Analyzing single-molecule time series via nonparametric Bayesian inference

Bayesian analysis and efficient algorithms for single-molecule fluorescence data and step counting

With the growing adoption of single-molecule fluorescence experiments, there is an increasing demand for efficient statistical methodologies and accurate analysis of the acquired measurements. Existing analysis frameworks, such as those that use kinetic models, often rely on strong assumptions on the dynamics of the molecules and fluorophores under study that render them inappropriate for general purpose step-counting applications, especially when the systems of study exhibit uncharac-terized dynamics. Here, we propose a novel Bayesian nonparametric framework to analyze single-molecule fluorescence data that is kinetic model independent. For the evaluation of our methods, we develop four MCMC samplers, ranging from elemental to highly sophisticated, and demonstrate that the added complexity is essential for accurate data analysis. We apply our methods to experimental data obtained from TIRF photobleaching assays of the EphA2 receptor tagged with GFP. In addition, we validate our approach with synthetic data mimicking realistic conditions and demonstrate its ability to recover ground truth under high- and low-signal-to-noise data, establishing it as a versatile tool for fluorescence data analysis.

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.

Efficient, nonparametric removal of noise and recovery of probability distributions from time series using nonlinear-correlation functions: Photon and photon-counting noise

A preceding paper [M. Dhar, J. A. Dickinson, and M. A. Berg, J. Chem. Phys. 159, 054110 (2023)] shows how to remove additive noise from an experimental time series, allowing both the equilibrium distribution of the system and its Green's function to be recovered. The approach is based on nonlinear-correlation functions and is fully nonparametric: no initial model of the system or of the noise is needed. However, single-molecule spectroscopy often produces time series with either photon or photon-counting noise. Unlike additive noise, photon noise is signal-size correlated and quantized. Photon counting adds the potential for bias. This paper extends noise-corrected-correlation methods to these cases and tests them on synthetic datasets. Neither signal-size correlation nor quantization is a significant complication. Analysis of the sampling error yields guidelines for the data quality needed to recover the properties of a system with a given complexity. We show that bias in photon-counting data can be corrected, even at the high count rates needed to optimize the time resolution. Using all these results, we discuss the factors that limit the time resolution of single-molecule spectroscopy and the conditions that would be needed to push measurements into the submicrosecond region.

Atomic force microscope kymograph analysis: A case study of two membrane proteins

Kymograph analysis is employed across the biological atomic force microscopy (AFM) community to boost temporal resolution. The method is well suited for revealing protein dynamics at the single molecule level in near-native conditions. Yet, kymograph analysis comes with limitations that depend on several factors including protein geometry and instrumental drift. This work focuses on conformational dynamics of difficult-to-study sparse distributions of membrane proteins. We compare and contrast AFM kymograph analysis for two proteins, one of which (SecDF) exhibits conformational dynamics primarily in the vertical direction (normal to the membrane surface) and the other (Pgp) exhibits a combination of lateral dynamics and vertical motion. Common experimental issues are analyzed including translational and rotational drift. Conformational transition detection is evaluated via kymograph simulations followed by state detection algorithms. We find that kymograph analysis is largely robust to lateral drift. Displacement of the AFM line scan trajectory away from the protein center of mass by a few nanometers, roughly half of the molecule diameter, does not significantly affect transition detection nor generate undue dwell time errors. On the other hand, for proteins like Pgp that exhibit significant azimuthal maximum height dependence, rotational drift can potentially produce artifactual transitions. Measuring the height of a membrane protein protrusion is generally superior to measurement of width, confirming intuition based on vertical resolution superiority. In low signal-to-noise scenarios, common state detection algorithms struggle with transition detection as opposed to infinite hidden Markov models. AFM kymography represents a valuable addition to the membrane biophysics toolkit; continued hardware and software improvements are poised to expand the method's impact in the field.

Trajectory Statistical Learning of the Potential Mean of Force and Diffusion Coefficient from Molecular Dynamics Simulations

Central to studying the conformational changes of a complex protein is understanding the dynamics and energetics involved. Phenomenologically, structural dynamics can be formulated using an overdamped Langevin model along an observable, e.g., the distance between two residues in the protein. The Langevin model is specified by the deterministic force (the potential of mean force, PMF) and stochastic force (characterized by the diffusion coefficient, D). It is therefore of great interest to be able to extract both PMF and D from an observable time series but under the same computational framework. Here, we approach this challenge in molecular dynamics (MD) simulations by treating it as a missing-data Bayesian estimation problem. An important distinction in our methodology is that the entire MD trajectory, as opposed to the individual data elements, is used as the statistical variable in Bayesian imputation. This idea is implemented through an eigen-decomposition procedure for a time-symmetrized Fokker-Planck equation, followed by maximizing the likelihood for parameter estimation. The mathematical expressions for the functional derivatives used in learning PMF and D also provide new physical insights for the manner by which the information on both the deterministic and stochastic forces is encoded in the dynamics data. An all-atom MD simulation of a nontrivial biomolecule case is used to illustrate the application of this approach. We show that, interestingly, the results of trajectory statistical learning can motivate new order parameters for an improved description of the kinetic bottlenecks in conformational changes. Complementing purely data-driven or black-box methods, this work underscores the advantages of physics-based machine learning in gaining chemical insights from quantitative parameter estimation.

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.

Fluorescence Microscopy: a statistics-optics perspective

Fundamental properties of light unavoidably impose features on images collected using fluorescence microscopes. Modeling these features is ever more important in quantitatively interpreting microscopy images collected at scales on par or smaller than light's wavelength. Here we review the optics responsible for generating fluorescent images, fluorophore properties, microscopy modalities leveraging properties of both light and fluorophores, in addition to the necessarily probabilistic modeling tools imposed by the stochastic nature of light and measurement.

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.

Resolving Molecular Heterogeneity with Single-Molecule Centrifugation

For many classes of biomolecules, population-level heterogeneity is an essential aspect of biological function─from antibodies produced by the immune system to post-translationally modified proteins that regulate cellular processes. However, heterogeneity is difficult to fully characterize for multiple reasons: (i) single-molecule approaches are needed to avoid information lost by ensemble-level averaging, (ii) sufficient statistics must be gathered on both a per-molecule and per-population level, and (iii) a suitable analysis framework is required to make sense of a potentially limited number of intrinsically noisy measurements. Here, we introduce an approach that overcomes these difficulties by combining three techniques: a DNA nanoswitch construct to repeatedly interrogate the same molecule, a benchtop centrifuge force microscope (CFM) to obtain thousands of statistics in a highly parallel manner, and a Bayesian nonparametric (BNP) inference method to resolve separate subpopulations with distinct kinetics. We apply this approach to characterize commercially available antibodies and find that polyclonal antibody from rabbit serum is well-modeled by a mixture of three subpopulations. Our results show how combining a spatially and temporally multiplexed nanoswitch-CFM assay with BNP analysis can help resolve complex biomolecular interactions in heterogeneous samples.

Permissive and nonpermissive channel closings in CFTR revealed by a factor graph inference algorithm

The closing of the gated ion channel in the cystic fibrosis transmembrane conductance regulator can be categorized as nonpermissive to reopening, which involves the unbinding of ADP or ATP, or permissive, which does not. Identifying the type of closing is of interest as interactions with nucleotides can be affected in mutants or by introducing agonists. However, all closings are electrically silent and difficult to differentiate. For single-channel patch-clamp traces, we show that the type of the closing can be accurately determined by an inference algorithm implemented on a factor graph, which we demonstrate using both simulated and lab-obtained patch-clamp traces.

Nonparametric Bayesian inference for meta-stable conformational dynamics

Analyses of structural dynamics of biomolecules hold great promise to deepen the understanding of and ability to construct complex molecular systems. To this end, both experimental and computational means are available, such as fluorescence quenching experiments or molecular dynamics simulations, respectively. We argue that while seemingly disparate, both fields of study have to deal with the same type of data about the same underlying phenomenon of conformational switching. Two central challenges typically arise in both contexts: (i) the amount of obtained data is large, and (ii) it is often unknown how many distinct molecular states underlie these data. In this study, we build on the established idea of Markov state modeling and propose a generative, Bayesian nonparametric hidden Markov state model that addresses these challenges. Utilizing hierarchical Dirichlet processes, we treat different meta-stable molecule conformations as distinct Markov states, the number of which we then do not have to set a priori. In contrast to existing approaches to both experimental as well as simulation data that are based on the same idea, we leverage a mean-field variational inference approach, enabling scalable inference on large amounts of data. Furthermore, we specify the model also for the important case of angular data, which however proves to be computationally intractable. Addressing this issue, we propose a computationally tractable approximation to the angular model. We demonstrate the method on synthetic ground truth data and apply it to known benchmark problems as well as electrophysiological experimental data from a conformation-switching ion channel to highlight its practical utility.

Bayesian inference of kinetic schemes for ion channels by Kalman filtering

Inferring adequate kinetic schemes for ion channel gating from ensemble currents is a daunting task due to limited information in the data. We address this problem by using a parallelized Bayesian filter to specify hidden Markov models for current and fluorescence data. We demonstrate the flexibility of this algorithm by including different noise distributions. Our generalized Kalman filter outperforms both a classical Kalman filter and a rate equation approach when applied to patch-clamp data exhibiting realistic open-channel noise. The derived generalization also enables inclusion of orthogonal fluorescence data, making unidentifiable parameters identifiable and increasing the accuracy of the parameter estimates by an order of magnitude. By using Bayesian highest credibility volumes, we found that our approach, in contrast to the rate equation approach, yields a realistic uncertainty quantification. Furthermore, the Bayesian filter delivers negligibly biased estimates for a wider range of data quality. For some data sets, it identifies more parameters than the rate equation approach. These results also demonstrate the power of assessing the validity of algorithms by Bayesian credibility volumes in general. Finally, we show that our Bayesian filter is more robust against errors induced by either analog filtering before analog-to-digital conversion or by limited time resolution of fluorescence data than a rate equation approach.

Molecular electronics sensors on a scalable semiconductor chip: A platform for single-molecule measurement of binding kinetics and enzyme activity

Detection of molecular interactions is the foundation for many important biotechnology applications in society and industry, such as drug discovery, diagnostics, and DNA sequencing. This report describes a broadly applicable platform for detecting molecular interactions at the single-molecule scale, in real-time, label-free, and potentially highly multiplexable fashion, using single-molecule sensors on a highly scalable semiconductor sensor array chip. Such chips are both practically manufacturable in the near term, and have a durable long-term scaling roadmap, thus providing an ideal way to bring the power of modern chip technology to the broad area of biosensing. This work also realizes a 50-year-old scientific vision of integrating single molecules into electronic chips to achieve the ultimate miniaturization of electronics.

For nearly 50 years, the vision of using single molecules in circuits has been seen as providing the ultimate miniaturization of electronic chips. An advanced example of such a molecular electronics chip is presented here, with the important distinction that the molecular circuit elements play the role of general-purpose single-molecule sensors. The device consists of a semiconductor chip with a scalable array architecture. Each array element contains a synthetic molecular wire assembled to span nanoelectrodes in a current monitoring circuit. A central conjugation site is used to attach a single probe molecule that defines the target of the sensor. The chip digitizes the resulting picoamp-scale current-versus-time readout from each sensor element of the array at a rate of 1,000 frames per second. This provides detailed electrical signatures of the single-molecule interactions between the probe and targets present in a solution-phase test sample. This platform is used to measure the interaction kinetics of single molecules, without the use of labels, in a massively parallel fashion. To demonstrate broad applicability, examples are shown for probe molecule binding, including DNA oligos, aptamers, antibodies, and antigens, and the activity of enzymes relevant to diagnostics and sequencing, including a CRISPR/Cas enzyme binding a target DNA, and a DNA polymerase enzyme incorporating nucleotides as it copies a DNA template. All of these applications are accomplished with high sensitivity and resolution, on a manufacturable, scalable, all-electronic semiconductor chip device, thereby bringing the power of modern chips to these diverse areas of biosensing.

Joint Detection of Change Points in Multichannel Single-Molecule Measurements

Recent developments in single-molecule measurement technology have expanded the capability to measure multiple parameters. These emergent modalities provide more holistic observations of complex biomolecular processes and call for new analysis methods to detect state changes in multichannel data. Here we develop an algorithm called MULLR (MUlti-channel Log-Likelihood Ratio test) to jointly identify change points in multichannel single-molecule measurements. MULLR is an extension of the popular single-channel implementation for change point detection based on a binary segmentation and log-likelihood ratio test framework. We validate the algorithm on simulated data and characterize the power of detection and false positive rate. We show that MULLR can identify change points in experimental multichannel data and naturally works with different noise statistics and time resolutions across channels. Further, we quantify the benefit of MULLR compared to single-channel analysis. We envision that the MULLR algorithm will be useful to a range of multiparameter single-molecule measurements.

Unsupervised selection of optimal single-molecule time series idealization criterion

Single-molecule (SM) approaches have provided valuable mechanistic information on many biophysical systems. As technological advances lead to ever-larger data sets, tools for rapid analysis and identification of molecules exhibiting the behavior of interest are increasingly important. In many cases the underlying mechanism is unknown, making unsupervised techniques desirable. The divisive segmentation and clustering (DISC) algorithm is one such unsupervised method that idealizes noisy SM time series much faster than computationally intensive approaches without sacrificing accuracy. However, DISC relies on a user-selected objective criterion (OC) to guide its estimation of the ideal time series. Here, we explore how different OCs affect DISC’s performance for data typical of SM fluorescence imaging experiments. We find that OCs differing in their penalty for model complexity each optimize DISC’s performance for time series with different properties such as signal/noise and number of sample points. Using a machine learning approach, we generate a decision boundary that allows unsupervised selection of OCs based on the input time series to maximize performance for different types of data. This is particularly relevant for SM fluorescence data sets, which often have signal/noise near the derived decision boundary and include time series of nonuniform length because of stochastic bleaching. Our approach, AutoDISC, allows unsupervised per-molecule optimization of DISC, which will substantially assist in the rapid analysis of high-throughput SM data sets with noisy samples and nonuniform time windows.

AutoStepfinder: A fast and automated step detection method for single-molecule analysis

Single-molecule techniques allow the visualization of the molecular dynamics of nucleic acids and proteins with high spatiotemporal resolution. Valuable kinetic information of biomolecules can be obtained when the discrete states within single-molecule time trajectories are determined. Here, we present a fast, automated, and bias-free step detection method, AutoStepfinder, that determines steps in large datasets without requiring prior knowledge on the noise contributions and location of steps. The analysis is based on a series of partition events that minimize the difference between the data and the fit. A dual-pass strategy determines the optimal fit and allows AutoStepfinder to detect steps of a wide variety of sizes. We demonstrate step detection for a broad variety of experimental traces. The user-friendly interface and the automated detection of AutoStepfinder provides a robust analysis procedure that enables anyone without programming knowledge to generate step fits and informative plots in less than an hour.

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Fast, automated, and bias-free detection of steps within single-molecule trajectories

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Robust step detection without any prior knowledge on the data

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A dual-pass strategy for the detection of steps over a wide variety of scales

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A user-friendly interface for a simplified step fitting procedure

Single-molecule techniques have made it possible to track individual protein complexes in real time with a nanometer spatial resolution and a millisecond timescale. Accurate determination of the dynamic states within single-molecule time traces provides valuable kinetic information that underlie the function of biological macromolecules. Here, we present a new automated step detection method called AutoStepfinder, a versatile, robust, and easy-to-use algorithm that allows researchers to determine the kinetic states within single-molecule time trajectories without any bias.

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.

Statistical Inference of Transport Mechanisms and Long Time Scale Behavior from Time Series of Solute Trajectories in Nanostructured Membranes

Appropriate time series modeling of complex diffusion in soft matter systems on the microsecond time scale can provide a path toward inferring transport mechanisms and predicting bulk properties characteristic of much longer time scales. In this work we apply nonparametric Bayesian time series analysis, more specifically the sticky hierarchical Dirichlet process autoregressive hidden Markov model (HDP-AR-HMM) to solute center-of-mass trajectories generated from long molecular dynamics (MD) simulations in a cross-linked inverted hexagonal phase lyotropic liquid crystal (LLC) membrane in order to automatically detect a variety of solute dynamical modes. We can better understand the mechanisms controlling these dynamical modes by grouping the states identified by the HDP-AR-HMM into clusters based on multiple metrics aimed at distinguishing solute behavior based on their fluctuations, dwell times in each state, and positions within the inhomogeneous membrane structure. We analyze predominant clusters in order to relate their dynamical parameters to physical interactions between solutes and the membrane. Along with parameters of individual states, the HDP-AR-HMM simultaneously infers a transition matrix which allows us to stochastically propagate solute behavior from all of the independent trajectories onto arbitrary length time scales while still preserving the qualitative behavior characteristic of the MD trajectories. This affords a direct connection to important macroscopic observables used to characterize performance like solute flux and selectivity. This work provides a promising way to simultaneously identify transport mechanisms in nanoporous materials and project complex diffusive behavior on long time scales. Our enhanced understanding of the diverse range of solute behavior allows us to hypothesize design changes to LLC monomers aimed toward controlling the rates of solute passage, thus improving the selective performance of LLC membranes.

Top-down machine learning approach for high-throughput single-molecule analysis

Single-molecule approaches provide enormous insight into the dynamics of biomolecules, but adequately sampling distributions of states and events often requires extensive sampling. Although emerging experimental techniques can generate such large datasets, existing analysis tools are not suitable to process the large volume of data obtained in high-throughput paradigms. Here, we present a new analysis platform (DISC) that accelerates unsupervised analysis of single-molecule trajectories. By merging model-free statistical learning with the Viterbi algorithm, DISC idealizes single-molecule trajectories up to three orders of magnitude faster with improved accuracy compared to other commonly used algorithms. Further, we demonstrate the utility of DISC algorithm to probe cooperativity between multiple binding events in the cyclic nucleotide binding domains of HCN pacemaker channel. Given the flexible and efficient nature of DISC, we anticipate it will be a powerful tool for unsupervised processing of high-throughput data across a range of single-molecule experiments.

SMAUG: Analyzing single-molecule tracks with nonparametric Bayesian statistics

Single-molecule fluorescence microscopy probes nanoscale, subcellular biology in real time. Existing methods for analyzing single-particle tracking data provide dynamical information, but can suffer from supervisory biases and high uncertainties. Here, we develop a method for the case of multiple interconverting species undergoing free diffusion and introduce a new approach to analyzing single-molecule trajectories: the Single-Molecule Analysis by Unsupervised Gibbs sampling (SMAUG) algorithm, which uses nonparametric Bayesian statistics to uncover the whole range of information contained within a single-particle trajectory dataset. Even in complex systems where multiple biological states lead to a number of observed mobility states, SMAUG provides the number of mobility states, the average diffusion coefficient of single molecules in that state, the fraction of single molecules in that state, the localization noise, and the probability of transitioning between two different states. In this paper, we provide the theoretical background for the SMAUG analysis and then we validate the method using realistic simulations of single-particle trajectory datasets as well as experiments on a controlled in vitro system. Finally, we demonstrate SMAUG on real experimental systems in both prokaryotes and eukaryotes to measure the motions of the regulatory protein TcpP in Vibrio cholerae and the dynamics of the B-cell receptor antigen response pathway in lymphocytes. Overall, SMAUG provides a mathematically rigorous approach to measuring the real-time dynamics of molecular interactions in living cells.

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.

Direct visualization of human myosin II force generation using DNA origami-based thick filaments

The sarcomere, the minimal mechanical unit of muscle, is composed of myosins, which self-assemble into thick filaments that interact with actin-based thin filaments in a highly-structured lattice. This complex imposes a geometric restriction on myosin in force generation. However, how single myosins generate force within the restriction remains elusive and conventional synthetic filaments do not recapitulate the symmetric bipolar filaments in sarcomeres. Here we engineered thick filaments using DNA origami that incorporate human muscle myosin to directly visualize the motion of the heads during force generation in a restricted space. We found that when the head diffuses, it weakly interacts with actin filaments and then strongly binds preferentially to the forward region as a Brownian ratchet. Upon strong binding, the two-step lever-arm swing dominantly halts at the first step and occasionally reverses direction. Our results illustrate the usefulness of our DNA origami-based assay system to dissect the mechanistic details of motor proteins.

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.

Fluorescence correlation spectroscopy is widely used for in vivo and in vitro applications, yet extracting information from experiments still requires long acquisition times. Here, the authors exploit Bayesian non-parametrics to directly analyze the output of confocal fluorescence experiments thereby probing physical processes on much faster timescales.

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.

Single molecule force spectroscopy at high data acquisition: A Bayesian nonparametric analysis

Bayesian nonparametrics (BNPs) are poised to have a deep impact in the analysis of single molecule data as they provide posterior probabilities over entire models consistent with the supplied data, not just model parameters of one preferred model. Thus they provide an elegant and rigorous solution to the difficult problem encountered when selecting an appropriate candidate model. Nevertheless, BNPs' flexibility to learn models and their associated parameters from experimental data is a double-edged sword. Most importantly, BNPs are prone to increasing the complexity of the estimated models due to artifactual features present in time traces. Thus, because of experimental challenges unique to single molecule methods, naive application of available BNP tools is not possible. Here we consider traces with time correlations and, as a specific example, we deal with force spectroscopy traces collected at high acquisition rates. While high acquisition rates are required in order to capture dwells in short-lived molecular states, in this setup, a slow response of the optical trap instrumentation (i.e., trapped beads, ambient fluid, and tethering handles) distorts the molecular signals introducing time correlations into the data that may be misinterpreted as true states by naive BNPs. Our adaptation of BNP tools explicitly takes into consideration these response dynamics, in addition to drift and noise, and makes unsupervised time series analysis of correlated single molecule force spectroscopy measurements possible, even at acquisition rates similar to or below the trap's response times.

ICON: An Adaptation of Infinite HMMs for Time Traces with Drift

Bayesian nonparametric methods have recently transformed emerging areas within data science. One such promising method, the infinite hidden Markov model (iHMM), generalizes the HMM that itself has become a workhorse in single molecule data analysis. The iHMM goes beyond the HMM by self-consistently learning all parameters learned by the HMM in addition to learning the number of states without recourse to any model selection steps. Despite its generality, simple features (such as drift), common to single molecule time traces, result in an overinterpretation of drift and the introduction of artifact states. Here we present an adaptation of the iHMM that can treat data with drift originating from one or many traces (e.g., Förster resonance energy transfer). Our fully Bayesian method couples the iHMM to a continuous control process (drift) self-consistently learned while learning all other quantities determined by the iHMM (including state numbers). A key advantage of this method is that all traces-regardless of drift or states visited across traces-may now be treated on an equal footing, thereby eliminating user-dependent trace selection (based on drift levels), preprocessing to remove drift, and postprocessing model selection based on state number.

An Introduction to Infinite HMMs for Single-Molecule Data Analysis

The hidden Markov model (HMM) has been a workhorse of single-molecule data analysis and is now commonly used as a stand-alone tool in time series analysis or in conjunction with other analysis methods such as tracking. Here, we provide a conceptual introduction to an important generalization of the HMM, which is poised to have a deep impact across the field of biophysics: the infinite HMM (iHMM). As a modeling tool, iHMMs can analyze sequential data without a priori setting a specific number of states as required for the traditional (finite) HMM. Although the current literature on the iHMM is primarily intended for audiences in statistics, the idea is powerful and the iHMM's breadth in applicability outside machine learning and data science warrants a careful exposition. Here, we explain the key ideas underlying the iHMM, with a special emphasis on implementation, and provide a description of a code we are making freely available. In a companion article, we provide an important extension of the iHMM to accommodate complications such as drift.

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

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

Decoding Single Molecule Time Traces with Dynamic Disorder

Single molecule time trajectories of biomolecules provide glimpses into complex folding landscapes that are difficult to visualize using conventional ensemble measurements. Recent experiments and theoretical analyses have highlighted dynamic disorder in certain classes of biomolecules, whose dynamic pattern of conformational transitions is affected by slower transition dynamics of internal state hidden in a low dimensional projection. A systematic means to analyze such data is, however, currently not well developed. Here we report a new algorithm—Variational Bayes-double chain Markov model (VB-DCMM)—to analyze single molecule time trajectories that display dynamic disorder. The proposed analysis employing VB-DCMM allows us to detect the presence of dynamic disorder, if any, in each trajectory, identify the number of internal states, and estimate transition rates between the internal states as well as the rates of conformational transition within each internal state. Applying VB-DCMM algorithm to single molecule FRET data of H-DNA in 100 mM-Na⁺ solution, followed by data clustering, we show that at least 6 kinetic paths linking 4 distinct internal states are required to correctly interpret the duplex-triplex transitions of H-DNA.

We have developed a new algorithm to better decode single molecule data with dynamic disorder. Our new algorithm, which represents a substantial improvement over other methodologies, can detect the presence of dynamic disorder in each trajectory and quantify the kinetic characteristics of underlying energy landscape. As a model system, we applied our algorithm to the single molecule FRET time traces of H-DNA. While duplex-triplex transitions of H-DNA are conventionally interpreted in terms of two-state kinetics, slowly varying dynamic patterns corresponding to hidden internal states can also be identified from the individual time traces. Our algorithm reveals that at least 4 distinct internal states are required to correctly interpret the data.

Measuring Membrane Protein Dimerization Equilibrium in Lipid Bilayers by Single-Molecule Fluorescence Microscopy

Dimerization of membrane protein interfaces occurs during membrane protein folding and cell receptor signaling. Here, we summarize a method that allows for measurement of equilibrium dimerization reactions of membrane proteins in lipid bilayers, by measuring the Poisson distribution of subunit capture into liposomes by single-molecule photobleaching analysis. This strategy is grounded in the fact that given a comparable labeling efficiency, monomeric or dimeric forms of a membrane protein will give rise to distinctly different photobleaching probability distributions. These methods have been used to verify the dimer stoichiometry of the Fluc F- ion channel and the dimerization equilibrium constant of the ClC-ec1 Cl-/H+ antiporter in lipid bilayers. This approach can be applied to any membrane protein system provided it can be purified, fluorescently labeled in a quantitative manner, and verified to be correctly folded by functional assays, even if the structure is not yet known.

Extraction of Protein Conformational Modes from Distance Distributions Using Structurally Imputed Bayesian Data Augmentation

Protein conformational changes are known to play important roles in assorted biochemical and biological processes. Driven by thermal motions of surrounding solvent molecules, such a structural remodeling often occurs stochastically. Yet, regardless of how random the conformational reconfiguration may appear, it could in principle be described by a linear combination of a set of orthogonal modes which, in turn, are contained in the intramolecular distance distributions. The central challenge is how to obtain the distribution. This contribution proposes a Bayesian data-augmentation scheme to extract the predominant modes from only few distance distributions, be they from computational sampling or directly from experiments such as single-molecule Förster-type resonance energy transfer (smFRET). The inference of the complete protein structure from insufficient data was recognized as isomorphic to the missing-data problem in Bayesian statistical learning. Using smFRET data as an example, the missing coordinates were deduced, given protein structural constraints and multiple but limited number of smFRET distances; the Boltzmann weighing of each inferred protein structure was then evaluated using computational modeling to numerically construct the posterior density for the global protein conformation. The conformational modes were then determined from the iteratively converged overall conformational distribution using principal component analysis. Two examples were presented to illustrate these basic ideas as well as their practical implementation. The scheme described herein was based on the theory behind the powerful Tanner-Wang algorithm that guarantees convergence to the true posterior density. However, instead of assuming a mathematical model to calculate the likelihood as in conventional statistical inference, here the protein structure was treated as a statistical parameter and was imputed from the numerical likelihood function based on structural information, a probability model-free method. The framework put forth here is anticipated to be generally applicable, offering a new way to articulate protein conformational changes in a quantifiable manner.

α-SNAP regulates dynamic, on-site assembly and calcium selectivity of Orai1 channels

Ion channel subunits typically assemble in the endoplasmic reticulum. α-SNAP orchestrates a unique assembly and calcium selectivity of Orai1 subunits into functional multimers. Dynamic assembly of Orai1 and its dependence on α-SNAP could enable localization of calcium signals and regulation of rate and amount of calcium influx.

Orai1 forms a highly calcium-selective pore of the calcium release activated channel, and α-SNAP is necessary for its function. Here we show that α-SNAP regulates on-site assembly of Orai1 dimers into calcium-selective multimers. We find that Orai1 is a dimer in resting primary mouse embryonic fibroblasts but displays variable stoichiometry in the plasma membrane of store-depleted cells. Remarkably, α-SNAP depletion induces formation of higher-order Orai1 oligomers, which permeate significant levels of sodium via Orai1 channels. Sodium permeation in α-SNAP–deficient cells cannot be corrected by tethering multiple Stim1 domains to Orai1 C-terminal tail, demonstrating that α-SNAP regulates functional assembly and calcium selectivity of Orai1 multimers independently of Stim1 levels. Fluorescence nanoscopy reveals sustained coassociation of α-SNAP with Stim1 and Orai1, and α-SNAP–depleted cells show faster and less constrained mobility of Orai1 within ER-PM junctions, suggesting Orai1 and Stim1 coentrapment without stable contacts. Furthermore, α-SNAP depletion significantly reduces fluorescence resonance energy transfer between Stim1 and Orai1 N-terminus but not C-terminus. Taken together, these data reveal a unique role of α-SNAP in the on-site functional assembly of Orai1 subunits and suggest that this process may, in part, involve enabling crucial low-affinity interactions between Orai1 N-terminus and Stim1.

Motion blur filtering: A statistical approach for extracting confinement forces and diffusivity from a single blurred trajectory

Single particle tracking (SPT) can aid in understanding a variety of complex spatiotemporal processes. However, quantifying diffusivity and confinement forces from individual live cell trajectories is complicated by inter- and intratrajectory kinetic heterogeneity, thermal fluctuations, and (experimentally resolvable) statistical temporal dependence inherent to the underlying molecule's time correlated confined dynamics experienced in the cell. The problem is further complicated by experimental artifacts such as localization uncertainty and motion blur. The latter is caused by the tagged molecule emitting photons at different spatial positions during the exposure time of a single frame. The aforementioned experimental artifacts induce spurious time correlations in measured SPT time series that obscure the information of interest (e.g., confinement forces and diffusivity). We develop a maximum likelihood estimation (MLE) technique that decouples the above noise sources and systematically treats temporal correlation via time series methods. This ultimately permits a reliable algorithm for extracting diffusivity and effective forces in confined or unconfined environments. We illustrate how our approach avoids complications inherent to mean square displacement or autocorrelation techniques. Our algorithm modifies the established Kalman filter (which does not handle motion blur artifacts) to provide a likelihood based time series estimation procedure. The result extends A. J. Berglund's motion blur model [Phys. Rev. E 82, 011917 (2010)PLEEE81539-375510.1103/PhysRevE.82.011917] to handle confined dynamics. The approach can also systematically utilize (possibly time dependent) localization uncertainty estimates afforded by image analysis if available. This technique, which explicitly treats confinement and motion blur within a time domain MLE framework, uses an exact likelihood (time domain methods facilitate analyzing nonstationary signals). Our estimator is demonstrated to be consistent over a wide range of exposure times (5 to 100 ms), diffusion coefficients (1×10^{-3} to 1μm^{2}/s), and confinement widths (100 nm to 2μm). We demonstrate that neglecting motion blur or confinement can substantially bias estimation of kinetic parameters of interest to researchers. The technique also permits one to check statistical model assumptions against measured individual trajectories without "ground truth." The ability to reliably and consistently extract motion parameters in trajectories exhibiting confined and/or non-stationary dynamics, without exposure time artifacts corrupting estimates, is expected to aid in directly comparing trajectories obtained from different experiments or imaging modalities. A Python implementation is provided (open-source code will be maintained on GitHub; see also the Supplemental Material with this paper).

A primer on Bayesian inference for biophysical systems

Bayesian inference is a powerful statistical paradigm that has gained popularity in many fields of science, but adoption has been somewhat slower in biophysics. Here, I provide an accessible tutorial on the use of Bayesian methods by focusing on example applications that will be familiar to biophysicists. I first discuss the goals of Bayesian inference and show simple examples of posterior inference using conjugate priors. I then describe Markov chain Monte Carlo sampling and, in particular, discuss Gibbs sampling and Metropolis random walk algorithms with reference to detailed examples. These Bayesian methods (with the aid of Markov chain Monte Carlo sampling) provide a generalizable way of rigorously addressing parameter inference and identifiability for arbitrarily complicated models.

Inferring Latent States and Refining Force Estimates via Hierarchical Dirichlet Process Modeling in Single Particle Tracking Experiments

Understanding the basis for intracellular motion is critical as the field moves toward a deeper understanding of the relation between Brownian forces, molecular crowding, and anisotropic (or isotropic) energetic forcing. Effective forces and other parameters used to summarize molecular motion change over time in live cells due to latent state changes, e.g., changes induced by dynamic micro-environments, photobleaching, and other heterogeneity inherent in biological processes. This study discusses limitations in currently popular analysis methods (e.g., mean square displacement-based analyses) and how new techniques can be used to systematically analyze Single Particle Tracking (SPT) data experiencing abrupt state changes in time or space. The approach is to track GFP tagged chromatids in metaphase in live yeast cells and quantitatively probe the effective forces resulting from dynamic interactions that reflect the sum of a number of physical phenomena. State changes can be induced by various sources including: microtubule dynamics exerting force through the centromere, thermal polymer fluctuations, and DNA-based molecular machines including polymerases and protein exchange complexes such as chaperones and chromatin remodeling complexes. Simulations aiming to show the relevance of the approach to more general SPT data analyses are also studied. Refined force estimates are obtained by adopting and modifying a nonparametric Bayesian modeling technique, the Hierarchical Dirichlet Process Switching Linear Dynamical System (HDP-SLDS), for SPT applications. The HDP-SLDS method shows promise in systematically identifying dynamical regime changes induced by unobserved state changes when the number of underlying states is unknown in advance (a common problem in SPT applications). We expand on the relevance of the HDP-SLDS approach, review the relevant background of Hierarchical Dirichlet Processes, show how to map discrete time HDP-SLDS models to classic SPT models, and discuss limitations of the approach. In addition, we demonstrate new computational techniques for tuning hyperparameters and for checking the statistical consistency of model assumptions directly against individual experimental trajectories; the techniques circumvent the need for "ground-truth" and/or subjective information.