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

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.Author Summary: The movement strategies of bacteria have received increasing attention over the past decade, in particular with respect to the tracking of individual cells and the mathematical description of the resulting trajectories. Bacteria typically move in almost straight runs interrupted by sharp turning events (run-and-tumble). In order to characterize their motion on a single cell level, the tumble events in individual trajectories have to be identified. Traditionally, tumble recognition relies on threshold values that are applied to the swimming speed and the reorientation angle. They are chosen in an ad hoc fashion and introduce a certain degree of arbitrariness to the results of statistical motion analyses. Here, we propose a new stochastic model for the orientation angle of a bacterium and formulate conditonal moments, which we determine both in theory and from experimental trajectories. This provides an alternative way of quantifying the bacterial run-and-tumble strategy and of recognizing tumble events. Our approach no longer relies on arbitrarily chosen segmentation thresholds and rigorously quantifies the uncertainty in tumble recognition. We successfully apply our method not only to the paradigmatic case of E. coli but also to trajectories of the soil bacterium Pseudomonas putida, demonstrating that our approach provides a novel way to reliably characterize the tumbling statistics and chemotaxis strategies of bacterial swimmers across different species.