= Solution
Write $m=[E_m]$, $r=V_R[R]$ and $b=V_B[B]$. Saturated <receptor methylation> provides a constant positive flux, so the complete balance is
$$
\dot m=r-\frac{bm}{K_B+m}.
$$
At a finite <steady state>, $r(K_B+m_*)=bm_*$ and therefore
$$
\boxed{m_* =\frac{K_Br}{b-r},\qquad b>r.}
$$
The derivative of the right side is $-bK_B/(K_B+m)^2<0$, so this fixed point is stable in the scalar model. With total <receptor> <concentration> $E_T$, it must additionally lie between zero and $E_T$ and leave sufficient <substrate> for the assumed saturated <receptor methylation>. If $r\geq b$, the formal equation has no finite positive fixed point: finite <receptor> capacity then matters.
For an <chemoattractant> step, activity and hence tumbling frequency fall rapidly before <receptor methylation> can change appreciably. Slower chemical modification can then restore <receptor> signaling and make the tumbling frequency rise toward a new steady value. \b[<Perfect adaptation>] means return of that output to its original baseline despite the persistent <chemoattractant> step; it does not mean that <receptor> <receptor methylation> also returns to its original level.
The balance written here alone does not specify how ligand changes activity or <receptor demethylation>, and does not yet establish <perfect adaptation>. If activity is simply multiplied by a ligand-dependent fraction while this $m$ balance remains unchanged, $m_*$ is unchanged and the activity remains reduced. The following two models add different mechanisms that can compensate for that direct activity reduction.
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