Signal-response curve (bifurcation diagram) of system (2) for the parameters k1 = 8, k2 = 1, k3 = 1, k4 = 1.5. Solid lines indicate locally stable steady states, the dashed line locally unstable steady states. The inset shows the signal-response curve if an additional small constant influx into X (here 0.6) is assumed (enabling a positive 'off' state, leaving the 'on' state and bifurcation point nearly unchanged). This is the classical toggle switch (terminology of Tyson et al. (6), others use the term toggle switch to describe a double negative (i.e. positive) feedback loop (4)) picture enabling the hysteresis cycle: starting with low values and increasing the signal continuously increases the response, until the saddle-node bifurcation at about S = 1.7 is reached. Further increase of the signal leads to a sudden jump of the response to the upper steady state. Decreasing the signal now leads to a continuous decrease of the response, the systems stays in the upper steady state until the left bifurcation point is reached where the response jumps back to the lower steady state.
Wilhelm BMC Systems Biology 2009 3:90 doi:10.1186/1752-0509-3-90