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

Analysis of Protein Folding Simulation with Moving Root Mean Square Deviation

We apply moving root-mean-square deviation (mRMSD), which does not require a reference structure, as a method for analyzing protein dynamics. This method can be used to calculate the root-mean-square deviation (RMSD) of structure between two specified time points and to analyze protein dynamics behavior through time series analysis. We applied this method to the Trp-cage trajectory calculated by the Anton supercomputer and found that it shows regions of stable states as well as the conventional RMSD. In addition, we extracted a characteristic structure in which the side chains of Asp1 and Arg16 form hydrogen bonds near the most stable structure of the Trp-cage. We also determined that ≥20 ns is an appropriate time interval to investigate protein dynamics using mRMSD. Applying this method to NuG2 protein, we found that mRMSD can be used to detect regions of metastable states in addition to the stable state. This method can be applied to molecular dynamics simulations of proteins whose stable structures are unknown.

Enhanced statistical sampling reveals microscopic complexity in the talin mechanosensor folding energy landscape

Statistical mechanics can describe the major conformational ensembles determining the equilibrium free-energy landscape of a folding protein. The challenge is to capture the full repertoire of low-occurrence conformations separated by high kinetic barriers that define complex landscapes. Computationally, enhanced sampling methods accelerate the exploration of molecular rare events. However, accessing the entire protein's conformational space in equilibrium experiments requires technological developments to enable extended observation times. We developed single-molecule magnetic tweezers to capture over a million individual transitions as a single talin protein unfolds and refolds under force in equilibrium. When observed at classically-probed timescales, talin folds in an apparently uncomplicated two-state manner. As the sampling time extends from minutes to days, the underlying energy landscape exhibits gradually larger signatures of complexity, involving a finite number of well-defined rare conformations. A fluctuation analysis allows us to propose plausible structures of each low-probability conformational state. The physiological relevance of each distinct conformation can be connected to the binding of the cytoskeletal protein vinculin, suggesting an extra layer of complexity in talin-mediated mechanotransduction. More generally, our experiments directly test the fundamental notion that equilibrium dynamics depend on the observation timescale.

Tuning Formation of Protein-DNA Coacervates by Sequence and Environment

The high concentration of nucleic acids and positively charged proteins in the cell nucleus provides many possibilities for complex coacervation. We consider a prototypical mixture of nucleic acids together with the polycationic C-terminus of histone H1 (CH1). Using a minimal coarse-grained model that captures the shape, flexibility, and charge distributions of the molecules, we find that coacervates are readily formed at physiological ionic strengths, in agreement with experiment, with a progressive increase in local ordering at low ionic strength. Variation of the positions of charged residues in the protein tunes phase separation: for well-mixed or only moderately blocky distributions of charge, there is a modest increase of local ordering with increasing blockiness that is also associated with an increased propensity to phase separate. This ordering is also associated with a slowdown of rotational and translational diffusion in the dense phase. However, for more extreme blockiness (and consequently higher local charge density), we see a qualitative change in the condensed phase to become a segregated structure with a dramatically increased ordering of the DNA. Naturally occurring proteins with these sequence properties, such as protamines in sperm cells, are found to be associated with very dense packing of nucleic acids.

Granger Causality Analysis of Chignolin Folding

Constantly advancing computer simulations of biomolecules provide huge amounts of data that are difficult to interpret. In particular, obtaining insights into functional aspects of macromolecular dynamics, often related to cascades of transient events, calls for methodologies that depart from the well-grounded framework of equilibrium statistical physics. One of the approaches toward the analysis of complex temporal data which has found applications in the fields of neuroscience and econometrics is Granger causality analysis. It allows determining which components of multidimensional time series are most influential for the evolution of the entire system, thus providing insights into causal relations within the dynamic structure of interest. In this work, we apply Granger analysis to a long molecular dynamics trajectory depicting repetitive folding and unfolding of a mini β-hairpin protein, CLN025. We find objective, quantitative evidence indicating that rearrangements within the hairpin turn region are determinant for protein folding and unfolding. On the contrary, interactions between hairpin arms score low on the causality scale. Taken together, these findings clearly favor the concept of zipperlike folding, which is one of two postulated β-hairpin folding mechanisms. More importantly, the results demonstrate the possibility of a conclusive application of Granger causality analysis to a biomolecular system.

Barrier Crossing in a Viscoelastic Bath

We investigate the hopping dynamics of a colloidal particle across a potential barrier and within a viscoelastic, i.e., non-Markovian, bath and report two clearly separated timescales in the corresponding waiting time distributions. While the longer timescale exponentially depends on the barrier height, the shorter one is similar to the relaxation time of the fluid. This short timescale is a signature of the storage and release of elastic energy inside the bath that strongly increases the hopping rate. Our results are in excellent agreement with numerical simulations of a simple Maxwell model.

Analysis of Molecular Dynamics Simulations of Protein Folding

Unbiased molecular dynamics simulations of proteins can now capture spontaneous folding events. This provides a wealth of data reflecting information on folding mechanism, but raises the challenge of interpreting it in a meaningful way. Here, I describe how such simulations can be used to identify reactive states and reaction coordinates for describing folding, and how folding dynamics can be captured by projection onto those coordinates. Methods are described for quantifying the interactions important for defining the folding mechanism, and for comparison of simulations with experimental mechanistic probes, such as ϕ-values.

Estimating transition path times and shapes from single-molecule photon trajectories: A simulation analysis

In a two-state molecular system, transition paths comprise the portions of trajectories during which the system transits from one stable state to the other. Because of their low population, it is essentially impossible to obtain information on transition paths from experiments on a large sample of molecules. However, single-molecule experiments such as laser optical tweezers or Förster resonance energy transfer (FRET) spectroscopy have allowed transition-path durations to be estimated. Here, we use molecular simulations to test the methodology for obtaining information on transition paths in single-molecule FRET by generating photon trajectories from the distance trajectories obtained in the simulation. Encouragingly, we find that this maximum likelihood analysis yields transition-path times within a factor of 2-4 of the values estimated using a good coordinate for folding, but tends to systematically underestimate them. The underestimation can be attributed partly to the fact that the large changes in the end-end distance occur mostly early in a folding trajectory. However, even if the transfer efficiency is a good reaction coordinate for folding, the assumption that the transition-path shape is a step function still leads to an underestimation of the transition-path time as defined here. We find that allowing more flexibility in the form of the transition path model allows more accurate transition-path times to be extracted and points the way toward further improvements in methods for estimating transition-path time and transition-path shape.

Barrier Crossing Dynamics from Single-Molecule Measurements

Chemists visualize chemical reactions as motion along one-dimensional "reaction coordinates" over free energy barriers. Various rate theories, such as transition state theory and the Kramers theory of diffusive barrier crossing, differ in their assumptions regarding the mathematical specifics of this motion. Direct experimental observation of the motion along reaction coordinates requires single-molecule experiments performed with unprecedented time resolution. Toward this goal, recent single-molecule studies achieved time resolution sufficient to catch biomolecules in the act of crossing free energy barriers as they fold, bind to their targets, or undergo other large structural changes, offering a window into the elusive reaction "mechanisms". This Perspective describes what we can learn (and what we have already learned) about barrier crossing dynamics through synergy of single-molecule experiments, theory, and molecular simulations. In particular, I will discuss how emerging experimental data can be used to answer several questions of principle. For example, is motion along the reaction coordinate diffusive, is there conformational memory, and is reduction to just one degree of freedom to represent the reaction mechanism justified? It turns out that these questions can be formulated as experimentally testable mathematical inequalities, and their application to experimental and simulated data has already led to a number of insights. I will also discuss open issues and current challenges in this fast evolving field of research.

Non-Markov bond model for dynamic force spectroscopy

Single-molecule force spectroscopy data are conventionally analyzed using a schematic model, wherein a molecular bond is represented as a virtual particle diffusing in a one-dimensional free-energy landscape. However, this simple and efficient approach is unable to account for the "anomalous" bond-breaking kinetics increasingly observed in force spectroscopy experiments and simulations, e.g., in the form of non-exponential distributions of bond lifetimes under constant load. Here, we show that such characteristic traits arise naturally in a rigorous extension of the one-dimensional theory that accounts for the transient dynamics of a generic set of coupled degrees of freedom. These "hidden modes" affect the reaction dynamics in various ways, depending on their relaxation spectrum and the loading protocol, giving rise, in particular, to apparent static and dynamic disorder. In two complementary asymptotic limits, we are able to find exact analytical expressions for pertinent experimental observables, such as the mean rupture force and the rupture-force distribution. Intriguingly, our asymptotic results become unconditionally exact at high loading rates, thus providing us with a microscopically consistent theory of rapid force spectroscopy that avoids the usual Markov assumption.

Learning dynamical information from static protein and sequencing data

Many complex processes, from protein folding to neuronal network dynamics, can be described as stochastic exploration of a high-dimensional energy landscape. Although efficient algorithms for cluster detection in high-dimensional spaces have been developed over the last two decades, considerably less is known about the reliable inference of state transition dynamics in such settings. Here we introduce a flexible and robust numerical framework to infer Markovian transition networks directly from time-independent data sampled from stationary equilibrium distributions. We demonstrate the practical potential of the inference scheme by reconstructing the network dynamics for several protein-folding transitions, gene-regulatory network motifs, and HIV evolution pathways. The predicted network topologies and relative transition time scales agree well with direct estimates from time-dependent molecular dynamics data, stochastic simulations, and phylogenetic trees, respectively. Owing to its generic structure, the framework introduced here will be applicable to high-throughput RNA and protein-sequencing datasets, and future cryo-electron microscopy (cryo-EM) data.

Diverse Folding Pathways of HIV-1 Protease Monomer on a Rugged Energy Landscape

The modern energy landscape theory of protein folding predicts multiple folding pathways connecting a myriad of unfolded conformations and a well-defined folded state. However, direct experimental observation of heterogeneous folding pathways is difficult. Naturally evolved proteins typically exhibit a smooth folding energy landscape for fast and efficient folding by avoiding unfavorable kinetic traps. In this case, rapid fluctuations between unfolded conformations result in apparent two-state behavior and make different pathways indistinguishable. However, the landscape roughness can be different, depending on the selection pressures during evolution. Here, we characterize the unusually rugged folding energy landscape of human immunodeficiency virus-1 protease monomer using single-molecule Förster resonance energy transfer spectroscopy. Our data show that fluctuations between unfolded conformations are slow, which enables the experimental observation of heterogeneous folding pathways as predicted by the landscape theory. Although the landscape ruggedness is sensitive to the mutations and fluorophore locations, the folding rate is similar for various protease constructs. The natural evolution of the protease to have a rugged energy landscape likely results from intrinsic pressures to maintain robust folding when human immunodeficiency virus-1 mutates frequently, which is essential for its survival.

Folding pathways of NuG2-a designed mutant of protein G-using relaxation mode analysis

Dynamic analysis methods are important for analyzing long simulations such as folding simulations. Relaxation mode analysis, which approximately extracts slow modes and rates, has been applied in molecular dynamics (MD) simulations of protein systems. Previously, we showed that slow modes are suitable for analyzing simulations in which large conformational changes occur. Here, we applied relaxation mode analysis to folding simulations of a designed mutant of protein G, NuG2, to investigate its folding pathways. The folding simulations of NuG2 were previously performed for this mutant with Anton. In the present study, the free energy surfaces were calculated by projecting the coordinates on the axis of the slow relaxation modes obtained from relaxation mode analysis. We classified various characteristic states such as native, nativelike, intermediate, and random states and clarified two main folding pathways. In the early folding process, the first and second β strands formed an N-terminal β-sheet. After the early folding process, the fourth β strand formed along the first β strand in the same or opposite direction as the native structure; two characteristic intermediate states were identified. Finally, the intermediate structures folded to the native structure in the folding process. Relaxation mode analysis can be applied to folding simulations of complex proteins to investigate their folding processes.

Position-, disorder-, and salt-dependent diffusion in binding-coupled-folding of intrinsically disordered proteins

Successful extensions of protein-folding energy landscape theory to intrinsically disordered proteins' (IDPs') binding-coupled-folding transition can enormously simplify this biomolecular process into diffusion along a limited number of reaction coordinates, and the dynamics subsequently is described by Kramers' rate theory. As the critical pre-factor, the diffusion coefficient D has direct implications on the binding kinetics. Here, we employ a structure-based model (SBM) to calculate D in the binding-folding of an IDP prototype. We identify a strong position-dependent D during binding by applying a reaction coordinate that directly measures the fluctuations in a Cartesian configuration space. Using the malleability of the SBM, we modulate the degree of conformational disorder in an isolated IDP and determine complex effects of intrinsic disorder on D varying for different binding stages. Here, D tends to increase with disorder during initial binding but shows a non-monotonic relationship with disorder in terms of a decrease-followed-by-increase in D during the late binding stage. The salt concentration, which correlates with electrostatic interactions via Debye-Hückel theory in our SBM, also modulates D in a stepwise way. The speeding up of diffusion by electrostatic interactions is observed during the formation of the encounter complex at the beginning of binding, while the last diffusive binding dynamics is hindered by non-native salt bridges. Because D describes the diffusive speed locally, which implicitly reflects the roughness of the energy landscape, we are eventually able to portray the binding energy landscape, including that from IDPs' binding, then to binding with partial folding, and finally to rigid docking, as well as that under different environmental salt concentrations. Our theoretical results provide key mechanistic insights into IDPs' binding-folding, which is internally conformation- and externally salt-controlled with respect to diffusion.

Instrumental Effects in the Dynamics of an Ultrafast Folding Protein under Mechanical Force

The analysis and interpretation of single molecule force spectroscopy (smFS) experiments is often complicated by hidden effects from the measuring device. Here we investigate these effects in our recent smFS experiments on the ultrafast folding protein gpW, which has been previously shown to fold without crossing a free energy barrier in the absence of force (i.e., downhill folding). Using atomic force microscopy (AFM) smFS experiments, we found that a very small force of ∼5 pN brings gpW near its unfolding midpoint and results in two-state (un)folding patterns that indicate the emergence of a force-induced free energy barrier. The change in the folding regime is concomitant with a 30,000-fold slowdown of the folding and unfolding times, from a few microseconds that it takes gpW to (un)fold at the midpoint temperature to seconds in the AFM. These results are puzzling because the barrier induced by force in the folding free energy landscape of gpW is far too small to account for such a difference in time scales. Here we use recently developed theoretical methods to resolve the origin of the strikingly slow dynamics of gpW under mechanical force. We find that, while the AFM experiments correctly capture the equilibrium distance distribution, the measured dynamics are entirely controlled by the response of the cantilever and polyprotein linker, which is much slower than the protein conformational dynamics. This interpretation is likely applicable to the folding of other small biomolecules in smFS experiments, and becomes particularly important in the case of systems with fast folding dynamics and small free energy barriers, and for instruments with slow response times.

Size and topology modulate the effects of frustration in protein folding

The presence of conflicting interactions, or frustration, determines how fast biomolecules can explore their configurational landscapes. Recent experiments have provided cases of systems with slow reconfiguration dynamics, perhaps arising from frustration. While it is well known that protein folding speed and mechanism are strongly affected by the protein native structure, it is still unknown how the response to frustration is modulated by the protein topology. We explore the effects of nonnative interactions in the reconfigurational and folding dynamics of proteins with different sizes and topologies. We find that structural correlations related to the folded state size and topology play an important role in determining the folding kinetics of proteins that otherwise have the same amount of nonnative interactions. In particular, we find that the reconfiguration dynamics of α-helical proteins are more susceptible to frustration than β-sheet proteins of the same size. Our results may explain recent experimental findings and suggest that attempts to measure the degree of frustration due to nonnative interactions might be more successful with α-helical proteins.

Memory-induced acceleration and slowdown of barrier crossing

We study the mean first-passage time τ_MFP for the barrier crossing of a single massive particle with non-Markovian memory by Langevin simulations in one dimension. In the Markovian limit of short memory time τ_Γ, the expected Kramers turnover between the overdamped (high-friction) and the inertial (low-friction) limits is recovered. Compared to the Markovian case, we find barrier crossing to be accelerated for intermediate memory time, while for long memory time, barrier crossing is slowed down and τ_MFP increases with τ_Γ as a power law τ_MFP∼τ_Γ². Both effects are derived from an asymptotic propagator analysis: while barrier crossing acceleration at intermediate memory can be understood as an effective particle mass reduction, slowing down for long memory is caused by the slow kinetics of energy diffusion. A simple and globally accurate heuristic formula for τ_MFP in terms of all relevant time scales of the system is presented and used to establish a scaling diagram featuring the Markovian overdamped and the Markovian inertial regimes, as well as the non-Markovian intermediate memory time regime where barrier crossing is accelerated and the non-Markovian long memory time regime where barrier crossing is slowed down.

A critical comparison of coarse-grained structure-based approaches and atomic models of protein folding

Structure-based coarse-grained Gō-like models have been used extensively in deciphering protein folding mechanisms because of their simplicity and tractability. Meanwhile, explicit-solvent molecular dynamics (MD) simulations with physics-based all-atom force fields have been applied successfully to simulate folding/unfolding transitions for several small, fast-folding proteins. To explore the degree to which coarse-grained Gō-like models and their extensions to incorporate nonnative interactions are capable of producing folding processes similar to those in all-atom MD simulations, here we systematically compare the computed unfolded states, transition states, and transition paths obtained using coarse-grained models and all-atom explicit-solvent MD simulations. The conformations in the unfolded state in common Gō models are more extended, and are thus more in line with experiment, than those from all-atom MD simulations. Nevertheless, the structural features of transition states obtained by the two types of models are largely similar. In contrast, the folding transition paths are significantly more sensitive to modeling details. In particular, when common Gō-like models are augmented with nonnative interactions, the predicted dimensions of the unfolded conformations become similar to those computed using all-atom MD. With this connection, the large deviations of all-atom MD from simple diffusion theory are likely caused in part by the presence of significant nonnative effects in folding processes modelled by current atomic force fields. The ramifications of our findings to the application of coarse-grained modeling to more complex biomolecular systems are discussed.

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.

Transition Path Times Measured by Single-Molecule Spectroscopy

The transition path is a tiny fraction of a molecular trajectory during which the free-energy barrier is crossed. It is a single-molecule property and contains all mechanistic information of folding processes of biomolecules such as proteins and nucleic acids. However, the transition path has been difficult to probe because it is short and rarely visited when transitions actually occur. Recent technical advances in single-molecule spectroscopy have made it possible to directly probe transition paths, which has opened up new theoretical and experimental approaches to investigating folding mechanisms. This article reviews recent single-molecule fluorescence and force spectroscopic measurements of transition path times and their connection to both theory and simulations.

Migration of small ligands in globins: Xe diffusion in truncated hemoglobin N

In heme proteins, the efficient transport of ligands such as NO or O₂ to the binding site is achieved via ligand migration networks. A quantitative assessment of ligand diffusion in these networks is thus essential for a better understanding of the function of these proteins. For this, Xe migration in truncated hemoglobin N (trHbN) of Mycobacterium Tuberculosis was studied using molecular dynamics simulations. Transitions between pockets of the migration network and intra-pocket relaxation occur on similar time scales (10 ps and 20 ps), consistent with low free energy barriers (1-2 kcal/mol). Depending on the pocket from where Xe enters a particular transition, the conformation of the side chains lining the transition region differs which highlights the coupling between ligand and protein degrees of freedom. Furthermore, comparison of transition probabilities shows that Xe migration in trHbN is a non-Markovian process. Memory effects arise due to protein rearrangements and coupled dynamics as Xe moves through it.

Binding and transport of ligands in proteins is essential, in particular in globular proteins which often exhibit internal cavities. In truncated Hemoglobin N (trHbN) these cavities are arranged as a network with particular connectivities. The present work supports the notion that ligand diffusion in trHbN is an active process and coupled to the protein dynamics. Furthermore, transition probabilities between neighboring pockets depend on the location from where the ligand entered the transition, which is typical for non-Markovian processes. Hence, ligand migration in trHbN exhibits memory effects due to dynamical coupling between the protein and ligand motion.

Modulation of Folding Internal Friction by Local and Global Barrier Heights

Recent experiments have revealed an unexpected deviation from a first power dependence of protein relaxation times on solvent viscosity, an effect that has been attributed to "internal friction". One clear source of internal friction in protein dynamics is the isomerization of dihedral angles. A key outstanding question is whether the global folding barrier height influences the measured internal friction, based on the observation that the folding rates of fast-folding proteins, with smaller folding free energy barriers, tend to exhibit larger internal friction. Here, by studying two alanine-based peptides, we find that systematic variation of global folding barrier heights has little effect on the internal friction for folding rates. On the other hand, increasing local torsion angle barriers leads to increased internal friction, which is consistent with solvent memory effects being the origin of the viscosity dependence. Thus, it appears that local torsion transitions determine the viscosity dependence of the diffusion coefficient on the global coordinate and, in turn, internal friction effects on the folding rate.

Topological constraints and modular structure in the folding and functional motions of GlpG, an intramembrane protease

We investigate the folding of GlpG, an intramembrane protease, using perfectly funneled structure-based models that implicitly account for the absence or presence of the membrane. These two models are used to describe, respectively, folding in detergent micelles and folding within a bilayer, which effectively constrains GlpG's topology in unfolded and partially folded states. Structural free-energy landscape analysis shows that although the presence of multiple folding pathways is an intrinsic property of GlpG's modular functional architecture, the large entropic cost of organizing helical bundles in the absence of the constraining bilayer leads to pathways that backtrack (i.e., local unfolding of previously folded substructures is required when moving from the unfolded to the folded state along the minimum free-energy pathway). This backtracking explains the experimental observation of thermodynamically destabilizing mutations that accelerate GlpG's folding in detergent micelles. In contrast, backtracking is absent from the model when folding is constrained within a bilayer, the environment in which GlpG has evolved to fold. We also characterize a near-native state with a highly mobile transmembrane helix 5 (TM5) that is significantly populated under folding conditions when GlpG is embedded in a bilayer. Unbinding of TM5 from the rest of the structure exposes GlpG's active site, consistent with studies of the catalytic mechanism of GlpG that suggest that TM5 serves as a substrate gate to the active site.