Canalizing Zhegalkin Polynomials as Models for Gene Expression Time Series Data

It has been observed that genetic regulatory networks share many characteristics with Boolean networks such as periodicity, self organization etc. Moreover it is also a known fact that in these networks, most genes are governed by canalizing Boolean functions. However, the actual gene expression level measurements are continuous valued. To combine discrete and continuous aspects, Zhegalkin polynomials can be used as continuous representations of Boolean functions. The requirement for the Boolean function to be canalizing can be extended to continuous functions by demanding monotonicity with respect to at least the canalizing variable. In this paper, it is proven that canalizing Zhegalkin polynomials observe this monotonicity property. Moreover, for correct handling of normalized data it is shown that the value of a Zhegalkin polynomial also lies within the unit interval as long as the values of its input variables also do so