Phase diagrams. (a) Symmetric feedback circuit as in Fig. 1d. (b) Asymmetric feedback circuit (from the fully phosphorylated J-state) as in Fig. 1c. The coupling parameter g = N/, the exogenous catalytic strength P/I, and the number of phosphoacceptor sites J are being varied. Symbols are simulation results (N = 1000) and solid lines are analytical results from the mean-field theory for J = 2, 3, 5 (solid, dash, dot). The discrepancies arise from fluctuations, which the mean-field theory ignores. In the circuit with symmetric feedback, there is one critical coupling strength g
for each ratio of phosphatase to kinase numbers. (The phase diagram for symmetric feedback is itself symmetric about 0 in log (I/P). Only half of the diagram is shown in panel (a); the symmetric half is obtained by reflection about the vertical axis.) As g is increased beyond g
, the system exhibits two stable distributions of phosphorylation states peaked at the end states of the chain. Below the critical number, the target molecules are mostly unphosphorylated (P/I > 1) and the system remains in this state as it becomes bistable. Within the bistable regime, the system could be prepared in the mostly phosphorylated state, in which it persists as the number of target molecules is increased. Yet, as the number of target molecules is decreased, the mostly phosphorylated state loses stability abruptly and the system shifts to the mostly unphosphorylated state. This is a first-order phase transition in statistical mechanics. In contrast, for the symmetric feedback circuit and P/I = 1 (the leftmost point on the abscissa in panel a), the transition is second-order, that is, continuous in the phosphorylation state. This is shown in Fig. 5 below. Panel (a) reveals that, once bistable, the symmetric circuit never loses bistability again as N is further increased. In the language of statistical mechanics, only a lower critical coupling exists. Notice that for a fixed coupling g and increasing P/I the symmetric system loses bistability again, as illustrated in Fig. 2. Unlike the symmetric case, the asymmetric circuit exhibits a window of bistability (lower and upper critical couplings) in N. For a suitable P/I, an increase in the number of target molecules N drives the system through a second threshold at which bistability disappears. If the system was mostly unphosphorylated, it now undergoes a (first-order) phase transition to the mostly phosphorylated state.