A formal approach to the hierarchical structures of microbial communities with negative interactions.

Microbial communities typically consist of numerous species that coexist through intricate mutual dependencies. Understanding the structure of these communities and the interactions among their species is essential for explaining their functions and predicting their behavior. In this study, we follow the idea that a community organizes itself into a hierarchy of potentially persistent sub-communities. Previously, this hierarchy was described using Chemical Organization Theory (COT). However, that approach did not account for negative interactions. Here, we enhance the theory by incorporating negative interactions through an inhibitory resource called a toxin. For simplicity, we assume that a taxon sensitive to a toxin cannot coexist with a taxon that produces that toxin. Our results demonstrate that introducing a toxin reduces the number of organizations, with the extent of this reduction depending on various modeling parameters. Further, we show that the usage of essential resources leads to a computationally NP-hard transformation problem into direct taxa interactions. Additionally, we demonstrate that the number of measurements required to infer all persistent subspaces increases. We determine which groups of species are mutually excluded due to toxin interactions. Besides toxic interactions, it is also possible to infer cross-feeding aspects of the microbial community, for which a potential algorithm is outlined and illustrated by an example.