Steady state ratios in bipartite population models emerge through obligate cross-feeding

Cross-feeding, the phenomenon where one organism consumes metabolites secreted by another, is a ubiquitious phenomenon in microbial communities. Obligate, mutually cross-feeding organisms are a special case of mutualism, where each organism produces a resource that the other organism requires for growth. Many obligate, mutually cross-feeding microbial systems have been studied, including naturally occurring as well as synthetically designed communities. Interestingly, in all these experimental systems it has been observed that the communities approach a fixed biomass ratio, independent of the initial biomass inoculation composition. The repeated reports of stable biomass ratios suggest that such ratios are actually a generic property of mutually obligate cross-feeding systems and that, if the system is stable, such a ratio must be approached. As a consequence, it should also be possible to explain and predict this ratio from measurable parameters. Here, we explore mutually obligate cross-feeding systems with mathematical models and show that a stable biomass ratio is a general feature of such systems. Moreover, we show how this ratio can be predicted from key parameters that can be interpreted as the value of the resource to the consuming organism and the cost of producing a resource to the producing organism.