Folding lambda-repressor at its speed limit

Chemical interactions modulate λ6-85 stability in cells

Because steric crowding is most effective when the crowding agent is similar in size to the molecule that it acts upon and the average macromolecule inside cells is much larger than a small protein or peptide, steric crowding is not predicted to affect their folding inside cells. On the other hand, chemical interactions should perturb in-cell structure and stability because they arise from interactions between the surface of the small protein or peptide and its environment. Indeed, previous in vitro measurements of the λ-repressor fragment, λ6-85 , in crowding matrices comprised of Ficoll or protein crowders support these predictions. Here, we directly quantify the in-cell stability of λ6-85 and distinguish the contribution of steric crowding and chemical interactions to its stability. Using a FRET-labeled λ6-85 construct, we find that the fragment is stabilized by 5°C in-cells compared to in vitro. We demonstrate that this stabilization cannot be explained by steric crowding because, as anticipated, Ficoll has no effect on λ6-85 stability. We find that the in-cell stabilization arises from chemical interactions, mimicked in vitro by mammalian protein extraction reagent (M-PER™). Comparison between FRET values in-cell and in Ficoll confirms that U-2 OS cytosolic crowding is reproduced at macromolecule concentrations of 15% w/v. Our measurements validate the cytomimetic of 15% Ficoll and 20% M-PER™ that we previously developed for protein and RNA folding studies. However, because the in-cell stability of λ6-85 is reproduced by 20% v/v M-PER™ alone, we predict that this simplified mixture could be a useful tool to predict the in-cell behaviors of other small proteins and peptides.

Direct Observation of Single-Protein Transition State Passage by Nanopore Ionic Current Jumps

Conformational transitions of proteins are governed by chemical kinetics, often toggled by passage through an activated state separating two conformational ensembles. The passage time of a protein through the activated state can be too fast to be detected by single-molecule experiments without the aid of viscogenic agents. Here, we use high-bandwidth nanopore measurements to resolve microsecond-duration transitions that occur between conformational states of individual protein molecules partly blocking pore current. We measure the transition state passage time between folded and unfolded states of a two-state λ_6-85 mutant and between metastable intermediates and the unfolded state of the multistate folder cytochrome c. Consistent with the principle of microscopic reversibility, the transition state passage time is the same for the forward and backward reactions. A passage time distribution whose tail is broader than a single exponential observed in cytochrome c suggests a multidimensional energy landscape for this protein.

DnaK response to expression of protein mutants is dependent on translation rate and stability

Chaperones play a central part in the quality control system in cells by clearing misfolded and aggregated proteins. The chaperone DnaK acts as a sensor for molecular stress by recognising short hydrophobic stretches of misfolded proteins. As the level of unfolded protein is a function of protein stability, we hypothesised that the level of DnaK response upon overexpression of recombinant proteins would be correlated to stability. Using a set of mutants of the λ-repressor with varying thermal stabilities and a fluorescent reporter system, the effect of stability on DnaK response and protein abundance was investigated. Our results demonstrate that the initial DnaK response is largely dependent on protein synthesis rate but as the recombinantly expressed protein accumulates and homeostasis is approached the response correlates strongly with stability. Furthermore, we observe a large degree of cell-cell variation in protein abundance and DnaK response in more stable proteins.

A Disordered Loop Mediates Heterogeneous Unfolding of an Ordered Protein by Altering the Native Ensemble

The high flexibility of long disordered or partially structured loops in folded proteins allows for entropic stabilization of native ensembles. Destabilization of such loops could alter the native ensemble or promote alternate conformations within the native ensemble if the ordered regions themselves are held together weakly. This is particularly true of downhill folding systems that exhibit weak unfolding cooperativity. Here, we combine experimental and computational methods to probe the response of the native ensemble of a helical, downhill folding domain PDD, which harbors an 11-residue partially structured loop, to perturbations. Statistical mechanical modeling points to continuous structural changes on both temperature and mutational perturbations driven by entropic stabilization of partially structured conformations within the native ensemble. Long time-scale simulations of the wild-type protein and two mutants showcase a remarkable conformational switching behavior wherein the parallel helices in the wild-type protein sample an antiparallel orientation in the mutants, with the C-terminal helix and the loop connecting the helices displaying high flexibility, disorder, and non-native interactions. We validate these computational predictions via the anomalous fluorescence of a native tyrosine located at the interface of the helices. Our observations highlight the role of long loops in determining the unfolding mechanisms, sensitivity of the native ensembles to mutational perturbations and provide experimentally testable predictions that can be explored in even two-state folding systems.

In-Cell Protein-Protein Contacts: Transient Interactions in the Crowd

Proteins in vivo are immersed in a crowded environment of water, ions, metabolites, and macromolecules. In-cell experiments highlight how transient weak protein-protein interactions promote (via functional "quinary structure") or hinder (via competitive binding or "sticking") complex formation. Computational models of the cytoplasm are expensive. We tackle this challenge with an all-atom model of a small volume of the E. coli cytoplasm to simulate protein-protein contacts up to the 5 μs time scale on the special-purpose supercomputer Anton 2. We use three CHARMM-derived force fields: C22*, C36m, and C36mCU (with CUFIX corrections). We find that both C36m and C36mCU form smaller contact surfaces than C22*. Although CUFIX was developed to reduce protein-protein sticking, larger contacts are observed with C36mCU than C36m. We show that the lifespan Δt of protein-protein contacts obeys a power law distribution between 0.03 and 3 μs, with ∼90% of all contacts lasting <1 μs (similar to the time scale for downhill folding).

Choice of Adaptive Sampling Strategy Impacts State Discovery, Transition Probabilities, and the Apparent Mechanism of Conformational Changes

Interest in atomically detailed simulations has grown significantly with recent advances in computational hardware and Markov state modeling (MSM) methods, yet outstanding questions remain that hinder their widespread adoption. Namely, how do alternative sampling strategies explore conformational space and how might this influence predictions generated from the data? Here, we seek to answer these questions for four commonly used sampling methods: (1) a single long simulation, (2) many short simulations run in parallel, (3) adaptive sampling, and (4) our recently developed goal-oriented sampling algorithm, FAST. We first develop a theoretical framework for analytically calculating the probability of discovering select states on simple landscapes, where we uncover the drastic effects of varying the number and length of simulations. We then use kinetic Monte Carlo simulations on a variety of physically inspired landscapes to characterize the probability of discovering particular states and transition pathways for each of the four methods. Consistently, we find that FAST simulations discover each target state with the highest probability, while traversing realistic pathways. Furthermore, we uncover the potential pathology that short parallel simulations sometimes predict an incorrect transition pathway by crossing large energy barriers that long simulations would typically circumnavigate. We refer to this pathology as "pathway tunneling". To protect against this phenomenon when using adaptive-sampling and FAST simulations, we introduce the FAST-string method. This method enhances sampling along the highest-flux transition paths to refine an MSMs transition probabilities and discriminate between competing pathways. Additionally, we compare the performance of a variety of MSM estimators in describing accurate thermodynamics and kinetics. For adaptive sampling, we recommend simply normalizing the transition counts out of each state after adding small pseudocounts to avoid creating sources or sinks. Lastly, we evaluate whether our insights from simple landscapes hold for all-atom molecular dynamics simulations of the folding of the λ-repressor protein. Remarkably, we find that FAST-contacts predicts the same folding pathway as a set of long simulations but with orders of magnitude less simulation time.

Roles of conformational disorder and downhill folding in modulating protein-DNA recognition

Transcription factors are thought to efficiently search for their target DNA site via a combination of conventional 3D diffusion and 1D diffusion along the DNA molecule mediated by non-specific electrostatic interactions. This process requires the DNA-binding protein to quickly exchange between a search competent and a target recognition mode, but little is known as to how these two binding modes are encoded in the conformational properties of the protein. Here, we investigate this issue on the engrailed homeodomain (EngHD), a DNA-binding domain that folds ultrafast and exhibits a complex conformational behavior consistent with the downhill folding scenario. We explore the interplay between folding and DNA recognition using a coarse-grained computational model that allows us to manipulate the folding properties of the protein and monitor its non-specific and specific binding to DNA. We find that conformational disorder increases the search efficiency of EngHD by promoting a fast gliding search mode in addition to sliding. When gliding, EngHD remains loosely bound to DNA moving linearly along its length. A partially disordered EngHD also binds more dynamically to the target site, reducing the half-life of the specific complex via a spring-loaded mechanism. These findings apply to all conditions leading to partial disorder. However, we also find that at physiologically relevant temperatures EngHD is well folded and can only obtain the conformational flexibility required to accelerate 1D diffusion when it folds/unfolds within the downhill scenario (crossing a marginal free energy barrier). In addition, the conformational flexibility of native downhill EngHD enables its fast reconfiguration to lock into the specific binding site upon arrival, thereby affording finer control of the on- and off-rates of the specific complex. Our results provide key mechanistic insights into how DNA-binding domains optimize specific DNA recognition through the control of their conformational dynamics and folding mechanism.

Protein unfolding mechanisms and their effects on folding experiments

In this review, I discuss the various methods researchers use to unfold proteins in the lab in order to understand protein folding both in vitro and in vivo. The four main techniques, chemical-, heat-, pressure- and force-denaturation, produce distinctly different unfolded conformational ensembles. Recent measurements have revealed different folding kinetics from different unfolding mechanisms. Thus, comparing these distinct unfolded ensembles sheds light on the underlying free energy landscape of 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.

Cooperative folding near the downhill limit determined with amino acid resolution by hydrogen exchange

The relationship between folding cooperativity and downhill, or barrier-free, folding of proteins under highly stabilizing conditions remains an unresolved topic, especially for proteins such as λ-repressor that fold on the microsecond timescale. Under aqueous conditions where downhill folding is most likely to occur, we measure the stability of multiple H bonds, using hydrogen exchange (HX) in a λYA variant that is suggested to be an incipient downhill folder having an extrapolated folding rate constant of 2 × 10(5) s(-1) and a stability of 7.4 kcal·mol(-1) at 298 K. At least one H bond on each of the three largest helices (α1, α3, and α4) breaks during a common unfolding event that reflects global denaturation. The use of HX enables us to both examine folding under highly stabilizing, native-like conditions and probe the pretransition state region for stable species without the need to initiate the folding reaction. The equivalence of the stability determined at zero and high denaturant indicates that any residual denatured state structure minimally affects the stability even under native conditions. Using our ψ analysis method along with mutational ϕ analysis, we find that the three aforementioned helices are all present in the folding transition state. Hence, the free energy surface has a sufficiently high barrier separating the denatured and native states that folding appears cooperative even under extremely stable and fast folding conditions.

Fast-folding proteins under stress

Proteins are subject to a variety of stresses in biological organisms, including pressure and temperature, which are the easiest stresses to simulate by molecular dynamics. We discuss the effect of pressure and thermal stress on very-fast-folding model proteins, whose in vitro folding can be fully simulated on computers and compared with experiments. We then discuss experiments that can be used to subject proteins to low- and high-temperature unfolding, as well as low- and high-pressure unfolding. Pressure and temperature are prototypical perturbations that illustrate how close many proteins are to instability, a property that cells can exploit to control protein function. We conclude by reviewing some recent in-cell experiments, and progress being made in simulating and measuring protein stability and function inside live cells.

The roughness of the protein energy landscape results in anomalous diffusion of the polypeptide backbone

Recombination of photolysed protein disulfide bonds confirms subdiffusional backbone motion and measures the roughness of the protein's energy landscape.

Probing the origins of two-state folding

Many protein systems fold in a two-state manner. Random models, however, rarely display two-state kinetics and thus such behavior should not be accepted as a default. While theories for the prevalence of two-state kinetics have been presented, none sufficiently explain the breadth of experimental observations. A model, making minimal assumptions, is introduced that suggests two-state behavior is likely for any system with an overwhelmingly populated native state. We show two-state folding is a natural consequence of such two-state thermodynamics, and is strengthened by increasing the population of the native state. Further, the model exhibits hub-like behavior, with slow interconversions between unfolded states. Despite this, the unfolded state equilibrates quickly relative to the folding time. This apparent paradox is readily understood through this model. Finally, our results compare favorable with measurements of folding rates as a function of chain length and Keq, providing new insight into these relations.

Observation of complete pressure-jump protein refolding in molecular dynamics simulation and experiment

Density is an easily adjusted variable in molecular dynamics (MD) simulations. Thus, pressure-jump (P-jump)-induced protein refolding, if it could be made fast enough, would be ideally suited for comparison with MD. Although pressure denaturation perturbs secondary structure less than temperature denaturation, protein refolding after a fast P-jump is not necessarily faster than that after a temperature jump. Recent P-jump refolding experiments on the helix bundle λ-repressor have shown evidence of a <3 μs burst phase, but also of a ~1.5 ms "slow" phase of refolding, attributed to non-native helical structure frustrating microsecond refolding. Here we show that a λ-repressor mutant is nonetheless capable of refolding in a single explicit solvent MD trajectory in about 19 μs, indicating that the burst phase observed in experiments on the same mutant could produce native protein. The simulation reveals that after about 18.5 μs of conformational sampling, the productive structural rearrangement to the native state does not occur in a single swift step but is spread out over a brief series of helix and loop rearrangements that take about 0.9 μs. Our results support the molecular time scale inferred for λ-repressor from near-downhill folding experiments, where transition-state population can be seen experimentally, and also agrees with the transition-state transit time observed in slower folding proteins by single-molecule spectroscopy.

Fast protein folding kinetics

Fast-folding proteins have been a major focus of computational and experimental study because they are accessible to both techniques: they are small and fast enough to be reasonably simulated with current computational power, but have dynamics slow enough to be observed with specially developed experimental techniques. This coupled study of fast-folding proteins has provided insight into the mechanisms, which allow some proteins to find their native conformation well <1 ms and has uncovered examples of theoretically predicted phenomena such as downhill folding. The study of fast folders also informs our understanding of even 'slow' folding processes: fast folders are small; relatively simple protein domains and the principles that govern their folding also govern the folding of more complex systems. This review summarizes the major theoretical and experimental techniques used to study fast-folding proteins and provides an overview of the major findings of fast-folding research. Finally, we examine the themes that have emerged from studying fast folders and briefly summarize their application to protein folding in general, as well as some work that is left to do.

How force unfolding differs from chemical denaturation

Single-molecule force spectroscopies are remarkable tools for studying protein folding and unfolding, but force unfolding explores protein configurations that are potentially very different from the ones traditionally explored in chemical or thermal denaturation. Understanding these differences is crucial because such configurations serve as starting points of folding studies, and thus can affect both the folding mechanism and the kinetics. Here we provide a detailed comparison of both chemically induced and force-induced unfolded state ensembles of ubiquitin based on extensive, all-atom simulations of the protein either extended by force or denatured by urea. As expected, the respective unfolded states are very different on a macromolecular scale, being fully extended under force with no contacts and partially extended in urea with many nonnative contacts. The amount of residual secondary structure also differs: A significant population of α-helices is found in chemically denatured configurations but such helices are absent under force, except at the lowest applied force of 30 pN where short helices form transiently. We see that typical-size helices are unstable above this force, and β-sheets cannot form. More surprisingly, we observe striking differences in the backbone dihedral angle distributions for the protein unfolded under force and the one unfolded by denaturant. A simple model based on the dialanine peptide is shown to not only provide an explanation for these striking differences but also illustrates how the force dependence of the protein dihedral angle distributions give rise to the worm-like chain behavior of the chain upon force.

Engineering folding dynamics from two-state to downhill: application to λ-repressor

One strategy for reaching the downhill folding regime, primarily exploited for the λ_6–85 protein fragment, consists of cumulatively introducing mutations that speed up folding. This is an experimentally demanding process where chemical intuition usually serves as a guide for the choice of amino acid residues to mutate. Such an approach can be aided by computational methods that screen for protein engineering hot spots. Here we present one such method that involves sampling the energy landscape of the pseudo-wild-type protein and investigating the effect of point mutations on this landscape. Using a novel metric for the cooperativity, we identify those residues leading to the least cooperative folding. The folding dynamics of the selected mutants are then directly characterized and the differences in the kinetics are analyzed within a Markov-state model framework. Although the method is general, here we present results for a coarse-grained topology-based simulation model of λ-repressor, whose barrier is reduced from an initial value of ∼4k_BT at the midpoint to ∼1k_BT, thereby reaching the downhill folding regime.

Using VIPT-jump to distinguish between different folding mechanisms: application to BBL and a Trpzip

Protein folding involves a large number of sequential molecular steps or conformational substates. Thus, experimental characterization of the underlying folding energy landscape for any given protein is difficult. Herein, we present a new method that can be used to determine the major characteristics of the folding energy landscape in question, e.g., to distinguish between activated and barrierless downhill folding scenarios. This method is based on the idea that the conformational relaxation kinetics of different folding mechanisms at a given final condition will show different dependences on the initial condition. We show, using both simulation and experiment, that it is possible to differentiate between disparate kinetic folding models by comparing temperature jump (T-jump) relaxation traces obtained with a fixed final temperature and varied initial temperatures, which effectively varies the initial potential (VIP) of the system of interest. We apply this method (hereafter refer to as VIPT-jump) to two model systems, tryptophan zipper (Trpzip)-2c and BBL, and our results show that BBL exhibits characteristics of barrierless downhill folding, whereas Trpzip-2c folding encounters a free energy barrier. In addition, using the T-jump data of BBL we are able to provide, via Langevin dynamics simulations, a realistic estimate of its conformational diffusion coefficient.

Atomic-level description of ubiquitin folding

Equilibrium molecular dynamics simulations, in which proteins spontaneously and repeatedly fold and unfold, have recently been used to help elucidate the mechanistic principles that underlie the folding of fast-folding proteins. The extent to which the conclusions drawn from the analysis of such proteins, which fold on the microsecond timescale, apply to the millisecond or slower folding of naturally occurring proteins is, however, unclear. As a first attempt to address this outstanding issue, we examine here the folding of ubiquitin, a 76-residue-long protein found in all eukaryotes that is known experimentally to fold on a millisecond timescale. Ubiquitin folding has been the subject of many experimental studies, but its slow folding rate has made it difficult to observe and characterize the folding process through all-atom molecular dynamics simulations. Here we determine the mechanism, thermodynamics, and kinetics of ubiquitin folding through equilibrium atomistic simulations. The picture emerging from the simulations is in agreement with a view of ubiquitin folding suggested from previous experiments. Our findings related to the folding of ubiquitin are also consistent, for the most part, with the folding principles derived from the simulation of fast-folding proteins, suggesting that these principles may be applicable to a wider range of proteins.

Microsecond folding experiments and simulations: a match is made

For the past two decades, protein folding experiments have been speeding up from the second or millisecond time scale to the microsecond time scale, and full-atom simulations have been extended from the nanosecond to the microsecond and even millisecond time scale. Where the two meet, it is now possible to compare results directly, allowing force fields to be validated and refined, and allowing experimental data to be interpreted in atomistic detail. In this perspective we compare recent experiments and simulations on the microsecond time scale, pointing out the progress that has been made in determining native structures from physics-based simulations, refining experiments and simulations to provide more quantitative underlying mechanisms, and tackling the problems of multiple reaction coordinates, downhill folding, and complex underlying structure of unfolded or misfolded states.

Equilibrium unfolding of the PDZ domain of β2-syntrophin

β2-syntrophin, a dystrophin-associated protein, plays a pivotal role in insulin secretion by pancreatic β-cells. It contains a PDZ domain (β2S-PDZ) that, in complex with protein-tyrosine phosphatase ICA512, anchors the dense insulin granules to actin filaments. The phosphorylation state of β2-syntrophin allosterically regulates the affinity of β2S-PDZ for ICA512, and the disruption of the complex triggers the mobilization of the insulin granule stores. Here, we investigate the thermal unfolding of β2S-PDZ at different pH and urea concentrations. Our results indicate that, unlike other PDZ domains, β2S-PDZ is marginally stable. Thermal denaturation experiments show broad transitions and cold denaturation, and a two-state model fit reveals a significant unfolded fraction under physiological conditions. Furthermore, T(m) and T(max) denaturant-dependent shifts and noncoincidence of melting curves monitored at different wavelengths suggest that two-state and three-state models fail to explain the equilibrium data properly and are in better agreement with a downhill scenario. Its higher stability at pH >9 and the results of molecular dynamics simulations indicate that this behavior of β2S-PDZ might be related to its charge distribution. All together, our results suggest a link between the conformational plasticity of the native ensemble of this PDZ domain and the regulation of insulin secretion.

Experiments and comprehensive simulations of the formation of a helical turn

We investigate the kinetics and thermodynamics of a helical turn formation in the peptide Ac-WAAAH-NH(2). NMR measurements indicate that this peptide has significant tendency to form a structure of a helical turn, while temperature dependent CD establishes the helix fraction at different temperatures. Molecular dynamics and milestoning simulations agree with experimental observables and suggest an atomically detailed picture for the turn formation. Using a network representation, two alternative mechanisms of folding are identified: (i) a direct co-operative mechanism from the unfolded to the folded state without intermediate formation of hydrogen bonds and (ii) an indirect mechanism with structural intermediates with two residues in a helical conformation. This picture is consistent with kinetic measurements that reveal two experimental time scales of sub-nanosecond and several nanoseconds.

Crowding effects on the small, fast-folding protein lambda6-85

The microsecond folder lambda6-85 is a small (9.2 kDa = 9200 amu) five helix bundle protein. We investigated the stability of lambda6-85 in two different low-fluorescence crowding matrices: the large 70 kDa carbohydrate Ficoll 70, and the small 14 kDa thermophilic protein SubL. The same thermal stability of secondary structure was measured by circular dichroism in aqueous buffer, and at a crowding fraction phi = 15 +/- 1% of Ficoll 70. Tryptophan fluorescence detection (probing a tertiary contact) yielded the same thermal stability in Ficoll, but 4 degrees C lower in aqueous buffer. Temperature-jump kinetics revealed that the relaxation rate, corrected for bulk viscosity, was very similar in Ficoll and in aqueous buffer. Thus viscosity, hydrodynamics and crowding seem to compensate one another. However, a new fast phase was observed in Ficoll, attributed to crowding-induced downhill folding. We also measured the stability of lambda6-85 in phi = 14 +/- 1% SubL, which acts as a smaller more rigid crowder. Significantly greater stabilization (7 to 13 degrees C depending on probe) was observed than in the Ficoll matrix. The results highlight the importance of crowding agent choice for studies of small, fast-folding proteins amenable to comparison with molecular dynamics simulations.

Geofold: topology-based protein unfolding pathways capture the effects of engineered disulfides on kinetic stability

Protein unfolding is modeled as an ensemble of pathways, where each step in each pathway is the addition of one topologically possible conformational degree of freedom. Starting with a known protein structure, GeoFold hierarchically partitions (cuts) the native structure into substructures using revolute joints and translations. The energy of each cut and its activation barrier are calculated using buried solvent accessible surface area, side chain entropy, hydrogen bonding, buried cavities, and backbone degrees of freedom. A directed acyclic graph is constructed from the cuts, representing a network of simultaneous equilibria. Finite difference simulations on this graph simulate native unfolding pathways. Experimentally observed changes in the unfolding rates for disulfide mutants of barnase, T4 lysozyme, dihydrofolate reductase, and factor for inversion stimulation were qualitatively reproduced in these simulations. Detailed unfolding pathways for each case explain the effects of changes in the chain topology on the folding energy landscape. GeoFold is a useful tool for the inference of the effects of disulfide engineering on the energy landscape of protein unfolding.

The fast and the slow: folding and trapping of λ6-85

Molecular dynamics simulations combining many microsecond trajectories have recently predicted that a very fast folding protein like lambda repressor fragment λ(6-85) D14A could have a slow millisecond kinetic phase. We investigated this possibility by detecting temperature-jump relaxation to 5 ms. While λ(6-85) D14A has no significant slow phase, two even more stable mutants do. A slow phase of λ(6-85) D14A does appear in mild denaturant. The experimental data and computational modeling together suggest the following hypothesis: λ(6-85) takes only microseconds to reach its native state from an extensively unfolded state, while the latter takes milliseconds to reach compact β-rich traps. λ(6-85) is not only thermodynamically but also kinetically protected from reaching such "intramolecular amyloids" while folding.

Atomistic folding simulations of the five-helix bundle protein λ(6−85)

Protein folding is a classic grand challenge that is relevant to numerous human diseases, such as protein misfolding diseases like Alzheimer’s disease. Solving the folding problem will ultimately require a combination of theory, simulation, and experiment, with theory and simulation providing an atomically detailed picture of both the thermodynamics and kinetics of folding and experimental tests grounding these models in reality. However, theory and simulation generally fall orders of magnitude short of biologically relevant time scales. Here we report significant progress toward closing this gap: an atomistic model of the folding of an 80-residue fragment of the λ repressor protein with explicit solvent that captures dynamics on a 10 milliseconds time scale. In addition, we provide a number of predictions that warrant further experimental investigation. For example, our model’s native state is a kinetic hub, and biexponential kinetics arises from the presence of many free-energy basins separated by barriers of different heights rather than a single low barrier along one reaction coordinate (the previously proposed incipient downhill folding scenario).

Unifying model for two-state and downhill protein folding

A protein-folding model is proposed at the amino acid level, in which the folding process is divided into two successive stages: the rate-determining step, dominated by the "stochastic interactions"of solvent molecules, and the rapid phase, dominated by the "order interactions"among atoms in polypeptide. The master equation approach is used to investigate the folding kinetics, and an analytical treatment of the master equation yields a simple three-parameter expression for folding time. It is found that both two-state and downhill protein-folding kinetics can be described by a unifying model.

Reducing lambda repressor to the core

Lambda repressor fragment λ(*)(6-85) is one of the fastest folding small protein fragments known to date. We hypothesized that removal of three out of five helices of λ(*)(6-85) would further reduce this protein to its smallest folding core. Molecular dynamics simulations singled out two energetically stable reduced structures consisting of only helices 1 and 4 connected by a short glycine/serine linker, as well as a less stable control. We investigated these three polypeptides and their fragments experimentally by using circular dichroism, fluorescence spectroscopy, and temperature jump relaxation spectroscopy to gain insight into their thermodynamic and kinetic properties. Based on the thermal melts, the order of peptide stability was in correspondence with theoretical predictions. The most stable two-helix bundle, λ(blue1), is a cooperatively folding miniprotein with the same melting temperature and folding rate as the full-length λ(*)(6-85) pseudo wild type and a well-defined computed structure.

Navigating the downhill protein folding regime via structural homologues

Proteins that fold over free-energy barriers <or= 3RT are classified as downhill folders. This regime is characterized by equilibrium unfolding that is proportionally broader and more complex the lower the folding barrier. Downhill proteins are also expected to fold up in a few microseconds. However, the relationship between rate and equilibrium signatures is affected by other factors such as protein size and folding topology. Here we perform a direct comparison of the kinetics and equilibrium unfolding of two structural homologues: BBL and PDD. BBL folds-unfolds in just approximately 1 micros at 335 K and displays the equilibrium signatures expected for a protein at the bottom of the downhill folding regime. PDD, which has the same 3D structure and size, folds-unfolds approximately 8 times more slowly and, concomitantly, exhibits all the downhill equilibrium signatures to a lesser degree. Our results demonstrate that the equilibrium signatures of downhill folding are proportional to the changes in folding rate once structural and size-scaling effects are factored out. This conclusion has two important implications: (1) it confirms that the quantitative analysis of equilibrium experiments in ultrafast folding proteins does provide direct information about free-energy barriers, a result that is incompatible with the conventional view of protein folding as a highly activated process, and (2) it advocates for equilibrium-kinetic studies of homologous proteins as a powerful tool to navigate the downhill folding regime via comparative analysis. The latter should prove extremely useful for the investigation of sequence, functional, and evolutionary determinants of protein folding barriers.

Dynamics of protein folding: probing the kinetic network of folding-unfolding transitions with experiment and theory

The problem of spontaneous folding of amino acid chains into highly organized, biologically functional three-dimensional protein structures continues to challenge the modern science. Understanding how proteins fold requires characterization of the underlying energy landscapes as well as the dynamics of the polypeptide chains in all stages of the folding process. In recent years, important advances toward these goals have been achieved owing to the rapidly growing interdisciplinary interest and significant progress in both experimental techniques and theoretical methods. Improvements in the experimental time resolution led to determination of the timescales of the important elementary events in folding, such as formation of secondary structure and tertiary contacts. Sensitive single molecule methods made possible probing the distributions of the unfolded and folded states and following the folding reaction of individual protein molecules. Discovery of proteins that fold in microseconds opened the possibility of atomic-level theoretical simulations of folding and their direct comparisons with experimental data, as well as of direct experimental observation of the barrier-less folding transition. The ultra-fast folding also brought new questions, concerning the intrinsic limits of the folding rates and experimental signatures of barrier-less "downhill" folding. These problems will require novel approaches for even more detailed experimental investigations of the folding dynamics as well as for the analysis of the folding kinetic data. For theoretical simulations of folding, a main challenge is how to extract the relevant information from overwhelmingly detailed atomistic trajectories. New theoretical methods have been devised to allow a systematic approach towards a quantitative analysis of the kinetic network of folding-unfolding transitions between various configuration states of a protein, revealing the transition states and the associated folding pathways at multiple levels, from atomistic to coarse-grained representations. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.

Efficient nonbonded interactions for molecular dynamics on a graphics processing unit

We describe an algorithm for computing nonbonded interactions with cutoffs on a graphics processing unit. We have incorporated it into OpenMM, a library for performing molecular simulations on high-performance computer architectures. We benchmark it on a variety of systems including boxes of water molecules, proteins in explicit solvent, a lipid bilayer, and proteins with implicit solvent. The results demonstrate that its performance scales linearly with the number of atoms over a wide range of system sizes, while being significantly faster than other published algorithms.

A natural missing link between activated and downhill protein folding scenarios

We propose protein PTB1 : 4W as a good candidate for engineering into a downhill folder. PTB1 : 4W has a probe-dependent thermal unfolding curve and sub-millisecond T-jump relaxation kinetics on more than one time scale. Its refolding rate in denaturant is a non-linear function of denaturant concentration (curved chevron plot). Yet at high denaturant concentration its unfolding is probe-independent, and the folding kinetics can be fitted to a single exponential decay. The domain appears to fold via a mechanism between downhill folding and activated folding over several small barriers, and when denaturant is added, one of these barriers greatly increases and simplifies the observed folding to apparent two-state kinetics. We predict the simplest free energy function consistent with the thermal denaturation and kinetics experiments by using the singular value Smoluchowski dynamics (SVSD) model. PTB1 : 4W is a natural 'missing link' between downhill and activated folding. We suggest mutations that could move the protein into the downhill folding limit.

CCMA: A Robust, Parallelizable Constraint Method for Molecular Simulations

We introduce a new algorithm, the Constant Constraint Matrix Approximation (CCMA), for constraining distances in molecular simulations. It combines the best features of many existing algorithms while avoiding their defects: it is fast, stable, can be applied to arbitrary constraint topologies, and can be efficiently implemented on modern parallel architectures. We test it on a protein with bond length and limited angle constraints, and find that it requires less than one sixth as many iterations as SHAKE to converge.

A survey of lambda repressor fragments from two-state to downhill folding

We survey the two-state to downhill folding transition by examining 20 lambda(6-85)* mutants that cover a wide range of stabilities and folding rates. We investigated four new lambda(6-85)* mutants designed to fold especially rapidly. Two were engineered using the core remodeling of Lim and Sauer, and two were engineered using Ferreiro et al.'s frustratometer. These proteins have probe-dependent melting temperatures as high as 80 degrees C and exhibit a fast molecular phase with the characteristic temperature dependence of the amplitude expected for downhill folding. The survey reveals a correlation between melting temperature and downhill folding previously observed for the beta-sheet protein WW domain. A simple model explains this correlation and predicts the melting temperature at which downhill folding becomes possible. An X-ray crystal structure with a 1.64-A resolution of a fast-folding mutant fragment shows regions of enhanced rigidity compared to the full wild-type protein.

Capillarity theory for the fly-casting mechanism

Biomolecular folding and function are often coupled. During molecular recognition events, one of the binding partners may transiently or partially unfold, allowing more rapid access to a binding site. We describe a simple model for this fly-casting mechanism based on the capillarity approximation and polymer chain statistics. The model shows that fly casting is most effective when the protein unfolding barrier is small and the part of the chain which extends toward the target is relatively rigid. These features are often seen in known examples of fly casting in protein-DNA binding. Simulations of protein-DNA binding based on well-funneled native-topology models with electrostatic forces confirm the trends of the analytical theory.

Direct observation of downhill folding of lambda-repressor in a microfluidic mixer

The protein lambda(6-85) has been implicated in barrierless folding by observations of kinetic relaxation after nanosecond T-jump. In this work we observed folding of this protein after dilution of a high denaturant in an ultrarapid microfluidic mixer at temperatures far below the thermal midpoint. The observations of total intensity and spectral shift of tryptophan fluorescence yielded distinctly different kinetics and activation energies. These results may be explained as diffusion on a low-barrier, one-dimensional, free-energy surface, with different probes having different sensitivities along the reaction coordinate. Additionally, we observed an extremely fast phase within the mixing time that was not observed by T-jump, suggesting that the ensemble of unfolded states populated at high denaturant is distinct from those accessible at high temperature.

Slowing down downhill folding: a three-probe study

The mutant Tyr22Trp/Glu33Tyr/Gly46Ala/Gly48Ala of lambda repressor fragment lambda(6-85) was previously assigned as an incipient downhill folder. We slow down its folding in a cryogenic water-ethylene-glycol solvent (-18 to -28 degrees C). The refolding kinetics are probed by small-angle x-ray scattering, circular dichroism, and fluorescence to measure the radius of gyration, the average secondary structure content, and the native packing around the single tryptophan residue. The main resolved kinetic phase of the mutant is probe independent and faster than the main phase observed for the pseudo-wild-type. Excess helical structure formed early on by the mutant may reduce the formation of turns and prevent the formation of compact misfolded states, speeding up the overall folding process. Extrapolation of our main cryogenic folding phase and previous T-jump measurements to 37 degrees C yields nearly the same refolding rate as extrapolated by Oas and co-workers from NMR line-shape data. Taken together, all the data consistently indicate a folding speed limit of approximately 4.5 micros for this fast folder.

Force field bias in protein folding simulations

Long timescale (>1 micros) molecular dynamics simulations of protein folding offer a powerful tool for understanding the atomic-scale interactions that determine a protein's folding pathway and stabilize its native state. Unfortunately, when the simulated protein fails to fold, it is often unclear whether the failure is due to a deficiency in the underlying force fields or simply a lack of sufficient simulation time. We examine one such case, the human Pin1 WW domain, using the recently developed deactivated morphing method to calculate free energy differences between misfolded and folded states. We find that the force field we used favors the misfolded states, explaining the failure of the folding simulations. Possible further applications of deactivated morphing and implications for force field development are discussed.

Analysis of the free-energy surface of proteins from reversible folding simulations

Computer generated trajectories can, in principle, reveal the folding pathways of a protein at atomic resolution and possibly suggest general and simple rules for predicting the folded structure of a given sequence. While such reversible folding trajectories can only be determined ab initio using all-atom transferable force-fields for a few small proteins, they can be determined for a large number of proteins using coarse-grained and structure-based force-fields, in which a known folded structure is by construction the absolute energy and free-energy minimum. Here we use a model of the fast folding helical λ-repressor protein to generate trajectories in which native and non-native states are in equilibrium and transitions are accurately sampled. Yet, representation of the free-energy surface, which underlies the thermodynamic and dynamic properties of the protein model, from such a trajectory remains a challenge. Projections over one or a small number of arbitrarily chosen progress variables often hide the most important features of such surfaces. The results unequivocally show that an unprojected representation of the free-energy surface provides important and unbiased information and allows a simple and meaningful description of many-dimensional, heterogeneous trajectories, providing new insight into the possible mechanisms of fast-folding proteins.

The process of protein folding is a complex transition from a disordered to an ordered state. Here, we simulate a specific fast-folding protein at the point at which the native and denatured states are at equilibrium and show that obtaining an accurate description of the mechanisms of folding and unfolding is far from trivial. Using simple quantities which quantify the degree of native order is, in the case of this protein, clearly misleading. We show that an unbiased representation of the free-energy surface can be obtained; using such a representation we are able to redesign the landscape and thus modify, upon site-specific “mutations”, the folding and unfolding rates. This leads us to formulate a hypothesis to explain the very fast folding of many proteins.

Evaluating beta-turn mimics as beta-sheet folding nucleators

Beta-turns are common conformations that enable proteins to adopt globular structures, and their formation is often rate limiting for folding. Beta-turn mimics, molecules that replace the i + 1 and i + 2 amino acid residues of a beta-turn, are envisioned to act as folding nucleators by preorganizing the pendant polypeptide chains, thereby lowering the activation barrier for beta-sheet formation. However, the crucial kinetic experiments to demonstrate that beta-turn mimics can act as strong nucleators in the context of a cooperatively folding protein have not been reported. We have incorporated 6 beta-turn mimics simulating varied beta-turn types in place of 2 residues in an engineered beta-turn 1 or beta-bulge turn 1 of the Pin 1 WW domain, a three-stranded beta-sheet protein. We present 2 lines of kinetic evidence that the inclusion of beta-turn mimics alters beta-sheet folding rates, enabling us to classify beta-turn mimics into 3 categories: those that are weak nucleators but permit Pin WW folding, native-like nucleators, and strong nucleators. Strong nucleators accelerate folding relative to WW domains incorporating all alpha-amino acid sequences. A solution NMR structure reveals that the native Pin WW beta-sheet structure is retained upon incorporating a strong E-olefin nucleator. These beta-turn mimics can now be used to interrogate protein folding transition state structures and the 2 kinetic analyses presented can be used to assess the nucleation capacity of other beta-turn mimics.

Accelerating molecular dynamic simulation on graphics processing units

We describe a complete implementation of all-atom protein molecular dynamics running entirely on a graphics processing unit (GPU), including all standard force field terms, integration, constraints, and implicit solvent. We discuss the design of our algorithms and important optimizations needed to fully take advantage of a GPU. We evaluate its performance, and show that it can be more than 700 times faster than a conventional implementation running on a single CPU core.

A one-dimensional free energy surface does not account for two-probe folding kinetics of protein alpha(3)D

We present fluorescence-detected measurements of the temperature-jump relaxation kinetics of the designed three-helix bundle protein alpha(3)D taken under solvent conditions identical to previous infrared-detected kinetics. The fluorescence-detected rate is similar to the IR-detected rate only at the lowest temperature where we could measure it (326 K). The fluorescence-detected rate decreases by a factor of 3 over the 326-344 K temperature range, whereas the IR-detected rate remains nearly constant over the same range. To investigate this probe dependence, we tested an extensive set of physically reasonable one-dimensional (1D) free energy surfaces by Langevin dynamics simulation. The simulations included coordinate- and temperature-dependent roughness, diffusion coefficients, and IR/fluorescence spectroscopic signatures. None of these can reproduce the IR and fluorescence data simultaneously, forcing us to the conclusion that a 1D free energy surface cannot accurately describe the folding of alpha(3)D. This supports the hypothesis that alpha(3)D has a multidimensional free energy surface conducive to downhill folding at 326 K, and that it is already an incipient downhill folder with probe-dependent kinetics near its melting point.