"

From 2014.igem.org

Form a growing snowflake with Cellular Automaton

A growing snowflake can be programmed with two-dimensional cellular automaton[]. The cellular space is defined as a two - dimensional square lattice of cells with size of n*n. Every cell has two states, on and off, represented by “1” or “0” respectively, and has four orthogonally adjacent cells as neighborhood. Its evolution rule directs how cells change states from one discrete time point to the next and it is that: if a cell is off, it can change state to be on at the next time point only when one out of its four neighborhood is on; if a cell is on, it will remain on forever.

For a cell with coordination of , at time point , its state can be shown as. Then the evolution rule is:

Let initial state as all cells off except for the center four “on” cells. As time increases, a growing snowflake forms with less than 10 steps (one step equals to one time point). Figure 1 shows how a snowflake grows up at time point 6. Figure 2 shows a dynamic pattern of it.

Figure 1

Figure 2

In this experiment, the reaction process can be divided into three parts: signal receiving and signal feedback and signal transmission. The spreading of signaling molecules in culture satisfies certain regularity. The purpose of this model is to describe the molecular diffusion process, the process of signal transmission. Fick's law is one of the models described in physics molecular thermal motion, based on the fick's law to build the molecular diffusion model. This model is based on the following assumptions.

Hypothesis 1

No intermolecular interaction. Each molecule movement is independent of each other. The interactions between molecules is simplified in this model, so molecules move randomly in the medium, and the average moving rate is constant. So all the spread of the signal source is only related to the initial state and time, thus the problem is simplified to single source.

Hypothesis 2

There are two kinds of motions for molecular including vibration and spin at some point. The vibration frequency is f. The movement direction is random and it moves at an average rate and spins the original position unchanged.

Hypothesis 3

Molecular motion is always located on the same horizontal plane. It is convenient to make the movement rate in the horizontal direction to stand for the average speed. And it doesn't reflect molecules’ movement in the vertical direction.

This model uses one-dimensional cellular automata. The signal source is located in the first cell, and the cell number i stands for the distance between molecules and the signal source. Cellular signal source stands for circle medium which is in a i distance. For each of the molecules in a cell, there are three motion state at a same time, spin, i - 1 direction vibration and i + 1 direction, and the probability of vibration in different directions is equal. The signal molecules from signal source follow the secretion function of P, an average rate v and molecular vibration frequency f. This model is built depending on all above.

Optimization and improvement for the model is in the following.

There is interaction between molecules and the movement probability in the direction of I - 1 and I + 1 is influenced by the concentration. It is hard to measure the average rate of molecular motion and vibration frequency. And it may be better to use diffusion coefficient D instead.