Decoding the pattern of photon colors in single-molecule FRET

Single-molecule fluorescence probes dynamics of barrier crossing

Kramers developed the theory on how chemical reaction rates are influenced by the viscosity of the medium. At the viscosity of water, the kinetics of unimolecular reactions are described by diffusion of a Brownian particle over a free-energy barrier separating reactants and products. For reactions in solution this famous theory extended Eyring's transition state theory, and is widely applied in physics, chemistry and biology, including to reactions as complex as protein folding. Because the diffusion coefficient of Kramers' theory is determined by the dynamics in the sparsely populated region of the barrier top, its properties have not been directly measured for any molecular system. Here we show that the Kramers diffusion coefficient and free-energy barrier can be characterized by measuring the temperature- and viscosity-dependence of the transition path time for protein folding. The transition path is the small fraction of an equilibrium trajectory for a single molecule when the free-energy barrier separating two states is actually crossed. Its duration, the transition path time, can now be determined from photon trajectories for single protein molecules undergoing folding/unfolding transitions. Our finding of a long transition path time with an unusually small solvent viscosity dependence suggests that internal friction as well as solvent friction determine the Kramers diffusion coefficient for α-helical proteins, as opposed to a breakdown of his theory, which occurs for many small-molecule reactions. It is noteworthy that the new and fundamental information concerning Kramers' theory and the dynamics of barrier crossings obtained here come from experiments on a protein rather than a much simpler chemical or physical system.

Conformational changes in complex macromolecules studied by single donor-acceptor pair fluorescence

A single donor-acceptor (D-A) pair with a fluctuating FRET rate F is studied theoretically in the frame of a "two-state model" in which the FRET rate F fluctuates taking two values, F1 and F2, with average rates B for the forward and b for the backward transitions. Theoretical expressions are derived for the autocorrelation function g((2))(DA)(τ) and for the Mandel parameter Q(D,A)(τ) allowing for background emission. Fluctuating intensities ID,A(t) and FRET efficiency E(t) = IA(t)/[IA(t) + ID(t)] are calculated with the help of the Monte Carlo technique. The probability w(D,A)(N)(T) of finding N photons in a time interval T, and the distribution of the FRET efficiency P(E) are found by statistical treating of the fluctuating intensities ID,A(t). The shape of the distribution w(D,A)(N)(T) enables one to find the values of the parameters: F1, F2, and b/B. The influence of the background light on g((2))(DA)(τ), Q(D,A)(τ) and w(D,A)(N)(T) is studied. It is shown how the background light influences the ratio b/B found from the analysis of w(D,A)(N)(T) and P(E).

Expectation-maximization of the potential of mean force and diffusion coefficient in Langevin dynamics from single molecule FRET data photon by photon

The dynamics of a protein along a well-defined coordinate can be formally projected onto the form of an overdamped Lagevin equation. Here, we present a comprehensive statistical-learning framework for simultaneously quantifying the deterministic force (the potential of mean force, PMF) and the stochastic force (characterized by the diffusion coefficient, D) from single-molecule Förster-type resonance energy transfer (smFRET) experiments. The likelihood functional of the Langevin parameters, PMF and D, is expressed by a path integral of the latent smFRET distance that follows Langevin dynamics and realized by the donor and the acceptor photon emissions. The solution is made possible by an eigen decomposition of the time-symmetrized form of the corresponding Fokker-Planck equation coupled with photon statistics. To extract the Langevin parameters from photon arrival time data, we advance the expectation-maximization algorithm in statistical learning, originally developed for and mostly used in discrete-state systems, to a general form in the continuous space that allows for a variational calculus on the continuous PMF function. We also introduce the regularization of the solution space in this Bayesian inference based on a maximum trajectory-entropy principle. We use a highly nontrivial example with realistically simulated smFRET data to illustrate the application of this new method.

Reconstruction of Calmodulin Single-Molecule FRET States, Dye-Interactions, and CaMKII Peptide Binding by MultiNest and Classic Maximum Entropy

We analyze single molecule FRET burst measurements using Bayesian nested sampling. The MultiNest algorithm produces accurate FRET efficiency distributions from single-molecule data. FRET efficiency distributions recovered by MultiNest and classic maximum entropy are compared for simulated data and for calmodulin labeled at residues 44 and 117. MultiNest compares favorably with maximum entropy analysis for simulated data, judged by the Bayesian evidence. FRET efficiency distributions recovered for calmodulin labeled with two different FRET dye pairs depended on the dye pair and changed upon Ca2+ binding. We also looked at the FRET efficiency distributions of calmodulin bound to the calcium/calmodulin dependent protein kinase II (CaMKII) binding domain. For both dye pairs, the FRET efficiency distribution collapsed to a single peak in the case of calmodulin bound to the CaMKII peptide. These measurements strongly suggest that consideration of dye-protein interactions is crucial in forming an accurate picture of protein conformations from FRET data.

Measuring ultrafast protein folding rates from photon-by-photon analysis of single molecule fluorescence trajectories

Folding and unfolding rates for the ultrafast folding villin subdomain were determined from a photon-by-photon analysis of fluorescence trajectories in single molecule FRET experiments. One of the obstacles to measuring fast kinetics in single molecule fluorescence experiments is blinking of the fluorophores on a timescale that is not well separated from the process of interest. By incorporating acceptor blinking into a two-state kinetics model, we show that it is possible to extract accurate rate coefficients on the microsecond time scale for folding and unfolding using the maximum likelihood method of I.V. Gopich and A. Szabo. This method yields the most likely parameters of a given model that can reproduce the observed photon trajectories. The extracted parameters agree with both the decay rate of the donor-acceptor cross correlation function and the results of ensemble equilibrium and kinetic experiments using nanosecond laser temperature jump.

Analyzing conformational dynamics of single P-glycoprotein transporters by Förster resonance energy transfer using hidden Markov models

Single-molecule Förster resonance energy (smFRET) transfer has become a powerful tool for observing conformational dynamics of biological macromolecules. Analyzing smFRET time trajectories allows to identify the state transitions occuring on reaction pathways of molecular machines. Previously, we have developed a smFRET approach to monitor movements of the two nucleotide binding domains (NBDs) of P-glycoprotein (Pgp) during ATP hydrolysis driven drug transport in solution. One limitation of this initial work was that single-molecule photon bursts were analyzed by visual inspection with manual assignment of individual FRET levels. Here a fully automated analysis of Pgp smFRET data using hidden Markov models (HMM) for transitions up to 9 conformational states is applied. We propose new estimators for HMMs to integrate the information of fluctuating intensities in confocal smFRET measurements of freely diffusing lipid bilayer bound membrane proteins in solution. HMM analysis strongly supports that under conditions of steady state turnover, conformational states with short NBD distances and short dwell times are more populated compared to conditions without nucleotide or transport substrate present.

Intramolecular distances and dynamics from the combined photon statistics of single-molecule FRET and photoinduced electron transfer

Single-molecule Förster resonance energy transfer (FRET) and photoinduced electron transfer (PET) have developed into versatile and complementary methods for probing distances and dynamics in biomolecules. Here we show that the two methods can be combined in one molecule to obtain both accurate distance information and the kinetics of intramolecular contact formation. In a first step, we show that the fluorescent dyes Alexa 488 and Alexa 594, which are frequently used as a donor and acceptor for single-molecule FRET, are also suitable as PET probes with tryptophan as a fluorescence quencher. We then performed combined FRET/PET experiments with FRET donor- and acceptor-labeled polyproline peptides. The placement of a tryptophan residue into the polyglycylserine tail incorporated in the peptides allowed us to measure both FRET efficiencies and the nanosecond dynamics of contact formation between one of the fluorescent dyes and the quencher. Variation of the linker length between the polyproline and the Alexa dyes and in the position of the tryptophan residue demonstrates the sensitivity of this approach. Modeling of the combined photon statistics underlying the combined FRET and PET process enables the accurate analysis of both the resulting transfer efficiency histograms and the nanosecond fluorescence correlation functions. This approach opens up new possibilities for investigating single biomolecules with high spatial and temporal resolution.

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.

Single-molecule nanometry for biological physics

Precision measurement is a hallmark of physics but the small length scale (∼nanometer) of elementary biological components and thermal fluctuations surrounding them challenge our ability to visualize their action. Here, we highlight the recent developments in single-molecule nanometry where the position of a single fluorescent molecule can be determined with nanometer precision, reaching the limit imposed by the shot noise, and the relative motion between two molecules can be determined with ∼0.3 nm precision at ∼1 ms time resolution, as well as how these new tools are providing fundamental insights into how motor proteins move on cellular highways. We will also discuss how interactions between three and four fluorescent molecules can be used to measure three and six coordinates, respectively, allowing us to correlate the movements of multiple components. Finally, we will discuss recent progress in combining angstrom-precision optical tweezers with single-molecule fluorescent detection, opening new windows for multi-dimensional single-molecule nanometry for biological physics.

Analyzing Förster resonance energy transfer with fluctuation algorithms

Fluorescence correlation spectroscopy (FCS) in combination with Förster resonance energy transfer (FRET) has been developed to a powerful statistical tool, which allows for the analysis of FRET fluctuations in the huge time of nanoseconds to seconds. FRET-FCS utilizes the strong distance dependence of the FRET efficiency on the donor (D)-acceptor (A) distance so that it developed to a perfect method for studying structural fluctuation in biomolecules involved in conformational flexibility, structural dynamics, complex formation, folding, and catalysis. Structural fluctuations thereby result in anticorrelated donor and acceptor signals, which are analyzed by FRET-FCS in order to characterize underlying structural dynamics. Simulated and experimental examples are discussed. First, we review experimental implementations of FRET-FCS and present theory for a two-state interconverting system. Additionally, we consider a very common case of FRET dynamics in the presence of donor-only labeled species. We demonstrate that the mean relaxation time for the structural dynamics can be easily obtained in most of cases, whereas extracting meaningful information from correlation amplitudes can be challenging. We present a strategy to avoid a fit with an underdetermined model function by restraining the D and A brightnesses of the at least one involved state, so that both FRET efficiencies and both rate constants (i.e., the equilibrium constant) can be determined. For samples containing several fluorescent species, the use of pulsed polarized excitation with multiparameter fluorescence detection allows for filtered FCS (fFCS), where species-specific correlation functions can be obtained, which can be directly interpreted. The species selection is achieved by filtering using fluorescence decays of individual species. Analytical functions for species auto- and cross-correlation functions are given. Moreover, fFCS is less affected by photophysical artifacts and often offers higher contrast, which effectively increases its time resolution and significantly enhances its capability to resolve multistate kinetics. fFCS can also differentiate between species even when their brightnesses are the same and thus opens up new possibilities to characterize complex dynamics. Alternative fluctuation algorithms to study FRET dynamics are also briefly reviewed.

Advances in quantitative FRET-based methods for studying nucleic acids

Förster resonance energy transfer (FRET) is a powerful tool for monitoring molecular distances and interactions at the nanoscale level. The strong dependence of transfer efficiency on probe separation makes FRET perfectly suited for "on/off" experiments. To use FRET to obtain quantitative distances and three-dimensional structures, however, is more challenging. This review summarises recent studies and technological advances that have improved FRET as a quantitative molecular ruler in nucleic acid systems, both at the ensemble and at the single-molecule levels.

Identification of intensity ratio break points from photon arrival trajectories in ratiometric single molecule spectroscopy

We describe a statistical method to analyze dual-channel photon arrival trajectories from single molecule spectroscopy model-free to identify break points in the intensity ratio. Photons are binned with a short bin size to calculate the logarithm of the intensity ratio for each bin. Stochastic photon counting noise leads to a near-normal distribution of this logarithm and the standard student t-test is used to find statistically significant changes in this quantity. In stochastic simulations we determine the significance threshold for the t-test's p-value at a given level of confidence. We test the method's sensitivity and accuracy indicating that the analysis reliably locates break points with significant changes in the intensity ratio with little or no error in realistic trajectories with large numbers of small change points, while still identifying a large fraction of the frequent break points with small intensity changes. Based on these results we present an approach to estimate confidence intervals for the identified break point locations and recommend a bin size to choose for the analysis. The method proves powerful and reliable in the analysis of simulated and actual data of single molecule reorientation in a glassy matrix.

Blinking fluorescence of single donor-acceptor pairs: important role of "dark'' states in resonance energy transfer via singlet levels

The influence of triplet levels on Förster resonance energy transfer via singlet levels in donor-acceptor (D-A) pairs is studied. Four types of D-A pair are considered: (i) two-level donor and two-level acceptor, (ii) three-level donor and two-level acceptor, (iii) two-level donor and three-level acceptor, and (iv) three-level donor and three-level acceptor. If singlet-triplet transitions in a three-level acceptor molecule are ineffective, the energy transfer efficiency E=I_{A}/(I_{A}+I_{D}), where I_{D} and I_{A} are the average intensities of donor and acceptor fluorescence, can be described by the simple theoretical equation E(F)=FT_{D}/(1+FT_{D}). Here F is the rate of energy transfer, and T_{D} is the donor fluorescence lifetime. In accordance with the last equation, 100% of the donor electronic energy can be transferred to an acceptor molecule at FT_{D}≫1. However, if singlet-triplet transitions in a three-level acceptor molecule are effective, the energy transfer efficiency is described by another theoretical equation, E(F)=F[over ¯](F)T_{D}/[1+F[over ¯](F)T_{D}]. Here F[over ¯](F) is a function of F depending on singlet-triplet transitions in both donor and acceptor molecules. Expressions for the functions F[over ¯](F) are derived. In this case the energy transfer efficiency will be far from 100% even at FT_{D}≫1. The character of the intensity fluctuations of donor and acceptor fluorescence indicates which of the two equations for E(F) should be used to find the value of the rate F. Therefore, random time instants of photon emission in both donor and acceptor fluorescence are calculated by the Monte Carlo method for all four types of D-A pair. Theoretical expressions for start-stop correlators (waiting time distributions) in donor and acceptor fluorescence are derived. The probabilities w_{N}^{D}(t) and w_{N}^{A}(t) of finding N photons of donor and acceptor fluorescence in the time interval t are calculated for various values of the energy transfer rate F and for all four types of D-A pair. Comparison of the calculated D and A fluorescence trajectories with those measured by Weiss and co-workers proves the important role of triplet levels in energy transfer via singlet levels.

Theory of the energy transfer efficiency and fluorescence lifetime distribution in single-molecule FRET

In single-molecule FRET experiments with pulsed lasers, not only the colors of the photons but also the fluorescence lifetimes can be monitored. Although these quantities appear to be random, they are modulated by conformational dynamics. In order to extract information about such dynamics, we develop the theory of the joint distribution of FRET efficiencies and fluorescence lifetimes determined from bins (or bursts) of photons. Our starting point is a rigorous formal expression for the distribution of the numbers of donor and acceptor photons and donor lifetimes in a bin that treats the influence of conformational dynamics on all timescales. This formula leads to an analytic result for a two-state system interconverting on a timescale slower than the interphoton time and to an efficient simulation algorithm for multistate dynamics. The shape of the joint distribution contains more information about conformational dynamics than the FRET efficiency histogram alone. In favorable cases, the connectivity of the underlying conformational states can be determined directly by simple inspection of the projection of the joint distribution on the efficiency-lifetime plane.

Classic maximum entropy recovery of the average joint distribution of apparent FRET efficiency and fluorescence photons for single-molecule burst measurements

We describe a method for analysis of single-molecule Förster resonance energy transfer (FRET) burst measurements using classic maximum entropy. Classic maximum entropy determines the Bayesian inference for the joint probability describing the total fluorescence photons and the apparent FRET efficiency. The method was tested with simulated data and then with DNA labeled with fluorescent dyes. The most probable joint distribution can be marginalized to obtain both the overall distribution of fluorescence photons and the apparent FRET efficiency distribution. This method proves to be ideal for determining the distance distribution of FRET-labeled biomolecules, and it successfully predicts the shape of the recovered distributions.

Disentangling subpopulations in single-molecule FRET and ALEX experiments with photon distribution analysis

Among the advantages of the single-molecule approach when used to study biomolecular structural dynamics and interaction is its ability to distinguish between and independently observe minor subpopulations. In a single-molecule Förster resonance energy transfer (FRET) and alternating laser excitation diffusion experiment, the various populations are apparent in the resultant histograms. However, because histograms are calculated based on the per-burst mean FRET and stoichiometry ratio and not on the internal photon distribution, much of the acquired information is lost, thereby reducing the capabilities of the method. Here we suggest what to our knowledge is a novel statistical analysis tool that significantly enhances these capabilities, and we use it to identify and isolate static and dynamic subpopulations. Based on a kernel density estimator and a proper photon distribution analysis, for each individual burst, we calculate scores that reflect properties of interest. Specifically, we determine the FRET efficiency and brightness ratio distributions and use them to reveal 1), the underlying structure of a two-state DNA-hairpin and a DNA hairpin that is bound to DNA origami; 2), a minor doubly labeled dsDNA subpopulation concealed in a larger singly labeled dsDNA; and 3), functioning DNA origami motors concealed within a larger subpopulation of defective motors. Altogether, these findings demonstrate the usefulness of the proposed approach. The method was developed and tested using simulations, its rationality is described, and a computer algorithm is provided.

Likelihood functions for the analysis of single-molecule binned photon sequences

We consider the analysis of a class of experiments in which the number of photons in consecutive time intervals is recorded. Sequence of photon counts or, alternatively, of FRET efficiencies can be studied using likelihood-based methods. For a kinetic model of the conformational dynamics and state-dependent Poisson photon statistics, the formalism to calculate the exact likelihood that this model describes such sequences of photons or FRET efficiencies is developed. Explicit analytic expressions for the likelihood function for a two-state kinetic model are provided. The important special case when conformational dynamics are so slow that at most a single transition occurs in a time bin is considered. By making a series of approximations, we eventually recover the likelihood function used in hidden Markov models. In this way, not only is insight gained into the range of validity of this procedure, but also an improved likelihood function can be obtained.

Single-molecule fluorescence experiments determine protein folding transition path times

The transition path is the tiny fraction of an equilibrium molecular trajectory when a transition occurs as the free-energy barrier between two states is crossed. It is a single-molecule property that contains all the mechanistic information on how a process occurs. As a step toward observing transition paths in protein folding, we determined the average transition-path time for a fast- and a slow-folding protein from a photon-by-photon analysis of fluorescence trajectories in single-molecule Förster resonance energy transfer experiments. Whereas the folding rate coefficients differ by a factor of 10,000, the transition-path times differ by a factor of less than 5, which shows that a fast- and a slow-folding protein take almost the same time to fold when folding actually happens. A very simple model based on energy landscape theory can explain this result.

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.

Diffusion models of protein folding

In theory and in the analysis of experiments, protein folding is often described as diffusion along a single coordinate. We explore here the application of a one-dimensional diffusion model to interpret simulations of protein folding, where the parameters of a model that "best" describes the simulation trajectories are determined using a Bayesian analysis. We discuss the requirements for such a model to be a good approximation to the global dynamics, and several methods for testing its accuracy. For example, one test considers the effect of an added bias potential on the fitted free energies and diffusion coefficients. Such a bias may also be used to extend our approach to determining parameters for the model to systems that would not normally explore the full coordinate range on accessible time scales. Alternatively, the propagators predicted from the model at different "lag" times may be compared with observations from simulation. We then present some applications of the model to protein folding, including Kramers-like turnover in folding rates of coarse-grained models, the effect of non-native interactions on folding, and the effect of the chosen coordinate on the observed position-dependence of the diffusion coefficients. Lastly, we consider how our results are useful for the interpretation of experiments, and how this type of Bayesian analysis may eventually be applied directly to analyse experimental data.

A distribution-based method to resolve single-molecule Förster resonance energy transfer observations

We introduce a new approach to analyze single-molecule Förster resonance energy transfer (FRET) data. The method recognizes that FRET efficiencies assumed by traditional ensemble methods are unobservable for single molecules. We propose instead a method to predict distributions of FRET parameters obtained directly from the data. Distributions of FRET rates, given the data, are precisely defined using Bayesian methods and increase the information derived from the data. Benchmark comparisons find that the response time of the new method outperforms traditional methods of averaging. Our approach makes no assumption about the number or distribution of underlying FRET states. The new method also yields information about joint parameter distributions going beyond the standard framework of FRET analysis. For example, the running distribution of FRET means contains more information than any conceivable single measure of FRET efficiency. The method is tested against simulated data and then applied to a pilot-study sample of calmodulin molecules immobilized in lipid vesicles, revealing evidence for multiple dynamical states.

Structural heterogeneity and quantitative FRET efficiency distributions of polyprolines through a hybrid atomistic simulation and Monte Carlo approach

Förster Resonance Energy Transfer (FRET) experiments probe molecular distances via distance dependent energy transfer from an excited donor dye to an acceptor dye. Single molecule experiments not only probe average distances, but also distance distributions or even fluctuations, and thus provide a powerful tool to study biomolecular structure and dynamics. However, the measured energy transfer efficiency depends not only on the distance between the dyes, but also on their mutual orientation, which is typically inaccessible to experiments. Thus, assumptions on the orientation distributions and averages are usually made, limiting the accuracy of the distance distributions extracted from FRET experiments. Here, we demonstrate that by combining single molecule FRET experiments with the mutual dye orientation statistics obtained from Molecular Dynamics (MD) simulations, improved estimates of distances and distributions are obtained. From the simulated time-dependent mutual orientations, FRET efficiencies are calculated and the full statistics of individual photon absorption, energy transfer, and photon emission events is obtained from subsequent Monte Carlo (MC) simulations of the FRET kinetics. All recorded emission events are collected to bursts from which efficiency distributions are calculated in close resemblance to the actual FRET experiment, taking shot noise fully into account. Using polyproline chains with attached Alexa 488 and Alexa 594 dyes as a test system, we demonstrate the feasibility of this approach by direct comparison to experimental data. We identified cis-isomers and different static local environments as sources of the experimentally observed heterogeneity. Reconstructions of distance distributions from experimental data at different levels of theory demonstrate how the respective underlying assumptions and approximations affect the obtained accuracy. Our results show that dye fluctuations obtained from MD simulations, combined with MC single photon kinetics, provide a versatile tool to improve the accuracy of distance distributions that can be extracted from measured single molecule FRET efficiencies.

Maximum likelihood-based analysis of single-molecule photon arrival trajectories

In this work we explore the statistical properties of the maximum likelihood-based analysis of one-color photon arrival trajectories. This approach does not involve binning and, therefore, all of the information contained in an observed photon strajectory is used. We study the accuracy and precision of parameter estimates and the efficiency of the Akaike information criterion and the Bayesian information criterion (BIC) in selecting the true kinetic model. We focus on the low excitation regime where photon trajectories can be modeled as realizations of Markov modulated Poisson processes. The number of observed photons is the key parameter in determining model selection and parameter estimation. For example, the BIC can select the true three-state model from competing two-, three-, and four-state kinetic models even for relatively short trajectories made up of 2 × 10(3) photons. When the intensity levels are well-separated and 10(4) photons are observed, the two-state model parameters can be estimated with about 10% precision and those for a three-state model with about 20% precision.

Changepoint analysis for single-molecule polarized total internal reflection fluorescence microscopy experiments

The experimental study of individual macromolecules has opened a door to determining the details of their mechanochemical operation. Motor enzymes such as the myosin family have been particularly attractive targets for such study, in part because some of them are highly processive and their "product" is spatial motion. But single-molecule resolution comes with its own costs and limitations. Often, the observations rest on single fluorescent dye molecules, which emit a limited number of photons before photobleaching and are subject to complex internal dynamics. Thus, it is important to develop methods that extract the maximum useful information from a finite set of detected photons. We have extended an experimental technique, multiple polarization illumination in total internal reflection fluorescence microscopy (polTIRF), to record the arrival time and polarization state of each individual detected photon. We also extended an analysis technique, previously applied to FRET experiments, that optimally determines times of changes in photon emission rates. Combining these improvements allows us to identify the structural dynamics of a molecular motor (myosin V) with unprecedented detail and temporal resolution.

Protein folding at single-molecule resolution

The protein folding reaction carries great significance for cellular function and hence continues to be the research focus of a large interdisciplinary protein science community. Single-molecule methods are providing new and powerful tools for dissecting the mechanisms of this complex process by virtue of their ability to provide views of protein structure and dynamics without associated ensemble averaging. This review briefly introduces common FRET and force methods, and then explores several areas of protein folding where single-molecule experiments have yielded insights. These include exciting new information about folding landscapes, dynamics, intermediates, unfolded ensembles, intrinsically disordered proteins, assisted folding and biomechanical unfolding. Emerging and future work is expected to include advances in single-molecule techniques aimed at such investigations, and increasing work on more complex systems from both the physics and biology standpoints, including folding and dynamics of systems of interacting proteins and of proteins in cells and organisms. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.

Quantifying heterogeneity and conformational dynamics from single molecule FRET of diffusing molecules: recurrence analysis of single particles (RASP)

Single molecule Förster resonance energy transfer (FRET) experiments are a versatile method for investigating the conformational distributions and dynamics of biological macromolecules. In a common type of experiment, the fluorescence bursts from individual molecules freely diffusing in solution are detected as they pass through the observation volume of a confocal microscope. Correlation analysis of the fluorescence bursts shows that under typical experimental conditions, for time scales up to several tens of milliseconds, the probability for a molecule to return to the confocal volume is greater than the probability of a new molecule being detected. Here we present RASP (recurrence analysis of single particles), a method that is based on this recurrence behavior and allows us to significantly extend the information that can be extracted from single molecule FRET experiments. The number and peak shapes of subpopulations within the sample can be identified essentially in a model-free way by constructing recurrence FRET efficiency histograms. These are obtained by first selecting photon bursts from a small transfer efficiency range (initial bursts), and then building the FRET efficiency histogram only from bursts detected within a short time (the recurrence interval) after the initial bursts. Systematic variation of the recurrence interval allows the kinetics of interconversion between subpopulations to be determined on time scales from ~50 μs up to ~100 ms from equilibrium measurements. We demonstrate the applicability of the method on measurements of several peptides and proteins with different degrees of conformational heterogeneity and folding dynamics. The concepts presented here can be extended to other observables available from single molecule fluorescence experiments.

Improved resolution of complex single-molecule FRET systems via wavelet shrinkage

The resolution of complex interactions found in single-molecule fluorescence resonance energy transfer (smFRET) experiments is hindered by noise. Wavelet shrinkage is proven to reduce noise, but traditional methods introduce artifacts when acting on discontinuous signals, such as those acquired in smFRET experiments. Modifications to the basic method that are specific to smFRET are developed and tested on simulated systems. Use of the Haar wavelet basis produces the most optimally denoised estimates. We also assess various thresholding methods, develop a time-localized noise estimator, and implement a translation-invariant wavelet transformation to reduce artifacts associated with discontinuities and inadequate distinction of noise. The time-local estimator enhances noise reduction by 5-20%, and translation-invariant transformation nearly eliminates the aforementioned artifacts. Kinetic parameters extracted from denoised estimates are accurate to within 5% of the simulated values. Overall, the improved resolution results in the complete and accurate characterization of both simple and complex smFRET systems.

Deconvolution of dynamic mechanical networks

Time-resolved single-molecule biophysical experiments yield data that contain a wealth of dynamic information, in addition to the equilibrium distributions derived from histograms of the time series. In typical force spectroscopic setups the molecule is connected via linkers to a readout device, forming a mechanically coupled dynamic network. Deconvolution of equilibrium distributions, filtering out the influence of the linkers, is a straightforward and common practice. We have developed an analogous dynamic deconvolution theory for the more challenging task of extracting kinetic properties of individual components in networks of arbitrary complexity and topology. Our method determines the intrinsic linear response functions of a given object in the network, describing the power spectrum of conformational fluctuations. The practicality of our approach is demonstrated for the particular case of a protein linked via DNA handles to two optically trapped beads at constant stretching force, which we mimic through Brownian dynamics simulations. Each well in the protein free energy landscape (corresponding to folded, unfolded, or possibly intermediate states) will have its own characteristic equilibrium fluctuations. The associated linear response function is rich in physical content, because it depends both on the shape of the well and its diffusivity-a measure of the internal friction arising from such processes as the transient breaking and reformation of bonds in the protein structure. Starting from the autocorrelation functions of the equilibrium bead fluctuations measured in this force clamp setup, we show how an experimentalist can accurately extract the state-dependent protein diffusivity using a straightforward two-step procedure.

FRET efficiency distributions of multistate single molecules

A simple analytic theory is developed to describe FRET efficiency histograms constructed from a photon trajectory generated by a molecule with multiple conformational states. The histograms are approximated by a sum of Gaussians with the parameters explicitly determined by the FRET efficiencies of the states and the rates of the transitions between the states. The theory, which has been tested against exact histograms for two conformational states and simulated data for three and four conformational states, accurately describes how the peaks in the histograms collapse as the bin time or the transition rates increase.

Bayesian Nonparametric Hidden Markov Models with application to the analysis of copy-number-variation in mammalian genomes

We consider the development of Bayesian Nonparametric methods for product partition models such as Hidden Markov Models and change point models. Our approach uses a Mixture of Dirichlet Process (MDP) model for the unknown sampling distribution (likelihood) for the observations arising in each state and a computationally efficient data augmentation scheme to aid inference. The method uses novel MCMC methodology which combines recent retrospective sampling methods with the use of slice sampler variables. The methodology is computationally efficient, both in terms of MCMC mixing properties, and robustness to the length of the time series being investigated. Moreover, the method is easy to implement requiring little or no user-interaction. We apply our methodology to the analysis of genomic copy number variation.

Extracting rate coefficients from single-molecule photon trajectories and FRET efficiency histograms for a fast-folding protein

Recently developed statistical methods by Gopich and Szabo were used to extract folding and unfolding rate coefficients from single-molecule Förster resonance energy transfer (FRET) data for proteins with kinetics too fast to measure waiting time distributions. Two types of experiments and two different analyses were performed. In one experiment bursts of photons were collected from donor and acceptor fluorophores attached to a 73-residue protein, α(3)D, freely diffusing through the illuminated volume of a confocal microscope system. In the second, the protein was immobilized by linkage to a surface, and photons were collected until one of the fluorophores bleached. Folding and unfolding rate coefficients and mean FRET efficiencies for the folded and unfolded subpopulations were obtained from a photon by photon analysis of the trajectories using a maximum likelihood method. The ability of the method to describe the data in terms of a two-state model was checked by recoloring the photon trajectories with the extracted parameters and comparing the calculated FRET efficiency histograms with the measured histograms. The sum of the rate coefficients for the two-state model agreed to within 30% with the relaxation rate obtained from the decay of the donor-acceptor cross-correlation function, confirming the high accuracy of the method. Interestingly, apparently reliable rate coefficients could be extracted using the maximum likelihood method, even at low (<10%) population of the minor component where the cross-correlation function was too noisy to obtain any useful information. The rate coefficients and mean FRET efficiencies were also obtained in an approximate procedure by simply fitting the FRET efficiency histograms, calculated by binning the donor and acceptor photons, with a sum of three-Gaussian functions. The kinetics are exposed in these histograms by the growth of a FRET efficiency peak at values intermediate between the folded and unfolded peaks as the bin size increases, a phenomenon with similarities to NMR exchange broadening. When comparable populations of folded and unfolded molecules are present, this method yields rate coefficients in very good agreement with those obtained with the maximum likelihood method. As a first step toward characterizing transition paths, the Viterbi algorithm was used to locate the most probable transition points in the photon trajectories.

Single-molecule Förster resonance energy transfer studies of RNA structure, dynamics and function

Single-molecule fluorescence microscopy experiments on RNA molecules brought to light the highly complex dynamics of key biological processes, including RNA folding, catalysis of ribozymes, ligand sensing of riboswitches and aptamers, and protein synthesis in the ribosome. By using highly advanced biophysical spectroscopy techniques in combination with sophisticated biochemical synthesis approaches, molecular dynamics of individual RNA molecules can be observed in real time and under physiological conditions in unprecedented detail that cannot be achieved with bulk experiments. Here, we review recent advances in RNA folding and functional studies of RNA and RNA-protein complexes addressed by using single-molecule Förster (fluorescence) resonance energy transfer (smFRET) technique.