Traveling concentration pulses of bacteria in a generalized Keller–Segel model

We formulate a Markovian response theory for the tumble rate of a bacterium moving in a chemical field and use it in the Smoluchowski equation. Based on a multipole expansion for the one-particle distribution function and a reaction-diffusion equation for the chemoattractant field, we derive a polarization extended model, which also includes the recently discovered angle bias. In the adiabatic limit we recover a generalized Keller–Segel equation with diffusion and chemotactic coefficients that depend on the microscopic swimming parameters. Requiring the tumble rate to be positive, our model introduces an upper bound for the chemotactic drift velocity, which is no longer singular as in the original Keller–Segel model. Solving the Keller–Segel equations numerically, we identify traveling bacterial concentration pulses, for which we do not need a second, signaling chemical field nor a singular chemotactic drift velocity as demanded in earlier publications. We present an extensive study of the traveling pulses and demonstrate how their speeds, widths, and heights depend on the microscopic parameters. Most importantly, we discover a maximum number of bacteria that the pulse can sustain—the maximum carrying capacity. Finally, by tuning our parameters, we are able to match the experimental realization of the traveling bacterial pulse.DFG, 87159868, GRK 1558: Kollektive Dynamik im Nichtgleichgewicht: in kondensierter Materie und biologischen SystemenDFG, 414044773, Open Access Publizieren 2019 - 2020 / Technische Universität Berli

Statistical parameter inference of bacterial swimming strategies

We provide a detailed stochastic description of the swimming motion of an E.coli bacterium in two dimension, where we resolve tumble events in time. For this purpose, we set up two Langevin equations for the orientation angle and speed dynamics. Calculating moments, distribution and autocorrelation functions from both Langevin equations and matching them to the same quantities determined from data recorded in experiments, we infer the swimming parameters of E.coli. They are the tumble rate λ, the tumble time r−1, the swimming speed v0, the strength of speed fluctuations σ, the relative height of speed jumps η, the thermal value for the rotational diffusion coefficient D0, and the enhanced rotational diffusivity during tumbling DT. Conditioning the observables on the swimming direction relative to the gradient of a chemoattractant, we infer the chemotaxis strategies of E.coli. We confirm the classical strategy of a lower tumble rate for swimming up the gradient but also a smaller mean tumble angle (angle bias). The latter is realized by shorter tumbles as well as a slower diffusive reorientation. We also find that speed fluctuations are increased by about 30% when swimming up the gradient compared to the reversed direction.DFG, 87159868, GRK 1558: Kollektive Dynamik im Nichtgleichgewicht: in kondensierter Materie und biologischen Systeme

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.DFG, 87159868, GRK 1558: Kollektive Dynamik im Nichtgleichgewicht: in kondensierter Materie und biologischen Systeme

Statistische Physik der bakteriellen Chemotaxis

Das Darmbakterium Escherichia coli (E.coli ) ist in den Bereichen der Molekularbiologie, der Zellbiologie und des Forschungsfeldes der Mikroschwimmer eines der am besten erforschten Lebewesen. Die vorliegende Arbeit leistet einen Beitrag zur Grundlagenforschung der Chemotaxis von E.coli (gerichtete Bewegung entlang eines externen chemischen Stimulus). Insbesondere werden neue theoretische Modelle sowohl für die individuelle als auch für die kollektive Chemotaxis von E.coli entwickelt. Im ersten Teil führen wir eine stochastische Beschreibung der Schwimmbewegung eines individuellen Bakteriums ein. Hierfür stellen wir zwei überdämpfte Langevin-Gleichungen für die Dynamiken des Orientierungswinkels sowie für den Betrag des Geschwindigkeitsvektors auf. Für beide Gleichungen berechnen wir die zugehörigen Momente, Wahrscheinlichkeitsverteilungen und Autokorrelationsfunktionen. Im Experiment erzeugen moderne Aufnahmetechniken große Datenmengen an bakteriellen Trajektorien. Der Vergleich der oben genannten stochastischen Größen aus Theorie und Experiment erlaubt die Bestimmung der Modellparameter. Im Einzelnen sind dies die Taumelrate, die Taumelzeit, der Betrag der Schwimmgeschwindigkeit, die Stärke mit der die Geschwindigkeit fluktuiert, die relative Höhe der Geschwindigkeitssprünge, der thermische Wert für die Rotationsdiffusionskonstante und der erhöhte Wert der Letzteren während eines Taumels. Gegenüber etablierten Analysemethoden hat unsere Methode den Vorteil, dass keinerlei a priori Parameter gewählt werden müssen. Durch Berechnung der bedingten Wahrscheinlichkeiten für verschiedene Schwimmrichtungen können wir die Chemotaxisstrategien von E.coli untersuchen. Wir bestätigen die klassische Strategie einer Reduzierung der Taumelrate und die kürzlich entdeckte Verringerung des mittleren Taumelwinkels (angle bias) bei Hinaufschwimmen eines Lockstoffgradienten. Wir zeigen, dass letztere Strategie sowohl durch kürzere Taumelzeiten als auch durch eine kleinere Diffusivität der Orientierung verursacht wird. Wir beobachten zudem, dass die Geschwindigkeitsfluktuationen in Richtung des Gradienten 30% höher sind als bei entgegengesetzter Schwimmrichtung. Zuletzt untersuchen wir die chemotaktische Antwortfunktion des Bodenbakteriums Pseudomonas putida. Im zweiten Teil formulieren wir eine Markov’sche lineare Antwort für die Taumelrate eines Bakteriums, das in einem chemischen Feld schwimmt. Diese Relation für die Taumelrate setzen wir in die Smoluchowski-Gleichung ein. Nach einer Multipol-Entwicklung erhalten wir ein verallgemeinertes Keller-Segel-Modell, das auch den angle bias beinhaltet. Wir bestimmen die Diffusions- und Chemotaxis-Koeffizienten als Funktion der mikroskopischen Schwimmparameter. Wir sehen, dass für eine positive Taumelrate die chemotaktische Driftgeschwindigkeit beschränkt ist. Damit beheben wir die Singularität des ursprünglichen Keller-Segel-Modells. Numerische Lösungen unserer beschränkten Keller-Segel-Gleichungen zeigen bakterielle Dichtepulse. Wir zeigen somit, dass man für einen bakteriellen Puls weder eine singuläre Driftgeschwindigkeit noch einen zweiten Lockstoff braucht. Wir führen eine quantitative Parameterstudie des Pulses durch und bestimmen wie Pulsgeschwindigkeit, -breite und -höhe von den mikroskopischen Parametern abhängen. Dabei entdecken wir eine maximale Anzahl an Bakterien, die sich im Puls befinden können. Zuletzt zeigen wir, dass unser Modell den experimentellen Dichtepuls quantitativ sehr gut beschreibt.This thesis introduces theoretical frameworks for the individual and collective chemotaxis of Escherichia coli (E.coli ). In the first part, we describe the swimming motion of an individual bacterium. For this purpose, we set up two Langevin equations for the orientation angle and speed dynamics. We calculate corresponding moments, distribution and autocorrelation functions for both equations. Matching them to the same quantities determined from experimentally recorded trajectories, we infer the swimming parameters of E.coli. These are the tumble rate, the tumble time, the swimming speed, the strength of speed fluctuations, the relative height of speed jumps, the thermal value for the rotational diffusion coefficient, and the enhanced rotational diffusivity during tumbling. In contrast to established analysis methods by a tumble recognizer algorithm, our inference method does not rely on a priori parameters. Conditioning the observables on the swimming direction relative to the gradient of a chemoattractant, we infer the chemotaxis strategies of E.coli. We confirm the classical strategy of a lower tumble rate for swimming up the gradient but also a smaller mean tumble angle (angle bias). The latter is realized by shorter tumbles as well as a slower diffusive reorientation. We also find that speed fluctuations are increased by about 30% when swimming up the gradient compared to the reversed direction. Finally, we investigate the chemotactic response function of the soil bacterium Pseudomonas putida. In the second part, we formulate a Markovian response theory for the tumble rate of a bacterium moving in a chemical field and use it in the Smoluchowski equation. Based on a multipole expansion for the one-particle distribution function and a reaction-diffusion equation for the chemoattractant field, we derive a polarization extended model, which also includes the recently discovered angle bias. In the adiabatic limit we recover a generalized Keller-Segel equation with diffusion and chemotactic coefficients that depend on the microscopic swimming parameters. Requiring the tumble rate to be positive, our model introduces an upper bound for the chemotactic drift velocity, which is no longer singular as in the original Keller-Segel model. Solving the Keller-Segel equations numerically, we identify traveling bacterial concentration pulses, for which we need neither a second, signaling chemical field nor a singular chemotactic drift velocity, as demanded by earlier publications. We present an extensive study of the traveling pulses and demonstrate how their speeds, widths, and heights depend on the microscopic parameters. Most importantly, we discover a maximum number of bacteria that the pulse can sustain - the maximum carrying capacity. Finally, by tuning our parameters, we are able to match the experimental realization of the traveling bacterial pulse.DFG, GRK 1558, Nonequilibrium Collective Dynamics in Condensed Matter and Biological SystemsDFG, SFB 910, Control of self-organizing nonlinear systems: Theoretical methods and concepts of applicatio

Experimental setup.

<p><b>Left:</b> 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. <b>Right:</b> 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 <i>α</i>-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.</p

Distribution of tumble angles, <i>P</i>(|<i>β</i>|).

<p>It is determined from experiments by the heuristic tumble recognizer (bar graph) and by the inference method with the gamma function <i>γ</i>(<i>σ</i>, <i>k</i>) 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 〈|<i>β</i>|〉 = 0.42<i>π</i> = 76.0°, Δ|<i>β</i>| = 0.27<i>π</i> = 48.7° (heuristic tumble recognizer) and 〈|<i>β</i>|〉 = 0.47<i>π</i> = 85.4°, Δ|<i>β</i>| = 0.23<i>π</i> = 41.8° (inference method). The inferred parameters of <i>γ</i>(<i>σ</i>, <i>k</i>) are <i>σ</i> = 0.64 and <i>k</i> = 2.73. The red dashed line refers to the inferred gamma distribution (<i>σ</i> = 0.78 and <i>k</i> = 2.15), when the original data is smoothed. The blue dashed line refers to the histogram values multiplied by sin(|<i>β</i>|) and then normalized to one, thus representing the tumble angle distribution in three dimensions.</p

Schematics of a bacterial tumble event.

<p><i>E. coli</i> moves in direction Θ(<i>t</i>), tumbles at time <i>t</i> + Δ<i>t</i>, and moves in the new direction Θ(<i>t</i> + Δ<i>t</i>). Thus, the turning angle becomes |Θ(<i>t</i> + Δ<i>t</i>) − Θ(<i>t</i>)|_<i>a</i>.</p

Tumbling statistics of <i>P. putida</i>.

<p>(a) Distribution of tumble angles, <i>P</i>(|<i>β</i>|), 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(|<i>β</i>|) thereby representing the tumble angle distribution in three dimensions. The mean tumble angle and the standard deviation are 〈|<i>β</i>|〉 = 0.75<i>π</i> = 135°, Δ|<i>β</i>| = 0.29<i>π</i> = 52.2° (heuristic tumble recognizer) and 〈|<i>β</i>|〉 = 0.72<i>π</i> = 130°, Δ|<i>β</i>| = 0.26<i>π</i> = 46.8° (inference method). (b) The mean tumble rate λ (red) and the mean tumble angle 〈|<i>β</i>|〉 (blue) plotted versus <i>θ</i>. The tumble rate is fitted by a cosine function.</p

Flow diagram of the CM method.

<p>As input one provides the model of bacterial motion summarized in Eqs (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005329#pcbi.1005329.e001" target="_blank">1</a>)–(<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005329#pcbi.1005329.e004" target="_blank">4</a>) and a sufficient number of experimental trajectories. Then, the theoretical CMs <b>m</b>_theo(<i>θ</i>,<b>p</b>) are calculated from the model as a function of the parameters <b>p</b> and the current orientation angle <i>θ</i> [see Eqs (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005329#pcbi.1005329.e016" target="_blank">13</a>)–(<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005329#pcbi.1005329.e021" target="_blank">18</a>)] The experimental CMs <b>m</b>_exp(<i>θ</i>) are determined using <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005329#pcbi.1005329.e025" target="_blank">Eq (21)</a>. Matching these moments with a least square fit yields as an output the parameters of the model: <b>p</b>(<i>θ</i>) = arg min_<b>p</b>|<b>m</b>_exp(<i>θ</i>) − <b>m</b>_theo(<b>p</b>)|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.</p