A bacterial swimmer with two alternating speeds of propagation

A new look at bubbles during biofilm inoculation reveals pronounced effects on growth and patterning

Specially designed microfluidic bioflow cells were used to temporarily trap microbubbles during different inoculation stages of Pseudomonas sp. biofilms. Despite being eliminated many hours before biofilm appearance, templated growth could occur at former bubble positions. Bubble-templated growth was either continuous or in ring patterns, depending on the stage of inoculation when the bubbles were introduced. Templated biofilms were strongly enhanced in terms of their growth kinetics and structural homogeneity. High resolution confocal imaging showed two separate bubble-induced bacterial trapping modes, which were responsible for the altered biofilm development. It is concluded that static bubbles can be exploited for fundamental improvements to bioreactor performance, as well as open new avenues to study isolated bacteria and small colonies.

Generalized run-and-turn motions: From bacteria to Lévy walks

Swimming bacteria exhibit a repertoire of motility patterns, in which persistent motion is interrupted by turning events. What are the statistical properties of such random walks? If some particular instances have long been studied, the general case where turning times do not follow a Poisson process has remained unsolved. We present a generic extension of the continuous time random walks formalism relying on operators and noncommutative calculus. The approach is first applied to a unimodal model of bacterial motion. We examine the existence of a minimum in velocity correlation function and discuss the maximum of diffusivity at an optimal value of rotational diffusion. The model is then extended to bimodal patterns and includes as particular cases all swimming strategies: run-and-tumble, run-stop, run-reverse and run-reverse-flick. We characterize their velocity correlation functions and investigate how bimodality affects diffusivity. Finally, the wider applicability of the method is illustrated by considering curved trajectories and Lévy walks. Our results are relevant for intermittent motion of living beings, be they swimming micro-organisms or crawling cells.

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.

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.

Run-and-pause dynamics of cytoskeletal motor proteins

Cytoskeletal motor proteins are involved in major intracellular transport processes which are vital for maintaining appropriate cellular function. When attached to cytoskeletal filaments, the motor exhibits distinct states of motility: active motion along the filaments, and pause phase in which it remains stationary for a finite time interval. The transition probabilities between motion and pause phases are asymmetric in general, and considerably affected by changes in environmental conditions which influences the efficiency of cargo delivery to specific targets. By considering the motion of individual non-interacting molecular motors on a single filament as well as a dynamic filamentous network, we present an analytical model for the dynamics of self-propelled particles which undergo frequent pause phases. The interplay between motor processivity, structural properties of filamentous network, and transition probabilities between the two states of motility drastically changes the dynamics: multiple transitions between different types of anomalous diffusive dynamics occur and the crossover time to the asymptotic diffusive or ballistic motion varies by several orders of magnitude. We map out the phase diagrams in the space of transition probabilities, and address the role of initial conditions of motion on the resulting dynamics.

Active Brownian motion of emulsion droplets: Coarsening dynamics at the interface and rotational diffusion

A micron-sized droplet of bromine water immersed in a surfactant-laden oil phase can swim (S. Thutupalli, R. Seemann, S. Herminghaus, New J. Phys. 13 073021 (2011). The bromine reacts with the surfactant at the droplet interface and generates a surfactant mixture. It can spontaneously phase-separate due to solutocapillary Marangoni flow, which propels the droplet. We model the system by a diffusion-advection-reaction equation for the mixture order parameter at the interface including thermal noise and couple it to fluid flow. Going beyond previous work, we illustrate the coarsening dynamics of the surfactant mixture towards phase separation in the axisymmetric swimming state. Coarsening proceeds in two steps: an initially slow growth of domain size followed by a nearly ballistic regime. On larger time scales thermal fluctuations in the local surfactant composition initiates random changes in the swimming direction and the droplet performs a persistent random walk, as observed in experiments. Numerical solutions show that the rotational correlation time scales with the square of the inverse noise strength. We confirm this scaling by a perturbation theory for the fluctuations in the mixture order parameter and thereby identify the active emulsion droplet as an active Brownian particle.

Singly Flagellated Pseudomonas aeruginosa Chemotaxes Efficiently by Unbiased Motor Regulation

Pseudomonas aeruginosa is an opportunistic human pathogen that has long been known to chemotax. More recently, it has been established that chemotaxis is an important factor in the ability of P. aeruginosa to make biofilms. Genes that allow P. aeruginosa to chemotax are homologous with genes in the paradigmatic model organism for chemotaxis, Escherichia coli. However, P. aeruginosa is singly flagellated and E. coli has multiple flagella. Therefore, the regulation of counterclockwise/clockwise flagellar motor bias that allows E. coli to efficiently chemotax by runs and tumbles would lead to inefficient chemotaxis by P. aeruginosa, as half of a randomly oriented population would respond to a chemoattractant gradient in the wrong sense. How P. aeruginosa regulates flagellar rotation to achieve chemotaxis is not known. Here, we analyze the swimming trajectories of single cells in microfluidic channels and the rotations of cells tethered by their flagella to the surface of a variable-environment flow cell. We show that P. aeruginosa chemotaxes by symmetrically increasing the durations of both counterclockwise and clockwise flagellar rotations when swimming up the chemoattractant gradient and symmetrically decreasing rotation durations when swimming down the chemoattractant gradient. Unlike the case for E. coli, the counterclockwise/clockwise bias stays constant for P. aeruginosa. We describe P. aeruginosa’s chemotaxis using an analytical model for symmetric motor regulation. We use this model to do simulations that show that, given P. aeruginosa’s physiological constraints on motility, its distinct, symmetric regulation of motor switching optimizes chemotaxis.

Chemotaxis has long been known to strongly affect biofilm formation by the opportunistic human pathogen P. aeruginosa, whose essential chemotaxis genes have homologues in E. coli, which achieves chemotaxis by biasing the relative probability of counterclockwise and clockwise flagellar rotation. However, the physiological difference between multiflagellated E. coli and singly flagellated P. aeruginosa implies that biased motor regulation should prevent P. aeruginosa populations from chemotaxing efficiently. Here, we used experiments, analytical modeling, and simulations to demonstrate that P. aeruginosa uses unbiased, symmetric regulation of the flagellar motor to maximize its chemotaxis efficiency. This mode of chemotaxis was not previously known and demonstrates a new variant of a paradigmatic signaling system in an important human pathogen.

Transient Superdiffusion and Long-Range Correlations in the Motility Patterns of Trypanosomatid Flagellate Protozoa

We report on a diffusive analysis of the motion of flagellate protozoa species. These parasites are the etiological agents of neglected tropical diseases: leishmaniasis caused by Leishmania amazonensis and Leishmania braziliensis, African sleeping sickness caused by Trypanosoma brucei, and Chagas disease caused by Trypanosoma cruzi. By tracking the positions of these parasites and evaluating the variance related to the radial positions, we find that their motions are characterized by a short-time transient superdiffusive behavior. Also, the probability distributions of the radial positions are self-similar and can be approximated by a stretched Gaussian distribution. We further investigate the probability distributions of the radial velocities of individual trajectories. Among several candidates, we find that the generalized gamma distribution shows a good agreement with these distributions. The velocity time series have long-range correlations, displaying a strong persistent behavior (Hurst exponents close to one). The prevalence of “universal” patterns across all analyzed species indicates that similar mechanisms may be ruling the motion of these parasites, despite their differences in morphological traits. In addition, further analysis of these patterns could become a useful tool for investigating the activity of new candidate drugs against these and others neglected tropical diseases.

The flagellar motor of Caulobacter crescentus generates more torque when a cell swims backward

Caulobacter crescentus, a monotrichous bacterium, swims by rotating a single right-handed helical filament. CW motor rotation thrusts the cell forward ¹, a mode of motility known as the pusher mode; CCW motor rotation pulls the cell backward, a mode of motility referred to as the puller mode ². The situation is opposite in E. coli, a peritrichous bacterium, where CCW rotation of multiple left-handed filaments drives the cell forward. The flagellar motor in E. coli generates more torque in the CCW direction than the CW direction in swimming cells ^3,4. However, monotrichous bacteria including C. crescentus swim forward and backward at similar speeds, prompting the assumption that motor torques in the two modes are the same ^5,6. Here, we present evidence that motors in C. crescentus develop higher torques in the puller mode than in the pusher mode, and suggest that the anisotropy in torque-generation is similar in two species, despite the differences in filament handedness and motor bias (probability of CW rotation).

Experimental and Mathematical-Modeling Characterization of Trypanosoma cruzi Epimastigote Motility

The present work is aimed at characterizing the motility of parasite T. cruzi in its epimastigote form. To that end, we recorded the trajectories of two strains of this parasite (a wild-type strain and a stable transfected strain, which contains an ectopic copy of LYT1 gene and whose motility is known to be affected). We further extracted parasite trajectories from the recorded videos, and statistically analysed the following trajectory-step features: step length, angular change of direction, longitudinal and transverse displacements with respect to the previous step, and mean square displacement. Based on the resulting observations, we developed a mathematical model to simulate parasite trajectories. The fact that the model predictions closely match most of the experimentally observed parasite-trajectory characteristics, allows us to conclude that the model is an accurate description of T. cruzi motility.

High-throughput 3D tracking of bacteria on a standard phase contrast microscope

Bacteria employ diverse motility patterns in traversing complex three-dimensional (3D) natural habitats. 2D microscopy misses crucial features of 3D behaviour, but the applicability of existing 3D tracking techniques is constrained by their performance or ease of use. Here we present a simple, broadly applicable, high-throughput 3D bacterial tracking method for use in standard phase contrast microscopy. Bacteria are localized at micron-scale resolution over a range of 350 × 300 × 200 μm by maximizing image cross-correlations between their observed diffraction patterns and a reference library. We demonstrate the applicability of our technique to a range of bacterial species and exploit its high throughput to expose hidden contributions of bacterial individuality to population-level variability in motile behaviour. The simplicity of this powerful new tool for bacterial motility research renders 3D tracking accessible to a wider community and paves the way for investigations of bacterial motility in complex 3D environments.

Microscopy techniques used to study the movement of swimming microbes are limited to two dimensions or require sophisticated devices. Here, Taute et al. present a simple method for high-throughput 3D tracking of bacteria using standard phase contrast microscopy.

A Non-Poissonian Flagellar Motor Switch Increases Bacterial Chemotactic Potential

We investigate bacterial chemotactic strategies using run-tumble and run-reverse-flick motility patterns. The former is typically observed in enteric bacteria such as Escherichia coli and Salmonella and the latter was recently observed in the marine bacteria Vibrio alginolyticus and is possibly exhibited by other polar flagellated species. It is shown that although the three-step motility pattern helps the bacterium to localize near hot spots, an exploitative behavior, its exploratory potential in short times can be significantly enhanced by employing a non-Poissonian regulation scheme for its flagellar motor switches.

Hysteretic dynamics of active particles in a periodic orienting field

Active motion of living organisms and artificial self-propelling particles has been an area of intense research at the interface of biology, chemistry and physics. Significant progress in understanding these phenomena has been related to the observation that dynamic self-organization in active systems has much in common with ordering in equilibrium condensed matter such as spontaneous magnetization in ferromagnets. The velocities of active particles may behave similar to magnetic dipoles and develop global alignment, although interactions between the individuals might be completely different. In this work, we show that the dynamics of active particles in external fields can also be described in a way that resembles equilibrium condensed matter. It follows simple general laws, which are independent of the microscopic details of the system. The dynamics is revealed through hysteresis of the mean velocity of active particles subjected to a periodic orienting field. The hysteresis is measured in computer simulations and experiments on unicellular organisms. We find that the ability of the particles to follow the field scales with the ratio of the field variation period to the particles' orientational relaxation time, which, in turn, is related to the particle self-propulsion power and the energy dissipation rate. The collective behaviour of the particles due to aligning interactions manifests itself at low frequencies via increased persistence of the swarm motion when compared with motion of an individual. By contrast, at high field frequencies, the active group fails to develop the alignment and tends to behave like a set of independent individuals even in the presence of interactions. We also report on asymptotic laws for the hysteretic dynamics of active particles, which resemble those in magnetic systems. The generality of the assumptions in the underlying model suggests that the observed laws might apply to a variety of dynamic phenomena from the motion of synthetic active particles to crowd or opinion dynamics.

Polar features in the flagellar propulsion of E. coli bacteria

E. coli bacteria swim following a run and tumble pattern. In the run state all flagella join in a single helical bundle that propels the cell body along approximately straight paths. When one or more flagellar motors reverse direction the bundle unwinds and the cell randomizes its orientation. This basic picture represents an idealization of a much more complex dynamical problem. Although it has been shown that bundle formation can occur at either pole of the cell, it is still unclear whether these two run states correspond to asymmetric propulsion features. Using holographic microscopy we record the 3D motions of individual bacteria swimming in optical traps. We find that most cells possess two run states characterized by different propulsion forces, total torque, and bundle conformations. We analyze the statistical properties of bundle reversal and compare the hydrodynamic features of forward and backward running states. Our method is naturally multi-particle and opens up the way towards controlled hydrodynamic studies of interacting swimming cells.

Adaptive settings for the nearest-neighbor particle tracking algorithm

BACKGROUND

The performance of the single particle tracking (SPT) nearest-neighbor algorithm is determined by parameters that need to be set according to the characteristics of the time series under study. Inhomogeneous systems, where these characteristics fluctuate spatially, are poorly tracked when parameters are set globally.

RESULTS

We present a novel SPT approach that adapts the well-known nearest-neighbor tracking algorithm to the local density of particles to overcome the problems of inhomogeneity.

CONCLUSIONS

We demonstrate the performance improvement provided by the proposed method using numerical simulations and experimental data and compare its performance with state of the art SPT algorithms.

AVAILABILITY AND IMPLEMENTATION

The algorithms proposed here, are released under the GNU General Public License and are freely available on the web at http://sourceforge.net/p/adaptivespt.

CONTACT

javier.mazzaferri@gmail.com

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Unimodal and bimodal random motions of independent exponential steps

We consider random walks that arise from the repetition of independent, statistically identical steps, whose nature may be arbitrary. Such unimodal motions appear in a variety of contexts, including particle propagation, cell motility, swimming of micro-organisms, animal motion and foraging strategies. Building on general frameworks, we focus on the case where step duration is exponentially distributed. We explore systematically unimodal processes whose steps are ballistic, diffusive, cyclic or governed by rotational diffusion, and give the exact propagator in Fourier-Laplace domain, from which the moments and the diffusion coefficient are obtained. We also address bimodal processes, where two kinds of step are taken in turn, and show that the mean square displacement, the quantity of prime importance in experiments, is simply related to those of unimodal motions.

Secondary bacterial flagellar system improves bacterial spreading by increasing the directional persistence of swimming

As numerous bacterial species, Shewanella putrefaciens CN-32 possesses a complete secondary flagellar system. A significant subpopulation of CN-32 cells induces expression of the secondary system under planktonic conditions, resulting in formation of one, sometimes two, filaments at lateral positions in addition to the primary polar flagellum. Mutant analysis revealed that the single chemotaxis system primarily or even exclusively addresses the main polar flagellar system. Cells with secondary filaments outperformed their monopolarly flagellated counterparts in spreading on soft-agar plates and through medium-filled channels despite having lower swimming speed. While mutant cells with only polar flagella navigate by a "run-reverse-flick" mechanism resulting in effective cell realignments of about 90°, wild-type cells with secondary filaments exhibited a range of realignment angles with an average value of smaller than 90°. Mathematical modeling and computer simulations demonstrated that the smaller realignment angle of wild-type cells results in the higher directional persistence, increasing spreading efficiency both with and without a chemical gradient. Taken together, we propose that in S. putrefaciens CN-32, cell propulsion and directional switches are mainly mediated by the polar flagellar system, while the secondary filament increases the directional persistence of swimming and thus of spreading in the environment.

Analytical modeling and experimental characterization of chemotaxis in Serratia marcescens

This paper presents a modeling and experimental framework to characterize the chemotaxis of Serratia marcescens (S. marcescens) relying on two-dimensional and three-dimensional tracking of individual bacteria. Previous studies mainly characterized bacterial chemotaxis based on population density analysis. Instead, this study focuses on single-cell tracking and measuring the chemotactic drift velocity V(C) from the biased tumble rate of individual bacteria on exposure to a concentration gradient of l-aspartate. The chemotactic response of S. marcescens is quantified over a range of concentration gradients (10^{-3} to 5 mM/mm) and average concentrations (0.5 × 10(-3) to 2.5 mM). Through the analysis of a large number of bacterial swimming trajectories, the tumble rate is found to have a significant bias with respect to the swimming direction. We also verify the relative gradient sensing mechanism in the chemotaxis of S. marcescens by measuring the change of V(C) with the average concentration and the gradient. The applied full pathway model with fitted parameters matches the experimental data. Finally, we show that our measurements based on individual bacteria lead to the determination of the motility coefficient μ (7.25 × 10(-6) cm(2)/s) of a population. The experimental characterization and simulation results for the chemotaxis of this bacterial species contribute towards using S. marcescens in chemically controlled biohybrid systems.

How the motility pattern of bacteria affects their dispersal and chemotaxis

Most bacteria at certain stages of their life cycle are able to move actively; they can swim in a liquid or crawl on various surfaces. A typical path of the moving cell often resembles the trajectory of a random walk. However, bacteria are capable of modifying their apparently random motion in response to changing environmental conditions. As a result, bacteria can migrate towards the source of nutrients or away from harmful chemicals. Surprisingly, many bacterial species that were studied have several distinct motility patterns, which can be theoretically modeled by a unifying random walk approach. We use this approach to quantify the process of cell dispersal in a homogeneous environment and show how the bacterial drift velocity towards the source of attracting chemicals is affected by the motility pattern of the bacteria. Our results open up the possibility of accessing additional information about the intrinsic response of the cells using macroscopic observations of bacteria moving in inhomogeneous environments.