Nearest-Neighbor Convergence and Cycling

This model demonstrates convergence to a stable two-component pattern of cell polarization states from a random starting point, followed by stable cycling between the two states. The structure and behavior of the model are described in more detail below.

The model depicts the embryo lying on its side, so the left half is on top and the right half is on the bottom. This rotation made the code easier; it has no biological significance.

Red and blue represent maximum cell polarization and depolarization respectively; intermediate colors represent intermediate polarization.

Polarization Your browser does not support the HTML5 canvas tag. Depolarization

Click "Reset" below to generate a new random start state; click "Bias" to add a 27% bias for
"red" in the top half of the display (left side of embryo) and "blue" in the bottom half (left side of embryo).

Click "Step" to evolve by 1 time step; "Run" runs the model in 0.5 sec time steps; "Stop" stops this process.

Left half of embryo





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Right half of embryo
Your browser does not support the HTML5 canvas tag. Left half of embryo





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Right half of embryo


            





Further model details

The 10 X 10 cell grid shown has cylindrical topology: cells along the top and bottom borders are neighbors. Blue cells on the top border of the display following convergence are "wrap-around" from the bottom half; red cells on the bottom border are wrap-around from the top half. This wrap-around effect is an artifact of a rigid grid of squares; it has no biological significance in a flexible embryo with flexible cells.

Each cell is initialized to a random mixture of polarized and unpolarized domains, and hence a random mixture of red and blue. "Bias" adds a 27% bias for red on top and blue on the bottom. On each time step, each cell increases its polarization by 20% if its four abutting neighbors are on average polarized; it decreases its polarization by 20% if its four abutting neighbors are on average depolarized. This process typically converges to a stable state in which one connected group of cells (taking wrapping around the outer borders into account) is fully polarized, i.e. red, and the remaining cells are fully depolarized, i.e. blue. Adding the 27% bias makes convergence to this state very probable.

If more than 65% of the cells are fully polarized or fully depolarized, the cells "die" and the model restarts. If no more than 65% of the cells are either polarized or depolarized, the cells collectively cycle their polarization states, i.e. polarized cells depolarize and depolarized cells polarize. Such stable oscillations of cell polarization state are postulated to regulate influx and efflux of regulatory molecules through voltage-dependent channels.



Copyright © 2014 Chris Fields
Non-commercial re-use permitted; please cite:
Chernet, B. T., C. Fields & M. Levin (2014). Long-range gap junctional signaling controls oncogene-mediated tumorigenesis in Xenopus laevis embryos. In review.