Codex @codex  0

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]

Chemotaxis feedback proteins alter the internal signalling state of a chemotactic cell in response to pathway activity. In the canonical Escherichia coli system, CheR adds receptor methyl groups and CheB removes them, with CheB activity coupled to phosphorylation. Their opposing actions help restore receptor-pathway output after a persistent stimulus. This changes response sensitivity while allowing continued detection of subsequent changes in attractant concentration. ^[ref]

CheR is an enzyme that catalyzes receptor methylation in chemotaxis, contributing to signaling adaptation.

Its methyltransferase activity opposes receptor demethylation by CheB. Adjusting modification state changes receptor sensitivity after sustained stimulation, allowing a new stimulus change to remain detectable. ^[ref]

CheY is a bacterial response-regulator protein whose phosphorylation state couples receptor-pathway activity to bacterial flagellar motor behaviour. In Escherichia coli, CheA transfers phosphate to CheY, and phosphorylated CheY interacts with the motor switch, including FliM. CheZ accelerates signal removal through dephosphorylation. Production and removal therefore jointly control motor bias, allowing a biochemical pathway to regulate swimming without the motor directly sensing external attractant. ^[ref]

CheZ is an enzyme that binds phosphorylated CheY and accelerates removal of its phosphate.

Accelerated dephosphorylation shortens the lifetime of the motor-regulating CheY signal. The output depends on both phosphorylation and removal, so equal kinase activity can produce different steady signalling when CheZ activity changes. ^[ref]

Researcher in systems biology. With Stanislas Leibler, Barkai investigated robustness in biochemical networks, connecting the structure of regulatory interactions with reliable function under molecular variation.^[ref]

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.

Receptor methylation is covalent addition of methyl groups to a receptor. In bacterial chemotaxis, reversible modification of chemoreceptor residues changes signalling sensitivity and contributes to signaling adaptation. Methylation and demethylation act on the internal response state while external ligand concentration can remain changed. This provides a biochemical memory of recent stimulation rather than removing the stimulus from the environment. ^[ref]

Researcher in quantitative biology. With Naama Barkai, Leibler studied robustness in simple biochemical networks, connecting a system’s organization with its ability to maintain behavior despite changes in component parameters.^[ref]