Cholesterol inhibits assembly and activation of the EphA2 receptor.

The receptor tyrosine kinase EphA2 drives cancer malignancy by facilitating metastasis. EphA2 can be found in different self-assembly states: as a monomer, dimer, and oligomer. However, our understanding remains limited regarding which EphA2 state is responsible for driving pro-metastatic signaling. To address this limitation, we have developed SiMPull-POP, a single-molecule method for accurate quantification of membrane protein self-assembly. Our experiments revealed that a reduction of plasma membrane cholesterol strongly promoted EphA2 self-assembly. Indeed, low cholesterol caused a similar effect to the EphA2 ligand ephrinA1-Fc. These results indicate that cholesterol inhibits EphA2 assembly. Phosphorylation studies in different cell lines revealed that low cholesterol increased phospho-serine levels, the signature of oncogenic signaling. Investigation of the mechanism that cholesterol uses to inhibit the assembly and activity of EphA2 indicate an in-trans effect, where EphA2 is phosphorylated by protein kinase A downstream of beta-adrenergic receptor activity, which cholesterol also inhibits. Our study not only provides new mechanistic insights on EphA2 oncogenic function, but also suggests that cholesterol acts as a molecular safeguard mechanism that prevents uncontrolled self-assembly and activation of EphA2.

An accurate probabilistic step finder for time-series analysis.

Noisy time-series data-from various experiments, including Förster resonance energy transfer, patch clamp, and force spectroscopy, among others-are commonly analyzed with either hidden Markov models or step-finding algorithms, both of which detect discrete transitions. Hidden Markov models, including their extensions to infinite state spaces, inherently assume exponential-or technically geometric-holding time distributions, biasing step locations toward steps with geometric holding times, especially in sparse and/or noisy data. In contrast, existing step-finding algorithms, while free of this restraint, often rely on ad hoc metrics to penalize steps recovered in time traces (by using various information criteria) and otherwise rely on approximate greedy algorithms to identify putative global optima. Here, instead, we devise a robust and general probabilistic (Bayesian) step-finding tool that neither relies on ad hoc metrics to penalize step numbers nor assumes geometric holding times in each state. As the number of steps themselves in a time-series are a priori unknown, we treat these within a Bayesian nonparametric (BNP) paradigm. We find that the method developed, BNP Step (BNP-Step), accurately determines the number and location of transitions between discrete states without any assumed kinetic model and learns the emission distribution characteristic of each state. In doing so, we verify that BNP-Step can analyze sparser data sets containing higher noise and more closely spaced states than otherwise resolved by current state-of-the-art methods. What is more, BNP-Step rigorously propagates measurement uncertainty into uncertainty over state transition locations, numbers, and emission levels as characterized by the posterior. We demonstrate the performance of BNP-Step on both synthetic data as well as data drawn from force spectroscopy experiments.

An accurate probabilistic step finder for time-series analysis.

Noisy time-series data is commonly collected from sources including Förster Resonance Energy Transfer experiments, patch clamp and force spectroscopy setups, among many others. Two of the most common paradigms for the detection of discrete transitions in such time-series data include: hidden Markov models (HMMs) and step-finding algorithms. HMMs, including their extensions to infinite state-spaces, inherently assume in analysis that holding times in discrete states visited are geometrically-or, loosely speaking in common language, exponentially-distributed. Thus the determination of step locations, especially in sparse and noisy data, is biased by HMMs toward identifying steps resulting in geometric holding times. In contrast, existing step-finding algorithms, while free of this restraint, often rely on ad hoc metrics to penalize steps recovered in time traces (by using various information criteria) and otherwise rely on approximate greedy algorithms to identify putative global optima. Here, instead, we devise a robust and general probabilistic (Bayesian) step-finding tool that neither relies on ad hoc metrics to penalize step numbers nor assumes geometric holding times in each state. As the number of steps themselves in a time-series are, a priori unknown, we treat these within a Bayesian nonparametric (BNP) paradigm. We find that the method developed, Bayesian Nonparametric Step (BNP-Step), accurately determines the number and location of transitions between discrete states without any assumed kinetic model and learns the emission distribution characteristic of each state. In doing so, we verify that BNP-Step can analyze sparser data sets containing higher noise and more closely-spaced states than otherwise resolved by current state-of-the-art methods. What is more, BNP-Step rigorously propagates measurement uncertainty into uncertainty over state transition locations, numbers, and emission levels as characterized by the posterior. We demonstrate the performance of BNP-Step on both synthetic data as well as data drawn from force spectroscopy experiments.

BNP-Track: A framework for multi-particle superresolved tracking.

When tracking fluorescently labeled molecules (termed "emitters") under widefield microscopes, point spread function overlap of neighboring molecules is inevitable in both dilute and especially crowded environments. In such cases, superresolution methods leveraging rare photophysical events to distinguish static targets nearby in space introduce temporal delays that compromise tracking. As we have shown in a companion manuscript, for dynamic targets, information on neighboring fluorescent molecules is encoded as spatial intensity correlations across pixels and temporal correlations in intensity patterns across time frames. We then demonstrated how we used all spatiotemporal correlations encoded in the data to achieve superresolved tracking. That is, we showed the results of full posterior inference over both the number of emitters and their associated tracks simultaneously and self-consistently through Bayesian nonparametrics. In this companion manuscript we focus on testing the robustness of our tracking tool, BNP-Track, across sets of parameter regimes and compare BNP-Track to competing tracking methods in the spirit of a prior Nature Methods tracking competition. We explore additional features of BNP-Track including how a stochastic treatment of background yields greater accuracy in emitter number determination and how BNP-Track corrects for point spread function blur (or "aliasing") introduced by intraframe motion in addition to propagating error originating from myriad sources (such as criss-crossing tracks, out-of-focus particles, pixelation, shot and camera artefact, stochastic background) in posterior inference over emitter numbers and their associated tracks. While head-to-head comparison with other tracking methods is not possible (as competitors cannot simultaneously learn molecule numbers and associated tracks), we can give competing methods some advantages in order to perform approximate head-to-head comparison. We show that even under such optimistic scenarios, BNP-Track is capable of tracking multiple diffraction-limited point emitters conventional tracking methods cannot resolve thereby extending the superresolution paradigm to dynamical targets.

BNP-Track: A framework for superresolved tracking.

Assessing dynamic processes at single molecule scales is key toward capturing life at the level of its molecular actors. Widefield superresolution methods, such as STORM, PALM, and PAINT, provide nanoscale localization accuracy, even when distances between fluorescently labeled single molecules ("emitters") fall below light's diffraction limit. However, as these superresolution methods rely on rare photophysical events to distinguish emitters from both each other and background, they are largely limited to static samples. In contrast, here we leverage spatiotemporal correlations of dynamic widefield imaging data to extend superresolution to simultaneous multiple emitter tracking without relying on photodynamics even as emitter distances from one another fall below the diffraction limit. We simultaneously determine emitter numbers and their tracks (localization and linking) with the same localization accuracy per frame as widefield superresolution does for immobilized emitters under similar imaging conditions (≈50nm). We demonstrate our results for both in cellulo data and, for benchmarking purposes, on synthetic data. To this end, we avoid the existing tracking paradigm relying on completely or partially separating the tasks of emitter number determination, localization of each emitter, and linking emitter positions across frames. Instead, we develop a fully joint posterior distribution over the quantities of interest, including emitter tracks and their total, otherwise unknown, number within the Bayesian nonparametric paradigm. Our posterior quantifies the full uncertainty over emitter numbers and their associated tracks propagated from origins including shot noise and camera artefacts, pixelation, stochastic background, and out-of-focus motion. Finally, it remains accurate in more crowded regimes where alternative tracking tools cannot be applied.

Single-photon smFRET: II. Application to continuous illumination.

Here we adapt the Bayesian nonparametrics (BNP) framework presented in the first companion article to analyze kinetics from single-photon, single-molecule Förster resonance energy transfer (smFRET) traces generated under continuous illumination. Using our sampler, BNP-FRET, we learn the escape rates and the number of system states given a photon trace. We benchmark our method by analyzing a range of synthetic and experimental data. Particularly, we apply our method to simultaneously learn the number of system states and the corresponding kinetics for intrinsically disordered proteins using two-color FRET under varying chemical conditions. Moreover, using synthetic data, we show that our method can deduce the number of system states even when kinetics occur at timescales of interphoton intervals.

Single-photon smFRET. I: Theory and conceptual basis.

We present a unified conceptual framework and the associated software package for single-molecule Förster resonance energy transfer (smFRET) analysis from single-photon arrivals leveraging Bayesian nonparametrics, BNP-FRET. This unified framework addresses the following key physical complexities of a single-photon smFRET experiment, including: 1) fluorophore photophysics; 2) continuous time kinetics of the labeled system with large timescale separations between photophysical phenomena such as excited photophysical state lifetimes and events such as transition between system states; 3) unavoidable detector artefacts; 4) background emissions; 5) unknown number of system states; and 6) both continuous and pulsed illumination. These physical features necessarily demand a novel framework that extends beyond existing tools. In particular, the theory naturally brings us to a hidden Markov model with a second-order structure and Bayesian nonparametrics on account of items 1, 2, and 5 on the list. In the second and third companion articles, we discuss the direct effects of these key complexities on the inference of parameters for continuous and pulsed illumination, respectively.

Single-photon smFRET. III. Application to pulsed illumination.

Förster resonance energy transfer (FRET) using pulsed illumination has been pivotal in leveraging lifetime information in FRET analysis. However, there remain major challenges in quantitative single-photon, single-molecule FRET (smFRET) data analysis under pulsed illumination including 1) simultaneously deducing kinetics and number of system states; 2) providing uncertainties over estimates, particularly uncertainty over the number of system states; and 3) taking into account detector noise sources such as cross talk and the instrument response function contributing to uncertainty; in addition to 4) other experimental noise sources such as background. Here, we implement the Bayesian nonparametric framework described in the first companion article that addresses all aforementioned issues in smFRET data analysis specialized for the case of pulsed illumination. Furthermore, we apply our method to both synthetic as well as experimental data acquired using Holliday junctions.

Computational Proposal for Tracking Multiple Molecules in a Multifocus Confocal Setup.

Tracking single molecules continues to provide new insights into the fundamental rules governing biological function. Despite continued technical advances in fluorescent and non-fluorescent labeling as well as data analysis, direct observations of trajectories and interactions of multiple molecules in dense environments remain aspirational goals. While confocal methods provide a means to deduce dynamical parameters with high temporal resolution, such as diffusion coefficients, they do so at the expense of spatial resolution. Indeed, on account of a confocal volume's symmetry, typically only distances from the center of the confocal spot can be deduced. Motivated by the need for true three dimensional high speed tracking in densely labeled environments, we propose a computational tool for tracking many fluorescent molecules traversing multiple, closely spaced, confocal measurement volumes providing independent observations. Various realizations of this multiple confocal volumes strategy have previously been used for long term, large area, tracking of one fluorescent molecule in three dimensions. What is more, we achieve tracking by directly using single photon arrival times to inform our likelihood and exploit Hamiltonian Monte Carlo to efficiently sample trajectories from our posterior within a Bayesian nonparametric paradigm. A nonparametric paradigm here is warranted as the number of molecules present are, themselves, a priori unknown. Taken together, we provide a computational framework to infer trajectories of multiple molecules at once, below the diffraction limit (the width of a confocal spot), in three dimensions at sub-millisecond or faster time scales.

Diffraction-Limited Molecular Cluster Quantification with Bayesian Nonparametrics.

Life's fundamental processes involve multiple molecules operating in close proximity within cells. To probe the composition and kinetics of molecular clusters confined within small (diffraction-limited) regions, experiments often report on the total fluorescence intensity simultaneously emitted from labeled molecules confined to such regions. Methods exist to enumerate total fluorophore numbers (e.g., step counting by photobleaching). However, methods aimed at step counting by photobleaching cannot treat photophysical dynamics in counting nor learn their associated kinetic rates. Here we propose a method to simultaneously enumerate fluorophores and determine their individual photophysical state trajectories. As the number of active (fluorescent) molecules at any given time is unknown, we rely on Bayesian nonparametrics and use specialized Monte Carlo algorithms to derive our estimates. Our formulation is benchmarked on synthetic and real data sets. While our focus here is on photophysical dynamics (in which labels transition between active and inactive states), such dynamics can also serve as a proxy for other types of dynamics such as assembly and disassembly kinetics of clusters. Similarly, while we focus on the case where all labels are initially fluorescent, other regimes, more appropriate to photoactivated localization microscopy, where fluorophores are instantiated in a non-fluorescent state, fall within the scope of the framework. As such, we provide a complete and versatile framework for the interpretation of complex time traces arising from the simultaneous activity of up to 100 fluorophores.

Modeling Non-additive Effects in Neighboring Chemically Identical Fluorophores.

Quantitative fluorescence analysis is often used to derive chemical properties, including stoichiometries, of biomolecular complexes. One fundamental underlying assumption in the analysis of fluorescence data─whether it be the determination of protein complex stoichiometry by super-resolution, or step-counting by photobleaching, or the determination of RNA counts in diffraction-limited spots in RNA fluorescence in situ hybridization (RNA-FISH) experiments─is that fluorophores behave identically and do not interact. However, recent experiments on fluorophore-labeled DNA origami structures such as fluorocubes have shed light on the nature of the interactions between identical fluorophores as these are brought closer together, thereby raising questions on the validity of the modeling assumption that fluorophores do not interact. Here, we analyze photon arrival data under pulsed illumination from fluorocubes where distances between dyes range from 2 to 10 nm. We discuss the implications of non-additivity of brightness on quantitative fluorescence analysis.

Extraction of rapid kinetics from smFRET measurements using integrative detectors.

Hidden Markov models (HMMs) are used to learn single-molecule kinetics across a range of experimental techniques. By their construction, HMMs assume that single-molecule events occur on slower timescales than those of data acquisition. To move beyond that HMM limitation and allow for single-molecule events to occur on any timescale, we must treat single-molecule events in continuous time as they occur in nature. We propose a method to learn kinetic rates from single-molecule Förster resonance energy transfer (smFRET) data collected by integrative detectors, even if those rates exceed data acquisition rates. To achieve that, we exploit our recently proposed "hidden Markov jump process" (HMJP), with which we learn transition kinetics from parallel measurements in donor and acceptor channels. HMJPs generalize the HMM paradigm in two critical ways: (1) they deal with physical smFRET systems as they switch between conformational states in continuous time, and (2) they estimate transition rates between conformational states directly without having recourse to transition probabilities or assuming slow dynamics. Our continuous-time treatment learns the transition kinetics and photon emission rates for dynamic regimes that are inaccessible to HMMs, which treat system kinetics in discrete time. We validate our framework's robustness on simulated data and demonstrate its performance on experimental data from FRET-labeled Holliday junctions.

Residence time analysis of RNA polymerase transcription dynamics: A Bayesian sticky HMM approach.

The time spent by a single RNA polymerase (RNAP) at specific locations along the DNA, termed "residence time," reports on the initiation, elongation, and termination stages of transcription. At the single-molecule level, this information can be obtained from dual ultrastable optical trapping experiments, revealing a transcriptional elongation of RNAP interspersed with residence times of variable duration. Successfully discriminating between long and short residence times was used by previous approaches to learn about RNAP's transcription elongation dynamics. Here, we propose an approach based on the Bayesian sticky hidden Markov model that treats all residence times for an Escherichia coli RNAP on an equal footing without a priori discriminating between long and short residence times. Furthermore, our method has two additional advantages: we provide full distributions around key point statistics and directly treat the sequence dependence of RNAP's elongation rate. By applying our approach to experimental data, we find assigned relative probabilities on long versus short residence times, force-dependent average residence time transcription elongation dynamics, ∼10% drop in the average backtracking durations in the presence of GreB, and ∼20% drop in the average residence time as a function of applied force in the presence of RNaseA.

Generalizing HMMs to Continuous Time for Fast Kinetics: Hidden Markov Jump Processes.

The hidden Markov model (HMM) is a framework for time series analysis widely applied to single-molecule experiments. Although initially developed for applications outside the natural sciences, the HMM has traditionally been used to interpret signals generated by physical systems, such as single molecules, evolving in a discrete state space observed at discrete time levels dictated by the data acquisition rate. Within the HMM framework, transitions between states are modeled as occurring at the end of each data acquisition period and are described using transition probabilities. Yet, whereas measurements are often performed at discrete time levels in the natural sciences, physical systems evolve in continuous time according to transition rates. It then follows that the modeling assumptions underlying the HMM are justified if the transition rates of a physical process from state to state are small as compared to the data acquisition rate. In other words, HMMs apply to slow kinetics. The problem is, because the transition rates are unknown in principle, it is unclear, a priori, whether the HMM applies to a particular system. For this reason, we must generalize HMMs for physical systems, such as single molecules, because these switch between discrete states in "continuous time". We do so by exploiting recent mathematical tools developed in the context of inferring Markov jump processes and propose the hidden Markov jump process. We explicitly show in what limit the hidden Markov jump process reduces to the HMM. Resolving the discrete time discrepancy of the HMM has clear implications: we no longer need to assume that processes, such as molecular events, must occur on timescales slower than data acquisition and can learn transition rates even if these are on the same timescale or otherwise exceed data acquisition rates.

Inferring effective forces for Langevin dynamics using Gaussian processes.

Effective forces derived from experimental or in silico molecular dynamics time traces are critical in developing reduced and computationally efficient descriptions of otherwise complex dynamical problems. This helps motivate why it is important to develop methods to efficiently learn effective forces from time series data. A number of methods already exist to do this when data are plentiful but otherwise fail for sparse datasets or datasets where some regions of phase space are undersampled. In addition, any method developed to learn effective forces from time series data should be minimally a priori committal as to the shape of the effective force profile, exploit every data point without reducing data quality through any form of binning or pre-processing, and provide full credible intervals (error bars) about the prediction for the entirety of the effective force curve. Here, we propose a generalization of the Gaussian process, a key tool in Bayesian nonparametric inference and machine learning, which meets all of the above criteria in learning effective forces for the first time.

Direct Photon-by-Photon Analysis of Time-Resolved Pulsed Excitation Data using Bayesian Nonparametrics.

Lifetimes of chemical species are typically estimated by either fitting time-correlated single-photon counting (TCSPC) histograms or phasor analysis from time-resolved photon arrivals. While both methods yield lifetimes in a computationally efficient manner, their performance is limited by choices made on the number of distinct chemical species contributing photons. However, the number of species is encoded in the photon arrival times collected for each illuminated spot and need not be set by hand a priori. Here, we propose a direct photon-by-photon analysis of data drawn from pulsed excitation experiments to infer, simultaneously and self-consistently, the number of species and their associated lifetimes from a few thousand photons. We do so by leveraging new mathematical tools within the Bayesian nonparametric. We benchmark our method for both simulated and experimental data for 1-4 species.

Pitching single-focus confocal data analysis one photon at a time with Bayesian nonparametrics.

Fluorescence time traces are used to report on dynamical properties of molecules. The basic unit of information in these traces is the arrival time of individual photons, which carry instantaneous information from the molecule, from which they are emitted, to the detector on timescales as fast as microseconds. Thus, it is theoretically possible to monitor molecular dynamics at such timescales from traces containing only a sufficient number of photon arrivals. In practice, however, traces are stochastic and in order to deduce dynamical information through traditional means-such as fluorescence correlation spectroscopy (FCS) and related techniques-they are collected and temporally autocorrelated over several minutes. So far, it has been impossible to analyze dynamical properties of molecules on timescales approaching data acquisition without collecting long traces under the strong assumption of stationarity of the process under observation or assumptions required for the analytic derivation of a correlation function. To avoid these assumptions, we would otherwise need to estimate the instantaneous number of molecules emitting photons and their positions within the confocal volume. As the number of molecules in a typical experiment is unknown, this problem demands that we abandon the conventional analysis paradigm. Here, we exploit Bayesian nonparametrics that allow us to obtain, in a principled fashion, estimates of the same quantities as FCS but from the direct analysis of traces of photon arrivals that are significantly smaller in size, or total duration, than those required by FCS.

An alternative framework for fluorescence correlation spectroscopy.

Fluorescence correlation spectroscopy (FCS), is a widely used tool routinely exploited for in vivo and in vitro applications. While FCS provides estimates of dynamical quantities, such as diffusion coefficients, it demands high signal to noise ratios and long time traces, typically in the minute range. In principle, the same information can be extracted from microseconds to seconds long time traces; however, an appropriate analysis method is missing. To overcome these limitations, we adapt novel tools inspired by Bayesian non-parametrics, which starts from the direct analysis of the observed photon counts. With this approach, we are able to analyze time traces, which are too short to be analyzed by existing methods, including FCS. Our new analysis extends the capability of single molecule fluorescence confocal microscopy approaches to probe processes several orders of magnitude faster and permits a reduction of photo-toxic effects on living samples induced by long periods of light exposure.

A method for single molecule tracking using a conventional single-focus confocal setup.

One way to achieve spatial resolution using fluorescence imaging-and track single molecules-is to use wide-field illumination and collect measurements over multiple sensors (camera pixels). Here we propose another way that uses confocal measurements and a single sensor. Traditionally, confocal microscopy has been used to achieve high temporal resolution at the expense of spatial resolution. This is because it utilizes very few, and commonly just one, sensors to collect data. Yet confocal data encode spatial information. Here we show that non-uniformities in the shape of the confocal excitation volume can be exploited to achieve spatial resolution. To achieve this, we formulate a specialized hidden Markov model and adapt a forward filtering-backward sampling Markov chain Monte Carlo scheme to efficiently handle molecular motion within a symmetric confocal volume characteristically used in fluorescence correlation spectroscopy. Our method can be used for single confocal volume applications or incorporated into larger computational schemes for specialized, multi-confocal volume, optical setups.

A Bayesian Nonparametric Approach to Single Molecule Förster Resonance Energy Transfer.

We develop a Bayesian nonparametric framework to analyze single molecule FRET (smFRET) data. This framework, a variation on infinite hidden Markov models, goes beyond traditional hidden Markov analysis, which already treats photon shot noise, in three critical ways: (1) it learns the number of molecular states present in a smFRET time trace (a hallmark of nonparametric approaches), (2) it accounts, simultaneously and self-consistently, for photophysical features of donor and acceptor fluorophores (blinking kinetics, spectral cross-talk, detector quantum efficiency), and (3) it treats background photons. Point 2 is essential in reducing the tendency of nonparametric approaches to overinterpret noisy single molecule time traces and so to estimate states and transition kinetics robust to photophysical artifacts. As a result, with the proposed framework, we obtain accurate estimates of single molecule properties even when the supplied traces are excessively noisy, subject to photoartifacts, and of short duration. We validate our method using synthetic data sets and demonstrate its applicability to real data sets from single molecule experiments on Holliday junctions labeled with conventional fluorescent dyes.