Fig. 2
A toy model illustrating invFBA. To illustrate the use of the invFBA algorithm, we applied it to a simple metabolic network with a single metabolite A and three reactions, described by the stoichiometric matrix S = [1, 2, -1]. This corresponds essentially to two reactions (with fluxes x1 and x2) producing A, and one reaction (with flux x3) consuming it. We additionally impose that all fluxes are non-negative and that x3 ≤ 3. Thus, the feasible space is represented by the polyhedron {x | Sx = 0, x ≥ 0, x3 ≤ 3}, corresponding to the triangle in the (x1, x2) plane shown in the figure. Given a specific metabolic flux vector (yellow dot), we use invFBA to identify an objective function that would give such a point as an FBA optimum. In this case, invFBA yields c = (1/3, 2/3) as the objective. This corresponds to the vector perpendicular to the optimal facet closest to the given flux point. Note that the alternative possibility of seeking the extreme point of the FBA polytope closest to the yellow dot (as done in [4, 5]), would yield the faraway extreme point (x1, x2) = (0, 0) and an objective function within the blue cone C that renders this point