Dynamics of Boolean Networks Controlled by Biologically Meaningful Functions

The remarkably stable dynamics displayed by randomly constructed Boolean networks is one of the most striking examples of the spontaneous emergence of self-organization in model systems composed of many interacting elements (Kauffman, S., J. theor. Biol.22, 437–467, 1969; The Origins of Order, Oxford University Press, Oxford, 1993). The dynamics of such networks is most stable for a connectivity of two inputs per element, and decreases dramatically with increasing number of connections. Whereas the simplicity of this model system allows the tracing of the dynamical trajectories, it leaves out many features of real biological connections. For instance, the dynamics has been studied in detail only for networks constructed by allowing all theoretically possible Boolean rules, whereas only a subset of them make sense in the material world. This paper analyses the effect on the dynamics of using only Boolean functions which are meaningful in a biological sense. This analysis is particularly relevant for nets with more than two inputs per element because biological networks generally appear to be more extensively interconnected. Sets of the meaningful functions were assembled for up to four inputs per element. The use of these rules results in a smaller number of distinct attractors which have a shorter length, with relatively little sensitivity to the size of the network and to the number of inputs per element. Forcing away the activator/inhibitor ratio from the expected value of 50% further enhances the stability. This effect is more pronounced for networks consisting of a majority of activators than for networks with a corresponding majority of inhibitors, indicating that the former allow the evolution of larger genetic networks. The data further support the idea of the usefulness of logical networks as a conceptual framework for the understanding of real-world phenomena.