Networks that enable response ( R ) to perfectly adapt to signal ( S ). (A) Incoherent feedforward loop: on one hand, S directly phsophorylates and hence activates R; on the other hand, S inactivates R proportionally via stimulating the expression of proportioner (P) that dephosphorylates R. The activation and inactivation effects cancel out for R in such a loop. As a result, R always resumes to the original state upon adaptation. In the mathematical model, R refers to the phosphorylated form of R (i.e. Ra in the diagram). The first equation dictates that the steady state expression level of P is proportional to S (this is why it is called “proportioner”). Plug this relationship into the second equation and take the steady state: S in the first term will cancel with P in the second term. This makes the steady state of R independent from S. (B) Incoherent feedforward loop. This slightly more complex example also helps demonstrate perfect adaptation. In this case, S phosphorylates and thereby activates both R and P. If S operates in the saturated regime (i.e. K
< <(1-P)) and the dephosphorylation of P operates in the unsaturated regime (i.e. K
> > P), then the first equation reduces to the same form as that in (A). By the same reasoning, Ra in this incoherent feedforward loop perfects adapts. (C) Negative feedback loop involving time-scale separation: S phosphorylates and activates R that subsequently simulates the expression of its own inhibitor E. In the model, R refers to the phosphorylated form of R (i.e. Ra in the diagram). When E disappears at a constant rate v
, the second equation ensures that the steady state R is independent of S, and is only determined by k
. In this case, E integrates over R, and this negative feedback loop constitutes an integral controller. As a special case, when the half life of E is much longer than the time scale of other reactions, v
is approximated to 0. Consequently, R always resumes 0 upon adaptation. (D) Negative feedback loop involving saturated enzyme kinetics: in this slightly more complex scenario, element (E) is the phosphatase of R and is activated by R via phosphorylation. If Ra and E’s phosphatases both operate in their saturated regimes, then the second equation reduces to the same form as that in (C). By the same reasoning, such a negative feedback loop is capable of perfect adaptation.