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.

Mixtures of active and passive colloids show an intriguing dynamics of self-assembling, which is driven by the active component. Self-phoretic active colloids generate sinks in a chemical concentration field that cause passive colloids to drift toward active colloids by diffusiophoresis. The strength of this effective attraction is governed by the diffusiophoretic parameter, which determines the drift velocity. Simulating the Langevin dynamics of the colloids, we determine the state diagram for increasing diffusiophoretic strength and fixed active velocity. Three main states are distinguished. For weak attraction, passive particles are first scattered in the simulation box and then form a colloidal cloud around its center. Increasing the diffusiophoretic parameter further, passive particles oscillate between the cloud and a compact cluster, which embeds active colloids. Ultimately, in the third state, all particles collapse into a single stable cluster. In the collapse regime, the clustering dynamics of the largest cluster follows a logistic function and the mean cluster velocity vs cluster size decays with a power law. Throughout this article, we discuss our simulation results with regard to the experiments of Singh et al., Adv. Mater. 29(32), 1701328 (2017).

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.

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 v 0 , the strength of speed fluctuations σ, the relative height of speed jumps η, the thermal value for the rotational diffusion coefficient D 0 , and the enhanced rotational diffusivity during tumbling D T . 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.

We present a plausible micro-founded model for the previously postulated power law finite time singular form of the crash hazard rate in the Johansen-Ledoit-Sornette model of rational expectation bubbles. The model is based on a percolation picture of the network of traders and the concept that clusters of connected traders share the same opinion. The key ingredient is the notion that a shift of position from buyer to seller of a sufficiently large group of traders can trigger a crash. This provides a formula to estimate the crash hazard rate by summation over percolation clusters above a minimum size of a power s a (with a > 1) of the cluster sizes s, similarly to a generalized percolation susceptibility. The power s a of cluster sizes emerges from the super-linear dependence of group activity as a function of group size, previously documented in the literature. The crash hazard rate exhibits explosive finite-time singular behaviors when the control parameter (fraction of occupied sites, or density of traders in the network) approaches the percolation threshold pc. Realistic dynamics are generated by modelling the density of traders on the percolation network by an Ornstein-Uhlenbeck process, whose memory controls the spontaneous excursion of the control parameter close to the critical region of bubble formation. Our numerical simulations recover the main stylized properties of the JLS model with intermittent explosive super-exponential bubbles interrupted by crashes.

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