The novel optimization framework implemented by BOSS. Panel (a) illustrates the bi-level optimization problem that forms the basis for BOSS. This problem involves minimizing the sum-squared error between experimentally-measured (in vivo) and framework-computed (in silico) fluxes (line 1) subject to the fundamental flux balance analysis (FBA) problem (lines 2–5), i.e., the putative objective reaction is maximized (line 2) subject to physico-chemical (line 3) and other constraints (lines 4 and 5). In this framework: (1) N corresponds to the set of metabolites; (2) M to the set of reactions; (3) P to the set of putative objective reactions, usually a new column inserted into the stoichiometric matrix S, Si, j, with flux v
where j ∈ P; (4) vframework to the set of framework-computed fluxes; and (5) vexperimental to the set of experimentally-measured fluxes. Additionally, to normalize the flux data, the "input flux" corresponding to the uptake of the carbon source (e.g., glucose) is set to a predetermined value called "uptake." Panel (b) illustrates the optimization problem in panel (a) reformulated as a single-level optimization problem via the duality theorem of linear programming (LP) [3, 40]. This novel framework for predicting objectives of biological systems is comprised of an objective that aims to minimize the sum-squared error between experimentally-measured and framework-computed fluxes (line 1) subject to a set of primal (lines 3 through 5) and dual constraints (lines 7 through 9), as well as two new constraints, one that sets the value of the primal and dual problems equivalent to one another (line 2) and another that normalizes the flux distribution by setting the flux corresponding to the new objective reaction to a specific value (line 6). The notations in panel (a) apply here as well. The decision variables in this optimization are: (1) the stoichiometric coefficients of the objective reaction, Si, jwhere i ∈ N and j ∈ P; (2) the framework-computed fluxes vexperimental; (3) the dual variable g associated with the uptake constraint; and (4) the dual variables u indicating shadow prices on the mass balance constraints for each metabolite in the system.