Theory of Single-Molecule FRET Efficiency Histograms. In: Komatsuzaki T, Kawakami M, Takahashi S, Yang H, Silbey RJ, editors. Single-Molecule Biophysics. Hoboken: John Wiley & Sons, Inc.; 2011. pp. 245-97. [DOI: 10.1002/9781118131374.ch10]

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

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

Unraveling Burst Selection Bias in Single-Molecule FRET of Species with Unequal Brightness and Diffusivity

Single-molecule free diffusion experiments enable accurate quantification of coexisting species or states. However, unequal brightness and diffusivity introduce a burst selection bias and affect the interpretation of experimental results. We address this issue with a photon-by-photon maximum likelihood method, burstML, which explicitly considers burst selection criteria. BurstML accurately estimates parameters, including photon count rates, diffusion times, Förster resonance energy transfer (FRET) efficiencies, and population, even in cases where species are poorly distinguished in FRET efficiency histograms. We develop a quantitative theory that determines the fraction of photon bursts corresponding to each species and thus obtain accurate species populations from the measured burst fractions. In addition, we provide a simple approximate formula for burst fractions and establish the range of parameters where unequal brightness and diffusivity can significantly affect the results obtained by conventional methods. The performance of the burstML method is compared with that of a maximum likelihood method that assumes equal species brightness and diffusivity, as well as standard Gaussian fitting of FRET efficiency histograms, using both simulated and real single-molecule data for cold-shock protein, protein L, and protein G. The burstML method enhances the accuracy of parameter estimation in single-molecule fluorescence studies.

Three-Color Single-Molecule FRET and Fluorescence Lifetime Analysis of Fast Protein Folding

We describe the theory, experiment, and analysis of three-color Förster resonance energy transfer (FRET) spectroscopy for probing conformational dynamics of a fast-folding protein, α₃D. In three-color FRET, site-specific labeling of fluorophores is required to avoid ambiguity resulting from various species with different combinations of labeling positions. To this end, we first attached two dyes to a cysteine residue and an unnatural amino acid and then appended a cysteine residue to the C-terminus of the protein by the sortase-mediated ligation for attaching the third dye. To determine all three FRET efficiencies, we used alternating excitation of the donor and acceptor 1 with two picosecond-pulsed lasers. Since the folded and unfolded states are not distinguishable in binned fluorescence trajectories due to fast-folding on a millisecond time scale, we used a maximum likelihood method that analyzes photon trajectories without binning the data. The extracted kinetic parameters agree very well with the previously measured parameters for the same protein with two-color FRET, suggesting that the addition of the third fluorophore does not affect the folding dynamics of the protein. From the extracted fractions of acceptor photon counts, the FRET efficiencies for all three dye pairs were calculated after various corrections. They were compared with the FRET efficiencies obtained from the global analysis of two-color segments collected in the same experiment. The FRET efficiencies of the folded state from the three-color segments agree with those from the two-color segments, whereas the three-color and two-color FRET efficiencies of the unfolded state are different. This happens because fluctuations of all three interdye distances contribute to the FRET efficiency measured in three-color FRET. We show that this difference can be accounted for by using the Gaussian chain model for the unfolded state with the parameters obtained from the analysis of two-color segments. This result shows that three-color FRET provides additional information on the flexibility of molecules that cannot be obtained from a combination of two-color FRET experiments with three dye pairs. Using the delay times of photons from the laser pulse, fluorescence lifetimes were determined using the maximum likelihood analysis. The correlation between FRET efficiencies and lifetimes of the donor, acceptor 1, and acceptor 2 was visualized in two-dimensional FRET efficiency-lifetime histograms. These histograms can be used to demonstrate the presence of conformational dynamics in a protein.

Linking time-series of single-molecule experiments with molecular dynamics simulations by machine learning

Single-molecule experiments and molecular dynamics (MD) simulations are indispensable tools for investigating protein conformational dynamics. The former provide time-series data, such as donor-acceptor distances, whereas the latter give atomistic information, although this information is often biased by model parameters. Here, we devise a machine-learning method to combine the complementary information from the two approaches and construct a consistent model of conformational dynamics. It is applied to the folding dynamics of the formin-binding protein WW domain. MD simulations over 400 μs led to an initial Markov state model (MSM), which was then "refined" using single-molecule Förster resonance energy transfer (FRET) data through hidden Markov modeling. The refined or data-assimilated MSM reproduces the FRET data and features hairpin one in the transition-state ensemble, consistent with mutation experiments. The folding pathway in the data-assimilated MSM suggests interplay between hydrophobic contacts and turn formation. Our method provides a general framework for investigating conformational transitions in other proteins.

Direct observation of multiple conformational states in Cytochrome P450 oxidoreductase and their modulation by membrane environment and ionic strength

Cytochrome P450 oxidoreductase (POR) is the primary electron donor in eukaryotic cytochrome P450 (CYP) containing systems. A wealth of ensemble biophysical studies of Cytochrome P450 oxidoreductase (POR) has reported a binary model of the conformational equilibrium directing its catalytic efficiency and biomolecular recognition. In this study, full length POR from the crop plant Sorghum bicolor was site-specifically labeled with Cy3 (donor) and Cy5 (acceptor) fluorophores and reconstituted in nanodiscs. Our single molecule fluorescence resonance energy transfer (smFRET) burst analyses of POR allowed the direct observation and quantification of at least three dominant conformational sub-populations, their distribution and occupancies. Moreover, the state occupancies were remodeled significantly by ionic strength and the nature of reconstitution environment, i.e. phospholipid bilayers (nanodiscs) composed of different lipid head group charges vs. detergent micelles. The existence of conformational heterogeneity in POR may mediate selective activation of multiple downstream electron acceptors and association in complexes in the ER membrane.

Dynamics of the Orientational Factor in Fluorescence Resonance Energy Transfer

A one-dimensional diffusion equation is derived for the time evolution of the orientational factor, κ², in the Förster energy transfer rate. The κ²-dependent diffusion coefficient is obtained in three different ways: (1) by requiring the κ² autocorrelation function, calculated using the κ² diffusion equation, to be single-exponential with the exact characteristic time; (2) by projecting the multidimensional diffusion equation for the transition dipoles onto κ² using the local equilibrium approximation; and (3) by requiring exact and approximate κ² trajectories to be as close as possible using a Bayesian approach. Within the framework of this simple theory, the distance dependence of the fluorescence resonance energy transfer (FRET) efficiency can be calculated for all values of the ratio of the rotational correlation time of the transition dipoles to the lifetime of the donor excited state. The theoretical predictions are compared to the exact values obtained from Brownian dynamics simulations of the reorientation of the donor and acceptor transition dipoles.

Single-molecule spectroscopy of protein conformational dynamics in live eukaryotic cells

Single-molecule methods have become widely used for quantifying the conformational heterogeneity and structural dynamics of biomolecules in vitro. Their application in vivo, however, has remained challenging owing to shortcomings in the design and reproducible delivery of labeled molecules, the range of applicable analysis methods, and suboptimal cell culture conditions. By addressing these limitations in an integrated approach, we demonstrate the feasibility of probing protein dynamics from milliseconds down to the nanosecond regime in live eukaryotic cells with confocal single-molecule Förster resonance energy transfer (FRET) spectroscopy. We illustrate the versatility of the approach by determining the dimensions and submicrosecond chain dynamics of an intrinsically disordered protein; by detecting even subtle changes in the temperature dependence of protein stability, including in-cell cold denaturation; and by quantifying the folding dynamics of a small protein. The methodology opens possibilities for assessing the effect of the cellular environment on biomolecular conformation, dynamics and function.

A Method for Extracting the Free Energy Surface and Conformational Dynamics of Fast-Folding Proteins from Single Molecule Photon Trajectories

Single molecule fluorescence spectroscopy holds the promise of providing direct measurements of protein folding free energy landscapes and conformational motions. However, fulfilling this promise has been prevented by technical limitations, most notably, the difficulty in analyzing the small packets of photons per millisecond that are typically recorded from individual biomolecules. Such limitation impairs the ability to accurately determine conformational distributions and resolve sub-millisecond processes. Here we develop an analytical procedure for extracting the conformational distribution and dynamics of fast-folding proteins directly from time-stamped photon arrival trajectories produced by single molecule FRET experiments. Our procedure combines the maximum likelihood analysis originally developed by Gopich and Szabo with a statistical mechanical model that describes protein folding as diffusion on a one-dimensional free energy surface. Using stochastic kinetic simulations, we thoroughly tested the performance of the method in identifying diverse fast-folding scenarios, ranging from two-state to one-state downhill folding, as a function of relevant experimental variables such as photon count rate, amount of input data, and background noise. The tests demonstrate that the analysis can accurately retrieve the original one-dimensional free energy surface and microsecond folding dynamics in spite of the sub-megahertz photon count rates and significant background noise levels of current single molecule fluorescence experiments. Therefore, our approach provides a powerful tool for the quantitative analysis of single molecule FRET experiments of fast protein folding that is also potentially extensible to the analysis of any other biomolecular process governed by sub-millisecond conformational dynamics.

smFRET studies of the 'encounter' complexes and subsequent intermediate states that regulate the selectivity of ligand binding

The selectivity with which a biomolecule can bind its cognate ligand when confronted by the vast array of structurally similar, competing ligands that are present in the cell underlies the fidelity of some of the most fundamental processes in biology. Because they collectively comprise one of only a few methods that can sensitively detect the 'encounter' complexes and subsequent intermediate states that regulate the selectivity of ligand binding, single-molecule fluorescence, and particularly single-molecule fluorescence resonance energy transfer (smFRET), approaches have revolutionized studies of ligand-binding reactions. Here, we describe a widely used smFRET strategy that enables investigations of a large variety of ligand-binding reactions, and discuss two such reactions, aminoacyl-tRNA selection during translation elongation and splice site selection during spliceosome assembly, that highlight both the successes and challenges of smFRET studies of ligand-binding reactions. We conclude by reviewing a number of emerging experimental and computational approaches that are expanding the capabilities of smFRET approaches for studies of ligand-binding reactions and that promise to reveal the mechanisms that control the selectivity of ligand binding with unprecedented resolution.