Observation of strange kinetics in protein folding

Connecting Protein Millisecond Conformational Dynamics to Protein Thermal Stability

The stability of protein folded states is crucial for its function, yet the relationship with the protein sequence remains poorly understood. Prior studies have focused on the amino acid composition and thermodynamic couplings within a single folded conformation, overlooking the potential contribution of protein dynamics. Here, we address this gap by systematically analyzing the impact of alanine mutations in the C-terminal β-strand (β5) of ubiquitin, a model protein exhibiting millisecond timescale interconversion between two conformational states differing in the β5 position. Our findings unveil a negative correlation between millisecond dynamics and thermal stability, with alanine substitutions at seemingly flexible C-terminal residues significantly enhancing thermostability. Integrating spectroscopic and computational approaches, we demonstrate that the thermally unfolded state retains a substantial secondary structure but lacks β5 engagement, recapitulating the transition state for millisecond dynamics. Thus, alanine mutations that modulate the stabilities of the folded states with respect to the partially unfolded state impact both the dynamics and stability. Our findings underscore the importance of conformational dynamics with implications for protein engineering and design.

Metastability of Protein Solution Structures in the Absence of a Solvent: Rugged Energy Landscape and Glass-like Behavior

Native ion mobility/mass spectrometry is well-poised to structurally screen proteomes but characterizes protein structures in the absence of a solvent. This raises long-standing unanswered questions about the biological significance of protein structures identified through ion mobility/mass spectrometry. Using newly developed computational and experimental ion mobility/ion mobility/mass spectrometry methods, we investigate the unfolding of the protein ubiquitin in a solvent-free environment. Our data suggest that the folded, solvent-free ubiquitin observed by ion mobility/mass spectrometry exists in a largely native fold with an intact β-grasp motif and α-helix. The ensemble of folded, solvent-free ubiquitin ions can be partitioned into kinetically stable subpopulations that appear to correspond to the structural heterogeneity of ubiquitin in solution. Time-resolved ion mobility/ion mobility/mass spectrometry measurements show that folded, solvent-free ubiquitin exhibits a strongly stretched-exponential time dependence, which simulations trace to a rugged energy landscape with kinetic traps. Unfolding rate constants are estimated to be approximately 800 to 20,000 times smaller than in the presence of water, effectively quenching the unfolding process on the time scale of typical ion mobility/mass spectrometry measurements. Our proposed unfolding pathway of solvent-free ubiquitin shares substantial characteristics with that established for the presence of solvent, including a polarized transition state with significant native content in the N-terminal β-hairpin and α-helix. Our experimental and computational data suggest that (1) the energy landscape governing the motions of folded, solvent-free proteins is rugged in analogy to that of glassy systems; (2) large-scale protein motions may at least partially be determined by the amino acid sequence of a polypeptide chain; and (3) solvent facilitates, rather than controls, protein motions.

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Appearance of de Gennes length in force-induced transitions

Using Langevin dynamic simulations, a simple coarse-grained model of a DNA protein construct is used to study the DNA rupture and the protein unfolding. We identify three distinct states: (i) zipped DNA and collapsed protein, (ii) unzipped DNA and stretched protein, and (iii) unzipped DNA and collapsed protein. Here, we find a phase diagram that shows these states depending on the size of the DNA handle and the protein. For a less stable protein, unfolding is solely governed by the size of the linker DNA, whereas if the protein's stability increases, complete unfolding becomes impossible because the rupture force for DNA has reached a saturation regime influenced by the de Gennes length. We show that unfolding occurs via a few intermediate states by monitoring the force-extension curve of the entire protein. We extend our study to a heterogeneous protein system, where similar intermediate states in two systems can lead to different protein unfolding paths.

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.

Dynamical Signatures of Multifunnel Energy Landscapes

Multifunctional systems, such as molecular switches, exhibit multifunnel energy landscapes associated with the alternative functional states. In this contribution the multifunnel organization is decoded from dynamical signatures in the first passage time distribution between reactants and products. Characteristic relaxation rates are revealed by analyzing the kinetics as a function of the observation time scale, which scans the underlying distribution. Extracting the corresponding dynamical signatures provides direct insight into the organization of the molecular energy landscape, which will facilitate a rational design of target functionality. Examples are illustrated for multifunnel landscapes in biomolecular systems and an atomic cluster.

Beyond the Average: Spatial and Temporal Fluctuations in Oxide Glass-Forming Systems

Atomic structure dictates the performance of all materials systems; the characteristic of disordered materials is the significance of spatial and temporal fluctuations on composition-structure-property-performance relationships. Glass has a disordered atomic arrangement, which induces localized distributions in physical properties that are conventionally defined by average values. Quantifying these statistical distributions (including variances, fluctuations, and heterogeneities) is necessary to describe the complexity of glass-forming systems. Only recently have rigorous theories been developed to predict heterogeneities to manipulate and optimize glass properties. This article provides a comprehensive review of experimental, computational, and theoretical approaches to characterize and demonstrate the effects of short-, medium-, and long-range statistical fluctuations on physical properties (e.g., thermodynamic, kinetic, mechanical, and optical) and processes (e.g., relaxation, crystallization, and phase separation), focusing primarily on commercially relevant oxide glasses. Rigorous investigations of fluctuations enable researchers to improve the fundamental understanding of the chemistry and physics governing glass-forming systems and optimize structure-property-performance relationships for next-generation technological applications of glass, including damage-resistant electronic displays, safer pharmaceutical vials to store and transport vaccines, and lower-attenuation fiber optics. We invite the reader to join us in exploring what can be discovered by going beyond the average.

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.

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.

Applying text mining to identify relevant literature in food science: Cold denaturation as a case study

In a research environment characterized by the five V's of big data, volume, velocity, variety, value, and veracity, the need to develop tools that quickly screen a large number of publications into relevant work is an increasing area of concern, and the data-rich food industry is no exception. Here, a combination of latent Dirichlet allocation and food keyword searches were employed to analyze and filter a dataset of 6102 publications about cold denaturation. After using the Python toolkit generated in this work, the approach yielded 22 topics that provide background and insight on the direction of research in this field, as well as identified the publications in this dataset which are most pertinent to the food industry with precision and recall of 0.419 and 0.949, respectively. Precision is related to the relevance of a paper in the filtered dataset and the recall represents papers which were not identified in the screening method. Lastly, gaps in the literature based on keyword trends are identified to improve the knowledge base of cold denaturation as it relates to the food industry. This approach is generalizable to any similarly organized dataset, and the code is available upon request. Practical Application: A common problem in research is that when you are an expert in one field, learning about another field is difficult, because you may lack the vocabulary and background needed to read cutting edge literature from a new discipline. The Python toolkit developed in this research can be applied by any researcher that is new to a field to identify what the key literature is, what topics they should familiarize themselves with, and what the current trends are in the field. Using this structure, researchers can greatly speed up how they identify new areas to research and find new projects.

Nonrefoldability is Pervasive Across the E. coli Proteome

Decades of research on protein folding have primarily focused on a subset of small proteins that can reversibly refold from a denatured state. However, these studies have generally not been representative of the complexity of natural proteomes, which consist of many proteins with complex architectures and domain organizations. Here, we introduce an experimental approach to probe protein refolding kinetics for whole proteomes using mass spectrometry-based proteomics. Our study covers the majority of the soluble E. coli proteome expressed during log-phase growth, and among this group, we find that one-third of the E. coli proteome is not intrinsically refoldable on physiological time scales, a cohort that is enriched with certain fold-types, domain organizations, and other biophysical features. We also identify several properties and fold-types that are correlated with slow refolding on the minute time scale. Hence, these results illuminate when exogenous factors and processes, such as chaperones or cotranslational folding, might be required for efficient protein folding.

Intrinsic Dynamics of Protein-Peptide Unbinding

The dynamics of peptide-protein binding and unbinding of a variant of the RNase S system has been investigated. To initiate the process, a photoswitchable azobenzene moiety has been covalently linked to the S-peptide, thereby switching its binding affinity to the S-protein. Transient fluorescence quenching was measured with the help of a time-resolved fluorometer, which has been specifically designed for these experiments and is based on inexpensive light-emitting diodes and laser diodes only. One mutant shows on-off behavior with no specific binding detectable in one of the states of the photoswitch. Unbinding is faster by at least 2 orders of magnitude, compared to that of other variants of the RNase S system. We conclude that unbinding is essentially barrier-less in that case, revealing the intrinsic dynamics of the unbinding event, which occurs on a time scale of a few hundred microseconds in a strongly stretched-exponential manner.

Lessons about Protein Folding and Binding from Archetypal Folds

The function of proteins as biological nanomachines relies on their ability to fold into complex 3D structures, bind selectively to partners, and undergo conformational changes on cue. The native functional structures, and the rates of interconversion between conformational states (folded-unfolded, bound-free), are all encoded in the physical chemistry of their amino acid sequence. However, despite extensive research over decades, this code has proven difficult to fully crack, in terms of both prediction and understanding the molecular mechanisms at play.Earlier work on single-domain proteins reported a commonality of slow rates (10^-2-10² s^-1) and simple behavior in both kinetic and thermodynamic unfolding experiments, which suggested the process was all-or-none and thereby analogous to a chemical reaction (e.g., A ⇄ B). In the absence of a first-principles pre-exponential factor for protein (un)folding dynamics, the rates could only be interpreted in relative terms, e.g., the changes induced by mutation, and hence, neither the height of nor the entropic contribution to the free energy barriers was known. The rates were also many orders of magnitude too slow for direct atomistic simulations, and the computational focus was on predicting rate changes induced by mutation via coarse grained simulations. However, even the effects of mutation proved to be strikingly homogeneous with all experimental data clustering at ∼1/3 of the free energy perturbation recovered on folding and ∼2/3 on unfolding.The implementation of ultrafast kinetic methods turned the field upside down because they allowed researchers to measure the time scales of elementary (un)folding motions, which set the pre-exponential factor for protein conformational transitions at ∼1 μs. In parallel, we and others set out to investigate the simplest possible protein structures capable of autonomous folding, which we defined as archetypal folds. The rationale was to recapitulate the hierarchical organization of protein structure, starting from the bottom up. The study of fold archetypes ended up opening new research avenues in protein (un)folding, but also making unexpected connections with the folding upon binding of intrinsically disordered proteins and suggesting their functioning as conformational rheostats.This Account describes our work on the kinetic, thermodynamic, mechanistic, and functional analysis of fold archetypes. We first discuss the kinetic studies, emphasizing their impact on our understanding of (un)folding rates, of barrierless (downhill) folding, and as benchmarks for atomistic simulations. We continue with the thermodynamic analysis, introducing the differential scanning calorimetry, multiprobe, and NMR approaches that we developed to dissect their gradual, minimally cooperative (un)folding transitions and to probe the underlying mechanisms with unprecedented detail. The last two sections cover single-molecule analyses and some recent, mostly computational, results on the exploration of possible biological and technological roles for the gradual conformational transitions of fold archetypes.

Real-time observation of ligand-induced allosteric transitions in a PDZ domain

While allostery is of paramount importance for protein regulation, the underlying dynamical process of ligand (un)binding at one site, resulting time evolution of the protein structure, and change of the binding affinity at a remote site are not well understood. Here the ligand-induced conformational transition in a widely studied model system of allostery, the PDZ2 domain, is investigated by transient infrared spectroscopy accompanied by molecular dynamics simulations. To this end, an azobenzene-derived photoswitch is linked to a peptide ligand in a way that its binding affinity to the PDZ2 domain changes upon switching, thus initiating an allosteric transition in the PDZ2 domain protein. The subsequent response of the protein, covering four decades of time, ranging from ∼1 ns to ∼μs, can be rationalized by a remodeling of its rugged free-energy landscape, with very subtle shifts in the populations of a small number of structurally well-defined states. It is proposed that structurally and dynamically driven allostery, often discussed as limiting scenarios of allosteric communication, actually go hand-in-hand, allowing the protein to adapt its free-energy landscape to incoming signals.

Temperature-Jump 2D IR Spectroscopy with Intensity-Modulated CW Optical Heating

Pulsed temperature-jump (T-jump) spectroscopy with infrared (IR) detection has been widely used to study biophysical processes occurring from nanoseconds to ∼1 ms with structural sensitivity. However, many systems exhibit structural dynamics on time scales longer than the millisecond barrier that is set by the time scale for thermal relaxation of the sample. We developed a linear and nonlinear infrared spectrometer coupled to an intensity-modulated continuous wave (CW) laser to probe T-jump-initiated chemical reactions from <1 ms to seconds. Time-dependent modulation of the CW laser leads to a <1 ms heating time as well as a constant final temperature (±3%) for the duration of the heating time. Temperature changes of up to 75 °C in D₂O are demonstrated, allowing for nonequilibrium measurements inaccessible to standard pulsed optical T-jump setups. T-jump linear absorption, pump-probe, and two-dimensional IR (2D IR) spectroscopy are applied to the unfolding and refolding of ubiquitin and a model intercalated motif (i-motif) DNA sequence, and analysis of the observed signals is used to demonstrate the limits and utility of each method. Overall, the ability to probe temperature-induced chemical processes from <1 ms to many seconds with 2D IR spectroscopy provides multiple new avenues for time-dependent spectroscopy in chemistry and biophysics.

Glassy Dynamics and Memory Effects in an Intrinsically Disordered Protein Construct

Glassy, nonexponential relaxations in globular proteins are typically attributed to conformational behaviors that are missing from intrinsically disordered proteins. Yet, we show that single molecules of a disordered-protein construct display two signatures of glassy dynamics, logarithmic relaxations and a Kovacs memory effect, in response to changes in applied tension. We attribute this to the presence of multiple independent local structures in the chain, which we corroborate with a model that correctly predicts the force dependence of the relaxation. The mechanism established here likely applies to other disordered proteins.

Protein folding from heterogeneous unfolded state revealed by time-resolved X-ray solution scattering

One of the most challenging tasks in biological science is to understand how a protein folds. In theoretical studies, the hypothesis adopting a funnel-like free-energy landscape has been recognized as a prominent scheme for explaining protein folding in views of both internal energy and conformational heterogeneity of a protein. Despite numerous experimental efforts, however, comprehensively studying protein folding with respect to its global conformational changes in conjunction with the heterogeneity has been elusive. Here we investigate the redox-coupled folding dynamics of equine heart cytochrome c (cyt-c) induced by external electron injection by using time-resolved X-ray solution scattering. A systematic kinetic analysis unveils a kinetic model for its folding with a stretched exponential behavior during the transition toward the folded state. With the aid of the ensemble optimization method combined with molecular dynamics simulations, we found that during the folding the heterogeneously populated ensemble of the unfolded state is converted to a narrowly populated ensemble of folded conformations. These observations obtained from the kinetic and the structural analyses of X-ray scattering data reveal that the folding dynamics of cyt-c accompanies many parallel pathways associated with the heterogeneously populated ensemble of unfolded conformations, resulting in the stretched exponential kinetics at room temperature. This finding provides direct evidence with a view to microscopic protein conformations that the cyt-c folding initiates from a highly heterogeneous unfolded state, passes through still diverse intermediate structures, and reaches structural homogeneity by arriving at the folded state.

A new way to recognize downhill folding based on generalized path length

The protein folding is an important scientific problem and many methods were designed to elucidate the protein folding and obtain insight into the molecular mechanism. A novel means is presented to identify the downhill pathways of protein folding in this paper. This method is based on barrier energy profile projected onto the generalized path length (GPL) with Breadth-first searching (BFS) algorithm. We show the effectiveness of this approach by constructing the barrier energy profile of trpzip2 and comparing with other methods.

Protein unfolding by SDS: the microscopic mechanisms and the properties of the SDS-protein assembly

The effects of detergent sodium dodecyl sulfate (SDS) on protein structure and dynamics are fundamental to the most common laboratory technique used to separate proteins and determine their molecular weights: polyacrylamide gel electrophoresis. However, the mechanism by which SDS induces protein unfolding and the microstructure of protein-SDS complexes remain largely unknown. Here, we report a detailed account of SDS-induced unfolding of two proteins-I27 domain of titin and β-amylase-obtained through all-atom molecular dynamics simulations. Both proteins were found to spontaneously unfold in the presence of SDS at boiling water temperature on the time scale of several microseconds. The protein unfolding was found to occur via two distinct mechanisms in which specific interactions of individual SDS molecules disrupt the protein's secondary structure. In the final state of the unfolding process, the proteins are found to wrap around SDS micelles in a fluid necklace-and-beads configuration, where the number and location of bound micelles changes dynamically. The global conformation of the protein was found to correlate with the number of SDS micelles bound to it, whereas the number of SDS molecules directly bound to the protein was found to define the relaxation time scale of the unfolded protein. Our microscopic characterization of SDS-protein interactions sets the stage for future refinement of SDS-enabled protein characterization methods, including protein fingerprinting and sequencing using a solid-state nanopore.

Studying biomolecular folding and binding using temperature-jump mass spectrometry

Characterizing folding and complex formation of biomolecules provides a view into their thermodynamics, kinetics and folding pathways. Deciphering kinetic intermediates is particularly important because they can often be targeted by drugs. The key advantage of native mass spectrometry over conventional methods that monitor a single observable is its ability to identify and quantify coexisting species. Here, we show the design of a temperature-jump electrospray source for mass spectrometry that allows one to perform fast kinetics experiments (0.16-32 s) at different temperatures (10-90 °C). The setup allows recording of both folding and unfolding kinetics by using temperature jumps from high to low, and low to high, temperatures. Six biological systems, ranging from peptides to proteins to DNA complexes, exemplify the use of this device. Using temperature-dependent experiments, the folding and unfolding of a DNA triplex are studied, providing detailed information on its thermodynamics and kinetics.

Understanding biochemical processes in the presence of sub-diffusive behavior of biomolecules in solution and living cells

In order to maintain cellular function, biomolecules like protein, DNA, and RNAs have to diffuse to the target spaces within the cell. Changes in the cytosolic microenvironment or in the nucleus during the fulfillment of these cellular processes affect their mobility, folding, and stability thereby impacting the transient or stable interactions with their adjacent neighbors in the organized and dynamic cellular interior. Using classical Brownian motion to elucidate the diffusion behavior of these biomolecules is hard considering their complex nature. The understanding of biomolecular diffusion inside cells still remains elusive due to the lack of a proper model that can be extrapolated to these cases. In this review, we have comprehensively addressed the progresses in this field, laying emphasis on the different aspects of anomalous diffusion in the different biochemical reactions in cell interior. These experiment-based models help to explain the diffusion behavior of biomolecules in the cytosolic and nuclear microenvironment. Moreover, since understanding of biochemical reactions within living cellular system is our main focus, we coupled the experimental observations with the concept of sub-diffusion from in vitro to in vivo condition. We believe that the pairing between the understanding of complex behavior and structure-function paradigm of biological molecules would take us forward by one step in order to solve the puzzle around diseases caused by cellular dysfunction.

Proteins: "Boil 'Em, Mash 'Em, Stick 'Em in a Stew"

Cells of the vast majority of organisms are subject to temperature, pressure, pH, ionic strength, and other stresses. We discuss these effects in the light of protein folding and protein interactions in vitro, in complex environments, in cells, and in vivo. Protein phase diagrams provide a way of organizing different structural ensembles that occur under stress and how one can move among ensembles. Experiments that perturb biomolecules in vitro or in cells by stressing them have revealed much about the underlying forces that are competing to control protein stability, folding, and function. Two phenomena that emerge and serve to broadly classify effects of the cellular environment are crowding (mainly due to repulsive forces) and sticking (mainly due to attractive forces). The interior of cells is closely balanced between these emergent effects, and stress can tip the balance one way or the other. The free energy scale involved is small but significant on the scale of the "on/off switches" that control signaling in cells or of protein-protein association with a favorable function such as increased enzyme processivity. Quantitative tools from biophysical chemistry will play an important role in elucidating the world of crowding and sticking under stress.

Critical phenomena in the temperature-pressure-crowding phase diagram of a protein

In the cell, proteins fold and perform complex functions through global structural rearrangements. Function requires a protein to be at the brink of stability to be susceptible to small environmental fluctuations, yet stable enough to maintain structural integrity. These apparently conflicting behaviors are exhibited by systems near a critical point, where distinct phases merge-a concept beyond previous studies indicating proteins have a well-defined folded/unfolded phase boundary in the pressure-temperature plane. Here, by modeling the protein phosphoglycerate kinase (PGK) on the temperature (T), pressure (P), and crowding volume-fraction (ϕ) phase diagram, we demonstrate a critical transition where phases merge, and PGK exhibits large structural fluctuations. Above the critical point, the difference between the intermediate and unfolded phases disappears. When ϕ increases, the critical point moves to lower T _c. We verify the calculations with experiments mapping the T-P-ϕ space, which likewise reveal a critical point at 305 K and 170 MPa that moves to lower T _c as ϕ increases. Crowding places PGK near a critical line in its natural parameter space, where large conformational changes can occur without costly free energy barriers. Specific structures are proposed for each phase based on simulation.

Unraveling the Mechanical Unfolding Pathways of a Multidomain Protein: Phosphoglycerate Kinase

Phosphoglycerate kinase (PGK) is a highly conserved enzyme that is crucial for glycolysis. PGK is a monomeric protein composed of two similar domains and has been the focus of many studies for investigating interdomain interactions within the native state and during folding. Previous studies used traditional biophysical methods (such as circular dichroism, tryptophan fluorescence, and NMR) to measure signals over a large ensemble of molecules, which made it difficult to observe transient changes in stability or structure during unfolding and refolding of single molecules. Here, we unfold single molecules of PGK using atomic force spectroscopy and steered molecular dynamic computer simulations to examine the conformational dynamics of PGK during its unfolding process. Our results show that after the initial forced separation of its domains, yeast PGK (yPGK) does not follow a single mechanical unfolding pathway; instead, it stochastically follows two distinct pathways: unfolding from the N-terminal domain or unfolding from the C-terminal domain. The truncated yPGK N-terminal domain unfolds via a transient intermediate, whereas the structurally similar isolated C-terminal domain has no detectable intermediates throughout its mechanical unfolding process. The N-terminal domain in the full-length yPGK displays a strong unfolding intermediate 13% of the time, whereas the truncated domain (yPGKNT) transitions through the intermediate 81% of the time. This effect indicates that the mechanical properties of yPGK cannot be simply deduced from the mechanical properties of its constituents. We also find that Escherichia coli PGK is significantly less mechanically stable as compared to yPGK, contrary to bulk unfolding measurements. Our results support the growing body of observations that the folding behavior of multidomain proteins is difficult to predict based solely on the studies of isolated domains.

Characterizing proteins in their cellular environment: Examples of recent advances in quantitative fluorescence microscopy

Quantitative characterization of protein interactions, both intramolecular and intermolecular, is crucial in understanding the mechanisms and regulation of their function. In recent years, it has become possible to obtain such information on protein systems in live cells, from bacteria to mammalian cell lines. This review discusses recent advances in measuring protein folding, absolute concentration, oligomerization, diffusion, transport, and organization at super-resolution.

Measuring the average shape of transition paths during the folding of a single biological molecule

Transition paths represent the parts of a reaction where the energy barrier separating products and reactants is crossed. They are essential to understanding reaction mechanisms, yet many of their properties remain unstudied. Here, we report measurements of the average shape of transition paths, studying the folding of DNA hairpins as a model system for folding reactions. Individual transition paths were detected in the folding trajectories of hairpins with different sequences held under tension in optical tweezers, and path shapes were computed by averaging all transitions in the time domain, 〈t(x)〉, or by averaging transitions of a given duration in the extension domain, 〈x(t|τ)〉 _τ Whereas 〈t(x)〉 was close to straight, with only a subtle curvature, 〈x(t|τ)〉 _τ had more pronounced curvature that fit well to theoretical expectations for the dominant transition path, returning diffusion coefficients similar to values obtained previously from independent methods. Simulations suggested that 〈t(x)〉 provided a less reliable representation of the path shape than 〈x(t|τ)〉 _τ , because it was far more sensitive to the effects of coupling the molecule to the experimental force probe. Intriguingly, the path shape variance was larger for some hairpins than others, indicating sequence-dependent changes in the diversity of transition paths reflective of differences in the character of the energy barriers, such as the width of the barrier saddle-point or the presence of parallel paths through multiple barriers between the folded and unfolded states. These studies of average path shapes point the way forward for probing the rich information contained in path shape fluctuations.

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.

A non-equilibrium approach to allosteric communication

While the theory of protein folding is well developed, including concepts such as rugged energy landscape, folding funnel, etc., the same degree of understanding has not been reached for the description of the dynamics of allosteric transitions in proteins. This is not only due to the small size of the structural change upon ligand binding to an allosteric site, but also due to challenges in designing experiments that directly observe such an allosteric transition. On the basis of recent pump-probe-type experiments (Buchli et al. 2013 Proc. Natl Acad. Sci. USA110, 11 725-11 730. (doi:10.1073/pnas.1306323110)) and non-equilibrium molecular dynamics simulations (Buchenberg et al. 2017 Proc. Natl Acad. Sci. USA114, E6804-E6811. (doi:10.1073/pnas.1707694114)) studying an photoswitchable PDZ2 domain as model for an allosteric transition, we outline in this perspective how such a description of allosteric communication might look. That is, calculating the dynamical content of both experiment and simulation (which agree remarkably well with each other), we find that allosteric communication shares some properties with downhill folding, except that it is an 'order-order' transition. Discussing the multiscale and hierarchical features of the dynamics, the validity of linear response theory as well as the meaning of 'allosteric pathways', we conclude that non-equilibrium experiments and simulations are a promising way to study dynamical aspects of allostery.This article is part of a discussion meeting issue 'Allostery and molecular machines'.

Hydrophobic Collapse of Ubiquitin Generates Rapid Protein-Water Motions

We report time-resolved measurements of the coupled protein-water modes of solvated ubiquitin during protein folding. Kinetic terahertz absorption (KITA) spectroscopy serves as a label-free technique for monitoring large scale conformational changes and folding of proteins subsequent to a sudden T-jump. We report here KITA measurements at an unprecedented time resolution of 500 ns, a resolution 2 orders of magnitude better than those of any previous KITA measurements, which reveal the coupled ubiquitin-solvent dynamics even in the initial phase of hydrophobic collapse. Complementary equilibrium experiments and molecular simulations of ubiquitin solutions are performed to clarify non-equilibrium contributions and reveal the molecular picture upon a change in structure, respectively. On the basis of our results, we propose that in the case of ubiquitin a rapid (<500 ns) initial phase of the hydrophobic collapse from the elongated protein to a molten globule structure precedes secondary structure formation. We find that these very first steps, including large-amplitude changes within the unfolded manifold, are accompanied by a rapid (<500 ns) pronounced change of the coupled protein-solvent response. The KITA response upon secondary structure formation exhibits an opposite sign, which indicates a distinct effect on the solvent-exposed surface.

Estimating the mean first passage time of protein misfolding

Most theoretical and experimental studies confirm that proteins fold in the time scale of microseconds to milliseconds, but the kinetics of the protein misfolding remains largely unexplored. The kinetics of unfolding-folding-misfolding equilibrium in proteins is formulated in the analytical framework of the Master equation. The folded, unfolded and the misfolded state are characterized in terms of their respective contacts. The Mean First Passage Time (MFPT) to acquire the misfolded conformation from the native or folded state is derived from this equation with different boundary conditions. The MFPT is found to be practically independent of the length of the protein, the number of native contacts and the rate constant for the misfolded to the folded state. The results obtained from the survival probability are directly correlated to the age of onset and appearance of misfolding diseases in humans.

Metadynamics Enhanced Markov Modeling of Protein Dynamics

Enhanced sampling techniques represent a versatile approach to account for rare conformational transitions in biomolecules. A particularly promising strategy is to combine massive parallel computing of short molecular dynamics (MD) trajectories (to sample the free energy landscape of the system) with Markov state modeling (to rebuild the kinetics from the sampled data). To obtain well-distributed initial structures for the short trajectories, it is proposed to employ metadynamics MD, which quickly sweeps through the entire free energy landscape of interest. Being only used to generate initial conformations, the implementation of metadynamics can be simple and fast. The conformational dynamics of helical peptide Aib₉ is adopted to discuss various technical issues of the approach, including metadynamics settings, minimal number and length of short MD trajectories, and the validation of the resulting Markov models. Using metadynamics to launch some thousands of nanosecond trajectories, several Markov state models are constructed that reveal that previous unbiased MD simulations of in total 16 μs length cannot provide correct equilibrium populations or qualitative features of the pathway distribution of the short peptide.

Site-specific time-resolved FRET reveals local variations in the unfolding mechanism in an apparently two-state protein unfolding transition

Using multi-site time-resolved FRET, it is shown that equilibrium unfolding of monellin is not only heterogeneous, but that the degree of non-cooperativity differs between the sole α-helix and different parts of the β-sheet.

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.

Globular Protein Folding In Vitro and In Vivo

In vitro, computational, and theoretical studies of protein folding have converged to paint a rich and complex energy landscape. This landscape is sensitively modulated by environmental conditions and subject to evolutionary pressure on protein function. Of these environments, none is more complex than the cell itself, where proteins function in the cytosol, in membranes, and in different compartments. A wide variety of kinetic and thermodynamics experiments, ranging from single-molecule studies to jump kinetics and from nuclear magnetic resonance to imaging on the microscope, have elucidated how protein energy landscapes facilitate folding and how they are subject to evolutionary constraints and environmental perturbation. Here we review some recent developments in the field and refer the reader to some original work and additional reviews that cover this broad topic in protein science.

Time-resolved observation of protein allosteric communication

Allostery represents a fundamental mechanism of biological regulation that is mediated via long-range communication between distant protein sites. Although little is known about the underlying dynamical process, recent time-resolved infrared spectroscopy experiments on a photoswitchable PDZ domain (PDZ2S) have indicated that the allosteric transition occurs on multiple timescales. Here, using extensive nonequilibrium molecular dynamics simulations, a time-dependent picture of the allosteric communication in PDZ2S is developed. The simulations reveal that allostery amounts to the propagation of structural and dynamical changes that are genuinely nonlinear and can occur in a nonlocal fashion. A dynamic network model is constructed that illustrates the hierarchy and exceeding structural heterogeneity of the process. In compelling agreement with experiment, three physically distinct phases of the time evolution are identified, describing elastic response ([Formula: see text] ns), inelastic reorganization ([Formula: see text] ns), and structural relaxation ([Formula: see text]s). Issues such as the similarity to downhill folding as well as the interpretation of allosteric pathways are discussed.

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.

On the polymer physics origins of protein folding thermodynamics

A remarkable feature of the spontaneous folding of many small proteins is the striking similarity in the thermodynamics of the folding process. This process is characterized by simple two-state thermodynamics with large and compensating changes in entropy and enthalpy and a funnel-like free energy landscape with a free-energy barrier that varies linearly with temperature. One might attribute the commonality of this two-state folding behavior to features particular to these proteins (e.g., chain length, hydrophobic/hydrophilic balance, attributes of the native state) or one might suspect that this similarity in behavior has a more general polymer-physics origin. Here we show that this behavior is also typical for flexible homopolymer chains with sufficiently short range interactions. Two-state behavior arises from the presence of a low entropy ground (folded) state separated from a set of high entropy disordered (unfolded) states by a free energy barrier. This homopolymer model exhibits a funneled free energy landscape that reveals a complex underlying dynamics involving competition between folding and non-folding pathways. Despite the presence of multiple pathways, this simple physics model gives the robust result of two-state thermodynamics for both the cases of folding from a basin of expanded coil states and from a basin of compact globule states.

How cooperative are protein folding and unfolding transitions?

A thermodynamically and kinetically simple picture of protein folding envisages only two states, native (N) and unfolded (U), separated by a single activation free energy barrier, and interconverting by cooperative two-state transitions. The folding/unfolding transitions of many proteins occur, however, in multiple discrete steps associated with the formation of intermediates, which is indicative of reduced cooperativity. Furthermore, much advancement in experimental and computational approaches has demonstrated entirely non-cooperative (gradual) transitions via a continuum of states and a multitude of small energetic barriers between the N and U states of some proteins. These findings have been instrumental towards providing a structural rationale for cooperative versus noncooperative transitions, based on the coupling between interaction networks in proteins. The cooperativity inherent in a folding/unfolding reaction appears to be context dependent, and can be tuned via experimental conditions which change the stabilities of N and U. The evolution of cooperativity in protein folding transitions is linked closely to the evolution of function as well as the aggregation propensity of the protein. A large activation energy barrier in a fully cooperative transition can provide the kinetic control required to prevent the accumulation of partially unfolded forms, which may promote aggregation. Nevertheless, increasing evidence for barrier-less "downhill" folding, as well as for continuous "uphill" unfolding transitions, indicate that gradual non-cooperative processes may be ubiquitous features on the free energy landscape of protein folding.

Path statistics, memory, and coarse-graining of continuous-time random walks on networks

Continuous-time random walks (CTRWs) on discrete state spaces, ranging from regular lattices to complex networks, are ubiquitous across physics, chemistry, and biology. Models with coarse-grained states (for example, those employed in studies of molecular kinetics) or spatial disorder can give rise to memory and non-exponential distributions of waiting times and first-passage statistics. However, existing methods for analyzing CTRWs on complex energy landscapes do not address these effects. Here we use statistical mechanics of the nonequilibrium path ensemble to characterize first-passage CTRWs on networks with arbitrary connectivity, energy landscape, and waiting time distributions. Our approach can be applied to calculating higher moments (beyond the mean) of path length, time, and action, as well as statistics of any conservative or non-conservative force along a path. For homogeneous networks, we derive exact relations between length and time moments, quantifying the validity of approximating a continuous-time process with its discrete-time projection. For more general models, we obtain recursion relations, reminiscent of transfer matrix and exact enumeration techniques, to efficiently calculate path statistics numerically. We have implemented our algorithm in PathMAN (Path Matrix Algorithm for Networks), a Python script that users can apply to their model of choice. We demonstrate the algorithm on a few representative examples which underscore the importance of non-exponential distributions, memory, and coarse-graining in CTRWs.

EF-G Activation by Phosphate Analogs

Elongation factor G (EF-G) is a universally conserved translational GTPase that promotes the translocation of tRNA and mRNA through the ribosome. EF-G binds to the ribosome in a GTP-bound form and subsequently catalyzes GTP hydrolysis. The contribution of the ribosome-stimulated GTP hydrolysis by EF-G to tRNA/mRNA translocation remains debated. Here, we show that while EF-G•GDP does not stably bind to the ribosome and induce translocation, EF-G•GDP in complex with phosphate group analogs BeF3(-) and AlF4(-) promotes the translocation of tRNA and mRNA. Furthermore, the rates of mRNA translocation induced by EF-G in the presence of GTP and a non-hydrolyzable analog of GTP, GDP•BeF3(-) are similar. Our results are consistent with the model suggesting that GTP hydrolysis is not directly coupled to mRNA/tRNA translocation. Hence, GTP binding is required to induce the activated, translocation-competent conformation of EF-G while GTP hydrolysis triggers EF-G release from the ribosome.

Reduction of All-Atom Protein Folding Dynamics to One-Dimensional Diffusion

Theoretical models have often modeled protein folding dynamics as diffusion on a low-dimensional free energy surface, a remarkable simplification. However, the accuracy of such an approximation and the number of dimensions required were not clear. For all-atom folding simulations of ten small proteins in explicit solvent we show that the folding dynamics can indeed be accurately described as diffusion on just a single coordinate, the fraction of native contacts (Q). The diffusion models reproduce both folding rates, and finer details such as transition-path durations and diffusive propagators. The Q-averaged diffusion coefficients decrease with chain length, as anticipated from energy landscape theory. Although the Q-diffusion model does not capture transition-path durations for the protein NuG2, we show that this can be accomplished by designing an improved coordinate Qopt. Overall, one-dimensional diffusion on a suitable coordinate turns out to be a remarkably faithful model for the dynamics of the proteins considered.

A hydrodynamic view of the first-passage folding of Trp-cage miniprotein

We study folding of Trp-cage miniprotein in the conditions when the native state of the protein is stable and unfolding events are improbable, which corresponds to physiological conditions. Using molecular dynamics simulations with an implicit solvent model, an ensemble of folding trajectories from unfolded (practically extended) states of the protein to the native state was generated. To get insight into the folding kinetics, the free energy surface and kinetic network projected on this surface were constructed. This, "conventional" analysis of the folding reaction was followed by a recently proposed hydrodynamic description of protein folding (Chekmarev et al. in Phys Rev Lett 100(1):018107, 2008), in which the process of the first-passage folding is viewed as a stationary flow of a folding "fluid" from the unfolded to native state. This approach is conceptually different from the previously used approaches and thus allows an alternative view of the folding dynamics and kinetics of Trp-cage, the conclusions about which are very diverse. In agreement with most previous studies, we observed two characteristic folding pathways: in one pathway (I), the collapse of the hydrophobic core precedes the formation of the [Formula: see text]-helix, and in the other pathway (II), these events occur in the reverse order. We found that although pathway II is complicated by a repeated partial protein unfolding, it contributes to the total folding flow as little as ≈10%, so that the folding kinetics remain essentially single-exponential.