Using Partition Information Entropy to Computationally Rank Order Critical Subreactions in a Petri Net Model of a...

Improved computational methods to analyze the mathematical structure and function of biochemical networks are needed when the biomolecular connectivity is known but when a complete set of the equilibrium and rate constants may not be available. We use Petri nets, which are equivalently bipartite digraphs, to analyze the rule-based flow of information through the network. We present several computational improvements to Petri net modeling as an aid to improve this approach, previously limited by the combinatorics of network size and complexity. The generation of Petri nets using equations for three elemental stencils (molecular reaction, synthesis complex formation, and decomposition complex formation) has been automated. A set of finite probability measures is defined in terms of a partition information entropy, where the complete listing of unique minimal cycles (UMCs) of the Petri net provides the natural partitioning. This enables the ranking of the UMC listing that covers all possible information flows in the reaction network; the information entropy measure enables the identification of which UMCs are more significant than others. In terms of the information entropy, forward cycles are less surprising and carry less information entropy, whereas backward cycles carry more information entropy and serve as regulators by providing feedback to control the network. As the systems analyzed increase in size and complexity, the automatic rank ordering of the UMCs provides a mechanism to highlight the globally most important information without the need to make local simplifying modeling choices. The information entropy metric is also used to compute source-to-sink information costs and is related to knockout analyses. The hybrid Petri net approach shows the most important species and where it is easiest to disrupt or otherwise affect the network. As exemplar, the enhanced methodology is applied to a model of the initial subnetwork in the EGFR network.