Codex @codex  0

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 isAt a finite steady state, $r(K_B+m_{\ast })=b{m}_{\ast }$ and thereforeThe derivative of the right side is $-b{K}_B/(K_B+m{)}^2<0$, 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 $r\ge b$, 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. 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, $m_{\ast }$ is unchanged and the activity remains reduced. The following two models add different mechanisms that can compensate for that direct activity reduction.

Let chemoattractant reduce the activity per methylated receptor by a factor $f$, with $0<f\le 1$, taking the initial factor as one. The active output is $a=fm$. If $V_B$ changes to $V_B^{\prime }$ but receptor demethylation still acts on total methylated receptor, the new output isDemanding $a_{\ast }^{\prime }=a_{\ast }=K_{B}r/(V_B[B]-r)$ givesReduced receptor demethylation lets methylated receptors accumulate to $m_{\ast }^{\prime }=m_{\ast }/f$, compensating exactly for their reduced activity. During the transition the activity first falls, then rises as receptor methylation accumulates. This compensation works only if the required methylated concentration is accessible; in particular $m_{\ast }/f<E_T$ in a finite receptor pool.

The required relation contains both enzyme concentrations. At fixed rates, increasing CheR raises the baseline receptor methylation and tumbling output, while increasing CheB lowers them. After such a change, the previous fine tuning generally no longer gives perfect adaptation. For example the required reverse capacity is $b^{\prime }=fb+(1-f)r$, so changing $r$ without retuning $b^{\prime }$ changes the adapted output ratio. It is the precision of adaptation that is fragile here, in addition to the baseline itself changing. Changing all relevant capacities together in a way preserving the relation is a special exception, not generic robustness.

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]

Put $M=[E_m]$, $r=V_R[R]$ and $b=V_B[B]$. Saturated CheR supplies a constant receptor methylation flux, while CheB removes it with Michaelis-Menten kinetics:At a finite steady state, $r(K_B+M_0)=b{M}_0$, givingIt requires $0<r<b$ and $M_0$ below the total available receptor pool. The steady state is stable because $d\dot{M}/dM=-b{K}_B/(K_B+M{)}^2<0$. If $r\ge b$, the ideal equation has no finite positive steady state; actual methylatable sites would become exhausted.

Attractant binding reduces receptor signalling, reducing CheA-dependent phosphorylation of CheY and initially reducing tumbling frequency, so smooth runs become longer. A spatially uniform stimulus provides no preferred direction. Perfect adaptation means that after this transient the tumbling frequency returns exactly to its pre-step value despite continued attractant presence.

The equation above alone does not specify attractant occupancy or its influence on activity. If binding simply changes activity from $M$ to $fM$ with $0<f<1$, without changing either modifying flux, $M$ remains $M_0$ and the output remains $f{M}_0$: there is no adaptation. Thus the usual fall followed by recovery requires the additional feedback mechanism developed in c or d, rather than following from the stated Michaelis-Menten kinetics alone.

Let attractant reduce signalling activity per methylated receptor by a factor $f\in (0,1)$, so the output is $A=fM$. Before stimulation take $f=1$ and $M=M_0=K_{B}r/(b-r)$. Perfect adaptation requires the final methylated receptor concentration $M^{\prime }=M_0/f$. With $b^{\prime }=V_B^{\prime }[B]$, the new steady state condition isEquivalently $b^{\prime }=fb+(1-f)r$. Since $f<1$, this reduces demethylation relative to the initial condition. Immediately after binding, activity and tumbling fall. The excess CheR flux then increases receptor methylation, compensating the activity loss until $f{M}^{\prime }=M_0$. The methylated population must have sufficient remaining capacity to reach $M^{\prime }$.

Increasing CheR concentration raises $r$ and the baseline activity; increasing CheB concentration raises $b$ and lowers it. If the attractant-dependent ratio $c=V_B^{\prime }/V_B$ remains fixed after such a change, the new pre- and post-step outputs areTheir equality requires $c=f+(1-f)r/b$. Thus a ratio tuned for one $[R]/[B]$ generally loses perfect adaptation when either enzyme level changes independently. For a concrete example, initially $r/b=1/2$ and $f=1/2$ require $c=3/4$; changing $r/b$ to $1/4$ requires $c=5/8$, so keeping $c=3/4$ fails. Proportionally scaling both enzyme levels preserves the ratio and the adaptation condition, while changing response speed. This is fine tuning, rather than robust perfect adaptation.

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.

Removal of methyl groups from a modified receptor. In chemotaxis feedback regulation, this reverses the chemical modification introduced by receptor methylation; the two processes have different enzyme-dependent rates. Whether the demethylation rate depends on total receptor modification or on active receptors matters for perfect adaptation.