Two states or not two states: Single-molecule folding studies of protein L

Information-theoretical limit on the estimates of dissipation by molecular machines using single-molecule fluorescence resonance energy transfer experiments

Single-molecule fluorescence resonance energy transfer (FRET) experiments are commonly used to study the dynamics of molecular machines. While in vivo molecular processes often break time-reversal symmetry, the temporal directionality of cyclically operating molecular machines is often not evident from single-molecule FRET trajectories, especially in the most common two-color FRET studies. Solving a more quantitative problem of estimating the energy dissipation/entropy production by a molecular machine from single-molecule data is even more challenging. Here, we present a critical assessment of several practical methods of doing so, including Markov-model-based methods and a model-free approach based on an information-theoretical measure of entropy production that quantifies how (statistically) dissimilar observed photon sequences are from their time reverses. The Markov model approach is computationally feasible and may outperform model free approaches, but its performance strongly depends on how well the assumed model approximates the true microscopic dynamics. Markov models are also not guaranteed to give a lower bound on dissipation. Meanwhile, model-free, information-theoretical methods systematically underestimate entropy production at low photoemission rates, and long memory effects in the photon sequences make these methods demanding computationally. There is no clear winner among the approaches studied here, and all methods deserve to belong to a comprehensive data analysis toolkit.

An accurate probabilistic step finder for time-series analysis

Noisy time-series data is commonly collected from sources including Förster Resonance Energy Transfer experiments, patch clamp and force spectroscopy setups, among many others. Two of the most common paradigms for the detection of discrete transitions in such time-series data include: hidden Markov models (HMMs) and step-finding algorithms. HMMs, including their extensions to infinite state-spaces, inherently assume in analysis that holding times in discrete states visited are geometrically-or, loosely speaking in common language, exponentially-distributed. Thus the determination of step locations, especially in sparse and noisy data, is biased by HMMs toward identifying steps resulting in geometric holding times. In contrast, existing step-finding algorithms, while free of this restraint, often rely on ad hoc metrics to penalize steps recovered in time traces (by using various information criteria) and otherwise rely on approximate greedy algorithms to identify putative global optima. Here, instead, we devise a robust and general probabilistic (Bayesian) step-finding tool that neither relies on ad hoc metrics to penalize step numbers nor assumes geometric holding times in each state. As the number of steps themselves in a time-series are, a priori unknown, we treat these within a Bayesian nonparametric (BNP) paradigm. We find that the method developed, Bayesian Nonparametric Step (BNP-Step), accurately determines the number and location of transitions between discrete states without any assumed kinetic model and learns the emission distribution characteristic of each state. In doing so, we verify that BNP-Step can analyze sparser data sets containing higher noise and more closely-spaced states than otherwise resolved by current state-of-the-art methods. What is more, BNP-Step rigorously propagates measurement uncertainty into uncertainty over state transition locations, numbers, and emission levels as characterized by the posterior. We demonstrate the performance of BNP-Step on both synthetic data as well as data drawn from force spectroscopy experiments.

Direct Observation and Real-Time Tracking of an Extraordinarily Stable Folding Intermediate in Mitotic Arrest Deficient Protein 2 Folding by Single-Molecule Fluorescence Resonance Energy Transfer

Although ensemble experiments have suggested that mitotic arrest deficient protein 2 (Mad2), a metamorphic protein, has folding intermediates, direct evidence and characterization are not available. It remains an outstanding challenge to capture the folding intermediates in real time, which is crucial to elucidate the folding mechanism, but the folding intermediates are normally unstable and only exist transiently. By combining confocal-microscopy-based and total internal reflection fluorescence (TIRF)-microscopy-based single-molecule Förster resonance energy transfer (sm-FRET) techniques, we have investigated the folding/unfolding process of Mad2 and captured its folding intermediate at the single-molecule level. This provides direct evidence for the existence of an intermediate along the folding pathway of Mad2. The folding intermediate proved to be extraordinarily stable, with an extremely long average dwell time of 2.3 s under the conditions of 3 M GdmCl at ambient temperature. The folding trajectories obtained from TIRF experiments further suggest that the intermediate is on-pathway to native Mad2.

Challenges in Inferring the Directionality of Active Molecular Processes from Single-Molecule Fluorescence Resonance Energy Transfer Trajectories

We discuss some of the practical challenges that one faces in using stochastic thermodynamics to infer directionality of molecular machines from experimental single-molecule trajectories. Because of the limited spatiotemporal resolution of single-molecule experiments and because both forward and backward transitions between the same pairs of states cannot always be detected, differentiating between the forward and backward directions of, e.g., an ATP-consuming molecular machine that operates periodically, turns out to be a nontrivial task. Using a simple extension of a Markov-state model that is commonly employed to analyze single-molecule transition-path measurements, we illustrate how irreversibility can be hidden from such measurements but in some cases can be uncovered when non-Markov effects in low-dimensional single-molecule trajectories are considered.

Coordinate-Dependent Drift-Diffusion Reveals the Kinetic Intermediate Traps of Top7-Based Proteins

The computer-designed Top7 served as a scaffold to produce immunoreactive proteins by grafting of the 2F5 HIV-1 antibody epitope (Top7-2F5) followed by biotinylation (Top7-2F5-biotin). The resulting nonimmunoglobulin affinity proteins were effective in inducing and detecting the HIV-1 antibody. However, the grafted Top7-2F5 design led to protein aggregation, as opposed to the soluble biotinylated Top7-2F5-biotin. The structure-based model predicted that the thermodynamic cooperativity of Top7 increases after grafting and biotin-labeling, reducing their intermediate state populations. In this work, the folding kinetic traps that might contribute to the aggregation propensity are investigated by the diffusion theory. Since the engineered proteins have similar sequence and structural homology, they served as protein models to study the kinetic intermediate traps that were uncovered by characterizing the position-dependent drift-velocity (v(Q)) and the diffusion (D(Q)) coefficients. These coordinate-dependent coefficients were taken into account to obtain the folding and transition path times over the free energy transition states containing the intermediate kinetic traps. This analysis may be useful to predict the aggregated kinetic traps of scaffold-epitope proteins that might compose novel diagnostic and therapeutic platforms.

Protein folding problem: enigma, paradox, solution

The ability of protein chains to spontaneously form their three-dimensional structures is a long-standing mystery in molecular biology. The most conceptual aspect of this mystery is how the protein chain can find its native, "working" spatial structure (which, for not too big protein chains, corresponds to the global free energy minimum) in a biologically reasonable time, without exhaustive enumeration of all possible conformations, which would take billions of years. This is the so-called "Levinthal's paradox." In this review, we discuss the key ideas and discoveries leading to the current understanding of protein folding kinetics, including folding landscapes and funnels, free energy barriers at the folding/unfolding pathways, and the solution of Levinthal's paradox. A special role here is played by the "all-or-none" phase transition occurring at protein folding and unfolding and by the point of thermodynamic (and kinetic) equilibrium between the "native" and the "unfolded" phases of the protein chain (where the theory obtains the simplest form). The modern theory provides an understanding of key features of protein folding and, in good agreement with experiments, it (i) outlines the chain length-dependent range of protein folding times, (ii) predicts the observed maximal size of "foldable" proteins and domains. Besides, it predicts the maximal size of proteins and domains that fold under solely thermodynamic (rather than kinetic) control. Complementarily, a theoretical analysis of the number of possible protein folding patterns, performed at the level of formation and assembly of secondary structures, correctly outlines the upper limit of protein folding times.

Differentiating between the sequence of structural events on alternative pathways of folding of a heterodimeric protein

Distinguishing between competing pathways of folding of a protein, on the basis of how they differ in their progress of structure acquisition, remains an important challenge in protein folding studies. A previous study had shown that the heterodimeric protein, double chain monellin (dcMN) switches between alternative folding pathways upon a change in guanidine hydrochloride (GdnHCl) concentration. In the current study, the folding of dcMN has been characterized by the pulsed hydrogen exchange (HX) labeling methodology used in conjunction with mass spectrometry. Quantification of the extent to which folding intermediates accumulate and then disappear with time of folding at both low and high GdnHCl concentrations, where the folding pathways are known to be different, shows that the folding mechanism is describable by a triangular three-state mechanism. Structural characterization of the productive folding intermediates populated on the alternative pathways has enabled the pathways to be differentiated on the basis of the progress of structure acquisition that occurs on them. The intermediates on the two pathways differ in the extent to which the α-helix and the rest of the β-sheet have acquired structure that is protective against HX. The major difference is, however, that β2 has not acquired any protective structure in the intermediate formed on one pathway, but it has acquired significant protective structure in the intermediate formed on the alternative pathway. Hence, the sequence of structural events is different on the two alternative pathways.

The Effect of Time Resolution on Apparent Transition Path Times Observed in Single-Molecule Studies of Biomolecules

Single-molecule experiments have now achieved a time resolution allowing observation of transition paths, the brief trajectory segments where the molecule undergoing an unfolding or folding transition enters the energetically or entropically unfavorable barrier region from the folded/unfolded side and exits to the unfolded/folded side, thereby completing the transition. This resolution, however, is yet insufficient to identify the precise entrance/exit events that mark the beginning and the end of a transition path: the nature of the diffusive dynamics is such that a molecular trajectory will recross the boundary between the barrier region and the folded/unfolded state, multiple times, at a time scale much shorter than that of the typical experimental resolution. Here we use theory and Brownian dynamics simulations to show that, as a result of such recrossings, the apparent transition path times are generally longer than the true ones. We quantify this effect using a simple model where the observed dynamics is a moving average of the true dynamics and discuss experimental implications of our results.

Deciphering the Folding Mechanism of Proteins G and L and Their Mutants

Much of our understanding of folding mechanisms comes from interpretations of experimental ϕ and ψ value analysis, relating the differences in stability of the transition state ensemble (TSE) and folded state. We introduce a unified approach combining simulations and Bayesian inference to provide atomistic detail for the folding mechanism of proteins G and L and their mutants. Proteins G and L fold to similar topologies despite low sequence similarity, but differ in their folding pathways. A fast folding redesign of protein G, NuG2, switches folding pathways and folds through a similar pathway with protein L. A redesign of protein L also leads to faster folding, respecting the original folding pathway. Our Bayesian inference approach starts from the same prior on all systems and correctly identifies the folding mechanism for each of the four proteins, a success of the force field and sampling strategy. The approach is computationally efficient and correctly identifies the TSE and intermediate structures along the folding pathway in good agreement with experiments. We complement our findings by using two orthogonal approaches that differ in computational cost and interpretability. Adaptive sampling MD combined with the Markov state model provides a kinetic model that confirms the more complex folding mechanism of protein G and its mutant. Finally, a novel fragment decomposition approach using AlphaFold identifies preferences for secondary structure element combinations that follow the order of events observed in the folding pathways.

Slow conformational dynamics of the human A2A adenosine receptor are temporally ordered

A more complete depiction of protein energy landscapes includes the identification of different function-related conformational states and the determination of the pathways connecting them. We used total internal reflection fluorescence (TIRF) imaging to investigate the conformational dynamics of the human A2A adenosine receptor (A2AAR), a class A G protein-coupled receptor (GPCR), at the single-molecule level. Slow, reversible conformational exchange was observed among three different fluorescence emission states populated for agonist-bound A2AAR. Transitions among these states predominantly occurred in a specific order, and exchange between inactive and active-like conformations proceeded through an intermediate state. Models derived from molecular dynamics simulations with available A2AAR structures rationalized the relative fluorescence emission intensities for the highest and lowest emission states but not the transition state. This suggests that the functionally critical intermediate state required to achieve activation is not currently visualized among available A2AAR structures.

Force-Dependent Folding Kinetics of Single Molecules with Multiple Intermediates and Pathways

Most single-molecule studies derive the kinetic rates of native, intermediate, and unfolded states from equilibrium hopping experiments. Here, we apply the Kramers kinetic diffusive model to derive the force-dependent kinetic rates of intermediate states from nonequilibrium pulling experiments. From the kinetic rates, we also extract the force-dependent kinetic barriers and the equilibrium folding energies. We apply our method to DNA hairpins with multiple folding pathways and intermediates. The experimental results agree with theoretical predictions. Furthermore, the proposed nonequilibrium single-molecule approach permits us to characterize kinetic and thermodynamic properties of native, unfolded, and intermediate states that cannot be derived from equilibrium hopping experiments.

Single-Molecule FRET of Membrane Transport Proteins

Uncovering the structure and function of biomolecules is a fundamental goal in structural biology. Membrane-embedded transport proteins are ubiquitous in all kingdoms of life. Despite structural flexibility, their mechanisms are typically studied by ensemble biochemical methods or by static high-resolution structures, which complicate a detailed understanding of their dynamics. Here, we review the recent progress of single molecule Förster Resonance Energy Transfer (smFRET) in determining mechanisms and timescales of substrate transport across membranes. These studies do not only demonstrate the versatility and suitability of state-of-the-art smFRET tools for studying membrane transport proteins but they also highlight the importance of membrane mimicking environments in preserving the function of these proteins. The current achievements advance our understanding of transport mechanisms and have the potential to facilitate future progress in drug design.

The road less traveled in protein folding: evidence for multiple pathways

Free Energy Landscape theory of Protein Folding, introduced over 20 years ago, implies that a protein has many paths to the folded conformation with the lowest free energy. Despite the knowledge in principle, it has been remarkably hard to detect such pathways. The lack of such observations is primarily due to the fact that no one experimental technique can detect many parts of the protein simultaneously with the time resolution necessary to see such differences in paths. However, recent technical developments and employment of multiple experimental probes and folding prompts have illuminated multiple folding pathways in a number of proteins that had all previously been described with a single path.

Single-molecule FRET methods to study the dynamics of proteins at work

Feynman commented that "Everything that living things do can be understood in terms of the jiggling and wiggling of atoms". Proteins can jiggle and wiggle large structural elements such as domains and subunits as part of their functional cycles. Single-molecule fluorescence resonance energy transfer (smFRET) is an excellent tool to study conformational dynamics and decipher coordinated large-scale motions within proteins. smFRET methods introduced in recent years are geared toward understanding the time scales and amplitudes of function-related motions. This review discusses the methodology for obtaining and analyzing smFRET temporal trajectories that provide direct dynamic information on transitions between conformational states. It also introduces correlation methods that are useful for characterizing intramolecular motions. This arsenal of techniques has been used to study multiple molecular systems, from membrane proteins through molecular chaperones, and we examine some of these studies here. Recent exciting methodological novelties permit revealing very fast, submillisecond dynamics, whose relevance to protein function is yet to be fully grasped.

A Double-Resonance CEST Experiment To Study Multistate Protein Conformational Exchange: An Application to Protein Folding

Despite the importance of protein dynamics to function, studying exchange between multiple conformational states remains a challenge because sparsely populated states are invisible to conventional techniques. CEST NMR experiments can detect minor states with lifetimes between 5 and 200 ms populated to a level of just ∼1%. However, CEST often cannot provide the exchange mechanism for processes involving three or more states, leaving the role of the detected minor states unknown. Here a double-resonance CEST experiment to determine the kinetics of multistate exchange is presented. The approach that involves irradiating resonances from two minor states simultaneously is used to study the exchange of T4 lysozyme (T4L) between the dominant native state and two minor states, the unfolded state and a second minor state (B), each populated to only ∼4%. Regular CEST does not provide the folding mechanism, but double-resonance CEST clearly shows that T4L can fold directly without going through B.

Stochastic diffusion framework determines the free-energy landscape and rate from single-molecule trajectory

A theoretical stochastic diffusion framework is developed that characterizes the position-dependent diffusion coefficient [D(Q)] and drift velocity [ v (Q)] by analysing single-molecule time traces [Q(t)]. The free-energy landscape [F(Q)] that governs the dynamics is reconstructed with the calculated D and v . There are many computational tools that perform this task in which some are computationaly demanding, difficult to run, and, most of the time, not directly available to the community. This is a first attempt to implement the simplified stochastic diffusion framework that is fast, easy to run in a Python environment, and available to be extended as needed. It does not require adjustable parameters, inference methods, or sampling bias such as Monte Carlo Bayesian estimators or umbrella samplings. The stochastic framework was applied in the protein-like lattice model with Monte Carlo simulations, which accurately predicted the folding rates with the coordinate-dependent D and F plugged into Kramers' theory. The results were compared with two other independently developed methodologies (the Bayesian analysis and fep1D algorithm) presenting a good match, which confirms its validity. This theoretical framework might be useful in determining the free-energy and rates by providing time series only from biological or condensed-phase systems. The code is freely available at https://github.com/ronaldolab/stochastic_diffusion.

Preface: Special Topic on Single-Molecule Biophysics

Single-molecule measurements are now almost routinely used to study biological systems and processes. The scope of this special topic emphasizes the physics side of single-molecule observations, with the goal of highlighting new developments in physical techniques as well as conceptual insights that single-molecule measurements bring to biophysics. This issue also comprises recent advances in theoretical physical models of single-molecule phenomena, interpretation of single-molecule signals, and fundamental areas of statistical mechanics that are related to single-molecule observations. A particular goal is to illustrate the increasing synergy between theory, simulation, and experiment in single-molecule biophysics.