Illustrative example for Corollary 2. This example shows a small network with 3 internal reactions (x2−4). The flux directions were chosen according to the direction of the arrows. The matrix A is the internal stoichiometric matrix. (a) A flux distribution is shown where all 3 internal reactions are active and form a loop. Therefore there is a solution (x2 = x3 = x4 = 1) for the mass balance equation Ax = 0. In this case, no solution exists for A⊤y > 0. Therefore this flux distribution will be eliminated by loopless-COBRA. (b) A loopless flux distribution, in this case x4is not active. There is no solution for Ax = 0(except for the trivial solution x = 0). Gordan’s theorem claims that there must be a solution for A⊤y > 0, e.g. the one shown in the figure. Thus, loopless-COBRA will not eliminate any such flux distributions.