Single-molecule Time Series Research Articles

Compression algorithms reveal memory effects and static disorder in single-molecule trajectories

Information-theory-based analysis of single-molecule time series, inspired by Shannon's studies of the information content of printed English, can differentiate between Markov (memoryless) and non-Markov single-molecule signals and between static and dynamic disorder.

Learned Reconstruction of Protein Folding Trajectories from Noisy Single-Molecule Time Series.

Single-molecule Förster resonance energy transfer (smFRET) is an experimental methodology to track the real-time dynamics of molecules using fluorescent probes to follow one or more intramolecular distances. These distances provide a low-dimensional representation of the full atomistic dynamics. Under mild technical conditions, Takens' Delay Embedding Theorem guarantees that the full three-dimensional atomistic dynamics of a system are diffeomorphic (i.e., related by a smooth and invertible transformation) to a time-delayed embedding of one or more scalar observables. Appealing to these theoretical guarantees, we employ manifold learning, artificial neural networks, and statistical mechanics to learn from molecular simulation training data the a priori unknown transformation between the atomic coordinates and delay-embedded intramolecular distances accessible to smFRET. This learned transformation may then be used to reconstruct atomistic coordinates from smFRET time series data. We term this approach Single-molecule TAkens Reconstruction (STAR). We have previously applied STAR to reconstruct molecular configurations of a C24H50 polymer chain and the mini-protein Chignolin with accuracies better than 0.2 nm from simulated smFRET data under noise free and high time resolution conditions. In the present work, we investigate the role of signal-to-noise ratio, data volume, and time resolution in simulated smFRET data to assess the performance of STAR under conditions more representative of experimental realities. We show that STAR can reconstruct the Chignolin and Villin mini-proteins to accuracies of 0.12 and 0.42 nm, respectively, and place bounds on these conditions for accurate reconstructions. These results demonstrate that it is possible to reconstruct dynamical trajectories of protein folding from time series in noisy, time binned, experimentally measurable observables and lay the foundations for the application of STAR to real experimental data.

SmBEVO: a computer vision approach to rapid baseline correction of single-molecule time series

For many biophysical systems, single-molecule time series provide valuable mechanistic and dynamic information which is often obscured in ensemble measures due to averaging over asynchronous events and heterogeneous conformations. Standard amplitude-based approaches for analyzing these data such as Hidden Markov Models or changepoint analyses assume that each underlying molecular state has a constant signal amplitude over the course of the time series. However, time series generated by real-world experiments are often corrupted by baseline drift which must be corrected prior to application of any amplitude-based technique.

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.

Nanoscale Patterning of Graphene Field-Effect Transistors for Single-Molecule Functionalization and Time Series

With the discovery of low-dimensional materials, the extreme miniaturization of field-effect transistors (FETs) has revealed such devices as powerful sensors for experiments at the single-molecule scale. In particular, FET devices functionalized with an individual molecule have been able to record, via fluctuations in the electrical conductance, various types of single-molecule dynamic changes, for example enzymatic activity1 DNA hybridization2,3 or DNA folding4. So far, such studies have mostly been carried out with FETs based on 1D materials such as carbon nanotubes (CNTs) or silicon nanowires (SiNWs). The 1D geometry of these materials facilitates the capture of a 0D molecule and provides enhanced sensitivity of the conductance to the functionalized site. However, FETs based on 1D materials often present scalability issues, due to challenges in controlling their growth and/or assembly.Here, we propose an approach based on nanoscale patterning of a 2D material to assemble FET arrays compatible for single-molecule capture and detection. First, we report wafer-scale microfabrication of arrays of graphene field-effect transistors (GFETs) with micron-scale channels. Like its 1D counterpart, graphene presents a high and sensitive electrical conductivity as well as carbon-based surface chemistry for single-molecule functionalization. GFET arrays were built from high-quality graphene synthesized by chemical vapor deposition (CVD) and transferred using a custom automated process on patterned electrodes. Then, we describe the nanoscale patterning of graphene channels in nanoconstrictions suitable for single-molecule functionalization and sensitivity. Using electron-beam lithography (EBL) and deep reactive ion etching (DRIE), we were able to produce a combination of high-resolution features in graphene (<100nm) as well as nanofluidics reactions chambers (<50nm), to promote single-molecule chemistry on the graphene. We will present our recent results in functionalizing these nanoconstrictions with single-stranded DNA and in collecting high-resolution electrical time series from these constructs. Finally, we will discuss the performance of such graphene-based devices for single-molecule experiments. Bibliography Choi, Y., Weiss, G. A. & Collins, P. G. Single molecule recordings of lysozyme activity. Physical Chemistry Chemical Physics (2013). doi:10.1039/c3cp51356d Sorgenfrei, S. & Shepard, K. L. Label-free field-effect-based single-molecule detection of DNA hybridization kinetics. Nat. Nanotechnol. 6, 126–132 (2011). Vernick, S. et al. Electrostatic melting in a single-molecule field-effect transistor with applications in genomic identification. Nat. Commun. 8, 1–9 (2017). Bouilly, D. et al. Single-molecule reaction chemistry in patterned nanowells. Nano Lett. 16, 4679–4685 (2016).

Reconstruction of protein structures from single-molecule time series.

Single-molecule experimental techniques track the real-time dynamics of molecules by recording a small number of experimental observables. Following these observables provides a coarse-grained, low-dimensional representation of the conformational dynamics but does not furnish an atomistic representation of the instantaneous molecular structure. Takens's delay embedding theorem asserts that, under quite general conditions, these low-dimensional time series can contain sufficient information to reconstruct the full molecular configuration of the system up to an a priori unknown transformation. By combining Takens's theorem with tools from statistical thermodynamics, manifold learning, artificial neural networks, and rigid graph theory, we establish an approach, Single-molecule TAkens Reconstruction, to learn this transformation and reconstruct molecular configurations from time series in experimentally measurable observables such as intramolecular distances accessible to single molecule Förster resonance energy transfer. We demonstrate the approach in applications to molecular dynamics simulations of a C24H50 polymer chain and the artificial mini-protein chignolin. The trained models reconstruct molecular configurations from synthetic time series data in the head-to-tail molecular distances with atomistic root mean squared deviation accuracies better than 0.2 nm. This work demonstrates that it is possible to accurately reconstruct protein structures from time series in experimentally measurable observables and establishes the theoretical and algorithmic foundations to do so in applications to real experimental data.

A Divisive Segmentation and Clustering Scheme for Accelerated and Improved Single-Molecule Time Series Idealization (DiSC)

Single-molecule methods offer unprecedented view into the heterogenous dynamics underlying chemical and biological interactions. As their popularity grows, so too does the demand for efficient and accurate methods to reliably extract kinetic parameters from large, often noisy, time series data sets. Conventional methods include change-point analysis and hidden Markov modeling; the former being computationally efficient and parameter free at the cost of lower accuracy, and the latter being the opposite. Here, we introduce DiSC, a new idealization scheme that combines the strengths of both these methods for parameter free acceleration of hidden Markov modeling. DiSC identifies segments and clusters simultaneously in a divisive manner, wherein the growth of each node is controlled by a specified information criterion. The identified clusters and change-points are then used as priors for hidden Markov modeling to improve the overall fit at a fraction of the standard computational cost. We validate the performance of DiSC using simulated data and determine DiSC is both faster and more accurate for state and change-point identification than the standard cutting-edge methods.

Recovery of Protein Folding Funnels from Single-Molecule Time Series by Delay Embeddings and Manifold Learning.

The stability and folding of proteins is governed by the underlying single-molecule free energy surface (smFES) mapping the free energy of the molecule as a function of configurational state. Ascertaining the smFES is of great value in understanding and engineering protein structure and function. By integrating tools from dynamical systems theory and nonlinear manifold learning, we describe an approach to reconstruct the multidimensional smFES for a protein from a time series in a single experimentally measurable observable. We employ Takens' delay embeddings to project the time series into a high-dimensional space in which the projected dynamics are C1-equivalent to the true system dynamics and employ diffusion maps to recover a low-dimensional reconstruction of the smFES that is equivalent to the true smFES up to a smooth and invertible transformation. We validate the approach in molecular dynamics simulations of Trp-cage, Villin, and BBA to demonstrate that landscapes recovered from univariate time series in the head-to-tail distance are topologically identical-they precisely preserve the metastable states and folding pathways-and topographically approximate-the free energy barrier heights and well depths are approximately preserved-to the true landscapes determined from complete knowledge of all atomic coordinates. We go on to show that the reconstructed landscapes reliably predict temperature denaturation and identify point mutations and groups of mutations critical to folding. These results demonstrate that protein folding funnels can be reconstructed from experimentally measurable time series and used to understand and engineer folding.

Refining Markov state models for conformational dynamics using ensemble-averaged data and time-series trajectories.

A data-driven modeling scheme is proposed for conformational dynamics of biomolecules based on molecular dynamics (MD) simulations and experimental measurements. In this scheme, an initial Markov State Model (MSM) is constructed from MD simulation trajectories, and then, the MSM parameters are refined using experimental measurements through machine learning techniques. The second step can reduce the bias of MD simulation results due to inaccurate force-field parameters. Either time-series trajectories or ensemble-averaged data are available as a training data set in the scheme. Using a coarse-grained model of a dye-labeled polyproline-20, we compare the performance of machine learning estimations from the two types of training data sets. Machine learning from time-series data could provide the equilibrium populations of conformational states as well as their transition probabilities. It estimates hidden conformational states in more robust ways compared to that from ensemble-averaged data although there are limitations in estimating the transition probabilities between minor states. We discuss how to use the machine learning scheme for various experimental measurements including single-molecule time-series trajectories.

Towards Revealing a Cooperative Mechanism of Camp Binding to HCN2 Cyclic Nucleotide Binding Domains at the Single-Molecule Level

Hyperpolarization-activated cyclic nucleotide-gated (HCN) ion channels are critical for oscillatory neuronal activity in the brain and pacemaking in the heart. Although activation of these tetrameric channels can be driven solely by hyperpolarizing voltages, the binding of cyclic nucleotides to their intracellular cyclic nucleotide binding domains (CNBDs) enhances their voltage-dependence of activation through a mechanism that remains unclear. Previous studies dissecting the cooperative nature of subunit activation upon cAMP binding have relied on macroscopic methods, such as ensemble channel currents or fluorescence. Although these studies provide pivotal information in determining the gating-mechanism of HCN channels, they both afford only ensemble-averaged data, which obscures the resolution of individual binding steps. Herein, we aim to circumvent this limitation and determine the cooperative mechanism of cAMP binding to HCN2 CNBDs at the single-molecule level. By tethering an artificially-linked tetrameric CNBD construct into nanofabricated devices called zero-mode waveguides, we resolve the stepwise binding of fluorescent cAMP molecules to all four binding sites of our complex. Using hidden Markov modeling of our idealized single-molecule fluorescence time series, we begin to reveal a cooperative mechanism of cAMP binding to HCN2 CNBDs.

Deciphering hierarchical features in the energy landscape of adenylate kinase folding/unfolding

Hierarchical features of the energy landscape of the folding/unfolding behavior of adenylate kinase, including its dependence on denaturant concentration, are elucidated in terms of single-molecule fluorescence resonance energy transfer (smFRET) measurements in which the proteins are encapsulated in a lipid vesicle. The core in constructing the energy landscape from single-molecule time-series across different denaturant concentrations is the application of rate-distortion theory (RDT), which naturally considers the effects of measurement noise and sampling error, in combination with change-point detection and the quantification of the FRET efficiency-dependent photobleaching behavior. Energy landscapes are constructed as a function of observation time scale, revealing multiple partially folded conformations at small time scales that are situated in a superbasin. As the time scale increases, these denatured states merge into a single basin, demonstrating the coarse-graining of the energy landscape as observation time increases. Because the photobleaching time scale is dependent on the conformational state of the protein, possible nonequilibrium features are discussed, and a statistical test for violation of the detailed balance condition is developed based on the state sequences arising from the RDT framework.

Microsecond resolved single-molecule FRET time series measurements based on the line confocal optical system combined with hybrid photodetectors.

Single-molecule (sm) fluorescence time series measurements based on the line confocal optical system are a powerful strategy for the investigation of the structure, dynamics, and heterogeneity of biological macromolecules. This method enables the detection of more than several thousands of fluorescence photons per millisecond from single fluorophores, implying that the potential time resolution for measurements of the fluorescence resonance energy transfer (FRET) efficiency is 10 μs. However, the necessity of using imaging photodetectors in the method limits the time resolution in the FRET efficiency measurements to approximately 100 μs. In this investigation, a new photodetector called a hybrid photodetector (HPD) was incorporated into the line confocal system to improve the time resolution without sacrificing the length of the time series detection. Among several settings examined, the system based on a slit width of 10 μm and a high-speed counting device made the best of the features of the line confocal optical system and the HPD. This method achieved a time resolution of 10 μs and an observation time of approximately 5 ms in the sm-FRET time series measurements. The developed device was used for the native state of the B domain of protein A.

Complexity of the folding transition of the B domain of protein A revealed by the high-speed tracking of single-molecule fluorescence time series.

The equilibrium unfolding transition of the B domain of protein A (BdpA) was investigated by using single-molecule fluorescence spectroscopy based on line-confocal detection of fast-flowing samples. The method achieved the time resolution of 120 μs and the observation time of a few milliseconds in the single-molecule time-series measurements of fluorescence resonance energy transfer (FRET). Two samples of BdpA doubly labeled with donor and acceptor fluorophores, the first possessing fluorophores at residues 22 and 55 (sample 1) and the second at residues 5 and 55 (sample 2), were prepared. The equilibrium unfolding transition induced by guanidium chloride (GdmCl) was monitored by bulk measurements and demonstrated that the both samples obey the apparent two-state unfolding. In the absence of GdmCl, the single-molecule FRET measurements for the both samples showed a single peak assignable to the native state (N). The FRET efficiency for N shifts to lower values as the increase of GdmCl concentration, suggesting the swelling of the native state structure. At the higher concentration of GdmCl, the both samples convert to the unfolded state (U). Near the unfolding midpoint for sample 1, the kinetic exchange between N and U causes the averaging of the two states and the higher values of the relative fluctuation. The time series for different molecules in U showed slightly different FRET efficiencies, suggesting the apparent heterogeneity. Thus, the high-speed tracking of fluorescence signals from single molecules revealed a complexity and heterogeneity hidden in the apparent two-state behavior of protein folding.

Error-based extraction of states and energy landscapes from experimental single-molecule time-series.

Characterization of states, the essential components of the underlying energy landscapes, is one of the most intriguing subjects in single-molecule (SM) experiments due to the existence of noise inherent to the measurements. Here we present a method to extract the underlying state sequences from experimental SM time-series. Taking into account empirical error and the finite sampling of the time-series, the method extracts a steady-state network which provides an approximation of the underlying effective free energy landscape. The core of the method is the application of rate-distortion theory from information theory, allowing the individual data points to be assigned to multiple states simultaneously. We demonstrate the method's proficiency in its application to simulated trajectories as well as to experimental SM fluorescence resonance energy transfer (FRET) trajectories obtained from isolated agonist binding domains of the AMPA receptor, an ionotropic glutamate receptor that is prevalent in the central nervous system.

Analyzing Single-Molecule Time Series via Nonparametric Bayesian Inference

The ability to measure the properties of proteins at the single-molecule level offers an unparalleled glimpse into biological systems at the molecular scale. The interpretation of single-molecule time series has often been rooted in statistical mechanics and the theory of Markov processes. While existing analysis methods have been useful, they are not without significant limitations including problems of model selection and parameter nonidentifiability. To address these challenges, we introduce the use of nonparametric Bayesian inference for the analysis of single-molecule time series. These methods provide a flexible way to extract structure from data instead of assuming models beforehand. We demonstrate these methods with applications to several diverse settings in single-molecule biophysics. This approach provides a well-constrained and rigorously grounded method for determining the number of biophysical states underlying single-molecule data.

Non-Markovian properties and multiscale hidden Markovian network buried in single molecule time series

We present a novel scheme to extract a multiscale state space network (SSN) from single-molecule time series. The multiscale SSN is a type of hidden Markov model that takes into account both multiple states buried in the measurement and memory effects in the process of the observable whenever they exist. Most biological systems function in a nonstationary manner across multiple timescales. Combined with a recently established nonlinear time series analysis based on information theory, a simple scheme is proposed to deal with the properties of multiscale and nonstationarity for a discrete time series. We derived an explicit analytical expression of the autocorrelation function in terms of the SSN. To demonstrate the potential of our scheme, we investigated single-molecule time series of dissociation and association kinetics between epidermal growth factor receptor (EGFR) on the plasma membrane and its adaptor protein Ash/Grb2 (Grb2) in an in vitro reconstituted system. We found that our formula successfully reproduces their autocorrelation function for a wide range of timescales (up to 3 s), and the underlying SSNs change their topographical structure as a function of the timescale; while the corresponding SSN is simple at the short timescale (0.033-0.1 s), the SSN at the longer timescales (0.1 s to ~3 s) becomes rather complex in order to capture multiscale nonstationary kinetics emerging at longer timescales. It is also found that visiting the unbound form of the EGFR-Grb2 system approximately resets all information of history or memory of the process.

1SCA-02 Inferring Kinetics Objectively from Single Molecule Time Series with Full Information Content(1SCA Information Dynamics in Biological Systems,Symposium,The 51th Annual Meeting of the Biophysical Society of Japan)

1SCA-02 Inferring Kinetics Objectively from Single Molecule Time Series with Full Information Content(1SCA Information Dynamics in Biological Systems,Symposium,The 51th Annual Meeting of the Biophysical Society of Japan)

Go-and-Back method: Effective estimation of the hidden motion of proteins from single-molecule time series

We present an effective method for estimating the motion of proteins from the motion of attached probe particles in single-molecule experiments. The framework naturally incorporates Langevin dynamics to compute the most probable trajectory of the protein. By using a perturbation expansion technique, we achieve computational costs more than 3 orders of magnitude smaller than the conventional gradient descent method without loss of simplicity in the computation algorithm. We present illustrative applications of the method using simple models of single-molecule experiments and confirm that the proposed method yields reasonable and stable estimates of the hidden motion in a highly efficient manner.

Extracting the underlying effective free energy landscape from single-molecule time series—local equilibrium states and their network

We present a new self-consistent procedure to construct a multidimensional effective free energy landscape from a scalar single molecule time series, when single molecules experience the landscape within a given timescale of "observation." The theory is based on a framework we recently developed to extract a set of local equilibrium states (LESs) and their network from a scalar time series, such as distance between dye molecules tagged in a biomolecule. We scrutinize the appropriateness of the assumptions of local equilibration and local detailed balance among LESs at the single molecule level within the given timescale, rather than postulating them a priori. The self-consistent procedure in this article incorporates the effect of local correlation of the system dynamics inside potential basins, and the effect of finiteness of the sampled data points in assigning the boundary between different LESs. We propose a new simple scheme to assign the dimensionality of the energy landscape from a single molecule time series. We also address the question of what the molecules actually "feel" from the underlying landscape at the single molecule level.

1J1412 1分子タイムスタンプFRET時系列データから分子状態変化ダイナミクスを推定する変分ベイズ-隠れマルコフモデル解析(計測,第49回日本生物物理学会年会)

1J1412 1分子タイムスタンプFRET時系列データから分子状態変化ダイナミクスを推定する変分ベイズ-隠れマルコフモデル解析(計測,第49回日本生物物理学会年会)