On canalizing Boolean functions

Boolean networks are an important model of gene regulatory networks in systems and computational biology. Such networks have been widely studied with respect to their stability and error tolerance. It has turned out that canalizing Boolean functions and their subclass, the nested canalizing functions, appear frequently in such networks. These classes have been shown to have a stabilizing effect on the network dynamics. One measure for the stability is the average sensitivity of Boolean functions. Using Fourier analysis, we provide upper bounds on the average sensitivity for canalizing and nested canalizing functions. The latter bound proves an open conjecture in the literature. Further, we state upper and lower bounds on the noise sensitivity, based on an inductive proof of the spectra of the functions. The noise sensitivity gives the error tolerance of functions with noisy inputs. We show that canalizing functions maximize the mutual information between an input variable and the outcome of the function. We provide relationships between the noise sensitivity and the mutual information of functions with noisy inputs. Using these relationships we prove upper and lower bounds on the mutual information for canalizing and nested canalizing functions. To prove our results we derive spectral properties of canalizing and nested canalizing functions, which can also be used to test membership of functions to these classes. Finally, we present two algorithms solving the preimage problem of Boolean networks. The first approach depends on the properties of canalizing functions, the second algorithm is based on the well-known sum-product algorithm (belief propagation).