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HW 7 : Due 12/2

1. Design a 1D stochastic cellular automata that uses on rules involving two adjacent cells at at time that (with high probability) do the following tasks:

a. If the majority of cells are initially 0, the final state is all zeros. If the majority of cells is initial 1, the final state is 1.
b. Starting with all cells at 0, reach a final state in which one cell is a 1 and the rest are 2.

Explain why these automata work.

2. Starting with three cells, make the reachability graph for each of the above automata and label the transitions with the rates. Form the master equation for the resulting system and simulate the equations. Show how you can trade off accuracy for speed by tuning the rates of the system.

3. In gro, make a simulation of leader election as follows. Start with 64 cells arranged in a square and all running the same program. Have each cell flip coins until one comes up heads. This is the leader. It should then send a signal to the other cells to become followers. Followers should also send the signal. Show, by simulation, that the rate at which the coin comes up heads and the rate of diffusion of the signal affect accuracy and speed.

4. Using the Kegg database, find three enzymes that (a) are not present in E. coli MG1655 but for which (b) the metabolite acted on by the enzyme is present in E. coli MG1655. Describe what the enzymes do, what organism they come from, and why that organism might need the enzyme.