Synergy and Complementarity: The Generative Basis of Chemical Organizations

Abstract

Identifying structural features ensuring the persistence of a reaction network is fundamental for understanding the evolution and complexification of biological entities. By defining organizations as subsets of species that are closed and self-maintaining, Chemical Organization Theory (COT) shows that organizations enable filtering out the regions of the phase space where attractors can exist. Despite various theoretical explorations current methods to compute the organizations face unclear combinatorial challenges. In this article, we first perform a systematic study of such combinatorial challenges and identify a criteria of productive novelty and irreducibility that separate relevant from redundant combinations. Second, we identify the minimal building blocks that shall be combined to build organizations, called elementary reaction closures (ERC), and characterize them as a hierarchy. Third, we show that all persistent modules can be generated as combination of ERCs only, and operationalize two properties among ERCs, synergy and complementarity, that define a sufficient criteria to build such generators in a minimal way. Fourth we show that every relevant persistent module can be built from such minimal sequences of ERCs. We next show that the synergies and complementarities we use to generate persistent modules not only scale radically slower than usual combinatorial methods. We quantify these structures across $438$ biological reaction networks from the BioModels and BiGG databases. The proportion of fundamental synergies and complementarities among all ERC pairs shrinks as networks grow even as raw counts increase, confirming that larger biological networks achieve persistence through a progressively smaller and more selective sets. We do not prescribe algorithms to compute organizations, discuss how algorithms based on our results could permit applying COT.