Quantitative analysis of single particle trajectories: mean maximal excursion method

Trajectory Analysis in Single-Particle Tracking: From Mean Squared Displacement to Machine Learning Approaches

Single-particle tracking is a powerful technique to investigate the motion of molecules or particles. Here, we review the methods for analyzing the reconstructed trajectories, a fundamental step for deciphering the underlying mechanisms driving the motion. First, we review the traditional analysis based on the mean squared displacement (MSD), highlighting the sometimes-neglected factors potentially affecting the accuracy of the results. We then report methods that exploit the distribution of parameters other than displacements, e.g., angles, velocities, and times and probabilities of reaching a target, discussing how they are more sensitive in characterizing heterogeneities and transient behaviors masked in the MSD analysis. Hidden Markov Models are also used for this purpose, and these allow for the identification of different states, their populations and the switching kinetics. Finally, we discuss a rapidly expanding field-trajectory analysis based on machine learning. Various approaches, from random forest to deep learning, are used to classify trajectory motions, which can be identified by motion models or by model-free sets of trajectory features, either previously defined or automatically identified by the algorithms. We also review free software available for some of the analysis methods. We emphasize that approaches based on a combination of the different methods, including classical statistics and machine learning, may be the way to obtain the most informative and accurate results.

Multifractal spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)2643-156410.1103/PhysRevResearch.5.023144], we established a comprehensive framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous-time random walk, annealed transient time motion, and Lévy walk. We generate extensive datasets comprising 10^{6} trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and p varation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

Ligand-Receptor Interaction Triggers Hopping and Sliding Motions on Living Cell Membranes

Exploring the surface-capturing and releasing processes of nanocargo on the living cell membrane is critical for understanding the membrane translocation process. In this work, we achieve total internal reflection scattering (TIRS) illumination on a commercial dark-field optical microscope without the introduction of any additional optical components. By gradually reducing the diaphragm size in the excitation light path, the angle of the incident beam can be well manipulated. Under optimal conditions, the excitation light can be totally reflected at the glass/water interface, resulting in a thin layer of evanescent field for TIRS illumination. Due to the exponential decay feature of the evanescent field, the displacement of the nanocargo along the vertical direction can be directly resolved in the intensity track. With this method, we selectively monitor the dynamics of the transferrin-modified nanocargo on the living cell membrane. Transition between confined diffusion and long-range searching is involved in the binding site recognition process, which exhibits non-Gaussian and nonergodic-like behavior. More interestingly, 2D fast sliding and 3D hopping motions are also distinguished on the fluidic cell membrane, which is essentially modulated by the strength of ligand-receptor interactions, as revealed by the free-energy profiles. These heterogeneous and dynamic interactions together control the diffusion mode of the nanocargo on the lipid membrane and, thus, determine the cellular translocation efficiency.

Parameter estimation of the fractional Ornstein-Uhlenbeck process based on quadratic variation

Modern experiments routinely produce extensive data of the diffusive dynamics of tracer particles in a large range of systems. Often, the measured diffusion turns out to deviate from the laws of Brownian motion, i.e., it is anomalous. Considerable effort has been put in conceiving methods to extract the exact parameters underlying the diffusive dynamics. Mostly, this has been done for unconfined motion of the tracer particle. Here, we consider the case when the particle is confined by an external harmonic potential, e.g., in an optical trap. The anomalous particle dynamics is described by the fractional Ornstein-Uhlenbeck process, for which we establish new estimators for the parameters. Specifically, by calculating the empirical quadratic variation of a single trajectory, we are able to recover the subordination process governing the particle motion and use it as a basis for the parameter estimation. The statistical properties of the estimators are evaluated from simulations.

Burst of hopping trafficking correlated reversible dynamic interactions between lipid droplets and mitochondria under starvation

Dynamic membrane contacts between lipid droplets (LDs) and mitochondria play key roles in lipid metabolism and energy homeostasis. Understanding the dynamics of LDs under energy stimulation is thereby crucial to disclosing the metabolic mechanism. Here, the reversible interactions between LDs and mitochondria are tracked in real-time using a robust LDs-specific fluorescent probe (LDs-Tags). Through tracking the dynamics of LDs at the single-particle level, spatiotemporal heterogeneity is revealed. LDs in starved cells communicate and integrate their activities (i.e., lipid exchange) through a membrane contact site-mediated mechanism. Thus the diffusion is intermittently alternated between active and confined states. Statistical analysis shows that the translocation of LDs in response to starvation stress is non-Gaussian, and obeys nonergodic-like behavior. These results provide deep understanding of the anomalous diffusion of LDs in living cells, and also afford guidance for rationally designing efficient transporter.

Dynamic switching of transcriptional regulators between two distinct low-mobility chromatin states

How chromatin dynamics relate to transcriptional activity remains poorly understood. Using single-molecule tracking, coupled with machine learning, we show that histone H2B and multiple chromatin-bound transcriptional regulators display two distinct low-mobility states. Ligand activation results in a marked increase in the propensity of steroid receptors to bind in the lowest-mobility state. Mutational analysis revealed that interactions with chromatin in the lowest-mobility state require an intact DNA binding domain and oligomerization domains. These states are not spatially separated as previously believed, but individual H2B and bound-TF molecules can dynamically switch between them on time scales of seconds. Single bound-TF molecules with different mobilities exhibit different dwell time distributions, suggesting that the mobility of TFs is intimately coupled with their binding dynamics. Together, our results identify two unique and distinct low-mobility states that appear to represent common pathways for transcription activation in mammalian cells.

Simulation-based inference for non-parametric statistical comparison of biomolecule dynamics

Numerous models have been developed to account for the complex properties of the random walks of biomolecules. However, when analysing experimental data, conditions are rarely met to ensure model identification. The dynamics may simultaneously be influenced by spatial and temporal heterogeneities of the environment, out-of-equilibrium fluxes and conformal changes of the tracked molecules. Recorded trajectories are often too short to reliably discern such multi-scale dynamics, which precludes unambiguous assessment of the type of random walk and its parameters. Furthermore, the motion of biomolecules may not be well described by a single, canonical random walk model. Here, we develop a two-step statistical testing scheme for comparing biomolecule dynamics observed in different experimental conditions without having to identify or make strong prior assumptions about the model generating the recorded random walks. We first train a graph neural network to perform simulation-based inference and thus learn a rich summary statistics vector describing individual trajectories. We then compare trajectories obtained in different biological conditions using a non-parametric maximum mean discrepancy (MMD) statistical test on their so-obtained summary statistics. This procedure allows us to characterise sets of random walks regardless of their generating models, without resorting to model-specific physical quantities or estimators. We first validate the relevance of our approach on numerically simulated trajectories. This demonstrates both the statistical power of the MMD test and the descriptive power of the learnt summary statistics compared to estimates of physical quantities. We then illustrate the ability of our framework to detect changes in α-synuclein dynamics at synapses in cultured cortical neurons, in response to membrane depolarisation, and show that detected differences are largely driven by increased protein mobility in the depolarised state, in agreement with previous findings. The method provides a means of interpreting the differences it detects in terms of single trajectory characteristics. Finally, we emphasise the interest of performing various comparisons to probe the heterogeneity of experimentally acquired datasets at different levels of granularity (e.g., biological replicates, fields of view, and organelles).

Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking

A panoply of new tools for tracking single particles and molecules has led to an explosion of experimental data, leading to novel insights into physical properties of living matter governing cellular development and function, health and disease. In this Perspective, we present tools to investigate the dynamics and mechanics of living systems from the molecular to cellular scale via single-particle techniques. In particular, we focus on methods to measure, interpret, and analyse complex data sets that are associated with forces, materials properties, transport, and emergent organisation phenomena within biological and soft-matter systems. Current approaches, challenges, and existing solutions in the associated fields are outlined in order to support the growing community of researchers at the interface of physics and the life sciences. Each section focuses not only on the general physical principles and the potential for understanding living matter, but also on details of practical data extraction and analysis, discussing limitations, interpretation, and comparison across different experimental realisations and theoretical frameworks. Particularly relevant results are introduced as examples. While this Perspective describes living matter from a physical perspective, highlighting experimental and theoretical physics techniques relevant for such systems, it is also meant to serve as a solid starting point for researchers in the life sciences interested in the implementation of biophysical methods.

Fluorescence microscopy imaging of a neurotransmitter receptor and its cell membrane lipid milieu

Hampered by the diffraction phenomenon, as expressed in 1873 by Abbe, applications of optical microscopy to image biological structures were for a long time limited to resolutions above the ∼200 nm barrier and restricted to the observation of stained specimens. The introduction of fluorescence was a game changer, and since its inception it became the gold standard technique in biological microscopy. The plasma membrane is a tenuous envelope of 4 nm-10 nm in thickness surrounding the cell. Because of its highly versatile spectroscopic properties and availability of suitable instrumentation, fluorescence techniques epitomize the current approach to study this delicate structure and its molecular constituents. The wide spectral range covered by fluorescence, intimately linked to the availability of appropriate intrinsic and extrinsic probes, provides the ability to dissect membrane constituents at the molecular scale in the spatial domain. In addition, the time resolution capabilities of fluorescence methods provide complementary high precision for studying the behavior of membrane molecules in the time domain. This review illustrates the value of various fluorescence techniques to extract information on the topography and motion of plasma membrane receptors. To this end I resort to a paradigmatic membrane-bound neurotransmitter receptor, the nicotinic acetylcholine receptor (nAChR). The structural and dynamic picture emerging from studies of this prototypic pentameric ligand-gated ion channel can be extrapolated not only to other members of this superfamily of ion channels but to other membrane-bound proteins. I also briefly discuss the various emerging techniques in the field of biomembrane labeling with new organic chemistry strategies oriented to applications in fluorescence nanoscopy, the form of fluorescence microscopy that is expanding the depth and scope of interrogation of membrane-associated phenomena.

Machine learning-informed predictions of nanoparticle mobility and fate in the mucus barrier

Nanomaterial diffusion through mucus is important to basic and applied areas of research such as drug delivery. However, it is often challenging to interpret nanoparticle dynamics within the mucus gel due to its heterogeneous microstructure and biochemistry. In this study, we measured the diffusion of polyethylene glycolylated nanoparticles (NPs) in human airway mucus ex vivo using multiple particle tracking and utilized machine learning to classify diffusive vs sub-diffusive NP movement. Using mathematic models that account for the mode of NP diffusion, we calculate the percentage of NPs that would cross the mucus barrier over time in airway mucus with varied total solids concentration. From this analysis, we predict rapidly diffusing NPs will cross the mucus barrier in a physiological timespan. Although less efficient, sub-diffusive "hopping" motion, a characteristic of a continuous time random walk, may also enable NPs to cross the mucus barrier. However, NPs exhibiting fractional Brownian sub-diffusion would be rapidly removed from the airways via mucociliary clearance. In samples with increased solids concentration (>5% w/v), we predict up to threefold reductions in the number of nanoparticles capable of crossing the mucus barrier. We also apply this approach to explore diffusion and to predict the fate of influenza A virus within human mucus. We predict only a small fraction of influenza virions will cross the mucus barrier presumably due to physical obstruction and adhesive interactions with mucin-associated glycans. These results provide new tools to evaluate the extent of synthetic and viral nanoparticle penetration through mucus in the lung and other tissues.

Thermal conductivity temperature dependence of water confined in nanoporous silicon

Recently, it has been shown that high density nanoconfined water was the reason of the important enhancement of the effective thermal conductivity up to a factor of 50% of a nanoporous silicon filled with water. In this work, using molecular dynamics simulations, we further investigate the role of the temperatureT(from 285 to 360 K) on the thermal conductivity enhancement of nanohybrid porous silicon and water system. Furthermore, by studying and analysing several structural and dynamical parameters of the nanoconfined water, we give physical insights of the observed phenomena. Upon increasing the temperature of the system, the thermal conductivity of the hybrid system increases reaching a maximum forT= 300 K. With this article, we prove the existence of new heat flux channels between a solid matrix and a nanoconfined liquid, with clear signatures both in the radial distribution function, mean square displacements, water molecules orientation, hydrogen bond networks and phonon density of states.

An overview of diffusion models for intracellular dynamics analysis

We present an overview of diffusion models commonly used for quantifying the dynamics of intracellular particles (e.g. biomolecules) inside eukaryotic living cells. It is established that inference on the modes of mobility of molecules is central in cell biology since it reflects interactions between structures and determines functions of biomolecules in the cell. In that context, Brownian motion is a key component in short distance transportation (e.g. connectivity for signal transduction). Another dynamical process that has been heavily studied in the past decade is the motor-mediated transport (e.g. dynein, kinesin and myosin) of molecules. Primarily supported by actin filament and microtubule network, it ensures spatial organization and temporal synchronization in the intracellular mechanisms and structures. Nevertheless, the complexity of internal structures and molecular processes in the living cell influence the molecular dynamics and prevent the systematic application of pure Brownian or directed motion modeling. On the one hand, cytoskeleton density will hinder the free displacement of the particle, a phenomenon called subdiffusion. On the other hand, the cytoskeleton elasticity combined with thermal bending can contribute a phenomenon called superdiffusion. This paper discusses the basics of diffusion modes observed in eukariotic cells, by introducing the essential properties of these processes. Applications of diffusion models include protein trafficking and transport and membrane diffusion.

Classification of particle trajectories in living cells: Machine learning versus statistical testing hypothesis for fractional anomalous diffusion

Single-particle tracking (SPT) has become a popular tool to study the intracellular transport of molecules in living cells. Inferring the character of their dynamics is important, because it determines the organization and functions of the cells. For this reason, one of the first steps in the analysis of SPT data is the identification of the diffusion type of the observed particles. The most popular method to identify the class of a trajectory is based on the mean-square displacement (MSD). However, due to its known limitations, several other approaches have been already proposed. With the recent advances in algorithms and the developments of modern hardware, the classification attempts rooted in machine learning (ML) are of particular interest. In this work, we adopt two ML ensemble algorithms, i.e., random forest and gradient boosting, to the problem of trajectory classification. We present a new set of features used to transform the raw trajectories data into input vectors required by the classifiers. The resulting models are then applied to real data for G protein-coupled receptors and G proteins. The classification results are compared to recent statistical methods going beyond MSD.

Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient

The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient's value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.

Detection of Anomalous Diffusion with Deep Residual Networks

Identification of the diffusion type of molecules in living cells is crucial to deduct their driving forces and hence to get insight into the characteristics of the cells. In this paper, deep residual networks have been used to classify the trajectories of molecules. We started from the well known ResNet architecture, developed for image classification, and carried out a series of numerical experiments to adapt it to detection of diffusion modes. We managed to find a model that has a better accuracy than the initial network, but contains only a small fraction of its parameters. The reduced size significantly shortened the training time of the model. Moreover, the resulting network has less tendency to overfitting and generalizes better to unseen data.

Modelling the effect of ribosome mobility on the rate of protein synthesis

Translation is one of the main steps in the synthesis of proteins. It consists of ribosomes that translate sequences of nucleotides encoded on mRNA into polypeptide sequences of amino acids. Ribosomes bound to mRNA move unidirectionally, while unbound ribosomes diffuse in the cytoplasm. It has been hypothesized that finite diffusion of ribosomes plays an important role in ribosome recycling and that mRNA circularization enhances the efficiency of translation, see e.g. Lodish et al. (Molecular cell biology, 8th edn, W.H. Freeman and Company, San Francisco, 2016). In order to estimate the effect of cytoplasmic diffusion on the rate of translation, we consider a totally asymmetric simple exclusion process coupled to a finite diffusive reservoir, which we call the ribosome transport model with diffusion. In this model, we derive an analytical expression for the rate of protein synthesis as a function of the diffusion constant of ribosomes, which is corroborated with results from continuous-time Monte Carlo simulations. Using a wide range of biological relevant parameters, we conclude that diffusion is not a rate limiting factor in translation initiation because diffusion is fast enough in biological cells.

Impact of Feature Choice on Machine Learning Classification of Fractional Anomalous Diffusion

The growing interest in machine learning methods has raised the need for a careful study of their application to the experimental single-particle tracking data. In this paper, we present the differences in the classification of the fractional anomalous diffusion trajectories that arise from the selection of the features used in random forest and gradient boosting algorithms. Comparing two recently used sets of human-engineered attributes with a new one, which was tailor-made for the problem, we show the importance of a thoughtful choice of the features and parameters. We also analyse the influence of alterations of synthetic training data set on the classification results. The trained classifiers are tested on real trajectories of G proteins and their receptors on a plasma membrane.

Shape of population interfaces as an indicator of mutational instability in coexisting cell populations

Cellular populations such as avascular tumors and microbial biofilms may 'invade' or grow into surrounding populations. The invading population is often comprised of a heterogeneous mixture of cells with varying growth rates. The population may also exhibit mutational instabilities, such as a heavy deleterious mutation load in a cancerous growth. We study the dynamics of a heterogeneous, mutating population competing with a surrounding homogeneous population, as one might find in a cancerous invasion of healthy tissue. We find that the shape of the population interface serves as an indicator for the evolutionary dynamics within the heterogeneous population. In particular, invasion front undulations become enhanced when the invading population is near a mutational meltdown transition or when the surrounding 'bystander' population is barely able to reinvade the mutating population. We characterize these interface undulations and the effective fitness of the heterogeneous population in one- and two-dimensional systems.

Comparing ion transport in ionic liquids and polymerized ionic liquids

Polymerized ionic liquids (polyILs) combine the unique properties of ionic liquids (ILs) with macromolecular polymers. But anion diffusivities in polyILs can be three orders of magnitude lower than that in ILs. Endeavors to improve ion transport in polyILs urgently need in-depth insights of ion transport in polyILs. As such in the work we compared ion transport in poly (1-butyl-3-vinylimidazolium-tetrafluoroborate) (poly ([BVIM]-[BF4])) polyIL and 1-butyl-3-methylimidazolium tetrafluoroborate ([BMIM]-[BF4]) IL. The diffusivities of ions in the polyIL and IL were measured and computed. According to the results of the molecular dynamics simulations performed, in the IL the coupling motion between an anion and the ions around determines the ion diffusivities, and the ion association lifetime gives the time scale of ion transport. But in the polyIL, the hopping of an anion among cages composed of cationic branch chains determines the diffusivity, and the associated anion transport time scale is the trap time, which is the time when an anion is caught inside a cage, not the ion association lifetime, as Mogurampelly et al. regarded. The calculation results of average displacements (ADs) of the polyIL chains show that, besides free volume fraction, average amplitudes of the oscillation of chains and chain translation speed lead to various diffusivities at various temperatures.

Measurement of anomalous diffusion using recurrent neural networks

Anomalous diffusion occurs in many physical and biological phenomena, when the growth of the mean squared displacement (MSD) with time has an exponent different from one. We show that recurrent neural networks (RNNs) can efficiently characterize anomalous diffusion by determining the exponent from a single short trajectory, outperforming the standard estimation based on the MSD when the available data points are limited, as is often the case in experiments. Furthermore, the RNNs can handle more complex tasks where there are no standard approaches, such as determining the anomalous diffusion exponent from a trajectory sampled at irregular times, and estimating the switching time and anomalous diffusion exponents of an intermittent system that switches between different kinds of anomalous diffusion. We validate our method on experimental data obtained from subdiffusive colloids trapped in speckle light fields and superdiffusive microswimmers.

Membrane Diffusion Occurs by Continuous-Time Random Walk Sustained by Vesicular Trafficking

Diffusion in cellular membranes is regulated by processes that occur over a range of spatial and temporal scales. These processes include membrane fluidity, interprotein and interlipid interactions, interactions with membrane microdomains, interactions with the underlying cytoskeleton, and cellular processes that result in net membrane movement. The complex, non-Brownian diffusion that results from these processes has been difficult to characterize, and moreover, the impact of factors such as membrane recycling on membrane diffusion remains largely unexplored. We have used a careful statistical analysis of single-particle tracking data of the single-pass plasma membrane protein CD93 to show that the diffusion of this protein is well described by a continuous-time random walk in parallel with an aging process mediated by membrane corrals. The overall result is an evolution in the diffusion of CD93: proteins initially diffuse freely on the cell surface but over time become increasingly trapped within diffusion-limiting membrane corrals. Stable populations of freely diffusing and corralled CD93 are maintained by an endocytic/exocytic process in which corralled CD93 is selectively endocytosed, whereas freely diffusing CD93 is replenished by exocytosis of newly synthesized and recycled CD93. This trafficking not only maintained CD93 diffusivity but also maintained the heterogeneous distribution of CD93 in the plasma membrane. These results provide insight into the nature of the biological and biophysical processes that can lead to significantly non-Brownian diffusion of membrane proteins and demonstrate that ongoing membrane recycling is critical to maintaining steady-state diffusion and distribution of proteins in the plasma membrane.

Classification of diffusion modes in single-particle tracking data: Feature-based versus deep-learning approach

Single-particle trajectories measured in microscopy experiments contain important information about dynamic processes occurring in a range of materials including living cells and tissues. However, extracting that information is not a trivial task due to the stochastic nature of the particles' movement and the sampling noise. In this paper, we adopt a deep-learning method known as a convolutional neural network (CNN) to classify modes of diffusion from given trajectories. We compare this fully automated approach working with raw data to classical machine learning techniques that require data preprocessing and extraction of human-engineered features from the trajectories to feed classifiers like random forest or gradient boosting. All methods are tested using simulated trajectories for which the underlying physical model is known. From the results it follows that CNN is usually slightly better than the feature-based methods, but at the cost of much longer processing times. Moreover, there are still some borderline cases in which the classical methods perform better than CNN.

Regulation of migration of chemotactic tumor cells by the spatial distribution of collagen fiber orientation

Collagen fibers, an important component of the extracellular matrix (ECM), can both inhibit and promote cellular migration. In vitro studies have revealed that the fibers' orientations are crucial to cellular invasion, while in vivo investigations have led to the development of tumor-associated collagen signatures (TACS) as an important prognostic factor. Studying biophysical regulation of cell invasion and the effect of the fibers' orientation not only deepens our understanding of the phenomenon, but also helps classify the TACSs precisely, which is currently lacking. We present a stochastic model for random or chemotactic migration of cells in fibrous ECM, and study the role of the various factors in it. The model provides a framework for quantitative classification of the TACSs, and reproduces quantitatively recent experimental data for cell motility. It also indicates that the spatial distribution of the fibers' orientations and extended correlations between them, hitherto ignored, as well as dynamics of cellular motion all contribute to regulation of the cells' invasion length, which represents a measure of metastatic risk. Although the fibers' orientations trivially affect randomly moving cells, their effect on chemotactic cells is completely nontrivial and unexplored, which we study in this paper.

Single-Particle Diffusion Characterization by Deep Learning

Diffusion plays a crucial role in many biological processes including signaling, cellular organization, transport mechanisms, and more. Direct observation of molecular movement by single-particle-tracking experiments has contributed to a growing body of evidence that many cellular systems do not exhibit classical Brownian motion but rather anomalous diffusion. Despite this evidence, characterization of the physical process underlying anomalous diffusion remains a challenging problem for several reasons. First, different physical processes can exist simultaneously in a system. Second, commonly used tools for distinguishing between these processes are based on asymptotic behavior, which is experimentally inaccessible in most cases. Finally, an accurate analysis of the diffusion model requires the calculation of many observables because different transport modes can result in the same diffusion power-law α, which is typically obtained from the mean-square displacements (MSDs). The outstanding challenge in the field is to develop a method to extract an accurate assessment of the diffusion process using many short trajectories with a simple scheme that is applicable at the nonexpert level. Here, we use deep learning to infer the underlying process resulting in anomalous diffusion. We implement a neural network to classify single-particle trajectories by diffusion type: Brownian motion, fractional Brownian motion and continuous time random walk. Further, we demonstrate the applicability of our network architecture for estimating the Hurst exponent for fractional Brownian motion and the diffusion coefficient for Brownian motion on both simulated and experimental data. These networks achieve greater accuracy than time-averaged MSD analysis on simulated trajectories while only requiring as few as 25 steps. When tested on experimental data, both net and ensemble MSD analysis converge to similar values; however, the net needs only half the number of trajectories required for ensemble MSD to achieve the same confidence interval. Finally, we extract diffusion parameters from multiple extremely short trajectories (10 steps) using our approach.

A Jump-Distance-Based Parameter Inference Scheme for Particulate Trajectories

This study presents an improved quantitative tool for the analysis of particulate trajectories. Particulate trajectory data appears in several different biological contexts, from the trajectory of chemotaxing bacteria to the nuclear mobility inferred from the trajectory of MS2 spots. Presently, the majority of analyses performed on particulate trajectory data have been limited to mean-squared displacement (MSD) analysis. Although simple, MSD analysis has several pitfalls, including difficulty in selecting between competing methods of motion, handling systems with multiple distinct subpopulations, and parameter extraction from limited time-series data. Here, we provide an alternative to MSD analysis using the jump distance distribution (JDD), which addresses the aforementioned issues. In particular, the method outperforms MSD analysis in the data-poor limit, thereby giving access to a larger range of temporal dynamics. In this work, we construct and validate a derivation of the JDD for different transportation modes and dimensions and implement a parameter estimation and model selection scheme. This scheme is validated, and direct improvements over MSD analysis are shown. Through an analysis of bacterial chemotaxis data, we highlight the JDD's ability to extract parameters at a variety of timescales, as well as extract underlying biological features of interest.

Rouse model with transient intramolecular contacts on a timescale of seconds recapitulates folding and fluctuation of yeast chromosomes

DNA folding and dynamics along with major nuclear functions are determined by chromosome structural properties, which remain, thus far, elusive in vivo. Here, we combine polymer modeling and single particle tracking experiments to determine the physico-chemical parameters of chromatin in vitro and in living yeast. We find that the motion of reconstituted chromatin fibers can be recapitulated by the Rouse model using mechanical parameters of nucleosome arrays deduced from structural simulations. Conversely, we report that the Rouse model shows some inconsistencies to analyze the motion and structural properties inferred from yeast chromosomes determined with chromosome conformation capture techniques (specifically, Hi-C). We hence introduce the Rouse model with Transient Internal Contacts (RouseTIC), in which random association and dissociation occurs along the chromosome contour. The parametrization of this model by fitting motion and Hi-C data allows us to measure the kinetic parameters of the contact formation reaction. Chromosome contacts appear to be transient; associated to a lifetime of seconds and characterized by an attractive energy of -0.3 to -0.5 kBT. We suggest attributing this energy to the occurrence of histone tail-DNA contacts and notice that its amplitude sets chromosomes in 'theta' conditions, in which they are poised for compartmentalization and phase separation.

Thermal skyrmion diffusion used in a reshuffler device

Magnetic skyrmions in thin films can be efficiently displaced with high speed by using spin-transfer torques^1,2 and spin-orbit torques^3-5 at low current densities. Although this favourable combination of properties has raised expectations for using skyrmions in devices^6,7, only a few publications have studied the thermal effects on the skyrmion dynamics^8-10. However, thermally induced skyrmion dynamics can be used for applications¹¹ such as unconventional computing approaches¹², as they have been predicted to be useful for probabilistic computing devices¹³. In our work, we uncover thermal diffusive skyrmion dynamics by a combined experimental and numerical study. We probed the dynamics of magnetic skyrmions in a specially tailored low-pinning multilayer material. The observed thermally excited skyrmion motion dominates the dynamics. Analysing the diffusion as a function of temperature, we found an exponential dependence, which we confirmed by means of numerical simulations. The diffusion of skyrmions was further used in a signal reshuffling device as part of a skyrmion-based probabilistic computing architecture. Owing to its inherent two-dimensional texture, the observation of a diffusive motion of skyrmions in thin-film systems may also yield insights in soft-matter-like characteristics (for example, studies of fluctuation theorems, thermally induced roughening and so on), which thus makes it highly desirable to realize and study thermal effects in experimentally accessible skyrmion systems.

Statistical testing approach for fractional anomalous diffusion classification

Taking advantage of recent single-particle tracking data, we compare the popular standard mean-squared displacement method with a statistical testing hypothesis procedure for three testing statistics and for two particle types: membrane receptors and the G proteins coupled to them. Each method results in different classifications. For this reason, more rigorous statistical tests are analyzed here in detail. The main conclusion is that the statistical testing approaches might provide good results even for short trajectories, but none of the proposed methods is "the best" for all considered examples; in other words, one needs to combine different approaches to get a reliable classification.

Bayesian analysis of single-particle tracking data using the nested-sampling algorithm: maximum-likelihood model selection applied to stochastic-diffusivity data

We employ Bayesian statistics using the nested-sampling algorithm to compare and rank multiple models of ergodic diffusion (including anomalous diffusion) as well as to assess their optimal parameters for in silico-generated and real time-series. We focus on the recently-introduced model of Brownian motion with "diffusing diffusivity"-giving rise to widely-observed non-Gaussian displacement statistics-and its comparison to Brownian and fractional Brownian motion, also for the time-series with some measurement noise. We conduct this model-assessment analysis using Bayesian statistics and the nested-sampling algorithm on the level of individual particle trajectories. We evaluate relative model probabilities and compute best-parameter sets for each diffusion model, comparing the estimated parameters to the true ones. We test the performance of the nested-sampling algorithm and its predictive power both for computer-generated (idealised) trajectories as well as for real single-particle-tracking trajectories. Our approach delivers new important insight into the objective selection of the most suitable stochastic model for a given time-series. We also present first model-ranking results in application to experimental data of tracer diffusion in polymer-based hydrogels.

Non-Markovian intracellular transport with sub-diffusion and run-length dependent detachment rate

Intracellular transport of organelles is fundamental to cell function and health. The mounting evidence suggests that this transport is in fact anomalous. However, the reasons for the anomaly is still under debate. We examined experimental trajectories of organelles inside a living cell and propose a mathematical model that describes the previously reported transition from sub-diffusive to super-diffusive motion. In order to explain super-diffusive behaviour at long times, we introduce non-Markovian detachment kinetics of the cargo: the rate of detachment is inversely proportional to the time since the last attachment. Recently, we observed the non-Markovian detachment rate experimentally in eukaryotic cells. Here we further discuss different scenarios of how this effective non-Markovian detachment rate could arise. The non-Markovian model is successful in simultaneously describing the time averaged variance (the time averaged mean squared displacement corrected for directed motion), the mean first passage time of trajectories and the multiple peaks observed in the distributions of cargo velocities. We argue that non-Markovian kinetics could be biologically beneficial compared to the Markovian kinetics commonly used for modelling, by increasing the average distance the cargoes travel when a microtubule is blocked by other filaments. In turn, sub-diffusion allows cargoes to reach neighbouring filaments with higher probability, which promotes active motion along the microtubules.

Statistical analysis of particle trajectories in living cells

Recent advances in molecular biology and fluorescence microscopy imaging have made possible the inference of the dynamics of molecules in living cells. Such inference allows us to understand and determine the organization and function of the cell. The trajectories of particles (e.g., biomolecules) in living cells, computed with the help of object tracking methods, can be modeled with diffusion processes. Three types of diffusion are considered: (i) free diffusion, (ii) subdiffusion, and (iii) superdiffusion. The mean-square displacement (MSD) is generally used to discriminate the three types of particle dynamics. We propose here a nonparametric three-decision test as an alternative to the MSD method. The rejection of the null hypothesis, i.e., free diffusion, is accompanied by claims of the direction of the alternative (subdiffusion or superdiffusion). We study the asymptotic behavior of the test statistic under the null hypothesis and under parametric alternatives which are currently considered in the biophysics literature. In addition, we adapt the multiple-testing procedure of Benjamini and Hochberg to fit with the three-decision-test setting, in order to apply the test procedure to a collection of independent trajectories. The performance of our procedure is much better than the MSD method as confirmed by Monte Carlo experiments. The method is demonstrated on real data sets corresponding to protein dynamics observed in fluorescence microscopy.

Protein motion in the nucleus: from anomalous diffusion to weak interactions

Understanding how transcription factors (TFs) regulate mammalian gene expression in space and time is a central topic in biology. To activate a gene, a TF has first to diffuse in the available space of the nucleus until it reaches a target DNA sequence or protein (target site). This eventually results in the recruitment of the whole transcriptional machinery. All these processes take place in the mammalian nucleoplasm, a highly organized and dynamic environment, in which some complexes transiently assemble and break apart, whereas others appear more stable. This diversity of dynamic behaviors arises from the number of biomolecules that make up the nucleoplasm and their pairwise interactions. Indeed, interactions energies that span several orders of magnitude, from covalent bounds to transient and dynamic interactions, can shape nuclear landscapes. Thus, the nuclear environment determines how frequently and how fast a TF contacts its target site, and it indirectly regulates gene expression. How exactly transient interactions are involved in the regulation of TF diffusion is unclear, but are reflected by live cell imaging techniques, including single-particle tracking (SPT). Overall, the macroscopic result of these microscopic interactions is almost always anomalous diffusion, a phenomenon widely studied and modeled. Here, we review the connections between the anomalous diffusion of a TF observed by SPT and the microscopic organization of the nucleus, including recently described topologically associated domains and dynamic phase-separated compartments. We propose that anomalous diffusion found in SPT data result from weak and transient interactions with dynamic nuclear substructures, and that SPT data analysis would benefit from a better description of such structures.

Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity

How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and out-of-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.

Generalized Langevin equation formulation for anomalous diffusion in the Ising model at the critical temperature

We consider the two- (2D) and three-dimensional (3D) Ising models on a square lattice at the critical temperature T_{c}, under Monte Carlo spin flip dynamics. The bulk magnetization and the magnetization of a tagged line in the 2D Ising model, and the bulk magnetization and the magnetization of a tagged plane in the 3D Ising model, exhibit anomalous diffusion. Specifically, their mean-square displacements increase as power laws in time, collectively denoted as ∼t^{c}, where c is the anomalous exponent. We argue that the anomalous diffusion in all these quantities for the Ising model stems from time-dependent restoring forces, decaying as power laws in time-also with exponent c -in striking similarity to anomalous diffusion in polymeric systems. Prompted by our previous work that has established a memory-kernel based generalized Langevin equation (GLE) formulation for polymeric systems, we show that a closely analogous GLE formulation holds for the Ising model as well. We obtain the memory kernels from spin-spin correlation functions, and the formulation allows us to consistently explain anomalous diffusion as well as anomalous response of the Ising model to an externally applied magnetic field in a consistent manner.

Exploration of bulk and interface behavior of gas molecules and 1-butyl-3-methylimidazolium tetrafluoroborate ionic liquid using equilibrium and nonequilibrium molecular dynamics simulation and quantum chemical calculation

Ionic liquids (ILs) show brilliant performance in separating gas impurities, but few researchers have performed an in-depth exploration of the bulk and interface behavior of penetrants and ILs thoroughly. In this research, we have performed a study on both molecular dynamics (MD) simulation and quantum chemical (QC) calculation to explore the transport of acetylene and ethylene in the bulk and interface regions of 1-butyl-3-methylimidazolium tetrafluoroborate ([BMIM]-[BF4]). The diffusivity, solubility and permeability of gas molecules in the bulk were researched with MD simulation first. The subdiffusion behavior of gas molecules is induced by coupling between the motion of gas molecules and the ions, and the relaxation processes of the ions after the disturbance caused by gas molecules. Then, QC calculation was performed to explore the optical geometry of ions, ion pairs and complexes of ions and penetrants, and interaction potential for pairs and complexes. Finally, nonequilibrium MD simulation was performed to explore the interface structure and properties of the IL-gas system and gas molecule behavior in the interface region. The research results may be used in the design of IL separation media.

Tackling the Challenges of Dynamic Experiments Using Liquid-Cell Transmission Electron Microscopy

Revolutions in science and engineering frequently result from the development, and wide adoption, of a new, powerful characterization or imaging technique. Beginning with the first glass lenses and telescopes in astronomy, to the development of visual-light microscopy, staining techniques, confocal microscopy, and fluorescence super-resolution microscopy in biology, and most recently aberration-corrected, cryogenic, and ultrafast (4D) electron microscopy, X-ray microscopy, and scanning probe microscopy in nanoscience. Through these developments, our perception and understanding of the physical nature of matter at length-scales beyond ordinary perception have been fundamentally transformed. Despite this progression in microscopy, techniques for observing nanoscale chemical processes and solvated/hydrated systems are limited, as the necessary spatial and temporal resolution presents significant technical challenges. However, the standard reliance on indirect or bulk phase characterization of nanoscale samples in liquids is undergoing a shift in recent times with the realization ( Williamson et al. Nat. Mater . 2003 , 2 , 532 - 536 ) of liquid-cell (scanning) transmission electron microscopy, LC(S)TEM, where picoliters of solution are hermetically sealed between electron-transparent "windows," which can be directly imaged or videoed at the nanoscale using conventional transmission electron microscopes. This Account seeks to open a discussion on the topic of standardizing strategies for conducting imaging experiments with a view to characterizing dynamics and motion of nanoscale materials. This is a challenge that could be described by critics and proponents alike, as analogous to doing chemistry in a lightning storm; where the nature of the solution, the nanomaterial, and the dynamic behaviors are all potentially subject to artifactual influence by the very act of our observation.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Comparing gas transport in three polymers via molecular dynamics simulation

Molecular dynamics (MD) simulation was employed to study the transport of methane and n-butane molecules in the bulk and interface region of polyethylene (PE), poly(4-methyl-2-pentyne) (PMP) and polydimethylsiloxane (PDMS).

Axonemal Lumen Dominates Cytosolic Protein Diffusion inside the Primary Cilium

Transport of membrane and cytosolic proteins in primary cilia is thought to depend on intraflagellar transport (IFT) and diffusion. However, the relative contribution and spatial routes of each transport mechanism are largely unknown. Although challenging to decipher, the details of these routes are essential for our understanding of protein transport in primary cilia, a critically affected process in many genetic diseases. By using a high-speed virtual 3D super-resolution microscopy, we have mapped the 3D spatial locations of transport routes for various cytosolic proteins in the 250-nm-wide shaft of live primary cilia with a spatiotemporal resolution of 2 ms and <16 nm. Our data reveal two spatially distinguishable transport routes for cytosolic proteins: an IFT-dependent path along the axoneme, and a passive-diffusion route in the axonemal lumen that escaped previous studies. While all cytosolic proteins tested primarily utilize the IFT path in the anterograde direction, differences are observed in the retrograde direction where IFT20 only utilizes IFT, and approximately half of KIF17 and one third of α–tubulin utilizes diffusion besides IFT.

Limitations of Qdot labelling compared to directly-conjugated probes for single particle tracking of B cell receptor mobility

Single-particle tracking (SPT) is a powerful method for exploring single-molecule dynamics in living cells with nanoscale spatiotemporal resolution. Photostability and bright fluorescence make quantum dots (Qdots) a popular choice for SPT. However, their large size could potentially alter the mobility of the molecule of interest. To test this, we labelled B cell receptors on the surface of B-lymphocytes with monovalent Fab fragments of antibodies that were either linked to Qdots via streptavidin or directly conjugated to the small organic fluorophore Cy3. Imaging of receptor mobility by total internal reflection fluorescence microscopy (TIRFM), followed by quantitative single-molecule diffusion and confinement analysis, definitively showed that Qdots sterically hinder lateral mobility regardless of the substrate to which the cells were adhered. Qdot labelling also drastically altered the frequency with which receptors transitioned between apparent slow- and fast-moving states and reduced the size of apparent confinement zones. Although we show that Qdot-labelled probes can detect large differences in receptor mobility, they fail to resolve subtle differences in lateral diffusion that are readily detectable using Cy3-labelled Fabs. Our findings highlight the utility and limitations of using Qdots for TIRFM and wide-field-based SPT, and have significant implications for interpreting SPT data.

Procedures for Model-Guided Data Analysis of Chromosomal Loci Dynamics at Short Time Scales

This chapter provides theoretical background and practical procedures for model-guided analysis of mobility of chromosomal loci from movies of many single trajectories. We guide the reader through existing physical models and measurable quantities, illustrating how this knowledge is useful for the interpretation of the measurements.

Unraveling intermittent features in single-particle trajectories by a local convex hull method

We propose a model-free method to detect change points between distinct phases in a single random trajectory of an intermittent stochastic process. The local convex hull (LCH) is constructed for each trajectory point, while its geometric properties (e.g., the diameter or the volume) are used as discriminators between phases. The efficiency of the LCH method is validated for six models of intermittent motion, including Brownian motion with different diffusivities or drifts, fractional Brownian motion with different Hurst exponents, and surface-mediated diffusion. We discuss potential applications of the method for detection of active and passive phases in the intracellular transport, temporal trapping or binding of diffusing molecules, alternating bulk and surface diffusion, run and tumble (or search) phases in the motion of bacteria and foraging animals, and instantaneous firing rates in neurons.

Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.

Ergodicity breaking on the neuronal surface emerges from random switching between diffusive states

Stochastic motion on the surface of living cells is critical to promote molecular encounters that are necessary for multiple cellular processes. Often the complexity of the cell membranes leads to anomalous diffusion, which under certain conditions it is accompanied by non-ergodic dynamics. Here, we unravel two manifestations of ergodicity breaking in the dynamics of membrane proteins in the somatic surface of hippocampal neurons. Three different tagged molecules are studied on the surface of the soma: the voltage-gated potassium and sodium channels Kv1.4 and Nav1.6 and the glycoprotein CD4. In these three molecules ergodicity breaking is unveiled by the confidence interval of the mean square displacement and by the dynamical functional estimator. Ergodicity breaking is found to take place due to transient confinement effects since the molecules alternate between free diffusion and confined motion.