Quantitative modeling of Escherichia coli chemotactic motion in environments varying in space and time

Abstract

Escherichia coli chemotactic motion in spatiotemporally varying environments is studied by using a computational model based on a coarse-grained description of the intracellular signaling pathway dynamics. We find that the cell's chemotaxis drift velocity v(d) is a constant in an exponential attractant concentration gradient [L] proportional, variantexp(Gx). v(d) depends linearly on the exponential gradient G before it saturates when G is larger than a critical value G(C). We find that G(C) is determined by the intracellular adaptation rate k(R) with a simple scaling law: G(C) infinity k(1/2)(R). The linear dependence of v(d) on G = d(ln[L])/dx directly demonstrates E. coli's ability in sensing the derivative of the logarithmic attractant concentration. The existence of the limiting gradient G(C) and its scaling with k(R) are explained by the underlying intracellular adaptation dynamics and the flagellar motor response characteristics. For individual cells, we find that the overall average run length in an exponential gradient is longer than that in a homogeneous environment, which is caused by the constant kinase activity shift (decrease). The forward runs (up the gradient) are longer than the backward runs, as expected; and depending on the exact gradient, the (shorter) backward runs can be comparable to runs in a spatially homogeneous environment, consistent with previous experiments. In (spatial) ligand gradients that also vary in time, the chemotaxis motion is damped as the frequency omega of the time-varying spatial gradient becomes faster than a critical value omega(c), which is controlled by the cell's chemotaxis adaptation rate k(R). Finally, our model, with no adjustable parameters, agrees quantitatively with the classical capillary assay experiments where the attractant concentration changes both in space and time. Our model can thus be used to study E. coli chemotaxis behavior in arbitrary spatiotemporally varying environments. Further experiments are suggested to test some of the model predictions.

Figure 1. Illustrations of the E. coli chemotaxis pathway and the SPECS (Signaling Pathway-based E. coli Chemotaxis Simulator) model.

(A) The E. coli chemotaxis signal transduction pathway. The MCP complex receptor-CheW-CheA is the sensor and can be active (green) or inactive (dark brown). Binding of attractant molecules (yellow) decreases the probability of the receptor to be active. Once activated, the histidine kinase CheA quickly autophosphorylates and then transfers the phosphate group to CheY. CheY-p, the response regulator, diffuses to the flagellar motor and modulates its switching probability between CW and CCW rotations. CheZ, the CheY-p phosphatase, serves as the sink of the signal. Adaptation of the system is carried out by methylation and demethylation of the receptor, which are facilitated by the enzymes, CheR and CheB-p, respectively. (B) Flow chart of the SPECS model (reproduced from Figure 3 in [21]). The input for the signaling pathway (inside the box) is the instantaneous ligand concentration . The internal signaling pathway dynamics is described at the coarse-grained level by the interactions between the average receptor methylation level and the kinase activity , which eventually determines the switching probability of the flagellar motor . The switching probability is then used to determine the cell motion (run or tumble), and direction of motion during run fluctuates due to rotational diffusion. The motion of the cell leads it to a new location with a new ligand concentration for the cell and the whole simulation process continues.

Figure 2. Comparison of chemotactic motions in exponential and linear ligand profiles.

(A) Cell motion and intracellular dynamics in exponential ligand concentration profiles: . (B) Cell motion and intracellular dynamics in linear ligand profiles: . In both (A) & (B), the dynamics of the (population) averaged position (); the average receptor methylation level () and the average kinase activity are shown for different decay lengths . . The population-averaged position increases linearly with time until the methylation level reaches saturation in exponential profiles; while it slows down continuously in the linear profiles. After a transient decrease, the kinase activity stays roughly constant in exponential profiles, while it varies continuously recovering to its pre-stimulus level in linear profiles. (C) Direct comparison of instantaneous velocities between exponential (solid lines) and linear (dotted lines) profiles for (black); 1.0 mm (purple). Within the chemosensitivity range , the instantaneous chemotaxis drift velocity is constant in exponential profiles, while it decreases continuously with [L] in the linear ligand profiles.

Figure 3. Dependence of chemotaxis motion on the adaptation rate.

(A) The chemotaxis drift velocity for different exponential gradient . Different symbols represent different adaptation rates . Note that first increases linearly (dashed line) with before reaching a saturation velocity at a critical gradient ,. We can fit with: , in which is the chemotaxis motility constant given by the linear fitting coefficient and the saturation drift velocity is . The dependences of and on are shown in (B) and (C) respectively. For the range of we studied, we found that is roughly independent of and depends on with a simple scaling relation: .

Figure 4. Single cell behaviors in exponential attractant gradients.

(A) Trajectory of a cell for 10 min for with . The random walk motion is biased towards the gradient (arrow). The forward runs up the gradient are in red, and the backward runs down the gradient are in black. (B) Run length distribution for forward runs (red), backward runs (black) in the presence of an exponential gradient and without a gradient (purple). The backward run length distribution is close to the run length distribution in the absence of a gradient, similar to the experimental results from Berg and Brown as reproduced in the inset. (C) Distributions of cell kinase activity for forward (red) and backward (black) runs. A time series of the activity of a single cell is shown: forward is in red and backward is in black. The average activity for backward (black dashed line) is closer to the adapted activity (purple dashed line, 0.5) compared with the average activity for forward (red dashed line). (D) Methylation level of different individual cells at different times and in different exponential gradients (represented by color symbols as in Figure 2) all increase with logarithmic ligand concentration along a universal line, despite large temporal fluctuations in methylation levels and positions for (two) individual cells as shown in the inset.

Figure 5. Responses to oscillating linear gradient.

(A) Time dependence of the average positions of cells for three oscillatory gradients, all with the same amplitude but different frequencies (ω). The responses have the same frequencies as their driving signals, but the response amplitude decreases with the driving frequency. (B) The amplitude of the response decreases with frequency ω. The cross-over at low frequency () is caused by boundary effects. (C) Upon decreasing adaptation rate, a transition to a steeper decay of the amplitude appears at frequencies higher than a transition frequency within the range of frequencies studied. Three cases with smaller values of are shown, and the dependence of on the adaptation rate is shown in the inset of (C).

Figure 6. Quantitative simulation of the classical capillary assay and comparison with experiments.

(A) Time-dependent ligand concentrations at three different positions in the suspension pool (see inset) from directly solving the ligand diffusion equation. The ligand concentration at a given position peaks at a given time, depending on its location. . (B) The exact ligand profile (solid line) along the center line of the capillary at different times, in comparison with the asymptotic solutions (dashed lines) by Furtelle and Berg . (C) Cell density in the rectangular coordinate is shown together with the contours of the logarithmic ligand concentration (in ) at different times. Three individual cell trajectories (starting from circles and ending at squares) are shown. Only the black cell ends in the capillary. (D) Probability distribution of the original positions of cells that end in the capillary. For a cell originally located at position , the probability of it ending in the capillary at a later time (45 min), , is shown. , . (E) Concentration-response curve for the capillary assay. The average number of bacteria in the capillary after 45–50 min subtracted by the number of bacteria in the capillary in the absence of attractant is defined as the response (ordinate). The results from our model with the exact ligand profile are labeled by solid symbols (fitted by a solid line). They agree well with the experimental measurements (hollow squares) of Mesibov et al ,. The results from using the asymptotic ligand profile by Furtelle and Berg are shown by the dashed lines. (F) Response curve for capillary assay with . The solid symbols (fitted with a solid line) represent the model, and the hollow symbols represent the experimental results (both collected at 60 min).