# American Institute of Mathematical Sciences

May  2019, 24(5): 2053-2071. doi: 10.3934/dcdsb.2019084

## Co-evolving cellular automata for morphogenesis

 1 University of Hawaii, Department of Mathematics, Honolulu, HI 96822, USA 2 University of Hawaii, Laboratory for Advanced Visualization and Applications, Honolulu, HI 96822, USA

Received  January 2018 Revised  January 2019 Published  May 2019 Early access  March 2019

Fund Project: This work is partially supported by the Simons Foundation, award # 359510.

Morphogenesis, the shaping of an organism, is a complex biological process accomplished through an well organized interplay between growth, differentiation and cell movement.It is still today one of the major outstanding problems in the biological sciences. Pattern formation has been well-addressed in the literature with the development of many mathematical models including the famous reaction-diffusion ones. We here take a different approach, introducing a controlled cellular automaton in order to model the signal molecules, known as growth factors, that convey information from one cell to another during an organism's development and help maintain the viability of the adult. This control represents extracellular structures that have been associated with the regulation of stem cell proliferation and are called fractones. In this paper we introduce two co-evolving automata, one describing the perturbed diffusion of growth factors and one accounting for the rules of basic cellular functions (proliferation, differentiation, migration and apoptosis). Fractones are introduced as an external input to control the shaping of multi-cellular organisms; we analyze their influence on the emerging shape. We illustrate our theory with 2 and 3 dimensional simulations. This work presents the foundation upon which to develop cellular automata as a tool to simulate the morphodynamics in embryonic development.

Citation: Achilles Beros, Monique Chyba, Kari Noe. Co-evolving cellular automata for morphogenesis. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2053-2071. doi: 10.3934/dcdsb.2019084
##### References:
 [1] T.-S. A., A Hybrid Control Model of Fractone-Dependent Morphogenesis, PhD thesis, University of Hawaii, 2015. [2] T. Alarc'on, H. M. Byrne and P. K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment, J. Theor. Biol., 225 (2003), 257-274.  doi: 10.1016/S0022-5193(03)00244-3. [3] H. Balzter, P. W. Braun and W. Köhler, Cellular automata models for vegetation dynamics, Ecological Modelling, 107 (1998), 113-125.  doi: 10.1016/S0304-3800(97)00202-0. [4] A. Beros, M. Chyba, A. Fronville and F. Mercie, A morphogenetic cellular automaton, in Proceedings of the 2018 American Control Conference Milwaukee, 2018 (To Appear). doi: 10.23919/ACC.2018.8431498. [5] 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. [6] M. K. L. Collins, G. R. Perkins, G. Rodriguez-Tarduchy, M. A. Nieto and A. López-Rivas, Growth factors as survival factors: Regulation of apoptosis, BioEssays, 16 (1994), 133-138.  doi: 10.1002/bies.950160210. [7] J. A. D. L. Vaux S. Cory, Bcl-2 gene promotes haemopoietic cell survival and cooperates with c-myc to immortalize pre-b cells, Nature, 335 (1988), 440-442. [8] 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. [9] A.-H. E. M. F. Douet V Kerever A, Fractone-heparan sulphates mediate fgf-2 stimulation of cell proliferation in the adult subventricular zone, Cell Prolif, 46 (2013), 137-145. [10] E.-K. L. Ermentrout G.B., Neurogenesis in the adult brain, J. Theor. Biol., 160. [11] M. Gardner, Mathematical games: The fantastic combinations of john conway's new solitaire game life, Scientific American, 223. [12] 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. [13] T. Maretzky, A. Evers, W. Zhou, S. L. Swendeman, P.-M. Wong, S. Rafii, K. Reiss and C. P. Blobel, Migration of growth factor-stimulated epithelial and endothelial cells depends on egfr transactivation by adam17, Nature Communications, 2 (2011), 229. doi: 10.1038/ncomms1232. [14] 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. [15] F. Mercier, Fractones: extracellular matrix niche structures controlling stem cell fate and growth factor activity in the brain in health and disease, Cell Mol Life Sci., 73 (2016), 4661-4674.  doi: 10.1007/s00018-016-2314-y. [16] F. Mercier and V. Douet, Bone morphogenetic protein-4 inhibits adult neurogenesis and is regulated by fractone-heparan sulfates in the subventricular zones, J Chem Neuroanat, 57/58 (2014), 54-61.  doi: 10.1016/j.jchemneu.2014.03.005. [17] 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. [18] F. Mercier, Y. Kwon and R. Kodama, Meningeal/vascular alterations and loss of extracellular matrix in the neurogenic zone of adult btbr t+ tf/j mice, animal model for autism, Neuroscience Letters, 498 (2011), 173-178.  doi: 10.1016/j.neulet.2011.05.014. [19] 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. [20] S. El Yacoubi, P. Jacewicz and N. Ammor, Analyse et contrôle par automates cellulaires, An. Univ. Craiova Ser. Mat. Inform., 30 (2003), 210-221. [21] P. J. S. El Yacoubi, Cellular automata and controllability problem, in Proceeding of the 14th International Symposium on Mathematical Theory of Networks and Systems MTNS 2000, Perpignan, 2000. [22] A. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, Biological Sciences, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [23] J. Von Neumann and A. W. Burks, Theory of Self-Reproducing Automata, Urbana, University of Illinois Press, 1996. [24] 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. [25] S. Wolfram, Statistical mechanics of cellular automata, Reviews of Modern Physics, 55 (1983), 601-644.  doi: 10.1103/RevModPhys.55.601. [26] 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.

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##### References:
 [1] T.-S. A., A Hybrid Control Model of Fractone-Dependent Morphogenesis, PhD thesis, University of Hawaii, 2015. [2] T. Alarc'on, H. M. Byrne and P. K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment, J. Theor. Biol., 225 (2003), 257-274.  doi: 10.1016/S0022-5193(03)00244-3. [3] H. Balzter, P. W. Braun and W. Köhler, Cellular automata models for vegetation dynamics, Ecological Modelling, 107 (1998), 113-125.  doi: 10.1016/S0304-3800(97)00202-0. [4] A. Beros, M. Chyba, A. Fronville and F. Mercie, A morphogenetic cellular automaton, in Proceedings of the 2018 American Control Conference Milwaukee, 2018 (To Appear). doi: 10.23919/ACC.2018.8431498. [5] 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. [6] M. K. L. Collins, G. R. Perkins, G. Rodriguez-Tarduchy, M. A. Nieto and A. López-Rivas, Growth factors as survival factors: Regulation of apoptosis, BioEssays, 16 (1994), 133-138.  doi: 10.1002/bies.950160210. [7] J. A. D. L. Vaux S. Cory, Bcl-2 gene promotes haemopoietic cell survival and cooperates with c-myc to immortalize pre-b cells, Nature, 335 (1988), 440-442. [8] 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. [9] A.-H. E. M. F. Douet V Kerever A, Fractone-heparan sulphates mediate fgf-2 stimulation of cell proliferation in the adult subventricular zone, Cell Prolif, 46 (2013), 137-145. [10] E.-K. L. Ermentrout G.B., Neurogenesis in the adult brain, J. Theor. Biol., 160. [11] M. Gardner, Mathematical games: The fantastic combinations of john conway's new solitaire game life, Scientific American, 223. [12] 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. [13] T. Maretzky, A. Evers, W. Zhou, S. L. Swendeman, P.-M. Wong, S. Rafii, K. Reiss and C. P. Blobel, Migration of growth factor-stimulated epithelial and endothelial cells depends on egfr transactivation by adam17, Nature Communications, 2 (2011), 229. doi: 10.1038/ncomms1232. [14] 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. [15] F. Mercier, Fractones: extracellular matrix niche structures controlling stem cell fate and growth factor activity in the brain in health and disease, Cell Mol Life Sci., 73 (2016), 4661-4674.  doi: 10.1007/s00018-016-2314-y. [16] F. Mercier and V. Douet, Bone morphogenetic protein-4 inhibits adult neurogenesis and is regulated by fractone-heparan sulfates in the subventricular zones, J Chem Neuroanat, 57/58 (2014), 54-61.  doi: 10.1016/j.jchemneu.2014.03.005. [17] 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. [18] F. Mercier, Y. Kwon and R. Kodama, Meningeal/vascular alterations and loss of extracellular matrix in the neurogenic zone of adult btbr t+ tf/j mice, animal model for autism, Neuroscience Letters, 498 (2011), 173-178.  doi: 10.1016/j.neulet.2011.05.014. [19] 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. [20] S. El Yacoubi, P. Jacewicz and N. Ammor, Analyse et contrôle par automates cellulaires, An. Univ. Craiova Ser. Mat. Inform., 30 (2003), 210-221. [21] P. J. S. El Yacoubi, Cellular automata and controllability problem, in Proceeding of the 14th International Symposium on Mathematical Theory of Networks and Systems MTNS 2000, Perpignan, 2000. [22] A. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, Biological Sciences, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [23] J. Von Neumann and A. W. Burks, Theory of Self-Reproducing Automata, Urbana, University of Illinois Press, 1996. [24] 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. [25] S. Wolfram, Statistical mechanics of cellular automata, Reviews of Modern Physics, 55 (1983), 601-644.  doi: 10.1103/RevModPhys.55.601. [26] 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.
Fractones are visible as the small puncta arranged around the aqueduct. The larger elongated shapes surrounding aqueduct are blood vessels. Image of the cerebral aqueduct in an adult mouse taken by 20x PlanApo objective lens. The aqueduct connects the third and fourth ventricles together. The image is from [1], see [18] for information about the experimental technique. It was obtained in Dr. Mercier's laboratory at the University of Hawaii
Illustration of the superposition of the grids on which the two co-evolving automata act. (a) Grid for the diffusion automaton. (b) Grid for the cell-functions automaton. The yellow displays the space occupied by a biological cell and green units represent fractones
Our definition of neighborhood is based on the Manhattan distance. Blue units are at a distance 1 from the central unit $u$, yellow at a distance 2 and red at a distance 3. $N_c(u)$ with $r = 3$ is the union of all colored units in this picture beside unit $u$
A representation of the 3D neighborhood. The colors are as in Figure 3. Picture (1) displays the sphere of radius 3, and a view with the front units made transparents to show the inside is provided in (2). In (3) we can see a cross section of the sphere that passes through the middle
Starfish-like shape. Starting from the top left, going right and down, the snapshots are at timesteps 1, 35,100,130,160 and 198. Each cell in the simulation is marked with its age
3D simulation of a uniform mass starting from a unique cell and using a unique fractone throughout the simulation. The cell color indicates age – a red cell is the product of a recent mitosis, a yellow cell is old enough to undergo mitosis. The color of the diffusion units indicate concentration with blue representing low concentration and red high concentration. Note that for 3D simulations, we have hidden the diffusion concentrations except close to each of the cells in order to make the shape of the cell mass visible
3D simulation of an elongated shape. (a) represents the original cell and fractone as well as the heterogenous concentration of growth factor around the cell. (b) corresponds to time step # 77. At this point in the simulation, there are 13 cells. The visible fractone on the image is becomes active at this time step #78 represented in (c) which explains the new cell daughter cell on the bottom. Finally (d) is a side view (left) and a top view (right) of the final mass of cells, it corresponds to time step #240 and there are 41 cells
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