STRUCTURAL PRINCIPLES FOR COMPLEX DYNAMICS IN GLASS NETWORKS

Gene-regulatory networks are potentially capable of more complex behavior than convergence to a stationary state, or even cycling through a simple sequence of expression patterns. The analysis of qualitative dynamics for these networks is facilitated by using piecewise-linear equations and its state transition diagram (an n-dimensional hypercube, in the case of n genes with a single effective threshold for the protein product of each). Our previous work has dealt with cycles of states in the state transition diagram that allow periodic solutions. Here, we study a particular kind of figure-8 pattern in the state transition diagram and determine conditions that allow complex behavior. Previous studies of complex behavior, such as chaos, in such networks have dealt only with specific examples. Our approach allows an appreciation of the design principles that give rise to complex dynamics, which may have application in assessing the dynamical possibilities of gene networks with poorly known parameters, or for synthesis or control of gene networks with complex behavior.