Inferring the Chemotactic Strategy of P. putida and E. coli Using Modified Kramers-Moyal Coefficients

Abstract

Many bacteria perform a run-and-tumble random walk to explore their surrounding and to perform chemotaxis. In this article we present a novel method to infer the relevant parameters of bacterial motion from experimental trajectories including the tumbling events. We introduce a stochastic model for the orientation angle, where a shot-noise process initiates tumbles, and analytically calculate conditional moments, reminiscent of Kramers-Moyal coefficients. Matching them with the moments calculated from experimental trajectories of the bacteria E. coli and Pseudomonas putida, we are able to infer their respective tumble rates, the rotational diffusion constants, and the distributions of tumble angles in good agreement with results from conventional tumble recognizers. We also define a novel tumble recognizer, which explicitly quantifies the error in recognizing tumbles. In the presence of a chemical gradient we condition the moments on the bacterial direction of motion and thereby explore the chemotaxis strategy. For both bacteria we recover and quantify the classical chemotactic strategy, where the tumble rate is smallest along the chemical gradient. In addition, for E. coli we detect some cells, which bias their mean tumble angle towards smaller values. Our findings are supported by a scaling analysis of appropriate ratios of conditional moments, which are directly calculated from experimental data.

Statistical parameter inference of bacterial swimming strategies

“…This validates our inference method. Moreover, our findings are in good agreement with previously measured tumble rates [3,9,22] and swimming speeds [38]. The inferred value for the thermal rotational diffusivity D 0 agrees with previously reported values in the literature, which range from 0.06 s −1 [20,22] to 0.18 s −1 [39].…”

“…The two inferred tumble rates λ are very close together and the inverse tumble times r agree within the error bars. Our results are in good agreement with tumble rate λ=0.84 s −1 and swimming velocity m = v 20.7 ms 0 1 determined with a heuristic tumble recognizer (see section 3.4 and [22]). This validates our inference method.…”

“…Instantaneous speed = D D v s t , direction of propagation θ, and turning rate w = q D Dt were evaluated on the smoothed tracks. The tumble events were detected as described previously [9,22,37]. Briefly, in the time series of speed and turning rate, local minima and local maxima were detected, respectively, to identify tumble events.…”

“…Briefly, in the time series of speed and turning rate, local minima and local maxima were detected, respectively, to identify tumble events. Four parameters, two for the speed and two for the turning rate, were adjusted such that the recognition of tumble events was correct as checked by visual examination (threshold parameters α=3 and β=6.5 and tumble duration parameters 0.55×Δv and 0.65×Δω, see the supporting information S5 in [22]).…”

“…Typically, the tumble phase is much shorter than the run phase [1]. Therefore, in theoretical models tumbling is considered as instantaneous and a single event is described by a tumble angle drawn from a distribution [8,[20][21][22]. However, a recent and instructive work by Saragosti et al showed that reorientation during tumbling can be modeled by enhanced rotational diffusion [23].…”