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

Morphomigrational description as a new approach connecting cell's migration with its morphology

The examination of morphology and migration of cells plays substantial role in understanding the cellular behaviour, being described by plethora of quantitative parameters and models. These descriptions, however, treat cell migration and morphology as independent properties of temporal cell state, while not taking into account their strong interdependence in adherent cells. Here we present the new and simple mathematical parameter called signed morphomigrational angle (sMM angle) that links cell geometry with translocation of cell centroid, considering them as one morphomigrational behaviour. The sMM angle combined with pre-existing quantitative parameters enabled us to build a new tool called morphomigrational description, used to assign the numerical values to several cellular behaviours. Thus, the cellular activities that until now were characterized using verbal description or by complex mathematical models, are described here by a set of numbers. Our tool can be further used in automatic analysis of cell populations as well as in studies focused on cellular response to environmental directional signals.

Evidence for a consistent use of external cues by marine fish larvae for orientation

The larval stage is the main dispersive process of most marine teleost species. The degree to which larval behavior controls dispersal has been a subject of debate. Here, we apply a cross-species meta-analysis, focusing on the fundamental question of whether larval fish use external cues for directional movement (i.e., directed movement). Under the assumption that directed movement results in straighter paths (i.e., higher mean vector lengths) compared to undirected, we compare observed patterns to those expected under undirected pattern of Correlated Random Walk (CRW). We find that the bulk of larvae exhibit higher mean vector lengths than those expected under CRW, suggesting the use of external cues for directional movement. We discuss special cases which diverge from our assumptions. Our results highlight the potential contribution of orientation to larval dispersal outcomes. This finding can improve the accuracy of larval dispersal models, and promote a sustainable management of marine resources.

Mind the Queue: A Case Study in Visualizing Heterogeneous Behavioral Patterns in Livestock Sensor Data Using Unsupervised Machine Learning Techniques

Sensor technologies allow ethologists to continuously monitor the behaviors of large numbers of animals over extended periods of time. This creates new opportunities to study livestock behavior in commercial settings, but also new methodological challenges. Densely sampled behavioral data from large heterogeneous groups can contain a range of complex patterns and stochastic structures that may be difficult to visualize using conventional exploratory data analysis techniques. The goal of this research was to assess the efficacy of unsupervised machine learning tools in recovering complex behavioral patterns from such datasets to better inform subsequent statistical modeling. This methodological case study was carried out using records on milking order, or the sequence in which cows arrange themselves as they enter the milking parlor. Data was collected over a 6-month period from a closed group of 200 mixed-parity Holstein cattle on an organic dairy. Cows at the front and rear of the queue proved more consistent in their entry position than animals at the center of the queue, a systematic pattern of heterogeneity more clearly visualized using entropy estimates, a scale and distribution-free alternative to variance robust to outliers. Dimension reduction techniques were then used to visualize relationships between cows. No evidence of social cohesion was recovered, but Diffusion Map embeddings proved more adept than PCA at revealing the underlying linear geometry of this data. Median parlor entry positions from the pre- and post-pasture subperiods were highly correlated (R = 0.91), suggesting a surprising degree of temporal stationarity. Data Mechanics visualizations, however, revealed heterogeneous non-stationary among subgroups of animals in the center of the group and herd-level temporal outliers. A repeated measures model recovered inconsistent evidence of a relationships between entry position and cow attributes. Mutual conditional entropy tests, a permutation-based approach to assessing bivariate correlations robust to non-independence, confirmed a significant but non-linear association with peak milk yield, but revealed the age effect to be potentially confounded by health status. Finally, queueing records were related back to behaviors recorded via ear tag accelerometers using linear models and mutual conditional entropy tests. Both approaches recovered consistent evidence of differences in home pen behaviors across subsections of the queue.

Movement-mediated community assembly and coexistence

Organismal movement is ubiquitous and facilitates important ecological mechanisms that drive community and metacommunity composition and hence biodiversity. In most existing ecological theories and models in biodiversity research, movement is represented simplistically, ignoring the behavioural basis of movement and consequently the variation in behaviour at species and individual levels. However, as human endeavours modify climate and land use, the behavioural processes of organisms in response to these changes, including movement, become critical to understanding the resulting biodiversity loss. Here, we draw together research from different subdisciplines in ecology to understand the impact of individual-level movement processes on community-level patterns in species composition and coexistence. We join the movement ecology framework with the key concepts from metacommunity theory, community assembly and modern coexistence theory using the idea of micro-macro links, where various aspects of emergent movement behaviour scale up to local and regional patterns in species mobility and mobile-link-generated patterns in abiotic and biotic environmental conditions. These in turn influence both individual movement and, at ecological timescales, mechanisms such as dispersal limitation, environmental filtering, and niche partitioning. We conclude by highlighting challenges to and promising future avenues for data generation, data analysis and complementary modelling approaches and provide a brief outlook on how a new behaviour-based view on movement becomes important in understanding the responses of communities under ongoing environmental change.

On Modeling Ensemble Transport of Metal Reducing Motile Bacteria

Many metal reducing bacteria are motile with their run-and-tumble behavior exhibiting series of flights and waiting-time spanning multiple orders of magnitude. While several models of bacterial processes do not consider their ensemble motion, some models treat motility using an advection diffusion equation (ADE). In this study, Geobacter and Pelosinus, two metal reducing species, are used in micromodel experiments for study of their motility characteristics. Trajectories of individual cells on the order of several seconds to few minutes in duration are analyzed to provide information on (1) the length of runs, and (2) time needed to complete a run (waiting or residence time). A Continuous Time Random Walk (CTRW) model to predict ensemble breakthrough plots is developed based on the motility statistics. The results of the CTRW model and an ADE model are compared with the real breakthrough plots obtained directly from the trajectories. The ADE model is shown to be insufficient, whereas a coupled CTRW model is found to be good at predicting breakthroughs at short distances and at early times, but not at late time and long distances. The inadequacies of the simple CTRW model can possibly be improved by accounting for correlation in run length and waiting time.

Tracking random walks

In empirical studies, trajectories of animals or individuals are sampled in space and time. Yet, it is unclear how sampling procedures bias the recorded data. Here, we consider the important case of movements that consist of alternating rests and moves of random durations and study how the estimate of their statistical properties is affected by the way we measure them. We first discuss the ideal case of a constant sampling interval and short-tailed distributions of rest and move durations, and provide an exact analytical calculation of the fraction of correctly sampled trajectories. Further insights are obtained with simulations using more realistic long-tailed rest duration distributions showing that this fraction is dramatically reduced for real cases. We test our results for real human mobility with high-resolution GPS trajectories, where a constant sampling interval allows one to recover at best 18% of the movements, while over-evaluating the average trip length by a factor of 2. Using a sampling interval extracted from real communication data, we recover only 11% of the moves, a value that cannot be increased above 16% even with ideal algorithms. These figures call for a more cautious use of data in quantitative studies of individuals' movements.

TumbleScore: Run and tumble analysis for low frame-rate motility videos

Scientists often exploit the motility of peritrichously flagellated bacteria for various applications. A common alteration is modifying the frequency of mid-movement changes in direction, known as tumbles. Such differences in bacterial swimming patterns can prove difficult to quantify, especially for those without access to high-speed optical equipment. Traditionally, scientists have resorted to less accurate techniques, such as soft agar plate assays, or have been forced to invest in costly equipment. Here, we present TumbleScore, software designed to track and quantify bacterial movies with slow, as well as fast, frame-rates. Developed and fully contained within MATLAB, TumbleScore processes motility videos and returns pertinent tumbling metrics, including: (i) linear speed, (ii) rotational speed, (iii) percentage of angle changes below a given threshold, and (iv) ratio of total path length to Euclidian distance, or arc-chord ratio (ACR). In addition, TumbleScore produces a "rose graph" visualization of bacterial paths. The software was validated using both fabricated and experimental motility videos.

How to connect time-lapse recorded trajectories of motile microorganisms with dynamical models in continuous time

We provide a tool for data-driven modeling of motility, data being time-lapse recorded trajectories. Several mathematical properties of a model to be found can be gleaned from appropriate model-independent experimental statistics, if one understands how such statistics are distorted by the finite sampling frequency of time-lapse recording, by experimental errors on recorded positions, and by conditional averaging. We give exact analytical expressions for these effects in the simplest possible model for persistent random motion, the Ornstein-Uhlenbeck process. Then we describe those aspects of these effects that are valid for any reasonable model for persistent random motion. Our findings are illustrated with experimental data and Monte Carlo simulations.

Automated single particle detection and tracking for large microscopy datasets

Recent advances in optical microscopy have enabled the acquisition of very large datasets from living cells with unprecedented spatial and temporal resolutions. Our ability to process these datasets now plays an essential role in order to understand many biological processes. In this paper, we present an automated particle detection algorithm capable of operating in low signal-to-noise fluorescence microscopy environments and handling large datasets. When combined with our particle linking framework, it can provide hitherto intractable quantitative measurements describing the dynamics of large cohorts of cellular components from organelles to single molecules. We begin with validating the performance of our method on synthetic image data, and then extend the validation to include experiment images with ground truth. Finally, we apply the algorithm to two single-particle-tracking photo-activated localization microscopy biological datasets, acquired from living primary cells with very high temporal rates. Our analysis of the dynamics of very large cohorts of 10 000 s of membrane-associated protein molecules show that they behave as if caged in nanodomains. We show that the robustness and efficiency of our method provides a tool for the examination of single-molecule behaviour with unprecedented spatial detail and high acquisition rates.

Robustness of movement models: can models bridge the gap between temporal scales of data sets and behavioural processes?

Discrete-time random walks and their extensions are common tools for analyzing animal movement data. In these analyses, resolution of temporal discretization is a critical feature. Ideally, a model both mirrors the relevant temporal scale of the biological process of interest and matches the data sampling rate. Challenges arise when resolution of data is too coarse due to technological constraints, or when we wish to extrapolate results or compare results obtained from data with different resolutions. Drawing loosely on the concept of robustness in statistics, we propose a rigorous mathematical framework for studying movement models' robustness against changes in temporal resolution. In this framework, we define varying levels of robustness as formal model properties, focusing on random walk models with spatially-explicit component. With the new framework, we can investigate whether models can validly be applied to data across varying temporal resolutions and how we can account for these different resolutions in statistical inference results. We apply the new framework to movement-based resource selection models, demonstrating both analytical and numerical calculations, as well as a Monte Carlo simulation approach. While exact robustness is rare, the concept of approximate robustness provides a promising new direction for analyzing movement models.

A framework for analyzing the robustness of movement models to variable step discretization

When sampling animal movement paths, the frequency at which location measurements are attempted is a critical feature for data analysis. Important quantities derived from raw data, e.g. travel distance or sinuosity, can differ largely based on the temporal resolution of the data. Likewise, when movement models are fitted to data, parameter estimates have been demonstrated to vary with sampling rate. Thus, biological statements derived from such analyses can only be made with respect to the resolution of the underlying data, limiting extrapolation of results and comparison between studies. To address this problem, we investigate whether there are models that are robust against changes in temporal resolution. First, we propose a mathematically rigorous framework, in which we formally define robustness as a model property. We then use the framework for a thorough assessment of a range of basic random walk models, in which we also show how robustness relates to other probabilistic concepts. While we found robustness to be a strong condition met by few models only, we suggest a new method to extend models so as to make them robust. Our framework provides a new systematic, mathematically founded approach to the question if, and how, sampling rate of movement paths affects statistical inference.

From Birds to Bacteria: Generalised Velocity Jump Processes with Resting States

There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long-distance movement state and a resting or local movement state. Here, a mathematical description of this process is formulated, adapted from Friedrich et al. (Phys Rev E, 74:041103, 2006b). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of integro-partial differential equations is tumultuous, and therefore, it is necessary to both simplify and derive summary statistics. An expression for the mean squared displacement is derived, which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally, a large time diffusive approximation is considered via a Cattaneo approximation (Hillen in Discrete Continuous Dyn Syst Ser B, 5:299–318, 2003). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution.

Modelling and analysis of bacterial tracks suggest an active reorientation mechanism in Rhodobacter sphaeroides

Most free-swimming bacteria move in approximately straight lines, interspersed with random reorientation phases. A key open question concerns varying mechanisms by which reorientation occurs. We combine mathematical modelling with analysis of a large tracking dataset to study the poorly understood reorientation mechanism in the monoflagellate species Rhodobacter sphaeroides. The flagellum on this species rotates counterclockwise to propel the bacterium, periodically ceasing rotation to enable reorientation. When rotation restarts the cell body usually points in a new direction. It has been assumed that the new direction is simply the result of Brownian rotation. We consider three variants of a self-propelled particle model of bacterial motility. The first considers rotational diffusion only, corresponding to a non-chemotactic mutant strain. Two further models incorporate stochastic reorientations, describing 'run-and-tumble' motility. We derive expressions for key summary statistics and simulate each model using a stochastic computational algorithm. We also discuss the effect of cell geometry on rotational diffusion. Working with a previously published tracking dataset, we compare predictions of the models with data on individual stopping events in R. sphaeroides. This provides strong evidence that this species undergoes some form of active reorientation rather than simple reorientation by Brownian rotation.