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 D ₁ = 0.68D ₀ − 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.     

Analysis of molecular diffusion by first-passage time variance identifies the size of confinement zones

The diffusion of receptors within the two-dimensional environment of the plasma membrane is a complex process. Although certain components diffuse according to a random walk model (Brownian diffusion), an overwhelming body of work has found that membrane diffusion is nonideal (anomalous diffusion). One of the most powerful methods for studying membrane diffusion is single particle tracking (SPT), which records the trajectory of a label attached to a membrane component of interest. One of the outstanding problems in SPT is the analysis of data to identify the presence of heterogeneity. We have adapted a first-passage time (FPT) algorithm, originally developed for the interpretation of animal movement, for the analysis of SPT data. We discuss the general application of the FPT analysis to molecular diffusion, and use simulations to test the method against data containing known regions of confinement. We conclude that FPT can be used to identify the presence and size of confinement within trajectories of the receptor LFA-1, and these results are consistent with previous reports on the size of LFA-1 clusters. The analysis of trajectory data for cell surface receptors by FPT provides a robust method to determine the presence and size of confined regions of diffusion.     

Analysis of Microscopic Parameters of Single-Particle Trajectories in Neurons

We performed a comparative study of the statistical uncertainties that arise when calculating the velocity and diffusion coefficients from single-particle trajectories. We show that a method where particle mean displacement is used to calculate velocity and mean square fluctuation is used to calculate diffusion coefficient offers greater accuracy than analysis of time-dependent mean square displacement. Our assessment of the performance of the two analysis strategies is conducted in two ways. First, we apply each of the methods to simulated trajectories where each parameter term is known. Second, we analyze the motion of previously uncharacterized EphB2 receptors in the membrane of hippocampal neurons. We find that EphB2 receptors display different types of motion mode and transition between these modes. We present our data as a distribution of microscopic diffusion coefficients for each particle trajectory, which we refer to as partial distributions. Partial distributions are summed to form a cumulative distribution of diffusion coefficients for EphB2 receptors in hippocampal neurons. The structure and interpretation of the EphB2 cumulative distribution are discussed.     

Statistical Analysis of Lateral Diffusion and Multistate Kinetics in Single-Molecule Imaging

Single-molecule trajectories of molecules on the membrane of living cells have indicated the possibility that the lateral mobility of individual molecules is variable with time. Such temporal variation in mobility may indicate intrinsic kinetics of multiple molecular states. To clarify the mechanisms of signal processing on the membrane, quantitative characterizations of such temporal variations are necessary. Here we propose a method to analyze and characterize the multiple states in lateral mobility and their transition kinetics from single-molecule trajectories based on a displacement probability density function and an autocorrelation function of squared displacements. We performed our method for three cases: a molecule with a single diffusion coefficient (D), a mixture of molecules in two states with different D-values, and a molecule switching between two states with different D-values. Our analysis of numerically generated trajectories successfully distinguished the three cases and estimated the characteristic parameters for mobility and the kinetics of state transitions. This method is applicable to single-molecule tracking analysis of molecules in multiple functional states with different lateral mobility on the membrane of living cells.     

Empirical Bayes Methods Enable Advanced Population-Level Analyses of Single-Molecule FRET Experiments

Many single-molecule experiments aim to characterize biomolecular processes in terms of kinetic models that specify the rates of transition between conformational states of the biomolecule. Estimation of these rates often requires analysis of a population of molecules, in which the conformational trajectory of each molecule is represented by a noisy, time-dependent signal trajectory. Although hidden Markov models (HMMs) may be used to infer the conformational trajectories of individual molecules, estimating a consensus kinetic model from the population of inferred conformational trajectories remains a statistically difficult task, as inferred parameters vary widely within a population. Here, we demonstrate how a recently developed empirical Bayesian method for HMMs can be extended to enable a more automated and statistically principled approach to two widely occurring tasks in the analysis of single-molecule fluorescence resonance energy transfer (smFRET) experiments: 1), the characterization of changes in rates across a series of experiments performed under variable conditions; and 2), the detection of degenerate states that exhibit the same FRET efficiency but differ in their rates of transition. We apply this newly developed methodology to two studies of the bacterial ribosome, each exemplary of one of these two analysis tasks. We conclude with a discussion of model-selection techniques for determination of the appropriate number of conformational states. The code used to perform this analysis and a basic graphical user interface front end are available as open source software.     

Particle image correlation spectroscopy (PICS): retrieving nanometer-scale correlations from high-density single-molecule position data

A new data analysis tool that resolves correlations on the nanometer length and millisecond timescale is derived. This tool, adapted from methods of spatiotemporal image correlation spectroscopy, exploits the high positional accuracy of single-particle tracking. While conventional tracking methods break down if multiple particle trajectories intersect, our method works in principle for arbitrarily large molecule densities and diffusion coefficients as long as individual molecules can be identified. The method is computationally cheap and robust and requires no a priori knowledge about the dynamical coefficients, as opposed to other methods. We demonstrate the validity of the method by Monte Carlo simulations and by application to single-molecule tracking data of membrane-anchored proteins in live cells. The results faithfully reproduce those obtained by conventional tracking. Upon activation, a fraction of the small GTPase H-Ras is confined to domains of <200 nm diameter, which further substantiates the prediction that membrane organization is a determinant in cellular signaling.     

Empirical Bayes Methods Enable Advanced Population-Level Analyses of Single-Molecule FRET Experiments

Many single-molecule experiments aim to characterize biomolecular processes in terms of kinetic models that specify the rates of transition between conformational states of the biomolecule. Estimation of these rates often requires analysis of a population of molecules, in which the conformational trajectory of each molecule is represented by a noisy, time-dependent signal trajectory. Although hidden Markov models (HMMs) may be used to infer the conformational trajectories of individual molecules, estimating a consensus kinetic model from the population of inferred conformational trajectories remains a statistically difficult task, as inferred parameters vary widely within a population. Here, we demonstrate how a recently developed empirical Bayesian method for HMMs can be extended to enable a more automated and statistically principled approach to two widely occurring tasks in the analysis of single-molecule fluorescence resonance energy transfer (smFRET) experiments: 1), the characterization of changes in rates across a series of experiments performed under variable conditions; and 2), the detection of degenerate states that exhibit the same FRET efficiency but differ in their rates of transition. We apply this newly developed methodology to two studies of the bacterial ribosome, each exemplary of one of these two analysis tasks. We conclude with a discussion of model-selection techniques for determination of the appropriate number of conformational states. The code used to perform this analysis and a basic graphical user interface front end are available as open source software.     

Fluorescence Fluctuation Spectroscopy of mCherry in Living Cells

The red fluorescent protein mCherry is of considerable interest for fluorescence fluctuation spectroscopy (FFS), because the wide separation in color between mCherry and green fluorescent protein provides excellent conditions for identifying protein interactions inside cells. This two-photon study reveals that mCherry exists in more than a single brightness state. Unbiased analysis of the data needs to account for the presence of multiple states. We introduce a two-state model that successfully describes the brightness and fluctuation amplitude of mCherry. The properties of the two states are characterized by FFS and fluorescence lifetime experiments. No interconversion between the two states was observed over the experimentally probed timescales. The effect of fluorescence resonance energy transfer between enhanced green fluorescent protein (EGFP) and mCherry is incorporated into the two-state model to describe protein hetero-oligomerization. The model is verified by comparing the predicted and measured brightness and fluctuation amplitude of several fusion proteins that contain mCherry and EGFP. In addition, hetero-fluorescence resonance energy transfer between mCherry molecules in different states is detected, but its influence on FFS parameters is small enough to be negligible. Finally, the two-state model is applied to study protein oligomerization in living cells. We demonstrate that the model successfully describes the homodimerization of nuclear receptors. In addition, we resolved a mixture of interacting and noninteracting proteins labeled with EGFP and mCherry. These results provide the foundation for quantitative applications of mCherry in FFS studies.     

3D single-particle tracking and optical trap measurements on adhesion proteins.

A three-dimensional single-particle tracking system was combined with an optical trap to investigate the behavior of transmembrane adhesion proteins. We exploited this setup to investigate which part of the cell adhesion protein LFA-1 forms a connection to the cytoskeleton after binding to its ligand ICAM-1. LFA-1 is an integrin consisting of an alpha and a beta chain. Thus far, only the cytoplasmic tail of the beta chain is known to form a connection to the cytoskeleton. We investigated cells that express a mutant form of LFA-1 that lacks the complete beta cytoplasmic tail and therefore is not thought to bind to the cytoskeleton. Interestingly, single-particle tracking measurements using beads coated with the ligand ICAM-1 indicate that this mutant form of LFA-1 does not move freely within the cell membrane, suggesting that LFA-1 is still connected to the cytoskeleton network. This finding is strongly supported by the observation that LFA-1 exhibits a more diffusive motion when the cytoskeleton network is disrupted and confirmed by the optical trap measurements used to force the proteins to move through the membrane. Collectively, our findings suggest that the interaction of LFA-1 with the cytoskeleton cannot solely be attributed to the cytoplasmic part of the beta chain.     

Extracting Transition Rates in Particle Tracking Using Analytical Diffusion Distribution Analysis

Single-particle tracking is an important technique in the life sciences to understand the kinetics of biomolecules. The analysis of apparent diffusion coefficients in vivo, for example, enables researchers to determine whether biomolecules are moving alone, as part of a larger complex, or are bound to large cellular components such as the membrane or chromosomal DNA. A remaining challenge has been to retrieve quantitative kinetic models, especially for molecules that rapidly switch between different diffusional states. Here, we present analytical diffusion distribution analysis (anaDDA), a framework that allows for extracting transition rates from distributions of apparent diffusion coefficients calculated from short trajectories that feature less than 10 localizations per track. Under the assumption that the system is Markovian and diffusion is purely Brownian, we show that theoretically predicted distributions accurately match simulated distributions and that anaDDA outperforms existing methods to retrieve kinetics, especially in the fast regime of 0.1–10 transitions per imaging frame. AnaDDA does account for the effects of confinement and tracking window boundaries. Furthermore, we added the option to perform global fitting of data acquired at different frame times to allow complex models with multiple states to be fitted confidently. Previously, we have started to develop anaDDA to investigate the target search of CRISPR-Cas complexes. In this work, we have optimized the algorithms and reanalyzed experimental data of DNA polymerase I diffusing in live Escherichia coli. We found that long-lived DNA interaction by DNA polymerase are more abundant upon DNA damage, suggesting roles in DNA repair. We further revealed and quantified fast DNA probing interactions that last shorter than 10 ms. AnaDDA pushes the boundaries of the timescale of interactions that can be probed with single-particle tracking and is a mathematically rigorous framework that can be further expanded to extract detailed information about the behavior of biomolecules in living cells.     

Using single-particle tracking to study nuclear trafficking of viral genes

The question of how genetic materials are trafficked in and out of the cell nucleus is a problem of great importance not only for understanding viral infections but also for advancing gene-delivery technology. Here we demonstrate a physical technique that allows gene trafficking to be studied at the single-gene level by combining sensitive fluorescence microscopy with microinjection. As a model system, we investigate the nuclear import of influenza genes, in the form of ribonucleoproteins (vRNPs), by imaging single vRNPs in living cells in real time. Our single-particle trajectories show that vRNPs are transported to the nuclear envelope by diffusion. We have observed heterogeneous interactions between the vRNPs and nuclear pore complexes with dissociation rate constants spanning two orders of magnitude. Our single-particle tracking experiments also provided new insights into the regulation mechanisms for the nuclear import of vRNPs: the influenza M1 protein, a regulatory protein for the import process, downregulates the nuclear import of vRNPs by inhibiting the interactions between vRNPs and nuclear pore complexes but has no significant effect on the transport properties of vRNPs. We expect this single-particle tracking approach to find broad application in investigations of genetic trafficking.     

Bayesian Approach to MSD-Based Analysis of Particle Motion in Live Cells

Quantitative tracking of particle motion using live-cell imaging is a powerful approach to understanding the mechanism of transport of biological molecules, organelles, and cells. However, inferring complex stochastic motion models from single-particle trajectories in an objective manner is nontrivial due to noise from sampling limitations and biological heterogeneity. Here, we present a systematic Bayesian approach to multiple-hypothesis testing of a general set of competing motion models based on particle mean-square displacements that automatically classifies particle motion, properly accounting for sampling limitations and correlated noise while appropriately penalizing model complexity according to Occam's Razor to avoid over-fitting. We test the procedure rigorously using simulated trajectories for which the underlying physical process is known, demonstrating that it chooses the simplest physical model that explains the observed data. Further, we show that computed model probabilities provide a reliability test for the downstream biological interpretation of associated parameter values. We subsequently illustrate the broad utility of the approach by applying it to disparate biological systems including experimental particle trajectories from chromosomes, kinetochores, and membrane receptors undergoing a variety of complex motions. This automated and objective Bayesian framework easily scales to large numbers of particle trajectories, making it ideal for classifying the complex motion of large numbers of single molecules and cells from high-throughput screens, as well as single-cell-, tissue-, and organism-level studies.     

Measuring and Overcoming Limits of the Saffman-Delbrück Model for Soap Film Viscosities

We observe tracer particles diffusing in soap films to measure the two-dimensional (2D) viscous properties of the films. Saffman-Delbrück type models relate the single-particle diffusivity to parameters of the film (such as thickness h) for thin films, but the relation breaks down for thicker films. Notably, the diffusivity is faster than expected for thicker films, with the crossover at h/d = 5.2 ± 0.9 using the tracer particle diameter d. This indicates a crossover from purely 2D diffusion to diffusion that is more three-dimensional. We demonstrate that measuring the correlations of particle pairs as a function of their separation overcomes the limitations of the Saffman-Delbrück model and allows one to measure the viscosity of a soap film for any thickness.     

Folding rates and low-entropy-loss routes of two-state proteins

We develop a simple model for computing the rates and routes of folding of two-state proteins from the contact maps of their native structures. The model is based on the graph-theoretical concept of effective contact order (ECO). The model predicts that proteins fold by "zipping up" in a sequence of small-loop-closure events, depending on the native chain fold. Using a simple equation, with a few physical rate parameters, we obtain a good correlation with the folding rates of 24 two-state folding proteins. The model rationalizes data from Phi-value analysis that have been interpreted in terms of delocalized or polarized transition states. This model indicates how much of protein folding may take place in parallel, not along a single reaction coordinate or with a single transition state.     

Structural basis of folding cooperativity in model proteins: insights from a microcanonical perspective

Two-state cooperativity is an important characteristic in protein folding. It is defined by a depletion of states that lie energetically between folded and unfolded conformations. There are different ways to test for two-state cooperativity; however, most of these approaches probe indirect proxies of this depletion. Generalized-ensemble computer simulations allow us to unambiguously identify this transition by a microcanonical analysis on the basis of the density of states. Here, we present a detailed characterization of several helical peptides obtained by coarse-grained simulations. The level of resolution of the coarse-grained model allowed to study realistic structures ranging from small α-helices to a de novo three-helix bundle without biasing the force field toward the native state of the protein. By linking thermodynamic and structural features, we are able to show that whereas short α-helices exhibit two-state cooperativity, the type of transition changes for longer chain lengths because the chain forms multiple helix nucleation sites, stabilizing a significant population of intermediate states. The helix bundle exhibits signs of two-state cooperativity owing to favorable helix-helix interactions, as predicted from theoretical models. A detailed analysis of secondary and tertiary structure formation fits well into the framework of several folding mechanisms and confirms features that up to now have been observed only in lattice models.     

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.     

A new method for inferring hidden markov models from noisy time sequences

We present a new method for inferring hidden Markov models from noisy time sequences without the necessity of assuming a model architecture, thus allowing for the detection of degenerate states. This is based on the statistical prediction techniques developed by Crutchfield et al. and generates so called causal state models, equivalent in structure to hidden Markov models. The new method is applicable to any continuous data which clusters around discrete values and exhibits multiple transitions between these values such as tethered particle motion data or Fluorescence Resonance Energy Transfer (FRET) spectra. The algorithms developed have been shown to perform well on simulated data, demonstrating the ability to recover the model used to generate the data under high noise, sparse data conditions and the ability to infer the existence of degenerate states. They have also been applied to new experimental FRET data of Holliday Junction dynamics, extracting the expected two state model and providing values for the transition rates in good agreement with previous results and with results obtained using existing maximum likelihood based methods. The method differs markedly from previous Markov-model reconstructions in being able to uncover truly hidden states.     

Agonist induction, conformational selection, and mutant receptors

Current models of receptor activation are based on either of two basic mechanisms: agonist induction or conformational selection. The importance of one pathway relative to the other is controversial. In this article, the impossibility of distinguishing between the two mechanisms under a thermodynamic approach is shown. The effect of receptor mutation on the constants governing ligand-receptor equilibria is discussed. The two-state model of agonism both in its original formulation (one cycle) and including multiple active states (multiple cycles) is used. Pharmacological equations for the double (two cycles) two-state model are derived. The simulations performed suggest that the double two-state model of agonism can be a useful model for assessing quantitatively the changes in pharmacological activity following receptor mutation.