Turning Angle Distribution Research Articles

Simultaneous estimation of the temporal and spatial extent of animal migration using step lengths and turning angles

BackgroundAnimals of many different species, trophic levels, and life history strategies migrate, and the improvement of animal tracking technology allows ecologists to collect increasing amounts of detailed data on these movements. Understanding when animals migrate is important for managing their populations, but is still difficult despite modelling advancements.MethodsWe designed a model that parametrically estimates the timing of migration from animal tracking data. Our model identifies the beginning and end of migratory movements as signaled by change-points in step length and turning angle distributions. To this end, we can also use the model to estimate how an animal’s movement changes when it begins migrating. In addition to a thorough simulation analysis, we tested our model on three datasets: migratory ferruginous hawks (Buteo regalis) in the Great Plains, barren-ground caribou (Rangifer tarandus groenlandicus) in northern Canada, and non-migratory brown bears (Ursus arctos) from the Canadian Arctic.ResultsOur simulation analysis suggests that our model is most useful for datasets where an increase in movement speed or directional autocorrelation is clearly detectable. We estimated the beginning and end of migration in caribou and hawks to the nearest day, while confirming a lack of migratory behaviour in the brown bears. In addition to estimating when caribou and ferruginous hawks migrated, our model also identified differences in how they migrated; ferruginous hawks achieved efficient migrations by drastically increasing their movement rates while caribou migration was achieved through significant increases in directional persistence.ConclusionsOur approach is applicable to many animal movement studies and includes parameters that can facilitate comparison between different species or datasets. We hope that rigorous assessment of migration metrics will aid understanding of both how and why animals move.

CoFee-L: A Model of Animal Displacement in Large Groups Combining Cohesion Maintenance, Feeding Area Search and Transient Leadership

Simple SummaryIn the current context of climate change and forest cover degradation, the dispersal potential of trees is an issue of great importance. This is particularly the case in the tropics, where trees often need animals to disperse their seeds, as this increases the chances of survival of the progeny and allows migration in the face of climate change. An accurate representation of animal movement is therefore necessary to study the colonization potential of trees in new areas. We have conceived an innovative mathematical model describing the individual movement of gregarious animals, integrating several fundamental features of collective behaviors: cohesion maintenance, feeding area search and leadership. The model was applied to simulate the movements of a wild-ranging troop of primates in a nature reserve. As a result, the model allowed us to simulate the movement of the troop, taking into account the movement and individual characteristics of each member, which is important to consider when the group is composed of many individuals. In the future, this model could be used to improve existing zoochoric seed dispersal models and can be coupled with dynamic vegetation models to predict the shifts of trees species distribution under future climate hypotheses.In the tropics, the conservation of tree species is closely linked to that of animals, as a large proportion of trees are zoochoric and therefore rely on the movement of animals to disperse their seeds in order to increase the chances of the survival of progeny and to allow migration in the face of climate change. Research into the prediction of animal movements is therefore critical but has so far focused only on particular features of collective behavior. In contrast, we included the concepts of cohesion maintenance, feeding area search and transient leadership in a single model, CoFee-L, and tested it to simulate the movement of a wild-ranging troop of primates (Macaca leonina). We analyzed and compared observations and simulations with a statistical physics tool (mean squared displacement) and with histograms and χ2 (for the step length and turning angle distributions). CoFee-L allowed us to simulate the physical properties of the troop’s center of mass trajectory as well as the step length and angle distributions of the field data. The parametrization of CoFee-L was rather straightforward, as it was sufficient to fix a set of parameters easily observable in the field and then to adjust the values of four parameters that have biological meaning.

Biased reorientation in the chemotaxis of peritrichous bacteria Salmonella enterica serovar Typhimurium

Many kinds of peritrichous bacteria that repeat runs and tumbles by using multiple flagella exhibit chemotaxis by sensing a difference in the concentration of the attractant or repellent between two adjacent time points. If a cell senses that the concentration of an attractant has increased, their flagellar motors decrease the switching frequency from counterclockwise to clockwise direction of rotation, which causes a longer run in swimming up the concentration gradient than swimming down. We investigated the turn angle in tumbles of peritrichous bacteria swimming across the concentration gradient of a chemoattractant because the change in the switching frequency in the rotational direction may affect the way tumbles. We tracked several hundreds of runs and tumbles of single cells of Salmonella enterica serovar Typhimurium in the concentration gradient of L-serine and found that the turn angle depends on the concentration gradient that the cell senses just before the tumble. The turn angle is biased toward a smaller value when the cells swim up the concentration gradient, whereas the distribution of the angle is almost uniform (random direction) when the cells swim down the gradient. The effect of the observed bias in the turn angle on the degree of chemotaxis was investigated by random walk simulation. In the concentration field where attractants diffuse concentrically from the point source, we found that this angular distribution clearly affects the reduction of the mean-square displacement of the cell that has started at the attractant source, that is, the bias in the turn angle distribution contributes to chemotaxis in peritrichous bacteria.

Trajectory assessment of the vulnerable marsupial Dromiciops gliroides in the Patagonian temperate forest

We studied the movement of the marsupial Dromiciops gliroides by means of an innovative approach that makes use of two complementary techniques, specifically devised for the monitoring of small animals that live in dense forests. Despite playing a key role in the Southern Temperate Forest of Patagonia, very little is known about the ecology and habitat use of this “almost threatened species” (according to IUCN), the only living representative of an entire order (Microbiotheria). We present here novel results about the movement and explored area of D. gliroides at different spatio-temporal scales using complementary approaches: the spool-and-line and a radiotelemetry techniques. Both are complemented in such a way that, while the first one allows to obtain trajectories at small spatial scale with very precise resolution for relatively short periods, the second one provides longer temporal records at larger spatial scales adding temporal resolution. We show in this work very precise nocturnal trajectories unknown so far of 41 individuals of D. gliroides using spool and line, analyzing several of their statistical properties. For instance, from the turning angle distribution we find that, after release, the animals followed paths that exhibited little angular deviation between steps. In a complementary way, using radiotelemetry, we were able to study the velocity distribution of their movements, assessing a most probable value of $$2.0\pm 0.8$$ m/min and a median value of $$7.2\pm 2.8$$ m/min. The combination of both approaches brings new opportunities for studying other cryptic and poorly known forest dwellers.

Slip slidin’ away: Demographic variation in movement behavior of the dune-dwelling lizard Meroles anchietae

The shovel-snouted sand lizard, Meroles anchietae, is a small lacertid that inhabits the aeolian sand dunes of the Namib Desert. We conducted focal observations to characterize the movement patterns of adult male, adult female, and juvenile M. anchietae in their preferred habitat: dune slip faces. Movement varied considerably among demographic classes. Both adult males and females made longer moves than juveniles, but males moved less frequently than females or juveniles. Males exhibited shorter path lengths and shorter net displacements than females or juveniles. Turn angle distributions varied among classes, with females differing from the other two classes. For all three classes, the distribution of movement directions differed from uniform, with lateral moves across the slip face being most common. Movement across the slip face reflects both the ease with which lizards can navigate the substrate and behavioral factors such as foraging strategy and social interactions. Distance and direction pairings differed both from random expectations and among the demographic classes. Movement characteristics indicate that demographic classes respond to the environment differently, even while using the same habitat.

Comparison of the abnormal diffusion characteristics of tumor cells

The random dynamics of two different tumor cell types on a substrate were observed under a microscope and analyzed based on three parameters—mean square displacement (MSD), turning angle distribution (TAD), and velocity autocorrelation function (VACF)—under well-controlled experimental conditions wherein only thermal energy existed. A two-dimensional analysis on the substrate showed linear MSD but a lowered diffusion coefficient compared to the prediction by Stokes–Einstein equation due to particle substrate interaction. Moreover, anomalous characteristics based on the MSD and TAD of the two different cells, A549 (adenocarcinomic human alveolar basal epithelial cells) and MCF-7 (breast cancer cell line; estrogen, progesterone receptors +, HER2 −), were observed, and the nonlinear MSD at a short time scale was detected. This result is attributed to the localization error from the bright-field image of a different cell taken, while the cell moved randomly on the substrate. Parameters related to random process and localization errors were employed to distinguish the two cell types. This study could be useful for the characterization of cellular properties and cell types.

Identifying the mechanism for superdiffusivity in mouse fibroblast motility.

We seek to characterize the motility of mouse fibroblasts on 2D substrates. Utilizing automated tracking techniques, we find that cell trajectories are super-diffusive, where displacements scale faster than t1/2 in all directions. Two mechanisms have been proposed to explain such statistics in other cell types: run and tumble behavior with Lévy-distributed run times, and ensembles of cells with heterogeneous speed and rotational noise. We develop an automated toolkit that directly compares cell trajectories to the predictions of each model and demonstrate that ensemble-averaged quantities such as the mean-squared displacements and velocity autocorrelation functions are equally well-fit by either model. However, neither model correctly captures the short-timescale behavior quantified by the displacement probability distribution or the turning angle distribution. We develop a hybrid model that includes both run and tumble behavior and heterogeneous noise during the runs, which correctly matches the short-timescale behaviors and indicates that the run times are not Lévy distributed. The analysis tools developed here should be broadly useful for distinguishing between mechanisms for superdiffusivity in other cells types and environments.

Animal movement tools (amt): R package for managing tracking data and conducting habitat selection analyses.

Advances in tracking technology have led to an exponential increase in animal location data, greatly enhancing our ability to address interesting questions in movement ecology, but also presenting new challenges related to data management and analysis. Step‐selection functions (SSFs) are commonly used to link environmental covariates to animal location data collected at fine temporal resolution. SSFs are estimated by comparing observed steps connecting successive animal locations to random steps, using a likelihood equivalent of a Cox proportional hazards model. By using common statistical distributions to model step length and turn angle distributions, and including habitat‐ and movement‐related covariates (functions of distances between points, angular deviations), it is possible to make inference regarding habitat selection and movement processes or to control one process while investigating the other. The fitted model can also be used to estimate utilization distributions and mechanistic home ranges. Here, we present the R package amt (animal movement tools) that allows users to fit SSFs to data and to simulate space use of animals from fitted models. The amt package also provides tools for managing telemetry data. Using fisher (Pekania pennanti) data as a case study, we illustrate a four‐step approach to the analysis of animal movement data, consisting of data management, exploratory data analysis, fitting of models, and simulating from fitted models.

Turning Angle Distribution of a Single Bacterial Cell around Attractants

Turning Angle Distribution of a Single Bacterial Cell around Attractants

How realistic are painted lightnings? Quantitative comparison of the morphology of painted and real lightnings: a psychophysical approach.

Inspired by the pioneer work of the nineteenth century photographer, William Nicholson Jennings, we studied quantitatively how realistic painted lightnings are. In order to answer this question, we examined 100 paintings and 400 photographs of lightnings. We used our software package to process and evaluate the morphology of lightnings. Three morphological parameters of the main lightning branch were analysed: (i) number of branches Nb, (ii) relative length r, and (iii) number of local maxima (peaks) Np of the turning angle distribution. We concluded: (i) Painted lightnings differ from real ones in Nb and Np. (ii) The r-values of painted and real lightnings vary in the same range. (iii) 67 and 22% of the studied painted and real lightnings were non-bifurcating (Nb = 1, meaning only the main branch), the maximum of Nb of painted and real lightnings is 11 and 51, respectively, and painted bifurcating lightnings possess mostly 2-4 branches, while real lightnings have mostly 2-10 branches. To understand these findings, we performed two psychophysical experiments with 10 test persons, whose task was to guess Nb on photographs of real lightnings which were flashed for short time periods Δt = 0.5, 0.75 and 1 s (characteristic to lightnings) on a monitor. We obtained that (i) test persons can estimate the number of lightning branches quite correctly if Nb ≤ 11. (ii) If Nb > 11, its value is strongly underestimated with exponentially increasing difference between the real and estimated numbers. (iii) The estimation is independent of the flashing period Δt of lightning photos/pictures. (iv) The estimation is more accurate, if skeletonized lightning pictures are flashed, rather than real lightning photos. These findings explain why artists usually illustrate lightnings with branches not larger than 11.

Swimming patterns of a polarly flagellated bacterium in environments of increasing complexity

The natural habitat of many bacterial swimmers is dominated by interfaces and narrow interstitial spacings where they frequently interact with the fluid boundaries in their vicinity. To quantify these interactions, we investigated the swimming behavior of the soil bacterium Pseudomonas putida in a variety of confined environments. Using microfluidic techniques, we fabricated structured microchannels with different configurations of cylindrical obstacles. In these environments, we analyzed the swimming trajectories for different obstacle densities and arrangements. Although the overall swimming pattern remained similar to movement in the bulk fluid, we observed a change in the turning angle distribution that could be attributed to collisions with the cylindrical obstacles. Furthermore, a comparison of the mean run length of the bacteria to the mean free path of a billiard particle in the same geometry indicated that, inside a densely packed environment, the trajectories of the bacterial swimmers are efficiently guided along the open spacings.

Persistent-random-walk approach to anomalous transport of self-propelled particles.

The motion of self-propelled particles is modeled as a persistent random walk. An analytical framework is developed that allows the derivation of exact expressions for the time evolution of arbitrary moments of the persistent walk's displacement. It is shown that the interplay of step length and turning angle distributions and self-propulsion produces various signs of anomalous diffusion at short time scales and asymptotically a normal diffusion behavior with a broad range of diffusion coefficients. The crossover from the anomalous short-time behavior to the asymptotic diffusion regime is studied and the parameter dependencies of the crossover time are discussed. Higher moments of the displacement distribution are calculated and analytical expressions for the time evolution of the skewness and the kurtosis of the distribution are presented.

Normal and tumoral melanocytes exhibit q-Gaussian random search patterns.

In multicellular organisms, cell motility is central in all morphogenetic processes, tissue maintenance, wound healing and immune surveillance. Hence, failures in its regulation potentiates numerous diseases. Here, cell migration assays on plastic 2D surfaces were performed using normal (Melan A) and tumoral (B16F10) murine melanocytes in random motility conditions. The trajectories of the centroids of the cell perimeters were tracked through time-lapse microscopy. The statistics of these trajectories was analyzed by building velocity and turn angle distributions, as well as velocity autocorrelations and the scaling of mean-squared displacements. We find that these cells exhibit a crossover from a normal to a super-diffusive motion without angular persistence at long time scales. Moreover, these melanocytes move with non-Gaussian velocity distributions. This major finding indicates that amongst those animal cells supposedly migrating through Lévy walks, some of them can instead perform q-Gaussian walks. Furthermore, our results reveal that B16F10 cells infected by mycoplasmas exhibit essentially the same diffusivity than their healthy counterparts. Finally, a q-Gaussian random walk model was proposed to account for these melanocytic migratory traits. Simulations based on this model correctly describe the crossover to super-diffusivity in the cell migration tracks.

The Correlated Random Walk and the Rise of Movement Ecology

Understanding why, how, and when animals move is essential to many areas of ecology and related fields. Indeed, key aspects of animal behavior, population genetics, biological control, predator–prey dynamics, ecosystem engineering, and conservation biology all hinge upon knowing what critters are moving from where to where in a landscape, including information on how quickly, how regularly, and by what route they travel. The complexities involved in such processes have spawned tremendous efforts in both field research (where goals include measuring and characterizing such movements) and theoretical research (where goals include exploring the nature and potential consequences of movement). Ecologists today routinely receive some training in both empirical and theoretical research, and in the role of statistical analyses and model fitting as a way of linking the two perspectives. However, that has not always been the case. Spatial questions in ecology were long an area where the gulf between theory and reality was particularly wide. In part, this was due to the additional mathematical challenges of spatial models, but it also due to the perhaps greater technological challenges of measuring and contextualizing animal movements.

The effectiveness of Bayesian state‐space models for estimating behavioural states from movement paths

Summary Bayesian state‐space movement models have been proposed as a method of inferring behavioural states from movement paths (Morales et al. 2004), thereby providing insight into the behavioural processes from which patterns of animal space use arise in heterogeneous environments. It is not clear, however, how effective state‐space models are at estimating behavioural states. We use stochastic simulations of two movement models to quantify how behavioural state movement characteristics affect classification error. State‐space movement models can be a highly effective approach to estimating behavioural states from movement paths. Classification accuracy was contingent upon the degree of separation between the distributions that characterize the states (e.g. step length and turn angle distributions) and the relative frequency of the behavioural states. In the best case scenarios classification accuracy approached 100%, but was close to 0% when step length and turn angle distributions of each state were similar, or when one state was rare. Mean classification accuracy was uncorrelated with path length, but the variance in classification accuracy was inversely related to path length. Importantly, we find that classification accuracy can be predicted based on the separation between distributions that characterize the movement paths, thereby providing a method of estimating classification accuracy for real movement paths. We demonstrate this approach using radiotelemetry relocation data of 34 moose (Alces alces). We conclude that Bayesian state‐space models offer powerful new opportunities for inferring behavioural states from relocation data.

Human Mammary Epithelial Cells Exhibit a Bimodal Correlated Random Walk Pattern

BackgroundOrganisms, at scales ranging from unicellular to mammals, have been known to exhibit foraging behavior described by random walks whose segments confirm to Lévy or exponential distributions. For the first time, we present evidence that single cells (mammary epithelial cells) that exist in multi-cellular organisms (humans) follow a bimodal correlated random walk (BCRW).Methodology/Principal FindingsCellular tracks of MCF-10A pBabe, neuN and neuT random migration on 2-D plastic substrates, analyzed using bimodal analysis, were found to reveal the BCRW pattern. We find two types of exponentially distributed correlated flights (corresponding to what we refer to as the directional and re-orientation phases) each having its own correlation between move step-lengths within flights. The exponential distribution of flight lengths was confirmed using different analysis methods (logarithmic binning with normalization, survival frequency plots and maximum likelihood estimation).Conclusions/SignificanceBecause of the presence of non-uniform turn angle distribution of move step-lengths within a flight and two different types of flights, we propose that the epithelial random walk is a BCRW comprising of two alternating modes with varying degree of correlations, rather than a simple persistent random walk. A BCRW model rather than a simple persistent random walk correctly matches the super-diffusivity in the cell migration paths as indicated by simulations based on the BCRW model.

Physical constraints and the evolution of different foraging strategies in aquatic space

Optimal balance between energy usage and feeding is crucial to an animal's fitness, and thus is a driving force in the evolution of species. Animals show varieties of different foraging strategies, each adapted to particular ecological and physical constraints. For example, while both the paddlefish and the zooplankton Daphnia filter-feed on patchy food sources, they differ greatly in size and in their methods of propulsion, affecting the degrees of influence of the viscous and inertial forces on these swimmers. To examine the effects of these physical constraints on the evolution of foraging strategies, we modified a recent evolution simulation designed for Daphnia by adding an energy penalty proportional to each turning angle used by modelled foraging agents. This modification accounts for the loss of linear momentum of larger swimmers as they change directions, directly contrasting with Daphnia, which, as smaller, rowing organisms, never build an appreciable momentum since they are halted by the surrounding medium after each forward stroke. While realistic random-walk-like trajectories were predicted by the simulation for Daphnia, distinct circling trajectories and nonzero peaks in the turning angle distributions were predicted for larger species, such as paddlefish. These results are strikingly similar to experimental data also reported here: the circling patterns and bimodal turning angle distributions observed in juvenile paddlefish.

Bimodal Analysis of Mammary Epithelial Cell Migration in Two Dimensions

Cell migration paths of mammary epithelial cells (expressing different versions of the promigratory tyrosine kinase receptor Her2/Neu) were analyzed within a bimodal framework that is a generalization of the run-and-tumble description applicable to bacterial migration. The mammalian cell trajectories were segregated into two types of alternating modes, namely, the "directional mode" (mode I, the more persistent mode, analogous to the bacterial run phase) and the "re-orientation mode" (mode II, the less persistent mode, analogous to the bacterial tumble phase). Higher resolution (more pixel information, relative to cell size) and smaller sampling intervals (time between images) were found to give a better estimate of the deduced single cell dynamics (such as directional-mode time and turn angle distribution) of the various cell types from the bimodal analysis. The bimodal analysis tool permits the deduction of short-time dynamics of cell motion such as the turn angle distributions and turn frequencies during the course of cell migration compared to standard methods of cell migration analysis. We find that the 2-h mammalian cell tracking data do not fall into the diffusive regime implying that the often-used random motility expressions for mammalian cell motion (based on assuming diffusive motion) are invalid over the time steps (fraction of minute) typically used in modeling mammalian cell migration.

The influence of turning angles on the success of non-oriented animal searches

Animal searches cover a full range of possibilities from highly deterministic to apparently completely random behaviors. However, even those stochastic components of animal movement can be adaptive, since not all random distributions lead to similar success in finding targets. Here we address the general problem of optimizing encounter rates in non-deterministic, non-oriented searches, both in homogeneous and patchy target landscapes. Specifically, we investigate how two different features related to turning angle distributions influence encounter success: (i) the shape (relative kurtosis) of the angular distribution and (ii) the correlations between successive relative orientations (directional memory). Such influence is analyzed in correlated random walk models using a proper choice of representative turning angle distributions of the recently proposed Jones and Pewsey class. We consider the cases of distributions with nearly the same shape but considerably distinct correlation lengths, and distributions with same correlation but with contrasting relative kurtosis. In homogeneous landscapes, we find that the correlation length has a large influence in the search efficiency. Moreover, similar search efficiencies can be reached by means of distinctly shaped turning angle distributions, provided that the resulting correlation length is the same. In contrast, in patchy landscapes the particular shape of the distribution also becomes relevant for the search efficiency, specially at high target densities. Excessively sharp distributions generate very inefficient searches in landscapes where local target density fluctuations are large. These results are of evolutionary interest. On the one hand, it is shown that equally successful directional memory can arise from contrasting turning behaviors, therefore increasing the likelihood of robust adaptive stochastic behavior. On the other hand, when target landscape is patchy, adequate tumbling may help to explore better local scale heterogeneities, being some details of the shape of the distribution also potentially adaptive.

Patch density determines movement patterns and foraging efficiency of large herbivores

Few experimental studies have tested theoretical predictions regarding the movement strategies of large herbivores and their consequences for foraging efficiency. We therefore analyze how the movement and foraging behavior of goats are related to patch density, with patches being trees and bushes. We show that their movements become slower and more tortuous when patch density increases, resulting in shorter steps, more acute turns, and a lower net displacement. Furthermore, the movements of the goats can be well described by Levy walks (LWs). In agreement with hypotheses generated by LW models, the goats move with µ 2 at low patch density but with µ 3 when patches are abundant. However, simplified statistical descriptors of movement patterns like the shape of the step/flight length and turn angle distributions become insufficient in predicting foraging efficiency when patch density is high because then the sequence of steps and turns becomes an important determinant of foraging efficiency. By changing their movements and behavior with increasing patch density, the goats intensify their utilization of resources and consequently are able to raise the efficiency of the foraging process more than proportional to the increase in patch density. This resembles the concept of area-restricted search, stating that animals concentrate their foraging effort in areas with high reward, thereby increasing the efficiency of foraging. The findings as presented in this paper provide support for theoretical expectations on the movement and foraging behavior of large herbivores in relation to resource density.