A traveling-wave solution for bacterial chemotaxis with growth

Abstract

Bacterial cells navigate their environment by directing their movement along chemical gradients. This process, known as chemotaxis, can promote the rapid expansion of bacterial populations into previously unoccupied territories. However, despite numerous experimental and theoretical studies on this classical topic, chemotaxis-driven population expansion is not understood in quantitative terms. Building on recent experimental progress, we here present a detailed analytical study that provides a quantitative understanding of how chemotaxis and cell growth lead to rapid and stable expansion of bacterial populations. We provide analytical relations that accurately describe the dependence of the expansion speed and density profile of the expanding population on important molecular, cellular, and environmental parameters. In particular, expansion speeds can be boosted by orders of magnitude when the environmental availability of chemicals relative to the cellular limits of chemical sensing is high. Analytical understanding of such complex spatiotemporal dynamic processes is rare. Our analytical results and the methods employed to attain them provide a mathematical framework for investigations of the roles of taxis in diverse ecological contexts across broad parameter regimes.

Keywords: Fisher wave; Keller–Segel model; bacterial chemotaxis; front propagation; range expansion.

Fig. 1.

Profiles of bacterial density (solid red line, in optical density measured at a wavelength of 600 nm [${\text{OD}}_{600}$]), drift velocity (dashed green line, in mm/h), and attractant concentration (dotted blue line, in mM) for a steadily expanding population 14.5 h after the inoculation. Arrows indicate the different regimes used in the analytical consideration. Model parameters used are adapted from those determined in ref. and are provided in SI Appendix, Table S1 (this simulation used the low-motility parameters).

Fig. 2.

Dependence on (A) growth rate r, (B) uptake rate μ, (C) relative attractant levels $a_0/a_m$, and (D) attractant diffusion D_a. Analytical relation for the expansion speed (Eq. 23) is shown by solid lines ($D_{\rho }=50 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s},1,000 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$ in red and blue, respectively). The corresponding Fisher speeds, $c_F=2\sqrt{D_{\rho }·r}$, are denoted by corresponding dashed lines. Numerical solutions of the GE model (Eqs. 4 and 5) are shown by corresponding symbols. Unless specified, all parameter values are the default values given in SI Appendix, Table S1.

Fig. 3.

Dependence of expansion speed on motility parameters. (A) Dependence on cellular motility $D_{\rho }$. Numerical solutions for $\varphi =1$ and $\varphi =5$ are shown by red circles and dark blue triangles, respectively. Analytical solutions following Eq. 23 are shown by corresponding solid red and blue lines. The green dashed line represents the stable Fisher speed, $c_F=2\sqrt{D_{\rho }r}$, the minimum expansion speed of our system. (B) Dependence on the chemotactic sensitivity, $\varphi$. Numerical solutions for $D_{\rho }=50 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$ and $D_{\rho }=1,000 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$ are shown by red and dark blue circles, respectively. Analytic solutions following Eq. 23 are shown by the corresponding solid lines. Thick yellow and cyan dashed lines are best fits for the respective values of $\varphi$ and $D_{\rho }$ to demonstrate that $c\propto D_{\rho }\varphi$ for $D_{\rho }\varphi \mathrm{\lesssim }{D}_a$ and that $c\propto \sqrt{D_{\rho }\varphi }$ if $D_{\rho }\varphi$ is large compared to D_a. Unless specified, all parameter values are the default values given in SI Appendix, Table S1.

Fig. 4.

Effect of carrying capacity. (A) Dependence of expansion speed on the ambient attractant concentration when the carrying capacity is finite (${\rho }_c=10 {\text{OD}}_{600}$). Markers (red circles and blue triangles) indicate numerical values, solid lines indicate analytical predictions as per Eq. 24, and dashed lines indicate analytical predictions with ${\rho }_c\to \infty$. All results in red are for $D_{\rho }=50 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$, and all results in blue are for $D_{\rho }=1,000 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$. (B) The ambient attractant concentration resulting in maximum expansion speed $a_0^{\text{max}}$ is shown depending on the dimensionless parameter $\mu {\rho }_c/(r{a}_m)$. The analytical solution, Eq. 24, is shown as corresponding solid lines. Dashed lines show the solutions ($c_{\infty }$) without a limiting carrying capacity (${\rho }_c\to \infty$, as shown in Fig. 3). Different symbols in B denote which model parameter was varied from its default value (square if μ, circle if ρ_c, triangle if r, and diamond if a_m) for $D_{\rho }=50 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$ (red) and $D_{\rho }=1,000 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$ (blue). For details, refer to SI Appendix, Supplemental Methods, and to SI Appendix, Table S2, for range of values used for each parameter. Parameters have the default values from SI Appendix, Table S1, unless specified.

Fig. 5.

Effect of varying Michaelis constant, a_k. (A) Dependence of expansion speed on the chemotactic sensitivity, $\varphi$, for different values of a_k and $D_{\rho }=50 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$. Solid lines indicate analytical solutions for corresponding best fit values of η, and markers denote the numerical solutions. Results for $a_k=0.1 \mathrm{\mu }\mathrm{M},\hspace{0.17em}1 \mathrm{\mu }\mathrm{M}$, and $10 \mathrm{\mu }\mathrm{M}$ are shown in yellow, red, and blue, respectively. (B) Dependence of the expansion speed on model parameter a_k. The numerical solutions obtained for $D_{\rho }=50 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s},\hspace{0.17em}\varphi =5$ are represented by yellow triangles, and the analytic solution found in Eq. 21 for $a_k=a_m={10}^{-3} \text{mM}$ is shown by the red line. Parameters have the default values from SI Appendix, Table S1, unless specified.

Fig. 6.

Transition from the chemotaxis to the growth regime. (A) Steady expansion profiles of $\rho (z)$ (solid red line) and a(z) (solid blue line) for the standard parameters (SI Appendix, Table S1; $D_{\rho }=50 \mathrm{\mu }{\mathrm{m}}^2/\mathrm{s},{\chi }_0=300\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^2/\mathrm{s}$). The profile of $\rho (z)$ as predicted by the ansatz Eq. 7 is shown using the dashed green line. Dashed horizontal lines indicate distinct values of a and ρ as indicated. (B–D) Numerically obtained values of $a(z_{\text{min}})$, $\rho (z_{\text{min}})$, and $z_m-z_{\text{min}}$ for a broad variation of parameters. Seven model parameters in Eqs. 4 and 5 (other than ρ_c, which was >1, 000 ${\text{OD}}_{600}$ for all results here) were varied across many decades (see SI Appendix, Supplemental Methods, for details of what was done and SI Appendix, Table S3, for the range of values investigated). Blue lines show y = x to demonstrate agreement with the predicted values of $a(z_{\text{min}}),\hspace{0.17em}\rho (z_{\text{min}})$, and $z_m-z_{\text{min}}$.

Fig. 7.

Schematic of the dynamics of the transition between chemotaxis and growth regimes. (A) In a short time $\delta t$, the density bulge shown near x₀ (dotted red line) moves forward to be near $x_0+c\delta t$ (solid red line). In that time, the density bulge grows by an amount $r{N}_0\delta t$ and is diminished by leakage given by an amount $J_0\delta t$. During steady expansion, these values match the expressions given by our ansatz (Eqs. 7 and 37). The leaked cells are deposited behind the density bulge where the bacterial density is roughly constant for a distance δx. Thus, $\rho (x_0,t_0+\delta t)\approx {\rho }_{\text{min}}$, and the total deposition over time δt, given by $\delta N_0$, is also equal to $J_0\delta t$. (B) After a long time $\mathrm{\Delta }t$, the density bulge moves to be near a position $x_0+c\mathrm{\Delta }t$ (dashed red line). Cells behind the density bulge grow at a rate r, and the density thus accumulates as $\rho (x_0,t)={\rho }_{\text{min}}\exp (r\mathrm{\Delta }t)$.

Fig. 8.

Illustrations of possible extensions of our analysis. (A) Chemotaxis in marine bacteria in the plume of sinking marine particles. The gray bead is the sinking particulate organic matter. A flux of nutrients (gray circles) and attractants (pink hexagons) diffuse into the wake of the particle. The cyan motile bacteria are able to direct their motion up the attractant gradient and thus move toward the sinking particle and keep up with it. (B) Chemotaxis in pursuit–evasion dynamics in a predator–prey system. A predator (shown as a blue amoeba) may pursue gradients of chemicals (attractant; yellow beads) left by a motile prey (a yellow bacterial cell). The motile prey may, in turn, evade the predator by performing chemotaxis and moving away from chemicals (repellent; blue beads) secreted by the predator.