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A chemoattractant is a substance whose spatial distribution can bias responsive biological cells to migrate towards higher concentration. A directional response to its gradient is chemotaxis; a change only in movement speed does not by itself establish a directional response.

Chemotaxis is directed movement of biological cells in response to a chemical concentration gradient. A chemoattractant biases movement toward higher concentration; other signals can bias movement away. At the population scale, chemotactic drift contributes a chemotactic flux that can compete with diffusion and fluid flow.

The directional bias need not mean that each cell points continuously toward the source. In swimming bacteria, changes in the durations of relatively straight movement and reorientation can bias an otherwise wandering trajectory. A motivating experiment changes the spatially uniform concentration of attractant rapidly while observing cell motion. Increased and decreased stimulation produce different changes in movement, demonstrating a temporal response without requiring a front-to-back gradient across the cell.^[ref] This temporal sensing in bacterial chemotaxis explains how successive samples during movement can guide migration through a spatial chemical landscape.

Haptotaxis is directed cell migration along spatial variation of an immobilized adhesive or guidance cue. It differs from guidance by a soluble chemoattractant field.

A nutrient supplies material or energy used by an organism. In mathematical biology, a nutrient concentration can be transported, produced and consumed, and can influence population dynamics or act as a chemoattractant.

The bacterium samples its surroundings in time. A sustained increase in chemoattractant makes a favorable run persist by reducing the chance of tumbling; declining chemoattractant makes reorientation more likely. Individual tumbles need not point up the gradient. Biasing run durations can generate net migration even if reorientation itself is random. This is temporal comparison, not a direct measurement of a concentration gradient across the short cell body.

Compare the swimming time over length $\ell$, $t_a=\ell /v$, with diffusion over the same scale, $t_D\sim {\ell }^2/D$. Their ratio is the Péclet number $\mathrm{Pe}=v\ell /D$. The crossover estimate isOrder-one factors depend on the chosen diffusion-distance convention. A hypothetical stirrer of extent $d$ must instead have $Ud/D$ of order one, givingThese are transport-scale estimates, not a proof of a particular increase in uptake. Around a one-micrometre body, $vd/D\simeq 0.03$: local diffusive supply is already rapid. Actual uptake enhancement depends on the nutrient field and the cell's uptake boundary condition, neither supplied. The long run can help the organism reach a better nutrient region without implying that it greatly increases the flux from a uniform bath. E. coli normally uses bacterial flagella; the cilium in this comparison is a hypothetical generic stirring appendage, not its actual locomotor organelle.

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]

Take $x=0$ at chamber A and $x=L$ at chamber B. At steady state, the diffusion equation with constant positive diffusion coefficient givesThe two Dirichlet boundary conditions give $A=c_0$ and $B=-c_0/L$. ThusThe concentration gradient is constant, $dc/dx=-c_0/L$. By Fick's first law, the diffusive flux is $J=-Ddc/dx=D{c}_0/L$, directed from A towards B. The profile is linear independently of $D$, while the flux and the time needed to approach it depend on $D$.

This result assumes no significant bulk flow, chemoattractant loss to the walls, degradation or cellular uptake. Experimentally, a uniform-concentration control with matched reservoir concentrations tests for directional cell migration caused by flow or apparatus asymmetry; reversing the reservoir concentrations tests whether the directional response follows the chemical gradient. Keeping both reservoirs well mixed and measuring the concentration profile checks the assumed boundary concentrations and whether uptake has distorted it. These controls distinguish the assumed passive diffusion field from other directional cues; the derived profile alone does not establish the mechanism of chemotaxis.

For the same concentration gradient, the front-to-back difference scales with cell length. A much smaller bacterial cell samples a correspondingly smaller spatial signal, while stochastic arrival and binding of individual chemoattractant molecules make receptor measurements noisy. For illustration, a $2\mu \mathrm{m}$ cell in the $0.5$ cm capillary would experience only $0.04\mathrm{nM}$ across its length, compared with $0.4\mathrm{nM}$ for the longer cell. This illustrates a size penalty; it does not assume that bacterial and human detection thresholds are identical.

Many swimming bacteria, including Escherichia coli, use temporal sensing in bacterial chemotaxis: they compare recent receptor stimulation with an adapted baseline as they move. Increasing attractant suppresses reorientation and prolongs runs; decreasing attractant increases tumbling. Repeated runs and reorientations therefore generate a biased random walk up the gradient, without requiring a simultaneous comparison between the front and back. Chemotaxis feedback regulation supplies the adaptation needed to remain sensitive to changes rather than responding only to absolute concentration.

A discriminating experiment changes concentration rapidly and uniformly around the bacteria: a response then occurs without a spatial difference across the cell. The classic temporal-step experiments found increased tumbling after attractant decreases and smoother swimming after increases.^[ref] In a new experiment, compare up-steps and down-steps with constant-concentration and sham-mixing controls, holding temperature and medium composition fixed. Measure run durations and tumble frequency, rather than only average swimming speed. Such a test supports temporal comparison but does not by itself identify every molecular component or show that all bacterial species use the same mechanism.

The term $\alpha {\nabla }^{2}q$ is cell diffusion arising from random movement. Chemotaxis up the concentration gradient gives the cell flux $\beta q\nabla c$, whose negative divergence is the transport contribution $-\beta \nabla \cdot (q\nabla c)$. The reaction term $q(1-q)$ gives growth governed by the logistic differential equation with dimensionless carrying capacity one.

The chemoattractant itself diffuses, is produced by cells at rate $\kappa \eta q$, and decays at rate $\kappa c$. The homogeneous Neumann boundary conditions suppress the normal diffusive flux of each quantity. They also suppress the normal chemotactic cell flux because the normal derivative of $c$ vanishes. Thus the rectangular boundary is impermeable, and the spatially integrated cell count changes through cell proliferation rather than transport across that boundary.

External chemical signals can guide cell migration, but the statement is too strong if “determined” means that they uniquely specify whether, how fast and in which direction a cell moves. A chemoattractant gradient biases a cell's internal polarity machinery; it does not directly supply the mechanical work of locomotion. Chemotaxis is directional bias, whereas a change in speed or persistence without directional preference is chemokinesis.

A chemical response depends on receptor abundance, signalling adaptation and the cell's existing cell polarity. The same external field can therefore yield different trajectories in different cell states. Uniform stimulation can increase motility without providing a direction, and spontaneous fluctuations of the actin system can initiate movement without a spatial chemical instruction. Persistent migration combines feedback within the cell with a cue that biases that feedback.

Mechanical and geometrical inputs also matter. Durotaxis describes a bias associated with substrate stiffness, while haptotaxis follows spatial variation in adhesive ligands. Contact guidance by fibre alignment can channel motion even when soluble signals are uniform. Narrow spaces constrain the available paths and can force a switch of protrusive mechanism. These effects are not wholly separate from chemistry: cells sense mechanical forces through molecular interactions. Nevertheless, their directional information resides in mechanics or geometry, not necessarily in an imposed soluble concentration gradient.

Contacts between cells supply another source of organization. Experiments on neural crest cells showed that collisions can cause protrusion collapse and repolarization, and that perturbing noncanonical Wnt signalling disrupts this behaviour and group directionality.^[ref] Contact inhibition of locomotion therefore helps generate collective trajectories through local interactions, rather than each cell independently reading a complete external chemical map. Cohesion and other attractive interactions can prevent such repulsion from simply dispersing the group.

A discriminating experiment independently varies soluble gradient, adhesive pattern and mechanical confinement. Measure direction, speed and persistence separately; reverse the gradient, remove it, and then block the relevant receptor while checking viability and the ability to move. If a directional bias follows the reversed chemical field, that supports chemical guidance. If speed rises under uniform stimulation, that supports chemokinesis. If trajectories remain aligned with fibres after chemical reversal, physical guidance is influential. Manipulating intracellular polarity can test whether a cue is instructive or merely permissive.

The useful conclusion is that motion emerges from an internally powered, state-dependent cell interacting with chemical, mechanical and cellular surroundings. External chemical signals are often decisive inputs, but the trajectory is a property of the coupled system.