A general method to find all attractors of multi-level discrete networks
Analyzing the long-term behaviors (attractors) of dynamic models of biological systems can provide valuable insight. We propose a general method that can find all attractors of discrete dynamical systems by extending a method that finds all attractors of a Boolean network model. The method is based on finding stable motifs, subgraphs whose nodes’ states can stabilize on their own. We extended this method from binary states to any finite discrete levels by establishing a multi-level formalism, where a virtual node is created for each level of a multi-level node, and describing each virtual node with a quasi-Boolean function. We then create an expanded representation of the multi-level network, find multi-level stable motifs, and identify attractors in a similar way as in the Boolean case.
We test and validate the algorithm on representative synthetic multi-level networks and on a published biological network. Multi-level stable motifs offer a way to find all attractors without constraints on the update scheme and suggest ways to control which attractor the multi-level network model evolves into.