Chemotaxis

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.

Chemotactic drift is the directed velocity associated with sensing a chemical gradient. In a simple continuum model the drift is $\chi \nabla c$; multiplying by cell density gives the chemotactic flux.

A chemotactic flux is the transport flux caused by directed cell movement in response to a chemical gradient. For a cell density $q$ and attractant concentration $c$, a common flux is $\chi q\nabla c$ with $\chi >0$.

For $n_t=D_n{n}_{xx}-\chi (n{a}_x{)}_x+n(1-n)$, $a_t=D_a{a}_{xx}+\alpha n-\beta a$, the equilibrium $(1,\alpha /\beta )$ is stable without transport. A spatial mode of wavenumber $k$ has negative trace and determinant $D_n{D}_a{k}^4+(D_a+\beta D_n-\alpha \chi )k^2+\beta$. A continuous unstable band exists exactly under the displayed inequality. On a finite interval with Neumann boundary conditions, an allowed discrete wavenumber must additionally fall in that band. The high-wavenumber positive term is diffusive smoothing; the negative chemotactic term drives aggregation.

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.

Temporal sensing in bacterial chemotaxis compares changing stimulation during movement with a recent adapted baseline. Attractant increases can lengthen runs and decreases can increase tumbling, producing biased movement up a chemical gradient. Rapid spatially uniform concentration steps distinguish a temporal response from a requirement for front-to-back spatial comparison.^[ref]

For a homogeneous state with linear decay rate $A>0$, chemical decay $\gamma >0$, chemical diffusivity $D>0$ and chemotactic coupling $\beta {\chi }_0{n}_0$, a spatial mode $q=k^2$ has determinant $Q(q)=D{q}^2-Bq+A\gamma$, where $B=\beta {\chi }_0{n}_0-AD-\gamma$. The trace is negative, so an instability requires $Q(q)<0$. Without a cutoff, the sharp continuous-wavenumber threshold is $B>2\sqrt{DA\gamma }$. With $q>q_0$, minimize $Dq+A\gamma /q$ over that range; its unconstrained minimizer is $\sqrt{A\gamma /D}$. Discrete geometries additionally require an actual allowed Laplacian eigenvalue in the open unstable band.

A Keller--Segel model couples the density of moving cells to a chemical field that biases their motion. A common cell conservation law is $b_t=D{b}_{xx}-{\partial }_x(\chi (c)b{c}_x)$. Chemical dynamics depend on the application: the attractant may be produced by cells, or a nutrient may instead be consumed. The nutrient-consumption model $c_t=-kb$ with no nutrient diffusion gives travelling bacterial bands rather than prescribing the chemical externally.

For cell diffusion, self-produced chemotactic agent and logistic proliferation, Neumann cosine modes reduce linear stability of the uniform state to a two-by-two matrix indexed by $s={\pi }^2(n^2/L_1^2+m^2/L_2^2)$. Its trace is negative and its determinant is $D(s)=\alpha s^2+(1+\alpha \kappa -\beta \kappa \eta )s+\kappa$. Chemotaxis can destabilize a mode precisely when $D(s)<0$, despite stability of the spatially uniform reaction system.

When $\alpha =\kappa =\eta =1$, the instability threshold of a spatial mode with squared wave number $s$ is $\beta =s+2+1/s$. On a rectangle $L_1=1$, $L_2=1/2$, modes varying in both coordinates have $s\ge 5{\pi }^2$. Since the threshold increases for $s>1$, the first such mode is $(n,m)=(1,1)$, at ${\beta }_{crit}=5{\pi }^2+2+(5{\pi }^2{)}^{-1}$. At equality it is neutral; strict instability begins above this value.

For a homogeneous Keller--Segel model with nonnegative chemical production, linearized cell diffusion $D_2>0$, attraction coefficient $D_1>0$, and chemical diffusion $D_{\rho }>0$, put $c=\kappa -a_0{f}_0^{\prime }$ and $\kappa =(\rho k(\rho ){)}^{\prime }{|}_{{\rho }_0}$. A Fourier mode with squared wavenumber $z$ has determinant $z[D_2(c+D_{\rho }z)-D_1{f}_0]$. If $c\ge 0$, a growing mode on the infinite plane exists exactly when $D_1{f}_0>D_{2}c$, and $0<z<(D_1{f}_0-D_{2}c)/(D_2{D}_{\rho })$ is the unstable band. If $c<0$, the homogeneous chemical mode is already unstable. Dividing the threshold by $D_2\kappa$ to obtain a sum of two ratios is valid only when $\kappa >0$. Positivity of $k$ does not guarantee positivity of $(\rho k{)}^{\prime }$.

For the two-field Keller--Segel model, put $a=D_2$, $b=D_{\rho }$, $C=D_1{f}_0\ge 0$, $c=\kappa -a_0{f}_0^{\prime }$. The upper growth rate is $s_{+}(z)=-[c+(a+b)z]/2+\sqrt{[c+(b-a)z{]}^2+4Cz}/2$. When $C>\max (ac,-bc)$ its unique maximum occurs at positive $z_{\ast }$ satisfying $s_{+}^{\prime }(z_{\ast })=0$. For equal diffusion coefficients $a=b$, this gives $z_{\ast }=(C^2/a^2-c^2)/(4C)$. For $c<0$ and $C\le -bc$, the fastest growth is homogeneous, with infinite wavelength. The finite-domain answer must maximize the growth rate over the permitted Fourier modes. Eliminating chemical dynamics instantaneously generally changes this selected scale.

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.

For a Keller--Segel model with $c_t=-kb$, a positive-speed travelling wave $B(z),C(z)$, $z=x-vt$, satisfies $v{C}^{\prime }=kB$ and $D{B}^{\prime }+vB-\alpha B{C}^{\prime }/C=0$ when the integrated bacterial flux constant is zero. Writing $\mu =\alpha /D>1$ and $\xi =v(z-z_0)/D$, its positive band solution isTo derive it, integrate $B^{\prime }/B=\mu C^{\prime }/C-v/D$ to obtain $B=K{C}^{\mu }{e}^{-vz/D}$, then separate $v{C}^{\prime }=kB$. The positive integration constant is a translation of the wave. Concentration rises monotonically from zero behind to $C_{\infty }$ ahead; $B$ vanishes at both ends and has a unique maximum at $\xi =-\log (\mu -1)$. The condition $\mu >1$ describes this exponential-tail branch; it is not asserted to exclude every limiting or weak travelling wave at other parameter values.

For a nutrient-consuming chemotactic travelling band in a tube of cross-sectional area $a$, let $N=a{\int }_{\mathbb{R}}B(z)dz$. Integrating $v{C}^{\prime }=kB$ between depleted nutrient behind and $C_{\infty }$ ahead gives $v{C}_{\infty }=kN/a$, hence $v=Nk/(a{C}_{\infty })$. This is a nutrient budget: advancing by a unit length supplies $a{C}_{\infty }$ nutrient, while the bacteria consume at total rate $kN$. Diffusion and chemotactic sensitivity affect the band shape and admissibility, but not this speed at fixed $N,k,a,C_{\infty }$.

A chemotactic pattern-forming instability occurs when cells produce an attractant and drift up its gradient strongly enough that positive aggregation feedback overcomes diffusion and turnover at a finite wavelength.

For $b_t=d_b{b}_{xx}-{\chi }_b{\partial }_x[b(1+n{)}^{-2}{n}_x]+b(1-b)$ and $n_t=d_n{n}_{xx}+\lambda b^2/(b_0^2+b^2)-\mu bn$, the positive homogeneous equilibrium is $(1,n_{\ast })$, where $n_{\ast }=\lambda /[\mu (1+b_0^2)]$. Set $\kappa ={\chi }_b/(1+n_{\ast }{)}^2$ and $\eta =\lambda (b_0^2-1)/(1+b_0^2{)}^2$. The linear stability matrix at squared wavenumber $q$ is $\left(\begin{array}{cc}-1-d_{b}q& \kappa q\\ \eta & -\mu -d_{n}q\end{array}\right)$. Its trace is negative and its determinant is $d_b{d}_n{q}^2+(d_n+\mu d_b-\kappa \eta )q+\mu$. Minimizing this quadratic polynomial over $q>0$ proves the displayed continuous-wavenumber growth criterion. It requires $b_0>1$. On a bounded domain an actual permitted nonzero Fourier mode must lie strictly inside the negative-determinant interval. This is a criterion for growth of infinitesimal perturbations, not a general exclusion of finite-amplitude patterns.

A chemotactic telegraph equation describes persistent motion with finite turning rate and a turning bias set by a chemical gradient. Its large-turning-rate limit is a drift--diffusion equation with chemotactic drift.