Boolean dynamics of biological networks with multiple coupled feedback loops

Abstract

Boolean networks have been frequently used to study the dynamics of biological networks. In particular, there have been various studies showing that the network connectivity and the update rule of logical functions affect the dynamics of Boolean networks. There has been, however, relatively little attention paid to the dynamical role of a feedback loop, which is a circular chain of interactions between Boolean variables. We note that such feedback loops are ubiquitously found in various biological systems as multiple coupled structures and they are often the primary cause of complex dynamics. In this article, we investigate the relationship between the multiple coupled feedback loops and the dynamics of Boolean networks. We show that networks have a larger proportion of basins corresponding to fixed-point attractors as they have more coupled positive feedback loops, and a larger proportion of basins for limit-cycle attractors as they have more coupled negative feedback loops.

FIGURE 1

An example of a Boolean network and a state transition network.

FIGURE 2

An example illustrating the relationship between feedback loops and Boolean dynamics.

FIGURE 3

The relationship between the number of feedback loops and the Derrida value where the number of networks is 2000.

FIGURE 4

The effects of negative feedbacks on the fixed-point basins.

FIGURE 5

The effects of negative feedback loops on the number of fixed-points.

FIGURE 6

The dynamics of networks with multiple-positive-feedback and multiple-negative-feedback loops.