The dynamical behavior of a class of biochemical control circuits that regulate enzyme or protein synthesis by end-product feedback is analyzed. Both inducible and repressible systems are studied and it is proven that in the former unique steady states are globally asymptotically stable. This precludes periodic solutions in these systems. A similar result holds for repressible systems under certain constraints on kinetic parameters and binding contants. However, when the reaction sequence is sufficiently long, or when a large enough number of effector molecules bind to each represser molecule, repressible systems can show zero-amplitude ("soft") bifurcations: these are predicted by Hopf's bifurcation theorem.