A sample network containing a hybrid metabolite. The cellular objective of the sample network (A) was to maximize Z and the system could be formulated as a linear programming problem (B). We attenuate the flux-sum of metabolite M1, which is also equal to x1, and examine the effects on the objective Z. When x1 was attenuated by decreasing the value of C, the maximum value of Z decreased as the objective function (red line) traversed the edges of the solution space (shaded region) in the direction shown by the blue arrow. As the objective function passes x1 = 1, the "rate" of decrease of Z changes due to the difference in gradients of the edges.