= Solution
In the simplified <Barkai-Leibler model>, let $M$ be the total methylated <receptor> <concentration> and $A=f(S)M$ the active, unoccupied part. Ligand-bound methylated <receptors> contribute zero activity. Because <CheB> acts only on active <receptors>,
$$
\dot M=r-\frac{bA}{K_B+A},\qquad r=V_R[R],\quad b=V_B[B].
$$
At a finite <steady state>,
$$
\boxed{A_* =\frac{K_Br}{b-r},\qquad M_* =\frac{A_*}{f(S)}.}
$$
The output $A_*$ is independent of attractant <concentration> $S$: attractant changes the necessary <receptor methylation> level, while the flux balance fixes activity. This proves <perfect adaptation> for $0<r<b$, $f(S)>0$ and enough methylatable <receptors>. At fixed $S$, $d\dot M/dM=-bf(S)K_B/[K_B+f(S)M]^2<0$, so this <steady state> is locally stable. If every <receptor> is permanently bound, $f=0$, or if the required $M_*$ exceeds the available pool, compensation fails.
This feedback couples demethylation directly to the output. Lower activity lowers demethylation; continuing <CheR> action restores it. The structural dependence on active <receptors> removes the separate adjustment of $V'_B$ required in <2/c>.<Robustness in simple biochemical networks by Naama Barkai and Stanislas Leibler>[Figure 1b and equation 1: saturated modifying flux and output-dependent reverse modification]{ref}
Changing <CheR> or <CheB> <concentration> changes $A_*$ and hence the baseline tumbling frequency, but the output still returns to that new baseline after an attractant step within the operating range. Raising <CheR> raises $A_*$; raising <CheB> lowers it. Adaptation time also changes: linearization gives $\tau=(K_B+A_*)^2/[bf(S)K_B]$. Thus robustness of the return property does not mean an invariant tumbling frequency or response time across different <enzyme> abundances.
If <CheR> instead has <Michaelis-Menten kinetics> on the unmethylated pool $U=E_T-M$, its flux is $r(M)=V_R[R](E_T-M)/(K_R+E_T-M)$. Then
$$
\frac{V_R[R](E_T-A/f)}{K_R+E_T-A/f}=\frac{bA}{K_B+A}.
$$
The left side depends on ligand through $f$, even at fixed output $A$; consequently the equilibrium output generally depends on attractant and <perfect adaptation> is lost. It is recovered approximately when $E_T-M\gg K_R$ throughout the response, so the <CheR> flux is effectively saturated.
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