Biologically Relevant Classes of Boolean Functions

A large influx of experimental data has prompted the development of innovative computational techniques for modeling and reverse engineering biological networks. While finite dynamical systems, in particular Boolean networks, have gained attention as relevant models of network dynamics, not all Boolean functions reflect the behaviors of real biological systems. In this work, we focus on two classes of Boolean functions and study their applicability as biologically relevant network models: the nested and partially nested canalyzing functions. We begin by analyzing the nested canalyzing functions (NCFs), which have been proposed as gene regulatory network models due to their stability properties. We introduce two biologically motivated measures of network stability, the average height and average cycle length on a state space graph and show that, on average, networks comprised of NCFs are more stable than general Boolean networks. Next, we introduce the partially nested canalyzing functions (PNCFs), a generalization of the NCFs, and the nested canalyzing depth, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and