Optimal diffusion coefficient estimation in single-particle tracking

Single-Molecule Fluorescence Microscopy Reveals Local Diffusion Coefficients in the Pore Network of an Individual Catalyst Particle

We used single-molecule fluorescence microscopy to study self-diffusion of a feedstock-like probe molecule with nanometer accuracy in the macropores of a micrometer-sized, real-life fluid catalytic cracking (FCC) particle. Movies of single fluorescent molecules allowed their movement through the pore network to be reconstructed. The observed tracks were classified into three different states by machine learning and all found to be distributed homogeneously over the particle. Most probe molecules (88%) were immobile, with the molecule most likely being physisorbed or trapped; the remainder was either mobile (8%), with the molecule moving inside the macropores, or showed hybrid behavior (4%). Mobile tracks had an average diffusion coefficient of D = 8 × 10-14 ± 1 × 10-13 m2 s-1, with the standard deviation thought to be related to the large range of pore sizes found in FCC particles. The developed methodology can be used to evaluate, quantify and map heterogeneities in diffusional properties within complex hierarchically porous materials.

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.

Unraveling the Thousand Word Picture: An Introduction to Super-Resolution Data Analysis

Super-resolution microscopy provides direct insight into fundamental biological processes occurring at length scales smaller than light's diffraction limit. The analysis of data at such scales has brought statistical and machine learning methods into the mainstream. Here we provide a survey of data analysis methods starting from an overview of basic statistical techniques underlying the analysis of super-resolution and, more broadly, imaging data. We subsequently break down the analysis of super-resolution data into four problems: the localization problem, the counting problem, the linking problem, and what we've termed the interpretation problem.

Single-molecule and super-resolution imaging of transcription in living bacteria

In vivo single-molecule and super-resolution techniques are transforming our ability to study transcription as it takes place in its native environment in living cells. This review will detail the methods for imaging single molecules in cells, and the data-analysis tools which can be used to extract quantitative information on the spatial organization, mobility, and kinetics of the transcription machinery from these experiments. Furthermore, we will highlight studies which have applied these techniques to shed new light on bacterial transcription.

Maximum likelihood estimations of force and mobility from single short Brownian trajectories

We describe a method to extract force and diffusion parameters from single trajectories of Brownian particles. The analysis, based on the principle of maximum likelihood, is well-suited for out-of-equilibrium trajectories, even when a limited amount of data is available and the dynamical parameters vary spatially. We substantiate this method with experimental and simulated data, and discuss its practical implementation, strengths, and limitations.

Decorrelation correction for nanoparticle tracking analysis of dilute polydisperse suspensions in bulk flow

Nanoparticle tracking analysis, a multiprobe single particle tracking technique, is a widely used method to quickly determine the concentration and size distribution of colloidal particle suspensions. Many popular tools remove non-Brownian components of particle motion by subtracting the ensemble-average displacement at each time step, which is termed dedrifting. Though critical for accurate size measurements, dedrifting is shown here to introduce significant biasing error and can fundamentally limit the dynamic range of particle size that can be measured for dilute heterogeneous suspensions such as biological extracellular vesicles. We report a more accurate estimate of particle mean-square displacement, which we call decorrelation analysis, that accounts for correlations between individual and ensemble particle motion, which are spuriously introduced by dedrifting. Particle tracking simulation and experimental results show that this approach more accurately determines particle diameters for low-concentration polydisperse suspensions when compared with standard dedrifting techniques.

Convergence of lateral dynamic measurements in the plasma membrane of live cells from single particle tracking and STED-FCS

Fluorescence correlation spectroscopy (FCS) in combination with the super-resolution imaging method STED (STED-FCS), and single-particle tracking (SPT) are able to directly probe the lateral dynamics of lipids and proteins in the plasma membrane of live cells at spatial scales much below the diffraction limit of conventional microscopy. However, a major disparity in interpretation of data from SPT and STED-FCS remains, namely the proposed existence of a very fast (unhindered) lateral diffusion coefficient, ⩾5 µm2 s-1, in the plasma membrane of live cells at very short length scales, ≈⩽ 100 nm, and time scales, ≈1-10 ms. This fast diffusion coefficient has been advocated in several high-speed SPT studies, for lipids and membrane proteins alike, but the equivalent has not been detected in STED-FCS measurements. Resolving this ambiguity is important because the assessment of membrane dynamics currently relies heavily on SPT for the determination of heterogeneous diffusion. A possible systematic error in this approach would thus have vast implications in this field. To address this, we have re-visited the analysis procedure for SPT data with an emphasis on the measurement errors and the effect that these errors have on the measurement outputs. We subsequently demonstrate that STED-FCS and SPT data, following careful consideration of the experimental errors of the SPT data, converge to a common interpretation which for the case of a diffusing phospholipid analogue in the plasma membrane of live mouse embryo fibroblasts results in an unhindered, intra-compartment, diffusion coefficient of  ≈0.7-1.0 µm2 s-1, and a compartment size of about 100-150 nm.

A Primer on the Bayesian Approach to High-Density Single-Molecule Trajectories Analysis

Tracking single molecules in living cells provides invaluable information on their environment and on the interactions that underlie their motion. New experimental techniques now permit the recording of large amounts of individual trajectories, enabling the implementation of advanced statistical tools for data analysis. In this primer, we present a Bayesian approach toward treating these data, and we discuss how it can be fruitfully employed to infer physical and biochemical parameters from single-molecule trajectories.

One-dimensional Gaussian-core fluid: ordering and crossover from normal diffusion to single-file dynamics

The peculiarity of a bounded pair potential in combination with strong confinement brings some quite interesting new phenomenology in the structure and dynamics of one-dimensional colloidal systems. Such behaviour is atypical in comparison with colloidal systems interacting with potentials that diverge at the origin. In this contribution, by means of molecular dynamics simulations, a confined one-dimensional model of particles interacting via a Gaussian-core pair potential is studied. We explore the effects of confinement, density and temperature on the structural and dynamical correlation functions. Our findings indicate that the static and dynamic liquid-state anomalies already reported in open systems are also present in this 1D model system. Using the radial distribution function and the static structure factor to characterise the spatial ordering, it is observed that the system remains fluid at all densities. However, when the reduced temperature is above 0.03, it displays typical features of a liquid regime, i.e., there exist short-range spatial correlations among particles. In contrast, at lower temperatures and densities, where the particle-particle interaction dominates, the system behaves structurally and dynamically similar to a hard-core repulsive system. In such a region, interestingly, there is a crossover from a liquid to a solid-like regime. At any given temperature, the system undergoes a sort of reentrant structural behaviour as the density increases. At either high densities or temperatures, particle correlations vanish, thus, the system exhibits structural and dynamical properties similar to those of an ideal gas. To examine a possible correlation between the structural anomalies and the diffusive behaviour, the mean-square displacement and the self-intermediate scattering function are also computed. From these observables, we establish the thermodynamic phase-space points where the dynamical behaviour is non-monotonic. In conjunction with the observed anomalous diffusion, we have found a dynamical crossover from single-file diffusion, which is characteristic of one-dimensional systems with a well-defined hard-core, to the ordinary Fickian diffusion present in open systems.

In vivo single-RNA tracking shows that most tRNA diffuses freely in live bacteria

Transfer RNA (tRNA) links messenger RNA nucleotide sequence with amino acid sequence during protein synthesis. Despite the importance of tRNA for translation, its subcellular distribution and diffusion properties in live cells are poorly understood. Here, we provide the first direct report on tRNA diffusion localization in live bacteria. We internalized tRNA labeled with organic fluorophores into live bacteria, applied single-molecule fluorescence imaging with single-particle tracking and localized and tracked single tRNA molecules over seconds. We observed two diffusive species: fast (with a diffusion coefficient of ∼8 μm²/s, consistent with free tRNA) and slow (consistent with tRNA bound to larger complexes). Our data indicate that a large fraction of internalized fluorescent tRNA (>70%) appears to diffuse freely in the bacterial cell. We also obtained the subcellular distribution of fast and slow diffusing tRNA molecules in multiple cells by normalizing for cell morphology. While fast diffusing tRNA is not excluded from the bacterial nucleoid, slow diffusing tRNA is localized to the cell periphery (showing a 30% enrichment versus a uniform distribution), similar to non-uniform localizations previously observed for mRNA and ribosomes.

Temporal Correlations of the Running Maximum of a Brownian Trajectory

We study the correlations between the maxima m and M of a Brownian motion (BM) on the time intervals [0,t_{1}] and [0,t_{2}], with t_{2}>t_{1}. We determine the exact forms of the distribution functions P(m,M) and P(G=M-m), and calculate the moments E{(M-m)^{k}} and the cross-moments E{m^{l}M^{k}} with arbitrary integers l and k. We show that correlations between m and M decay as sqrt[t_{1}/t_{2}] when t_{2}/t_{1}→∞, revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient ρ(m,M) and the power spectrum of M_{t}, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process.

Characterization of the diffusive dynamics of particles with time-dependent asymmetric microscopy intensity profiles

We put forth an algorithm to track isolated micron-size solid and liquid particles that produce time-dependent asymmetric intensity patterns. This method quantifies the displacement of a particle in the image plane from the peak of a spatial cross-correlation function with a reference image. The peak sharpness results in subpixel resolution. We demonstrate the utility of the method for tracking liquid droplets with changing shapes and micron-size particles producing images with exaggerated asymmetry. We compare the accuracy of diffusivity determination with particles of known size by this method to that by common tracking techniques and demonstrate that our algorithm is superior. We address several open questions on the characterization of diffusive behaviors. We show that for particles, diffusing with a root-mean-square displacement of 0.6 pixel widths in the time between two successive recorded frames, more accurate diffusivity determinations result from mean squared displacement (MSD) for lag times up to 5 time intervals and that MSDs determined from non-overlapping displacements do not yield more accurate diffusivities. We discuss the optimal length of image sequences and demonstrate that lower frame rates do not affect the accuracy of the estimated diffusivity.

Optimizing experimental parameters for tracking of diffusing particles

We describe how a single-particle tracking experiment should be designed in order for its recorded trajectories to contain the most information about a tracked particle's diffusion coefficient. The precision of estimators for the diffusion coefficient is affected by motion blur, limited photon statistics, and the length of recorded time series. We demonstrate for a particle undergoing free diffusion that precision is negligibly affected by motion blur in typical experiments, while optimizing photon counts and the number of recorded frames is the key to precision. Building on these results, we describe for a wide range of experimental scenarios how to choose experimental parameters in order to optimize the precision. Generally, one should choose quantity over quality: experiments should be designed to maximize the number of frames recorded in a time series, even if this means lower information content in individual frames.

Quantification of Multivalent Interactions by Tracking Single Biological Nanoparticle Mobility on a Lipid Membrane

Macromolecular association commonly occurs via dynamic engagement of multiple weak bonds referred to as multivalent interactions. The distribution of the number of bonds, combined with their strong influence on the residence time, makes it very demanding to quantify this type of interaction. To address this challenge in the context of virology, we mimicked the virion association to a cell membrane by attaching lipid vesicles (100 nm diameter) to a supported lipid bilayer via multiple, identical cholesterol-based DNA linker molecules, each mimicking an individual virion-receptor link. Using total internal reflection microscopy to track single attached vesicles combined with a novel filtering approach, we show that histograms of the vesicle diffusion coefficient D exhibit a spectrum of distinct peaks, which are associated with vesicles differing in the number, n, of linking DNA tethers. These peaks are only observed if vesicles with transient changes in n are excluded from the analysis. D is found to be proportional to 1/n, in excellent agreement with the free draining model, allowing to quantify transient changes of n on the single vesicle level and to extract transition rates between individual linking states. Necessary imaging conditions to extend the analysis to multivalent interactions in general are also reported.

Fisher information theory for parameter estimation in single molecule microscopy: tutorial

Estimation of a parameter of interest from image data represents a task that is commonly carried out in single molecule microscopy data analysis. The determination of the positional coordinates of a molecule from its image, for example, forms the basis of standard applications such as single molecule tracking and localization-based super-resolution image reconstruction. Assuming that the estimator used recovers, on average, the true value of the parameter, its accuracy, or standard deviation, is then at best equal to the square root of the Cramér-Rao lower bound. The Cramér-Rao lower bound can therefore be used as a benchmark in the evaluation of the accuracy of an estimator. Additionally, as its value can be computed and assessed for different experimental settings, it is useful as an experimental design tool. This tutorial demonstrates a mathematical framework that has been specifically developed to calculate the Cramér-Rao lower bound for estimation problems in single molecule microscopy and, more broadly, fluorescence microscopy. The material includes a presentation of the photon detection process that underlies all image data, various image data models that describe images acquired with different detector types, and Fisher information expressions that are necessary for the calculation of the lower bound. Throughout the tutorial, examples involving concrete estimation problems are used to illustrate the effects of various factors on the accuracy of parameter estimation and, more generally, to demonstrate the flexibility of the mathematical framework.

Quiescent Saccharomyces cerevisiae forms telomere hyperclusters at the nuclear membrane vicinity through a multifaceted mechanism involving Esc1, the Sir complex, and chromatin condensation

Upon quiescence entry, yeast cells assemble telomere hyperclusters. These structures localize to the nuclear membrane in an Esc1-dependent manner and assemble through the combined action of the Sir complex, deacetylation of H4K16, the binding of the linker histone H1, and condensin.

Like other eukaryotes, Saccharomyces cerevisiae spatially organizes its chromosomes within the nucleus. In G₁ phase, the yeast’s 32 telomeres are clustered into 6–10 foci that dynamically interact with the nuclear membrane. Here we show that, when cells leave the division cycle and enter quiescence, telomeres gather into two to three hyperclusters at the nuclear membrane vicinity. This localization depends on Esc1 but not on the Ku proteins. Telomere hypercluster formation requires the Sir complex but is independent of the nuclear microtubule bundle that specifically assembles in quiescent cells. Importantly, mutants deleted for the linker histone H1 Hho1 or defective in condensin activity or affected for histone H4 Lys-16 deacetylation are impaired, at least in part, for telomere hypercluster formation in quiescence, suggesting that this process involves chromosome condensation. Finally, we establish that telomere hypercluster formation is not necessary for quiescence establishment, maintenance, and exit, raising the question of the physiological raison d’être of this nuclear reorganization.

Estimation of the diffusion constant from intermittent trajectories with variable position uncertainties

The movement of a particle described by Brownian motion is quantified by a single parameter, D, the diffusion constant. The estimation of D from a discrete sequence of noisy observations is a fundamental problem in biological single-particle tracking experiments since it can provide information on the environment and/or the state of the particle itself via the hydrodynamic radius. Here, we present a method to estimate D that takes into account several effects that occur in practice, important for the correct estimation of D, and that have hitherto not been combined together for an estimation of D. These effects are motion blur from the finite integration time of the camera, intermittent trajectories, and time-dependent localization uncertainty. Our estimation procedure, a maximum-likelihood estimation with an information-based confidence interval, follows directly from the likelihood expression for a discretely observed Brownian trajectory that explicitly includes these effects. We begin with the formulation of the likelihood expression and then present three methods to find the exact solution. Each method has its own advantages in either computational robustness, theoretical insight, or the estimation of hidden variables. The Fisher information for this likelihood distribution is calculated and analyzed to show that localization uncertainties impose a lower bound on the estimation of D. Confidence intervals are established and then used to evaluate our estimator on simulated data with experimentally relevant camera effects to demonstrate the benefit of incorporating variable localization errors.

Optical tracking of nanoscale particles in microscale environments

The trajectories of nanoscale particles through microscale environments record useful information about both the particles and the environments. Optical microscopes provide efficient access to this information through measurements of light in the far field from nanoparticles. Such measurements necessarily involve trade-offs in tracking capabilities. This article presents a measurement framework, based on information theory, that facilitates a more systematic understanding of such trade-offs to rationally design tracking systems for diverse applications. This framework includes the degrees of freedom of optical microscopes, which determine the limitations of tracking measurements in theory. In the laboratory, tracking systems are assemblies of sources and sensors, optics and stages, and nanoparticle emitters. The combined characteristics of such systems determine the limitations of tracking measurements in practice. This article reviews this tracking hardware with a focus on the essential functions of nanoparticles as optical emitters and microenvironmental probes. Within these theoretical and practical limitations, experimentalists have implemented a variety of tracking systems with different capabilities. This article reviews a selection of apparatuses and techniques for tracking multiple and single particles by tuning illumination and detection, and by using feedback and confinement to improve the measurements. Prior information is also useful in many tracking systems and measurements, which apply across a broad spectrum of science and technology. In the context of the framework and review of apparatuses and techniques, this article reviews a selection of applications, with particle diffusion serving as a prelude to tracking measurements in biological, fluid, and material systems, fabrication and assembly processes, and engineered devices. In so doing, this review identifies trends and gaps in particle tracking that might influence future research.

Nanoscopic substructures of raft-mimetic liquid-ordered membrane domains revealed by high-speed single-particle tracking

Lipid rafts are membrane nanodomains that facilitate important cell functions. Despite recent advances in identifying the biological significance of rafts, nature and regulation mechanism of rafts are largely unknown due to the difficulty of resolving dynamic molecular interaction of rafts at the nanoscale. Here, we investigate organization and single-molecule dynamics of rafts by monitoring lateral diffusion of single molecules in raft-containing reconstituted membranes supported on mica substrates. Using high-speed interferometric scattering (iSCAT) optical microscopy and small gold nanoparticles as labels, motion of single lipids is recorded via single-particle tracking (SPT) with nanometer spatial precision and microsecond temporal resolution. Processes of single molecules partitioning into and escaping from the raft-mimetic liquid-ordered (L_o) domains are directly visualized in a continuous manner with unprecedented clarity. Importantly, we observe subdiffusion of saturated lipids in the L_o domain in microsecond timescale, indicating the nanoscopic heterogeneous molecular arrangement of the L_o domain. Further analysis of the diffusion trajectory shows the presence of nano-subdomains of the L_o phase, as small as 10 nm, which transiently trap the lipids. Our results provide the first experimental evidence of non-uniform molecular organization of the L_o phase, giving a new view of how rafts recruit and confine molecules in cell membranes.

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

Optical microscopy has for centuries been a key tool to study living cells with minimum invasiveness. The advent of single molecule techniques over the past two decades has revolutionized the field of cell biology by providing a more quantitative picture of the complex and highly dynamic organization of living systems. Amongst these techniques, single particle tracking (SPT) has emerged as a powerful approach to study a variety of dynamic processes in life sciences. SPT provides access to single molecule behavior in the natural context of living cells, thereby allowing a complete statistical characterization of the system under study. In this review we describe the foundations of SPT together with novel optical implementations that nowadays allow the investigation of single molecule dynamic events with increasingly high spatiotemporal resolution using molecular densities closer to physiological expression levels. We outline some of the algorithms for the faithful reconstruction of SPT trajectories as well as data analysis, and highlight biological examples where the technique has provided novel insights into the role of diffusion regulating cellular function. The last part of the review concentrates on different theoretical models that describe anomalous transport behavior and ergodicity breaking observed from SPT studies in living cells.

Method for simultaneous localization and parameter estimation in particle tracking experiments

We present a numerical method for the simultaneous localization and parameter estimation of a fluorescent particle undergoing a discrete-time continuous-state Markov process. In particular, implementation of the method proposed in this work yields an approximation to the posterior density of the particle positions over time in addition to maximum likelihood estimates of fixed, unknown parameters. The method employs sequential Monte Carlo methods and can take into account complex, potentially nonlinear noise models, including shot noise and camera-specific readout noise, as well as a wide variety of motion models and observation models, including those representing recent engineered point spread functions. We demonstrate the technique by applying it to four scenarios, including a particle undergoing free, confined, and tethered diffusions.

Extracting Diffusive States of Rho GTPase in Live Cells: Towards In Vivo Biochemistry

Resolving distinct biochemical interaction states when analyzing the trajectories of diffusing proteins in live cells on an individual basis remains challenging because of the limited statistics provided by the relatively short trajectories available experimentally. Here, we introduce a novel, machine-learning based classification methodology, which we call perturbation expectation-maximization (pEM), that simultaneously analyzes a population of protein trajectories to uncover the system of diffusive behaviors which collectively result from distinct biochemical interactions. We validate the performance of pEM in silico and demonstrate that pEM is capable of uncovering the proper number of underlying diffusive states with an accurate characterization of their diffusion properties. We then apply pEM to experimental protein trajectories of Rho GTPases, an integral regulator of cytoskeletal dynamics and cellular homeostasis, in vivo via single particle tracking photo-activated localization microcopy. Remarkably, pEM uncovers 6 distinct diffusive states conserved across various Rho GTPase family members. The variability across family members in the propensities for each diffusive state reveals non-redundant roles in the activation states of RhoA and RhoC. In a resting cell, our results support a model where RhoA is constantly cycling between activation states, with an imbalance of rates favoring an inactive state. RhoC, on the other hand, remains predominantly inactive.

Single particle tracking is a powerful tool that captures the diffusive dynamics of proteins as they undergo various interactions in living cells. Uncovering different biochemical interactions by analyzing the diffusive behaviors of individual protein trajectories, however, is challenging due to the limited statistics provided by short trajectories and experimental noise sources which are intimately coupled into each protein’s localization. Here, we introduce a novel, unsupervised, machine-learning based classification methodology, which we call perturbation expectation-maximization (pEM), that simultaneously analyzes a population of protein trajectories to uncover the system of diffusive behaviors which collectively result from distinct biochemical interactions. We validate the performance of pEM in silico and in vivo on the biological system of Rho GTPase, a signal transduction protein responsible for regulating cytoskeletal dynamics. We envision that the presented methodology will be applicable to a wide range of single protein tracking data where different biochemical interactions result in distinct diffusive behaviors. More generally, this study brings us an important step closer to the possibility of monitoring the endogenous biochemistry of diffusing proteins within live cells with single molecule resolution.

Detection of Diffusion Heterogeneity in Single Particle Tracking Trajectories Using a Hidden Markov Model with Measurement Noise Propagation

We develop a Bayesian analysis framework to detect heterogeneity in the diffusive behaviour of single particle trajectories on cells, implementing model selection to classify trajectories as either consistent with Brownian motion or with a two-state (diffusion coefficient) switching model. The incorporation of localisation accuracy is essential, as otherwise false detection of switching within a trajectory was observed and diffusion coefficient estimates were inflated. Since our analysis is on a single trajectory basis, we are able to examine heterogeneity between trajectories in a quantitative manner. Applying our method to the lymphocyte function-associated antigen 1 (LFA-1) receptor tagged with latex beads (4 s trajectories at 1000 frames s⁻¹), both intra- and inter-trajectory heterogeneity were detected; 12–26% of trajectories display clear switching between diffusive states dependent on condition, whilst the inter-trajectory variability is highly structured with the diffusion coefficients being related by D₁ = 0.68D₀ − 1.5 × 10⁴ nm² s⁻¹, suggestive that on these time scales we are detecting switching due to a single process. Further, the inter-trajectory variability of the diffusion coefficient estimates (1.6 × 10² − 2.6 × 10⁵ nm² s⁻¹) is very much larger than the measurement uncertainty within trajectories, suggesting that LFA-1 aggregation and cytoskeletal interactions are significantly affecting mobility, whilst the timescales of these processes are distinctly different giving rise to inter- and intra-trajectory variability. There is also an ‘immobile’ state (defined as D < 3.0 × 10³ nm² s⁻¹) that is rarely involved in switching, immobility occurring with the highest frequency (47%) under T cell activation (phorbol-12-myristate-13-acetate (PMA) treatment) with enhanced cytoskeletal attachment (calpain inhibition). Such ‘immobile’ states frequently display slow linear drift, potentially reflecting binding to a dynamic actin cortex. Our methods allow significantly more information to be extracted from individual trajectories (ultimately limited by time resolution and time-series length), and allow statistical comparisons between trajectories thereby quantifying inter-trajectory heterogeneity. Such methods will be highly informative for the construction and fitting of molecule mobility models within membranes incorporating aggregation, binding to the cytoskeleton, or traversing membrane microdomains.

Live-cell superresolution microscopy reveals the organization of RNA polymerase in the bacterial nucleoid

Despite the fundamental importance of transcription, a comprehensive analysis of RNA polymerase (RNAP) behavior and its role in the nucleoid organization in vivo is lacking. Here, we used superresolution microscopy to study the localization and dynamics of the transcription machinery and DNA in live bacterial cells, at both the single-molecule and the population level. We used photoactivated single-molecule tracking to discriminate between mobile RNAPs and RNAPs specifically bound to DNA, either on promoters or transcribed genes. Mobile RNAPs can explore the whole nucleoid while searching for promoters, and spend 85% of their search time in nonspecific interactions with DNA. On the other hand, the distribution of specifically bound RNAPs shows that low levels of transcription can occur throughout the nucleoid. Further, clustering analysis and 3D structured illumination microscopy (SIM) show that dense clusters of transcribing RNAPs form almost exclusively at the nucleoid periphery. Treatment with rifampicin shows that active transcription is necessary for maintaining this spatial organization. In faster growth conditions, the fraction of transcribing RNAPs increases, as well as their clustering. Under these conditions, we observed dramatic phase separation between the densest clusters of RNAPs and the densest regions of the nucleoid. These findings show that transcription can cause spatial reorganization of the nucleoid, with movement of gene loci out of the bulk of DNA as levels of transcription increase. This work provides a global view of the organization of RNA polymerase and transcription in living cells.

Single-molecule imaging of Hedgehog pathway protein Smoothened in primary cilia reveals binding events regulated by Patched1

Accumulation of the signaling protein Smoothened (Smo) in the membrane of primary cilia is an essential step in Hedgehog (Hh) signal transduction, yet the molecular mechanisms of Smo movement and localization are poorly understood. Using ultrasensitive single-molecule tracking with high spatial/temporal precision (30 nm/10 ms), we discovered that binding events disrupt the primarily diffusive movement of Smo in cilia at an array of sites near the base. The affinity of Smo for these binding sites was modulated by the Hh pathway activation state. Activation, by either a ligand or genetic loss of the negatively acting Hh receptor Patched-1 (Ptch), reduced the affinity and frequency of Smo binding at the base. Our findings quantify activation-dependent changes in Smo dynamics in cilia and highlight a previously unknown step in Hh pathway activation.

Chromosomal locus tracking with proper accounting of static and dynamic errors

The mean-squared displacement (MSD) and velocity autocorrelation (VAC) of tracked single particles or molecules are ubiquitous metrics for extracting parameters that describe the object's motion, but they are both corrupted by experimental errors that hinder the quantitative extraction of underlying parameters. For the simple case of pure Brownian motion, the effects of localization error due to photon statistics ("static error") and motion blur due to finite exposure time ("dynamic error") on the MSD and VAC are already routinely treated. However, particles moving through complex environments such as cells, nuclei, or polymers often exhibit anomalous diffusion, for which the effects of these errors are less often sufficiently treated. We present data from tracked chromosomal loci in yeast that demonstrate the necessity of properly accounting for both static and dynamic error in the context of an anomalous diffusion that is consistent with a fractional Brownian motion (FBM). We compare these data to analytical forms of the expected values of the MSD and VAC for a general FBM in the presence of these errors.

Determination of localization accuracy based on experimentally acquired image sets: applications to single molecule microscopy

Fluorescence microscopy is a photon-limited imaging modality that allows the study of subcellular objects and processes with high specificity. The best possible accuracy (standard deviation) with which an object of interest can be localized when imaged using a fluorescence microscope is typically calculated using the Cramér-Rao lower bound, that is, the inverse of the Fisher information. However, the current approach for the calculation of the best possible localization accuracy relies on an analytical expression for the image of the object. This can pose practical challenges since it is often difficult to find appropriate analytical models for the images of general objects. In this study, we instead develop an approach that directly uses an experimentally collected image set to calculate the best possible localization accuracy for a general subcellular object. In this approach, we fit splines, i.e. smoothly connected piecewise polynomials, to the experimentally collected image set to provide a continuous model of the object, which can then be used for the calculation of the best possible localization accuracy. Due to its practical importance, we investigate in detail the application of the proposed approach in single molecule fluorescence microscopy. In this case, the object of interest is a point source and, therefore, the acquired image set pertains to an experimental point spread function.

Fluorogen-activating proteins provide tunable labeling densities for tracking FcεRI independent of IgE

Crosslinking of IgE bound FcεRI on mast cells and basophils by multivalent antigen leads to degranulation and the release of key inflammatory mediators that stimulate the allergic response. Here, we present and characterize the use of fluorogen-activating proteins (FAPs) for single particle tracking of FcεRI to investigate how receptor mobility is influenced after IgE-induced changes in mast cell behavior. FAPs are genetically encoded tags that bind a fluorogen dye and increase its brightness upon binding up to 20,000-fold. We demonstrate that, by titrating fluorogen concentration, labeling densities from ensemble to single particle can be achieved, independent of expression level and without the need for wash steps or photobleaching. The FcεRI γ-subunit fused to a FAP (FAP-γ) provides, for the first time, an IgE-independent probe for tracking this signaling subunit of FcεRI at the single molecule level. We show that the FcεRI γ-subunit dynamics are controlled by the IgE-binding α-subunit and that the cytokinergic IgE, SPE-7, induces mast cell activation without altering FcεRI mobility or promoting internalization. We take advantage of the far-red emission of the malachite green (MG) fluorogen to track FcεRI relative to dynamin-GFP and find that immobilized receptors readily correlate with locations of dynamin recruitment only under conditions that promote rapid endocytosis. These studies demonstrate the usefulness of the FAP system for single molecule studies and have provided new insights into the relationship among FcεRI structure, activity, and mobility.

Guidelines for the fitting of anomalous diffusion mean square displacement graphs from single particle tracking experiments

Single particle tracking is an essential tool in the study of complex systems and biophysics and it is commonly analyzed by the time-averaged mean square displacement (MSD) of the diffusive trajectories. However, past work has shown that MSDs are susceptible to significant errors and biases, preventing the comparison and assessment of experimental studies. Here, we attempt to extract practical guidelines for the estimation of anomalous time averaged MSDs through the simulation of multiple scenarios with fractional Brownian motion as a representative of a large class of fractional ergodic processes. We extract the precision and accuracy of the fitted MSD for various anomalous exponents and measurement errors with respect to measurement length and maximum time lags. Based on the calculated precision maps, we present guidelines to improve accuracy in single particle studies. Importantly, we find that in some experimental conditions, the time averaged MSD should not be used as an estimator.

Quantitative Aspects of Single Molecule Microscopy

Single molecule microscopy is a relatively new optical microscopy technique that allows the detection of individual molecules such as proteins in a cellular context. This technique has generated significant interest among biologists, biophysicists and biochemists, as it holds the promise to provide novel insights into subcellular processes and structures that otherwise cannot be gained through traditional experimental approaches. Single molecule experiments place stringent demands on experimental and algorithmic tools due to the low signal levels and the presence of significant extraneous noise sources. Consequently, this has necessitated the use of advanced statistical signal and image processing techniques for the design and analysis of single molecule experiments. In this tutorial paper, we provide an overview of single molecule microscopy from early works to current applications and challenges. Specific emphasis will be on the quantitative aspects of this imaging modality, in particular single molecule localization and resolvability, which will be discussed from an information theoretic perspective. We review the stochastic framework for image formation, different types of estimation techniques and expressions for the Fisher information matrix. We also discuss several open problems in the field that demand highly non-trivial signal processing algorithms.

Refractive index determination of nanoparticles in suspension using nanoparticle tracking analysis

The refractive index (RI) dictates interaction between light and nanoparticles and therefore is important to health, environmental, and materials sciences. Using nanoparticle tracking analysis, we have determined the RI of heterogeneous particles <500 nm in suspension. We demonstrate feasibility of distinguishing silica and polystyrene beads based on their RI. The hitherto unknown RI of extracellular vesicles from human urine was determined at 1.37 (mean). This method enables differentiation of single nanoparticles based on their RI.

Quantification of DNA-associated proteins inside eukaryotic cells using single-molecule localization microscopy

Development of single-molecule localization microscopy techniques has allowed nanometre scale localization accuracy inside cells, permitting the resolution of ultra-fine cell structure and the elucidation of crucial molecular mechanisms. Application of these methodologies to understanding processes underlying DNA replication and repair has been limited to defined in vitro biochemical analysis and prokaryotic cells. In order to expand these techniques to eukaryotic systems, we have further developed a photo-activated localization microscopy-based method to directly visualize DNA-associated proteins in unfixed eukaryotic cells. We demonstrate that motion blurring of fluorescence due to protein diffusivity can be used to selectively image the DNA-bound population of proteins. We designed and tested a simple methodology and show that it can be used to detect changes in DNA binding of a replicative helicase subunit, Mcm4, and the replication sliding clamp, PCNA, between different stages of the cell cycle and between distinct genetic backgrounds.

Single-molecule diffusion and conformational dynamics by spatial integration of temporal fluctuations

Single-molecule localization and tracking has been used to translate spatiotemporal information of individual molecules to map their diffusion behaviours. However, accurate analysis of diffusion behaviours and including other parameters, such as the conformation and size of molecules, remain as limitations to the method. Here, we report a method that addresses the limitations of existing single-molecular localization methods. The method is based on temporal tracking of the cumulative area occupied by molecules. These temporal fluctuations are tied to molecular size, rates of diffusion and conformational changes. By analysing fluorescent nanospheres and double-stranded DNA molecules of different lengths and topological forms, we demonstrate that our cumulative-area method surpasses the conventional single-molecule localization method in terms of the accuracy of determined diffusion coefficients. Furthermore, the cumulative-area method provides conformational relaxation times of structurally flexible chains along with diffusion coefficients, which together are relevant to work in a wide spectrum of scientific fields.

Single-molecule localization and tracking technique is widely used to visualize molecular dynamics in life science, yet it fails to detect molecular conformation. Serag et al. address this limitation via spatial quantization of temporal fluctuations in the cumulative area occupied by molecules.

Long-lived intracellular single-molecule fluorescence using electroporated molecules

Studies of biomolecules in vivo are crucial to understand their function in a natural, biological context. One powerful approach involves fusing molecules of interest to fluorescent proteins to study their expression, localization, and action; however, the scope of such studies would be increased considerably by using organic fluorophores, which are smaller and more photostable than their fluorescent protein counterparts. Here, we describe a straightforward, versatile, and high-throughput method to internalize DNA fragments and proteins labeled with organic fluorophores into live Escherichia coli by employing electroporation. We studied the copy numbers, diffusion profiles, and structure of internalized molecules at the single-molecule level in vivo, and were able to extend single-molecule observation times by two orders of magnitude compared to green fluorescent protein, allowing continuous monitoring of molecular processes occurring from seconds to minutes. We also exploited the desirable properties of organic fluorophores to perform single-molecule Förster resonance energy transfer measurements in the cytoplasm of live bacteria, both for DNA and proteins. Finally, we demonstrate internalization of labeled proteins and DNA into yeast Saccharomyces cerevisiae, a model eukaryotic system. Our method should broaden the range of biological questions addressable in microbes by single-molecule fluorescence.

Imaging live cells at the nanometer-scale with single-molecule microscopy: obstacles and achievements in experiment optimization for microbiology

Single-molecule fluorescence microscopy enables biological investigations inside living cells to achieve millisecond- and nanometer-scale resolution. Although single-molecule-based methods are becoming increasingly accessible to non-experts, optimizing new single-molecule experiments can be challenging, in particular when super-resolution imaging and tracking are applied to live cells. In this review, we summarize common obstacles to live-cell single-molecule microscopy and describe the methods we have developed and applied to overcome these challenges in live bacteria. We examine the choice of fluorophore and labeling scheme, approaches to achieving single-molecule levels of fluorescence, considerations for maintaining cell viability, and strategies for detecting single-molecule signals in the presence of noise and sample drift. We also discuss methods for analyzing single-molecule trajectories and the challenges presented by the finite size of a bacterial cell and the curvature of the bacterial membrane.

In vivo single-molecule imaging of bacterial DNA replication, transcription, and repair

In vivo single-molecule experiments offer new perspectives on the behaviour of DNA binding proteins, from the molecular level to the length scale of whole bacterial cells. With technological advances in instrumentation and data analysis, fluorescence microscopy can detect single molecules in live cells, opening the doors to directly follow individual proteins binding to DNA in real time. In this review, we describe key technical considerations for implementing in vivo single-molecule fluorescence microscopy. We discuss how single-molecule tracking and quantitative super-resolution microscopy can be adapted to extract DNA binding kinetics, spatial distributions, and copy numbers of proteins, as well as stoichiometries of protein complexes. We highlight experiments which have exploited these techniques to answer important questions in the field of bacterial gene regulation and transcription, as well as chromosome replication, organisation and repair. Together, these studies demonstrate how single-molecule imaging is transforming our understanding of DNA-binding proteins in cells.

Robust hypothesis tests for detecting statistical evidence of two-dimensional and three-dimensional interactions in single-molecule measurements

Experimental advances have improved the two- (2D) and three-dimensional (3D) spatial resolution that can be extracted from in vivo single-molecule measurements. This enables researchers to quantitatively infer the magnitude and directionality of forces experienced by biomolecules in their native environment. Situations where such force information is relevant range from mitosis to directed transport of protein cargo along cytoskeletal structures. Models commonly applied to quantify single-molecule dynamics assume that effective forces and velocity in the x,y (or x,y,z) directions are statistically independent, but this assumption is physically unrealistic in many situations. We present a hypothesis testing approach capable of determining if there is evidence of statistical dependence between positional coordinates in experimentally measured trajectories; if the hypothesis of independence between spatial coordinates is rejected, then a new model accounting for 2D (3D) interactions can and should be considered. Our hypothesis testing technique is robust, meaning it can detect interactions, even if the noise statistics are not well captured by the model. The approach is demonstrated on control simulations and on experimental data (directed transport of intraflagellar transport protein 88 homolog in the primary cilium).

Dark field nanoparticle tracking analysis for size characterization of plasmonic and non-plasmonic particles

Dark field microscopy is a widely unknown method to measure the particle size distribution of diffusing nanoparticles by particle tracking. Here we demonstrate that by using the surface plasmonic resonance of Au nanoparticles, size differences of ca. 20 nm can be identified within the particle size distribution. For that purpose, we developed a software tool which helps to analyze color videos of diffusing nanoparticles retrieved from CCD or CMOS cameras. Polystyrene beads with a diameter of 100 and 200 nm were used to compare the results to those obtained with a well-established laser-based particle tracking system. The methodology will be discussed in the light of recent developments in the emerging field of optical nanoparticle tracking.

TrackArt: the user friendly interface for single molecule tracking data analysis and simulation applied to complex diffusion in mica supported lipid bilayers

Background

Single molecule tracking (SMT) analysis of fluorescently tagged lipid and protein probes is an attractive alternative to ensemble averaged methods such as fluorescence correlation spectroscopy (FCS) or fluorescence recovery after photobleaching (FRAP) for measuring diffusion in artificial and plasma membranes. The meaningful estimation of diffusion coefficients and their errors is however not straightforward, and is heavily dependent on sample type, acquisition method, and equipment used. Many approaches require advanced computing and programming skills for their implementation.

Findings

Here we present TrackArt software, an accessible graphic interface for simulation and complex analysis of multiple particle paths. Imported trajectories can be filtered to eliminate spurious or corrupted tracks, and are then analyzed using several previously described methodologies, to yield single or multiple diffusion coefficients, their population fractions, and estimated errors. We use TrackArt to analyze the single-molecule diffusion behavior of a sphingolipid analog SM-Atto647N, in mica supported DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine) bilayers. Fitting with a two-component diffusion model confirms the existence of two separate populations of diffusing particles in these bilayers on mica. As a demonstration of the TrackArt workflow, we characterize and discuss the effective activation energies required to increase the diffusion rates of these populations, obtained from Arrhenius plots of temperature-dependent diffusion. Finally, TrackArt provides a simulation module, allowing the user to generate models with multiple particle trajectories, diffusing with different characteristics. Maps of domains, acting as impermeable or permeable obstacles for particles diffusing with given rate constants and diffusion coefficients, can be simulated or imported from an image. Importantly, this allows one to use simulated data with a known diffusion behavior as a comparison for results acquired using particular algorithms on actual, “natural” samples whose diffusion behavior is to be extracted. It can also serve as a tool for demonstrating diffusion principles.

Conclusions

TrackArt is an open source, platform-independent, Matlab-based graphical user interface, and is easy to use even for those unfamiliar with the Matlab programming environment. TrackArt can be used for accurate simulation and analysis of complex diffusion data, such as diffusion in lipid bilayers, providing publication-quality formatted results.

Local characterization of hindered Brownian motion by using digital video microscopy and 3D particle tracking

In this article we present methods for measuring hindered Brownian motion in the confinement of complex 3D geometries using digital video microscopy. Here we discuss essential features of automated 3D particle tracking as well as diffusion data analysis. By introducing local mean squared displacement-vs-time curves, we are able to simultaneously measure the spatial dependence of diffusion coefficients, tracking accuracies and drift velocities. Such local measurements allow a more detailed and appropriate description of strongly heterogeneous systems as opposed to global measurements. Finite size effects of the tracking region on measuring mean squared displacements are also discussed. The use of these methods was crucial for the measurement of the diffusive behavior of spherical polystyrene particles (505 nm diameter) in a microfluidic chip. The particles explored an array of parallel channels with different cross sections as well as the bulk reservoirs. For this experiment we present the measurement of local tracking accuracies in all three axial directions as well as the diffusivity parallel to the channel axis while we observed no significant flow but purely Brownian motion. Finally, the presented algorithm is suitable also for tracking of fluorescently labeled particles and particles driven by an external force, e.g., electrokinetic or dielectrophoretic forces.

Optimal estimation of diffusion coefficients from single-particle trajectories

How does one optimally determine the diffusion coefficient of a diffusing particle from a single-time-lapse recorded trajectory of the particle? We answer this question with an explicit, unbiased, and practically optimal covariance-based estimator (CVE). This estimator is regression-free and is far superior to commonly used methods based on measured mean squared displacements. In experimentally relevant parameter ranges, it also outperforms the analytically intractable and computationally more demanding maximum likelihood estimator (MLE). For the case of diffusion on a flexible and fluctuating substrate, the CVE is biased by substrate motion. However, given some long time series and a substrate under some tension, an extended MLE can separate particle diffusion on the substrate from substrate motion in the laboratory frame. This provides benchmarks that allow removal of bias caused by substrate fluctuations in CVE. The resulting unbiased CVE is optimal also for short time series on a fluctuating substrate. We have applied our estimators to human 8-oxoguanine DNA glycolase proteins diffusing on flow-stretched DNA, a fluctuating substrate, and found that diffusion coefficients are severely overestimated if substrate fluctuations are not accounted for.

Lung gene therapy with highly compacted DNA nanoparticles that overcome the mucus barrier

Inhaled gene carriers must penetrate the highly viscoelastic and adhesive mucus barrier in the airway in order to overcome rapid mucociliary clearance and reach the underlying epithelium; however, even the most widely used viral gene carriers are unable to efficiently do so. We developed two polymeric gene carriers that compact plasmid DNA into small and highly stable nanoparticles with dense polyethylene glycol (PEG) surface coatings. These highly compacted, densely PEG-coated DNA nanoparticles rapidly penetrate human cystic fibrosis (CF) mucus ex vivo and mouse airway mucus ex situ. Intranasal administration of the mucus penetrating DNA nanoparticles greatly enhanced particle distribution, retention and gene transfer in the mouse lung airways compared to conventional gene carriers. Successful delivery of a full-length plasmid encoding the cystic fibrosis transmembrane conductance regulator protein was achieved in the mouse lungs and airway cells, including a primary culture of mucus-covered human airway epithelium grown at air-liquid interface, without causing acute inflammation or toxicity. Highly compacted mucus penetrating DNA nanoparticles hold promise for lung gene therapy.

Troika of single particle tracking programing: SNR enhancement, particle identification, and mapping

Single particle tracking (SPT) techniques provide a microscopic approach to probe in vivo and in vitro structure and reactions. Automatic analysis of SPT data with high efficiency and accuracy spurs the development of SPT algorithms. In this perspective, we review a range of available techniques used in SPT analysis programs. In addition, we present an example SPT program step-by-step to provide a guide so that researchers can use, modify, and/or write a SPT program for their own purposes.

Statistics and Related Topics in Single-Molecule Biophysics

Since the universal acceptance of atoms and molecules as the fundamental constituents of matter in the early twentieth century, molecular physics, chemistry and molecular biology have all experienced major theoretical breakthroughs. To be able to actually "see" biological macromolecules, one at a time in action, one has to wait until the 1970s. Since then the field of single-molecule biophysics has witnessed extensive growth both in experiments and theory. A distinct feature of single-molecule biophysics is that the motions and interactions of molecules and the transformation of molecular species are necessarily described in the language of stochastic processes, whether one investigates equilibrium or nonequilibrium living behavior. For laboratory measurements following a biological process, if it is sampled over time on individual participating molecules, then the analysis of experimental data naturally calls for the inference of stochastic processes. The theoretical and experimental developments of single-molecule biophysics thus present interesting questions and unique opportunity for applied statisticians and probabilists. In this article, we review some important statistical developments in connection to single-molecule biophysics, emphasizing the application of stochastic-process theory and the statistical questions arising from modeling and analyzing experimental data.