= Solution
Separate total methylated <receptor> $m$ from its active portion $a=f(c)m$, where ligand-bound <receptor> is inactive and the unbound fraction supplies $f(c)>0$ at finite ligand <concentration>. In the <Barkai-Leibler model>, <receptor demethylation> depends on the \b[active] portion:
$$
\dot m=r-\frac{ba}{K_B+a}.
$$
Every accessible stable <steady state> therefore obeys
$$
\boxed{a_* =\frac{K_Br}{b-r},\qquad m_* =a_*/f(c).}
$$
The output is independent of <chemoattractant> even though the methylated pool changes. A ligand step first lowers $a$, reducing its own reverse modification. Constant <receptor methylation> continues until $a$ recovers. Local stability follows from $\partial\dot m/\partial m=-bfK_B/(K_B+a)^2<0$. This is output-dependent <negative feedback>. <Robustness in simple biochemical networks by Naama Barkai and Stanislas Leibler>[1997, model mechanism and adaptation precision]{ref}
Changing <CheR> or <CheB> changes $a_*$ and the recovery time, but the return to each new pre-step baseline remains exact over the model's operating range. <Robust perfect adaptation> does \b[not] mean invariant tumbling frequency under changes in <enzyme> abundance. Nor does it extend to $f=0$, $b\leq r$ or saturation of the finite <receptor methylation> capacity. If every <receptor> is permanently inactive, $a=0$ and the reverse flux cannot restore a positive baseline. The literal phrase that binding reduces activity to zero applies to bound <receptors>, not to every <receptor> under every finite stimulus.
If <CheR> instead follows <Michaelis-Menten kinetics> on unmethylated <substrate>, $u=E_T-m$, then
$$
\dot m=\frac{r_0(E_T-m)}{K_R+E_T-m}-\frac{ba}{K_B+a},\qquad a=f(c)m.
$$
At <steady state> the forward flux becomes $r_0(E_T-a/f)/(K_R+E_T-a/f)$ and generally depends on ligand through $f$. Thus ligand-independent output is lost. Near-perfect adaptation can remain when $E_T-m\gg K_R$ so that <CheR> is effectively saturated, but Michaelis–Menten behavior outside that limit does not guarantee <perfect adaptation>. <Robust perfect adaptation in bacterial chemotaxis through integral feedback control by Tau-Mu Yi and others>[2000, analysis of ligand-independent bound CheR and the saturation assumption]{ref}
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2016-chemotaxis-adaptation-models.png]
{title=Fine tuning and output-dependent adaptation}
{description=Original numerical solutions of the stated mathematical models. The ligand activity factor changes from one to one half; saturated output-dependent feedback restores activity, while a total-pool reverse reaction requires fine tuning.}
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