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

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

Variational Bayes Analysis of a Photon-Based Hidden Markov Model for Single-Molecule FRET Trajectories

Single-molecule fluorescence resonance energy transfer (smFRET) measurement is a powerful technique for investigating dynamics of biomolecules, for which various efforts have been made to overcome significant stochastic noise. Time stamp (TS) measurement has been employed experimentally to enrich information within the signals, while data analyses such as the hidden Markov model (HMM) have been successfully applied to recover the trajectories of molecular state transitions from time-binned photon counting signals or images. In this article, we introduce the HMM for TS-FRET signals, employing the variational Bayes (VB) inference to solve the model, and demonstrate the application of VB-HMM-TS-FRET to simulated TS-FRET data. The same analysis using VB-HMM is conducted for other models and the previously reported change point detection scheme. The performance is compared to other analysis methods or data types and we show that our VB-HMM-TS-FRET analysis can achieve the best performance and results in the highest time resolution. Finally, an smFRET experiment was conducted to observe spontaneous branch migration of Holliday-junction DNA. VB-HMM-TS-FRET was successfully applied to reconstruct the state transition trajectory with the number of states consistent with the nucleotide sequence. The results suggest that a single migration process frequently involves rearrangement of multiple basepairs.     

Distinguishing between Protein Dynamics and Dye Photophysics in Single-Molecule FRET Experiments

Förster resonance energy transfer (FRET) efficiency distributions in single-molecule experiments contain both structural and dynamical information. Extraction of this information from these distributions requires a careful analysis of contributions from dye photophysics. To investigate how mechanisms other than FRET affect the distributions obtained by counting donor and acceptor photons, we have measured single-molecule fluorescence trajectories of a small α/β protein, i.e., protein GB1, undergoing two-state, folding/unfolding transitions. Alexa 488 donor and Alexa 594 acceptor dyes were attached to cysteines at positions 10 and 57 to yield two isomers—donor¹⁰/acceptor⁵⁷ and donor⁵⁷/acceptor¹⁰—which could not be separated in the purification. The protein was immobilized via binding of a histidine tag added to a linker sequence at the N-terminus to cupric ions embedded in a polyethylene-glycol-coated glass surface. The distribution of FRET efficiencies assembled from the trajectories is complex with widths for the individual peaks in large excess of that caused by shot noise. Most of this complexity can be explained by two interfering photophysical effects—a photoinduced red shift of the donor dye and differences in the quantum yield of the acceptor dye for the two isomers resulting from differences in quenching rate by the cupric ion. Measurements of steady-state polarization, calculation of the donor-acceptor cross-correlation function from photon trajectories, and comparison of the single molecule and ensemble kinetics all indicate that conformational distributions and dynamics do not contribute to the complexity.     

BOBA FRET: Bootstrap-Based Analysis of Single-Molecule FRET Data

Time-binned single-molecule Förster resonance energy transfer (smFRET) experiments with surface-tethered nucleic acids or proteins permit to follow folding and catalysis of single molecules in real-time. Due to the intrinsically low signal-to-noise ratio (SNR) in smFRET time traces, research over the past years has focused on the development of new methods to extract discrete states (conformations) from noisy data. However, limited observation time typically leads to pronounced cross-sample variability, i.e., single molecules display differences in the relative population of states and the corresponding conversion rates. Quantification of cross-sample variability is necessary to perform statistical testing in order to assess whether changes observed in response to an experimental parameter (metal ion concentration, the presence of a ligand, etc.) are significant. However, such hypothesis testing has been disregarded to date, precluding robust biological interpretation. Here, we address this problem by a bootstrap-based approach to estimate the experimental variability. Simulated time traces are presented to assess the robustness of the algorithm in conjunction with approaches commonly used in thermodynamic and kinetic analysis of time-binned smFRET data. Furthermore, a pair of functionally important sequences derived from the self-cleaving group II intron Sc.ai5γ (d3''EBS1*/IBS1*) is used as a model system. Through statistical hypothesis testing, divalent metal ions are shown to have a statistically significant effect on both thermodynamic and kinetic aspects of their interaction. The Matlab source code used for analysis (bootstrap-based analysis of smFRET data, BOBA FRET), as well as a graphical user interface, is available via http://www.aci.uzh.ch/rna/.     

Bayesian-Estimated Hierarchical HMMs Enable Robust Analysis of Single-Molecule Kinetic Heterogeneity

Single-molecule kinetic experiments allow the reaction trajectories of individual biomolecules to be directly observed, eliminating the effects of population averaging and providing a powerful approach for elucidating the kinetic mechanisms of biomolecular processes. A major challenge to the analysis and interpretation of these experiments, however, is the kinetic heterogeneity that almost universally complicates the recorded single-molecule signal versus time trajectories (i.e., signal trajectories). Such heterogeneity manifests as changes and/or differences in the transition rates that are observed within individual signal trajectories or across a population of signal trajectories. Because characterizing kinetic heterogeneity can provide critical mechanistic information, we have developed a computational method that effectively and comprehensively enables such analysis. To this end, we have developed a computational algorithm and software program, hFRET, that uses the variational approximation for Bayesian inference to estimate the parameters of a hierarchical hidden Markov model, thereby enabling robust identification and characterization of kinetic heterogeneity. Using simulated signal trajectories, we demonstrate the ability of hFRET to accurately and precisely characterize kinetic heterogeneity. In addition, we use hFRET to analyze experimentally recorded signal trajectories reporting on the conformational dynamics of ribosomal pre-translocation (PRE) complexes. The results of our analyses demonstrate that PRE complexes exhibit kinetic heterogeneity, reveal the physical origins of this heterogeneity, and allow us to expand the current model of PRE complex dynamics. The methods described here can be applied to signal trajectories generated using any type of signal and can be easily extended to the analysis of signal trajectories exhibiting more complex kinetic behaviors. Moreover, variations of our approach can be easily developed to integrate kinetic data obtained from different experimental constructs and/or from molecular dynamics simulations of a biomolecule of interest.     

Single-Molecule and FRET Fluorescence Correlation Spectroscopy Analyses of Phage DNA Packaging: Colocalization of Packaged Phage T4 DNA Ends within the Capsid

Linear DNAs of any sequence can be packaged into empty viral procapsids by the phage T4 terminase with high efficiency in vitro. Packaging substrates of 5 kbp and 50 kbp, terminated by energy transfer dye pairs, were constructed from plasmid and λ phage DNAs. Nuclease and fluorescence correlation spectroscopy (FCS) assays showed that ∼ 20% of the substrate DNA was packaged and that the DNA dye ends of the packaged DNA were protected from nuclease digestion. Upon packaging, both 5-kbp and  50-kbp DNAs produced comparable fluorescence resonance energy transfer (FRET) between Cy5 and Cy5.5 double-dye terminated DNAs. Single-molecule FRET (sm-FRET) and photobleaching analysis shows that FRET is intramolecular rather than intermolecular upon packaging of most procapsids and demonstrates that single-molecule detection allows mechanistic analysis of packaging in vitro. FRET-FCS and sm-FRET measurements are comparable and show that both the 5-kbp and the  50-kbp packaged DNA ends are held within 8-9 nm of each other, within the dimensions of the long axis of the procapsid portal. The calculated distribution of FRET distances is relatively narrow for both FRET-FCS and sm-FRET, suggesting that the two packaged DNA ends are held at the same fixed distance relative to each other in most capsids. Because one DNA end is known to be positioned for ejection through the portal, it can be inferred that both DNAs ends are held in proximity to the portal entrance and ejection channel. The analysis suggests that a DNA loop, rather than a DNA end, is translocated by the packaging motor to fill the procapsid.     

Solution-Based Single-Molecule FRET Studies of K+ Channel Gating in a Lipid Bilayer

Ion channels are dynamic multimeric proteins that often undergo multiple unsynchronized structural movements as they switch between their open and closed states. Such structural changes are difficult to measure within the context of a native lipid bilayer and have often been monitored via macroscopic changes in Förster resonance energy transfer (FRET) between probes attached to different parts of the protein. However, the resolution of this approach is limited by ensemble averaging of structurally heterogeneous subpopulations. These problems can be overcome by measurement of FRET in single molecules, but this presents many challenges, in particular the ability to control labeling of subunits within a multimeric protein with acceptor and donor fluorophores, as well as the requirement to image large numbers of individual molecules in a membrane environment. To address these challenges, we randomly labeled tetrameric KirBac1.1 potassium channels, reconstituted them into lipid nanodiscs, and performed single-molecule FRET confocal microscopy with alternating-laser excitation as the channels diffused in solution. These solution-based single-molecule FRET measurements of a multimeric ion channel in a lipid bilayer have allowed us to probe the structural changes that occur upon channel activation and inhibition. Our results provide direct evidence of the twist-to-shrink movement of the helix bundle crossing during channel gating and demonstrate how this method might be applied to real-time structural studies of ion channel gating.     

Alternating-Site Mechanism of Kinesin-1 Characterized by Single-Molecule FRET Using Fluorescent ATP Analogues

Kinesin-1 motor proteins move along microtubules in repetitive steps of 8 nm at the expense of ATP. To determine nucleotide dwell times during these processive runs, we used a Förster resonance energy transfer method at the single-molecule level that detects nucleotide binding to kinesin motor heads. We show that the fluorescent ATP analog used produces processive motility with kinetic parameters altered <2.5-fold compared with normal ATP. Using our confocal fluorescence kinesin motility assay, we obtained fluorescence intensity time traces that we then analyzed using autocorrelation techniques, yielding a time resolution of ∼1 ms for the intensity fluctuations due to fluorescent nucleotide binding and release. To compare these experimental autocorrelation curves with kinetic models, we used Monte-Carlo simulations. We find that the experimental data can only be described satisfactorily on the basis of models assuming an alternating-site mechanism, thus supporting the view that kinesin's two motor domains hydrolyze ATP and step in a sequential way.     

Increasing the Time Resolution of Single-Molecule Experiments with Bayesian Inference

Many time-resolved single-molecule biophysics experiments seek to characterize the kinetics of biomolecular systems exhibiting dynamics that challenge the time resolution of the given technique. Here, we present a general, computational approach to this problem that employs Bayesian inference to learn the underlying dynamics of such systems, even when they are much faster than the time resolution of the experimental technique being used. By accurately and precisely inferring rate constants, our Bayesian inference for the analysis of subtemporal resolution dynamics approach effectively enables the experimenter to super-resolve the poorly resolved dynamics that are present in their data.     

Dynamics of Membrane-Bound G12V-KRAS from Simulations and Single-Molecule FRET in Native Nanodiscs

Recent studies have shown that the small GTPase KRAS adopts multiple orientations with respect to the plane of anionic model membranes, whereby either the three C-terminal helices or the three N-terminal β-strands of the catalytic domain face the membrane. This has functional implications because, in the latter, the membrane occludes the effector-interacting surface. However, it remained unclear how membrane reorientation occurs and, critically, whether it occurs in the cell in which KRAS operates as a molecular switch in signaling pathways. Herein, using data from a 20 μs-long atomistic molecular dynamics simulation of the oncogenic G12V-KRAS mutant in a phosphatidylcholine/phosphatidylserine bilayer, we first show that internal conformational fluctuations of flexible regions in KRAS result in three distinct membrane orientations. We then show, using single-molecule fluorescence resonance energy transfer measurements in native lipid nanodiscs derived from baby hamster kidney cells, that G12V-KRAS samples three conformational states that correspond to the predicted orientations. The combined results suggest that relatively small energy barriers separate orientation states and that signaling-competent conformations dominate the overall population.     

Single-Molecule FRET Reveals Three Conformations for the TLS Domain of Brome Mosaic Virus Genome

Metabolite-dependent conformational switching in RNA riboswitches is now widely accepted as a critical regulatory mechanism for gene expression in bacterial systems. More recently, similar gene regulation mechanisms have been found to be important for viral systems as well. One of the most abundant and best-studied systems is the tRNA-like structure (TLS) domain, which has been found to occur in many plant viruses spread across numerous genera. In this work, folding dynamics for the TLS domain of Brome Mosaic Virus have been investigated using single-molecule fluorescence resonance energy transfer techniques. In particular, burst fluorescence methods are exploited to observe metal-ion ([Mⁿ⁺])-induced folding in freely diffusing RNA constructs resembling the minimal TLS element of brome mosaic virus RNA3. The results of these experiments reveal a complex equilibrium of at least three distinct populations. A stepwise, or consecutive, thermodynamic model for TLS folding is developed, which is in good agreement with the [Mⁿ⁺]-dependent evolution of conformational populations and existing structural information in the literature. Specifically, this folding pathway explains the metal-ion dependent formation of a functional TLS domain from unfolded RNAs via two consecutive steps: 1) hybridization of a long-range stem interaction, followed by 2) formation of a 3′-terminal pseudoknot. These two conformational transitions are well described by stepwise dissociation constants for [Mg²⁺] (K₁ = 328 ± 30 μM and K₂ = 1092 ± 183 μM) and [Na⁺] (K₁ = 74 ± 6 mM and K₂ = 243 ± 52 mM)-induced folding. The proposed thermodynamic model is further supported by inhibition studies of the long-range stem interaction using a complementary DNA oligomer, which effectively shifts the dynamic equilibrium toward the unfolded conformation. Implications of this multistep conformational folding mechanism are discussed with regard to regulation of virus replication.     

The Conformational Dynamics of Cas9 Governing DNA Cleavage Are Revealed by Single-Molecule FRET

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Action of the Chaperonin GroEL/ES on a Non-native Substrate Observed with Single-Molecule FRET

The double ring-shaped chaperonin GroEL binds a wide range of non-native polypeptides within its central cavity and, together with its cofactor GroES, assists their folding in an ATP-dependent manner. The conformational cycle of GroEL/ES has been studied extensively but little is known about how the environment in the central cavity affects substrate conformation. Here, we use the von Hippel-Lindau tumor suppressor protein VHL as a model substrate for studying the action of the GroEL/ES system on a bound polypeptide. Fluorescent labeling of pairs of sites on VHL for fluorescence (Förster) resonant energy transfer (FRET) allows VHL to be used to explore how GroEL binding and GroEL/ES/nucleotide binding affect the substrate conformation. On average, upon binding to GroEL, all pairs of labeling sites experience compaction relative to the unfolded protein while single-molecule FRET distributions show significant heterogeneity. Upon addition of GroES and ATP to close the GroEL cavity, on average further FRET increases occur between the two hydrophobic regions of VHL, accompanied by FRET decreases between the N- and C-termini. This suggests that ATP- and GroES-induced confinement within the GroEL cavity remodels bound polypeptides by causing expansion (or racking) of some regions and compaction of others, most notably, the hydrophobic core. However, single-molecule observations of the specific FRET changes for individual proteins at the moment of ATP/GroES addition reveal that a large fraction of the population shows the opposite behavior; that is, FRET decreases between the hydrophobic regions and FRET increases for the N- and C-termini. Our time-resolved single-molecule analysis reveals the underlying heterogeneity of the action of GroES/EL on a bound polypeptide substrate, which might arise from the random nature of the specific binding to the various identical subunits of GroEL, and might help explain why multiple rounds of binding and hydrolysis are required for some chaperonin substrates.     

Effect of Phosphorylation on EGFR Dimer Stability Probed by Single-Molecule Dynamics and FRET/FLIM

Deregulation of epidermal growth factor receptor (EGFR) signaling has been correlated with the development of a variety of human carcinomas. EGF-induced receptor dimerization and consequent trans- auto-phosphorylation are among the earliest events in signal transduction. Binding of EGF is thought to induce a conformational change that consequently unfolds an ectodomain loop required for dimerization indirectly. It may also induce important allosteric changes in the cytoplasmic domain. Despite extensive knowledge on the physiological activation of EGFR, the effect of targeted therapies on receptor conformation is not known and this particular aspect of receptor function, which can potentially be influenced by drug treatment, may in part explain the heterogeneous clinical response among cancer patients. Here, we used Förster resonance energy transfer/fluorescence lifetime imaging microscopy (FRET/FLIM) combined with two-color single-molecule tracking to study the effect of ATP-competitive small molecule tyrosine kinase inhibitors (TKIs) and phosphatase-based manipulation of EGFR phosphorylation on live cells. The distribution of dimer on-times was fitted to a monoexponential to extract dimer off-rates (k_off). Our data show that pretreatment with gefitinib (active conformation binder) stabilizes the EGFR ligand-bound homodimer. Overexpression of EGFR-specific DEP-1 phosphatase was also found to have a stabilizing effect on the homodimer. No significant difference in the k_off of the dimer could be detected when an anti-EGFR antibody (425 Snap single-chain variable fragment) that allows for dimerization of ligand-bound receptors, but not phosphorylation, was used. These results suggest that both the conformation of the extracellular domain and phosphorylation status of the receptor are involved in modulating the stability of the dimer. The relative fractions of these two EGFR subpopulations (interacting versus free) were obtained by a fractional-intensity analysis of ensemble FRET/FLIM images. Our combined imaging approach showed that both the fraction and affinity (surrogate of conformation at a single-molecule level) increased after gefitinib pretreatment or DEP-1 phosphatase overexpression. Using an EGFR mutation (I706Q, V948R) that perturbs the ability of EGFR to dimerize intracellularly, we showed that a modest drug-induced increase in the fraction/stability of the EGFR homodimer may have a significant biological impact on the tumor cell’s proliferation potential.     

Characterization of Photoactivated Singlet Oxygen Damage in Single-Molecule Optical Trap Experiments

Optical traps or “tweezers” use high-power, near-infrared laser beams to manipulate and apply forces to biological systems, ranging from individual molecules to cells. Although previous studies have established that optical tweezers induce photodamage in live cells, the effects of trap irradiation have yet to be examined in vitro, at the single-molecule level. In this study, we investigate trap-induced damage in a simple system consisting of DNA molecules tethered between optically trapped polystyrene microspheres. We show that exposure to the trapping light affects the lifetime of the tethers, the efficiency with which they can be formed, and their structure. Moreover, we establish that these irreversible effects are caused by oxidative damage from singlet oxygen. This reactive state of molecular oxygen is generated locally by the optical traps in the presence of a sensitizer, which we identify as the trapped polystyrene microspheres. Trap-induced oxidative damage can be reduced greatly by working under anaerobic conditions, using additives that quench singlet oxygen, or trapping microspheres lacking the sensitizers necessary for singlet state photoexcitation. Our findings are relevant to a broad range of trap-based single-molecule experiments—the most common biological application of optical tweezers—and may guide the development of more robust experimental protocols.     

Implementation of Transportation Distance for Analyzing FLIM and FRET Experiments

Analysis of fluorescence lifetime imaging microscopy (FLIM) and Förster resonance energy transfer (FRET) experiments in living cells is usually based on mean lifetimes computations. However, these mean lifetimes can induce misinterpretations. We propose in this work the implementation of the transportation distance for FLIM and FRET experiments in vivo. This non-fitting indicator, which is easy to compute, reflects the similarity between two distributions and can be used for pixels clustering to improve the estimation of the FRET parameters. We study the robustness and the discriminating power of this transportation distance, both theoretically and numerically. In addition, a comparison study with the largely used mean lifetime differences is performed. We finally demonstrate practically the benefits of the transportation distance over the usual mean lifetime differences for both FLIM and FRET experiments in living cells.     

Empirical Bayes analysis of single nucleotide polymorphisms

Background  

An important goal of whole-genome studies concerned with single nucleotide polymorphisms (SNPs) is the identification of SNPs associated with a covariate of interest such as the case-control status or the type of cancer. Since these studies often comprise the genotypes of hundreds of thousands of SNPs, methods are required that can cope with the corresponding multiple testing problem. For the analysis of gene expression data, approaches such as the empirical Bayes analysis of microarrays have been developed particularly for the detection of genes associated with the response. However, the empirical Bayes analysis of microarrays has only been suggested for binary responses when considering expression values, i.e. continuous predictors.