Perfect adaptation ($A_{\ast }(S)=A_0$)

A signalling output returns to its original steady state after a persistent change of input. This concerns the final output, not absence of an initial response or invariance of the internal state. Robust perfect adaptation additionally requires that the return property persist over a range of parameters.

For $\dot{m}=r-bm/(K+m)$ with activity $a=fm$, preserving the original output after f changes from one requires $b^{\prime }=fb+(1-f)r$. This relation depends on modification capacities and is generally disrupted by changing enzyme abundance. It differs from output-dependent reverse modification in the Barkai-Leibler model.