Phase transition of heterogeneous Boolean networks

Boolean networks have frequently and successfully been used to model many biological processes, from the time of their creation to model genetic networks to later developments on the same subject, signal transduction, immunology, and more. However, previous work has focused on only one particular class of Boolean function for any given case, or on rules generated for a specific network. Of particular note when analyzing a network's dynamics is its phase: order, chaos, or the edge-of chaos, which separates the other two phases. This describes the nature of the network's behavior with respect to the extent to which it is sensitive to perturbations in initial conditions. In this paper, we create a heterogeneous network model which includes four types of Boolean functions that govern the dynamics of the nodes: canalizing with one canalizing input, canalizing with two canalizing inputs, threshold, and bias functions---the proportion and parameters of which can be modified in order to study the average long-term behavior of the desired network. We develop a mathematical model for identifying the critical condition that leads to the edge-of-chaos by using an aggregated formula that takes into account the individual sensitivities of the four types of Boolean functions to minor perturbations. We also take into account that not all the states of the network are equally likely to occur in the long-run, thus expanding previous results in the literature to non-ergodic networks. Then, we use the model to create phase transition diagrams. The Boolean model contains numerous parameters, such as bias, proportions of activators or inhibitors, or parameters linked to canalization, whose modification may lead to different types of dynamics. We select several particular sets of parameters in order to study the impact of their change on the behavior of various networks, such as how easily the network tends toward order or chaos with given parameter values. Based on those simulations we find that the bias of the canalizing functions and the specific type of canalizing function that is more prevalent in the network seem to be of significant importance to the network dynamics. By increasing or decreasing their values we can widen or narrow the chaotic regions versus the order regions of the parameter space, thus shifting the critical curve within the given parameter scenario. Further extensive numerical investigations could provide an exhaustive classification of the parameter space into dynamical phases. Moreover, this model can be further honed by the addition of more function types, such as nested canalizing functions, and then tested against real biological networks, with the eventual goal that the ability to discover which parameter changes yield which dynamics may guide biological research in such areas as drug therapy.