Regulated bacterial interaction networks: A mathematical framework to describe competitive growth under inclusion of metabolite cross-feeding
Authors: Guex Isaline , Mazza Christian , Dubey Manupriyam , Batsch Maxime , Li Renyi , van der Meer Jan Roelof 
Published: February 2023

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 mono-culture 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 speciesPseudomonas putidaandPseudomonas veroniigrow 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 mono-cultures, 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 loops 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 over 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 mono-culture growth experiments.

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Author summary

Correctly predicting growth of communities of diverse bacterial taxa remains a challenge, because of the very different growth properties of individual members and their myriads of interactions that can influence growth. Here we tried to improve and empirically validate mathematical models that combine theory of bacterial growth kinetics (i.e., Monod models) with mathematical definition of interaction parameters. We focused in particular on common cases of shared primary substrates (i.e., competition) and independent substrates (i.e., indifference) in an experimental system consisting of one fast-growing and one slower growing Pseudomonas species. Growth kinetic parameters derived from mono-culture experiments included in a Monod-type consumer-resource model explained some 75% of biomass formation of either species in co-culture, but underestimated the observed growth improvement when either of the species started as a minority compared to the other. This suggested an in important role of cross-feeding, whereby released metabolites from one of the partners is utilized by the other. Inclusion of cross-feeding feedback in the two-species Monod growth model largely explained empirical data at all species-starting ratios, in particular when cross-feeding is activated in almost binary manner as a function of metabolite concentration. Our results also indicate the importance of cross-feeding for minority taxa, which can explain their survival despite being poorly competitive.