SHREDing a Biochemical Network Into Hierarchical Modules

Hierarchical modularity has emerged as an organizational principle of biochemical networks, providing insights into the coordinated regulation of reactions within and across pathways. In principle, the modularity of a biochemical network should allow the system to be partitioned into minimally interdependent parts, which in turn can facilitate detailed analysis of each part in the context of the overall system. In practice, modularity analysis has often relied on ad hoc decisions to mainly corroborate existing knowledge. While there is general agreement that a module should consist of a biologically meaningful group of connected components in the network, there is little consensus on the metric needed to quantitatively evaluate the quality of a partition. The goal of this study is to investigate a novel metric that can be used to systematically partition a biochemical network into functionally relevant groups of reactions. The metric, termed the Shortest Retroactive Distance (ShReD), characterizes the retroactive connectivity between any two reactions in a network arising from potential feedback interactions, thereby grouping togther network components which mutually influence each other. We evaluate the metric using two test networks: epidermal growth factor receptor (EGFR) signaling and liver drug transformation.    

Each test network was abstracted as a reaction-centric directed. An edge was drawn between two reaction nodes existed if one reaction produced either a reactant or allosteric effector of the other reaction. The ShReD between two reactions was calculated as the path-length of the shortest cycle that involves both reaction nodes. Using ShReD as the measure of connectivity, we adapted Newman's algorithm¹ to obtain a set of hierarchical partitions. Briefly, Newman's original algorithm divides an undirected network such that the resulting partition maximizes the number of connections within each sub-network relative to the number of expected connections between two randomly chosen nodes in the network. In this study, the partitions were performed based on the ShReD metric, rather than the undirected connectivity used in Newman's work.

For the EGFR network, the partitions obtained using Newman's original algorithm and ShReD were largely similar (not shown). Importantly, the ShReD partition generated reaction modules with larger numbers of cyclical interactions. For the metabolic network, which included cofactors, the ShReD partition again generated hierarchical modules whose compositions compared favorably with canonical associations based on textbook biochemistry (Fig. 1). The Newman partition was unable to generate any hierarchy. Interestingly, the ShReD partition revealed a 'redox' module involving reactions of xenobiotic transformation, glucose metabolism, pyruvate metabolism, and lipid metabolism interacting through shared production and consumption of NADPH.

Fig. 1. ShReD-based hierarchical partition of a hepatocyte metabolic network. The pie charts represent modules of reactions. Each pie represents the fraction of  

In conclusion, our novel metric ShReD, combined with Newman's algorithm, is to our knowledge the first modularity analysis technique that partitions a biochemical network to preserve cylical interactions between reactions.

1. Newman, ME (2006). Proc Natl Acad Sci U S A. 103(23): 8577-82.