Pathway-Based Mean-Field Model for Escherichia coli Chemotaxis

We develop a mean-field theory for Escherichia coli chemotaxis based on the coupled spatiotemporal dynamics of the cell population and the mean receptor methylation level field. This multiscale model connects the cells’ population level motility behavior with the molecular level pathway dynamics. It reveals a simple scaling dependence of the chemotaxis velocity on the adaptation rate in exponential gradients. It explains the molecular origin of a maximum chemotaxis velocity. Simulations of our model in various spatiotemporal stimuli profiles show quantitative agreements with experiments. Moreover, it predicts a surprising reversal of chemotaxis group velocity in traveling wave environments. Our approach may be used to bridge molecular level pathway dynamics with cellular behavior in other biological systems.

Figure 1

Chemotaxis velocity in exponential concentration gradient. (a) The relationship between $v_d$ and $G$ for different $k_R$. (b) Collapse of the $v_d$ dependence on $G$ (lines) with the scaling $v_d\to v_d/\sqrt{k_R}$, $G\to G/\sqrt{k_R}$. The symbols are results from SPECS for different $k_R$ values. Parameters used are: $z_{\theta }=0.14{\mathrm{s}}^{-1}$, $H=10$, $\tau =0.8\mathrm{s}$, $\alpha =1.7$, $m_0=1$, $K_I=18.2\mu \mathrm{M}$, $K_A=3\mathrm{mM}$, $N=6$, $a_0=0.5$; $v_0=16.5/\sqrt{2}\mu \mathrm{m}/\mathrm{s}$ is the dimension-adjusted run velocity used in one-dimensional PBMFT, whereas $v_0=16.5\mu \mathrm{m}/\mathrm{s}$ is used in two-dimensional SPECS. The default value of $k_R$ is $0.005{\mathrm{s}}^{-1}$.

Figure 2

Chemotactic behavior in traveling wave attractant. (a) The average velocity of bacteria $v_d$ in traveling wave environment for different wave speeds $u$. $v_d$ reverses its direction at $u=u_c$. The parameters used are: $[L{]}_0=500\mu \mathrm{M}$, $[L{]}_A=100\mu \mathrm{M}$, $\lambda =800\mu \mathrm{m}$. (b) The $M$ and $M^{*}$ profiles in the comoving frame for $u=8\mu \mathrm{m}/\mathrm{s}$ labeled in (a). (c) The corresponding $dM/dx$ and $\chi$ profiles. The regions with $dM/dx<0$ are shaded in (b) and (c).