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.

Fig 1. Flow diagram of the CM method.

As input one provides the model of bacterial motion summarized in Eqs (1)–(4) and a sufficient number of experimental trajectories. Then, the theoretical CMs m_theo(θ,p) are calculated from the model as a function of the parameters p and the current orientation angle θ [see Eqs (13)–(18)] The experimental CMs m_exp(θ) are determined using Eq (21). Matching these moments with a least square fit yields as an output the parameters of the model: p(θ) = arg min_p|m_exp(θ) − m_theo(p)|². Furthermore, starting from the model and using the inferred parameters, we introduce a new tumble recognizer and test it against a heuristic tumble recognizer. Finally, the CM ratios are obtained directly from the experimental CMs and are used to identify the angle bias.

Fig 2. Schematics of a bacterial tumble event.

E. coli moves in direction Θ(t), tumbles at time t + Δt, and moves in the new direction Θ(t + Δt). Thus, the turning angle becomes |Θ(t + Δt) − Θ(t)|_a.

Fig 3. Experimental setup.

Left: Layout of the chemotaxis chamber. Attractant reservoir is on the left, cell reservoir on the right. The central gradient region is marked in blue, its height is 70 μm, much less than the height of the reservoir chambers. Because of the significantly larger volume in the chambers, a linear gradient establishes after filling and is maintained for several hours. Marked in red is the field of view imaged by the microscope. Right: Temporal evolution of the chemical gradient profile after filling the channel, measured from the spatial profile of fluorescein. Since fluorescein has about twice the molecular weight of α-methyl-aspartate and thus a larger diffusion coefficient, we assume that the gradient evolution measured for fluorescein is similar or slightly slower than the gradient of the chemoattractant.

Fig 4. Distribution of tumble angles, P(|β|).

It is determined from experiments by the heuristic tumble recognizer (bar graph) and by the inference method with the gamma function γ(σ, k) as an ansatz (solid red line). All recorded trajectories at 30, 45, 60, and 95 min with at least one tumble are used. The mean tumble angle and the standard deviation are 〈|β|〉 = 0.42π = 76.0°, Δ|β| = 0.27π = 48.7° (heuristic tumble recognizer) and 〈|β|〉 = 0.47π = 85.4°, Δ|β| = 0.23π = 41.8° (inference method). The inferred parameters of γ(σ, k) are σ = 0.64 and k = 2.73. The red dashed line refers to the inferred gamma distribution (σ = 0.78 and k = 2.15), when the original data is smoothed. The blue dashed line refers to the histogram values multiplied by sin(|β|) and then normalized to one, thus representing the tumble angle distribution in three dimensions.

Fig 5. Definition and analysis of systematic tumble recognizer.

(a) Distributions for tumble angles, P(|dΘ|) (blue line), and for Brownian displacements, $2\mathcal{N}\left(\right|d\Theta \left|\right)$ (green line), as inferred from experimental data. The shaded blue area corresponds to the type-I error α₁, while the green area refers to the type-II error α₂. The dashed line marks the threshold dΘ_crit. (b) Smoothed sample trajectory with tumbles marked as green circles. They are obtained by a heuristic tumble recognizer (see S5 Text). Endpoints are not classified and therefore black. (c) Rational tumble recognition on the basis of a hypothesis test with the likelihood ratio using the inferred distributions P(|β|) and $\mathcal{N}(d\Theta )$. Faint orange points are the original trajectory points, colored fat points are trajectory points with time gap Δt = 0.5.

Fig 6. Tumbling statistics with chemotaxis.

(a), (c) The mean tumble rate λ (red) and the mean tumble angle 〈|β|〉 (blue) plotted versus the orientation angle θ of the bacterium prior to tumbling for (a) the late tracks at T = 30, 45, 60, and 95 min and for (c) the early tracks at T = 7 and 12 min. The tumble rate is fitted by a cosine function. The dashed blue line marks 〈|β|〉(θ) averaged over all directions. (b), (d) Ratios of CMs, $m_{\Delta t}^n\left(\theta \right)/m_{\Delta t}^n\left(\pi \right)$, plotted versus power n for different orientation angles θ for (b) the late tracks and for (d) the early tracks.

Fig 7. Early bacterial tracks analyzed separately in different parts of the channel.

Left column: The mean tumble rate λ (red) and the mean tumble angle 〈|β|〉 (blue) plotted versus the orientation angle θ prior to tumbling for (a) the left, (c) the middle, and (e) the right part. The blue dashed line marks 〈|β|〉(θ) averaged over all directions. Right column: Ratios of CMs, $m_{\Delta t}^n\left(\theta \right)/m_{\Delta t}^n\left(\pi \right)$, plotted versus power n for different orientation angles θ for (b) the left, (d) the middle, and (f) the right part.

Fig 8. Tumbling statistics of P. putida.

(a) Distribution of tumble angles, P(|β|), determined by the heuristic tumble recognizer (bar graph) and by the inference method (red line). The blue dashed line refers to the histogram values multiplied by sin(|β|) thereby representing the tumble angle distribution in three dimensions. The mean tumble angle and the standard deviation are 〈|β|〉 = 0.75π = 135°, Δ|β| = 0.29π = 52.2° (heuristic tumble recognizer) and 〈|β|〉 = 0.72π = 130°, Δ|β| = 0.26π = 46.8° (inference method). (b) The mean tumble rate λ (red) and the mean tumble angle 〈|β|〉 (blue) plotted versus θ. The tumble rate is fitted by a cosine function.