The bacterial chemotactic response reflects a compromise between transient and steady-state behavior

Abstract

Swimming bacteria detect chemical gradients by performing temporal comparisons of recent measurements of chemical concentration. These comparisons are described quantitatively by the chemotactic response function, which we expect to optimize chemotactic behavioral performance. We identify two independent chemotactic performance criteria: In the short run, a favorable response function should move bacteria up chemoattractant gradients; in the long run, bacteria should aggregate at peaks of chemoattractant concentration. Surprisingly, these two criteria conflict, so that when one performance criterion is most favorable, the other is unfavorable. Because both types of behavior are biologically relevant, we include both behaviors in a composite optimization that yields a response function that closely resembles experimental measurements. Our work suggests that the bacterial chemotactic response function can be derived from simple behavioral considerations and sheds light on how the response function contributes to chemotactic performance.

Fig. 1.

Comparison of the two performance calculations. (Upper) The integration in the expression for 𝒯 (Eq. 6). Two bacteria that have both just tumbled are considered as they move in different directions along the gradient until they tumble again at position x(t_f). (Lower) The integration in the expression for 𝒮 (Eq. 13). In this case, two bacteria meet that last tumbled at points x(t₀). One finds the expectation of their respective tumbling probabilities, P̄^±, by averaging over possible histories.

Fig. 2.

Cartoon showing origin of transient velocity. Solid lines indicate possible paths taken by bacteria that all execute exactly average paths; line thickness gives a sense of the probability weighting of each path segment. The chemoattractant gradient in this case is positive, and t̄⁺ > t̄^-. The dotted line shows the average position over time: It moves to the right, indicating an expected velocity up the gradient. Note that after the time elapsed in this figure, more bacteria on average will have reached the farthest right point than the farthest left point, because they have tumbled less frequently.

Fig. 3.

Simulations of the model. We performed discrete time simulations of the model on a positive concentration gradient with reflective boundary conditions to see the result of different R(t) on transient and steady-state behaviors. (a) Bacteria were released from the center of the gradient (Left) and evolved until they arrived at a steady-state distribution. R(t) was chosen to weight positively only at θ seconds before the current time, t (Right); that is, it weights only c(t - θ). It was further chosen so that the maximum perturbation from the average tumbling probability was 30%. (b) In a gradient of length , bacterial distributions and the mean position of bacteria were found by using a response function with , where is the run duration averaged over the box. At early times, bacteria are clustered and have a mean velocity up the gradient. After the bacteria hit the boundary, they approach a steady state peaked at low c. Note that more bacteria have reached the right-hand wall than the left-hand wall at . For this response function, and ; both results are reflected in the bacterial behavior. (c) We varied θ and calculated 𝒯 from the initial slope of the lower plot in b. The result shows the contribution of R(θ) to 𝒯. The solid line is the transient performance kernel, , derived in the text. (d) In a short-length scale gradient (), we varied θ and calculated 𝒮 from the bacterial distributions at long times. The contribution of R(θ) to 𝒮 is shown. The solid line is the steady-state performance kernel, , derived for a similar situation (see the supporting information). Error bars in c and d are 1 SEM.

Fig. 4.

Optimized response functions and comparison with data. (a) Response functions that optimize the performance measures 𝒯, 𝒮, and , where A = 1/2. Note that all three functions are normalized such that . (b) The points are data from figure 1 of ref. showing the counterclockwise bias in flagellar motor rotation after a very short impulse of chemoattractant at time t = 0. The bias response is linear in this experiment's regime. The solid line is a best fit of to the data, using a 10-Hz low-pass Gaussian filter to realistically smooth discontinuities. The fitting parameters were A, τ, and an overall amplitude, and the least-squares fit was A = 0.56 and τ = 0.9 s. The bias of a single flagellum is related to the tumbling probability P[x(t′); t] but is not identical, because multiple flagella are involved in running and tumbling (4), and cooperative effects could be involved.