Regulated bacterial interaction networks: A mathematical framework to describe competitive growth under inclusion of metabolite cross-feeding.

When bacterial species with the same resource preferences share the same growth environment, it is commonly believed that direct competition will arise. A large variety of competition and more general 'interaction' models have been formulated, but what is currently lacking are models that link monoculture growth kinetics and community growth under inclusion of emerging biological interactions, such as metabolite cross-feeding. In order to understand and mathematically describe the nature of potential cross-feeding interactions, we design experiments where two bacterial species Pseudomonas putida and Pseudomonas veronii grow in liquid medium either in mono- or as co-culture in a resource-limited environment. We measure population growth under single substrate competition or with double species-specific substrates (substrate 'indifference'), and starting from varying cell ratios of either species. Using experimental data as input, we first consider a mean-field model of resource-based competition, which captures well the empirically observed growth rates for monocultures, but fails to correctly predict growth rates in co-culture mixtures, in particular for skewed starting species ratios. Based on this, we extend the model by cross-feeding interactions where the consumption of substrate by one consumer produces metabolites that in turn are resources for the other consumer, thus leading to positive feedback in the species system. Two different cross-feeding options were considered, which either lead to constant metabolite cross-feeding, or to a regulated form, where metabolite utilization is activated with rates according to either a threshold or a Hill function, dependent on metabolite concentration. Both mathematical proof and experimental data indicate regulated cross-feeding to be the preferred model to constant metabolite utilization, with best co-culture growth predictions in case of high Hill coefficients, close to binary (on/off) activation states. This suggests that species use the appearing metabolite concentrations only when they are becoming high enough; possibly as a consequence of their lower energetic content than the primary substrate. Metabolite sharing was particularly relevant at unbalanced starting cell ratios, causing the minority partner to proliferate more than expected from the competitive substrate because of metabolite release from the majority partner. This effect thus likely quells immediate substrate competition and may be important in natural communities with typical very skewed relative taxa abundances and slower-growing taxa. In conclusion, the regulated bacterial interaction network correctly describes species substrate growth reactions in mixtures with few kinetic parameters that can be obtained from monoculture growth experiments.