Elementary vectors reveal minimal interactions in microbial communities

Understanding microbial communities is essential for progress in ecology, biotechnology, and human health. Recently, constraint-based metabolic models of individual organisms have been combined to study microbial consortia. In this work, we present a geometric framework for characterizing all feasible microbial interactions. We project community models onto the key variables of interaction: exchange fluxes and community compositions. Based on this projection, we compute elementary composition/exchange fluxes (ECXs), extending the concept of minimal metabolic pathways from individual species to entire communities. Every feasible metabolic state of a community can be expressed as a combination of these elementary vectors. Notably, each ECX corresponds to a distinct ecological interaction type, such as specialization, commensalism, or mutualism. Finally, our geometric formulation enables the direct application of existing constraint-based methods, such as flux variability analysis and minimal cut sets, to microbial communities, providing a foundation for rational community design.