Figure 4

ξ ( g ) as obtained from inverting Eq. 9 for J = 2. At a fixed ratio of kinase to phosphatase numbers, exp λ, the abscissa measures the total number of substrate molecules N in terms of g. The ordinate, ξ = log x, expresses (at fixed λ and J) the same information as the order parameter ⟨n _J - n₀⟩ /N by virtue of Eq. 8. The colors indentify different values of λ, indicated at the top on the right hand side. The negative sign means that there is more phosphatase than kinase in the system, and the only stable distribution below the critical point has ⟨n _J ⟩ < ⟨n₀⟩. i.e. x < 1. The blue curve for λ = 0 depicts the second-order phase transition (also rendered in Fig. 5, where it is compared against simulations for different J). At g _c = 3 the x = 1 solution becomes unstable (dotted continuation), branching off continuously into two stable stationary states. Parameter asymmetry (λ ≠ 0) leads to a discontinous change of the order parameter (first-order phase transition), which becomes apparent when the system is prepared in the upper branch and the control parameter g is decreased below critical. At that point, the system jumps to the stationary state on the lower curve of the corresponding color. Again, dotted segments indicate the unstable solution.