Fig. 3

a Average growth rate (relative to maximal achievable rate) as a function of βλ_max. Inferred value for data in Fig. 2a is shown as a blue point (thick black line = STD for \(\bar \lambda /\lambda _{{\mathrm{max}}}\) and the corresponding range for βλ_max). b Log fold change of average fluxes as a function of βλ_max, relative to the uniform distribution; fluxes are sorted by their change in the FBA (β → ∞) limit. c Dependence of three selected fluxes (legend, highlighted with correspondingly colored arrows in b) on the average growth rate, \(\bar \lambda\): ICL is turned off in the FBA limit (but not in the maximum entropy solution), ICDH remains nearly unchanged with \(\bar \lambda\), and GLUDy is the only flux that switches sign. d Flux variabilities (each black line = SD of one flux according to maximum entropy distribution, three selected fluxes from c highlighted in color) scale linearly with distance to maximal growth rate and vanish in the FBA limit. e, f Correlation coefficient matrix (Pearson correlation, color scale) between 23 selected fluxes in the uniform (e) and FBA (f) limits, computed within maximum entropy framework. Fluxes have been grouped into four clusters according to the correlation in the FBA limit and reordered accordingly in both plots; clusters are strongly enriched for fluxes belonging to pathways denoted at bottom. Note the flip in correlation sign between the glycolysis and glyoxylate shunt pathways between the two limits. g Achieving a particular growth rate (y-axis) requires reducing the entropy of the joint distribution of fluxes at least by I bits below the entropy of the uniform distribution (green region). Points in the hashed (forbidden) region are not achievable. h A simple model in which tight regulation of fluxes (higher I) enables higher growth rates, as in g, but also entails metabolic cost (see text). For a given cost α, the effective growth rate \(\bar \lambda _{{\mathrm{eff}}}\) is maximized at an intermediate value of βλ_max