A hidden Markov model for single particle tracks quantifies dynamic interactions between LFA-1 and the actin cytoskeleton

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

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

Robust Quantification of Live-Cell Single-Molecule Tracking Data for Fluorophores with Different Photophysical Properties

High-speed single-molecule tracking in live cells is becoming an increasingly popular method for quantifying the spatiotemporal behavior of proteins in vivo. The method provides a wealth of quantitative information, but users need to be aware of biases that can skew estimates of molecular mobilities. The range of suitable fluorophores for live-cell single-molecule imaging has grown substantially over the past few years, but it remains unclear to what extent differences in photophysical properties introduce biases. Here, we tested two fluorophores with entirely different photophysical properties, one that photoswitches frequently between bright and dark states (TMR) and one that shows exceptional photostability without photoswitching (JFX650). We used a fusion of the Escherichia coli DNA repair enzyme MutS to the HaloTag and optimized sample preparation and imaging conditions for both types of fluorophore. We then assessed the reliability of two common data analysis algorithms, mean-square displacement (MSD) analysis and Hidden Markov Modeling (HMM), to estimate the diffusion coefficients and fractions of MutS molecules in different states of motion. We introduce a simple approach that removes discrepancies in the data analyses and show that both algorithms yield consistent results, regardless of the fluorophore used. Nevertheless, each dye has its own strengths and weaknesses, with TMR being more suitable for sampling the diffusive behavior of many molecules, while JFX650 enables prolonged observation of only a few molecules per cell. These characterizations and recommendations should help to standardize measurements for increased reproducibility and comparability across studies.

Functional Assessment of Cerebral Capillaries using Single Capillary Reporters in Ultrasound Localization Microscopy

The brain's microvascular cerebral capillary network plays a vital role in maintaining neuronal health, yet capillary dynamics are still not well understood due to limitations in existing imaging techniques. Here, we present Single Capillary Reporters (SCaRe) for transcranial Ultrasound Localization Microscopy (ULM), a novel approach enabling non-invasive, whole-brain mapping of single capillaries and estimates of their transit-time as a neurovascular biomarker. We accomplish this first through computational Monte Carlo and ultrasound simulations of microbubbles flowing through a fully-connected capillary network. We unveil distinct capillary flow behaviors which informs methodological changes to ULM acquisitions to better capture capillaries in vivo. Subsequently, applying SCaRe-ULM in vivo, we achieve unprecedented visualization of single capillary tracks across brain regions, analysis of layer-specific capillary heterogeneous transit times (CHT), and characterization of whole microbubble trajectories from arterioles to venules. Lastly, we evaluate capillary biomarkers using injected lipopolysaccharide to induce systemic neuroinflammation and track the increase in SCaRe-ULM CHT, demonstrating the capability to detect subtle capillary functional changes. SCaRe-ULM represents a significant advance in studying microvascular dynamics, offering novel avenues for investigating capillary patterns in neurological disorders and potential diagnostic applications.

Hidden Markov modeling of single-particle diffusion with stochastic tethering

The statistics of the diffusive motion of particles often serve as an experimental proxy for their interaction with the environment. However, inferring the physical properties from the observed trajectories is challenging. Inspired by a recent experiment, here we analyze the problem of particles undergoing two-dimensional Brownian motion with transient tethering to the surface. We model the problem as a hidden Markov model where the physical position is observed and the tethering state is hidden. We develop an alternating maximization algorithm to infer the hidden state of the particle and estimate the physical parameters of the system. The crux of our method is a saddle-point-like approximation, which involves finding the most likely sequence of hidden states and estimating the physical parameters from it. Extensive numerical tests demonstrate that our algorithm reliably finds the model parameters and is insensitive to the initial guess. We discuss the different regimes of physical parameters and the algorithm's performance in these regimes. We also provide a free software implementation of our algorithm.

Single-molecule analysis of intracellular insulin granule behavior and its application to analyzing cytoskeletal dependence and pathophysiological implications

Introduction: Mobilization of intracellular insulin granules to the plasma membrane plays a crucial role in regulating insulin secretion. However, the regulatory mechanisms of this mobilization process have been poorly understood due to technical limitations. In this study, we propose a convenient approach for assessing intracellular insulin granule behavior based on single-molecule analysis of insulin granule membrane proteins labeled with Quantum dot fluorescent nanocrystals. Methods: This approach allows us to analyze intracellular insulin granule movement with subpixel accuracy at 33 fps. We tracked two insulin granule membrane proteins, phogrin and zinc transporter 8, fused to HaloTag in rat insulinoma INS-1 cells and, by evaluating the tracks with mean-square displacement, demonstrated the characteristic behavior of insulin granules. Results and discussion: Pharmacological perturbations of microtubules and F-actin affected insulin granule behavior on distinct modalities. Specifically, microtubule dynamics and F-actin positively and negatively regulate insulin granule behavior, respectively, presumably by modulating each different behavioral mode. Furthermore, we observed impaired insulin granule behavior and cytoskeletal architecture under chronic treatment of high concentrations of glucose and palmitate. Our approach provides detailed information regarding intracellular insulin granule mobilization and its pathophysiological implications. This study sheds new light on the regulatory mechanisms of intracellular insulin granule mobilization and has important implications for understanding the pathogenesis of diabetes.

Extract latent features of single-particle trajectories with historical experience learning

Single-particle tracking has enabled real-time, in situ quantitative studies of complex systems. However, inferring dynamic state changes from noisy and undersampling trajectories encounters challenges. Here, we introduce a data-driven method for extracting features of subtrajectories with historical experience learning (Deep-SEES), where a single-particle tracking analysis pipeline based on a self-supervised architecture automatically searches for the latent space, allowing effective segmentation of the underlying states from noisy trajectories without prior knowledge on the particle dynamics. We validated our method on a variety of noisy simulated and experimental data. Our results showed that the method can faithfully capture both stable states and their dynamic switch. In highly random systems, our method outperformed commonly used unsupervised methods in inferring motion states, which is important for understanding nanoparticles interacting with living cell membranes, active enzymes, and liquid-liquid phase separation. Self-generating latent features of trajectories could potentially improve the understanding, estimation, and prediction of many complex systems.

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.

Uncovering diffusive states of the yeast membrane protein, Pma1, and how labeling method can change diffusive behavior

We present and analyze video-microscopy-based single-particle-tracking measurements of the budding yeast (Saccharomyces cerevisiae) membrane protein, Pma1, fluorescently labeled either by direct fusion to the switchable fluorescent protein, mEos3.2, or by a novel, light-touch, labeling scheme, in which a 5 amino acid tag is directly fused to the C-terminus of Pma1, which then binds mEos3.2. The track diffusivity distributions of these two populations of single-particle tracks differ significantly, demonstrating that labeling method can be an important determinant of diffusive behavior. We also applied perturbation expectation maximization (pEMv2) (Koo and Mochrie in Phys Rev E 94(5):052412, 2016), which sorts trajectories into the statistically optimum number of diffusive states. For both TRAP-labeled Pma1 and Pma1-mEos3.2, pEMv2 sorts the tracks into two diffusive states: an essentially immobile state and a more mobile state. However, the mobile fraction of Pma1-mEos3.2 tracks is much smaller ([Formula: see text]) than the mobile fraction of TRAP-labeled Pma1 tracks ([Formula: see text]). In addition, the diffusivity of Pma1-mEos3.2's mobile state is several times smaller than the diffusivity of TRAP-labeled Pma1's mobile state. Thus, the two different labeling methods give rise to very different overall diffusive behaviors. To critically assess pEMv2's performance, we compare the diffusivity and covariance distributions of the experimental pEMv2-sorted populations to corresponding theoretical distributions, assuming that Pma1 displacements realize a Gaussian random process. The experiment-theory comparisons for both the TRAP-labeled Pma1 and Pma1-mEos3.2 reveal good agreement, bolstering the pEMv2 approach.

ExTrack characterizes transition kinetics and diffusion in noisy single-particle tracks

Single-particle tracking microscopy is a powerful technique to investigate how proteins dynamically interact with their environment in live cells. However, the analysis of tracks is confounded by noisy molecule localization, short tracks, and rapid transitions between different motion states, notably between immobile and diffusive states. Here, we propose a probabilistic method termed ExTrack that uses the full spatio-temporal information of tracks to extract global model parameters, to calculate state probabilities at every time point, to reveal distributions of state durations, and to refine the positions of bound molecules. ExTrack works for a wide range of diffusion coefficients and transition rates, even if experimental data deviate from model assumptions. We demonstrate its capacity by applying it to slowly diffusing and rapidly transitioning bacterial envelope proteins. ExTrack greatly increases the regime of computationally analyzable noisy single-particle tracks. The ExTrack package is available in ImageJ and Python.

ExTrack characterizes transition kinetics and diffusion in noisy single-particle tracks

Single-particle tracking microscopy is a powerful technique to investigate how proteins dynamically interact with their environment in live cells. However, the analysis of tracks is confounded by noisy molecule localization, short tracks, and rapid transitions between different motion states, notably between immobile and diffusive states. Here, we propose a probabilistic method termed ExTrack that uses the full spatio-temporal information of tracks to extract global model parameters, to calculate state probabilities at every time point, to reveal distributions of state durations, and to refine the positions of bound molecules. ExTrack works for a wide range of diffusion coefficients and transition rates, even if experimental data deviate from model assumptions. We demonstrate its capacity by applying it to slowly diffusing and rapidly transitioning bacterial envelope proteins. ExTrack greatly increases the regime of computationally analyzable noisy single-particle tracks. The ExTrack package is available in ImageJ and Python.

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.

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

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

Effect of St. John's wort extract Ze 117 on the lateral mobility of β1-adrenergic receptors in C6 cells

Depression has been associated with altered signal transduction of serotonergic, dopaminergic and adrenergic neurotransmitter systems in the brain. Signaling relies on receptor-ligand interactions and subsequent regulatory processes, but also on lateral receptor mobility. The aim of this study was to investigate the effect of the St. John's wort extract Ze 117 on the lateral mobility of SNAP-tagged β₁-adrenergic receptors (β₁AR) in the plasma membrane of C6 cells under both, non-stimulating and isoprenaline-stimulating conditions. Single particle tracking (SPT) was used, whereby the registered trajectories were evaluated by variational Bayesian treatment of a hidden Markov model (vbSPT) and packing coefficient (Pc) analysis with respect to diffusion coefficients, receptor state occupancies and confinement. Three different diffusion states were identified, differing in their diffusion coefficients. Treatment with Ze 117 [25 µg/ml] decreased the mobility of the β₁AR, which was manifested by a relative increase in the slow-diffusing state S1 (0.21-0.30) compared to control and by an increase in receptor confinement (79.4-68.1 nm). After isoprenaline stimulation of control cells, the slow-diffusing state was more pronounced, whereas confinement was not affected. In summary, SPT has been shown to be a powerful method to analyze lateral receptor mobility. Furthermore, the present study identified a correlation between Ze 117 treatment and β₁AR mobility.

Stochastically switching diffusion with partially reactive surfaces

In this paper we develop a hybrid version of the encounter-based approach to diffusion-mediated absorption at a reactive surface, which takes into account stochastic switching of a diffusing particle's conformational state. For simplicity, we consider a two-state model in which the probability of surface absorption depends on the current particle state and the amount of time the particle has spent in a neighborhood of the surface in each state. The latter is determined by a pair of local times ℓ_{n,t}, n=0,1, which are Brownian functionals that keep track of particle-surface encounters over the time interval [0,t]. We proceed by constructing a differential Chapman-Kolmogorov equation for a pair of generalized propagators P_{n}(x,ℓ_{0},ℓ_{1},t), where P_{n} is the joint probability density for the set (X_{t},ℓ_{0,t},ℓ_{1,t}) when N_{t}=n, where X_{t} denotes the particle position and N_{t} is the corresponding conformational state. Performing a double Laplace transform with respect to ℓ_{0},ℓ_{1} yields an effective system of equations describing diffusion in a bounded domain Ω, in which there is switching between two Robin boundary conditions on ∂Ω. The corresponding constant reactivities are κ_{j}=Dz_{j} and j=0,1, where z_{j} is the Laplace variable corresponding to ℓ_{j} and D is the diffusivity. Given the solution for the propagators in Laplace space, we construct a corresponding probabilistic model for partial absorption, which requires finding the inverse Laplace transform with respect to z_{0},z_{1}. We illustrate the theory by considering diffusion of a particle on the half-line with the boundary at x=0 effectively switching between a totally reflecting and a partially absorbing state. We calculate the flux due to absorption and use this to compute the resulting MFPT in the presence of a renewal-based stochastic resetting protocol. The latter resets the position and conformational state of the particle as well as the corresponding local times. Finally, we indicate how to extend the analysis to higher spatial dimensions using the spectral theory of Dirichlet-to-Neumann operators.

Receptor tyrosine kinase MET ligand-interaction classified via machine learning from single-particle tracking data

Internalin B-mediated activation of the membrane-bound receptor tyrosine kinase MET is accompanied by a change in receptor mobility. Conversely, it should be possible to infer from receptor mobility whether a cell has been treated with internalin B. Here, we propose a method based on hidden Markov modeling and explainable artificial intelligence that machine-learns the key differences in MET mobility between internalin B-treated and -untreated cells from single-particle tracking data. Our method assigns receptor mobility to three diffusion modes (immobile, slow, and fast). It discriminates between internalin B-treated and -untreated cells with a balanced accuracy of >99% and identifies three parameters that are most affected by internalin B treatment: a decrease in the mobility of slow molecules (1) and a depopulation of the fast mode (2) caused by an increased transition of fast molecules to the slow mode (3). Our approach is based entirely on free software and is readily applicable to the analysis of other membrane receptors.

Triple-color single-molecule imaging for analysis of the role of receptor oligomers in signal transduction

Membrane receptors provide interfaces of various extracellular stimuli to transduce the signal into the cell. Receptors are required to possess such conflicting properties as high sensitivity and noise reduction for the cell to keep its homeostasis and appropriate responses. To understand the mechanisms by which these functions are achieved, single-molecule monitoring of the motilities of receptors and signaling molecules on the plasma membrane is one of the most direct approaches. This review article introduces several recent single-molecule imaging studies of receptors, including the author’s recent work on triple-color single-molecule imaging of G protein-coupled receptors. Based on these researches, advantages and perspectives of the single-molecule imaging approach to solving the mechanisms of receptor functions are illustrated.

Mechanism of Cyanine5 to Cyanine3 Photoconversion and Its Application for High-Density Single-Particle Tracking in a Living Cell

Cyanine (Cy) dyes are among the most useful organic fluorophores that have found a wide range of applications in single-molecule and super-resolution imaging as well as in other biophysical studies. However, recent observations that blueshifted derivatives of Cy dyes are formed via photoconversion have raised concerns as to the potential artifacts in multicolor imaging. Here, we report the mechanism for the photoconversion of Cy5 to Cy3 that occurs upon photoexcitation during fluorescent imaging. Our studies show that the formal C₂H₂ excision from Cy5 occurs mainly through an intermolecular pathway involving a combination of bond cleavage and reconstitution while unambiguously confirming the identity of the fluorescent photoproduct of Cy5 to be Cy3 using various spectroscopic tools. The carbonyl products generated from singlet oxygen-mediated photooxidation of Cy5 undergo a sequence of carbon-carbon bond-breaking and -forming events to bring about the novel dye-to-dye transformation. We also show that the deletion of a two-methine unit from the polymethine chain, which results in the formation of blueshifted products, commonly occurs in other cyanine dyes, such as Alexa Fluor 647 (AF647) and Cyanine5.5. The formation of a blueshifted congener dye can obscure the multicolor fluorescence imaging, leading to misinterpretation of the data. We demonstrate that the potentially deleterious photoconversion, however, can be exploited to develop a new photoactivation method for high-density single-particle tracking in a living cell without using UV illumination and cell-toxic additives.

A pairwise distance distribution correction (DDC) algorithm to eliminate blinking-caused artifacts in SMLM

Single-molecule localization microscopy (SMLM) relies on the blinking behavior of a fluorophore, which is the stochastic switching between fluorescent and dark states. Blinking creates multiple localizations belonging to the same fluorophore, confounding quantitative analyses and interpretations. Here we present a method, termed distance distribution correction (DDC), to eliminate blinking-caused repeat localizations without any additional calibrations. The approach relies on obtaining the true pairwise distance distribution of different fluorophores naturally from the imaging sequence by using distances between localizations separated by a time much longer than the average fluorescence survival time. We show that, using the true pairwise distribution, we can define and maximize the likelihood, obtaining a set of localizations void of blinking artifacts. DDC results in drastic improvements in obtaining the closest estimate of the true spatial organization and number of fluorescent emitters in a wide range of applications, enabling accurate reconstruction and quantification of SMLM images.

A pairwise distance distribution correction (DDC) algorithm to eliminate blinking-caused artifacts in SMLM

Single-molecule localization microscopy (SMLM) relies on the blinking behavior of a fluorophore, which is the stochastic switching between fluorescent and dark states. Blinking creates multiple localizations belonging to the same fluorophore, confounding quantitative analyses and interpretations. Here we present a method, termed distance distribution correction (DDC), to eliminate blinking-caused repeat localizations without any additional calibrations. The approach relies on obtaining the true pairwise distance distribution of different fluorophores naturally from the imaging sequence by using distances between localizations separated by a time much longer than the average fluorescence survival time. We show that, using the true pairwise distribution, we can define and maximize the likelihood, obtaining a set of localizations void of blinking artifacts. DDC results in drastic improvements in obtaining the closest estimate of the true spatial organization and number of fluorescent emitters in a wide range of applications, enabling accurate reconstruction and quantification of SMLM images.

Detection of Anomalous Diffusion with Deep Residual Networks

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

Fundamentals of the logarithmic measure for revealing multimodal diffusion

We develop a theoretical foundation for a time-series analysis method suitable for revealing the spectrum of diffusion coefficients in mixed Brownian systems, for which no prior knowledge of particle distinction is required. This method is directly relevant for particle tracking in biological systems, in which diffusion processes are often nonuniform. We transform Brownian data onto the logarithmic domain, in which the coefficients for individual modes of diffusion appear as distinct spectral peaks in the probability density. We refer to the method as the logarithmic measure of diffusion, or simply as the logarithmic measure. We provide a general protocol for deriving analytical expressions for the probability densities on the logarithmic domain. The protocol is applicable for any number of spatial dimensions with any number of diffusive states. The analytical form can be fitted to data to reveal multiple diffusive modes. We validate the theoretical distributions and benchmark the accuracy and sensitivity of the method by extracting multimodal diffusion coefficients from two-dimensional Brownian simulations of polydisperse filament bundles. Bundling the filaments allows us to control the system nonuniformity and hence quantify the sensitivity of the method. By exploiting the anisotropy of the simulated filaments, we generalize the logarithmic measure to rotational diffusion. By fitting the analytical forms to simulation data, we confirm the method's theoretical foundation. An error analysis in the single-mode regime shows that the proposed method is comparable in accuracy to the standard mean-squared displacement approach for evaluating diffusion coefficients. For the case of multimodal diffusion, we compare the logarithmic measure against other, more sophisticated methods, showing that both model selectivity and extraction accuracy are comparable for small data sets. Therefore, we suggest that the logarithmic measure, as a method for multimodal diffusion coefficient extraction, is ideally suited for small data sets, a condition often confronted in the experimental context. Finally, we critically discuss the proposed benefits of the method and its information content.

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

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

Subdiffusion-limited fractional reaction-subdiffusion equations with affine reactions: Solution, stochastic paths, and applications

In contrast to normal diffusion, there is no canonical model for reactions between chemical species which move by anomalous subdiffusion. Indeed, the type of mesoscopic equation describing reaction-subdiffusion systems depends on subtle assumptions about the microscopic behavior of individual molecules. Furthermore, the correspondence between mesoscopic and microscopic models is not well understood. In this paper, we study the subdiffusion-limited model, which is defined by mesoscopic equations with fractional derivatives applied to both the movement and the reaction terms. Assuming that the reaction terms are affine functions, we show that the solution to the fractional system is the expectation of a random time change of the solution to the corresponding integer order system. This result yields a simple and explicit algebraic relationship between the fractional and integer order solutions in Laplace space. We then find the microscopic Langevin description of individual molecules that corresponds to such mesoscopic equations and give a computer simulation method to generate their stochastic trajectories. This analysis identifies some precise microscopic conditions that dictate when this type of mesoscopic model is or is not appropriate. We apply our results to several scenarios in cell biology which, despite the ubiquity of subdiffusion in cellular environments, have been modeled almost exclusively by normal diffusion. Specifically, we consider subdiffusive models of morphogen gradient formation, fluctuating mobility, and fluorescence recovery after photobleaching (FRAP) experiments. We also apply our results to fractional ordinary differential equations.

LPS-binding IgG arrests actively motile Salmonella Typhimurium in gastrointestinal mucus

The gastrointestinal (GI) mucosa is coated with a continuously secreted mucus layer that serves as the first line of defense against invading enteric bacteria. We have previously shown that antigen-specific immunoglobulin G (IgG) can immobilize viruses in both human airway and genital mucus secretions through multiple low-affinity bonds between the array of virion-bound IgG and mucins, thereby facilitating their rapid elimination from mucosal surfaces and preventing mucosal transmission. Nevertheless, it remains unclear whether weak IgG-mucin crosslinks could reinforce the mucus barrier against the permeation of bacteria driven by active flagella beating, or in predominantly MUC2 mucus gel. Here, we performed high-resolution multiple particle tracking to capture the real-time motion of hundreds of individual fluorescent Salmonella Typhimurium in fresh, undiluted GI mucus from Rag1^-/- mice, and analyzed the motion using a hidden Markov model framework. In contrast to control IgG, the addition of anti-lipopolysaccharide IgG to GI mucus markedly reduced the progressive motility of Salmonella by lowering the swim speed and retaining individual bacteria in an undirected motion state. Effective crosslinking of Salmonella to mucins was dependent on Fc N-glycans. Our findings implicate IgG-mucin crosslinking as a broadly conserved function that reduces mucous penetration of both bacterial and viral pathogens.

Quantitative Analysis of Single Quantum Dot Trajectories

Single quantum dot tracking (SQDT) is a powerful technique for interrogating biomolecular dynamics in living cells and tissue. SQDT has particularly excelled in driving discovery at the single-molecule level in the fields of neuronal communication, plasma membrane organization, viral infection, and immune system response. Here, we briefly characterize various elements of the SQDT analytical framework and provide the reader with a detailed set of executable commands to implement commonly used algorithms for SQDT data processing.

Complex Diffusion in Bacteria

Diffusion within bacteria is often thought of as a "simple" random process by which molecules collide and interact with each other. New research however shows that this is far from the truth. Here we shed light on the complexity and importance of diffusion in bacteria, illustrating the similarities and differences of diffusive behaviors of molecules within different compartments of bacterial cells. We first describe common methodologies used to probe diffusion and the associated models and analyses. We then discuss distinct diffusive behaviors of molecules within different bacterial cellular compartments, highlighting the influence of metabolism, size, crowding, charge, binding, and more. We also explicitly discuss where further research and a united understanding of what dictates diffusive behaviors across the different compartments of the cell are required, pointing out new research avenues to pursue.

Mapping spatio-temporal dynamics of single biomolecules in living cells

We present a Bayesian framework for inferring spatio-temporal maps of diffusivity and potential fields from recorded trajectories of single molecules inside living cells. The framework naturally lets us regularise the high-dimensional inference problem using prior distributions in order to obtain robust results. To overcome the computational complexity of inferring thousands of map parameters from large single particle tracking datasets, we developed a stochastic optimisation method based on local mini-batches and parsimonious gradient calculation. We quantified the gain in convergence speed on numerical simulations, and we demonstrated for the first time temporal regularisation and aligned values of the inferred potential fields across multiple time segments. As a proof-of-concept, we mapped the dynamics of HIV-1 Gag proteins involved in the formation of virus-like particles (VLPs) on the plasma membrane of live T cells at high spatial and temporal resolutions. We focused on transient aggregation events lasting only on tenth of the time required for full VLP formation. The framework and optimisation methods are implemented in the TRamWAy open-source software platform for analysing single biomolecule dynamics.

Particle Mobility Analysis Using Deep Learning and the Moment Scaling Spectrum

Quantitative analysis of dynamic processes in living cells using time-lapse microscopy requires not only accurate tracking of every particle in the images, but also reliable extraction of biologically relevant parameters from the resulting trajectories. Whereas many methods exist to perform the tracking task, there is still a lack of robust solutions for subsequent parameter extraction and analysis. Here a novel method is presented to address this need. It uses for the first time a deep learning approach to segment single particle trajectories into consistent tracklets (trajectory segments that exhibit one type of motion) and then performs moment scaling spectrum analysis of the tracklets to estimate the number of mobility classes and their associated parameters, providing rich fundamental knowledge about the behavior of the particles under study. Experiments on in-house datasets as well as publicly available particle tracking data for a wide range of proteins with different dynamic behavior demonstrate the broad applicability of the method.

Direct observation of Thermomyces lanuginosus lipase diffusional states by Single Particle Tracking and their remodeling by mutations and inhibition

Lipases are interfacially activated enzymes that catalyze the hydrolysis of ester bonds and constitute prime candidates for industrial and biotechnological applications ranging from detergent industry, to chiral organic synthesis. As a result, there is an incentive to understand the mechanisms underlying lipase activity at the molecular level, so as to be able to design new lipase variants with tailor-made functionalities. Our understanding of lipase function primarily relies on bulk assay averaging the behavior of a high number of enzymes masking structural dynamics and functional heterogeneities. Recent advances in single molecule techniques based on fluorogenic substrate analogues revealed the existence of lipase functional states, and furthermore so how they are remodeled by regulatory cues. Single particle studies of lipases on the other hand directly observed diffusional heterogeneities and suggested lipases to operate in two different modes. Here to decipher how mutations in the lid region controls Thermomyces lanuginosus lipase (TLL) diffusion and function we employed a Single Particle Tracking (SPT) assay to directly observe the spatiotemporal localization of TLL and rationally designed mutants on native substrate surfaces. Parallel imaging of thousands of individual TLL enzymes and HMM analysis allowed us to observe and quantify the diffusion, abundance and microscopic transition rates between three linearly interconverting diffusional states for each lipase. We proposed a model that correlate diffusion with function that allowed us to predict that lipase regulation, via mutations in lid region or product inhibition, primarily operates via biasing transitions to the active states.

Analysis of the Diffusivity Change from Single-Molecule Trajectories on Living Cells

With the wide application of live-cell single-molecule imaging and tracking of biomolecules at work, deriving diffusion state changes of individual molecules is of particular interest as these changes reflect molecular oligomerization or interaction with other cellular components and thus correlate with functional changes. We have developed a Rayleigh mixture distribution-based hidden Markov model method to analyze time-lapse diffusivity change of single molecules, especially membrane proteins, with unknown dynamic states in living cells. With this method, the diffusion parameters, including diffusion state number, state transition probability, diffusion coefficient, and state mixture ratio, can be extracted from the single-molecule diffusion trajectories accurately via easy computation. The validity of our method has been demonstrated with not only experiments on synthetic trajectories but also single-molecule fluorescence imaging data of two typical membrane receptors. Our method offers a new analytical tool for the investigation of molecular interaction kinetics at the single-molecule level.

A global sampler of single particle tracking solutions for single molecule microscopy

The dependence on model-fitting to evaluate particle trajectories makes it difficult for single particle tracking (SPT) to resolve the heterogeneous molecular motions typical of cells. We present here a global spatiotemporal sampler for SPT solutions using a Metropolis-Hastings algorithm. The sampler does not find just the most likely solution but also assesses its likelihood and presents alternative solutions. This enables the estimation of the tracking error. Furthermore the algorithm samples the parameters that govern the tracking process and therefore does not require any tweaking by the user. We demonstrate the algorithm on synthetic and single molecule data sets. Metrics for the comparison of SPT are generalised to be applied to a SPT sampler. We illustrate using the example of the diffusion coefficient how the distribution of the tracking solutions can be propagated into a distribution of derived quantities. We also discuss the major challenges that are posed by the realisation of a SPT sampler.

Modeling Barrier Properties of Intestinal Mucus Reinforced with IgG and Secretory IgA against Motile Bacteria

The gastrointestinal (GI) tract is lined with a layer of viscoelastic mucus gel, characterized by a dense network of entangled and cross-linked mucins together with an abundance of antibodies (Ab). Secretory IgA (sIgA), the predominant Ab isotype in the GI tract, is a dimeric molecule with 4 antigen-binding domains capable of inducing efficient clumping of bacteria, or agglutination. IgG, another common Ab at mucosal surfaces, can cross-link individual viruses to the mucin mesh through multiple weak bonds between IgG-Fc and mucins, a process termed muco-trapping. Relative contributions by agglutination versus muco-trapping in blocking permeation of motile bacteria through mucus remain poorly understood. Here, we developed a mathematical model that takes into account physiologically relevant spatial dimensions and time scales, binding and unbinding rates between Ab and bacteria as well as between Ab and mucins, the diffusivities of Ab, and run-tumble motion of active bacteria. Our model predicts both sIgA and IgG can accumulate on the surface of individual bacteria at sufficient quantities and rates to enable trapping individual bacteria in mucins before they penetrate the mucus layer. Furthermore, our model predicts that agglutination only modestly improves the ability for antibodies to block bacteria permeation through mucus. These results suggest that while sIgA is the most potent Ab isotype overall at stopping bacterial penetration, IgG may represent a practical alternative for mucosal prophylaxis and therapy. Our work improves the mechanistic understanding of Ab-enhanced barrier properties of mucus and highlights the ability for muco-trapping Ab to protect against motile pathogens at mucosal surfaces.

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

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

Protein concentration gradients and switching diffusions

Morphogen gradients play a vital role in developmental biology by enabling embryonic cells to infer their spatial location and determine their developmental fate accordingly. The standard mechanism for generating a morphogen gradient involves a morphogen being produced from a localized source and subsequently degrading. While this mechanism is effective over the length and time scales of tissue development, it fails over typical subcellular length scales due to the rapid dissipation of spatial asymmetries. In a recent theoretical work, we found an alternative mechanism for generating concentration gradients of diffusing molecules, in which the molecules switch between spatially constant diffusivities at switching rates that depend on the spatial location of a molecule. Independently, an experimental and computational study later found that Caenorhabditis elegans zygotes rely on this mechanism for cell polarization. In this paper, we extend our analysis of switching diffusivities to determine its role in protein concentration gradient formation. In particular, we determine how switching diffusivities modifies the standard theory and show how space-dependent switching diffusivities can yield a gradient in the absence of a localized source. Our mathematical analysis yields explicit formulas for the intracellular concentration gradient which closely match the results of previous experiments and numerical simulations.

Multistep Track Segmentation and Motion Classification for Transient Mobility Analysis

Molecular interactions are often transient and might change within the window of observation, leading to changes in molecule movement. Therefore, accurate motion analysis often requires transient motion classification. Here we present an accurate and computationally efficient transient mobility analysis framework, termed "divide-and-conquer moment scaling spectrum" (DC-MSS). DC-MSS works in a multistep fashion: 1) it utilizes a local movement descriptor throughout a track to divide it into initial segments of putatively different motion classes; 2) it classifies these segments via moment scaling spectrum (MSS) analysis of molecule displacements; and 3) it uses the MSS analysis results to refine the track segmentation. This strategy uncouples the initial identification of motion switches from motion classification, allowing DC-MSS to circumvent the sensitivity-accuracy tradeoff of classic rolling window approaches for transient motion analysis, while at the same time harnessing the classification power of MSS analysis. Testing of DC-MSS demonstrates that it detects switches among free diffusion, confined diffusion, directed diffusion, and immobility with great sensitivity. To illustrate the utility of DC-MSS, we have applied it to single-particle tracks of the transmembrane protein CD44 on the surface of macrophages, revealing actin cortex-dependent transient mobility changes.

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

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

Abl2 is recruited to ventral actin waves through cytoskeletal interactions to promote lamellipodium extension

Abl family nonreceptor tyrosine kinases regulate changes in cell shape and migration. Abl2 localizes to dynamic actin-rich protrusions, such as lamellipodia in fibroblasts and dendritic spines in neurons. Abl2 interactions with cortactin, an actin filament stabilizer, are crucial for the formation and stability of actin-rich structures, but Abl2:cortactin-positive structures have not been characterized with high spatiotemporal resolution in cells. Using total internal reflection fluorescence microscopy, we demonstrate that Abl2 colocalizes with cortactin at wave-like structures within lamellum and lamellipodium tips. Abl2 and cortactin within waves are focal and transient, extend to the outer edge of lamella, and serve as the base for lamellipodia protrusions. Abl2-positive foci colocalize with integrin β3 and paxillin, adhesive markers of the lamellum-lamellipodium interface. Cortactin-positive waves still form in Abl2 knockout cells, but the lamellipodium size is significantly reduced. This deficiency is restored following Abl2 reexpression. Complementation analyses revealed that the Abl2 C-terminal half, which contains domains that bind actin and microtubules, is necessary and sufficient for recruitment to the wave-like structures and to support normal lamellipodium size, while the kinase domain-containing N-terminal half does not impact lamellipodium size. Together, this work demonstrates that Abl2 is recruited with cortactin to actin waves through cytoskeletal interactions to promote lamellipodium extension.

Single-Molecule Analysis Reveals Linked Cycles of RSC Chromatin Remodeling and Ace1p Transcription Factor Binding in Yeast

It is unknown how the dynamic binding of transcription factors (TFs) is molecularly linked to chromatin remodeling and transcription. Using single-molecule tracking (SMT), we show that the chromatin remodeler RSC speeds up the search process of the TF Ace1p for its response elements (REs) at the CUP1 promoter. We quantified smFISH mRNA data using a gene bursting model and demonstrated that RSC regulates transcription bursts of CUP1 only by modulating TF occupancy but does not affect initiation and elongation rates. We show by SMT that RSC binds to activated promoters transiently, and based on MNase-seq data, that RSC does not affect the nucleosomal occupancy at CUP1. Therefore, transient binding of Ace1p and rapid bursts of transcription at CUP1 may be dependent on short repetitive cycles of nucleosome mobilization. This type of regulation reduces the transcriptional noise and ensures a homogeneous response of the cell population to heavy metal stress.

A Hidden Markov Model for Detecting Confinement in Single-Particle Tracking Trajectories

State-of-the-art single-particle tracking (SPT) techniques can generate long trajectories with high temporal and spatial resolution. This offers the possibility of mechanistically interpreting particle movements and behavior in membranes. To this end, a number of statistical techniques have been developed that partition SPT trajectories into states with distinct diffusion signatures, allowing a statistical analysis of diffusion state dynamics and switching behavior. Here, we develop a confinement model, within a hidden Markov framework, that switches between phases of free diffusion and confinement in a harmonic potential well. By using a Markov chain Monte Carlo algorithm to fit this model, automated partitioning of individual SPT trajectories into these two phases is achieved, which allows us to analyze confinement events. We demonstrate the utility of this algorithm on a previously published interferometric scattering microscopy data set, in which gold-nanoparticle-tagged ganglioside GM1 lipids were tracked in model membranes. We performed a comprehensive analysis of confinement events, demonstrating that there is heterogeneity in the lifetime, shape, and size of events, with confinement size and shape being highly conserved within trajectories. Our observations suggest that heterogeneity in confinement events is caused by both individual nanoparticle characteristics and the binding-site environment. The individual nanoparticle heterogeneity ultimately limits the ability of interferometric scattering microscopy to resolve molecule dynamics to the order of the tag size; homogeneous tags could potentially allow the resolution to be taken below this limit by deconvolution methods. In a wider context, the presented harmonic potential well confinement model has the potential to detect and characterize a wide variety of biological phenomena, such as hop diffusion, receptor clustering, and lipid rafts.

Detection of Velocity and Diffusion Coefficient Change Points in Single-Particle Trajectories

The position-time trajectory of a biological subject moving in a complex environment contains rich information about how it interacts with the local setting. Whether the subject be an animal or an intracellular endosomal vesicle, the two primary modes of biological locomotion are directional movement and random walk, respectively characterized by velocity and diffusion coefficient. This contribution introduces a method to quantitatively divide a single-particle trajectory into segments that exhibit changes in the diffusion coefficient, velocity, or both. With the determination of these two physical parameters given by the maximum likelihood estimators, the relative precisions are given as explicit functions of the number of data points and total trajectory time. The method is based on rigorous statistical tests and does not require any presumed kinetics scheme. Results of extensive characterizations, extensions to 2D and 3D trajectories, and applications to common scenarios are also discussed.

Bayesian inference with information content model check for Langevin equations

The Bayesian data analysis framework has been proven to be a systematic and effective method of parameter inference and model selection for stochastic processes. In this work, we introduce an information content model check that may serve as a goodness-of-fit, like the χ^{2} procedure, to complement conventional Bayesian analysis. We demonstrate this extended Bayesian framework on a system of Langevin equations, where coordinate-dependent mobilities and measurement noise hinder the normal mean-squared displacement approach.

Galectin-3 alters the lateral mobility and clustering of β1-integrin receptors

Glycoprotein receptors are influenced by myriad intermolecular interactions at the cell surface. Specific glycan structures may interact with endogenous lectins that enforce or disrupt receptor-receptor interactions. Glycoproteins bound by multivalent lectins may form extended oligomers or lattices, altering the lateral mobility of the receptor and influencing its function through endocytosis or changes in activation. In this study, we have examined the interaction of Galectin-3 (Gal-3), a human lectin, with adhesion receptors. We measured the effect of recombinant Gal-3 added exogenously on the lateral mobility of the α5β1 integrin on HeLa cells. Using single-particle tracking (SPT) we detected increased lateral mobility of the integrin in the presence of Gal-3, while its truncated C-terminal domain (Gal-3C) showed only minor reductions in lateral mobility. Treatment of cells with Gal-3 increased β1-integrin mediated migration with no apparent changes in viability. In contrast, Gal-3C decreased both cell migration and viability. Fluorescence microscopy allowed us to confirm that exogenous Gal-3 resulted in reorganization of the integrin into larger clusters. We used a proteomics analysis to confirm that cells expressed endogenous Gal-3, and found that addition of competitive oligosaccharide ligands for the lectin altered the lateral mobility of the integrin. Together, our results are consistent with a Gal-3-integrin lattice model of binding and confirm that the lateral mobility of integrins is natively regulated, in part, by galectins.

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.

Hybrid colored noise process with space-dependent switching rates

A fundamental issue in the theory of continuous stochastic process is the interpretation of multiplicative white noise, which is often referred to as the Itô-Stratonovich dilemma. From a physical perspective, this reflects the need to introduce additional constraints in order to specify the nature of the noise, whereas from a mathematical perspective it reflects an ambiguity in the formulation of stochastic differential equations (SDEs). Recently, we have identified a mechanism for obtaining an Itô SDE based on a form of temporal disorder. Motivated by switching processes in molecular biology, we considered a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state, the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast switching limit one obtains Brownian motion with a space-dependent diffusivity of the Itô form. In this paper, we extend our theory to include colored additive noise. We show that the nature of the effective multiplicative noise process obtained by taking both the white-noise limit (κ→0) and fast switching limit (ε→0) depends on the order the two limits are taken. If the white-noise limit is taken first, then we obtain Itô, and if the fast switching limit is taken first, then we obtain Stratonovich. Moreover, the form of the effective diffusion coefficient differs in the two cases. The latter result holds even in the case of space-independent transition rates, where one obtains additive noise processes with different diffusion coefficients. Finally, we show that yet another form of multiplicative noise is obtained in the simultaneous limit ε,κ→0 with ε/κ^{2} fixed.