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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.

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 active portion:Every accessible stable steady state therefore obeysThe 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 $\partial \dot{m}/\partial m=-bf{K}_B/(K_B+a{)}^2<0$. This is output-dependent negative feedback. ^[ref]

Changing CheR or CheB changes $a_{\ast }$ 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 not mean invariant tumbling frequency under changes in enzyme abundance. Nor does it extend to $f=0$, $b\le r$ 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, $u=E_T-m$, thenAt 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. ^[ref]

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,At a finite steady state,The output $A_{\ast }$ 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 $M_{\ast }$ 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 $V_B^{\prime }$ required in c.^[ref]

Changing CheR or CheB concentration changes $A_{\ast }$ 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 $A_{\ast }$; raising CheB lowers it. Adaptation time also changes: linearization gives $\tau =(K_B+A_{\ast }{)}^2/[bf(S)K_B]$. 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 $r(M)=V_R[R](E_T-M)/(K_R+E_T-M)$. ThenThe 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.