Figure 1.
Dynamical coupling between the sensor and the actuator in the bacterial chemotaxis system.
A. The bacterial chemotaxis system is composed of a sensor module (receptor-kinase complexes) and an actuator module (flagellar motors) coupled through the phosphorylated form of CheY. Both modules are ultra-sensitive and adapt to their respective input signals. Maintaining the output of the sensor within the right range relative to the actuator is critical for chemotaxis performance. B. Diagrams of the CheY-P concentration response to different signals. First line: when cells are immobilized onto a slide, a step stimulus of attractant (e.g. methylaspartate) causes a sudden decrease in CheY-P concentration followed by a slower adaptation. Because of the negative integral feedback architecture of the sensor module, CheY-P adapts back to its pre-stimulus level, the adapted CheY-P concentration, Y0. Second line: when immobilized cells are exposed to an exponential ramp in time of the same stimulus, the system, which is log sensing, experiences a constant “force” and adapts towards an operational CheY-P concentration, Ym, lower than the adapted level Y0. This deviation of CheY-P activity from Y0 to Ym changes the coupling between sensor and actuator. Third line: when cells are swimming in a gradient of attractant, their biased random walk causes them to climb the gradient. The average drift velocity of the cell up a chemical gradient affects the average input signal experienced by the cell. This creates a feedback of the behavior onto the input signal, which in turn can significantly affect the operating concentration of CheY-P and thus the coupling between sensor and actuator.
Figure 2.
Simulated and theoretical drift velocity VD in exponential gradient of aspartate L0egx.
A. VD as a function of the adapted CheY-P concentration Y0, in a shallow gradient (L0 = 200 µM and g−1 = 5,000 µm) for cells with adaptation times τ = 5 (blue), 10 (green), and 30 seconds (red). VD is the average velocity of 10,000 identical cells between t = 60 and 300 seconds (dots: stochastic simulations; lines: analytical solution from Eq. (3); grey: motor CW bias response curve. B. Expected trajectories of CheY-P concentration Y(F(t)) for cells running in one dimension up (green) or down (red) in a gradient (integration of Eqs. (2) and (5), see Text S1; τ = 30 s, g−1 = 5,000 µm, Y(Fi) = 2.4 µM and 3 µM). Expected run, (dotted line), and tumble,
(dashed line), durations as a function of Y0. Expected run duration along a given direction τR0 = (2Dr+λR0)−1 (solid black line) is limited by rotational diffusion (Dr = 0.062 rad2 s−1). Grey: motor CW bias. C. Same as A (τ = 10 s) but with the rotational diffusion constant Dr = 0.031 (red), 0.062 (green), and 0.124 (blue) rad2 s−1. Dotted lines: expected run duration in a given direction. D. Same as A (τ = 10 s) but with the motor switching rate ω = 2.6 (red), 1.3 (green), and 0.65 (blue) s−1. Dotted lines: expected run duration in a given direction.
Figure 3.
Feedback of the behavior of cells swimming in exponential gradients onto the operational CheY-P concentration.
A. Temporal profiles of the average methyl-aspartate concentration encountered by cells swimming in a steep exponential gradient (g−1 = 1,000 µm). Different phenotypes are considered (solid black: Y0 = 2.4 µM, τ = 10 s, solid gray: Y0 = 2.4 µM, τ = 30 s, dotted black: Y0 = 3 µM, τ = 10 s) (the y-axis is on a log scale). B. Corresponding average CheY-P concentration as a function of time in these same cells C. Magnitude of the drop in average CheY-P activity (difference between adapted and operational CheY-P concentrations Ym -Y0) as a function of the drift velocity. Two different adaptation times are considered (black: τ = 10 s, grey: τ = 30 s). The gradient is the same gradient as in panel A. Dots are averages over 10,000 stochastic simulations for populations with different adapted CheY-P concentrations (Y0>2.4 µM in both cases). Lines are from Eq. (4). D. Drift velocity VD as a function of adapted CheY-P concentration, Y0 (filled circles), and operational CheY-P concentration, Ym (open circles) in stochastic simulations (average over 10,000 replicates for each circle, τ = 10 s). Ym is instantaneous CheY-P concentration averaged over the population while drifting between t = 60 and 300 s). Two exponential gradients of methyl-aspartate are considered (g−1 = 1,000 µm (black), 5,000 µm (grey)). Black arrow: cell population in blue in Figure 4C.
Figure 4.
Behavioral feedback can create a chemotactic “trap”.
A. Analytical drift velocity, VD as a function of Ym (Eq. (3) with 0→m; solid line) and feedback of VD on Ym (Eq. (4), dashed line) (τ = 5 s, g−1 = 1,000 µm, Y0 = 2.7 µM). Steady state drift velocity (Ym = 2.48 µM, black circle). B. Same as panel A, but for cells with longer adaptation time and higher adapted CheY-P (τ = 30 s, Y0 = 3.5 µM). VD has three possible steady states: two stable (Ym2 = 2.1 µM and Ym1 = 3.49 µM (black dots)), and one unstable (Ym = 2.97 µM, white dot). C. Individual drift velocities (in the direction of the gradient) and root mean square displacements (perpendicular to the gradient) of two different populations of 10,000 simulated cells (blue: τ = 10 s, Y0 = 2.6 µM, red: τ = 30 s, Y0 = 3 µM). D. Average VD as a function of Y0 (filled circles) and Ym (open circles) for cells with a long adaptation time (τ = 30 s). Black arrow: cell population plotted in panel C (red).
Figure 5.
Effect of motor adaptation on drift velocity VD in exponential gradients.
A. Motor CW bias response curve as function of CheY-P concentration when the motor is allowed to adapt (solid line) fitted to data from [19] (circles; derivation in Materials and Methods). B. Average drift velocity as a function of operational CheY-P concentration Ym, in a shallow gradient. Same adaptation times and gradient steepness as Figure 2A. Lines: analytical solutions; circles: stochastic simulations (averages between t = 10 and 15 min are used to calculate VD (Ym)). C. Same as Figure 4B but with motor adaptation. The drift velocity has only one stable steady sate (Ym = 1.6 µM, black dot). Motor adaptation eliminated the other states present in Fig. 4B.
Figure 6.
Performance trade-off in bacterial chemotaxis.
A. Optimal adapted CheY-P concentrations Y0 (solution of Eqs (4) with 0→m and (5)) as a function of the chemoreceptor adaptation time in different exponential gradients (g−1 = 1,000 (red), 2,000 (green), and 5,000 (blue) µm). Dots indicate when the maximal theoretical drift velocities cross the bifurcation point (dotted lines represent the inaccessible optimal state). The optimal operational CheY-P concentration Ym is identical for all gradient length scales (black dashed line). B. Contour plot of drift velocities as a function of adaptation time and the adapted cell tumble bias in different exponential gradients (same colors as A). 75%, 90%, and 95% contours of the maximal theoretical drift velocities for each gradient (colors intensities from light to dark). Black dot: the best cell phenotype that achieves equal relative drift velocities in all three gradients (60% of the maximal VD with τ = 7.5 s and TB0 = 0.044).