A biased random walk model for the trajectories of swimming micro-organisms

Directional swimming patterns in jellyfish aggregations

Having a profound influence on marine and coastal environments worldwide, jellyfish hold significant scientific, economic, and public interest.1,2,3,4,5 The predictability of outbreaks and dispersion of jellyfish is limited by a fundamental gap in our understanding of their movement. Although there is evidence that jellyfish may actively affect their position,6,7,8,9,10 the role of active swimming in controlling jellyfish movement, and the characteristics of jellyfish swimming behavior, are not well understood. Consequently, jellyfish are often regarded as passively drifting or randomly moving organisms, both conceptually2,11 and in process studies.12,13,14 Here we show that the movement of jellyfish is modulated by distinctly directional swimming patterns that are oriented away from the coast and against the direction of surface gravity waves. Taking a Lagrangian viewpoint from drone videos that allows the tracking of multiple adjacent jellyfish, and focusing on the scyphozoan jellyfish Rhopilema nomadica as a model organism, we show that the behavior of individual jellyfish translates into a synchronized directional swimming of the aggregation as a whole. Numerical simulations show that this counter-wave swimming behavior results in biased correlated random-walk movement patterns that reduce the risk of stranding, thus providing jellyfish with an adaptive advantage critical to their survival. Our results emphasize the importance of active swimming in regulating jellyfish movement and open the way for a more accurate representation in model studies, thus improving the predictability of jellyfish outbreaks and their dispersion and contributing to our ability to mitigate their possible impact on coastal infrastructure and populations.

A self-generated Toddler gradient guides mesodermal cell migration

The sculpting of germ layers during gastrulation relies on the coordinated migration of progenitor cells, yet the cues controlling these long-range directed movements remain largely unknown. While directional migration often relies on a chemokine gradient generated from a localized source, we find that zebrafish ventrolateral mesoderm is guided by a self-generated gradient of the initially uniformly expressed and secreted protein Toddler/ELABELA/Apela. We show that the Apelin receptor, which is specifically expressed in mesodermal cells, has a dual role during gastrulation, acting as a scavenger receptor to generate a Toddler gradient, and as a chemokine receptor to sense this guidance cue. Thus, we uncover a single receptor-based self-generated gradient as the enigmatic guidance cue that can robustly steer the directional migration of mesoderm through the complex and continuously changing environment of the gastrulating embryo.

Investigating microscale patchiness of motile microbes under turbulence in a simulated convective mixed layer

Microbes play a primary role in aquatic ecosystems and biogeochemical cycles. Spatial patchiness is a critical factor underlying these activities, influencing biological productivity, nutrient cycling and dynamics across trophic levels. Incorporating spatial dynamics into microbial models is a long-standing challenge, particularly where small-scale turbulence is involved. Here, we combine a fully 3D direct numerical simulation of convective mixed layer turbulence, with an individual-based microbial model to test the key hypothesis that the coupling of gyrotactic motility and turbulence drives intense microscale patchiness. The fluid model simulates turbulent convection caused by heat loss through the fluid surface, for example during the night, during autumnal or winter cooling or during a cold-air outbreak. We find that under such conditions, turbulence-driven patchiness is depth-structured and requires high motility: Near the fluid surface, intense convective turbulence overpowers motility, homogenising motile and non-motile microbes approximately equally. At greater depth, in conditions analogous to a thermocline, highly motile microbes can be over twice as patch-concentrated as non-motile microbes, and can substantially amplify their swimming velocity by efficiently exploiting fast-moving packets of fluid. Our results substantiate the predictions of earlier studies, and demonstrate that turbulence-driven patchiness is not a ubiquitous consequence of motility but rather a delicate balance of motility and turbulent intensity.

Coordination of two opposite flagella allows high-speed swimming and active turning of individual zoospores

Phytophthora species cause diseases in a large variety of plants and represent a serious agricultural threat, leading, every year, to multibillion dollar losses. Infection occurs when their biflagellated zoospores move across the soil at their characteristic high speed and reach the roots of a host plant. Despite the relevance of zoospore spreading in the epidemics of plant diseases, individual swimming of zoospores have not been fully investigated. It remains unknown about the characteristics of two opposite beating flagella during translation and turning, and the roles of each flagellum on zoospore swimming. Here, combining experiments and modeling, we show how these two flagella contribute to generate thrust when beating together, and identify the mastigonemes-attached anterior flagellum as the main source of thrust. Furthermore, we find that turning involves a complex active process, in which the posterior flagellum temporarily stops, while the anterior flagellum keeps on beating and changes its gait from sinusoidal waves to power and recovery strokes, similar to Chlamydomonas's breaststroke, to reorient its body to a new direction. Our study is a fundamental step toward a better understanding of the spreading of plant pathogens' motile forms, and shows that the motility pattern of these biflagellated zoospores represents a distinct eukaryotic version of the celebrated 'run-and-tumble' motility class exhibited by peritrichous bacteria.

Rotational Locomotion of an Active Gel Driven by Internal Chemical Signals

Chemical waves arising from coupled reaction and transport can serve as biomimetic "nerve signals" to study the underlying origin and regulation of active locomotion. During wave propagation in more than one spatial dimension, the propagation direction of spiral and pulse waves in a nanogel-based PAAm self-oscillating gel, i.e., the orientation of the driving force, may deviate from the normal direction to the wave fronts. Alternating forward and backward retrograde wave locomotion along the normal and tangential kinematic vectors with a phase difference leads to a curved path, i.e., rotational locomotion. This work indicates that appendages in an organism are not required for this type of locomotion. This locomotion mechanism reveals a general principle underlying the dynamical origin of biological helical locomotion and also suggests design approaches for complex locomotion of soft robots and smart materials.

Theory of branching morphogenesis by local interactions and global guidance

Branching morphogenesis governs the formation of many organs such as lung, kidney, and the neurovascular system. Many studies have explored system-specific molecular and cellular regulatory mechanisms, as well as self-organizing rules underlying branching morphogenesis. However, in addition to local cues, branched tissue growth can also be influenced by global guidance. Here, we develop a theoretical framework for a stochastic self-organized branching process in the presence of external cues. Combining analytical theory with numerical simulations, we predict differential signatures of global vs. local regulatory mechanisms on the branching pattern, such as angle distributions, domain size, and space-filling efficiency. We find that branch alignment follows a generic scaling law determined by the strength of global guidance, while local interactions influence the tissue density but not its overall territory. Finally, using zebrafish innervation as a model system, we test these key features of the model experimentally. Our work thus provides quantitative predictions to disentangle the role of different types of cues in shaping branched structures across scales.

First passage under restart for discrete space and time: Application to one-dimensional confined lattice random walks

First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to outperform the completion of many first-passage processes which otherwise would take longer time to finish. However, most of the studies so far assumed continuous time underlying first-passage time processes and moreover considered continuous time resetting restricting out restart processes broken up into synchronized time steps. To bridge this gap, in this paper, we study discrete space and time first-passage processes under discrete time resetting in a general setup without specifying their forms. We sketch out the steps to compute the moments and the probability density function which is often intractable in the continuous time restarted process. A criterion that dictates when restart remains beneficial is then derived. We apply our results to a symmetric and a biased random walker in one-dimensional lattice confined within two absorbing boundaries. Numerical simulations are found to be in excellent agreement with the theoretical results. Our method can be useful to understand the effect of restart on the spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions.

Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions

Biased lattice random walks (BLRW) are used to model random motion with drift in a variety of empirical situations in engineering and natural systems such as phototaxis, chemotaxis, or gravitaxis. When motion is also affected by the presence of external borders resulting from natural barriers or experimental apparatuses, modelling biased random movement in confinement becomes necessary. To study these scenarios, confined BLRW models have been employed but so far only through computational techniques due to the lack of an analytic framework. Here, we lay the groundwork for such an analytical approach by deriving the Green's functions, or propagators, for the confined BLRW in arbitrary dimensions and arbitrary boundary conditions. By using these propagators we construct explicitly the time-dependent first-passage probability in one dimension for reflecting and periodic domains, while in higher dimensions we are able to find its generating function. The latter is used to find the mean first-passage passage time for a d-dimensional box, d-dimensional torus or a combination of both. We show the appearance of surprising characteristics such as the presence of saddles in the spatiotemporal dynamics of the propagator with reflecting boundaries, bimodal features in the first-passage probability in periodic domains and the minimization of the mean first-return time for a bias of intermediate strength in rectangular domains. Furthermore, we quantify how in a multitarget environment with the presence of a bias shorter mean first-passage times can be achieved by placing fewer targets close to boundaries in contrast to many targets away from them.

Diurnal Variations in the Motility of Populations of Biflagellate Microalgae

The motility of microalgae has been studied extensively, particularly in model microorganisms such as Chlamydomonas reinhardtii. For this and other microalgal species, diurnal cycles are well known to control the metabolism, growth, and cell division. Diurnal variations, however, have been largely neglected in quantitative studies of motility. Here, we demonstrate using tracking microscopy how the motility statistics of C. reinhardtii are modulated by diurnal cycles. With nine independently inoculated cultures synchronized to the light-dark cycle at the exponential growth phase, we repeatedly observed that the mean swimming speed is greater during the dark period of a diurnal cycle. From this measurement, using a hydrodynamic power balance, we infer the mean flagellar beat frequency and conjecture that its diurnal variation reflects modulation of intracellular ATP. Our measurements also quantify the diurnal variations of the orientational and gravitactic transport of C. reinhardtii. We use this to explore the population-level consequences of diurnal variations of motility statistics by evaluating a prediction for how the gravitactic steady state changes with time during a diurnal cycle. Finally, we discuss the consequences of diurnal variations of microalgal motility in soil and pelagic environments.

Path Meander of Male Codling Moths (Cydia pomonella) Foraging for Sex Pheromone Plumes: Field Validation of a Novel Method for Quantifying Path Meander of Random Movers Developed Using Computer Simulations

Simple Summary

Measures of insect movement patterns are key to understanding how insects forage for resources and mating opportunities in their environment. Directly observing large numbers of these small organisms can be extremely challenging, especially for flying insects in low light conditions such as codling moth (Cydia pomonella), a key pest of apple. Here we provide a novel approach to indirectly measure the path meander of randomly moving organisms. Computer simulations were used to simulate insect movement across a wide range of possible movement patterns, measured in circular standard deviation (c.s.d.) of turn angles between track segments. For each c.s.d., the pattern of catch across a rectangular grid of traps was plotted and the resulting exponential decay constant (k) of the fitted lines were used to generate a standard curve describing this linear relationship. Using this standard curve, field data from target organisms caught in the described trapping grid can reveal the pattern of movement employed by these movers. Here we have demonstrated methodology for indirect measure of the movement patterns employed by random walkers such as C. pomonella. While we employed codling moth as our model system, we suggest this approach could prove useful in a wide range of other systems.

Abstract

Measures of path meander are highly relevant to studies of optimal foraging by animals. However, directly recording paths of small animals such as insects can be difficult because of small size or crepuscular activity. Computer simulations of correlated random walkers demonstrated that the rates of decay in captures across a rectangular grid of traps when movers were released at its corner can be used to produce calibration curves for quantifying path meander indirectly. Simulations using spatial parameters matching those previously documented for male codling moths (Cydia pomonella (L.)) foraging for female pheromone plumes in the field predicted that meander, as measured in circular standard deviation (c.s.d.) of turn angles between track segments, should be ca. 50° and 30° when the target population density is high vs. low, respectively. Thus, if optimized, the mean value measured for C. pomonella populations encountering an unknown target density should fall between these limits. We recorded decay in C. pomonella catch across a 5 × 5 grid of pheromone-baited traps each separated by 15 m on 39 occasions where batches of ca. 800 males were released 10 m outside the corner of trapping grids arranged in five large Michigan apple orchards. This decay constant was translated into mean c.s.d value for path meander using the standard curve generated by the computer simulations. The measured decay constant for C. pomonella males was negative 0.99 ± 0.02 (S.E.M.), which translates to a path meander of 37 ± 2° c.s.d. Thus, the measured path meander of 37° fell between the 50° and 30° values optimal for dense and sparse populations, respectively. In addition to providing a rare documented example of optimal foraging for odor plumes, this research offers proof-of-concept for a novel approach to quantifying path meander of movers that could prove useful across diverse taxa.

Gyrotactic phytoplankton in laminar and turbulent flows: A dynamical systems approach

Gyrotactic algae are bottom heavy, motile cells whose swimming direction is determined by a balance between a buoyancy torque directing them upwards and fluid velocity gradients. Gyrotaxis has, in recent years, become a paradigmatic model for phytoplankton motility in flows. The essential attractiveness of this peculiar form of motility is the availability of a mechanistic description which, despite its simplicity, revealed predictive, rich in phenomenology, easily complemented to include the effects of shape, feedback on the fluid and stochasticity (e.g., in cell orientation). In this review we consider recent theoretical, numerical and experimental results to discuss how, depending on flow properties, gyrotaxis can produce inhomogeneous phytoplankton distributions on a wide range of scales, from millimeters to kilometers, in both laminar and turbulent flows. In particular, we focus on the phenomenon of gyrotactic trapping in nonlinear shear flows and in fractal clustering in turbulent flows. We shall demonstrate the usefulness of ideas and tools borrowed from dynamical systems theory in explaining and interpreting these phenomena.

Stochastic neural field model of stimulus-dependent variability in cortical neurons

We use stochastic neural field theory to analyze the stimulus-dependent tuning of neural variability in ring attractor networks. We apply perturbation methods to show how the neural field equations can be reduced to a pair of stochastic nonlinear phase equations describing the stochastic wandering of spontaneously formed tuning curves or bump solutions. These equations are analyzed using a modified version of the bivariate von Mises distribution, which is well-known in the theory of circular statistics. We first consider a single ring network and derive a simple mathematical expression that accounts for the experimentally observed bimodal (or M-shaped) tuning of neural variability. We then explore the effects of inter-network coupling on stimulus-dependent variability in a pair of ring networks. These could represent populations of cells in two different layers of a cortical hypercolumn linked via vertical synaptic connections, or two different cortical hypercolumns linked by horizontal patchy connections within the same layer. We find that neural variability can be suppressed or facilitated, depending on whether the inter-network coupling is excitatory or inhibitory, and on the relative strengths and biases of the external stimuli to the two networks. These results are consistent with the general observation that increasing the mean firing rate via external stimuli or modulating drives tends to reduce neural variability.

Theoretical description of the thermomolecular orientation of anisotropic colloids

We describe theoretically the orientation of anisotropic colloids under a thermal field.

Navigational efficiency in a biased and correlated random walk model of individual animal movement

Understanding how an individual animal is able to navigate through its environment is a key question in movement ecology that can give insight into observed movement patterns and the mechanisms behind them. Efficiency of navigation is important for behavioral processes at a range of different spatio-temporal scales, including foraging and migration. Random walk models provide a standard framework for modeling individual animal movement and navigation. Here we consider a vector-weighted biased and correlated random walk (BCRW) model for directed movement (taxis), where external navigation cues are balanced with forward persistence. We derive a mathematical approximation of the expected navigational efficiency for any BCRW of this form and confirm the model predictions using simulations. We demonstrate how the navigational efficiency is related to the weighting given to forward persistence and external navigation cues, and highlight the counter-intuitive result that for low (but realistic) levels of error on forward persistence, a higher navigational efficiency is achieved by giving more weighting to this indirect navigation cue rather than direct navigational cues. We discuss and interpret the relevance of these results for understanding animal movement and navigation strategies.

Rotational Diffusion of Soft Vesicles Filled by Chiral Active Particles

We investigate the dynamics of two-dimensional soft vesicles filled with chiral active particles by employing the overdamped Langevin dynamics simulation. The unidirectional rotation is observed for soft vesicles, and the rotational angular velocity of vesicles depends mainly on the area fraction (ρ) and angular velocity (ω) of chiral active particles. There exists an optimal parameter for ω at which the rotational angular velocity of vesicle takes its maximal value. Meanwhile, at low concentration the continuity of curvature is destroyed seriously by chiral active particles, especially for large ω, and at high concentration the chiral active particles cover the vesicle almost uniformly. In addition, the center-of-mass mean square displacement for vesicles is accompanied by oscillations at short timescales, and the oscillation period of diffusion for vesicles is consistent with the rotation period of chiral active particles. The diffusion coefficient of vesicle decreases monotonously with increasing the angular velocity ω of chiral active particles. Our investigation can provide a few designs for nanofabricated devices that can be driven in a unidirectional rotation by chiral active particles or could be used as drug-delivery agent.

Modeling of active swimmer suspensions and their interactions with the environment

In this article, we review mathematical models used to study the behaviour of suspensions of micro-swimmers and the accompanying biophysical phenomena, with specific focus on stimulus response. The methods discussed encompass a range of interactions exhibited by the micro-swimmers; including passive hydrodynamic (gyrotaxis) and gravitational (gravitaxis) effects, and active responses to chemical cues (chemotaxis) and light intensities (phototaxis). We introduce the simplest models first, and then build towards more sophisticated recent developments, in the process, identifying the limitations of the former and the new results obtained by the latter. We comment on the accuracy/validity of the models adopted, based on the agreement between theoretical results and experimental observations. We conclude by identifying some of the open problems and associated challenges faced by researchers in the realm of active suspensions.

A stochastic micro-machine inspired by bacterial chemotaxis

Inspired by bacterial chemotaxis, we propose and study the dynamics of a 3D hydrodynamical search-machine at micrometer scale. Chemotactic memory of the proposed system that is borrowed from bacteria, allows it to search the fluid medium and find the required target. As the motion in micron size length scale is dominated by random forces, we analyze the statistical properties of the model. Mean square displacements, orientational correlation functions and also the chemotactic index (CI) of the system are investigated in detail. Because of the chemotactic memory, the system shows superdiffusing displacements in all directions and the diffusion exponents are anisotropic for the directions along or perpendicular to a preferred direction given by the gradient of attractant molecules transmitted by the target.

Centripetal focusing of gyrotactic phytoplankton

A suspension of gyrotactic microalgae Chlamydomonas augustae swimming in a cylindrical water vessel in solid-body rotation is studied. Our experiments show that swimming algae form an aggregate around the axis of rotation, whose intensity increases with the rotation speed. We explain this phenomenon by the centripetal orientation of the swimming direction towards the axis of rotation. This centripetal focusing is contrasted by diffusive fluxes due to stochastic reorientation of the cells. The competition of the two effects lead to a stationary distribution, which we analytically derive from a refined mathematical model of gyrotactic swimmers. The temporal evolution of the cell distribution, obtained via numerical simulations of the stochastic model, is in quantitative agreement with the experimental measurements in the range of parameters explored.

Microalgae Scatter off Solid Surfaces by Hydrodynamic and Contact Forces

Interactions between microorganisms and solid boundaries play an important role in biological processes, such as egg fertilization, biofilm formation, and soil colonization, where microswimmers move within a structured environment. Despite recent efforts to understand their origin, it is not clear whether these interactions can be understood as being fundamentally of hydrodynamic origin or hinging on the swimmer's direct contact with the obstacle. Using a combination of experiments and simulations, here we study in detail the interaction of the biflagellate green alga Chlamydomonas reinhardtii, widely used as a model puller microorganism, with convex obstacles, a geometry ideally suited to highlight the different roles of steric and hydrodynamic effects. Our results reveal that both kinds of forces are crucial for the correct description of the interaction of this class of flagellated microorganisms with boundaries.

Phytoplankton's motion in turbulent ocean

We study the influence of turbulence on upward motion of phytoplankton. Interaction with the flow is described by the Pedley-Kessler model considering spherical microorganisms. We find a range of parameters when the upward drift is only weakly perturbed or when turbulence completely randomizes the drift direction. When the perturbation is small, the drift is either determined by the local vorticity or is Gaussian. We find a range of parameters where the phytoplankton interaction with the flow can be described consistently as diffusion of orientation in effective potential. By solving the corresponding Fokker-Planck equation we find exponential steady-state distribution of phytoplankton's propulsion orientation. We further identify the range of parameters where phytoplankton's drift velocity with respect to the flow is determined uniquely by its position. In this case, one can describe phytoplankton's motion by a smooth flow and phytoplankton concentrates on fractal. We find fractal dimensions and demonstrate that phytoplankton forms vertical stripes in space with a nonisotropic pair-correlation function of concentration increased in the vertical direction. The probability density function of the distance between two particles obeys power law with the negative exponent given by the ratio of integrals of the turbulent energy spectrum. We find the regime of strong clustering where the exponent is of order one so that turbulence increases the rate of collisions by a large factor. The predictions hold for Navier-Stokes turbulence and stand for testing.

Wavelength Selection in Gyrotactic Bioconvection

We investigate pattern formation by swimming micro-organisms (bioconvection), when their orientation is determined by balance between gravitational and viscous torques (gyrotaxis), due to being bottom heavy. The governing equations, which consist of the Navier-Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation, are solved numerically in a large cross section chamber with periodic boundary conditions in the horizontal directions. The influence of key parameters on wavelength selection in bioconvection patterns is investigated numerically. For realistic ranges of parameter values, the computed wavelengths are in good agreement with the experimental observations provided that the diffusion due to randomness in cell swimming behaviour is small, refuting a recently published claim that the mathematical model becomes inaccurate at long times. We also provide the first computational evidence of "bottom-standing" plumes in a three-dimensional simulation.

Effects of surface asymmetry on neuronal growth

Detailed knowledge of how the surface physical properties, such as mechanics, topography and texture influence axonal outgrowth and guidance is essential for understanding the processes that control neuron development, the formation of functional neuronal connections and nerve regeneration. Here we synthesize asymmetric surfaces with well-controlled topography and texture and perform a systematic experimental and theoretical investigation of axonal outgrowth on these substrates. We demonstrate unidirectional axonal bias imparted by the surface ratchet-based topography and quantify the topographical guidance cues that control neuronal growth. We describe the growth cone dynamics using a general stochastic model (Fokker-Planck formalism) and use this model to extract two key dynamical parameters: diffusion (cell motility) coefficient and asymmetric drift coefficient. The drift coefficient is identified with the torque caused by the asymmetric ratchet topography. We relate the observed directional bias in axonal outgrowth to cellular contact guidance behavior, which results in an increase in the cell-surface coupling with increased surface anisotropy. We also demonstrate that the disruption of cytoskeletal dynamics through application of Taxol (stabilizer of microtubules) and Blebbistatin (inhibitor of myosin II activity) greatly reduces the directional bias imparted by these asymmetric surfaces. These results provide new insight into the role played by topographical cues in neuronal growth and could lead to new methods for stimulating neuronal regeneration and the engineering of artificial neuronal tissue.

Immersive audiomotor game play enhances neural and perceptual salience of weak signals in noise

All sensory systems face the fundamental challenge of encoding weak signals in noisy backgrounds. Although discrimination abilities can improve with practice, these benefits rarely generalize to untrained stimulus dimensions. Inspired by recent findings that action video game training can impart a broader spectrum of benefits than traditional perceptual learning paradigms, we trained adult humans and mice in an immersive audio game that challenged them to forage for hidden auditory targets in a 2D soundscape. Both species learned to modulate their angular search vectors and target approach velocities based on real-time changes in the level of a weak tone embedded in broadband noise. In humans, mastery of this tone in noise task generalized to an improved ability to comprehend spoken sentences in speech babble noise. Neural plasticity in the auditory cortex of trained mice supported improved decoding of low-intensity sounds at the training frequency and an enhanced resistance to interference from background masking noise. These findings highlight the potential to improve the neural and perceptual salience of degraded sensory stimuli through immersive computerized games.

Random walks of colloidal probes in viscoelastic materials

To overcome limitations of using a single fixed time step in random walk simulations, such as those that rely on the classic Wiener approach, we have developed an algorithm for exploring random walks based on random temporal steps that are uniformly distributed in logarithmic time. This improvement enables us to generate random-walk trajectories of probe particles that span a highly extended dynamic range in time, thereby facilitating the exploration of probe motion in soft viscoelastic materials. By combining this faster approach with a Maxwell-Voigt model (MVM) of linear viscoelasticity, based on a slowly diffusing harmonically bound Brownian particle, we rapidly create trajectories of spherical probes in soft viscoelastic materials over more than 12 orders of magnitude in time. Appropriate windowing of these trajectories over different time intervals demonstrates that random walk for the MVM is neither self-similar nor self-affine, even if the viscoelastic material is isotropic. We extend this approach to spatially anisotropic viscoelastic materials, using binning to calculate the anisotropic mean square displacements and creep compliances along different orthogonal directions. The elimination of a fixed time step in simulations of random processes, including random walks, opens up interesting possibilities for modeling dynamics and response over a highly extended temporal dynamic range.

Novel methods for analysing bacterial tracks reveal persistence in Rhodobacter sphaeroides

Tracking bacteria using video microscopy is a powerful experimental approach to probe their motile behaviour. The trajectories obtained contain much information relating to the complex patterns of bacterial motility. However, methods for the quantitative analysis of such data are limited. Most swimming bacteria move in approximately straight lines, interspersed with random reorientation phases. It is therefore necessary to segment observed tracks into swimming and reorientation phases to extract useful statistics. We present novel robust analysis tools to discern these two phases in tracks. Our methods comprise a simple and effective protocol for removing spurious tracks from tracking datasets, followed by analysis based on a two-state hidden Markov model, taking advantage of the availability of mutant strains that exhibit swimming-only or reorientating-only motion to generate an empirical prior distribution. Using simulated tracks with varying levels of added noise, we validate our methods and compare them with an existing heuristic method. To our knowledge this is the first example of a systematic assessment of analysis methods in this field. The new methods are substantially more robust to noise and introduce less systematic bias than the heuristic method. We apply our methods to tracks obtained from the bacterial species Rhodobacter sphaeroides and Escherichia coli. Our results demonstrate that R. sphaeroides exhibits persistence over the course of a tumbling event, which is a novel result with important implications in the study of this and similar species.

Many species of planktonic bacteria are able to propel themselves through a liquid medium by the use of one or more helical flagella. Commonly, the observed motile behaviour consists of a series of approximately straight-line movements, interspersed with random, approximately stationary, reorientation events. This phenomenon is of current interest as it is known to be linked to important bacterial processes such as pathogenicity and biofilm formation. An accepted experimental approach for studying bacterial motility in approximately indigenous conditions is the tracking of cells using a microscope. However, there are currently no validated methods for the analysis of such tracking data. In particular, the identification of reorientation phases, which is complicated by various sources of noise in the data, remains an open challenge. In this paper we present novel methods for analysing large bacterial tracking datasets. We assess the performance of our new methods using computational simulations, and show that they are more reliable than a previously published method. We proceed to analyse previously unpublished tracks from the bacterial species Rhodobacter sphaeroides, an emerging model organism in the field of bacterial motility, and Escherichia coli, a well-studied model bacterium. The analysis demonstrates the novel result that R. sphaeroides exhibits directional persistence over the course of a reorientation event.

The effect of sampling rate on observed statistics in a correlated random walk

Tracking the movement of individual cells or animals can provide important information about their motile behaviour, with key examples including migrating birds, foraging mammals and bacterial chemotaxis. In many experimental protocols, observations are recorded with a fixed sampling interval and the continuous underlying motion is approximated as a series of discrete steps. The size of the sampling interval significantly affects the tracking measurements, the statistics computed from observed trajectories, and the inferences drawn. Despite the widespread use of tracking data to investigate motile behaviour, many open questions remain about these effects. We use a correlated random walk model to study the variation with sampling interval of two key quantities of interest: apparent speed and angle change. Two variants of the model are considered, in which reorientations occur instantaneously and with a stationary pause, respectively. We employ stochastic simulations to study the effect of sampling on the distributions of apparent speeds and angle changes, and present novel mathematical analysis in the case of rapid sampling. Our investigation elucidates the complex nature of sampling effects for sampling intervals ranging over many orders of magnitude. Results show that inclusion of a stationary phase significantly alters the observed distributions of both quantities.

Bioconvection in spatially extended domains

We numerically explore gyrotactic bioconvection in large spatially extended domains of finite depth using parameter values from available experiments with the unicellular alga Chlamydomonas nivalis. We numerically integrate the three-dimensional, time-dependent continuum model of Pedley et al. [J. Fluid Mech. 195, 223 (1988)] using a high-order, parallel, spectral-element approach. We explore the long-time nonlinear patterns and dynamics found for layers with an aspect ratio of 10 over a range of Rayleigh numbers. Our results yield the pattern wavelength and pattern dynamics which we compare with available theory and experimental measurement. There is good agreement for the pattern wavelength at short times between numerics, experiment, and a linear stability analysis. At long times we find that the general sequence of patterns given by the nonlinear evolution of the governing equations correspond qualitatively to what has been described experimentally. However, at long times the patterns in numerics grow to larger wavelengths, in contrast to what is observed in experiment where the wavelength is found to decrease with time.

Differential dynamic microscopy: a high-throughput method for characterizing the motility of microorganisms

We present a fast, high-throughput method for characterizing the motility of microorganisms in three dimensions based on standard imaging microscopy. Instead of tracking individual cells, we analyze the spatiotemporal fluctuations of the intensity in the sample from time-lapse images and obtain the intermediate scattering function of the system. We demonstrate our method on two different types of microorganisms: the bacterium Escherichia coli (both smooth swimming and wild type) and the biflagellate alga Chlamydomonas reinhardtii. We validate the methodology using computer simulations and particle tracking. From the intermediate scattering function, we are able to extract the swimming speed distribution, fraction of motile cells, and diffusivity for E. coli, and the swimming speed distribution, and amplitude and frequency of the oscillatory dynamics for C. reinhardtii. In both cases, the motility parameters were averaged over ∼10(4) cells and obtained in a few minutes.

Universality, limits and predictability of gold-medal performances at the olympic games

Inspired by the Games held in ancient Greece, modern Olympics represent the world’s largest pageant of athletic skill and competitive spirit. Performances of athletes at the Olympic Games mirror, since 1896, human potentialities in sports, and thus provide an optimal source of information for studying the evolution of sport achievements and predicting the limits that athletes can reach. Unfortunately, the models introduced so far for the description of athlete performances at the Olympics are either sophisticated or unrealistic, and more importantly, do not provide a unified theory for sport performances. Here, we address this issue by showing that relative performance improvements of medal winners at the Olympics are normally distributed, implying that the evolution of performance values can be described in good approximation as an exponential approach to an a priori unknown limiting performance value. This law holds for all specialties in athletics–including running, jumping, and throwing–and swimming. We present a self-consistent method, based on normality hypothesis testing, able to predict limiting performance values in all specialties. We further quantify the most likely years in which athletes will breach challenging performance walls in running, jumping, throwing, and swimming events, as well as the probability that new world records will be established at the next edition of the Olympic Games.

Origin of polar order in dense suspensions of phototactic micro-swimmers

A main question for the study of collective motion in living organisms is the origin of orientational polar order, i.e., how organisms align and what are the benefits of such collective behaviour. In the case of micro-organisms swimming at a low Reynolds number, steric repulsion and long-range hydrodynamic interactions are not sufficient to explain a homogeneous polar order state in which the direction of motion is aligned. An external symmetry-breaking guiding field such as a mechanism of taxis appears necessary to understand this phonemonon. We have investigated the onset of polar order in the velocity field induced by phototaxis in a suspension of a motile micro-organism, the algae Chlamydomonas reinhardtii, for density values above the limit provided by the hydrodynamic approximation of a force dipole model. We show that polar order originates from a combination of both the external guiding field intensity and the population density. In particular, we show evidence for a linear dependence of a phototactic guiding field on cell density to determine the polar order for dense suspensions and demonstrate the existence of a density threshold for the origin of polar order. This threshold represents the density value below which cells undergoing phototaxis are not able to maintain a homogeneous polar order state and marks the transition to ordered collective motion. Such a transition is driven by a noise dominated phototactic reorientation where the noise is modelled as a normal distribution with a variance that is inversely proportional to the guiding field strength. Finally, we discuss the role of density in dense suspensions of phototactic micro-swimmers.

Human movement is both diffusive and directed

Understanding the influence of the built environment on human movement requires quantifying spatial structure in a general sense. Because of the difficulty of this task, studies of movement dynamics often ignore spatial heterogeneity and treat movement through journey lengths or distances alone. This study analyses public bicycle data from central London to reveal that, although journey distances, directions, and frequencies of occurrence are spatially variable, their relative spatial patterns remain largely constant, suggesting the influence of a fixed spatial template. A method is presented to describe this underlying space in terms of the relative orientation of movements toward, away from, and around locations of geographical or cultural significance. This produces two fields: one of convergence and one of divergence, which are able to accurately reconstruct the observed spatial variations in movement. These two fields also reveal categorical distinctions between shorter journeys merely serving diffusion away from significant locations, and longer journeys intentionally serving transport between spatially distinct centres of collective importance. Collective patterns of human movement are thus revealed to arise from a combination of both diffusive and directed movement, with aggregate statistics such as mean travel distances primarily determined by relative numbers of these two kinds of journeys.

Thin phytoplankton layers: characteristics, mechanisms, and consequences

For over four decades, aggregations of phytoplankton known as thin layers have been observed to harbor large amounts of photosynthetic cells within narrow horizontal bands. Field observations have revealed complex linkages among thin phytoplankton layers, the physical environment, cell behavior, and higher trophic levels. Several mechanisms have been proposed to explain layer formation and persistence, in the face of the homogenizing effect of turbulent dispersion. The challenge ahead is to connect mechanistic hypotheses with field observations to gain better insight on the phenomena that shape layer dynamics. Only through a mechanistic understanding of the relevant biological and physical processes can we begin to predict the effect of thin layers on the ecology of phytoplankton and higher organisms.

A tale of three taxes: photo-gyro-gravitactic bioconvection

The term bioconvection encapsulates the intricate patterns in concentration, due to hydrodynamic instabilities, that may arise in suspensions of non-neutrally buoyant, biased swimming microorganisms. The directional bias may be due to light (phototaxis), gravity (gravitaxis), a combination of viscous and gravitational torques (gyrotaxis) or other taxes. The aim of this study is to quantify experimentally the wavelength of the initial pattern to form from an initially well-mixed suspension of unicellular, swimming green algae as a function of concentration and illumination. As this is the first such study, it is necessary to develop a robust and meticulous methodology to achieve this end. The phototactic, gyrotactic and gravitactic alga Chlamydomonas augustae was employed, with various red or white light intensities from above or below, as the three not altogether separable taxes were probed. Whilst bioconvection was found to be unresponsive to changes in red light, intriguing trends were found for pattern wavelength as a function of white light intensity, depending critically on the orientation of the illumination. These trends are explored to help unravel the mechanisms. Furthermore, comparisons are made with theoretical predictions of initial wavelengths from a recent model of photo-gyrotaxis, encouragingly revealing good qualitative agreement.

Assessing Lévy walks as models of animal foraging

The hypothesis that the optimal search strategy is a Lévy walk (LW) or Lévy flight, originally suggested in 1995, has generated an explosion of interest and controversy. Long-standing empirical evidence supporting the LW hypothesis has been overturned, while new models and data are constantly being published. Statistical methods have been criticized and new methods put forward. In parallel with the empirical studies, theoretical search models have been developed. Some theories have been disproved while others remain. Here, we gather together the current state of the art on the role of LWs in optimal foraging theory. We examine the body of theory underpinning the subject. Then we present new results showing that deviations from the idealized one-dimensional search model greatly reduce or remove the advantage of LWs. The search strategy of an LW with exponent μ = 2 is therefore not as robust as is widely thought. We also review the available techniques, and their potential pitfalls, for analysing field data. It is becoming increasingly recognized that there is a wide range of mechanisms that can lead to the apparent observation of power-law patterns. The consequence of this is that the detection of such patterns in field data implies neither that the foragers in question are performing an LW, nor that they have evolved to do so. We conclude that LWs are neither a universal optimal search strategy, nor are they as widespread in nature as was once thought.

Microbial alignment in flow changes ocean light climate

The growth of microbial cultures in the laboratory often is assessed informally with a quick flick of the wrist: dense suspensions of microorganisms produce translucent "swirls" when agitated. Here, we rationalize the mechanism behind this phenomenon and show that the same process may affect the propagation of light through the upper ocean. Analogous to the shaken test tubes, the ocean can be characterized by intense fluid motion and abundant microorganisms. We demonstrate that the swirl patterns arise when elongated microorganisms align preferentially in the direction of fluid flow and alter light scattering. Using a combination of experiments and mathematical modeling, we find that this phenomenon can be recurrent under typical marine conditions. Moderate shear rates (0.1 s(-1)) can increase optical backscattering of natural microbial assemblages by more than 20%, and even small shear rates (0.001 s(-1)) can increase backscattering from blooms of large phytoplankton by more than 30%. These results imply that fluid flow, currently neglected in models of marine optics, may exert an important control on light propagation, influencing rates of global carbon fixation and how we estimate these rates via remote sensing.

Inherent high correlation of individual motility enhances population dispersal in a heterotrophic, planktonic protist

Quantitative linkages between individual organism movements and the resulting population distributions are fundamental to understanding a wide range of ecological processes, including rates of reproduction, consumption, and mortality, as well as the spread of diseases and invasions. Typically, quantitative data are collected on either movement behaviors or population distributions, rarely both. This study combines empirical observations and model simulations to gain a mechanistic understanding and predictive ability of the linkages between both individual movement behaviors and population distributions of a single-celled planktonic herbivore. In the laboratory, microscopic 3D movements and macroscopic population distributions were simultaneously quantified in a 1L tank, using automated video- and image-analysis routines. The vertical velocity component of cell movements was extracted from the empirical data and used to motivate a series of correlated random walk models that predicted population distributions. Validation of the model predictions with empirical data was essential to distinguish amongst a number of theoretically plausible model formulations. All model predictions captured the essence of the population redistribution (mean upward drift) but only models assuming long correlation times (minute), captured the variance in population distribution. Models assuming correlation times of 8 minutes predicted the least deviation from the empirical observations. Autocorrelation analysis of the empirical data failed to identify a de-correlation time in the up to 30-second-long swimming trajectories. These minute-scale estimates are considerably greater than previous estimates of second-scale correlation times. Considerable cell-to-cell variation and behavioral heterogeneity were critical to these results. Strongly correlated random walkers were predicted to have significantly greater dispersal distances and more rapid encounters with remote targets (e.g. resource patches, predators) than weakly correlated random walkers. The tendency to disperse rapidly in the absence of aggregative stimuli has important ramifications for the ecology and biogeography of planktonic organisms that perform this kind of random walk.

Organism movement is fundamental to how organisms interact with each other and the environment. Such movements are also important on the population level and determine the spread of disease and invasion, reproduction, consumption, and mortality. Theoretical ecologists have sought to predict population dispersal rates, which are often hard to measure, from individual movement behaviors, which are often easier to measure. This problem has been non-trivial. This manuscript contributes seldom available, simultaneously measured movement behaviors and population distributions of a single celled planktonic organism. The empirical data are used to distinguish amongst a set of plausible theoretical modeling approaches to suggest that organism movements are highly correlated, meaning movement direction and speed is consistent over several minutes. Previous estimates suggested persistence only lasted several seconds. Minute-scale correlations result in much more rapid organism dispersal and greater dispersal distance, indicating that organisms encounter and impact a greater portion of their surrounding habitat than previously suspected.