Polynomial Models of time series over (Z/p)n

Micah Leamer

In recent years mathematical modeling has become an integral part of understanding biological systems. It is the hope that particularly elusive branches of biology can be brought into better understanding through new mathematical techniques. The methods in this paper were inspired by a need to understand gene regulatory networks. The methods are relevant to data that has been collected for the interactions of proteins and DNA over time. Generally a gene regulatory network will have thousands of components. Usually, however, only a few observations can be made as to the state that these various components are in over time. The states of these n components at each time step are represented as points in Rn. We then discretize these points and represent them as points in (Z/q)n, where q is a prime. This paper shows a method for finding particular functions that map each point to the next point for all points. Further more we offer the constraint that one may choose what components affect various other components at the next time step and offer a way to find simple functions of this type. Also we show how any particular function that maps one point to the point at the next time step can be used to determine all such functions from (Z/q)n to (Z/q)n. The paper then explains an attached computer program, which determines the simplest such function with the ability to constrain what components each of the components at the next time step are dependent upon. (Advisor Ed Green)