Categorizing Attractor-Effective Canalyzing Functions in Boolean Networks

Abstract

Canalyzing Boolean functions have shown their popularity in various biological networks and established themselves to be biologically meaningful at the system level, marking their importance in the analysis of stability and robustness of such complex systems. Based on a matrix representation of Boolean networks due to the recently developed tool called semi-tensor product, we categorize canalyzing functions in terms of their capabilities of affecting the number of attractors in the Boolean network, which is one key index for the stability and robustness of Boolean networks. We show that there exist only three categories of attractor-effective canalyzing functions for any network size larger than 1, while the number of all the interested canalyzing functions is proportional to the square of the network size. We also give the explicit expression of the mean number of attractors with any length for Boolean networks with a single canalyzing function. Compared with Boolean networks without any canalyzing functions, we are able to show quantitatively how canalyzing functions can affect the mean number and length of attractors in Boolean networks for the first time.