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

Reliability and accuracy of single-molecule FRET studies for characterization of structural dynamics and distances in proteins

Single-molecule Förster-resonance energy transfer (smFRET) experiments allow the study of biomolecular structure and dynamics in vitro and in vivo. We performed an international blind study involving 19 laboratories to assess the uncertainty of FRET experiments for proteins with respect to the measured FRET efficiency histograms, determination of distances, and the detection and quantification of structural dynamics. Using two protein systems with distinct conformational changes and dynamics, we obtained an uncertainty of the FRET efficiency ≤0.06, corresponding to an interdye distance precision of ≤2 Å and accuracy of ≤5 Å. We further discuss the limits for detecting fluctuations in this distance range and how to identify dye perturbations. Our work demonstrates the ability of smFRET experiments to simultaneously measure distances and avoid the averaging of conformational dynamics for realistic protein systems, highlighting its importance in the expanding toolbox of integrative structural biology.

Extraction of kinetics from equilibrium distributions of states using the Metropolis Monte Carlo method

The Metropolis Monte Carlo (MMC) method is used to extract reaction kinetics from a given equilibrium distribution of states of a complex system. The approach is illustrated by the folding/unfolding reaction for two proteins: a model β-hairpin and α-helical protein α_{3}D. For the β-hairpin, the free energy surfaces (FESs) and free energy profiles (FEPs) are employed as the equilibrium distributions of states, playing a role of the potentials of mean force to determine the acceptance probabilities of new states in the MMC simulations. Based on the FESs and PESs for a set of temperatures that were simulated with the molecular dynamics (MD) method, the MMC simulations are performed to extract folding/unfolding rates. It has been found that the rate constants and first-passage time (FPT) distributions obtained in the MMC simulations change with temperature in good agreement with those from the MD simulations. For α_{3}D protein, whose equilibrium folding/unfolding was studied with the single-molecule FRET method [Chung et al., J. Phys. Chem. A 115, 3642 (2011)1089-563910.1021/jp1009669], the FRET-efficiency histograms at different denaturant concentrations were used as the equilibrium distributions of protein states. It has been found that the rate constants for folding and unfolding obtained in the MMC simulations change with denaturant concentration in reasonable agreement with the constants that were extracted from the photon trajectories on the basis of theoretical models. The simulated FPT distributions are single-exponential, which is consistent with the assumption of two-state kinetics that was made in the theoretical models. The promising feature of the present approach is that it is based solely on the equilibrium distributions of states, without introducing any additional parameters to perform simulations, which suggests its applicability to other complex systems.

Theory and Analysis of Single-Molecule FRET Experiments

Inter-dye distances and conformational dynamics can be studied using single-molecule FRET measurements. We consider two approaches to analyze sequences of photons with recorded photon colors and arrival times. The first approach is based on FRET efficiency histograms obtained from binned photon sequences. The experimental histograms are compared with the theoretical histograms obtained using the joint distribution of acceptor and donor photons or the Gaussian approximation. In the second approach, a photon sequence is analyzed without binning. The parameters of a model describing conformational dynamics are found by maximizing the appropriate likelihood function. The first approach is simpler, while the second one is more accurate, especially when the population of species is small and transition rates are fast. The likelihood-based analysis as well as the recoloring method has the advantage that diffusion of molecules through the laser focus can be rigorously handled.

Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories

The rates of protein (un)folding are often described as diffusion on the projection of a hyperdimensional energy landscape onto a few (ideally one) order parameters. Testing such an approximation by experiment requires resolving the reactive transition paths of individual molecules, which is now becoming feasible with advanced single-molecule spectroscopic techniques. This has also sparked the interest of theorists in better understanding reactive transition paths. Here we focus on these issues aiming to establish (i) practical guidelines for the mechanistic interpretation of transition path times (TPT) and (ii) methods to extract the free energy surface and protein dynamics from the maximum likelihood analysis of photon trajectories (MLA-PT). We represent the (un)folding rates as diffusion on a 1D free energy surface with the FRET efficiency as a reaction coordinate proxy. We then perform diffusive kinetic simulations on surfaces with two minima and a barrier, but with different shapes (curvatures, barrier height, and symmetry), coupled to stochastic simulations of photon emissions that reproduce current SM-FRET experiments. From the analysis of transition paths, we find that the TPT is inversely proportional to the barrier height (difference in free energy between minimum and barrier top) for any given surface shape, and that dividing the TPT into climb and descent segments provides key information about the barrier's symmetry. We also find that the original MLA-PT procedure used to determine the TPT from experiments underestimates its value, particularly for the cases with smaller barriers (e.g., fast folders), and we suggest a simple strategy to correct for this bias. Importantly, we also demonstrate that photon trajectories contain enough information to extract the 1D free energy surface's shape and dynamics (if TPT is >4-5-fold longer than the interphoton time) using the MLA-PT directly implemented with a diffusive free energy surface model. When dealing with real (unknown) experimental data, the comparison between the likelihoods of the free energy surface and discrete kinetic three-state models can be used to evaluate the statistical significance of the estimated free energy surface.

Ultrafast pore-loop dynamics in a AAA+ machine point to a Brownian-ratchet mechanism for protein translocation

AAA+ ring–shaped machines, such as the disaggregation machines ClpB and Hsp104, mediate ATP-driven substrate translocation through their central channel by a set of pore loops. Recent structural studies have suggested a universal hand-over-hand translocation mechanism with slow and rigid subunit motions. However, functional and biophysical studies are in discord with this model. Here, we directly measure the real-time dynamics of the pore loops of ClpB during substrate threading, using single-molecule FRET spectroscopy. All pore loops undergo large-amplitude fluctuations on the microsecond time scale and change their conformation upon interaction with substrate proteins in an ATP-dependent manner. Conformational dynamics of two of the pore loops strongly correlate with disaggregation activity, suggesting that they are the main contributors to substrate pulling. This set of findings is rationalized in terms of an ultrafast Brownian-ratchet translocation mechanism, which likely acts in parallel to the much slower hand-over-hand process in ClpB and other AAA+ machines.

FRET-based dynamic structural biology: Challenges, perspectives and an appeal for open-science practices

Single-molecule FRET (smFRET) has become a mainstream technique for studying biomolecular structural dynamics. The rapid and wide adoption of smFRET experiments by an ever-increasing number of groups has generated significant progress in sample preparation, measurement procedures, data analysis, algorithms and documentation. Several labs that employ smFRET approaches have joined forces to inform the smFRET community about streamlining how to perform experiments and analyze results for obtaining quantitative information on biomolecular structure and dynamics. The recent efforts include blind tests to assess the accuracy and the precision of smFRET experiments among different labs using various procedures. These multi-lab studies have led to the development of smFRET procedures and documentation, which are important when submitting entries into the archiving system for integrative structure models, PDB-Dev. This position paper describes the current 'state of the art' from different perspectives, points to unresolved methodological issues for quantitative structural studies, provides a set of 'soft recommendations' about which an emerging consensus exists, and lists openly available resources for newcomers and seasoned practitioners. To make further progress, we strongly encourage 'open science' practices.

Downhill, Ultrafast and Fast Folding Proteins Revised

Research on the protein folding problem differentiates the protein folding process with respect to the duration of this process. The current structure encoded in sequence dogma seems to be clearly justified, especially in the case of proteins referred to as fast-folding, ultra-fast-folding or downhill. In the present work, an attempt to determine the characteristics of this group of proteins using fuzzy oil drop model is undertaken. According to the fuzzy oil drop model, a protein is a specific micelle composed of bi-polar molecules such as amino acids. Protein folding is regarded as a spherical micelle formation process. The presence of covalent peptide bonds between amino acids eliminates the possibility of free mutual arrangement of neighbors. An example would be the construction of co-micelles composed of more than one type of bipolar molecules. In the case of fast folding proteins, the amino acid sequence represents the optimal bipolarity system to generate a spherical micelle. In order to achieve the native form, it is enough to have an external force field provided by the water environment which directs the folding process towards the generation of a centric hydrophobic core. The influence of the external field can be expressed using the 3D Gaussian function which is a mathematical model of the folding process orientation towards the concentration of hydrophobic residues in the center with polar residues exposed on the surface. The set of proteins under study reveals a hydrophobicity distribution compatible with a 3D Gaussian distribution, taken as representing an idealized micelle-like distribution. The structure of the present hydrophobic core is also discussed in relation to the distribution of hydrophobic residues in a partially unfolded form.

Predicting the viability of beta-lactamase: How folding and binding free energies correlate with beta-lactamase fitness

One of the long-standing holy grails of molecular evolution has been the ability to predict an organism's fitness directly from its genotype. With such predictive abilities in hand, researchers would be able to more accurately forecast how organisms will evolve and how proteins with novel functions could be engineered, leading to revolutionary advances in medicine and biotechnology. In this work, we assemble the largest reported set of experimental TEM-1 β-lactamase folding free energies and use this data in conjunction with previously acquired fitness data and computational free energy predictions to determine how much of the fitness of β-lactamase can be directly predicted by thermodynamic folding and binding free energies. We focus upon β-lactamase because of its long history as a model enzyme and its central role in antibiotic resistance. Based upon a set of 21 β-lactamase single and double mutants expressly designed to influence protein folding, we first demonstrate that modeling software designed to compute folding free energies such as FoldX and PyRosetta can meaningfully, although not perfectly, predict the experimental folding free energies of single mutants. Interestingly, while these techniques also yield sensible double mutant free energies, we show that they do so for the wrong physical reasons. We then go on to assess how well both experimental and computational folding free energies explain single mutant fitness. We find that folding free energies account for, at most, 24% of the variance in β-lactamase fitness values according to linear models and, somewhat surprisingly, complementing folding free energies with computationally-predicted binding free energies of residues near the active site only increases the folding-only figure by a few percent. This strongly suggests that the majority of β-lactamase's fitness is controlled by factors other than free energies. Overall, our results shed a bright light on to what extent the community is justified in using thermodynamic measures to infer protein fitness as well as how applicable modern computational techniques for predicting free energies will be to the large data sets of multiply-mutated proteins forthcoming.

Modeling of Multicolor Single-Molecule Förster Resonance Energy-Transfer Experiments on Protein Folding

Using a coarse-grained, C_α-model of BBL protein, a multicolor single-molecule Förster resonance energy transfer (FRET) experiment is modeled. Three fluorophores are introduced, which, for simplicity, are associated with C_α beads. Two fluorophores are placed at the ends of protein chain and the third one at the middle of the chain. The free-energy surfaces (FESs) depending on the interfluorophore distances and on the FRET efficiencies corresponding to these distances have been constructed and compared with the FESs depending on the conventional collective variables, such as the fraction of native contacts and radius of gyration. It has been found that multicolor experiments can successfully resolve all essential BBL states that are revealed by the conventional FESs. The resolution of these states with the FRET-efficiency histogram is found to be successful if the energy transfer is measured between the fluorophores at the BBL ends. We also show that, although the present model construct of BBL is very simple, it captures some characteristic features of the single-molecule FRET experiments, such as the pattern of the FRET-efficiency histograms and their evolution with the denaturant concentration.

Perspective: Chain dynamics of unfolded and intrinsically disordered proteins from nanosecond fluorescence correlation spectroscopy combined with single-molecule FRET

The dynamics of unfolded proteins are important both for the process of protein folding and for the behavior of intrinsically disordered proteins. However, methods for investigating the global chain dynamics of these structurally diverse systems have been limited. A versatile experimental approach is single-molecule spectroscopy in combination with Förster resonance energy transfer and nanosecond fluorescence correlation spectroscopy. The concepts of polymer physics offer a powerful framework both for interpreting the results and for understanding and classifying the properties of unfolded and intrinsically disordered proteins. This information on long-range chain dynamics can be complemented with spectroscopic techniques that probe different length scales and time scales, and integration of these results greatly benefits from recent advances in molecular simulations. This increasing convergence between the experiment, theory, and simulation is thus starting to enable an increasingly detailed view of the dynamics of disordered proteins.

Protein folding transition path times from single molecule FRET

The transition path is the tiny segment of a single molecule trajectory when the free energy barrier between states is crossed and for protein folding contains all of the information about the self-assembly mechanism. As a first step toward obtaining structural information during the transition path from experiments, single molecule FRET spectroscopy has been used to determine average transition path times from a photon-by-photon analysis of fluorescence trajectories. These results, obtained for several different proteins, have already provided new and demanding tests that support both the accuracy of all-atom molecular dynamics simulations and the basic postulates of energy landscape theory of protein folding.

When fast is better: protein folding fundamentals and mechanisms from ultrafast approaches

Protein folding research stalled for decades because conventional experiments indicated that proteins fold slowly and in single strokes, whereas theory predicted a complex interplay between dynamics and energetics resulting in myriad microscopic pathways. Ultrafast kinetic methods turned the field upside down by providing the means to probe fundamental aspects of folding, test theoretical predictions and benchmark simulations. Accordingly, experimentalists could measure the timescales for all relevant folding motions, determine the folding speed limit and confirm that folding barriers are entropic bottlenecks. Moreover, a catalogue of proteins that fold extremely fast (microseconds) could be identified. Such fast-folding proteins cross shallow free energy barriers or fold downhill, and thus unfold with minimal co-operativity (gradually). A new generation of thermodynamic methods has exploited this property to map folding landscapes, interaction networks and mechanisms at nearly atomic resolution. In parallel, modern molecular dynamics simulations have finally reached the timescales required to watch fast-folding proteins fold and unfold in silico. All of these findings have buttressed the fundamentals of protein folding predicted by theory, and are now offering the first glimpses at the underlying mechanisms. Fast folding appears to also have functional implications as recent results connect downhill folding with intrinsically disordered proteins, their complex binding modes and ability to moonlight. These connections suggest that the coupling between downhill (un)folding and binding enables such protein domains to operate analogically as conformational rheostats.

Photon-by-Photon Hidden Markov Model Analysis for Microsecond Single-Molecule FRET Kinetics

The function of biological macromolecules involves large-scale conformational dynamics spanning multiple time scales, from microseconds to seconds. Such conformational motions, which may involve whole domains or subunits of a protein, play a key role in allosteric regulation. There is an urgent need for experimental methods to probe the fastest of these motions. Single-molecule fluorescence experiments can in principle be used for observing such dynamics, but there is a lack of analysis methods that can extract the maximum amount of information from the data, down to the microsecond time scale. To address this issue, we introduce H²MM, a maximum likelihood estimation algorithm for photon-by-photon analysis of single-molecule fluorescence resonance energy transfer (FRET) experiments. H²MM is based on analytical estimators for model parameters, derived using the Baum-Welch algorithm. An efficient and effective method for the calculation of these estimators is introduced. H²MM is shown to accurately retrieve the reaction times from ∼1 s to ∼10 μs and even faster when applied to simulations of freely diffusing molecules. We further apply this algorithm to single-molecule FRET data collected from Holliday junction molecules and show that at low magnesium concentrations their kinetics are as fast as ∼10⁴ s^-1. The new algorithm is particularly suitable for experiments on freely diffusing individual molecules and is readily incorporated into existing analysis packages. It paves the way for the broad application of single-molecule fluorescence to study ultrafast functional dynamics of biomolecules.

Analysis of Fluorescence Lifetime and Energy Transfer Efficiency in Single-Molecule Photon Trajectories of Fast-Folding Proteins

In single-molecule Förster resonance energy transfer (FRET) spectroscopy, the dynamics of molecular processes are usually determined by analyzing the fluorescence intensity of donor and acceptor dyes. Since FRET efficiency is related to fluorescence lifetimes, additional information can be extracted by analyzing fluorescence intensity and lifetime together. For fast processes where individual states are not well separated in a trajectory, it is not easy to obtain the lifetime information. Here, we present analysis methods to utilize fluorescence lifetime information from single-molecule FRET experiments, and apply these methods to three fast-folding, two-state proteins. By constructing 2D FRET efficiency-lifetime histograms, the correlation can be visualized between the FRET efficiency and fluorescence lifetimes in the presence of the submicrosecond to millisecond dynamics. We extend the previously developed method for analyzing delay times of donor photons to include acceptor delay times. To determine the kinetics and lifetime parameters accurately, we used a maximum likelihood method. We found that acceptor blinking can lead to inaccurate parameters in the donor delay time analysis. This problem can be solved by incorporating acceptor blinking into a model. While the analysis of acceptor delay times is not affected by acceptor blinking, it is more sensitive to the shape of the delay time distribution resulting from a broad conformational distribution in the unfolded state.