Protein folding studied by single-molecule FRET

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.

Theory for proton-coupled energy transfer

In the recently discovered proton-coupled energy transfer (PCEnT) mechanism, the transfer of electronic excitation energy between donor and acceptor chromophores is coupled to a proton transfer reaction. Herein, we develop a general theory for PCEnT and derive an analytical expression for the nonadiabatic PCEnT rate constant. This theory treats the transferring hydrogen nucleus quantum mechanically and describes the PCEnT process in terms of nonadiabatic transitions between reactant and product electron-proton vibronic states. The rate constant is expressed as a summation over these vibronic states, and the contribution of each pair of vibronic states depends on the square of the vibronic coupling as well as the spectral convolution integral, which can be viewed as a generalization of the Förster-type spectral overlap integral for vibronic rather than electronic states. The convolution integral also accounts for the common vibrational modes shared by the donor and acceptor chromophores for intramolecular PCEnT. We apply this theory to model systems to investigate the key features of PCEnT processes. The excited vibronic states can contribute significantly to the total PCEnT rate constant, and the common modes can either slow down or speed up the process. Because the pairs of vibronic states that contribute the most to the PCEnT rate constant may correspond to spectroscopically dark states, PCEnT could occur even when there is no apparent overlap between the donor emission and acceptor absorption spectra. This theory will assist in the interpretation of experimental data and will guide the design of additional PCEnT systems.

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.

Milestoning estimators of dissipation in systems observed at a coarse resolution

Many nonequilibrium, active processes are observed at a coarse-grained level, where different microscopic configurations are projected onto the same observable state. Such "lumped" observables display memory, and in many cases, the irreversible character of the underlying microscopic dynamics becomes blurred, e.g., when the projection hides dissipative cycles. As a result, the observations appear less irreversible, and it is very challenging to infer the degree of broken time-reversal symmetry. Here we show, contrary to intuition, that by ignoring parts of the already coarse-grained state space we may-via a process called milestoning-improve entropy-production estimates. We present diverse examples where milestoning systematically renders observations "closer to underlying microscopic dynamics" and thereby improves thermodynamic inference from lumped data assuming a given range of memory, and we hypothesize that this effect is quite general. Moreover, whereas the correct general physical definition of time reversal in the presence of memory remains unknown, we here show by means of physically relevant examples that at least for semi-Markov processes of first and second order, waiting-time contributions arising from adopting a naive Markovian definition of time reversal generally must be discarded.

Time-heterogeneity of the Förster Radius from Dipole Orientational Dynamics Explains Observed Dynamic Shift

Förster resonance energy transfer (FRET) is a quantum mechanical phenomenon involving the non-radiative transfer of energy between coupled electric dipoles. Due to the strong dependence of FRET on the distance between the dipoles, it is frequently used as a "molecular ruler" in biology, chemistry, and physics. This is done by placing dipolar molecules called dyes on molecules of interest. In time-resolved confocal single-molecule FRET (smFRET) experiments, the joint distribution of the FRET efficiency and the donor fluorescence lifetime can reveal underlying molecular conformational dynamics via deviation from their theoretical Förster relationship. This deviation is referred to as a dynamic shift. Quantifying the dynamic shift caused by the motion of the fluorescent dyes is essential to decoupling the dynamics of the studied molecules and the dyes. We develop novel Langevin models for the dye linker dynamics, including rotational dynamics, based on first physics principles and proper dye linker chemistry to match accessible volumes predicted by molecular dynamics simulations. By simulating the dyes' stochastic translational and rotational dynamics, we show that the observed dynamic shift can largely be attributed to the mutual orientational dynamics of the electric dipole moments associated with the dyes, not their accessible volume.

An optofluidic antenna for enhancing the sensitivity of single-emitter measurements

Many single-molecule investigations are performed in fluidic environments, for example, to avoid unwanted consequences of contact with surfaces. Diffusion of molecules in this arrangement limits the observation time and the number of collected photons, thus, compromising studies of processes with fast or slow dynamics. Here, we introduce a planar optofluidic antenna (OFA), which enhances the fluorescence signal from molecules by about 5 times per passage, leads to about 7-fold more frequent returns to the observation volume, and significantly lengthens the diffusion time within one passage. We use single-molecule multi-parameter fluorescence detection (sm-MFD), fluorescence correlation spectroscopy (FCS) and Förster resonance energy transfer (FRET) measurements to characterize our OFAs. The antenna advantages are showcased by examining both the slow (ms) and fast (50 μs) dynamics of DNA four-way (Holliday) junctions with real-time resolution. The FRET trajectories provide evidence for the absence of an intermediate conformational state and introduce an upper bound for its lifetime. The ease of implementation and compatibility with various microscopy modalities make OFAs broadly applicable to a diverse range of studies.

Quantifying distortion in time-correlated single photon counting: a universal parameter

One major drawback of the classic time-correlated single photon counting (TCSPC) technique is pileup-related distortion. To keep it under a reasonable level, the maximum count rate has to be reduced, posing a serious limitation to the overall measurement speed. This means that there is an intrinsic trade-off between speed and distortion: either count rate is raised, but distortion is worsened, or distortion is minimized at the expense of speed. In both cases, it is impossible to precisely evaluate the degree of distortion introduced. Here comes our new, to the best of our knowledge, figure of merit, which is able to provide a numerical estimate of the distortion whatever the signal shape is, marking a real turning point in the way of doing TCSPC. In this article, this new parameter will be defined and its effectiveness will be demonstrated by means of mathematical computations.

Hierarchical Assembly of Single-Stranded RNA

Single-stranded RNA (ssRNA) plays a major role in the flow of genetic information-most notably, in the form of messenger RNA (mRNA)-and in the regulation of biological processes. The highly dynamic nature of chains of unpaired nucleobases challenges structural characterizations of ssRNA by experiments or molecular dynamics (MD) simulations alike. Here, we use hierarchical chain growth (HCG) to construct ensembles of ssRNA chains. HCG assembles the structures of protein and nucleic acid chains from fragment libraries created by MD simulations. Applied to homo- and heteropolymeric ssRNAs of different lengths, we find that HCG produces structural ensembles that overall are in good agreement with diverse experiments, including nuclear magnetic resonance (NMR), small-angle X-ray scattering (SAXS), and single-molecule Förster resonance energy transfer (FRET). The agreement can be further improved by ensemble refinement using Bayesian inference of ensembles (BioEn). HCG can also be used to assemble RNA structures that combine base-paired and base-unpaired regions, as illustrated for the 5' untranslated region (UTR) of SARS-CoV-2 RNA.

Single-Molecule FRET at 10 MHz Count Rates

A bottleneck in many studies utilizing single-molecule Förster resonance energy transfer is the attainable photon count rate, as it determines the temporal resolution of the experiment. As many biologically relevant processes occur on time scales that are hardly accessible with currently achievable photon count rates, there has been considerable effort to find strategies to increase the stability and brightness of fluorescent dyes. Here, we use DNA nanoantennas to drastically increase the achievable photon count rates and observe fast biomolecular dynamics in the small volume between two plasmonic nanoparticles. As a proof of concept, we observe the coupled folding and binding of two intrinsically disordered proteins, which form transient encounter complexes with lifetimes on the order of 100 μs. To test the limits of our approach, we also investigated the hybridization of a short single-stranded DNA to its complementary counterpart, revealing a transition path time of 17 μs at photon count rates of around 10 MHz, which is an order-of-magnitude improvement compared to the state of the art. Concomitantly, the photostability was increased, enabling many seconds long megahertz fluorescence time traces. Due to the modular nature of the DNA origami method, this platform can be adapted to a broad range of biomolecules, providing a promising approach to study previously unobservable ultrafast biophysical processes.

Exploring the Synergies of Single-Molecule Fluorescence and 2D Materials Coupled by DNA

The world of 2D materials is steadily growing, with numerous researchers attempting to discover, elucidate, and exploit their properties. Approaches relying on the detection of single fluorescent molecules offer a set of advantages, for instance, high sensitivity and specificity, that allow the drawing of conclusions with unprecedented precision. Herein, it is argued how the study of 2D materials benefits from fluorescence-based single-molecule modalities, and vice versa. A special focus is placed on DNA, serving as a versatile adaptor when anchoring single dye molecules to 2D materials. The existing literature on the fruitful combination of the two fields is reviewed, and an outlook on the additional synergies that can be created between them provided.

Einstein Model of a Graph to Characterize Protein Folded/Unfolded States

The folded structures of proteins can be accurately predicted by deep learning algorithms from their amino-acid sequences. By contrast, in spite of decades of research studies, the prediction of folding pathways and the unfolded and misfolded states of proteins, which are intimately related to diseases, remains challenging. A two-state (folded/unfolded) description of protein folding dynamics hides the complexity of the unfolded and misfolded microstates. Here, we focus on the development of simplified order parameters to decipher the complexity of disordered protein structures. First, we show that any connected, undirected, and simple graph can be associated with a linear chain of atoms in thermal equilibrium. This analogy provides an interpretation of the usual topological descriptors of a graph, namely the Kirchhoff index and Randić resistance, in terms of effective force constants of a linear chain. We derive an exact relation between the Kirchhoff index and the average shortest path length for a linear graph and define the free energies of a graph using an Einstein model. Second, we represent the three-dimensional protein structures by connected, undirected, and simple graphs. As a proof of concept, we compute the topological descriptors and the graph free energies for an all-atom molecular dynamics trajectory of folding/unfolding events of the proteins Trp-cage and HP-36 and for the ensemble of experimental NMR models of Trp-cage. The present work shows that the local, nonlocal, and global force constants and free energies of a graph are promising tools to quantify unfolded/disordered protein states and folding/unfolding dynamics. In particular, they allow the detection of transient misfolded rigid states.

Enhancing the performance of a mutant pyrrolysyl-tRNA synthetase to create a highly versatile eukaryotic cell-free protein synthesis tool

Modification of proteins with a broad range of chemical functionalities enables the investigation of protein structure and activity by manipulating polypeptides at single amino acid resolution. Indeed, various functional groups including bulky non-canonical amino acids like strained cyclooctenes could be introduced by the unique features of the binding pocket of the double mutant pyrrolysyl-tRNA synthetase (Y306A, Y384F), but the instable nature of the enzyme limits its application in vivo. Here, we constructed a cell-free protein production system, which increased the overall enzyme stability by combining different reaction compartments. Moreover, a co-expression approach in a one-pot reaction allowed straightforward site-specific fluorescent labeling of the functional complex membrane protein cystic fibrosis transmembrane conductance regulator. Our work provides a versatile platform for introducing various non-canonical amino acids into difficult-to-express proteins for structural and fluorescence based investigation of proteins activity.

Single-Molecule Reaction-Diffusion

We propose to capture reaction-diffusion on a molecule-by-molecule basis from the fastest acquirable timescale, namely individual photon arrivals. We illustrate our method on intrinsically disordered human proteins, the linker histone H1.0 as well as its chaperone prothymosin α , as these diffuse through an illuminated confocal spot and interact forming larger ternary complexes on millisecond timescales. Most importantly, single-molecule reaction-diffusion, smRD, reveals single molecule properties without trapping or otherwise confining molecules to surfaces. We achieve smRD within a Bayesian paradigm and term our method Bayes-smRD. Bayes-smRD is further free of the average, bulk, results inherent to the analysis of long photon arrival traces by fluorescence correlation spectroscopy. In learning from thousands of photon arrivals continuous spatial positions and discrete conformational and photophysical state changes, Bayes-smRD estimates kinetic parameters on a molecule-by-molecule basis with two to three orders of magnitude less data than tools such as fluorescence correlation spectroscopy thereby also dramatically reducing sample photodamage.

Development of Single Molecule Techniques for Sensing and Manipulation of CRISPR and Polymerase Enzymes

Clustered regularly interspaced short palindromic repeats (CRISPR) and polymerases are powerful enzymes and their diverse applications in genomics, proteomics, and transcriptomics have revolutionized the biotechnology industry today. CRISPR has been widely adopted for genomic editing applications and Polymerases can efficiently amplify genomic transcripts via polymerase chain reaction (PCR). Further investigations into these enzymes can reveal specific details about their mechanisms that greatly expand their use. Single-molecule techniques are an effective way to probe enzymatic mechanisms because they may resolve intermediary conformations and states with greater detail than ensemble or bulk biosensing techniques. This review discusses various techniques for sensing and manipulation of single biomolecules that can help facilitate and expedite these discoveries. Each platform is categorized as optical, mechanical, or electronic. The methods, operating principles, outputs, and utility of each technique are briefly introduced, followed by a discussion of their applications to monitor and control CRISPR and Polymerases at the single molecule level, and closing with a brief overview of their limitations and future prospects.

Features of Protein Unfolding Transitions and Their Relation to Domain Topology Probed by Single-Molecule FRET

A protein fold is defined as a structural arrangement of a secondary structure in a three-dimensional space. It would be interesting to know whether a particular fold can be assigned to certain features of the corresponding folding/unfolding transitions. To understand the underlying principles of the manifold folding transitions in more detail, single-molecule FRET is the method of choice. Taking the two-domain protein phosphoglycerate kinase (PGK) as an example, we investigated denaturant-induced unfolded states of PGK using the above method. For this purpose, different intramolecular distances within the two domains were measured. In addition to the known two-state transition, a transition with a compact folding intermediate was also identified in each of the two domains. Based on the structural homology of the domains (characterized by a Rossmann fold) and the striking similarity in the features of the measured distance changes during unfolding, clear evidence emerged that the underlying domain topology plays an important role in determining the observed structural changes.

Refinement of multiconformer ensemble models from multi-temperature X-ray diffraction data

Conformational ensembles underlie all protein functions. Thus, acquiring atomic-level ensemble models that accurately represent conformational heterogeneity is vital to deepen our understanding of how proteins work. Modeling ensemble information from X-ray diffraction data has been challenging, as traditional cryo-crystallography restricts conformational variability while minimizing radiation damage. Recent advances have enabled the collection of high quality diffraction data at ambient temperatures, revealing innate conformational heterogeneity and temperature-driven changes. Here, we used diffraction datasets for Proteinase K collected at temperatures ranging from 313 to 363 K to provide a tutorial for the refinement of multiconformer ensemble models. Integrating automated sampling and refinement tools with manual adjustments, we obtained multiconformer models that describe alternative backbone and sidechain conformations, their relative occupancies, and interconnections between conformers. Our models revealed extensive and diverse conformational changes across temperature, including increased bound peptide ligand occupancies, different Ca2+ binding site configurations and altered rotameric distributions. These insights emphasize the value and need for multiconformer model refinement to extract ensemble information from diffraction data and to understand ensemble-function relationships.

Analysis of photon trajectories from diffusing single molecules

In single-molecule free diffusion experiments, molecules spend most of the time outside a laser spot and generate bursts of photons when they diffuse through the focal spot. Only these bursts contain meaningful information and, therefore, are selected using physically reasonable criteria. The analysis of the bursts must take into account the precise way they were chosen. We present new methods that allow one to accurately determine the brightness and diffusivity of individual molecule species from the photon arrival times of selected bursts. We derive analytical expressions for the distribution of inter-photon times (with and without burst selection), the distribution of the number of photons in a burst, and the distribution of photons in a burst with recorded arrival times. The theory accurately treats the bias introduced due to the burst selection criteria. We use a Maximum Likelihood (ML) method to find the molecule's photon count rate and diffusion coefficient from three kinds of data, i.e., the bursts of photons with recorded arrival times (burstML), inter-photon times in bursts (iptML), and the numbers of photon counts in a burst (pcML). The performance of these new methods is tested on simulated photon trajectories and on an experimental system, the fluorophore Atto 488.

Conformational Changes in Surface-Immobilized Proteins Measured Using Combined Atomic Force and Fluorescence Microscopy

Biological organisms rely on proteins to perform the majority of their functions. Most protein functions are based on their physical motions (conformational changes), which can be described as transitions between different conformational states in a multidimensional free-energy landscape. A comprehensive understanding of this free-energy landscape is therefore of paramount importance for understanding the biological functions of proteins. Protein dynamics includes both equilibrium and nonequilibrium motions, which typically exhibit a wide range of characteristic length and time scales. The relative probabilities of various conformational states in the energy landscape, the energy barriers between them, their dependence on external parameters such as force and temperature, and their connection to the protein function remain largely unknown for most proteins. In this paper, we present a multimolecule approach in which the proteins are immobilized at well-defined locations on Au substrates using an atomic force microscope (AFM)-based patterning method called nanografting. This method enables precise control over the protein location and orientation on the substrate, as well as the creation of biologically active protein ensembles that self-assemble into well-defined nanoscale regions (protein patches) on the gold substrate. We performed AFM-force compression and fluorescence experiments on these protein patches and measured the fundamental dynamical parameters such as protein stiffness, elastic modulus, and transition energies between distinct conformational states. Our results provide new insights into the processes that govern protein dynamics and its connection to protein function.

Refinement of Multiconformer Ensemble Models from Multi-temperature X-ray Diffraction Data

Conformational ensembles underlie all protein functions. Thus, acquiring atomic-level ensemble models that accurately represent conformational heterogeneity is vital to deepen our understanding of how proteins work. Modeling ensemble information from X-ray diffraction data has been challenging, as traditional cryo-crystallography restricts conformational variability while minimizing radiation damage. Recent advances have enabled the collection of high quality diffraction data at ambient temperatures, revealing innate conformational heterogeneity and temperature-driven changes. Here, we used diffraction datasets for Proteinase K collected at temperatures ranging from 313 to 363K to provide a tutorial for the refinement of multiconformer ensemble models. Integrating automated sampling and refinement tools with manual adjustments, we obtained multiconformer models that describe alternative backbone and sidechain conformations, their relative occupancies, and interconnections between conformers. Our models revealed extensive and diverse conformational changes across temperature, including increased bound peptide ligand occupancies, different Ca2+ binding site configurations and altered rotameric distributions. These insights emphasize the value and need for multiconformer model refinement to extract ensemble information from diffraction data and to understand ensemble-function relationships.

FRETing about the details: Case studies in the use of a genetically encoded fluorescent amino acid for distance-dependent energy transfer

Förster resonance energy transfer (FRET) is a valuable method for monitoring protein conformation and biomolecular interactions. Intrinsically fluorescent amino acids that can be genetically encoded, such as acridonylalanine (Acd), are particularly useful for FRET studies. However, quantitative interpretation of FRET data to derive distance information requires careful use of controls and consideration of photophysical effects. Here we present two case studies illustrating how Acd can be used in FRET experiments to study small molecule induced conformational changes and multicomponent biomolecular complexes.

Lysozyme-phenolics bioconjugates with antioxidant and antibacterial bifunctionalities: Structural basis underlying the dual-function

This work aims at adopting an Electron Paramagnetic Resonance (EPR) spectroscopic technique to help understanding protein-phenolic conjugation and final functionalities relationship as well as the underlying structural basis of antioxidant and antibacterial dual functionalities. Specifically, lysozyme (Lys) was conjugated with two natural phenolic acids, i.e. rosmarinic acid (RA) and gentisic acid (GA, our previous work) with obviously different molecular features. Lys-RA displayed 8.6- and 4.0-times enhanced antioxidant stoichiometry compared to the native Lys and ones with GA, respectively, due to the stronger antioxidant activity of RA. However, RA conjugation mitigated both enzymatic and antibacterial activities of Lys-RA conjugates. Such inhibition effect is attributed to the greater structural and surface property changes of Lys upon conjugating with RA. Furthermore, the polyphenol conjugation related structural basis of disturbance, reactivity and selectivity were explored via site-directed spin labeling (SDSL)-EPR. A dynamic picture of reactivity and selectivity of phenolics conjugation on Lys was proposed.

Hysteresis behavior in the unfolding/refolding processes of a protein trapped in metallo-cages

Confinement of molecules in a synthetic host can physically isolate even their unstable temporary structures, which has potential for application to protein transient structure analysis. Here we report the NMR snapshot observation of protein unfolding and refolding processes by confining a target protein in a self-assembled coordination cage. With increasing acetonitrile content in CD₃CN/H₂O media (50 to 90 vol%), the folding structure of a protein sharply denatured at 83 vol%, clearly revealing the regions of initial unfolding. Unfavorable aggregation of the protein leading to irreversible precipitation is completely prevented because of the spatial isolation of the single protein molecule in the cage. When the acetonitrile content reversed (84 to 70 vol%), the once-denatured protein started to regain its original folded structure at 80 vol%, showing that the protein folding/unfolding process can be referred to as a phase transition with hysteresis behavior.

Learning continuous potentials from smFRET

Potential energy landscapes are useful models in describing events such as protein folding and binding. While single-molecule fluorescence resonance energy transfer (smFRET) experiments encode information on continuous potentials for the system probed, including rarely visited barriers between putative potential minima, this information is rarely decoded from the data. This is because existing analysis methods often model smFRET output assuming, from the onset, that the system probed evolves in a discretized state space to be analyzed within a hidden Markov model (HMM) paradigm. By contrast, here, we infer continuous potentials from smFRET data without discretely approximating the state space. We do so by operating within a Bayesian nonparametric paradigm by placing priors on the family of all possible potential curves. As our inference accounts for a number of required experimental features raising computational cost (such as incorporating discrete photon shot noise), the framework leverages a structured-kernel-interpolation Gaussian process prior to help curtail computational cost. We show that our structured-kernel-interpolation priors for potential energy reconstruction from smFRET analysis accurately infers the potential energy landscape from a smFRET binding experiment. We then illustrate advantages of structured-kernel-interpolation priors for potential energy reconstruction from smFRET over standard HMM approaches by providing information, such as barrier heights and friction coefficients, that is otherwise inaccessible to HMMs.

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.

Development of hidden Markov modeling method for molecular orientations and structure estimation from high-speed atomic force microscopy time-series images

High-speed atomic force microscopy (HS-AFM) is a powerful technique for capturing the time-resolved behavior of biomolecules. However, structural information in HS-AFM images is limited to the surface geometry of a sample molecule. Inferring latent three-dimensional structures from the surface geometry is thus important for getting more insights into conformational dynamics of a target biomolecule. Existing methods for estimating the structures are based on the rigid-body fitting of candidate structures to each frame of HS-AFM images. Here, we extend the existing frame-by-frame rigid-body fitting analysis to multiple frames to exploit orientational correlations of a sample molecule between adjacent frames in HS-AFM data due to the interaction with the stage. In the method, we treat HS-AFM data as time-series data, and they are analyzed with the hidden Markov modeling. Using simulated HS-AFM images of the taste receptor type 1 as a test case, the proposed method shows a more robust estimation of molecular orientations than the frame-by-frame analysis. The method is applicable in integrative modeling of conformational dynamics using HS-AFM data.

Cryogenic Fluorescence Spectroscopy of Ionic Fluorones in Gaseous and Condensed Phases: New Light on Their Intrinsic Photophysics

Fluorescence spectroscopy of gas-phase ions generated through electrospray ionization is an emerging technique able to probe intrinsic molecular photophysics directly without perturbations from solvent interactions. While there is ample scope for the ongoing development of gas-phase fluorescence techniques, the recent expansion into low-temperature operating conditions accesses a wealth of data on intrinsic fluorophore photophysics, offering enhanced spectral resolution compared with room-temperature measurements, without matrix effects hindering the excited-state dynamics. This perspective reviews current progress on understanding the photophysics of anionic fluorone dyes, which exhibit an unusually large Stokes shift in the gas phase, and discusses how comparison of gas- and condensed-phase fluorescence spectra can fingerprint structural dynamics. The capacity for temperature-dependent measurements of both fluorescence emission and excitation spectra helps establish the foundation for the use of fluorone dyes as fluorescent tags in macromolecular structure determination. We suggest ideas for technique development.

The Protein Unfolded State: One, No One and One Hundred Thousand

Many in vitro studies, in which proteins have been unfolded by the action of a variety of physical or chemical agents, have led to the definition of a folded versus an unfolded state and to the question of what is the nature of the unfolded state. The unstructured nature of this state could suggest that "the" unfolded state is a unique entity which holds true for all kinds of unfolding processes. This assumption has to be questioned because the unfolding processes under different stress conditions are dictated by entirely different mechanisms. As a consequence, it can be easily understood that the final state, generically referred to as "the unfolded state", can be completely different for each of the unfolding processes. The present review examines recent data on the characteristics of the unfolded states emerging from experiments under different conditions, focusing specific attention to the level of compaction of the unfolded species.

Effects of External Perturbations on Protein Systems: A Microscopic View

Protein folding can be viewed as the origami engineering of biology resulting from the long process of evolution. Even decades after its recognition, research efforts worldwide focus on demystifying molecular factors that underlie protein structure-function relationships; this is particularly relevant in the era of proteopathic disease. A complex co-occurrence of different physicochemical factors such as temperature, pressure, solvent, cosolvent, macromolecular crowding, confinement, and mutations that represent realistic biological environments are known to modulate the folding process and protein stability in unique ways. In the current review, we have contextually summarized the substantial efforts in unveiling individual effects of these perturbative factors, with major attention toward bottom-up approaches. Moreover, we briefly present some of the biotechnological applications of the insights derived from these studies over various applications including pharmaceuticals, biofuels, cryopreservation, and novel materials. Finally, we conclude by summarizing the challenges in studying the combined effects of multifactorial perturbations in protein folding and refer to complementary advances in experiment and computational techniques that lend insights to the emergent challenges.

Single-photon smFRET. I: Theory and conceptual basis

We present a unified conceptual framework and the associated software package for single-molecule Förster resonance energy transfer (smFRET) analysis from single-photon arrivals leveraging Bayesian nonparametrics, BNP-FRET. This unified framework addresses the following key physical complexities of a single-photon smFRET experiment, including: 1) fluorophore photophysics; 2) continuous time kinetics of the labeled system with large timescale separations between photophysical phenomena such as excited photophysical state lifetimes and events such as transition between system states; 3) unavoidable detector artefacts; 4) background emissions; 5) unknown number of system states; and 6) both continuous and pulsed illumination. These physical features necessarily demand a novel framework that extends beyond existing tools. In particular, the theory naturally brings us to a hidden Markov model with a second-order structure and Bayesian nonparametrics on account of items 1, 2, and 5 on the list. In the second and third companion articles, we discuss the direct effects of these key complexities on the inference of parameters for continuous and pulsed illumination, respectively.

Evolutionary adaptation of the protein folding pathway for secretability

Secretory preproteins of the Sec pathway are targeted post-translationally and cross cellular membranes through translocases. During cytoplasmic transit, mature domains remain non-folded for translocase recognition/translocation. After translocation and signal peptide cleavage, mature domains fold to native states in the bacterial periplasm or traffic further. We sought the structural basis for delayed mature domain folding and how signal peptides regulate it. We compared how evolution diversified a periplasmic peptidyl-prolyl isomerase PpiA mature domain from its structural cytoplasmic PpiB twin. Global and local hydrogen-deuterium exchange mass spectrometry showed that PpiA is a slower folder. We defined at near-residue resolution hierarchical folding initiated by similar foldons in the twins, at different order and rates. PpiA folding is delayed by less hydrophobic native contacts, frustrated residues and a β-turn in the earliest foldon and by signal peptide-mediated disruption of foldon hierarchy. When selected PpiA residues and/or its signal peptide were grafted onto PpiB, they converted it into a slow folder with enhanced in vivo secretion. These structural adaptations in a secretory protein facilitate trafficking.

A review on application of molecular simulation technology in food molecules interaction

Molecular simulation is a new technology to analyze the interaction between molecules. This review mainly summarizes the application of molecular simulation technology in the food industry. This technology has been employed to assess structural changes of biomolecules, the interaction between components, and the mechanism of physical and chemical property alterations. These conclusions provide a deeper understanding of the molecular interaction mechanism in foods, break through the limitations of scientific experiments and avoid blind and time-consuming scientific research. In this paper, the advantages and development trends of molecular simulation technology in the food research field are described. This methodology can be used to contribute to further studies of the mechanism of molecular interactions in food, confirm experimental results and provide new ideas for research in the field of food sciences.

Mars, a molecule archive suite for reproducible analysis and reporting of single-molecule properties from bioimages

The rapid development of new imaging approaches is generating larger and more complex datasets, revealing the time evolution of individual cells and biomolecules. Single-molecule techniques, in particular, provide access to rare intermediates in complex, multistage molecular pathways. However, few standards exist for processing these information-rich datasets, posing challenges for wider dissemination. Here, we present Mars, an open-source platform for storing and processing image-derived properties of biomolecules. Mars provides Fiji/ImageJ2 commands written in Java for common single-molecule analysis tasks using a Molecule Archive architecture that is easily adapted to complex, multistep analysis workflows. Three diverse workflows involving molecule tracking, multichannel fluorescence imaging, and force spectroscopy, demonstrate the range of analysis applications. A comprehensive graphical user interface written in JavaFX enhances biomolecule feature exploration by providing charting, tagging, region highlighting, scriptable dashboards, and interactive image views. The interoperability of ImageJ2 ensures Molecule Archives can easily be opened in multiple environments, including those written in Python using PyImageJ, for interactive scripting and visualization. Mars provides a flexible solution for reproducible analysis of image-derived properties, facilitating the discovery and quantitative classification of new biological phenomena with an open data format accessible to everyone.

The role of structural dynamics in the thermal adaptation of hyperthermophilic enzymes

Proteins from hyperthermophilic organisms are evolutionary optimised to adopt functional structures and dynamics under conditions in which their mesophilic homologues are generally inactive or unfolded. Understanding the nature of such adaptation is of crucial interest to clarify the underlying mechanisms of biological activity in proteins. Here we measured NMR residual dipolar couplings of a hyperthermophilic acylphosphatase enzyme at 80°C and used these data to generate an accurate structural ensemble representative of its native state. The resulting energy landscape was compared to that obtained for a human homologue at 37°C, and additional NMR experiments were carried out to probe fast (¹⁵N relaxation) and slow (H/D exchange) backbone dynamics, collectively sampling fluctuations of the two proteins ranging from the nanosecond to the millisecond timescale. The results identified key differences in the strategies for protein-protein and protein-ligand interactions of the two enzymes at the respective physiological temperatures. These include the dynamical behaviour of a β-strand involved in the protection against aberrant protein aggregation and concerted motions of loops involved in substrate binding and catalysis. Taken together these results elucidate the structure-dynamics-function relationship associated with the strategies of thermal adaptation of protein molecules.

Visualizing the Conformational Dynamics of Membrane Receptors Using Single-Molecule FRET

The ability of cells to respond to external signals is essential for cellular development, growth, and survival. To respond to a signal from the environment, a cell must be able to recognize and process it. This task mainly relies on the function of membrane receptors, whose role is to convert signals into the biochemical language of the cell. G protein-coupled receptors (GPCRs) constitute the largest family of membrane receptor proteins in humans. Among GPCRs, metabotropic glutamate receptors (mGluRs) are a unique subclass that function as obligate dimers and possess a large extracellular domain that contains the ligand-binding site. Recent advances in structural studies of mGluRs have improved the understanding of their activation process. However, the propagation of large-scale conformational changes through mGluRs during activation and modulation is poorly understood. Single-molecule fluorescence resonance energy transfer (smFRET) is a powerful technique to visualize and quantify the structural dynamics of biomolecules at the single-protein level. To visualize the dynamic process of mGluR2 activation, fluorescent conformational sensors based on unnatural amino acid (UAA) incorporation were developed that allowed site-specific protein labeling without perturbation of the native structure of receptors. The protocol described here explains how to perform these experiments, including the novel UAA labeling approach, sample preparation, and smFRET data acquisition and analysis. These strategies are generalizable and can be extended to investigate the conformational dynamics of a variety of membrane proteins.

Multisite phosphorylation and binding alter conformational dynamics of the 4E-BP2 protein

Intrinsically disordered proteins (IDPs) play critical roles in regulatory protein interactions, but detailed structural/dynamic characterization of their ensembles remain challenging, both in isolation and when they form dynamic "fuzzy" complexes. Such is the case for mRNA cap-dependent translation initiation, which is regulated by the interaction of the predominantly folded eukaryotic initiation factor 4E (eIF4E) with the intrinsically disordered eIF4E binding proteins (4E-BPs) in a phosphorylation-dependent manner. Single-molecule Förster resonance energy transfer showed that the conformational changes of 4E-BP2 induced by binding to eIF4E are non-uniform along the sequence; while a central region containing both motifs that bind to eIF4E expands and becomes stiffer, the C-terminal region is less affected. Fluorescence anisotropy decay revealed a non-uniform segmental flexibility around six different labeling sites along the chain. Dynamic quenching of these fluorescent probes by intrinsic aromatic residues measured via fluorescence correlation spectroscopy report on transient intra- and inter-molecular contacts on nanosecond-to-microsecond timescales. Upon hyperphosphorylation, which induces folding of ∼40 residues in 4E-BP2, the quenching rates decreased at most labeling sites. The chain dynamics around sites in the C-terminal region far away from the two binding motifs significantly increased upon binding to eIF4E, suggesting that this region is also involved in the highly dynamic 4E-BP2:eIF4E complex. Our time-resolved fluorescence data paint a sequence-level rigidity map of three states of 4E-BP2 differing in phosphorylation or binding status and distinguish regions that form contacts with eIF4E. This study adds complementary structural and dynamics information to recent studies of 4E-BP2, and it constitutes an important step toward a mechanistic understanding of this important IDP via integrative modeling.

Inferring potential landscapes from noisy trajectories of particles within an optical feedback trap

While particle trajectories encode information on their governing potentials, potentials can be challenging to robustly extract from trajectories. Measurement errors may corrupt a particle’s position, and sparse sampling of the potential limits data in higher energy regions such as barriers. We develop a Bayesian method to infer potentials from trajectories corrupted by Markovian measurement noise without assuming prior functional form on the potentials. As an alternative to Gaussian process priors over potentials, we introduce structured kernel interpolation to the Natural Sciences which allows us to extend our analysis to large datasets. Structured-Kernel-Interpolation Priors for Potential Energy Reconstruction (SKIPPER) is validated on 1D and 2D experimental trajectories for particles in a feedback trap.

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A feedback trap was used to generate noisy Langevin microbead trajectories

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The potential energy surface is recovered using a Bayesian formulation

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The formulation uses a structured-kernel-interpolation Gaussian process (SKI-GP) to tractably approximate Gaussian process regression for larger datasets

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Thanks to our adaptation of SKI-GP, we have broadened the use of Gaussian processes for natural science applications

Cooperativity and Folding Kinetics in a Multidomain Protein with Interwoven Chain Topology

Although a large percentage of eukaryotic proteomes consist of proteins with multiple domains, not much is known about their assembly mechanism, especially those with intricate native state architectures. Some have a complex topology in which the structural elements along the sequence are interwoven in such a manner that the domains cannot be separated by cutting at any location along the sequence. Such proteins are multiply connected multidomain proteins (MMPs) with the three-domain (NMP, LID, and CORE) phosphotransferase enzyme adenylate kinase (ADK) being an example. We devised a coarse-grained model to simulate ADK folding initiated by changing either the temperature or guanidinium chloride (GdmCl) concentration. The simulations reproduce the experimentally measured melting temperatures (associated with two equilibrium transitions), FRET efficiency as a function of GdmCl concentration, and the folding times quantitatively. Although the NMP domain orders independently, cooperative interactions between the LID and the CORE domains are required for complete assembly of the enzyme. Kinetic simulations show that, on the collapse time scale, multiple interconnected metastable states are populated, attesting to the folding heterogeneity. The network of kinetically connected states reveals that the CORE domain folds only after the NMP and LID domains, reflecting the interwoven nature of the chain topology.

Combining High-Pressure NMR and Geometrical Sampling to Obtain a Full Topological Description of Protein Folding Landscapes: Application to the Folding of Two MAX Effectors from Magnaporthe oryzae

Despite advances in experimental and computational methods, the mechanisms by which an unstructured polypeptide chain regains its unique three-dimensional structure remains one of the main puzzling questions in biology. Single-molecule techniques, ultra-fast perturbation and detection approaches and improvement in all-atom and coarse-grained simulation methods have greatly deepened our understanding of protein folding and the effects of environmental factors on folding landscape. However, a major challenge remains the detailed characterization of the protein folding landscape. Here, we used high hydrostatic pressure 2D NMR spectroscopy to obtain high-resolution experimental structural information in a site-specific manner across the polypeptide sequence and along the folding reaction coordinate. We used this residue-specific information to constrain Cyana3 calculations, in order to obtain a topological description of the entire folding landscape. This approach was used to describe the conformers populating the folding landscape of two small globular proteins, AVR-Pia and AVR-Pib, that belong to the structurally conserved but sequence-unrelated MAX effectors superfamily. Comparing the two folding landscapes, we found that, in spite of their divergent sequences, the folding pathway of these two proteins involves a similar, inescapable, folding intermediate, even if, statistically, the routes used are different.

Progress and perspectives in single-molecule optical spectroscopy

We review some of the progress of single-molecule optical experiments in the past 20 years and propose some perspectives for the coming years. We particularly focus on methodological advances in fluorescence, super-resolution, photothermal contrast, and interferometric scattering and briefly discuss a few of the applications. These advances have enabled the exploration of new emitters and quantum optics; the chemistry and biology of complex heterogeneous systems, nanoparticles, and plasmonics; and the detection and study of non-fluorescing and non-absorbing nano-objects. We conclude by proposing some ideas for future experiments. The field will move toward more and better signals of a broader variety of objects and toward a sharper view of the surprising complexity of the nanoscale world of single (bio-)molecules, nanoparticles, and their nano-environments.

NMR Provides Unique Insight into the Functional Dynamics and Interactions of Intrinsically Disordered Proteins

Intrinsically disordered proteins are ubiquitous throughout all known proteomes, playing essential roles in all aspects of cellular and extracellular biochemistry. To understand their function, it is necessary to determine their structural and dynamic behavior and to describe the physical chemistry of their interaction trajectories. Nuclear magnetic resonance is perfectly adapted to this task, providing ensemble averaged structural and dynamic parameters that report on each assigned resonance in the molecule, unveiling otherwise inaccessible insight into the reaction kinetics and thermodynamics that are essential for function. In this review, we describe recent applications of NMR-based approaches to understanding the conformational energy landscape, the nature and time scales of local and long-range dynamics and how they depend on the environment, even in the cell. Finally, we illustrate the ability of NMR to uncover the mechanistic basis of functional disordered molecular assemblies that are important for human health.

Generating Ensembles of Dynamic Misfolding Proteins

The early stages of protein misfolding and aggregation involve disordered and partially folded protein conformers that contain a high degree of dynamic disorder. These dynamic species may undergo large-scale intra-molecular motions of intrinsically disordered protein (IDP) precursors, or flexible, low affinity inter-molecular binding in oligomeric assemblies. In both cases, generating atomic level visualization of the interconverting species that captures the conformations explored and their physico-chemical properties remains hugely challenging. How specific sub-ensembles of conformers that are on-pathway to aggregation into amyloid can be identified from their aggregation-resilient counterparts within these large heterogenous pools of rapidly moving molecules represents an additional level of complexity. Here, we describe current experimental and computational approaches designed to capture the dynamic nature of the early stages of protein misfolding and aggregation, and discuss potential challenges in describing these species because of the ensemble averaging of experimental restraints that arise from motions on the millisecond timescale. We give a perspective of how machine learning methods can be used to extract aggregation-relevant sub-ensembles and provide two examples of such an approach in which specific interactions of defined species within the dynamic ensembles of α-synuclein (αSyn) and β2-microgloblulin (β2m) can be captured and investigated.

Impact of Molecule Concentration, Diffusion Rates and Surface Passivation on Single-Molecule Fluorescence Studies in Solution

For single-molecule studies in solution, very small concentrations of dye-labelled molecules are employed in order to achieve single-molecule sensitivity. In typical studies with confocal microscopes, often concentrations in the pico-molar regime are required. For various applications that make use of single-molecule Förster resonance energy transfer (smFRET) or two-color coincidence detection (TCCD), the molecule concentration must be set explicitly to targeted values and furthermore needs to be stable over a period of several hours. As a consequence, specific demands must be imposed on the surface passivation of the cover slides during the measurements. The aim of having only one molecule in the detection volume at the time is not only affected by the absolute molecule concentration, but also by the rate of diffusion. Therefore, we discuss approaches to control and to measure absolute molecule concentrations. Furthermore, we introduce an approach to calculate the probability of chance coincidence events and demonstrate that measurements with challenging smFRET samples require a strict limit of maximal sample concentrations in order to produce meaningful results.

Heterogeneity in Protein Folding and Unfolding Reactions

Proteins have dynamic structures that undergo chain motions on time scales spanning from picoseconds to seconds. Resolving the resultant conformational heterogeneity is essential for gaining accurate insight into fundamental mechanistic aspects of the protein folding reaction. The use of high-resolution structural probes, sensitive to population distributions, has begun to enable the resolution of site-specific conformational heterogeneity at different stages of the folding reaction. Different states populated during protein folding, including the unfolded state, collapsed intermediate states, and even the native state, are found to possess significant conformational heterogeneity. Heterogeneity in protein folding and unfolding reactions originates from the reduced cooperativity of various kinds of physicochemical interactions between various structural elements of a protein, and between a protein and solvent. Heterogeneity may arise because of functional or evolutionary constraints. Conformational substates within the unfolded state and the collapsed intermediates that exchange at rates slower than the subsequent folding steps give rise to heterogeneity on the protein folding pathways. Multiple folding pathways are likely to represent distinct sequences of structure formation. Insight into the nature of the energy barriers separating different conformational states populated during (un)folding can also be obtained by resolving heterogeneity.

Structural Stability Analysis of Proteins Using End-to-End Distance: A 3D-RISM Approach

The stability of a protein is determined from its properties and surrounding solvent. In our previous study, the total energy as a sum of the conformational and solvation free energies was demonstrated to be an appropriate energy function for evaluating the stability of a protein in a protein folding system. We plotted the various energies against the root mean square deviation, required as a reference structure. Herein, we replotted the various energies against the end-to-end distance between the N- and C-termini, which is not a required reference and is experimentally measurable. The solvation free energies for all proteins tend to be low as the end-to-end distance increases, whereas the conformational energies tend to be low as the end-to-end distance decreases. The end-to-end distance is one of interesting measures to study the behavior of proteins.

Single-molecule FRET combined with electrokinetic trapping reveals real-time enzyme kinetics of individual F-ATP synthases

Single-molecule Förster resonance energy transfer (smFRET) is a key technique to observe conformational changes in molecular motors and to access the details of single-molecule static and dynamic disorder during catalytic processes. However, studying freely diffusing molecules in solution is limited to a few tens of milliseconds, while surface attachment often bears the risk to restrict their natural motion. In this paper we combine smFRET and electrokinetic trapping (ABEL trap) to non-invasively hold single F_OF₁-ATP synthases for up to 3 s within the detection volume, thereby extending the observation time by a factor of 10 as compared to Brownian diffusion without surface attachment. In addition, we are able to monitor complete reaction cycles and to selectively trap active molecules based on their smFRET signal, thus speeding up the data acquisition process. We demonstrate the capability of our method to study the dynamics of single molecules by recording the ATP-hydrolysis driven rotation of individual F_OF₁-ATP synthase molecules over numerous reaction cycles and extract their kinetic rates. We argue that our method is not limited to motor proteins. Instead, it can be applied to monitor conformational changes with millisecond time resolution for a wide range of enzymes, thereby making it a versatile tool for studying protein dynamics.

Above pile-up fluorescence microscopy with a 32 Mc/s single-channel time-resolved SPAD system

One of the major drawbacks of time-correlated single-photon counting (TCSPC) is generally represented by pile-up distortion, which strongly bounds the maximum acquisition speed to a few percent of the laser excitation rate. Based on a previous theoretical analysis, recently we presented the first, to the best of our knowledge, low-distortion and high-speed TCSPC system capable of overcoming the pile-up limitation by perfectly matching the single-photon avalanche diode (SPAD) dead time to the laser period. In this work, we validate the proposed system in a standard fluorescence measurement by comparing experimental data with the reference theoretical framework. As a result, a count rate of 32 Mc/s was achieved with a single-channel system still observing a negligible lifetime distortion.

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.

Single-Molecule Fluorescence Spectroscopy Approaches for Probing Fast Biomolecular Dynamics and Interactions

Single-molecule fluorescence spectroscopy, and particularly its Förster resonance energy transfer implementation (SM-FRET), provides the opportunity to resolve the stochastic conformational fluctuations undergone by individual protein molecules while they fold-unfold, bind to their partners, or carry out catalysis. Such information is key to resolve the microscopic pathways and mechanisms underlying such processes, and cannot be obtained from bulk experiments. To fully resolve protein conformational dynamics, SM-FRET experiments need to reach microsecond, and even sub-microsecond, time resolutions. The key to reach such resolution lies in increasing the efficiency at which photons emitted by a single molecule are collected and detected by the instrument (photon count rates). In this chapter, we describe basic procedures that an end user can follow to optimize the confocal microscope optics in order to maximize the photon count rates. We also discuss the use of photoprotection cocktails specifically designed to reduce fluorophore triplet buildup at high irradiance (the major cause of limiting photon emission rates) while improving the mid-term photostability of the fluorophores. Complementary strategies based on the data analysis are discussed in depth by other authors in Chap. 14 .

On distributions of barrier crossing times as observed in single-molecule studies of biomolecules

Single-molecule experiments that monitor time evolution of molecular observables in real time have expanded beyond measuring transition rates toward measuring distributions of times of various molecular events. Of particular interest is the first-passage time for making a transition from one molecular configuration ( a ) to another ( b ) and conditional first-passage times such as the transition path time, which is the first-passage time from a to b conditional upon not leaving the transition region intervening between a and b . Another experimentally accessible (but not yet studied experimentally) observable is the conditional exit time, i.e., the time to leave the transition region through a specified boundary. The distributions of such times contain a wealth of mechanistic information about the transitions in question. Here, we use the first and the second (and, if desired, higher) moments of these distributions to characterize their relative width for the model in which the experimental observable undergoes Brownian motion in a potential of mean force. We show that although the distributions of transition path times are always narrower than exponential (in that the ratio of the standard deviation to the distribution's mean is always less than 1), distributions of first-passage times and of conditional exit times can be either narrow or broad, in some cases displaying long power-law tails. The conditional exit time studied here provides a generalization of the transition path time that also allows one to characterize the temporal scales of failed barrier crossing attempts.

Quantitative Description of Intrinsically Disordered Proteins Using Single-Molecule FRET, NMR, and SAXS

Studying the conformational landscape of intrinsically disordered and partially folded proteins is challenging and only accessible to a few solution state techniques, such as nuclear magnetic resonance (NMR), small-angle scattering techniques, and single-molecule Förster resonance energy transfer (smFRET). While each of the techniques is sensitive to different properties of the disordered chain, such as local structural propensities, overall dimension, or intermediate- and long-range contacts, conformational ensembles describing intrinsically disordered proteins (IDPs) accurately should ideally respect all of these properties. Here we develop an integrated approach using a large set of FRET efficiencies and fluorescence lifetimes, NMR chemical shifts, and paramagnetic relaxation enhancements (PREs), as well as small-angle X-ray scattering (SAXS) to derive quantitative conformational ensembles in agreement with all parameters. Our approach is tested using simulated data (five sets of PREs and 15 FRET efficiencies) and validated experimentally on the example of the disordered domain of measles virus phosphoprotein, providing new insights into the conformational landscape of this viral protein that comprises transient structural elements and is more compact than an unfolded chain throughout its length. Rigorous cross-validation using FRET efficiencies, fluorescence lifetimes, and SAXS demonstrates the predictive nature of the calculated conformational ensembles and underlines the potential of this strategy in integrative dynamic structural biology.