Unraveling multi-state molecular dynamics in single-molecule FRET experiments. II. Quantitative analysis of multi-state kinetic networks.

Single-molecule Förster Resonance Energy Transfer (smFRET) experiments are ideally suited to resolve the structural dynamics of biomolecules. A significant challenge to date is capturing and quantifying the exchange between multiple conformational states, mainly when these dynamics occur on the sub-millisecond timescale. Many methods for quantitative analysis are challenged if more than two states are involved, and the appropriate choice of the number of states in the kinetic network is difficult. An additional complication arises if dynamically active molecules coexist with pseudo-static molecules in similar conformational states with undistinguishable Förster Resonance Energy Transfer (FRET) efficiencies. To address these problems, we developed a quantitative integrative analysis framework that combines the information from FRET-lines that relate average fluorescence lifetimes and intensities in two-dimensional burst frequency histograms, fluorescence decays obtained by time-correlated single-photon-counting, photon distribution analysis of the intensities, and fluorescence correlation spectroscopy. Individually, these methodologies provide ambiguous results for the characterization of dynamics in complex kinetic networks. However, the global analysis approach enables accurate determination of the number of states, their kinetic connectivity, the transition rate constants, and species fractions. To challenge the potential of smFRET experiments for studying multi-state kinetic networks, we apply our integrative framework using a set of synthetic data for three-state systems with different kinetic connectivity and exchange rates. Our methodology paves the way toward an integrated analysis of multiparameter smFRET experiments that spans all dimensions of the experimental data. Finally, we propose a workflow for the analysis and show examples that demonstrate the usefulness of this toolkit for dynamic structural biology.

Unraveling multi-state molecular dynamics in single-molecule FRET experiments. I. Theory of FRET-lines.

Conformational dynamics of biomolecules are of fundamental importance for their function. Single-molecule studies of Förster Resonance Energy Transfer (smFRET) between a tethered donor and acceptor dye pair are a powerful tool to investigate the structure and dynamics of labeled molecules. However, capturing and quantifying conformational dynamics in intensity-based smFRET experiments remains challenging when the dynamics occur on the sub-millisecond timescale. The method of multiparameter fluorescence detection addresses this challenge by simultaneously registering fluorescence intensities and lifetimes of the donor and acceptor. Together, two FRET observables, the donor fluorescence lifetime τD and the intensity-based FRET efficiency E, inform on the width of the FRET efficiency distribution as a characteristic fingerprint for conformational dynamics. We present a general framework for analyzing dynamics that relates average fluorescence lifetimes and intensities in two-dimensional burst frequency histograms. We present parametric relations of these observables for interpreting the location of FRET populations in E-τD diagrams, called FRET-lines. To facilitate the analysis of complex exchange equilibria, FRET-lines serve as reference curves for a graphical interpretation of experimental data to (i) identify conformational states, (ii) resolve their dynamic connectivity, (iii) compare different kinetic models, and (iv) infer polymer properties of unfolded or intrinsically disordered proteins. For a simplified graphical analysis of complex kinetic networks, we derive a moment-based representation of the experimental data that decouples the motion of the fluorescence labels from the conformational dynamics of the biomolecule. Importantly, FRET-lines facilitate exploring complex dynamic models via easily computed experimental observables. We provide extensive computational tools to facilitate applying FRET-lines.

Publisher Correction: Precision and accuracy of single-molecule FRET measurements-a multi-laboratory benchmark study.

This paper was originally published under standard Springer Nature copyright. As of the date of this correction, the Analysis is available online as an open-access paper with a CC-BY license. No other part of the paper has been changed.

Precision and accuracy of single-molecule FRET measurements-a multi-laboratory benchmark study.

Single-molecule Förster resonance energy transfer (smFRET) is increasingly being used to determine distances, structures, and dynamics of biomolecules in vitro and in vivo. However, generalized protocols and FRET standards to ensure the reproducibility and accuracy of measurements of FRET efficiencies are currently lacking. Here we report the results of a comparative blind study in which 20 labs determined the FRET efficiencies (E) of several dye-labeled DNA duplexes. Using a unified, straightforward method, we obtained FRET efficiencies with s.d. between ±0.02 and ±0.05. We suggest experimental and computational procedures for converting FRET efficiencies into accurate distances, and discuss potential uncertainties in the experiment and the modeling. Our quantitative assessment of the reproducibility of intensity-based smFRET measurements and a unified correction procedure represents an important step toward the validation of distance networks, with the ultimate aim of achieving reliable structural models of biomolecular systems by smFRET-based hybrid methods.

Combining Graphical and Analytical Methods with Molecular Simulations To Analyze Time-Resolved FRET Measurements of Labeled Macromolecules Accurately.

Förster resonance energy transfer (FRET) measurements from a donor, D, to an acceptor, A, fluorophore are frequently used in vitro and in live cells to reveal information on the structure and dynamics of DA labeled macromolecules. Accurate descriptions of FRET measurements by molecular models are complicated because the fluorophores are usually coupled to the macromolecule via flexible long linkers allowing for diffusional exchange between multiple states with different fluorescence properties caused by distinct environmental quenching, dye mobilities, and variable DA distances. It is often assumed for the analysis of fluorescence intensity decays that DA distances and D quenching are uncorrelated (homogeneous quenching by FRET) and that the exchange between distinct fluorophore states is slow (quasistatic). This allows us to introduce the FRET-induced donor decay, εD(t), a function solely depending on the species fraction distribution of the rate constants of energy transfer by FRET, for a convenient joint analysis of fluorescence decays of FRET and reference samples by integrated graphical and analytical procedures. Additionally, we developed a simulation toolkit to model dye diffusion, fluorescence quenching by the protein surface, and FRET. A benchmark study with simulated fluorescence decays of 500 protein structures demonstrates that the quasistatic homogeneous model works very well and recovers for single conformations the average DA distances with an accuracy of < 2%. For more complex cases, where proteins adopt multiple conformations with significantly different dye environments (heterogeneous case), we introduce a general analysis framework and evaluate its power in resolving heterogeneities in DA distances. The developed fast simulation methods, relying on Brownian dynamics of a coarse-grained dye in its sterically accessible volume, allow us to incorporate structural information in the decay analysis for heterogeneous cases by relating dye states with protein conformations to pave the way for fluorescence and FRET-based dynamic structural biology. Finally, we present theories and simulations to assess the accuracy and precision of steady-state and time-resolved FRET measurements in resolving DA distances on the single-molecule and ensemble level and provide a rigorous framework for estimating approximation, systematic, and statistical errors.

On the analyses of fluorescence depolarisation data in the presence of electronic energy migration. Part II: Applying and evaluating two-photon excited fluorescence.

Electronic energy migration within a bifluorophoric molecule has been studied by time-resolved two-photon excited (TPE) fluorescence depolarisation experiments. Data were analysed by using a recently developed quantitative approach [O. Opanasyuk and L. B.-Å. Johansson, On the Analyses of Fluorescence Depolarisation Data in the Presence of Electronic Energy Migration. Part I: Theory and General Description, Phys. Chem. Chem. Phys., submitted]. The energy migration occurs between the 9-anthrylmethyl groups of the bifluorophoric molecule, bis-(9-anthrylmethylphosphonate) bisteroid. These groups undergo local reorientations, while overall tumbling of the bisteroid is strongly hampered in the used viscous solvent, 1,2-propanediol. To solely obtain information about local reorientations of the 9-anthrylmethyl group, also the mono-(9-anthrylmethylphosphonate) bisteroid was studied, which enabled modelling of the ordering potential shape. The analysis of data is partly performed in the Fourier domain and the best-fit parameters are determined by using an approach based on a Genetic Algorithm. The energy migration process was described by an extended Förster theory (EFT). A reasonable value of the distance between the 9-anthrylmethyl groups, as well as for the mutual orientation of the ordering potentials, is found. Furthermore, values of the two-photon tensor components were obtained.

On the analyses of fluorescence depolarisation data in the presence of electronic energy migration. Part I: Theory and general description.

A new and general procedure is described for a detailed analysis of time-resolved fluorescence depolarisation data in the presence of electronic energy migration. An isotropic ensemble of bifluorophoric molecules (D(1)-R-D(2)) has been studied to demonstrate its utility. Intramolecular donor-donor energy migration occurs between the two donor groups (D), which are covalently connected to a rigid linker group (R). These groups undergo restricted reorientational motions with respect to the R group. The analysis of depolarisation data basically involves the search for best-fit parameters which describe the local reorientational motions, the intermolecular D(1)-D(2) distance, as well as the mutual orientations of the donors. For this, the analysis is partly performed on the Fourier domain and the best-fit parameters are determined by using an approach based on a Genetic Algorithm. The energy migration process has been described by using Monte Carlo simulations and an extended Förster theory (EFT). It is found that the EFT provides the least time-consuming computational method. Since one-photon and two-photon excited fluorescence experiments can be applied for energy migration studies, a general and unified theoretical formulation is given.

Extended Förster theory: a quantitative approach to the determination of inter-chromophore distances in biomacromolecules.

This review highlights recent theoretical and experimental advances in the study of biomacromolecular structure by using electronic transfer. The considered electronic transport in the extended Förster theory occurs within donor-acceptor pairs, donor-donor pairs, as well as within regular arrangements of many donors which may undergo reorienting and translational dynamics. The classical and the extended Förster theory are compared. Applications concern the determination of structural properties of proteins and non-covalent protein polymers. Studies of energy migration by means of two-photon excited fluorescence spectroscopy, as well as the relevant extension of the Förster theory are presented.

Two-photon excited fluorescence depolarisation and electronic energy migration within donor-donor pairs.

A unified theoretical description is presented for one- and two-photon excited fluorescence depolarisation and electronic energy migration within pairs of chromophores. Two weakly coupled donor groups are linked via a rigid macromolecule with the ability to undergo restricted reorienting motions. Describing these reorienting motions as well as their influence on the coupling is rather complex, but can be accounted for by using the extended Förster theory. Here explicit expressions have been derived for chromophores belonging to the point groups D(2h), D(2) and C(2v) when residing in uniaxial potentials (i.e. C(infinity v) symmetry). From the given basic equations, it is possible however, to derive the relevant equations for molecules of arbitrary symmetry in any uniaxial orienting potential. The expected time-resolved fluorescence anisotropy for different two-photon absorption tensors are compared for reorienting fluorophores in liquids, as well as in anisotropic systems. Simulated fluorescence depolarisation data are also displayed that mimic energy migration within pairs of two-photon excited donor molecules, which simultaneously undergo reorienting motions within effectively isotropic and uniaxially anisotropic environments. The obtained results demonstrate that the time-resolved fluorescence anisotropy strongly depends on the properties of the two-photon absorption tensor, as well as on using a linear or a circular polarisation of the excitation field.