COCIS: Markov processes in single molecule fluorescence

Principles of Resonance Energy Transfer

This unit describes the basic principles of Förster resonance energy transfer (FRET). Beginning with a brief summary of the history of FRET applications, the theory of FRET is introduced in detail using figures to explain all the important parameters of the FRET process. After listing various approaches for measuring FRET efficiency, several pieces of advice are given on choosing the appropriate instrumentation. The unit concludes with a discussion of the limitations of FRET measurements followed by a few examples of the latest FRET applications, including new developments such as spectral flow cytometric FRET, single-molecule FRET, and combinations of FRET with super-resolution or lifetime imaging microscopy and with molecular dynamics simulations. © 2022 The Authors. Current Protocols published by Wiley Periodicals LLC.

Gold Ion Beam Milled Gold Zero-Mode Waveguides

Zero-mode waveguides (ZMWs) are widely used in single molecule fluorescence microscopy for their enhancement of emitted light and the ability to study samples at physiological concentrations. ZMWs are typically produced using photo or electron beam lithography. We report a new method of ZMW production using focused ion beam (FIB) milling with gold ions. We demonstrate that ion-milled gold ZMWs with 200 nm apertures exhibit similar plasmon-enhanced fluorescence seen with ZMWs fabricated with traditional techniques such as electron beam lithography.

Nonlinear measurements of kinetics and generalized dynamical modes. I. Extracting the one-dimensional Green's function from a time series

Often, a single correlation function is used to measure the kinetics of a complex system. In contrast, a large set of k-vector modes and their correlation functions are commonly defined for motion in free space. This set can be transformed to the van Hove correlation function, which is the Green's function for molecular diffusion. Here, these ideas are generalized to other observables. A set of correlation functions of nonlinear functions of an observable is used to extract the corresponding Green's function. Although this paper focuses on nonlinear correlation functions of an equilibrium time series, the results are directly connected to other types of nonlinear kinetics, including perturbation-response experiments with strong fields. Generalized modes are defined as the orthogonal polynomials associated with the equilibrium distribution. A matrix of mode-correlation functions can be transformed to the complete, single-time-interval (1D) Green's function. Diagonalizing this matrix finds the eigendecays. To understand the advantages and limitation of this approach, Green's functions are calculated for a number of models of complex dynamics within a Gaussian probability distribution. Examples of non-diffusive motion, rate heterogeneity, and range heterogeneity are examined. General arguments are made that a full set of nonlinear 1D measurements is necessary to extract all the information available in a time series. However, when a process is neither dynamically Gaussian nor Markovian, they are not sufficient. In those cases, additional multidimensional measurements are needed.

Extracting Transition Rates in Particle Tracking Using Analytical Diffusion Distribution Analysis

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

Non-Markovian Methods in Glass Transition

A model for the heterogeneity of local dynamics in polymer and other glass-forming materials is provided here. The fundamental characteristics of the glass transition phenomenology emerge when simulating a condensed matter open cluster that has a strong interaction with its heterogeneous environment. General glass transition features, such as non-exponential structural relaxations, the slowing down of relaxation times with temperature and specific off-equilibrium glassy dynamics can be reproduced by non-Markovian dynamics simulations with the minimum computer resources. Non-Markovian models are shown to be useful tools for obtaining insights into the complex dynamics involved in the glass transition phenomenon, including whether or not there is a need for a growing correlation length or the relationship between the non-exponentiality of structural relaxations and dynamic heterogeneity.

Quo vadis FRET? Förster's method in the era of superresolution

Although the theoretical foundations of Förster resonance energy transfer (FRET) were laid in the 1940s as part of the quantum physical revolution of the 20th century, it was only in the 1970s that it made its way to biology as a result of the availability of suitable measuring and labeling technologies. Thanks to its ease of application, FRET became widely used for studying molecular associations on the nanometer scale. The development of superresolution techniques at the turn of the millennium promised an unprecedented insight into the structure and function of molecular complexes. Without downplaying the significance of superresolution microscopies this review expresses our view that FRET is still a legitimate tool in the armamentarium of biologists for studying molecular associations since it offers distinct advantages and overcomes certain limitations of superresolution approaches.

Single-molecule FRET methods to study the dynamics of proteins at work

Feynman commented that "Everything that living things do can be understood in terms of the jiggling and wiggling of atoms". Proteins can jiggle and wiggle large structural elements such as domains and subunits as part of their functional cycles. Single-molecule fluorescence resonance energy transfer (smFRET) is an excellent tool to study conformational dynamics and decipher coordinated large-scale motions within proteins. smFRET methods introduced in recent years are geared toward understanding the time scales and amplitudes of function-related motions. This review discusses the methodology for obtaining and analyzing smFRET temporal trajectories that provide direct dynamic information on transitions between conformational states. It also introduces correlation methods that are useful for characterizing intramolecular motions. This arsenal of techniques has been used to study multiple molecular systems, from membrane proteins through molecular chaperones, and we examine some of these studies here. Recent exciting methodological novelties permit revealing very fast, submillisecond dynamics, whose relevance to protein function is yet to be fully grasped.

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.

Analyzing the Dynamics of Single TBP-DNA-NC2 Complexes Using Hidden Markov Models

Single-pair Förster resonance energy transfer (spFRET) has become an important tool for investigating conformational dynamics in biological systems. To extract dynamic information from the spFRET traces measured with total internal reflection fluorescence microscopy, we extended the hidden Markov model (HMM) approach. In our extended HMM analysis, we incorporated the photon-shot noise from camera-based systems into the HMM. Thus, the variance in Förster resonance energy transfer (FRET) efficiency of the various states, which is typically a fitted parameter, is explicitly included in the analysis estimated from the number of detected photons. It is also possible to include an additional broadening of the FRET state, which would then only reflect the inherent flexibility of the dynamic biological systems. This approach is useful when comparing the dynamics of individual molecules for which the total intensities vary significantly. We used spFRET with the extended HMM analysis to investigate the dynamics of TATA-box-binding protein (TBP) on promoter DNA in the presence of negative cofactor 2 (NC2). We compared the dynamics of two promoters as well as DNAs of different length and labeling location. For the adenovirus major late promoter, four FRET states were observed; three states correspond to different conformations of the DNA in the TBP-DNA-NC2 complex and a four-state model in which the complex has shifted along the DNA. The HMM analysis revealed that the states are connected via a linear, four-well model. For the H2B promoter, more complex dynamics were observed. By clustering the FRET states detected with the HMM analysis, we could compare the general dynamics observed for the two promoter sequences. We observed that the dynamics from a stretched DNA conformation to a bent conformation for the two promoters were similar, whereas the bent conformation of the TBP-DNA-NC2 complex for the H2B promoter is approximately three times more stable than for the adenovirus major late promoter.

Application of Hidden Markov Models in Biomolecular Simulations

Hidden Markov models (HMMs) provide a framework to analyze large trajectories of biomolecular simulation datasets. HMMs decompose the conformational space of a biological molecule into finite number of states that interconvert among each other with certain rates. HMMs simplify long timescale trajectories for human comprehension, and allow comparison of simulations with experimental data. In this chapter, we provide an overview of building HMMs for analyzing bimolecular simulation datasets. We demonstrate the procedure for building a Hidden Markov model for Met-enkephalin peptide simulation dataset and compare the timescales of the process.

Quantifying the Assembly of Multicomponent Molecular Machines by Single-Molecule Total Internal Reflection Fluorescence Microscopy

Large, dynamic macromolecular complexes play essential roles in many cellular processes. Knowing how the components of these complexes associate with one another and undergo structural rearrangements is critical to understanding how they function. Single-molecule total internal reflection fluorescence (TIRF) microscopy is a powerful approach for addressing these fundamental issues. In this article, we first discuss single-molecule TIRF microscopes and strategies to immobilize and fluorescently label macromolecules. We then review the use of single-molecule TIRF microscopy to study the formation of binary macromolecular complexes using one-color imaging and inhibitors. We conclude with a discussion of the use of TIRF microscopy to examine the formation of higher-order (i.e., ternary) complexes using multicolor setups. The focus throughout this article is on experimental design, controls, data acquisition, and data analysis. We hope that single-molecule TIRF microscopy, which has largely been the province of specialists, will soon become as common in the tool box of biophysicists and biochemists as structural approaches have become today.

Single-molecule FRET studies of HIV TAR-DNA hairpin unfolding dynamics

We directly measure the dynamics of the HIV trans-activation response (TAR)-DNA hairpin with multiple loops using single-molecule Förster resonance energy transfer (smFRET) methods. Multiple FRET states are identified that correspond to intermediate melting states of the hairpin. The stability of each intermediate state is calculated from the smFRET data. The results indicate that hairpin unfolding obeys a "fraying and peeling" mechanism, and evidence for the collapse of the ends of the hairpin during folding is observed. These results suggest a possible biological function for hairpin loops serving as additional fraying centers to increase unfolding rates in otherwise stable systems. The experimental and analytical approaches developed in this article provide useful tools for studying the mechanism of multistate DNA hairpin dynamics and of other general systems with multiple parallel pathways of chemical reactions.

Application of confocal single-molecule FRET to intrinsically disordered proteins

Intrinsically disordered proteins (IDPs) are characterized by a large degree of conformational heterogeneity. In such cases, classical experimental methods often yield only mean values, averaged over the entire ensemble of molecules. The microscopic distributions of conformations, trajectories, or sequences of events often remain unknown, and with them the underlying molecular mechanisms. Signal averaging can be avoided by observing individual molecules. A particularly versatile method is highly sensitive fluorescence detection. In combination with Förster resonance energy transfer (FRET), distances and conformational dynamics can be investigated in single molecules. This chapter introduces the practical aspects of applying confocal single-molecule FRET experiments to the study of IDPs.

Near-infrared fluorescent sensors based on single-walled carbon nanotubes for life sciences applications

Many properties of single-walled carbon nanotubes (SWCNTs) make them ideal candidates for sensors, particularly for biological systems. Both their fluorescence in the near-infrared range of 820-1600 nm, where absorption by biological tissues is often minimal, and their inherent photostability are desirable attributes for the design of in vitro and in vivo sensors. The mechanisms by which a target molecule can selectively alter the fluorescent emission include primarily changes in emission wavelength (i.e., solvatochromism) and intensity, including effects such as charge-transfer transition bleaching and exciton quenching. The central challenge lies in engineering the nanotube interface to be selective for the analyte of interest. In this work, we review the recent development in this area over the past few years, and describe the design rules that we have developed for detecting various analytes, ranging from stable small molecules and reactive oxygen species (ROS) or reactive nitrogen species (RNS) to macromolecules. Applications to in vivo sensor measurements using these sensors are also described. In addition, the emerging field of SWCNT-based single-molecule detection using band gap fluorescence and the recent efforts to accurately quantify and utilize this unique class of stochastic sensors are also described in this article.

Structural landscape of isolated agonist-binding domains from single AMPA receptors

α-amino-3-hydroxy-5-methyl-4-isoxazole propionate (AMPA) receptors mediate fast excitatory neurotransmission by converting chemical signals into electrical signals. Thus, it is important to understand the relationship between their chemical biology and their function. Single molecule fluorescence resonance energy transfer (smFRET) was used to examine the conformations explored by the agonist binding domain of the AMPA receptor for wild type and T686 mutant proteins. Each form of the agonist binding domain exhibited a dynamic, multi-state sequential equilibrium, which could only be identified using wavelet shrinkage, a signal processing technique that removes experimental shot-noise. These results illustrate that the extent of activation is dependent not on a rigid closed cleft, but instead on the probability that a given subunit will occupy a closed cleft conformation, which in turn is not only determined by the lowest energy state but by the range of states that the protein explores.

Deciphering subcellular processes in live imaging datasets via dynamic probabilistic networks

MOTIVATION

Designing mathematical tools that can formally describe the dynamics of complex intracellular processes remains a challenge. Live cell imaging reveals changes in the cellular states, but current simple approaches extract only minimal information of a static snapshot.

RESULTS

We implemented a novel approach for analyzing organelle behavior in live cell imaging data based on hidden Markov models (HMMs) and showed that it can determine the number and evolution of distinct cellular states involved in a biological process. We analyzed insulin-mediated exocytosis of single Glut4-vesicles, a process critical for blood glucose homeostasis and impaired in type II diabetes, by using total internal reflection fluorescence microscopy (TIRFM). HMM analyses of movie sequences of living cells reveal that insulin controls spatial and temporal dynamics of exocytosis via the exocyst, a putative tethering protein complex. Our studies have validated the proof-of-principle of HMM for cellular imaging and provided direct evidence for the existence of complex spatial-temporal regulation of exocytosis in non-polarized cells. We independently confirmed insulin-dependent spatial regulation by using static spatial statistics methods.

CONCLUSION

We propose that HMM-based approach can be exploited in a wide avenue of cellular processes, especially those where the changes of cellular states in space and time may be highly complex and non-obvious, such as in cell polarization, signaling and developmental processes.

Single-biomolecule kinetics: the art of studying a single enzyme

The potential of single-enzyme studies to unravel the complex energy landscape of these polymeric catalysts is the next critical step in enzymology. From its inception in Rotman's emulsion experiments in the 1960s, the field of single-molecule enzymology has now advanced into the time-resolved age. Technological advances have enabled individual enzymatic turnover reactions to be observed with a millisecond time resolution. A number of initial studies have revealed the underlying static and dynamic disorder in the catalytic rates originating from conformational fluctuations. Although these experiments are still in their infancy, they may be able to relate the topography of the energy landscape to the biological function and regulation of enzymes. This review summarizes some of the experimental techniques and data-analysis methods that have been used to study individual enzyme molecules in search of a deeper understanding of their kinetics.

Analysis of complex single-molecule FRET time trajectories

Single-molecule methods have given researchers the ability to investigate the structural dynamics of biomolecules at unprecedented resolution and sensitivity. One of the preferred methods of studying single biomolecules is single-molecule fluorescence resonance energy transfer (smFRET). The popularity of smFRET stems from its ability to report on dynamic, either intra- or intermolecular interactions in real-time. For example, smFRET has been successfully used to characterize the role of dynamics in functional RNAs and their protein complexes, including ribozymes, the ribosome, and more recently the spliceosome. Being able to reliably extract quantitative kinetic and conformational parameters from smFRET experiments is crucial for the interpretation of their results. The need for efficient, unbiased analysis routines becomes more evident as the systems studied become more complex. In this chapter, we focus on the practical utility of statistical algorithms, particularly hidden Markov models, to aid in the objective quantification of complex smFRET trajectories with three or more discrete states, and to extract kinetic information from the trajectories. Additionally, we present a method for systematically eliminating transitions associated with uncorrelated fluorophore behavior that may occur due to dye anisotropy and quenching effects. We also highlight the importance of data condensation through the use of various transition density plots to fully understand the underlying conformational dynamics and kinetic behavior of the biological macromolecule of interest under varying conditions. Finally, the application of these techniques to studies of pre-mRNA conformational changes during eukaryotic splicing is discussed.

Hidden markov analysis of short single molecule intensity trajectories

Photon trajectories from single molecule experiments can report on biomolecule structural changes and motions. Hidden Markov models (HMM) facilitate extraction of the sequence of hidden states from noisy data through construction of probabilistic models. Typically, the true number of states is determined by the Bayesian information criteria (BIC); however, constraints resulting from short data sets and Poisson-distributed photons in radiative processes like fluorescence can limit successful application of goodness-of-fit statistics. For single molecule intensity trajectories, additional information criteria such as peak localization error (LE) and chi-square probabilities can incorporate theoretical constraints on experimental data while modifying normal HMM. Chi-square minimization also serves as a stopping point of the iteration in which the system parameters are trained. Peak LE enables exclusion of overfitted and overlapped states. These constraints and criteria are tested against BIC on simulated single molecule trajectories to best identify the true number of emissive levels in any sequence.

Stochastic analysis of stepwise fluorescence quenching reactions on single-walled carbon nanotubes: single molecule sensors

The 1D quantum confinement of photogenerated excitons in single-walled carbon nanotubes (SWNT) can amplify the detection of molecular adsorption to where single-molecule discrimination is realizable, even from within living cells and tissues. Toward this aim, we present a type 1 collagen film, similar to those used as 3D cell scaffolds for tissue engineering, containing embedded SWNT capable of reporting single-molecule adsorption of quenching molecules. We utilize hidden Markov modeling to link single-molecule adsorption events to rate constants for H2O2, H+, and Fe(CN)6(3-). Among the three kinds of reactant molecules studied, H2O2 has the highest quenching equilibrium constant of 1.59 at 20 microM, whereas H+ is so insensitive that a similar equilibrium constant is achieved only with a concentration of 0.1 M (pH 1). The results were self-consistent because reverse (unquenching) rate constants (600 micros(-1) for H2O2, 1130 micros(-1) for H+ and 4000 micros(-1) for Fe(CN)6(3-)) were observed to be concentration-independent and the forward (quenching) rate constants varied monotonically with concentration. The quenching rate constants also increased with an increase in the redox potential of the quencher, indicating that electron transfer increases the adsorption equilibrium constant on the nanotube surface and, hence, the dwell time of the quencher. These developments provide the material, analytical, and mechanistic groundwork for SWNT to function as single-molecule stochastic biosensors.

Fluctuations as a source of information in fluorescence microscopy

Fluctuations in fluorescence spectroscopy and microscopy have traditionally been regarded as noise—they lower the resolution and contrast and do not permit high acquisition rates. However, fluctuations can also be used to gain additional information about a system. This fact has been exploited in single-point microscopic techniques, such as fluorescence correlation spectroscopy and analysis of single molecule trajectories, and also in the imaging field, e.g. in spatio-temporal image correlation spectroscopy. Here, we discuss how fluctuations are used to obtain more quantitative information from the data than that given by average values, while minimizing the effects of noise due to stochastic photon detection.

A general statistical test for correlations in a finite-length time series

The statistical properties of the autocorrelation function from a time series composed of independently and identically distributed stochastic variables has been studied. Analytical expressions for the autocorrelation function's variance have been derived. It has been found that two common ways of calculating the autocorrelation, moving-average and Fourier transform, exhibit different uncertainty characteristics. For periodic time series, the Fourier transform method is preferred because it gives smaller uncertainties that are uniform through all time lags. Based on these analytical results, a statistically robust method has been proposed to test the existence of correlations in a time series. The statistical test is verified by computer simulations and an application to single-molecule fluorescence spectroscopy is discussed.