Figure 2

Ultrasensitive and bistable behavior in the symmetric feedback circuit with J = 2. The ordinate is the order-parameter ⟨n _J - n₀⟩/N measuring the macroscopic variable of interest. The abscissa measures the relative difference between kinase and phosphatase numbers. In this representation, all kinase to phosphatase proportions are compressed symmetrically between -1 and 1. As a result, the hyperbolic behavior of a single phosphorylation/dephosphorylation loop (J = 1) without feedback is a straight line, the dotted diagonal. All other curves are for J = 2, which is the minimal ultrasensitive case capable of bistability. The blue curve shows the ultrasensitivity of the chain without feedback. As J → ∞, the slope of the sigmoidal approaches 2 at the midpoint I = P (not shown). The blue curve already covers nearly 50% of the ultrasensitivity attainable at J = ∞. Ultrasensitivity is enhanced by feedback (green curve), when approaching the threshold for bistability from below. The strength of feedback is reported in terms of g = N/$\sqrt{IP}$, where N is the number of substrate molecules (see text for details). In the case of the green curve g = 2.85. The slope at the midpoint is always 8 at the critical point for the onset of bistability in the presence of feedback. Above the threshold (red curve; g = 5) there is an intermediate range of (I - P)/(I + P) with two stable solutions. The dots connecting the two branches indicate a sudden change in stationary phosphorylation state as one of the branches ceases to exist when (I - P)/(I + P) passes outside the bistable range. This creates hysteresis.