Study of Cellular Automata - The Kauffman-Model

Stephan Schuh, Niclas Wohlleben, Marc Zimmermann

We study the development of a chaotic system over several generations using a computer program that is intended to simulate the interaction among genes as well as ferromagnetic interactions. The program is based on a model by Stuart Kauffman from 1969.

Kauffman's model is based on findings of molecular biology. Every cell in our body has about 1,000,000 genes and thus - considering that genes are binary carriers of information - there could in principle be 21,000,000 different types of cells. However, only about 1000 types of cells have been found. It would be impossible to organise an organism with the substantially larger number of 21,000,000 types of cells. Kauffman explains the limited range of variation in nature with interactions among genes.

The genes are represented by positions in a two-dimensional L×L lattice, which can have one of two values (zero or one). In each iteration, the state of each position depends on the states of its four neighbours. A mutation can be introduced by reversing the binary value at the centre of the L×L lattice. We study the development of the mutated lattice in comparison to a lattice without mutation. We get statistical results describing the evolution of the mutation, e.g. the time over which the mutation spreads, and we attempt to identify patterns or typical behaviour. Our programming effort was supported by Prof. Dr. Stauffer from the Institute for Theoretical Physics at the University of Cologne.

We wrote our first program using Basic on a home PC, then we programmed the next version using Pascal on a school computer to get results for our project report. Finally, we conducted further numerical experiments at the Computing Centre of the University of Cologne using Fortran as the computing power of our computers was insufficient for our more demanding calculations. The simulations at our school suggested a result that we were able to confirm with our first simulations in Cologne. We are expecting further results as the set of experiments in Cologne has not been fully analysed yet and these results will need to be confirmed through a more targeted set of numerical experiments thereafter.

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