Predicting Escherichia coli's chemotactic drift under exponential gradient

Bacterial species are known to show chemotaxis, i.e., the directed motions in the presence of certain chemicals, whereas the motion is random in the absence of those chemicals. The bacteria modulate their run time to induce chemotactic drift towards the attractant chemicals and away from the repellent chemicals. However, the existing theoretical knowledge does not exhibit a proper match with experimental validation, and hence there is a need for developing alternate models and validating experimentally. In this paper a more robust theoretical model is proposed to investigate chemotactic drift of peritrichous Escherichia coli under an exponential nutrient gradient. An exponential gradient is used to understand the steady state behavior of drift because of the logarithmic functionality of the chemosensory receptors. Our theoretical estimations are validated through the experimentation and simulation results. Thus, the developed model successfully delineates the run time, run trajectory, and drift velocity as measured from the experiments.

Figure 1

Schematic details of the bacterial chemosensory network [24, 25]. (Gold solid lines, dashed lines, dotted lines, and shaded areas): elements and reactions that tag on CW flagellar rotation; (red solid lines, dotted lines, and shaded areas): counterclockwise-enhancing components. The response of the regulatory proteins (CheB and CheY) are inactive (gray) until phosphorylated.

Figure 2

Schematic of the bacterial run. In the absence of a chemical gradient, the average run and tumble time are typically 1 and 0.1 s, respectively. Biased random movement of E. coli exhibits in the presence of a chemical gradient $\left[L\right(x,y\left)\right]$. Temporal comparisons of the chemoattractant concentrations are made during the run [26] with constant instantaneous velocity ($v\sim 20\mu \mathrm{m}/\mathrm{s}$). The run time is modulated based on the gradient of the chemoattractant before the chemotactic response is reset by methylation adaptation.

Figure 3

A block diagram of the chemotaxis network for $K_{ia}\ll L\left[x\left(t\right),y\left(t\right)\right]\ll K_a$. Here, $w\left(t\right)=u\left(t\right)+\alpha N\left[m_o-m\left(t\right)\right]$.

Figure 4

Probability density of the initial condition of state ($a_o$) for different values of $r$. The ${10}^6$ runs of ${10}^5$ bacteria for every $r$ are simulated for estimating the probability density.

Figure 5

Schematic of the premixing microchannel in which "$D$" and "$B$" reservoirs represent the inlets of the dextrose and bacterial solutions, respectively. The inset shows that an exponential ligand concentration profile is generated along the width of the microchannel. An image of the experimental settings is shown in Fig. 12 of the Appendix.

Figure 6

Exponential ligand concentration in the premixing microchannel. (a) Microbead concentrations in the premixing microchannel for different values of ramp $r$. The marked region (i.e., oval-shaped region) shows an increasing concentration of beads and (b) ligand (dextrose) concentration in the premixing microchannel: The spatiolocalizations of the dextrose molecules are given by dots, and the least-squares fitted curves also are shown to represent the smooth spatial exponential ramp ligand concentration $L(x,y)=L_o{e}^{rx}$ for different values of ramp ($r$).

Figure 7

Bacterial trajectory for different values of $r$. (a) Experimental [(b) simulated] trajectories with time lapses of 233 s (235 s) and 152 s (137 s) are considered for $r=2$ and $r=6$, respectively. Algorithm 2 in the Supplemental Material [34] is followed to generate a simulated bacterial trajectory.

Figure 8

Theoretical and experimental run-time (in seconds) contours with respect to ${\theta }_k$ for different values of $r$ ($a_o=a^{*}$). Here, the means of the experimental run times are presented in this contour at every ${\theta }_k$ for particular values of $r$. The untethered E. coli movement is observed in a microfluidic chemical gradient by a microscope. The tumbling angle and position are measured from the untethered bacterial trajectory using image processing. The run-time $\tau (a_o^k,{\theta }_k,r)=\frac{v}{\sqrt{{(x_{k+1}-x_k)}^2+{(y_{k+1}-y_k)}^2}}$ is calculated from the trajectory, and the polar plot of the run time is drawn for particular values of $r$. The theoretically derived run-time contour with respect to ${\theta }_k$ for different values of $a_o$ (for constant $r$) is shown in Fig. 11 of the Appendix.

Figure 9

Drift velocity ($v_d$) for different values of $r$. The theoretical and simulated estimates $v_d$ are compared with experimental estimated $v_d$. The error bars of the graph are defined by μ ± σ, where, μ and σ are the mean and the standard deviation of drift velocities as collected from the repeated sets of experiments. For this analysis, 57 experimental observations and 500 simulation observations are considered for every $r$. For simulation, only the mean results (•) are displayed. The results also are compared with the two empirical models of Jiang et al. [17] and Reneaux and Gopalakrishnan [32] (for details please refer to Sec. 2 of the Appendix).

Figure 10

The probability distribution function of tumbling angle (θ) in 2D space [33].

Figure 11

Theoretical run-time (τ in seconds) contours for different values of $a_o\left(r=4\right)$.

Figure 12

(a) An image of the experimental settings in which the glass-polydimethylsiloxane based microchannel is connected with an external tubing for the controlled pressure-driven flow experiments. (b) A zoomed image of the microarchitecture of the premixing channel.