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

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

In Receptor sensitivity in bacterial chemotaxis by Victor Sourjik and others, Victor Sourjik and Howard C. Berg used a cyan fluorescent tag on CheZ as FRET donor and a yellow tag on CheY as FRET acceptor. Escherichia coli biological cells adhered to a polylysine-coated coverslip in a continuously perfused chamber. A fluorescence microscope excited them with a $442$-nanometre laser; split emission bands fed two photon-counting detectors. The yellow/cyan emission ratio fell when attractant was added and rose when it was removed.^[ref]

Saturating attractant increased cyan emission by an amount matching selective bleaching of the yellow tag: removing the FRET acceptor supported relief of FRET donor transfer. Strains carrying the tagged pair but lacking other chemotaxis machinery showed no stimulus-dependent emission ratio change. Weak illumination reduced bleaching drift.^[ref]

The inference is a change in the population of nearby tagged CheY/CheZ pairs associated with CheY phosphorylation. It neither measures each pair's separation nor directly observes motor motion. The readout also averages many biological cells, and perfusion takes finite time. The pathway interpretation belongs to the chemotaxis FRET assay; the general measurement lesson is that molecular association can alter FRET while the geometry of an individual associated pair stays the same.

For positive chemical concentration $c$, taking $\chi (c)=\alpha /c$ makes the chemotaxis drift velocity $\chi (c)c_x=\alpha {\partial }_x\log c$. Cells then respond to a relative chemical gradient. This singular response can remain finite where the concentration tends to zero if its logarithmic derivative has a finite limit.

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.

For a Laplacian mode with ${\nabla }^2=-k^2$, the linearized chemotaxis system isIts trace is negative for all $k$. Writing $s=k^2$, its determinant isInstability occurs when this determinant is negative, because then one real eigenvalue is positive. A quadratic with positive leading and constant coefficients is negative for some $s>0$ precisely when its middle coefficient is negative and its discriminant positive. These conditions combine toThis proves the printed threshold. In a bounded spatial domain, only its permitted Laplacian eigenmodes are available; the continuous-$k$ threshold describes the onset when wavenumbers can vary freely.

The term $\gamma n(n_0-n)$ is logistic population growth, with carrying density $n_0$ and low-density growth rate $\gamma n_0$. The cell flux is $-D_n\nabla n+\chi n\nabla C$: its first part is diffusion down the density gradient, and its second is chemotaxis toward higher chemical concentration. Taking minus its divergence gives the two transport terms. Chemical production is proportional to cell density at rate $\alpha n$, degradation is first order at rate $\beta C$, and the chemical diffuses with coefficient $D_c$. All the given positive signs correspond to growth, attraction, production, degradation and ordinary diffusion respectively.

For a positive-speed travelling wave, the nutrient equation gives $a^{\prime }=kb/c$. Integrating the bacterial equation once, with zero flux in the uncolonized limit, givesHenceIn the region where $b>0$, divide the two equations. With $\gamma =\chi /D$,The condition $b(1)=0$ fixes the integration constant, givingAt $\chi =D$ the limiting expression is $b=-(c^2/kD)a\log a$. These formulas are positive between zero and one. For $\chi <D$ the nutrient reaches zero at a finite rear coordinate, so the strictly positive smooth whole-line ansatz with logarithmic chemotactic sensitivity needs a separate zero-nutrient interpretation there; for $\chi \ge D$ it reaches zero only as $z\to -\infty$.

For $\chi =2D$, the nutrient equation reduces to $a^{\prime }=(c/D)a(1-a)$ and integration givesThe factor $K$ in the bacterial numerator is required by $a^{\prime }=kb/c$; it is absent in the printed expression unless $K=1$. A translation of $z$ sets $K=1$, which is the version plotted below. The nutrient rises monotonically from zero behind the band to one ahead, while bacteria form a localized pulse with maximum $c^2/(4kD)$ at the nutrient half-height. Its total population is $\int bdz=c/k$. The band advances into fresh nutrient, consumes it and leaves a depleted region behind; chemotaxis draws bacteria up the nutrient gradient while diffusion spreads them.

With no chemotaxis, the stationary oxygen equation is $D_c{c}^{\prime \prime }=k\bar{n}$. Symmetry and the two prescribed surface values giveBy Fick's first law, the vertical oxygen flux is $j_c=-D_c{c}^{\prime }=-k\bar{n}z$. Thus each face supplies an inward flux $k\bar{n}d/2$ per unit area, and the total influx is $k\bar{n}d$, equal to the integrated consumption across the layer. The profile is physically nonnegative only if $c_0\ge k\bar{n}{d}^2/(8D_c)$; otherwise the constant-consumption approximation must be modified.

For bacteria retained in the fluid, impose zero bacterial flux at the faces. The stationary bacterial equation says that $j_n=-D_n{n}^{\prime }+\mu n{c}^{\prime }$ is constant, hence zero. Integrating $n^{\prime }/n=(\mu /D_n)c^{\prime }$ gives $n(z)=n_0\exp (\mu c(z)/D_n)$, with $n_0$ fixed by the total bacterial population. Substitution into the oxygen equation provesZero bacterial through-flux is the boundary assumption behind this equilibrium relation.

The reaction–diffusion system for amoebae contains local reproduction $\alpha n_0^{2}n$ and density-dependent loss $-\alpha n^3$, giving carrying density $n_0$. The term ${\nabla }^{2}n$ describes diffusion with coefficient one. The chemotaxis flux is $J_{\mathrm{chem}}=\chi (n)n\nabla a$, so its contribution to the conservation equation is $-\nabla \cdot J_{\mathrm{chem}}$. Attraction towards greater chemical concentration requires $\chi (n)>0$ at the relevant positive densities; negative values would mean repulsion, and zero removes the chemotactic response.

In the chemical equation, $\beta n$ is secretion proportional to amoeba density, $-\gamma a$ is first-order chemical degradation, and $D{\nabla }^{2}a$ is chemical diffusion. The cell reaction term permits births and deaths, so the total number of amoebae is not conserved.

Use the homogeneous equilibrium point condition $k({\rho }_0){\rho }_0=a_{0}f({\rho }_0)$. Evaluate $D_1,D_2$ at $(a_0,{\rho }_0)$ and abbreviateThe quantity $c$ is the chemical relaxation rate with the cell density held fixed. Derivatives of $D_1,D_2$ do not enter this linear stability analysis: they multiply spatial derivatives of the homogeneous background or products of perturbations. For a Fourier mode with time dependence $e^{st}$, put $z=|\mathbf{q}{|}^2$ and writeConsequentlyAt $z=0$, the eigenvalues are $0$ and $-c$. The neutral cell-density mode reflects mass conservation, not decay of every homogeneous perturbation.

Write $a=D_2>0$, $b=D_{\rho }>0$, $C=D_1{f}_0$, and interpret production physically as $f_0\ge 0$. If $c\ge 0$, the trace-determinant stability criterion shows that an unstable nonzero wavenumber exists precisely when $C>ac$. The unstable band isIf $c<0$, the chemical field is already unstable at zero wavenumber; arbitrarily small positive wavenumbers are unstable too. Since $C\ge 0$, this also implies $C>ac$. Thus, on the infinite plane, the undivided Keller--Segel aggregation threshold isFor $\kappa >0$, dividing by $D_2\kappa$ gives the requested formAt equality there is no strictly growing mode when $c\ge 0$; nonzero spatial modes decay. On a finite domain, a permitted nonzero wavenumber must actually lie in the unstable band. If one allows a signed production function, the displayed matrix and trace-determinant stability criterion remain valid, but the simplification using $C\ge 0$ must be revisited.

The printed hypotheses do not ensure $\kappa >0$. Positivity of the degradation rate $k$ alone is insufficient. For a concrete counterexample, take $k(\rho )=e^{-2\rho }$, $f(\rho )=e^{-2}$, $a_0={\rho }_0=1$, and $D_1=D_2=D_{\rho }=1$. The homogeneous equilibrium point condition holds, all rates and transport coefficients are positive, but $\kappa =c=-e^{-2}$. The printed left-hand side equals $-1$, although $\det M=z(z-2e^{-2})<0$ for $0<z<2e^{-2}$. These spatial modes grow. If $\kappa =0$, the printed expression is undefined. The undivided criterion and the matrix above resolve both cases.

To find the fastest-growing Keller--Segel mode, use the larger eigenvalueFor $C>0$ the square root is real. Put $p=a+b$ and $d=b-a$. Differentiation givesA positive maximizing wavenumber exists in either of two growing cases: $c\ge 0$, $C>ac$, or $c<0$, $C>-bc$. Equivalently, $C>\max (ac,-bc)$. In this range $C+dc>0$, andThus the stationary point is the unique maximum. The derivative condition yieldsSolving it, with the root that satisfies the unsquared derivative equation, givesFor $c=0$, the unequal-diffusivity expression simplifies to $z_{\ast }=C/[\sqrt{ab}(\sqrt{a}+\sqrt{b}{)}^2]$, also agreeing with the equal-diffusivity limit. The formula maximizes the full two-field growth rate; it makes no instantaneous-chemical approximation. Every direction of $\mathbf{q}$ with this length is equivalent by rotational symmetry.

If $c<0$ but $0\le C\le -bc$, the maximum instead occurs at $z=0$: the fastest mode is homogeneous, with $s_{\ast }=-c$ and infinite wavelength. For $C>0$, its initial derivative is $-b+C/(-c)\le 0$. If $C+dc\ge 0$ the derivative thereafter decreases, while if $C+dc<0$ it increases towards the still-negative large-$z$ limit; either way no positive-$z$ maximum is missed. At $C=0$, the two eigenvalues are simply $-az$ and $-c-bz$, giving the same conclusion. In a finite box, maximize $s_{+}$ over the allowed Fourier modes; excluding the homogeneous mode can change the selected length. In a stable parameter range there is no fastest-growing mode.

The first ratio compares the positive feedback loop “more cells produce more attractant, which draws in more cells” with spreading by cell diffusion and removal of chemical perturbations. Its numerator $D_1{f}_0$ measures chemotaxis together with attractant production; its denominator $D_2\kappa$ measures dispersal together with incremental degradation. This competition between directed chemotaxis and cell diffusion produces spatial aggregation. The second ratio compares the concentration dependence of chemical production, $a_0{f}_0^{\prime }$, with incremental degradation $\kappa$. Positive $f_0^{\prime }$ amplifies chemical fluctuations, whereas negative $f_0^{\prime }$ suppresses them. It changes the chemical relaxation available to the chemotaxis feedback loop and can itself destabilize the homogeneous chemical field. Chemical diffusion $D_{\rho }$ suppresses short scales and sets the selected wavelength, but does not change the infinite-plane long-wave threshold. These interpretations as ratios of stabilizing and destabilizing processes assume $\kappa >0$.

Let $a$ denote tube cross-sectional area in this question, not a filament or sphere radius. The bacterial density is uniform across that section in this one-dimensional Keller--Segel model, so the total bacterial number isIntegrating the nutrient balance $v{C}^{\prime }=kB$ across the complete band givesHence the speed of a nutrient-consuming chemotactic band isThis is a nutrient budget: the whole population consumes at rate $Nk$, while advancement of the band at speed $v$ brings fresh nutrient at rate $av{C}_{\infty }$. The conservation of mass equation for bacteria ensures $N$ stays constant when there is no flux at infinity and no cell birth or death.

At fixed total bacterial number, the speed grows linearly with $N$ and the per-cell consumption rate $k$, and decreases with tube area and nutrient concentration ahead. More nutrient takes longer to deplete, giving the inverse dependence on $C_{\infty }$. Although $D$ and $\alpha$ determine the width, skewness, peak density and admissibility of the smooth band, they do not appear in this speed when $N,k,a,C_{\infty }$ are fixed. This independence relies on neglecting nutrient diffusion and using the specified concentration-independent consumption law; it is not a general prediction for all chemotaxis models. Dimensionally $Nk$ is nutrient amount per time and $a{C}_{\infty }$ nutrient amount per length, so their ratio is a velocity.

A local excess of cells produces extra attractant. The resulting concentration gradient drives chemotaxis toward that excess, which further raises cell density and attractant production. This positive feedback causes aggregation when $\chi$ is large enough. Cell and chemical diffusion suppress short wavelengths, while death and chemical degradation suppress very long and homogeneous perturbations. Their competition selects the finite $k_c$, producing a chemotactic pattern-forming instability even though the homogeneous kinetic system is stable.

Signaling adaptation is a decline of a response toward its earlier level while the initiating signal remains present. Specify which response is measured and its baseline: complete signaling adaptation means eventual return to that baseline; partial signaling adaptation leaves a lasting displacement. The continuing signal distinguishes signaling adaptation from simply switching off the stimulus. Signaling adaptation describes behavior, while signaling feedback describes a causal arrangement, so neither implies the other.

By reducing a persistent response to background ligand concentration, signaling adaptation can make a biological cell sensitive to changes rather than only to absolute levels. Resetting the response helps leave room for another change over a useful range of ligand concentrations. This is valuable in chemotaxis: swimming bacteria encounter changing chemical conditions, and their signal transduction influences whether a run continues or a tumble occurs. Signaling adaptation helps separate a change encountered during motion from a lasting background. This usefulness does not imply exact return or equal sensitivity at every possible concentration. ^{[ref] [ref]}

Here is a solvable incoherent feedforward loop toy that exhibits complete signaling adaptation without signaling feedback. The input $s(t)$ jumps from $0$ to $1$ at $t=0$ and remains $1$, a Heaviside step function. Let $x$ be a fast activating branch, $a$ a slower inhibitory branch, and $r=x-a$ a readout. The activities $s,x,a,r$ are dimensionless variables; physical time is in seconds. Choose time constants ${\tau }_x=1$ second and ${\tau }_a=5$ seconds, with initial conditions $x(0)=a(0)=0$:Both branches read $s$ directly. Neither branch depends on $r$, and $a$ does not change $x$: the subtraction only defines the readout. Thus the slow opposing influence is feedforward, with no closed feedback control loop.

For $t\ge 0$, solving the two ordinary differential equations givesWriting $t$ for the numerical time in seconds, these are $x=1-e^{-t}$, $a=1-e^{-t/5}$ and $r=e^{-t/5}-e^{-t}$. Initially $r=0$, then $x$ rises faster than $a$, producing a pulse. At long times $x$ and $a$ both approach $1$, so $r$ returns to $0$ despite sustained $s=1$. For this upward step $e^{-t/5}\ge e^{-t}$, hence $r(t)\ge 0$. The striking feature of this signaling adaptation toy is that both branches remain active while the readout becomes quiet: the continuing background and the response to its arrival are separable.

Find the pulse maximum by differentiating the exponential functions. With numerical time in seconds, the derivative isThe stationary condition is $e^{-4t/5}=1/5$, so the natural logarithm givesBefore this time the derivative is positive; afterwards it is negative. Together with $r(0)=0$ and ${\lim }_{t\to \infty }r(t)=0$, this proves that it is the unique pulse maximum. As another numerical check, at $t=10$ seconds $x\simeq 0.999955$, $a\simeq 0.864665$ and $r\simeq 0.135290$: the slow branch is still catching up.

The subtraction $r=x-a$ is a mathematical readout, not a claim that an actual chemical concentration equals this difference for arbitrary inputs. Nonnegativity was established only for the specified upward step; other input histories can give a negative signed readout. Complete signaling adaptation here also relies on equal eventual branch gains: changing the readout to $r=x-0.9a$ leaves $r_{\ast }=0.1$ for $s=1$. This toy illustrates cancellation and different time constants; it does not establish molecular implementation or robustness to arbitrary parameter changes. Ma and colleagues' 2009 study separately analyzed biochemical mathematical models with negative feedback and incoherent feedforward loops, identifying additional conditions for signaling adaptation in those models. ^[ref]

A historical connection to feedback control comes from Yi, Huang, Simon and Doyle's 2000 analysis of a bacterial chemotaxis model. They identified integral action in its regulation of receptor activity: accumulated deviations alter a slower adjustment, which in turn changes the measured activity. At a stationary state, an adjustment that integrates a nonzero error cannot remain constant, explaining how zero error can follow. Their conclusion depended on stated biochemical assumptions and stable behavior; it was not a claim that every biological cell uses that mechanism. A transient pulse alone, as the incoherent feedforward loop toy shows, cannot identify the underlying signaling feedback architecture. ^[ref]