is

$0$

concentration units, and at outer boundary of the diffusion space, it is

$20$

concentration units.

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Figure 9. The interaction of the domain, guard conditions, and edges of our model. Left to right, we begin in the domain of biological structure $q_1$. The guard condition for edge

$(q_1,q_2)$

is then met when the positive fractone captures

$100$

concentration units of growth factor. This triggers growth and we instantaneously switch discrete states to biological structure $q_2$.

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Figure 10. An example of the growth algorithm. (a) A mother cell (grey) about to undergo mitosis. The position of the new cell is determined by the direction of the tangent vector where the fractone (black circle) touches the cell. (b) The two daughter cells. Note that one of the new cells is in the same position as the mother cell, and the other is created a distance

$d_m$

away along the tangent axis. Both cells are assigned a cell-birth parameter

$\lambda$

equal to the time at which the growth occurs. In addition, the fractone is relocated to be on the displaced new cell.

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Figure 11. An example biological structure

$q$

with cells, $c_1$, $c_2$, $c_3$, in dark gray; positive fractones, $f^+_1$, $f^+_2$, in black; negative fractone, $f^-_3$, is shaded; and diffusion space in light gray. Corresponding cell-birth parameters $\lambda_{c_1}$, $\lambda_{c_2}$, $\lambda_{c_3}$ for $c_1$, $c_2$, $c_3$, respectively, are shown and given in minutes.

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Figure 12. (a) Two-dimensional example of an initial cell set (black) and a final target set (white). (b) The adjusted target set (grey) formed by adding cells in the proposition algorithm. Each grey cell is 4.5

$\mu$

m away from a neighboring cell.

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Figure 13. Top view of the neurulation simulation. In black, the cells. In grey, the fractones. We start with a U-shape configuration and wish to end with a closed ring of cells.

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Figure 14. Progression of the growth of the tube. Shown is the mass of cells at selected times. Cells (black), fractones (light grey), and immature cells (dark grey). Meninges is not shown in the image for clarity -it is present surrounding the mass, however. From

$t=0$

to $t=210$, we simply lengthen the sides of the U-shape. Until $t=560$, we continue to extend upwards, but with staggered growth to create the curved edge. Around

$t=1000$

we have closed the ring, and at

$t=1150$

we have completed the top of the ring.

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Figure 15. Left: Final target cell configuration. Center: Initial cell configuration (

$9 \times 9 \times 9$

cube of cells). Right: Transparency of target cell configuration showing relative location of initial cell configuration in black. Positive axes are as shown.

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Figure 16. Progression of growth of cells (black) compared to target space (light grey) over time. Images shown correspond to times at which the control exerted a change on the position of fractones (dark grey).

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