Uncovering Dynamic Structures Within Cyclic Attractors of Asynchronous Boolean Networks with Spectral Clustering

Boolean models provide an intuitive framework for the investigation of complex biological networks. Dynamics that implement asynchronous update rules, in particular, can help embody the complexity arising from non-deterministic behavior. These transition systems allow for the emergence of complex attractors, cyclic subgraphs that capture oscillating asymptotic behavior. Techniques that explore and attempt to describe the structures of these attractors have received limited attention. In this context, the incorporation of process rate information may yield additional insights into dynamical patterns. Here, we propose to use a spectral clustering algorithm on the kinetic rate matrix of time-continuous Boolean networks to uncover dynamic structures within cyclic attractors. The Robust Perron Cluster Analysis (PCCA+) can be used to unravel metastable sets in Markov jump processes, i.e. sets in which a system remains for a long time before it switches to another metastable set. As a proof-of-concept, we apply this method to Boolean models of the mammalian cell cycle. By considering the categorization of transitions as either slow or fast, we investigate the impact of time information on the emergence of significant sub-structures.