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A conceptual chemotaxis adaptation model in which saturated CheR action supplies receptor methylation while CheB preferentially demethylates active receptors. If $\dot{M}=r-bA/(K+A)$, the steady state output is $A_{\ast }=Kr/(b-r)$, independent of ligand whenever a physically accessible stable steady state exists. Changing enzyme abundance can change this baseline without destroying adaptation to a subsequent ligand step.^[ref]

CheA is the signaling enzyme that promotes phosphorylation of CheY and CheB in the chemotaxis feedback regulation pathway.

CheB is an enzyme that reverses receptor methylation in chemotaxis; its regulation contributes to negative feedback.

In Escherichia coli, chemotaxis feedback regulation couples receptor activity to modification of the receptors by CheR and CheB. It is a molecular example of negative feedback supporting signaling adaptation, with precision and kinetics assessed by the chemotaxis FRET assay above.

Negative feedback closes a causal loop whose net effect opposes an initiating change. In signal transduction, activation followed by a downstream inhibition of the earlier activating stage is one example of negative feedback. Loop sign alone does not establish asymptotic stability or complete signaling adaptation; see the worked signaling feedback example.

A useful biological example is chemotaxis feedback regulation: an Escherichia coli biological cell adjusts its response to a sustained attractant instead of remaining at its immediate response. Increased CheA activity promotes CheB phosphorylation; CheB reverses receptor methylation, opposing their activation. After attractant addition, reduced activity instead favors receptor methylation by CheR, restoring activity. This is a return from receptor activity to the machinery regulating that activity. The response need not return exactly at every dose. ^[ref]

A motivating measurement is the chemotaxis FRET assay. CheY and CheZ carrying yellow and cyan fluorescent tags were observed in immobilized Escherichia coli biological cells on a coverslip in a flow chamber. Adding and removing attractant changed the yellow-to-cyan emission ratio, reporting the association of phosphorylated CheY with CheZ. This measures a molecular readout upstream of motor behavior. Comparisons with biological cells lacking CheR, CheB, or both tested the dependence on receptor modification; these perturbations also change starting activity and sensitivity, so they are not pure changes of one feedback gain. ^[ref]

The later chemotaxis feedback regulation experiment used only the Tar receptor and nonmetabolizable methylaspartate, supplied and removed by flow. Long recordings separated slow recovery from incomplete signaling adaptation; correction for bleaching addressed drift in the optical readout. Activity recovered incompletely after sufficiently large additions, with detectable imprecision already at $10\mu \mathrm{M}$ in that preparation. Restricting the receptor type and preventing consumption of the stimulus help interpret the perturbation, but limit transfer to other receptors and conditions. The readout infers CheA activity through CheY-CheZ association, rather than directly measuring every molecular step. ^[ref]

Negative feedback can also overshoot or sustain oscillations if its return arrives too late. The worked signaling feedback example gives the contrasting instantaneous case; the delayed negative feedback model below isolates the effect of timing. Loop sign describes causation, while asymptotic stability describes what happens to displacements over time.

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.