Optimizing Cellular Metabolism through Quadratic Programming

The set of metabolite concentrations ([met]) and reaction free energies (ΔG) associated with central carbon metabolism (CCM; see Supplementary Fig. 4) that best matched i) the directly observed concentrations for measured metabolites ([met]_exp) and ii) the observed cellular ΔG for measured reactions (ΔG_exp) were computed. An important consideration in this calculation is that literature values for ΔG°′ may themselves contain error. In addition, these ΔG°′ values are interrelated, such that ΔG°′ for any sequence of metabolic reactions must be given by the sum of the formation energies of the products minus the formation energies of the reactants. Accordingly, we set out to optimize both metabolite concentrations and formation energies so as to maximize consistency with prior estimates of Δ_fG°′ based on the component contribution method (which itself incorporates prior literature data on ΔG°′) and our experimental observations of metabolite concentrations and cellular reaction free energies. To this end, a quadratic programming problem was formulated with independent variables ln[met] and Δ_fG°′, with the optimization objective of minimizing the departure from the expected Δ_fG°′ and measured ln[met] and ΔG:
$$\underset{ln[\mathit{met}],{\mathrm{\Delta }}_f{G^{\circ }}^{\prime }\in \mathit{CCM}}{min}\frac{1}{N_{\mathit{exp}}^{\mathit{met}}}\sum {\left(\frac{ln{[\mathit{met}]}_{\mathit{exp}}-ln[\mathit{met}]}{s_{\mathit{met}}}\right)}^2+\frac{1}{N_{\mathit{exp}}^{\mathit{for}}}\sum {\left(\frac{{\mathrm{\Delta }}_f{G}_{\mathit{exp}}^{{\circ }^{\prime }}-{\mathrm{\Delta }}_f{G^{\circ }}^{\prime }}{s_{{\mathrm{\Delta }}_{f}G}}\right)}^2+\frac{1}{N_{\mathit{exp}}^{\mathit{rxn}}}\sum {\left(\frac{\mathrm{\Delta }{G}_{\mathit{exp}}-\mathrm{\Delta }G}{s_{\mathrm{\Delta }G}}\right)}^2,$$
S is the stoichiometric matrix with rows and columns representing individual metabolites and reactions, respectively. ΔG, Δ_fG°′, and ln[met] are vectors of free energy of reaction, standard free energy of formation, and log-concentrations. s refer to the standard errors of the measurements or component contribution estimates.
$N_{\mathit{exp}}^{\mathit{for}},N_{\mathit{exp}}^{\mathit{met}}$ , and
$N_{\mathit{exp}}^{\mathit{rxn}}$ are the number of input metabolite formation energies, experimentally measured metabolite concentrations and ΔG.
For reactions whose ΔG were not precisely determined, ΔG was constrained to be negative in the direction of net flux. For the eukaryotic cells, Δ_fG°′ of TCA metabolites depended on whether they are in cytosol or mitochondria due to pH difference across compartments, and the values of Δ_fG°′ calculated for mitochondria were used. Inorganic phosphate concentrations were input as follows: 20 mM in E. coli^{58 (link)}; 50 mM in yeast^{5 (link)}; and 5 mM in mammalian iBMK cells^{59 (link)}. Mitochondrial coenzyme A concentration was input as 5 mM^{60 (link)}. The inorganic phosphate and coenzyme A concentrations were allowed to vary within 20% of these values.