Fig. 6: Model of acid-induced cross-feeding between 1A01 and 3B05 and the resulting population dynamics in 24-h growth-dilution cycles.

a We describe the cross-feeding dynamics by a consumer-resource model outside of steady-state growth. The model involves the densities of 1A01 and 3B05 cells and the concentrations of GlcNAc (G), acetate (E), and acetate-induced metabolites (M, to be interpreted as the sum of pyruvate and lactate concentrations in the medium). We also introduce an additional variable \({\sigma }_{A}\) to describe the internal state of 1A01 due to the depletion of other metabolites such as aspartate and glutamate (Supp. Figs. 8, 10). The key feature of our model is that the growth/death rate of the two species and the rates of uptake/excretion of the metabolites G, E, and M by the two species are dependent on the degree of acetate stress (E) and 1A01’s internal state (\({\sigma }_{A}\)). As illustrated in the schematic, we approximate these dependences by switching between several distinct forms of the rate functions depending on the values of \({\sigma }_{A}\) and E. The rate functions corresponding to each of the four regimes for 1A01 (\({A}_{{{\mbox{I}}}}^{+},\,{A}_{{{{{{\rm{I}}}}}}}^{-},\,{A}_{{{\mbox{II}}}},\,{A}_{{{\mbox{III}}}}\)) depend on both \({\sigma }_{A}\) and E, while the three regimes for 3B05 (\({B}_{{{\mbox{I}}}},\,{B}_{{{\mbox{II}}}},\,{B}_{{{\mbox{III}}}}\)) depend on E. These regimes are separated by the vertical red lines and horizontal dashed magenta line; see Supplementary Note 3 for a detailed description. In the schematic, black arrows with solid and dashed lines indicate effective and ineffective interactions in each regime. Red arrows indicate the change in the acetate concentration E, and the magenta arrows indicate the change of \({\sigma }_{A}\). Thick purple arrows indicate the crucial cross-feeding of acetate-induced metabolites allowing 3B05 to grow during acetate stress (and hence reduce the acetate concentration in the medium). b and c show numerical results of the model in (a) for the density of live 1A01 and 3B05 cells (top) and the concentration of GlcNAc, acetate, and acetate-induced metabolites (bottom) for the 24-h growth-dilution cycles with 1:1 initial species ratio. b Numerical results of the model during the stable cycle, which leads to two coordinated paths, \({A}_{{{\mbox{I}}}}^{+}\to \,{A}_{{{{{{\rm{I}}}}}}}^{-}\to {A}_{{{\mbox{II}}}}\to {A}_{{{\mbox{I}}}}^{+}\) for 1A01 (solid line) and \({B}_{{{\mbox{I}}}}\to \,{B}_{{{\mbox{II}}}}\to \,{B}_{{{\mbox{I}}}}\) for 3B05 (dashed line) over time. c Numerical results of the model at the end of each 24-h cycle.