A review of progress in single particle tracking: from methods to biophysical insights

Hour-Long, Kilohertz Sampling Rate Three-Dimensional Single-Virus Tracking in Live Cells Enabled by StayGold Fluorescent Protein Fusions

A viral infection process covers a large range of spatiotemporal scales. Tracking the viral infection process with fluorescent labels over long durations while maintaining a fast sampling rate requires bright and highly photostable labels. StayGold is a recently identified green fluorescent protein that has a greater photostability and higher signal intensity under identical illumination conditions compared to existing fluorescence protein variants. Here, StayGold protein fusions were used to generate virus-like particles (StayGold-VLPs) to achieve hour-long 3D single-virus tracking (SVT) with 1000 localizations per second (kHz sampling rate) in live cells. The expanded photon budget from StayGold protein fusions prolonged the tracking duration, facilitating a comprehensive study of viral trafficking dynamics with high temporal resolution over long time scales. The development of StayGold-VLPs presents a simple and general VLP labeling strategy for better performance in SVT, enabling exponentially more information to be collected from single trajectories and allowing for the future possibility of observing the entire life cycle of a single virus.

Nanoparticle Anisotropy Increases Targeting Interactions on Live-Cell Membranes

This paper describes how branch lengths of anisotropic nanoparticles can affect interactions between grafted ligands and cell-membrane receptors. Using live-cell, single-particle tracking, we found that DNA aptamer-gold nanostar nanoconstructs with longer branches showed improved binding efficacy to human epidermal growth factor receptor 2 (HER2) on cancer cell membranes. Inhibiting nanoconstruct-HER2 binding promoted nonspecific interactions, which increased the rotational speed of long-branched nanoconstructs but did not affect that of short-branched constructs. Bivariate analysis of the rotational and translational dynamics showed that longer branch lengths increased the ratio of targeting to nontargeting interactions. We also found that longer branches increased the nanoconstruct-cell interaction times before internalization and decreased intracellular trafficking velocities. Differences in binding efficacy revealed by single-particle dynamics can be attributed to the distinct protein corona distributions on short- and long-branched nanoconstructs, as validated by transmission electron microscopy. Minimal protein adsorption at the high positive curvature tips of long-branched nanoconstructs facilitated binding of DNA aptamer ligands to HER2. Our study reveals the significance of nanoparticle branch length in regulating local chemical environment and interactions with live cells at the single-particle level.

Inferring Stochastic Rates from Heterogeneous Snapshots of Particle Positions

Many imaging techniques for biological systems-like fixation of cells coupled with fluorescence microscopy-provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. These snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.

Hour-long, Kilohertz Sampling Rate 3D Single-virus Tracking in Live Cells Enabled by StayGold Fluorescent Protein Fusions

The viral infection process covers a large range of spatiotemporal scales. Tracking the viral infection process with fluorescent labels over long durations while maintaining a fast sampling rate requires bright and highly photostable labels. StayGold is a recently identified green fluorescent protein that has a greater photostability and higher signal intensity under identical illumination conditions as compared to existing fluorescence protein variants. Here, StayGold protein fusions were used to generate virus-like particles (StayGold-VLPs) to achieve hour-long 3D single-virus tracking (SVT) with one thousand localizations per second (kHz sampling rate) in live cells. The expanded photon budget from StayGold protein fusions prolonged the tracking duration, facilitating a comprehensive study of viral trafficking dynamics with high temporal resolution over long timescales. The development of StayGold-VLPs presents a simple and general VLP labeling strategy for better performance in SVT, enabling exponentially more information to be collected from single trajectories and allowing for the future possibility of observing the whole life cycle of a single virus.

Abstract Figure

Impact of temporal resolution in single particle tracking analysis

Temporal resolution is a key parameter in the observation of dynamic processes, as in the case of single molecules motions visualized in real time in two-dimensions by wide field (fluorescence) microscopy, but a systematic investigation of its effects in all the single particle tracking analysis steps is still lacking. Here we present tools to quantify its impact on the estimation of diffusivity and of its distribution using one of the most popular tracking software for biological applications on simulated data and movies. We found important shifts and different widths for diffusivity distributions, depending on the interplay of temporal sampling conditions with various parameters, such as simulated diffusivity, density of spots, signal-to-noise ratio, lengths of trajectories, and kind of boundaries in the simulation. We examined conditions starting from the ones of experiments on the fluorescently labelled receptor p75NTR, a relatively fast-diffusing membrane receptor (diffusivity around 0.5-1 µm2/s), visualized by TIRF microscopy on the basal membrane of living cells. From the analysis of the simulations, we identified the best conditions in cases similar to these ones; considering also the experiments, we could confirm a range of values of temporal resolution suitable for obtaining reliable diffusivity results. The procedure we present can be exploited in different single particle/molecule tracking applications to find an optimal temporal resolution.

OneFlowTraX: a user-friendly software for super-resolution analysis of single-molecule dynamics and nanoscale organization

Super-resolution microscopy (SRM) approaches revolutionize cell biology by providing insights into the nanoscale organization and dynamics of macromolecular assemblies and single molecules in living cells. A major hurdle limiting SRM democratization is post-acquisition data analysis which is often complex and time-consuming. Here, we present OneFlowTraX, a user-friendly and open-source software dedicated to the analysis of single-molecule localization microscopy (SMLM) approaches such as single-particle tracking photoactivated localization microscopy (sptPALM). Through an intuitive graphical user interface, OneFlowTraX provides an automated all-in-one solution for single-molecule localization, tracking, as well as mobility and clustering analyses. OneFlowTraX allows the extraction of diffusion and clustering parameters of millions of molecules in a few minutes. Finally, OneFlowTraX greatly simplifies data management following the FAIR (Findable, Accessible, Interoperable, Reusable) principles. We provide a detailed step-by-step manual and guidelines to assess the quality of single-molecule analyses. Applying different fluorophores including mEos3.2, PA-GFP, and PATagRFP, we exemplarily used OneFlowTraX to analyze the dynamics of plant plasma membrane-localized proteins including an aquaporin, the brassinosteroid receptor Brassinosteroid Insensitive 1 (BRI1) and the Receptor-Like Protein 44 (RLP44).

Targeted Single Particle Tracking with Upconverting Nanoparticles

Single particle tracking (SPT) is a powerful technique for real-time microscopic visualization of the movement of individual biomolecules within or on the surface of living cells. However, SPT often suffers from the suboptimal performance of the photon-emitting labels used to tag the biomolecules of interest. For example, fluorescent dyes have poor photostability, while quantum dots suffer from blinking that hampers track acquisition and interpretation. Upconverting nanoparticles (UCNPs) have recently emerged as a promising anti-Stokes luminescent label for SPT. In this work, we demonstrated targeted SPT using UCNPs. For this, we synthesized 30 nm diameter doped UCNPs and coated them with amphiphilic polymers decorated with polyethylene glycol chains to make them water-dispersible and minimize their nonspecific interactions with cells. Coated UCNPs highly homogeneous in brightness (as confirmed by a single particle investigation) were functionalized by immunoglobulin E (IgE) using a biotin-streptavidin strategy. Using these IgE-UCNP SPT labels, we tracked high-affinity IgE receptors (FcεRI) on the membrane of living RBL-2H3 mast cells at 37 °C in the presence and absence of antigen and obtained good agreement with the literature. Moreover, we used the FcεRI-IgE receptor-antibody system to directly compare the performance of UCNP-based SPT labels to organic dyes (AlexaFluor647) and quantum dots (QD655). Due to their photostability as well as their backgroundless and continuous luminescence, SPT trajectories obtained with UCNP labels are no longer limited by the photophysics of the label but only by the dynamics of the system and, in particular, the movement of the label out of the field of view and/or focal plane.

Packaging Quantum Dots in Viral Particles via a Strep-tag II/Streptavidin System for Single-Virus Tracking

Single-virus tracking provides a powerful tool for studying virus infection with high spatiotemporal resolution. Quantum dots (QDs) are used to label and track viral particles due to their brightness and photostability. However, labeling viral particles with QDs is not easy. We developed a new method for labeling viral particles with QDs by using the Strep-tag II/streptavidin system. In this method, QDs were site-specifically ligated to viral proteins in live cells and then packaged into viral-like particles (VLPs) of tick-borne encephalitis virus (TBEV) and Ebola virus during viral assembly. With TBEV VLP-QDs, we tracked the clathrin-mediated endocytic entry of TBEV and studied its intracellular dynamics at the single-particle level. Our Strep-tag II/streptavidin labeling procedure eliminates the need for BirA protein expression or biotin addition, providing a simple and general method for site-specifically labeling viral particles with QDs for single-virus tracking.

Coherent light scattering from cellular dynamics in living tissues

This review examines the biological physics of intracellular transport probed by the coherent optics of dynamic light scattering from optically thick living tissues. Cells and their constituents are in constant motion, composed of a broad range of speeds spanning many orders of magnitude that reflect the wide array of functions and mechanisms that maintain cellular health. From the organelle scale of tens of nanometers and upward in size, the motion inside living tissue is actively driven rather than thermal, propelled by the hydrolysis of bioenergetic molecules and the forces of molecular motors. Active transport can mimic the random walks of thermal Brownian motion, but mean-squared displacements are far from thermal equilibrium and can display anomalous diffusion through Lévy or fractional Brownian walks. Despite the average isotropic three-dimensional environment of cells and tissues, active cellular or intracellular transport of single light-scattering objects is often pseudo-one-dimensional, for instance as organelle displacement persists along cytoskeletal tracks or as membranes displace along the normal to cell surfaces, albeit isotropically oriented in three dimensions. Coherent light scattering is a natural tool to characterize such tissue dynamics because persistent directed transport induces Doppler shifts in the scattered light. The many frequency-shifted partial waves from the complex and dynamic media interfere to produce dynamic speckle that reveals tissue-scale processes through speckle contrast imaging and fluctuation spectroscopy. Low-coherence interferometry, dynamic optical coherence tomography, diffusing-wave spectroscopy, diffuse-correlation spectroscopy, differential dynamic microscopy and digital holography offer coherent detection methods that shed light on intracellular processes. In health-care applications, altered states of cellular health and disease display altered cellular motions that imprint on the statistical fluctuations of the scattered light. For instance, the efficacy of medical therapeutics can be monitored by measuring the changes they induce in the Doppler spectra of livingex vivocancer biopsies.

Variance sum rule for entropy production

Entropy production is the hallmark of nonequilibrium physics, quantifying irreversibility, dissipation, and the efficiency of energy transduction processes. Despite many efforts, its measurement at the nanoscale remains challenging. We introduce a variance sum rule (VSR) for displacement and force variances that permits us to measure the entropy production rate σ in nonequilibrium steady states. We first illustrate it for directly measurable forces, such as an active Brownian particle in an optical trap. We then apply the VSR to flickering experiments in human red blood cells. We find that σ is spatially heterogeneous with a finite correlation length, and its average value agrees with calorimetry measurements. The VSR paves the way to derive σ using force spectroscopy and time-resolved imaging in living and active matter.

Fluorescent-Probe Characterization for Pore-Space Mapping with Single-Particle Tracking

Porous solids often contain complex pore networks with pores of various sizes. Tracking individual fluorescent probes as they diffuse through porous materials can be used to characterize pore networks at tens of nanometers resolution. However, understanding the motion behavior of fluorescent probes in confinement is crucial to reliably derive pore network properties. Here, we introduce well-defined lithography-made model pores developed to study probe behavior in confinement. We investigated the influence of probe-host interactions on diffusion and trapping of confined single-emitter quantum-dot probes. Using the pH-responsiveness of the probes, we were able to largely suppress trapping at the pore walls. This enabled us to define experimental conditions for mapping of the accessible pore space of a one-dimensional pore array as well as a real-life polymerization-catalyst-support particle.

Anomalous diffusion, non-Gaussianity, nonergodicity, and confinement in stochastic-scaled Brownian motion with diffusing diffusivity dynamics

Scaled Brownian motions (SBMs) with power-law time-dependent diffusivity have been used to describe various types of anomalous diffusion yet Gaussian observed in granular gases kinetics, turbulent diffusion, and molecules mobility in cells, to name a few. However, some of these systems may exhibit non-Gaussian behavior which can be described by SBM with diffusing diffusivity (DD-SBM). Here, we numerically investigate both free and confined DD-SBM models characterized by fixed or stochastic scaling exponent of time-dependent diffusivity. The effects of distributed scaling exponent, random diffusivity, and confinement are considered. Different regimes of ultraslow diffusion, subdiffusion, normal diffusion, and superdiffusion are observed. In addition, weak ergodic and non-Gaussian behaviors are also detected. These results provide insights into diffusion in time-fluctuating diffusivity landscapes with potential applications to time-dependent temperature systems spreading in heterogeneous environments.

Measuring the electrophoretic mobility and size of single particles using microfluidic transverse AC electrophoresis (TrACE)

The ability to measure the charge and size of single particles is essential to understanding particle adhesion and interaction with their environment. Characterizing the physical properties of biological particles, like cells, can be a powerful tool in studying the association between the changes in physical properties and disease development. Currently, measuring charge via the electrophoretic mobility (μep) of individual particles remains challenging, and there is only one prior report of simultaneously measuring μep and size. We introduce microfluidic transverse AC electrophoresis (TrACE), a novel technique that combines particle tracking velocimetry (PTV) and AC electrophoresis. In TrACE, electric waves with 0.75 to 1.5 V amplitude are applied transversely to the bulk flow and cause the particles to oscillate. PTV records the particles' oscillating trajectories as pressure drives bulk flow through the microchannel. A simple quasi-equilibrium model agrees well with experimental measurements of frequency, amplitude, and phase, indicating that particle motion is largely described by DC electrophoresis. The measured μep of polystyrene particles (0.53, 0.84, 1, and 2 μm diameter) are consistent with ELS measurements, and precision is enhanced by averaging ∼100 measurements per particle. Particle size is simultaneously measured from Brownian motion quantified from the trajectory for particles <2 μm or image analysis for particles ≥2 μm. Lastly, the ability to analyze intact mammalian cells is demonstrated with B cells. TrACE systems are expected to be highly suitable as fieldable tools to measure the μep and size of a broad range of individual particles.

u-track3D: Measuring, navigating, and validating dense particle trajectories in three dimensions

We describe u-track3D, a software package that extends the versatile u-track framework established in 2D to address the specific challenges of 3D particle tracking. First, we present the performance of the new package in quantifying a variety of intracellular dynamics imaged by multiple 3D microcopy platforms and on the standard 3D test dataset of the particle tracking challenge. These analyses indicate that u-track3D presents a tracking solution that is competitive to both conventional and deep-learning-based approaches. We then present the concept of dynamic region of interest (dynROI), which allows an experimenter to interact with dynamic 3D processes in 2D views amenable to visual inspection. Third, we present an estimator of trackability that automatically defines a score for every trajectory, thereby overcoming the challenges of trajectory validation by visual inspection. With these combined strategies, u-track3D provides a complete framework for unbiased studies of molecular processes in complex volumetric sequences.

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.

Inferring pointwise diffusion properties of single trajectories with deep learning

To characterize the mechanisms governing the diffusion of particles in biological scenarios, it is essential to accurately determine their diffusive properties. To do so, we propose a machine-learning method to characterize diffusion processes with time-dependent properties at the experimental time resolution. Our approach operates at the single-trajectory level predicting the properties of interest, such as the diffusion coefficient or the anomalous diffusion exponent, at every time step of the trajectory. In this way, changes in the diffusive properties occurring along the trajectory emerge naturally in the prediction and thus allow the characterization without any prior knowledge or assumption about the system. We first benchmark the method on synthetic trajectories simulated under several conditions. We show that our approach can successfully characterize both abrupt and continuous changes in the diffusion coefficient or the anomalous diffusion exponent. Finally, we leverage the method to analyze experiments of single-molecule diffusion of two membrane proteins in living cells: the pathogen-recognition receptor DC-SIGN and the integrin α5β1. The analysis allows us to characterize physical parameters and diffusive states with unprecedented accuracy, shedding new light on the underlying mechanisms.

Single-molecule tracking of myelin basic protein during oligodendrocyte differentiation

This study aimed to expand our understanding of myelin basic protein (MBP), a key component of central nervous system myelin, by developing a protocol to track and quantifying individual MBP particles during oligodendrocyte (OL) differentiation. MBP particle directionality, confinement, and diffusion were tracked by rapid TIRF and HILO imaging of Dendra2 tagged MBP in three stages of mouse oligodendroglia: OL precursors, early myelinating OLs, and mature myelinating OLs. The directionality and confinement of MBP particles increased at each stage consistent with progressive transport toward, and recruitment into, emerging myelin structures. Unexpectedly, diffusion data presented a more complex pattern with subpopulations of the most diffusive particles disappearing at the transition between the precursor and early myelinating stage, before reemerging in the membrane sheets of mature OLs. This diversity of particle behaviors, which would be undetectable by conventional ensemble-averaged methods, are consistent with a multifunctional view of MBP involving roles in myelin expansion and compaction.

Quantitative analysis of peroxisome tracks using a Hidden Markov Model

Diffusion and mobility are essential for cellular functions, as molecules are usually distributed throughout the cell and have to meet to interact and perform their function. This also involves the cytosolic migration of cellular organelles. However, observing such diffusion and interaction dynamics is challenging due to the high spatial and temporal resolution required and the accurate analysis of the diffusional tracks. The latter is especially important when identifying anomalous diffusion events, such as directed motions, which are often rare. Here, we investigate the migration modes of peroxisome organelles in the cytosol of living cells. Peroxisomes predominantly migrate randomly, but occasionally they bind to the cell's microtubular network and perform directed migration, which is difficult to quantify, and so far, accurate analysis of switching between these migration modes is missing. We set out to solve this limitation by experiments and analysis with high statistical accuracy. Specifically, we collect temporal diffusion tracks of thousands of individual peroxisomes in the HEK 293 cell line using two-dimensional spinning disc fluorescence microscopy at a high acquisition rate of 10 frames/s. We use a Hidden Markov Model with two hidden states to (1) automatically identify directed migration segments of the tracks and (2) quantify the migration properties for comparison between states and between different experimental conditions. Comparing different cellular conditions, we show that the knockout of the peroxisomal membrane protein PEX14 leads to a decrease in the directed movement due to a lowered binding probability to the microtubule. However, it does not eradicate binding, highlighting further microtubule-binding mechanisms of peroxisomes than via PEX14. In contrast, structural changes of the microtubular network explain perceived eradication of directed movement by disassembly of microtubules by Nocodazole-treatment.

Compartmentalization of androgen receptors at endogenous genes in living cells

A wide range of nuclear proteins are involved in the spatio-temporal organization of the genome through diverse biological processes such as gene transcription and DNA replication. Upon stimulation by testosterone and translocation to the nucleus, multiple androgen receptors (ARs) accumulate in microscopically discernable foci which are irregularly distributed in the nucleus. Here, we investigated the formation and physical nature of these foci, by combining novel fluorescent labeling techniques to visualize a defined chromatin locus of AR-regulated genes-PTPRN2 or BANP-simultaneously with either AR foci or individual AR molecules. Quantitative colocalization analysis showed evidence of AR foci formation induced by R1881 at both PTPRN2 and BANP loci. Furthermore, single-particle tracking (SPT) revealed three distinct subdiffusive fractional Brownian motion (fBm) states: immobilized ARs were observed near the labeled genes likely as a consequence of DNA-binding, while the intermediate confined state showed a similar spatial behavior but with larger displacements, suggesting compartmentalization by liquid-liquid phase separation (LLPS), while freely mobile ARs were diffusing in the nuclear environment. All together, we show for the first time in living cells the presence of AR-regulated genes in AR foci.

Langevin picture of subdiffusion in nonuniformly expanding medium

Anomalous diffusion phenomena have been observed in many complex physical and biological systems. One significant advance recently is the physical extension of particle's motion in a static medium to a uniformly and even nonuniformly expanding medium. The dynamic mechanism of the anomalous diffusion in the nonuniformly expanding medium has only been investigated by the approach of continuous-time random walk. To study more physical observables and to supplement the physical models of the anomalous diffusion in the expanding mediums, we characterize the nonuniformly expanding medium with a spatiotemporal dependent scale factor a(x,t) and build the Langevin picture describing the particle's motion in the nonuniformly expanding medium. Besides the existing comoving and physical coordinates, by introducing a new coordinate and assuming that a(x,t) is separable at a long-time limit, we build the relation between the nonuniformly expanding medium and the uniformly expanding one and further obtain the moments of the comoving and physical coordinates. Different forms of the scale factor a(x,t) are considered to uncover the combined effects of the particle's intrinsic diffusion and the nonuniform expansion of medium. The theoretical analyses and simulations provide the foundation for studying more anomalous diffusion phenomena in the expanding mediums.

Enhanced diffusion of tracer particles in nonreciprocal mixtures

We study the diffusivity of a tagged particle in a binary mixture of Brownian particles with nonreciprocal interactions. Numerical simulations reveal that, for a broad class of interaction potentials, nonreciprocity can significantly increase the long-time diffusion coefficient of tracer particles and that this diffusion enhancement is associated with a breakdown of the Einstein relation. These observations are quantified and confirmed via two different and complementary analytical approaches: (i) a linearized stochastic density field theory, which is particularly accurate in the limit of soft interactions, and (ii) a reduced two-body description, which is exact at leading order in the density of particles. The latter reveals that diffusion enhancement can be attributed to the formation of transiently propelled dimers of particles, whose cohesion and speed are controlled by the nonreciprocal interactions.

Heterodyne dynamic light scattering for the characterization of particle dispersions

Particle self-diffusivities in unimodal and bimodal aqueous dispersions are characterized by dynamic light scattering (DLS) applying a heterodyne detection scheme. For unimodal dispersions close to infinite dilution, it could be evidenced that pure homodyne conditions cannot be realized, leading to an increasing underestimation of diffusivity with a decreasing concentration. Even for bimodal dispersions and neglecting any local oscillator field, the coherent superposition of scattered light from different particle species hinders a clear assignment of the measured signals and their evaluation for diffusivity. In this case, the impact of a cross term on the determined diffusivities cannot be neglected. The results emphasize that the use of a heterodyne detection scheme in DLS experiments is a key aspect for an accurate determination of particle diffusivities in low-concentrated unimodal and bimodal dispersions.

Transcription Factor Dynamics: One Molecule at a Time

Cells must tightly regulate their gene expression programs and yet rapidly respond to acute biochemical and biophysical cues within their environment. This information is transmitted to the nucleus through various signaling cascades, culminating in the activation or repression of target genes. Transcription factors (TFs) are key mediators of these signals, binding to specific regulatory elements within chromatin. While live-cell imaging has conclusively proven that TF-chromatin interactions are highly dynamic, how such transient interactions can have long-term impacts on developmental trajectories and disease progression is still largely unclear. In this review, we summarize our current understanding of the dynamic nature of TF functions, starting with a historical overview of early live-cell experiments. We highlight key factors that govern TF dynamics and how TF dynamics, in turn, affect downstream transcriptional bursting. Finally, we conclude with open challenges and emerging technologies that will further our understanding of transcriptional regulation.

Sorzano CO. TrackAnalyzer: A Fiji/ImageJ toolbox for a holistic analysis of tracks

Current live-cell imaging techniques make possible the observation of live events and the acquisition of large datasets to characterize the different parameters of the visualized events. They provide new insights into the dynamics of biological processes with unprecedented spatial and temporal resolutions. Here we describe the implementation and application of a new tool called TrackAnalyzer, accessible from Fiji and ImageJ. Our tool allows running semi-automated single-particle tracking (SPT) and subsequent motion classification, as well as quantitative analysis of diffusion and intensity for selected tracks relying on the graphical user interface (GUI) for large sets of temporal images (X-Y-T or X-Y-C-T dimensions). TrackAnalyzer also allows 3D visualization of the results as overlays of either spots, cells or end-tracks over time, along with corresponding feature extraction and further classification according to user criteria. Our analysis workflow automates the following steps: (1) spot or cell detection and filtering, (2) construction of tracks, (3) track classification and analysis (diffusion and chemotaxis), and (4) detailed analysis and visualization of all the outputs along the pipeline. All these analyses are automated and can be run in batch mode for a set of similar acquisitions.

Live-cell imaging uncovers the relationship between histone acetylation, transcription initiation, and nucleosome mobility

Histone acetylation and RNA polymerase II phosphorylation are associated with transcriptionally active chromatin, but their spatiotemporal relationship in live cells remains poorly understood. To address this problem, we combine Fab-based labeling of endogenous protein modifications with single-molecule tracking to quantify the dynamics of chromatin enriched with histone H3 lysine-27 acetylation (H3K27ac) and RNA polymerase II serine-5 phosphorylation (RNAP2-Ser5ph). Our analysis reveals that chromatin enriched with these two modifications is generally separate. In these separated sites, we show that the two modifications are inversely correlated with one another on the minutes time scale and that single nucleosomes within each region display distinct and opposing dynamics on the subsecond time scale. While nucleosomes diffuse ~15% faster in chromatin enriched with H3K27ac, they diffuse ~15% slower in chromatin enriched with RNAP2-Ser5ph. These results argue that high levels of H3K27ac and RNAP2-Ser5ph are not often present together at the same place and time, but rather each marks distinct transcriptionally poised or active sites, respectively.

Confined modes of single-particle trajectories induced by stochastic resetting

Random trajectories of single particles in living cells contain information about the interaction between particles, as well as with the cellular environment. However, precise consideration of the underlying stochastic properties, beyond normal diffusion, remains a challenge as applied to each particle trajectory separately. In this paper, we show how positions of confined particles in living cells can obey not only the Laplace distribution, but the Linnik one. This feature is detected in experimental data for the motion of G proteins and coupled receptors in cells, and its origin is explained in terms of stochastic resetting. This resetting process generates power-law waiting times, giving rise to the Linnik statistics in confined motion, and also includes exponentially distributed times as a limit case leading to the Laplace one. The stochastic process, which is affected by the resetting, can be Brownian motion commonly found in cells. Other possible models producing similar effects are discussed.

The LDL receptor is regulated by membrane cholesterol as revealed by fluorescence fluctuation analysis

Membrane cholesterol-rich domains have been shown to be important for regulating a range of membrane protein activities. Low-density lipoprotein receptor (LDLR)-mediated internalization of cholesterol-rich LDL particles is tightly regulated by feedback mechanisms involving intracellular sterol sensors. Since LDLR plays a role in maintaining cellular cholesterol homeostasis, we explore the role that membrane domains may have in regulating LDLR activity. We expressed a fluorescent LDLR-mEGFP construct in HEK293T cells and imaged the unligated receptor or bound to an LDL/DiI fluorescent ligand using total internal reflection fluorescence microscopy. We studied the receptor's spatiotemporal dynamics using fluorescence fluctuation analysis methods. Image cross correlation spectroscopy reveals a lower LDL-to-LDLR binding fraction when membrane cholesterol concentrations are augmented using cholesterol esterase, and a higher binding fraction when the cells are treated with methyl-β-cyclodextrin) to lower membrane cholesterol. This suggests that LDLR's ability to metabolize LDL particles is negatively correlated to membrane cholesterol concentrations. We then tested if a change in activity is accompanied by a change in membrane localization. Image mean-square displacement analysis reveals that unligated LDLR-mEGFP and ligated LDLR-mEGFP/LDL-DiI constructs are transiently confined on the cell membrane, and the size of their confinement domains increases with augmented cholesterol concentrations. Receptor diffusion within the domains and their domain-escape probabilities decrease upon treatment with methyl-β-cyclodextrin, consistent with a change in receptor populations to more confined domains, likely clathrin-coated pits. We propose a feedback model to account for regulation of LDLR within the cell membrane: when membrane cholesterol concentrations are high, LDLR is sequestered in cholesterol-rich domains. These LDLR populations are attenuated in their efficacy to bind and internalize LDL. However, when membrane cholesterol levels drop, LDL has a higher binding affinity to its receptor and the LDLR transits to nascent clathrin-coated domains, where it diffuses at a slower rate while awaiting internalization.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Switch of cell migration modes orchestrated by changes of three-dimensional lamellipodium structure and intracellular diffusion

Cell migration plays important roles in many biological processes, but how migrating cells orchestrate intracellular molecules and subcellular structures to regulate their speed and direction is still not clear. Here, by characterizing the intracellular diffusion and the three-dimensional lamellipodium structures of fish keratocyte cells, we observe a strong positive correlation between the intracellular diffusion and cell migration speed and, more importantly, discover a switching of cell migration modes with reversible intracellular diffusion variation and lamellipodium structure deformation. Distinct from the normal fast mode, cells migrating in the newly-found slow mode have a deformed lamellipodium with swollen-up front and thinned-down rear, reduced intracellular diffusion and compartmentalized macromolecule distribution in the lamellipodium. Furthermore, in turning cells, both lamellipodium structure and intracellular diffusion dynamics are also changed, with left-right symmetry breaking. We propose a mechanism involving the front-localized actin polymerization and increased molecular crowding in the lamellipodium to explain how cells spatiotemporally coordinate the intracellular diffusion dynamics and the lamellipodium structure in regulating their migrations.

Anomalous diffusion, non-Gaussianity, and nonergodicity for subordinated fractional Brownian motion with a drift

The stochastic motion of a particle with long-range correlated increments (the moving phase) which is intermittently interrupted by immobilizations (the trapping phase) in a disordered medium is considered in the presence of an external drift. In particular, we consider trapping events whose times follow a scale-free distribution with diverging mean trapping time. We construct this process in terms of fractional Brownian motion with constant forcing in which the trapping effect is introduced by the subordination technique, connecting "operational time" with observable "real time." We derive the statistical properties of this process such as non-Gaussianity and nonergodicity, for both ensemble and single-trajectory (time) averages. We demonstrate nice agreement with extensive simulations for the probability density function, skewness, kurtosis, as well as ensemble and time-averaged mean-squared displacements. We place a specific emphasis on the comparisons between the cases with and without drift.

Image Analysis Techniques for In Vivo Quantification of Cerebrospinal Fluid Flow

Over the last decade, there has been a tremendously increased interest in understanding the neurophysiology of cerebrospinal fluid (CSF) flow, which plays a crucial role in clearing metabolic waste from the brain. This growing interest was largely initiated by two significant discoveries: the glymphatic system (a pathway for solute exchange between interstitial fluid deep within the brain and the CSF surrounding the brain) and meningeal lymphatic vessels (lymphatic vessels in the layer of tissue surrounding the brain that drain CSF). These two CSF systems work in unison, and their disruption has been implicated in several neurological disorders including Alzheimer's disease, stoke, and traumatic brain injury. Here, we present experimental techniques for in vivo quantification of CSF flow via direct imaging of fluorescent microspheres injected into the CSF. We discuss detailed image processing methods, including registration and masking of stagnant particles, to improve the quality of measurements. We provide guidance for quantifying CSF flow through particle tracking and offer tips for optimizing the process. Additionally, we describe techniques for measuring changes in arterial diameter, which is an hypothesized CSF pumping mechanism. Finally, we outline how these same techniques can be applied to cervical lymphatic vessels, which collect fluid downstream from meningeal lymphatic vessels. We anticipate that these fluid mechanical techniques will prove valuable for future quantitative studies aimed at understanding mechanisms of CSF transport and disruption, as well as for other complex biophysical systems.

Dual-color DNA-PAINT single-particle tracking enables extended studies of membrane protein interactions

DNA-PAINT based single-particle tracking (DNA-PAINT-SPT) has recently significantly enhanced observation times in in vitro SPT experiments by overcoming the constraints of fluorophore photobleaching. However, with the reported implementation, only a single target can be imaged and the technique cannot be applied straight to live cell imaging. Here we report on leveraging this technique from a proof-of-principle implementation to a useful tool for the SPT community by introducing simultaneous live cell dual-color DNA-PAINT-SPT for quantifying protein dimerization and tracking proteins in living cell membranes, demonstrating its improved performance over single-dye SPT.

Recent Advances in Single-Molecule Tracking and Imaging Techniques

Since the early 1990s, single-molecule detection in solution at room temperature has enabled direct observation of single biomolecules at work in real time and under physiological conditions, providing insights into complex biological systems that the traditional ensemble methods cannot offer. In particular, recent advances in single-molecule tracking techniques allow researchers to follow individual biomolecules in their native environments for a timescale of seconds to minutes, revealing not only the distinct pathways these biomolecules take for downstream signaling but also their roles in supporting life. In this review, we discuss various single-molecule tracking and imaging techniques developed to date, with an emphasis on advanced three-dimensional (3D) tracking systems that not only achieve ultrahigh spatiotemporal resolution but also provide sufficient working depths suitable for tracking single molecules in 3D tissue models. We then summarize the observables that can be extracted from the trajectory data. Methods to perform single-molecule clustering analysis and future directions are also discussed.

Quantification of cell migration: metrics selection to model application

Cell migration plays an essential role in physiological and pathological states, such as immune response, tissue generation and tumor development. This phenomenon can occur spontaneously or it can be triggered by an external stimuli, including biochemical, mechanical, or electrical cues that induce or direct cells to migrate. The migratory response to these cues is foundational to several fields including neuroscience, cancer and regenerative medicine. Various platforms are available to qualitatively and quantitatively measure cell migration, making the measurements of cell motility straight-forward. Migratory behavior must be analyzed by multiple metrics and then models to connect the measurements to physiological meaning. This review will focus on describing and quantifying cell movement for individual cell migration.

Spatially Resolving Size Effects on Diffusivity in Nanoporous Extracellular Matrix-like Materials with Fluorescence Correlation Spectroscopy Super-Resolution Optical Fluctuation Imaging

It is well documented that the nanoscale structures within porous microenvironments greatly impact the diffusion dynamics of molecules. However, how the interaction between the environment and molecules influences the diffusion dynamics has not been thoroughly explored. Here, we show that fluorescence correlation spectroscopy super-resolution optical fluctuation imaging (fcsSOFI) can be used to accurately measure the diffusion dynamics of molecules within varying matrices such as nanopatterned surfaces and porous agarose hydrogels. Our data demonstrate the robustness of fcsSOFI, where it is possible not only to quantify the diffusion speeds of molecules in heterogeneous media but also to recover the matrix structure with resolution on the order of 100 nm. Using dextran molecules of varying sizes, we show that the diffusion coefficient is sensitive to the change in the molecular hydrodynamic radius. fcsSOFI images further reveal that smaller dextran molecules can freely move through the small pores of the hydrogel and report the detailed porous structure and local diffusion heterogeneities not captured by the average diffusion coefficient. Conversely, bigger dextran molecules are confined and unable to freely move through the hydrogel, highlighting only the larger pore structures. These findings establish fcsSOFI as a powerful tool to characterize spatial and diffusion information of diverse macromolecules within biorelevant matrices.

Quantitative Single-Molecule Localization Microscopy

Super-resolution fluorescence microscopy allows the investigation of cellular structures at nanoscale resolution using light. Current developments in super-resolution microscopy have focused on reliable quantification of the underlying biological data. In this review, we first describe the basic principles of super-resolution microscopy techniques such as stimulated emission depletion (STED) microscopy and single-molecule localization microscopy (SMLM), and then give a broad overview of methodological developments to quantify super-resolution data, particularly those geared toward SMLM data. We cover commonly used techniques such as spatial point pattern analysis, colocalization, and protein copy number quantification but also describe more advanced techniques such as structural modeling, single-particle tracking, and biosensing. Finally, we provide an outlook on exciting new research directions to which quantitative super-resolution microscopy might be applied.

★Track: Inferred counting and tracking of replicating DNA loci

Fluorescent microscopy is the primary method to study DNA organization within cells. However, the variability and low signal/noise commonly associated with live-cell time-lapse imaging challenges quantitative measurements. In particular, obtaining quantitative or mechanistic insight often depends on the accurate tracking of fluorescent particles. Here, we present ★Track, an inference method that determines the most likely temporal tracking of replicating intracellular particles such DNA loci while accounting for missing, merged, and spurious detections. It allows the accurate prediction of particle copy numbers as well as the timing of replication events. We demonstrate ★Track's abilities and gain new insight into plasmid copy number control and the volume dependence of bacterial chromosome replication initiation. By enabling the accurate tracking of DNA loci, ★Track can help to uncover the mechanistic principles of chromosome organization and dynamics across a range of systems.

The magic of unraveling genome architecture and function

Over the last decades, technological breakthroughs in super-resolution microscopy have allowed us to reach molecular resolution and design experiments of unprecedented complexity. Investigating how chromatin is folded in 3D, from the nucleosome level up to the entire genome, is becoming possible by "magic" (imaging genomic), i.e., the combination of imaging and genomic approaches. This offers endless opportunities to delve into the relationship between genome structure and function. Here, we review recently achieved objectives and the conceptual and technical challenges the field of genome architecture is currently undertaking. We discuss what we have learned so far and where we are heading. We elucidate how the different super-resolution microscopy approaches and, more specifically, live-cell imaging have contributed to the understanding of genome folding. Moreover, we discuss how future technical developments could address remaining open questions.

Single-Molecule Tracking of Reagent Diffusion during Chemical Reactions

Recent experiments have shown that the diffusion of reagent molecules is inconsistent with what the Stokes-Einstein equation predicts during a chemical reaction. Here, we used single-molecule tracking to observe the diffusion of reactive reagent molecules during click and Diels-Alder (DA) reactions. We found that the diffusion coefficient of the reagents remained unchanged within the experimental uncertainty upon the DA reaction. Yet, diffusion of reagent molecules is faster than predicted during the click reaction when the reagent concentration and catalyst concentration exceed a threshold. A stepwise analysis suggested that the fast diffusion scenario is due to the reaction but not the involvement of the tracer with the reaction itself. The present results provide experimental evidence on the faster-than-expected reagent diffusion during a CuAAC reaction in specific conditions and propose new insights into understanding this unexpected behavior.

Identification of fluorescently-barcoded nanoparticles using machine learning

Barcoding of nano- and micro-particles allows distinguishing multiple targets at the same time within a complex mixture and is emerging as a powerful tool to increase the throughput of many assays. Fluorescent barcoding is one of the most used strategies, where microparticles are labeled with dyes and classified based on fluorescence color, intensity, or other features. Microparticles are ideal targets due to their relative ease of detection, manufacturing, and higher homogeneity. Barcoding is considerably more challenging in the case of nanoparticles (NPs), where their small size results in a lower signal and greater heterogeneity. This is a significant limitation since many bioassays require the use of nano-sized carriers. In this study, we introduce a machine-learning-assisted workflow to write, read, and classify barcoded PLGA-PEG NPs at a single-particle level. This procedure is based on the encapsulation of fluorescent markers without modifying their physicochemical properties (writing), the optimization of their confocal imaging (reading), and the implementation of a machine learning-based barcode reader (classification). We found nanoparticle heterogeneity as one of the main factors that challenges barcode separation, and that information extracted from the dyes' nanoscale confinement effects (such as Förster Resonance Energy Transfer, FRET) can aid barcode identification. Moreover, we provide a guide to reaching the optimal trade-off between the number of simultaneous barcodes and classification accuracy supporting the use of this workflow for a variety of bioassays.

Single-Molecule Displacement Mapping Indicates Unhindered Intracellular Diffusion of Small (≲1 kDa) Solutes

While fundamentally important, the intracellular diffusion of small (≲1 kDa) solutes has been difficult to elucidate due to challenges in both labeling and measurement. Here we quantify and spatially map the translational diffusion patterns of small solutes in mammalian cells by integrating several recent advances. In particular, by executing tandem stroboscopic illumination pulses down to 400 μs separation, we extend single-molecule displacement/diffusivity mapping (SMdM), a super-resolution diffusion quantification tool, to small solutes with high diffusion coefficients D of >300 μm2/s. We thus show that for multiple water-soluble dyes and dye-tagged nucleotides, intracellular diffusion is dominated by vast regions of high diffusivity ∼60-70% of that in vitro, up to ∼250 μm2/s in the fastest cases. Meanwhile, we also visualize sub-micrometer foci of substantial slowdowns in diffusion, thus underscoring the importance of spatially resolving the local diffusion behavior. Together, these results suggest that the intracellular diffusion of small solutes is only modestly scaled down by the slightly higher viscosity of the cytosol over water but otherwise not further hindered by macromolecular crowding. We thus lift a paradoxically low speed limit for intracellular diffusion suggested by previous experiments.

Palmitate-mediated disruption of the endoplasmic reticulum decreases intracellular vesicle motility

Essential cellular processes such as metabolism, protein synthesis, and autophagy require the intracellular transport of membrane-bound vesicles. The importance of the cytoskeleton and associated molecular motors for transport is well documented. Recent research has suggested that the endoplasmic reticulum (ER) may also play a role in vesicle transport through a tethering of vesicles to the ER. We use single-particle tracking fluorescence microscopy and a Bayesian change-point algorithm to characterize vesicle motility in response to the disruption of the ER, actin, and microtubules. This high-throughput change-point algorithm allows us to efficiently analyze thousands of trajectory segments. We find that palmitate-mediated disruption of the ER leads to a significant decrease in vesicle motility. A comparison with the disruption of actin and microtubules shows that disruption of the ER has a significant impact on vesicle motility, greater than the disruption of actin. Vesicle motility was dependent on cellular region, with greater motility in the cell periphery than the perinuclear region, possibly due to regional differences in actin and the ER. Overall, these results suggest that the ER is an important factor in vesicle transport.

A machine learning approach to discover migration modes and transition dynamics of heterogeneous dendritic cells

Dendritic cell (DC) migration is crucial for mounting immune responses. Immature DCs (imDCs) reportedly sense infections, while mature DCs (mDCs) move quickly to lymph nodes to deliver antigens to T cells. However, their highly heterogeneous and complex innate motility remains elusive. Here, we used an unsupervised machine learning (ML) approach to analyze long-term, two-dimensional migration trajectories of Granulocyte-macrophage colony-stimulating factor (GMCSF)-derived bone marrow-derived DCs (BMDCs). We discovered three migratory modes independent of the cell state: slow-diffusive (SD), slow-persistent (SP), and fast-persistent (FP). Remarkably, imDCs more frequently changed their modes, predominantly following a unicyclic SD→FP→SP→SD transition, whereas mDCs showed no transition directionality. We report that DC migration exhibits a history-dependent mode transition and maturation-dependent motility changes are emergent properties of the dynamic switching of the three migratory modes. Our ML-based investigation provides new insights into studying complex cellular migratory behavior.

Correlative light-electron microscopy using small gold nanoparticles as single probes

Correlative light-electron microscopy (CLEM) requires the availability of robust probes which are visible both in light and electron microscopy. Here we demonstrate a CLEM approach using small gold nanoparticles as a single probe. Individual gold nanoparticles bound to the epidermal growth factor protein were located with nanometric precision background-free in human cancer cells by light microscopy using resonant four-wave mixing (FWM), and were correlatively mapped with high accuracy to the corresponding transmission electron microscopy images. We used nanoparticles of 10 nm and 5 nm radius, and show a correlation accuracy below 60 nm over an area larger than 10 µm size, without the need for additional fiducial markers. Correlation accuracy was improved to below 40 nm by reducing systematic errors, while the localisation precision is below 10 nm. Polarisation-resolved FWM correlates with nanoparticle shapes, promising for multiplexing by shape recognition in future applications. Owing to the photostability of gold nanoparticles and the applicability of FWM microscopy to living cells, FWM-CLEM opens up a powerful alternative to fluorescence-based methods.

Machine-learning-powered extraction of molecular diffusivity from single-molecule images for super-resolution mapping

While critical to biological processes, molecular diffusion is difficult to quantify, and spatial mapping of local diffusivity is even more challenging. Here we report a machine-learning-enabled approach, pixels-to-diffusivity (Pix2D), to directly extract the diffusion coefficient D from single-molecule images, and consequently enable super-resolved D spatial mapping. Working with single-molecule images recorded at a fixed framerate under typical single-molecule localization microscopy (SMLM) conditions, Pix2D exploits the often undesired yet evident motion blur, i.e., the convolution of single-molecule motion trajectory during the frame recording time with the diffraction-limited point spread function (PSF) of the microscope. Whereas the stochastic nature of diffusion imprints diverse diffusion trajectories to different molecules diffusing at the same given D, we construct a convolutional neural network (CNN) model that takes a stack of single-molecule images as the input and evaluates a D-value as the output. We thus validate robust D evaluation and spatial mapping with simulated data, and with experimental data successfully characterize D differences for supported lipid bilayers of different compositions and resolve gel and fluidic phases at the nanoscale.

Trap-and-Track for Characterizing Surfactants at Interfaces

Understanding the behavior of surfactants at interfaces is crucial for many applications in materials science and chemistry. Optical tweezers combined with trajectory analysis can become a powerful tool for investigating surfactant characteristics. In this study, we perform trap-and-track analysis to compare the behavior of cetyltrimethylammonium bromide (CTAB) and cetyltrimethylammonium chloride (CTAC) at water-glass interfaces. We use optical tweezers to trap a gold nanoparticle and statistically analyze the particle's movement in response to various surfactant concentrations, evidencing the rearrangement of surfactants adsorbed on glass surfaces. Our results show that counterions have a significant effect on surfactant behavior at the interface. The greater binding affinity of bromide ions to CTA+ micelle surfaces reduces the repulsion among surfactant head groups and enhances the mobility of micelles adsorbed on the interface. Our study provides valuable insights into the behavior of surfactants at interfaces and highlights the potential of optical tweezers for surfactant research. The development of this trap-and-track approach can have important implications for various applications, including drug delivery and nanomaterials.

Gramian angular fields for leveraging pretrained computer vision models with anomalous diffusion trajectories

Anomalous diffusion is present at all scales, from atomic to large ones. Some exemplary systems are ultracold atoms, telomeres in the nucleus of cells, moisture transport in cement-based materials, arthropods' free movement, and birds' migration patterns. The characterization of the diffusion gives critical information about the dynamics of these systems and provides an interdisciplinary framework with which to study diffusive transport. Thus, the problem of identifying underlying diffusive regimes and inferring the anomalous diffusion exponent α with high confidence is critical to physics, chemistry, biology, and ecology. Classification and analysis of raw trajectories combining machine learning techniques with statistics extracted from them have widely been studied in the Anomalous Diffusion Challenge [Muñoz-Gil et al., Nat. Commun. 12, 6253 (2021)2041-172310.1038/s41467-021-26320-w]. Here we present a new data-driven method for working with diffusive trajectories. This method utilizes Gramian angular fields (GAF) to encode one-dimensional trajectories as images (Gramian matrices), while preserving their spatiotemporal structure for input to computer-vision models. This allows us to leverage two well-established pretrained computer-vision models, ResNet and MobileNet, to characterize the underlying diffusive regime and infer the anomalous diffusion exponent α. Short raw trajectories of lengths between 10 and 50 are commonly encountered in single-particle tracking experiments and are the most difficult ones to characterize. We show that GAF images can outperform the current state-of-the-art while increasing accessibility to machine learning methods in an applied setting.

Langevin picture of anomalous diffusion processes in expanding medium

The expanding medium is very common in many different fields, such as biology and cosmology. It brings a nonnegligible influence on particle's diffusion, which is quite different from the effect of an external force field. The dynamic mechanism of a particle's motion in an expanding medium has only been investigated in the framework of a continuous-time random walk. To focus on more diffusion processes and physical observables, we build the Langevin picture of anomalous diffusion in an expanding medium, and conduct detailed analyses in the framework of the Langevin equation. With the help of a subordinator, both subdiffusion process and superdiffusion process in the expanding medium are discussed. We find that the expanding medium with different changing rate (exponential form and power-law form) leads to quite different diffusion phenomena. The particle's intrinsic diffusion behavior also plays an important role. Our detailed theoretical analyses and simulations present a panoramic view of investigating anomalous diffusion in an expanding medium under the framework of the Langevin equation.

Anomalous fluctuations of extremes in many-particle diffusion

In many-particle diffusions, particles that move the furthest and fastest can play an outsized role in physical phenomena. A theoretical understanding of the behavior of such extreme particles is nascent. A classical model, in the spirit of Einstein's treatment of single-particle diffusion, has each particle taking independent homogeneous random walks. This, however, neglects the fact that all particles diffuse in a common and often inhomogeneous environment that can affect their motion. A more sophisticated model treats this common environment as a space-time random biasing field which influences each particle's independent motion. While the bulk (or typical particle) behavior of these two models has been found to match to high degree, recent theoretical work of G. Barraquand and I. Corwin, Probab. Theory Relat. Fields 167, 1057 (2017)0178-805110.1007/s00440-016-0699-z and G. Barraquand and P. Le Doussal, J. Phys. A: Math. Theor. 53, 215002 (2020)1751-811310.1088/1751-8121/ab8b39 on a one-dimensional exactly solvable version of this random environment model suggests that the extreme behavior is quite different between the two models. We transform these asymptotic (in system size and time) results into physically applicable predictions. Using high-precision numerical simulations we reconcile different asymptotic phases in a manner that matches numerics down to realistic system sizes, amenable to experimental confirmation. We characterize the behavior of extreme diffusion in the random environment model by the presence of a new phase with anomalous fluctuations related to the Kardar-Parisi-Zhang universality class and equation.

Single-molecule displacement mapping indicates unhindered intracellular diffusion of small (<~1 kDa) solutes

While fundamentally important, the intracellular diffusion of small (<~1 kDa) solutes has been difficult to elucidate due to challenges in both labeling and measurement. Here we quantify and spatially map the translational diffusion patterns of small solutes in mammalian cells by integrating several recent advances. In particular, by executing tandem stroboscopic illumination pulses down to 400-μs separation, we extend single-molecule displacement/diffusivity mapping (SM d M), a super-resolution diffusion quantification tool, to small solutes with high diffusion coefficients D of >300 μm ² /s. We thus show that for multiple water-soluble dyes and dye-tagged nucleotides, intracellular diffusion is dominated by vast regions of high diffusivity ~60-70% of that in vitro , up to ~250 μm ² /s in the fastest cases. Meanwhile, we also visualize sub-micrometer foci of substantial slowdowns in diffusion, thus underscoring the importance of spatially resolving the local diffusion behavior. Together, these results suggest that the intracellular diffusion of small solutes is only modestly scaled down by the slightly higher viscosity of the cytosol over water, but otherwise not further hindered by macromolecular crowding. We thus lift a paradoxically low speed limit for intracellular diffusion suggested by previous experiments.

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