In situations where the qualitative behavior of the regulatory system changes in response to shifts in the control input, the corresponding bifurcation diagram is of importance. In particular, the proper working of the regulatory system may depend on the locations, shapes and sizes of the various regions of qualitative behavior. For instance, in the cell cycle model [8] the correct spatial relationship (in parameter space) of the transitions points is crucial for ensuring the survival of cells. In fact, the model predicts that "dynamically challenged" mutants with shifted bifurcation diagrams may suffer irrecoverably. The resulting effects include difficulties exiting mitosis and decreases in cell mass after each cell division. For this particular model, the inverse bifurcation problem is to map geometric as well as topological relationships in the bifurcation diagrams to conditions on the parameters. Another application of inverse bifurcation in the cell cycle model is for the so-called "Pinocchio effect" involving check points. This effect occur when the necessary conditions for safe progression to the next cell phase are not satisfied hence the regulatory system should delay the state transition as far as possible. Here, the inverse bifurcation problem is to find out how the system may be constructed so that, when triggered by the presence or absence of certain chemicals, the range of bistability is maximized.