Emergent metabolic cooperation in communities of microbes that maximise growth rate by optimal protein expression

Many microbial communities involve metabolic exchanges between their members. We present a mathematical modelling formalism for studying the adaptation of metabolic strategies by microbes to external metabolites produced by themselves and their community partners. Each microbial species is assumed to maximise its growth rate by optimal allocation of finite biosynthetic resources for protein synthesis in response to their current environment. Each species therefore acts as a selfish optimiser. Because the environment is shaped by all species, their strategies are interdependent. The optimal state of the community is the set of metabolic strategies of its members that outcompetes (outgrows) all other possible sets of metabolic strategies---including the ones in which some species may not be present at all. Our mathematical formalism compacts the coupled optimisations by the single species into one optimisation problem that also infers the optimal community composition. We show with simple examples how it can be used to study the emergence of positive species-species interactions such as commensalism, mutualism, syntrophy, and division of labour. The formalism is a direct generalisation of the well-known resource allocation models for single species, but includes reaction kinetics; we show that observed positive ecological interactions emerge from ecological feedback, in which producers are inhibited by the byproducts they excrete. This paper illustrates that (metabolism-dominated) microbial communities may be viewed as one coupled system in which the optimal pathway choice of each member depends on that of all others. It is directly generalisable to multiple species, and should pave the way for studies of larger communities.