Single Particle Trajectories Research Articles

Unraveling trajectories of diffusive particles on networks

The analysis of single-particle trajectories plays an important role in elucidating dynamics within complex environments such as those found in living cells. However, the characterization of intracellular particle motion is often confounded by confinement of the particles within nontrivial subcellular geometries. Here we focus specifically on the case of particles undergoing Brownian motion within a network of narrow tubules, as found in some cellular organelles. A computational unraveling algorithm is developed to uncouple particle motion from the confining network structure, allowing for an accurate extraction of the underlying one-dimensional diffusion coefficient, as well as differentiating between Brownian and fractional Langevin motion. We validate the algorithm with simulated trajectories and then highlight its application to an example system: analyzing the motion of membrane proteins confined in the tubules of the peripheral endoplasmic reticulum in mammalian cells. We show that these proteins undergo diffusive motion and provide a quantitative estimate of their diffusion coefficient. Our algorithm provides a generally applicable approach for disentangling geometric morphology and particle dynamics in networked architectures.

Dynamics of transient cages in a model 2D supercooled liquid

Supercooled state of a liquid is characterized by extremely slow and spatially heterogeneous dynamics attributed to caging of particles by their neighbors. The dynamics of cages in a model 2D supercooled liquid is studied by analyzing single particle trajectories. The temperature dependence of cage times and cage sizes is discussed. The average cage time is found to show power-law divergence as the critical temperature is approached. A correlation between cage sizes and cage survival time is identified. At higher temperatures, bigger cages survive for longer. However, this behavior changes at low enough temperatures, where a crossover timescale is observed beyond which the smallest cages are more likely to survive. This crossover time originates mainly due to different initial dynamics of the bigger and the smaller cages, which shows strong temperature dependence for smaller cages.

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift–diffusion processes with resetting are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. Our results can be verified in experimental set-ups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting.

TRAIT2D: a Software for Quantitative Analysis of Single Particle Diffusion Data.

Single particle tracking (SPT) is one of the most widely used tools in optical microscopy to evaluate particle mobility in a variety of situations, including cellular and model membrane dynamics. Recent technological developments, such as Interferometric Scattering microscopy, have allowed recording of long, uninterrupted single particle trajectories at kilohertz framerates. The resulting data, where particles are continuously detected and do not displace much between observations, thereby do not require complex linking algorithms. Moreover, while these measurements offer more details into the short-term diffusion behaviour of the tracked particles, they are also subject to the influence of localisation uncertainties, which are often underestimated by conventional analysis pipelines. we thus developed a Python library, under the name of TRAIT2D (Tracking Analysis Toolbox - 2D version), in order to track particle diffusion at high sampling rates, and analyse the resulting trajectories with an innovative approach. The data analysis pipeline introduced is more localisation-uncertainty aware, and also selects the most appropriate diffusion model for the data provided on a statistical basis. A trajectory simulation platform also allows the user to handily generate trajectories and even synthetic time-lapses to test alternative tracking algorithms and data analysis approaches. A high degree of customisation for the analysis pipeline, for example with the introduction of different diffusion modes, is possible from the source code. Finally, the presence of graphical user interfaces lowers the access barrier for users with little to no programming experience.

A Protocol for Studying Transcription Factor Dynamics Using Fast Single-Particle Tracking and Spot-On Model-Based Analysis.

Single-particle tracking (SPT) makes it possible to directly observe single protein diffusion dynamics in living cells over time. Thus, SPT has emerged as a powerful method to quantify the dynamics of nuclear proteins such as transcription factors (TFs). Here, we provide a protocol for conducting and analyzing SPT experiments with a focus on fast tracking ("fastSPT") of TFs in mammalian cells. First, we explore how to engineer and prepare cells for SPT experiments. Next, we examine how to optimize SPT experiments by imaging at low densities to minimize tracking errors and by using stroboscopic excitation to minimize motion-blur. Next, we discuss how to convert raw SPT data into single-particle trajectories. Finally, we illustrate how to analyze these trajectories using the kinetic modeling package Spot-On. We discuss how to use Spot-On to fit histograms of displacements and extract useful information such as the fraction of TFs that are bound and freely diffusing, and their associated diffusion coefficients.

Study on Transmission Channel and Pollution Sources Region of O3 in Qingyuan City.

Based on the Lagrange mixed single-particle trajectory model and NCEP global reanalysis meteorological data, the 72 h backward airflow trajectory in Qingyuan City in different seasons from 2018 to 2020 was analyzed by cluster analysis. Combined with the hourly average concentration data of O3, the potential source contribution factor (PSCF) analysis and concentration weighted trajectory (CWT) analysis were used to study the regional transport and possible source area of O3 in Qingyuan City and analyzed the relationship among O3 and wind speed, wind direction, NO2, and CO. The results showed that from 2018 to 2020, the most significant proportion of primary pollutants in Qingyuan City was ozone. The annual average concentration reached the highest value since monitoring in 2019. In 2020, the impact of epidemic prevention and control decreased. The daily average concentration change characteristics showed a single peak, with the highest concentration in the afternoon, the highest peak concentration in summer, followed by spring, and the lowest concentration in winter. There are differences in the concentration of O3 between different sources of airflow in Qingyuan City. The potential source contribution factor shows that the high-value covered areas are mainly in Guangzhou, Foshan, and Zhongshan, which can be considered the main potential source areas. These areas can be regarded as the main potential source areas. The concentration weight trajectory showed that external and local sources affected the O3 pollution in Qingyuan during the four seasons. The high ozone concentration in Qingyuan mainly appeared in the south wind direction, indicating that the high ozone concentration in Qingyuan was greatly affected by the external transmission of the southern Pearl River Delta. The correlation between ozone concentration and CO concentration is poor, and the effect on ozone concentration is less than that of NO2.

TRAIT2D: a Software for Quantitative Analysis of Single Particle Diffusion Data

Single particle tracking (SPT) is one of the most widely used tools in optical microscopy to evaluate particle mobility in a variety of situations, including cellular and model membrane dynamics. Recent technological developments, such as Interferometric Scattering microscopy, have allowed recording of long, uninterrupted single particle trajectories at kilohertz framerates. The resulting data, where particles are continuously detected and do not displace much between observations, thereby do not require complex linking algorithms. Moreover, while these measurements offer more details into the short-term diffusion behaviour of the tracked particles, they are also subject to the influence of localisation uncertainties, which are often underestimated by conventional analysis pipelines. we thus developed a Python library, under the name of TRAIT2D (Tracking Analysis Toolbox – 2D version), in order to track particle diffusion at high sampling rates, and analyse the resulting trajectories with an innovative approach. The data analysis pipeline introduced is more localisation-uncertainty aware, and also selects the most appropriate diffusion model for the data provided on a statistical basis. A trajectory simulation platform also allows the user to handily generate trajectories and even synthetic time-lapses to test alternative tracking algorithms and data analysis approaches. A high degree of customisation for the analysis pipeline, for example with the introduction of different diffusion modes, is possible from the source code. Finally, the presence of graphical user interfaces lowers the access barrier for users with little to no programming experience.

Experimental research on the motion behavior of particles in a rotating drum based on similarity

The large-scale industrial drums have a considerable number of material particles rotating in them, reaching 109 or more. This causes a large particle system; it is difficult to carry out a simulation study of equal ratios on such systems. In this study, the similarity criterion of the particle motion behavior in the drum with lifting strips was established using a dimensional analysis method, and the mapping relationship between the inner diameter and the rotational speed of the drum, the particle size, and the rotational speed of the drum under the same gravity field was obtained. By comparing and analyzing the movement trajectories of the particles in drums of different sizes, it was found that the movement trajectories of single and multiple particles in the rotating drums met the established similarity criterion. The results of this study can provide an effective solution to the huge computational problem of industrial applications.

Diffusion State Transitions in Single-Particle Trajectories of MET Receptor Tyrosine Kinase Measured in Live Cells

Single-particle tracking enables the analysis of the dynamics of biomolecules in living cells with nanometer spatial and millisecond temporal resolution. This technique reports on the mobility of membrane proteins and is sensitive to the molecular state of a biomolecule and to interactions with other biomolecules. Trajectories describe the mobility of single particles over time and provide information such as the diffusion coefficient and diffusion state. Changes in particle dynamics within single trajectories lead to segmentation, which allows to extract information on transitions of functional states of a biomolecule. Here, mean-squared displacement analysis is developed to classify trajectory segments into immobile, confined diffusing, and freely diffusing states, and to extract the occurrence of transitions between these modes. We applied this analysis to single-particle tracking data of the membrane receptor MET in live cells and analyzed state transitions in single trajectories of the un-activated receptor and the receptor bound to the ligand internalin B. We found that internalin B-bound MET shows an enhancement of transitions from freely and confined diffusing states into the immobile state as compared to un-activated MET. Confined diffusion acts as an intermediate state between immobile and free, as this state is most likely to change the diffusion state in the following segment. This analysis can be readily applied to single-particle tracking data of other membrane receptors and intracellular proteins under various conditions and contribute to the understanding of molecular states and signaling pathways.

Feedback and reactive flow effects on living crystal formation

Active spherical particles in a water film form aggregate on two neighboring water bubbles, where one of the bubbles’ half surface is heated by a laser source, creating a convective flow field. The specific geometry is similar to the experimental setup we used for colloidal particles (Ilday et al., 2017). Water flow is considered reactive during the simulation due to changing flow field, as well as temperature field. A feedback interaction is introduced, which may be due to the concentration of particles in some region of fluid, effectively a time-dependent accumulation. We show the effect of attractive and repulsive feedback forces, and observe, under attractive feedback, the formation of patches of aggregate distributed all over the space, and eventually a greater aggregate formation on the bubble surface. Reactive flow effects, with and without feedback effects, are analyzed, by examining the pair correlation and mean square displacement indicators, as well as the single-particle trajectories.

Astrophysical Explosions Revisited: Collisionless Coupling of Debris to Magnetized Plasma

AbstractThe coupling between a rapidly expanding cloud of ionized debris and an ambient magnetized plasma is revisited with a hybrid (kinetic ion/fluid electron) simulation code that allows a study over a wide range of plasma parameters. Over a specified range of hypothetical conditions, simple scaling laws in terms of the total debris mass and explosion speed are derived and verified for the maximal size of the debris cloud and the fraction of debris that free‐streams from the burst along the magnetic field. The amount of debris that escapes from the burst with minimal coupling to the background magnetic field increases with the debris gyroradius. Test cases with two different debris species—including a heavy minority species with a relatively large gyroradius—highlight how the collisionless coupling of the debris depends on the single particle trajectories as well as the overall conservation of energy and momentum.

TRAIT2D: a Software for Quantitative Analysis of Single Particle Diffusion Data.

Single particle tracking (SPT) is one of the most widely used tools in optical microscopy to evaluate particle mobility in a variety of situations, including cellular and model membrane dynamics. Recent technological developments, such as Interferometric Scattering microscopy, have allowed recording of long, uninterrupted single particle trajectories at kilohertz framerates. The resulting data, where particles are continuously detected and do not displace much between observations, thereby do not require complex linking algorithms. Moreover, while these measurements offer more details into the short-term diffusion behaviour of the tracked particles, they are also subject to the influence of localisation uncertainties, which are often underestimated by conventional analysis pipelines. we thus developed a Python library, under the name of TRAIT2D (Tracking Analysis Toolbox - 2D version), in order to track particle diffusion at high sampling rates, and analyse the resulting trajectories with an innovative approach. The data analysis pipeline introduced is more localisation-uncertainty aware, and also selects the most appropriate diffusion model for the data provided on a statistical basis. A trajectory simulation platform also allows the user to handily generate trajectories and even synthetic time-lapses to test alternative tracking algorithms and data analysis approaches. A high degree of customisation for the analysis pipeline, for example with the introduction of different diffusion modes, is possible from the source code. Finally, the presence of graphical user interfaces lowers the access barrier for users with little to no programming experience.

Detecting Transient Trapping from a Single Trajectory: A Structural Approach.

In this article, we introduce a new method to detect transient trapping events within a single particle trajectory, thus allowing the explicit accounting of changes in the particle’s dynamics over time. Our method is based on new measures of a smoothed recurrence matrix. The newly introduced set of measures takes into account both the spatial and temporal structure of the trajectory. Therefore, it is adapted to study short-lived trapping domains that are not visited by multiple trajectories. Contrary to most existing methods, it does not rely on using a window, sliding along the trajectory, but rather investigates the trajectory as a whole. This method provides useful information to study intracellular and plasma membrane compartmentalisation. Additionally, this method is applied to single particle trajectory data of -adrenergic receptors, revealing that receptor stimulation results in increased trapping of receptors in defined domains, without changing the diffusion of free receptors.

Learning physical properties of anomalous random walks using graph neural networks

Single particle tracking allows probing how biomolecules interact physically with their natural environments. A fundamental challenge when analysing recorded single particle trajectories is the inverse problem of inferring the physical model or class of models of the underlying random walks. Reliable inference is made difficult by the inherent stochastic nature of single particle motion, by experimental noise, and by the short duration of most experimental trajectories. Model identification is further complicated by the fact that main physical properties of random walk models are only defined asymptotically, and are thus degenerate for short trajectories. Here, we introduce a new, fast approach to inferring random walk properties based on graph neural networks (GNNs). Our approach consists in associating a vector of features with each observed position, and a sparse graph structure with each observed trajectory. By performing simulation-based supervised learning on this construct [], we show that we can reliably learn models of random walks and their anomalous exponents. The method can naturally be applied to trajectories of any length. We show its efficiency in analysing various anomalous random walks of biological relevance that were proposed in the AnDi challenge []. We explore how information is encoded in the GNN, and we show that it learns relevant physical features of the random walks. We furthermore evaluate its ability to generalize to types of trajectories not seen during training, and we show that the GNN retains high accuracy even with few parameters. We finally discuss the possibility to leverage these networks to analyse experimental data.

Disentangling Rotational Dynamics and Ordering Transitions in a System of Self-Organizing Protein Nanorods via Rotationally Invariant Latent Representations.

The dynamics of complex ordering systems with active rotational degrees of freedom exemplified by protein self-assembly is explored using a machine learning workflow that combines deep learning-based semantic segmentation and rotationally invariant variational autoencoder-based analysis of orientation and shape evolution. The latter allows for disentanglement of the particle orientation from other degrees of freedom and compensates for lateral shifts. The disentangled representations in the latent space encode the rich spectrum of local transitions that can now be visualized and explored via continuous variables. The time dependence of ensemble averages allows insight into the time dynamics of the system and, in particular, illustrates the presence of the potential ordering transition. Finally, analysis of the latent variables along the single-particle trajectory allows tracing these parameters on a single-particle level. The proposed approach is expected to be universally applicable for the description of the imaging data in optical, scanning probe, and electron microscopy seeking to understand the dynamics of complex systems where rotations are a significant part of the process.

Shear-driven rolling of DNA-adhesive microspheres

Many biologically important cell binding processes, such as the rolling of leukocytes in the vasculature, are multivalent, being mediated by large numbers of weak binding ligands. Quantitative agreement between experiments and models of rolling has been elusive and often limited by the poor understanding of the binding and unbinding kinetics of the ligands involved. Here, we present a cell-free experimental model for such rolling, consisting of polymer microspheres whose adhesion to a glass surface is mediated by ligands with well-understood force-dependent binding free energy—short complementary DNA strands. We observe robust rolling activity for certain values of the shear rate and the grafted DNA strands’ binding free energy and force sensitivity. The simulation framework developed to model leukocyte rolling, adhesive dynamics, quantitatively captures the mean rolling velocity and lateral diffusivity of the experimental particles using known values of the experimental parameters. Moreover, our model captures the velocity variations seen within the trajectories of single particles. Particle-to-particle variations can be attributed to small, plausible differences in particle characteristics. Overall, our findings confirm that state-of-the-art adhesive dynamics simulations are able to capture the complex physics of particle rolling, boding well for their extension to modeling more complex systems of rolling cells.

Reconstruction of Diffusion Coefficients and Power Exponents from Single Lagrangian Trajectories

Using discrete wavelets, a novel technique is developed to estimate turbulent diffusion coefficients and power exponents from single Lagrangian particle trajectories. The technique differs from the classical approach (Davis (1991)’s technique) because averaging over a statistical ensemble of the mean square displacement (<X2>) is replaced by averaging along a single Lagrangian trajectory X(t) = {X(t), Y(t)}. Metzler et al. (2014) have demonstrated that for an ergodic (for example, normal diffusion) flow, the mean square displacement is <X2> = limT→∞τX2(T,s), where τX2 (T, s) = 1/(T − s) ∫0T−s(X(t+Δt) − X(t))2 dt, T and s are observational and lag times but for weak non-ergodic (such as super-diffusion and sub-diffusion) flows <X2> = limT→∞≪τX2(T,s)≫, where ≪…≫ is some additional averaging. Numerical calculations for surface drifters in the Black Sea and isobaric RAFOS floats deployed at mid depths in the California Current system demonstrated that the reconstructed diffusion coefficients were smaller than those calculated by Davis (1991)’s technique. This difference is caused by the choice of the Lagrangian mean. The technique proposed here is applied to the analysis of Lagrangian motions in the Black Sea (horizontal diffusion coefficients varied from 105 to 106 cm2/s) and for the sub-diffusion of two RAFOS floats in the California Current system where power exponents varied from 0.65 to 0.72. RAFOS float motions were found to be strongly non-ergodic and non-Gaussian.

Falling balls in a viscous fluid with contact: Comparing numerical simulations with experimental data

We evaluate a number of different finite-element approaches for fluid–structure (contact) interaction problems against data from physical experiments. This consists of trajectories of single particles falling through a highly viscous fluid and rebounding off the bottom fluid tank wall. The resulting flow is in the transitional regime between creeping and turbulent flows. This type of configuration is particularly challenging for numerical methods due to the large change in the fluid domain and the contact between the wall and the particle. In the finite-element simulations, we consider both rigid body and linear elasticity models for the falling particles. In the first case, we compare the results obtained with the well-established Arbitrary Lagrangian–Eulerian (ALE) approach and an unfitted moving domain method together with a simple and common approach for contact avoidance. For the full fluid–structure interaction (FSI) problem with contact, we use a fully Eulerian approach in combination with a unified FSI-contact treatment using Nitsche's method. For higher computational efficiency, we use the geometrical symmetry of the experimental setup to reformulate the FSI system into two spatial dimensions. Finally, we show full three-dimensional ALE computations to study the effects of small perturbations in the initial state of the particle to investigate deviations from a perfectly vertical fall observed in the experiment. The methods are implemented in open-source finite element libraries, and the results are made freely available to aid reproducibility.

Study of Wound Healing Dynamics by Single Pseudo-Particle Tracking in Phase Contrast Images Acquired in Time-Lapse.

Cellular contacts modify the way cells migrate in a cohesive group with respect to a free single cell. The resulting motion is persistent and correlated, with cells’ velocities self-aligning in time. The presence of a dense agglomerate of cells makes the application of single particle tracking techniques to define cells dynamics difficult, especially in the case of phase contrast images. Here, we propose an original pipeline for the analysis of phase contrast images of the wound healing scratch assay acquired in time-lapse, with the aim of extracting single particle trajectories describing the dynamics of the wound closure. In such an approach, the membrane of the cells at the border of the wound is taken as a unicum, i.e., the wound edge, and the dynamics is described by the stochastic motion of an ensemble of points on such a membrane, i.e., pseudo-particles. For each single frame, the pipeline of analysis includes: first, a texture classification for separating the background from the cells and for identifying the wound edge; second, the computation of the coordinates of the ensemble of pseudo-particles, chosen to be uniformly distributed along the length of the wound edge. We show the results of this method applied to a glioma cell line (T98G) performing a wound healing scratch assay without external stimuli. We discuss the efficiency of the method to assess cell motility and possible applications to other experimental layouts, such as single cell motion. The pipeline is developed in the Python language and is available upon request.

A local-realistic quantum mechanical model of spin and spin entanglement

This paper aims at reproducing quantum mechanical (QM) spin and spin entanglement results using a realist, stochastic, and local approach, without the standard QM mathematical formulation. The concrete model proposed includes the description of Stern–Gerlach apparatuses and of Bell test experiments. Single particle trajectories are explicitly evaluated as a function of a few stochastic variables that they assumedly carry on. QM predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. Notably, it is shown that the proposed model, despite being both local and realist, is able to violate the Bell–CHSH inequalities by exploiting the coincidence loophole and thus intrinsically renouncing to one of the Bell’s assumptions.