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 that NCFs are not significantly better than PNCFs of sufficient depth for many applications to biological networks. Finally, we propose a method for the reverse engineering of networks of PNCFs using techniques from computational algebra. Given discretized time series data, this method finds a network model using PNCFs. Our ability to use these functions in reverse engineering applications further establishes their relevance as biological network models

Symmetry in Critical Random Boolean Network Dynamics

Using Boolean networks as prototypical examples, the role of symmetry in the dynamics of heterogeneous complex systems is explored. We show that symmetry of the dynamics, especially in critical states, is a controlling feature that can be used both to greatly simplify analysis and to characterize different types of dynamics. Symmetry in Boolean networks is found by determining the frequency at which the various Boolean output functions occur. There are classes of functions that consist of Boolean functions that behave similarly. These classes are orbits of the controlling symmetry group. We find that the symmetry that controls the critical random Boolean networks is expressed through the frequency by which output functions are utilized by nodes that remain active on dynamical attractors. This symmetry preserves canalization, a form of network robustness. We compare it to a different symmetry known to control the dynamics of an evolutionary process that allows Boolean networks to organize into a critical state. Our results demonstrate the usefulness and power of using the symmetry of the behavior of the nodes to characterize complex network dynamics, and introduce a novel approach to the analysis of heterogeneous complex systems

Entropy of complex relevant components of Boolean networks

Boolean network models of strongly connected modules are capable of capturing the high regulatory complexity of many biological gene regulatory circuits. We study numerically the previously introduced basin entropy, a parameter for the dynamical uncertainty or information storage capacity of a network as well as the average transient time in random relevant components as a function of their connectivity. We also demonstrate that basin entropy can be estimated from time-series data and is therefore also applicable to non-deterministic networks models.Comment: 8 pages, 6 figure