# American Institute of Mathematical Sciences

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

## Morphogenesis modelization of a fractone-based model

 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.

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:

show all references

##### References:
Fractones in the fourth ventricle of a rat brain. The dot-like structures arrayed around the boundary of the 4th ventricle are the fractones.
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.
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.
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.
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.
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.
(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).
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.
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 $q_2$.
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.
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.
(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.
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.
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.
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.
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).
 [1] Achilles Beros, Monique Chyba, Kari Noe. Co-evolving cellular automata for morphogenesis. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2053-2071. doi: 10.3934/dcdsb.2019084 [2] José Ignacio Tello. Mathematical analysis of a model of morphogenesis. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 343-361. doi: 10.3934/dcds.2009.25.343 [3] Elena Izquierdo-Kulich, Margarita Amigó de Quesada, Carlos Manuel Pérez-Amor, José Manuel Nieto-Villar. Morphogenesis and aggressiveness of cervix carcinoma. Mathematical Biosciences & Engineering, 2011, 8 (4) : 987-997. doi: 10.3934/mbe.2011.8.987 [4] Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Morphogenesis of the tumor patterns. Mathematical Biosciences & Engineering, 2008, 5 (2) : 299-313. doi: 10.3934/mbe.2008.5.299 [5] T.K. Subrahmonian Moothathu. Homogeneity of surjective cellular automata. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 195-202. doi: 10.3934/dcds.2005.13.195 [6] Achilles Beros, Monique Chyba, Oleksandr Markovichenko. Controlled cellular automata. Networks & Heterogeneous Media, 2019, 14 (1) : 1-22. doi: 10.3934/nhm.2019001 [7] Marcus Pivato. Invariant measures for bipermutative cellular automata. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 723-736. doi: 10.3934/dcds.2005.12.723 [8] Ana I. Muñoz, José Ignacio Tello. Mathematical analysis and numerical simulation of a model of morphogenesis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1035-1059. doi: 10.3934/mbe.2011.8.1035 [9] Bernard Host, Alejandro Maass, Servet Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1423-1446. doi: 10.3934/dcds.2003.9.1423 [10] Marcelo Sobottka. Right-permutative cellular automata on topological Markov chains. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1095-1109. doi: 10.3934/dcds.2008.20.1095 [11] Xinxin Tan, Shujuan Li, Sisi Liu, Zhiwei Zhao, Lisa Huang, Jiatai Gang. Dynamic simulation of a SEIQR-V epidemic model based on cellular automata. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 327-337. doi: 10.3934/naco.2015.5.327 [12] P. Alonso Ruiz, Y. Chen, H. Gu, R. S. Strichartz, Z. Zhou. Analysis on hybrid fractals. Communications on Pure & Applied Analysis, 2020, 19 (1) : 47-84. doi: 10.3934/cpaa.2020004 [13] Thomas I. Seidman, Olaf Klein. Periodic solutions of isotone hybrid systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 483-493. doi: 10.3934/dcdsb.2013.18.483 [14] David Kinderlehrer, Michał Kowalczyk. The Janossy effect and hybrid variational principles. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 153-176. doi: 10.3934/dcdsb.2009.11.153 [15] Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793 [16] Shin-Ichiro Ei, Kei Nishi, Yasumasa Nishiura, Takashi Teramoto. Annihilation of two interfaces in a hybrid system. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 857-869. doi: 10.3934/dcdss.2015.8.857 [17] William Clark, Anthony Bloch, Leonardo Colombo. A Poincaré-Bendixson theorem for hybrid systems. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019028 [18] Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953 [19] Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201 [20] Roger Brockett. Controllability with quadratic drift. Mathematical Control & Related Fields, 2013, 3 (4) : 433-446. doi: 10.3934/mcrf.2013.3.433

2018 Impact Factor: 1.008