Reconstructing heterogeneous metabolic trajectories of E. coli diauxie via a dynamical Maximum Entropy Principle

The glucose-acetate diauxic shift in E. coli is classically described as an abrupt, population-wide switch from glucose to acetate consumption. Recent experiments challenge this view, revealing a robust intermediate regime of co-consumption whose single-cell basis remains unresolved: does it reflect coexisting specialized subpopulations, or genuine mixed metabolic states within individual cells? We first develop a two-state consumer-resource model in which cells optimally grow on either glucose or acetate, and show that observed co-consumption trajectories cannot be decomposed into convex combinations of the two subpopulations -- ruling out discrete metabolic states as a sufficient explanation. Physico-chemical constraints of the metabolic network instead enforce genuine single-cell co-consumption across a continuous spectrum of phenotypes. To resolve this, we apply a dynamical maximum entropy (maximum caliber) framework constrained by batch and chemostat experiments, inferring time-resolved distributions of metabolic fluxes that naturally predict a continuum of single-cell phenotypes spanning glucose overflow, mixed substrate utilization, and acetate consumption -- revealing co-consumption as a dominant, persistent feature of single-cell metabolism around the switch rather than an artifact of population averaging. Finally, we formulate a continuous consumer-resource model over metabolic state space, in which selection, phenotypic diffusion, and moving metabolic boundaries driven by environmental feedback reproduce single-cell co-consumption trajectories and complex dynamical trends inaccessible to discrete models. Together, our results recast diauxic adaptation as a continuous redistribution of single-cell metabolic states rather than a discrete switch.