Dominant folding pathways of a WW domain

Information Bottleneck Approach for Markov Model Construction

Markov state models (MSMs) have proven valuable in studying the dynamics of protein conformational changes via statistical analysis of molecular dynamics simulations. In MSMs, the complex configuration space is coarse-grained into conformational states, with dynamics modeled by a series of Markovian transitions among these states at discrete lag times. Constructing the Markovian model at a specific lag time necessitates defining states that circumvent significant internal energy barriers, enabling internal dynamics relaxation within the lag time. This process effectively coarse-grains time and space, integrating out rapid motions within metastable states. Thus, MSMs possess a multiresolution nature, where the granularity of states can be adjusted according to the time-resolution, offering flexibility in capturing system dynamics. This work introduces a continuous embedding approach for molecular conformations using the state predictive information bottleneck (SPIB), a framework that unifies dimensionality reduction and state space partitioning via a continuous, machine learned basis set. Without explicit optimization of the VAMP-based scores, SPIB demonstrates state-of-the-art performance in identifying slow dynamical processes and constructing predictive multiresolution Markovian models. Through applications to well-validated mini-proteins, SPIB showcases unique advantages compared to competing methods. It autonomously and self-consistently adjusts the number of metastable states based on a specified minimal time resolution, eliminating the need for manual tuning. While maintaining efficacy in dynamical properties, SPIB excels in accurately distinguishing metastable states and capturing numerous well-populated macrostates. This contrasts with existing VAMP-based methods, which often emphasize slow dynamics at the expense of incorporating numerous sparsely populated states. Furthermore, SPIB's ability to learn a low-dimensional continuous embedding of the underlying MSMs enhances the interpretation of dynamic pathways. With these benefits, we propose SPIB as an easy-to-implement methodology for end-to-end MSM construction.

An Information Bottleneck Approach for Markov Model Construction

Markov state models (MSMs) have proven valuable in studying dynamics of protein conformational changes via statistical analysis of molecular dynamics (MD) simulations. In MSMs, the complex configuration space is coarse-grained into conformational states, with dynamics modeled by a series of Markovian transitions among these states at discrete lag times. Constructing the Markovian model at a specific lag time necessitates defining states that circumvent significant internal energy barriers, enabling internal dynamics relaxation within the lag time. This process effectively coarse-grains time and space, integrating out rapid motions within metastable states. Thus, MSMs possess a multi-resolution nature, where the granularity of states can be adjusted according to the time-resolution, offering flexibility in capturing system dynamics. This work introduces a continuous embedding approach for molecular conformations using the state predictive information bottleneck (SPIB), a framework that unifies dimensionality reduction and state space partitioning via a continuous, machine learned basis set. Without explicit optimization of the VAMP-based scores, SPIB demonstrates state-of-the-art performance in identifying slow dynamical processes and constructing predictive multi-resolution Markovian models. Through applications to well-validated mini-proteins, SPIB showcases unique advantages compared to competing methods. It autonomously and self-consistently adjusts the number of metastable states based on specified minimal time resolution, eliminating the need for manual tuning. While maintaining efficacy in dynamical properties, SPIB excels in accurately distinguishing metastable states and capturing numerous well-populated macrostates. This contrasts with existing VAMP-based methods, which often emphasize slow dynamics at the expense of incorporating numerous sparsely populated states. Furthermore, SPIB's ability to learn a low-dimensional continuous embedding of the underlying MSMs enhances the interpretation of dynamic pathways. With these benefits, we propose SPIB as an easy-to-implement methodology for end-to-end MSMs construction.

An Efficient Path Classification Algorithm Based on Variational Autoencoder to Identify Metastable Path Channels for Complex Conformational Changes

Conformational changes (i.e., dynamic transitions between pairs of conformational states) play important roles in many chemical and biological processes. Constructing the Markov state model (MSM) from extensive molecular dynamics (MD) simulations is an effective approach to dissect the mechanism of conformational changes. When combined with transition path theory (TPT), MSM can be applied to elucidate the ensemble of kinetic pathways connecting pairs of conformational states. However, the application of TPT to analyze complex conformational changes often results in a vast number of kinetic pathways with comparable fluxes. This obstacle is particularly pronounced in heterogeneous self-assembly and aggregation processes. The large number of kinetic pathways makes it challenging to comprehend the molecular mechanisms underlying conformational changes of interest. To address this challenge, we have developed a path classification algorithm named latent-space path clustering (LPC) that efficiently lumps parallel kinetic pathways into distinct metastable path channels, making them easier to comprehend. In our algorithm, MD conformations are first projected onto a low-dimensional space containing a small set of collective variables (CVs) by time-structure-based independent component analysis (tICA) with kinetic mapping. Then, MSM and TPT are constructed to obtain the ensemble of pathways, and a deep learning architecture named the variational autoencoder (VAE) is used to learn the spatial distributions of kinetic pathways in the continuous CV space. Based on the trained VAE model, the TPT-generated ensemble of kinetic pathways can be embedded into a latent space, where the classification becomes clear. We show that LPC can efficiently and accurately identify the metastable path channels in three systems: a 2D potential, the aggregation of two hydrophobic particles in water, and the folding of the Fip35 WW domain. Using the 2D potential, we further demonstrate that our LPC algorithm outperforms the previous path-lumping algorithms by making substantially fewer incorrect assignments of individual pathways to four path channels. We expect that LPC can be widely applied to identify the dominant kinetic pathways underlying complex conformational changes.

Evolutionary conservation of amino acids contributing to the protein folding transition state

The question of whether amino acids critical to protein folding kinetics are evolutionarily conserved has been investigated intensively in the past, but no consensus has yet been reached. Recently, we have demonstrated that the transition state, dictating folding kinetics, is characterized as the state of maximum dynamic cooperativity, i.e., the state of maximum correlations between amino acid contact formations. Here, we investigate the evolutionary conservation of those amino acids contributing significantly to the dynamic cooperativity. We find a strong indication of a new kind of relationship-necessary but not sufficient causality-between the evolutionary conservation and the dynamic cooperativity: larger contributions to the dynamic cooperativity arise from more conserved residues, but not vice versa. This holds for all the protein systems for which long folding simulation trajectories are available. To our knowledge, this is the first systematic demonstration of any kind of evolutionary conservation of amino acids relevant to folding kinetics.

Long-range allostery mediates the regulation of plasminogen activator inhibitor-1 by cell adhesion factor vitronectin

The serpin plasminogen activator inhibitor 1 (PAI-1) spontaneously undergoes a massive structural change from a metastable and active conformation, with a solvent-accessible reactive center loop (RCL), to a stable, inactive, or latent conformation, with the RCL inserted into the central β-sheet. Physiologically, conversion to the latent state is regulated by the binding of vitronectin, which hinders the latency transition rate approximately twofold. The molecular mechanisms leading to this rate change are unclear. Here, we investigated the effects of vitronectin on the PAI-1 latency transition using all-atom path sampling simulations in explicit solvent. In simulated latency transitions of free PAI-1, the RCL is quite mobile as is the gate, the region that impedes RCL access to the central β-sheet. This mobility allows the formation of a transient salt bridge that facilitates the transition; this finding rationalizes existing mutagenesis results. Vitronectin binding reduces RCL and gate mobility by allosterically rigidifying structural elements over 40 Å away from the binding site, thus blocking transition to the latent conformation. The effects of vitronectin are propagated by a network of dynamically correlated residues including a number of conserved sites that were previously identified as important for PAI-1 stability. Simulations also revealed a transient pocket populated only in the vitronectin-bound state, corresponding to a cryptic drug-binding site identified by crystallography. Overall, these results shed new light on PAI-1 latency transition regulation by vitronectin and illustrate the potential of path sampling simulations for understanding functional protein conformational changes and for facilitating drug discovery.

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.

Site-Specific Backbone and Side-Chain Contributions to Thermodynamic Stabilizing Forces of the WW Domain

The native structure of a protein is stabilized by a number of interactions such as main-chain hydrogen bonds and side-chain hydrophobic contacts. However, it has been challenging to determine how these interactions contribute to protein stability at single amino acid resolution. Here, we quantified site-specific thermodynamic stability at the molecular level to extend our understanding of the stabilizing forces in protein folding. We derived the free energy components of individual amino acid residues separately for the folding of the human Pin WW domain based on simulated structures. A further decomposition of the thermodynamic properties into contributions from backbone and side-chain groups enabled us to identify the critical residues in the secondary structure and hydrophobic core formation, without introducing physical modifications to the system as in site-directed mutagenesis methods. By relating the structural and thermodynamic changes upon folding for each residue, we find that the simultaneous formation of the backbone hydrogen bonds and side-chain contacts cooperatively stabilizes the folded structure. The identification of stabilizing interactions in a folding protein at atomic resolution will provide molecular insights into understanding the origin of the protein structure and into engineering a more stable protein.

Local Interactions That Contribute Minimal Frustration Determine Foldability

Earlier experiments suggest that the evolutionary information (conservation and coevolution) encoded in protein sequences is necessary and sufficient to specify the fold of a protein family. However, there is no computational work to quantify the effect of such evolutionary information on the folding process. Here we explore the role of early folding steps for sequences designed using coevolution and conservation through a combination of computational and experimental methods. We simulated a repertoire of native and designed WW domain sequences to analyze early local contact formation and found that the N-terminal β-hairpin turn would not form correctly due to strong non-native local contacts in unfoldable sequences. Through a maximum likelihood approach, we identified five local contacts that play a critical role in folding, suggesting that a small subset of amino acid pairs can be used to solve the "needle in the haystack" problem to design foldable sequences. Thus, using the contact probability of those five local contacts that form during the early stage of folding, we built a classification model that predicts the foldability of a WW sequence with 81% accuracy. This classification model was used to redesign WW domain sequences that could not fold due to frustration and make them foldable by introducing a few mutations that led to the stabilization of these critical local contacts. The experimental analysis shows that a redesigned sequence folds and binds to polyproline peptides with a similar affinity as those observed for native WW domains. Overall, our analysis shows that evolutionary-designed sequences should not only satisfy the folding stability but also ensure a minimally frustrated folding landscape.

Combining Virtual Reality Visualization with Ensemble Molecular Dynamics to Study Complex Protein Conformational Changes

Molecular dynamics (MD) simulations are increasingly used to elucidate relationships between protein structure, dynamics, and their biological function. Currently, it is extremely challenging to perform MD simulations of large-scale structural rearrangements in proteins that occur on millisecond timescales or beyond, as this requires very significant computational resources, or the use of cumbersome "collective variable" enhanced sampling protocols. Here, we describe a framework that combines ensemble MD simulations and virtual reality visualization (eMD-VR) to enable users to interactively generate realistic descriptions of large amplitude, millisecond timescale protein conformational changes in proteins. Detailed tests demonstrate that eMD-VR substantially decreases the computational cost of folding simulations of a WW domain, without the need to define collective variables a priori. We further show that eMD-VR generated pathways can be combined with Markov state models to describe the thermodynamics and kinetics of large-scale loop motions in the enzyme cyclophilin A. Our results suggest eMD-VR is a powerful tool for exploring protein energy landscapes in bioengineering efforts.

New Insights into Folding, Misfolding, and Nonfolding Dynamics of a WW Domain

Intermediate states in protein folding are associated with formation of amyloid fibrils, which are responsible for a number of neurodegenerative diseases. Therefore, prevention of the aggregation of folding intermediates is one of the most important problems to overcome. Recently, we studied the origins and prevention of formation of intermediate states with the example of the Formin binding protein 28 (FBP28) WW domain. We demonstrated that the replacement of Leu26 by Asp26 or Trp26 (in ∼15% of the folding trajectories) can alter the folding scenario from three-state folding, a major folding scenario for the FBP28 WW domain (WT) and its mutants, toward two-state or downhill folding at temperatures below the melting point. Here, for a better understanding of the physics of the formation/elimination of intermediates, (i) the dynamics and energetics of formation of β-strands in folding, misfolding, and nonfolding trajectories of these mutants (L26D and L26W) is investigated; (ii) the experimental structures of WT, L26D, and L26W are analyzed in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger equation. We show that the formation of each β-strand in folding trajectories is accompanied by the emergence of kinks in internal coordinate space as well as a decrease in local free energy. In particular, the decrease in downhill folding trajectory is ∼7 kcal/mol, while it varies between 31 and 48 kcal/mol for the three-state folding trajectory. The kink analyses of the experimental structures give new insights into formation of intermediates, which may become a useful tool for preventing aggregation.

Successes and challenges in simulating the folding of large proteins

Computational simulations of protein folding can be used to interpret experimental folding results, to design new folding experiments, and to test the effects of mutations and small molecules on folding. However, whereas major experimental and computational progress has been made in understanding how small proteins fold, research on larger, multidomain proteins, which comprise the majority of proteins, is less advanced. Specifically, large proteins often fold via long-lived partially folded intermediates, whose structures, potentially toxic oligomerization, and interactions with cellular chaperones remain poorly understood. Molecular dynamics based folding simulations that rely on knowledge of the native structure can provide critical, detailed information on folding free energy landscapes, intermediates, and pathways. Further, increases in computational power and methodological advances have made folding simulations of large proteins practical and valuable. Here, using serpins that inhibit proteases as an example, we review native-centric methods for simulating the folding of large proteins. These synergistic approaches range from Gō and related structure-based models that can predict the effects of the native structure on folding to all-atom-based methods that include side-chain chemistry and can predict how disease-associated mutations may impact folding. The application of these computational approaches to serpins and other large proteins highlights the successes and limitations of current computational methods and underscores how computational results can be used to inform experiments. These powerful simulation approaches in combination with experiments can provide unique insights into how large proteins fold and misfold, expanding our ability to predict and manipulate protein folding.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

All-Atom Simulations Reveal How Single-Point Mutations Promote Serpin Misfolding

Protein misfolding is implicated in many diseases, including serpinopathies. For the canonical inhibitory serpin α₁-antitrypsin, mutations can result in protein deficiencies leading to lung disease, and misfolded mutants can accumulate in hepatocytes, leading to liver disease. Using all-atom simulations based on the recently developed bias functional algorithm, we elucidate how wild-type α₁-antitrypsin folds and how the disease-associated S (Glu264Val) and Z (Glu342Lys) mutations lead to misfolding. The deleterious Z mutation disrupts folding at an early stage, whereas the relatively benign S mutant shows late-stage minor misfolding. A number of suppressor mutations ameliorate the effects of the Z mutation, and simulations on these mutants help to elucidate the relative roles of steric clashes and electrostatic interactions in Z misfolding. These results demonstrate a striking correlation between atomistic events and disease severity and shine light on the mechanisms driving chains away from their correct folding routes.

A Hybrid Hamiltonian for the Accelerated Sampling along Experimental Restraints

In this article, we present an enhanced sampling method based on a hybrid Hamiltonian which combines experimental distance restraints with a bias dependent from multiple path-dependent variables. This simulation method determines the bias-coordinates on the fly and does not require a priori knowledge about reaction coordinates. The hybrid Hamiltonian accelerates the sampling of proteins, and, combined with experimental distance information, the technique considers the restraints adaptively and in dependency of the system's intrinsic dynamics. We validate the methodology on the dipole relaxation of two water models and the conformational landscape of dialanine. Using experimental NMR-restraint data, we explore the folding landscape of the TrpCage mini-protein and in a second example apply distance restraints from chemical crosslinking/mass spectrometry experiments for the sampling of the conformation space of the Killer Cell Lectin-like Receptor Subfamily B Member 1A (NKR-P1A). The new methodology has the potential to adaptively introduce experimental restraints without affecting the conformational space of the system along an ergodic trajectory. Since only a limited number of input- and no-order parameters are required for the setup of the simulation, the method is broadly applicable and has the potential to be combined with coarse-graining methods.

Exploring Configuration Space and Path Space of Biomolecules Using Enhanced Sampling Techniques-Searching for Mechanism and Kinetics of Biomolecular Functions

To understand functions of biomolecules such as proteins, not only structures but their conformational change and kinetics need to be characterized, but its atomistic details are hard to obtain both experimentally and computationally. Here, we review our recent computational studies using novel enhanced sampling techniques for conformational sampling of biomolecules and calculations of their kinetics. For efficiently characterizing the free energy landscape of a biomolecule, we introduce the multiscale enhanced sampling method, which uses a combined system of atomistic and coarse-grained models. Based on the idea of Hamiltonian replica exchange, we can recover the statistical properties of the atomistic model without any biases. We next introduce the string method as a path search method to calculate the minimum free energy pathways along a multidimensional curve in high dimensional space. Finally we introduce novel methods to calculate kinetics of biomolecules based on the ideas of path sampling: one is the Onsager⁻Machlup action method, and the other is the weighted ensemble method. Some applications of the above methods to biomolecular systems are also discussed and illustrated.

Transition path theory from biased simulations

Transition Path Theory (TPT) provides a rigorous framework to investigate the dynamics of rare thermally activated transitions. In this theory, a central role is played by the forward committor function q⁺(x), which provides the ideal reaction coordinate. Furthermore, the reactive dynamics and kinetics are fully characterized in terms of two time-independent scalar and vector distributions. In this work, we develop a scheme which enables all these ingredients of TPT to be efficiently computed using the short non-equilibrium trajectories generated by means of a specific combination of enhanced path sampling techniques. In particular, first we further extend the recently introduced self-consistent path sampling algorithm in order to compute the committor q⁺(x). Next, we show how this result can be exploited in order to define efficient algorithms which enable us to directly sample the transition path ensemble.

Frequency adaptive metadynamics for the calculation of rare-event kinetics

The ability to predict accurate thermodynamic and kinetic properties in biomolecular systems is of both scientific and practical utility. While both remain very difficult, predictions of kinetics are particularly difficult because rates, in contrast to free energies, depend on the route taken. For this reason, specific enhanced sampling methods are needed to calculate long-time scale kinetics. It has recently been demonstrated that it is possible to recover kinetics through the so-called "infrequent metadynamics" simulations, where the simulations are biased in a way that minimally corrupts the dynamics of moving between metastable states. This method, however, requires the bias to be added slowly, thus hampering applications to processes with only modest separations of time scales. Here we present a frequency-adaptive strategy which bridges normal and infrequent metadynamics. We show that this strategy can improve the precision and accuracy of rate calculations at fixed computational cost and should be able to extend rate calculations for much slower kinetic processes.

Self-consistent calculation of protein folding pathways

We introduce an iterative algorithm to efficiently simulate protein folding and other conformational transitions, using state-of-the-art all-atom force fields. Starting from the Langevin equation, we obtain a self-consistent stochastic equation of motion, which directly yields the reaction pathways. From the solution of this set of equations we derive a stochastic estimate of the reaction coordinate. We validate this approach against the results of plain MD simulations of the folding of a small protein, which were performed on the Anton supercomputer. In order to explore the computational efficiency of this algorithm, we apply it to generate a folding pathway of a protein that consists of 130 amino acids and has a folding rate of the order of s^-1.

All-atom calculation of protein free-energy profiles

The Bias Functional (BF) approach is a variational method which enables one to efficiently generate ensembles of reactive trajectories for complex biomolecular transitions, using ordinary computer clusters. For example, this scheme was applied to simulate in atomistic detail the folding of proteins consisting of several hundreds of amino acids and with experimental folding time of several minutes. A drawback of the BF approach is that it produces trajectories which do not satisfy microscopic reversibility. Consequently, this method cannot be used to directly compute equilibrium observables, such as free energy landscapes or equilibrium constants. In this work, we develop a statistical analysis which permits us to compute the potential of mean-force (PMF) along an arbitrary collective coordinate, by exploiting the information contained in the reactive trajectories calculated with the BF approach. We assess the accuracy and computational efficiency of this scheme by comparing its results with the PMF obtained for a small protein by means of plain molecular dynamics.

Parallel folding pathways of Fip35 WW domain explained by infrared spectra and their computer simulation

We present a calculation of the amide I' infrared (IR) spectra of the folded, unfolded, and intermediate states of the WW domain Fip35, a model system for β-sheet folding. Using an all-atom molecular dynamics simulation in which multiple folding and unfolding events take place we identify six conformational states and then apply perturbed matrix method quantum-mechanical calculations to determine their amide I' IR spectra. Our analysis focuses on two states previously identified as Fip35 folding intermediates and suggests that a three-stranded core similar to the folded state core is the main source of the spectroscopic differences between the two intermediates. In particular, we propose a hypothesis for why folding via one of these intermediates was not experimentally observed by IR T-jump.

Intrinsic map dynamics exploration for uncharted effective free-energy landscapes

We describe and implement a computer-assisted approach for accelerating the exploration of uncharted effective free-energy surfaces (FESs). More generally, the aim is the extraction of coarse-grained, macroscopic information from stochastic or atomistic simulations, such as molecular dynamics (MD). The approach functionally links the MD simulator with nonlinear manifold learning techniques. The added value comes from biasing the simulator toward unexplored phase-space regions by exploiting the smoothness of the gradually revealed intrinsic low-dimensional geometry of the FES.

Finding multiple reaction pathways via global optimization of action

Global searching for reaction pathways is a long-standing challenge in computational chemistry and biology. Most existing approaches perform only local searches due to computational complexity. Here we present a computational approach, Action-CSA, to find multiple diverse reaction pathways connecting fixed initial and final states through global optimization of the Onsager-Machlup action using the conformational space annealing (CSA) method. Action-CSA successfully overcomes large energy barriers via crossovers and mutations of pathways and finds all possible pathways of small systems without initial guesses on pathways. The rank order and the transition time distribution of multiple pathways are in good agreement with those of long Langevin dynamics simulations. The lowest action folding pathway of FSD-1 is consistent with recent experiments. The results show that Action-CSA is an efficient and robust computational approach to study the multiple pathways of complex reactions and large-scale conformational changes.

Perspective: Computer simulations of long time dynamics

Atomically detailed computer simulations of complex molecular events attracted the imagination of many researchers in the field as providing comprehensive information on chemical, biological, and physical processes. However, one of the greatest limitations of these simulations is of time scales. The physical time scales accessible to straightforward simulations are too short to address many interesting and important molecular events. In the last decade significant advances were made in different directions (theory, software, and hardware) that significantly expand the capabilities and accuracies of these techniques. This perspective describes and critically examines some of these advances.

Folding Mechanism of Proteins Im7 and Im9: Insight from All-Atom Simulations in Implicit and Explicit Solvent

Im7 and Im9 are evolutionary related proteins with almost identical native structures. In spite of their structural similarity, experiments show that Im7 folds through a long-lived on-pathway intermediate, while Im9 folds according to two-state kinetics. In this work, we use a recently developed enhanced path sampling method to generate many folding trajectories for these proteins, using realistic atomistic force fields, in both implicit and explicit solvent. Overall, our results are in good agreement with the experimental ϕ values and with the result of ϕ-value-restrained molecular dynamics (MD) simulations. However, our implicit solvent simulations fail to predict a qualitative difference in the folding pathways of Im7 and Im9. In contrast, our simulations in explicit solvent correctly reproduce the fact that only protein Im7 folds through a on-pathway intermediate. By analyzing our atomistic trajectories, we provide a physical picture which explains the observed difference in the folding kinetics of these chains.

High-Resolution Mapping of the Folding Transition State of a WW Domain

Fast-folding WW domains are among the best-characterized systems for comparing experiments and simulations of protein folding. Recent microsecond-resolution experiments and long duration (totaling milliseconds) single-trajectory modeling have shown that even mechanistic changes in folding kinetics due to mutation can now be analyzed. Thus, a comprehensive set of experimental data would be helpful to benchmark the predictions made by simulations. Here, we use T-jump relaxation in conjunction with protein engineering and report mutational Φ-values (Φ(M)) as indicators for folding transition-state structure of 65 side chain, 7 backbone hydrogen bond, and 6 deletion and /or insertion mutants within loop 1 of the 34-residue hPin1 WW domain. Forty-five cross-validated consensus mutants could be identified that provide structural constraints for transition-state structure within all substructures of the WW domain fold (hydrophobic core, loop 1, loop 2, β-sheet). We probe the robustness of the two hydrophobic clusters in the folding transition state, discuss how local backbone disorder in the native-state can lead to non-classical Φ(M)-values (Φ(M) > 1) in the rate-determining loop 1 substructure, and conclusively identify mutations and positions along the sequence that perturb the folding mechanism from loop 1-limited toward loop 2-limited folding.

The Role of Electrostatic Interactions in Folding of β-Proteins

Atomic-level molecular dynamic simulations are capable of fully folding structurally diverse proteins; however, they are limited in their ability to accurately represent electrostatic interactions. Here we have experimentally tested the role of charged residues on stability and folding kinetics of one of the most widely simulated β-proteins, the WW domain. The folding of wild type Pin1 WW domain, which has two positively charged residues in the first turn, was compared to the fast folding mutant FiP35 Pin1, which introduces a negative charge into the first turn. A combination of FTIR spectroscopy and laser-induced temperature-jump coupled with infrared spectroscopy was used to probe changes in the amide I region. The relaxation dynamics of the peptide backbone, β-sheets and β-turns, and negatively charged aspartic acid side chain of FiP35 were measured independently by probing the corresponding bands assigned in the amide I region. Folding is initiated in the turns and the β-sheets form last. While the global folding mechanism is in good agreement with simulation predictions, we observe changes in the protonation state of aspartic acid during folding that have not been captured by simulation methods. The protonation state of aspartic acid is coupled to protein folding; the apparent pKa of aspartic acid in the folded protein is 6.4. The dynamics of the aspartic acid follow the dynamics of the intermediate phase, supporting assignment of this phase to formation of the first hairpin. These results demonstrate the importance of electrostatic interactions in turn stability and formation of extended β-sheet structures.

Preventing fibril formation of a protein by selective mutation

The origins of formation of an intermediate state involved in amyloid formation and ways to prevent it are illustrated with the example of the Formin binding protein 28 (FBP28) WW domain, which folds with biphasic kinetics. Molecular dynamics of protein folding trajectories are used to examine local and global motions and the time dependence of formation of contacts between C(α)s and C(β)s of selected pairs of residues. Focus is placed on the WT FBP28 WW domain and its six mutants (L26D, L26E, L26W, E27Y, T29D, and T29Y), which have structures that are determined by high-resolution NMR spectroscopy. The origins of formation of an intermediate state are elucidated, viz. as formation of hairpin 1 by a hydrophobic collapse mechanism causing significant delay of formation of both hairpins, especially hairpin 2, which facilitates the emergence of an intermediate state. It seems that three-state folding is a major folding scenario for all six mutants and WT. Additionally, two-state and downhill folding scenarios were identified in ∼ 15% of the folding trajectories for L26D and L26W, in which both hairpins are formed by the Matheson-Scheraga mechanism much faster than in three-state folding. These results indicate that formation of hairpins connecting two antiparallel β-strands determines overall folding. The correlations between the local and global motions identified for all folding trajectories lead to the identification of the residues making the main contributions in the formation of the intermediate state. The presented findings may provide an understanding of protein folding intermediates in general and lead to a procedure for their prevention.

Computing Reaction Pathways of Rare Biomolecular Transitions using Atomistic Force-Fields

The Dominant Reaction Pathway (DRP) method is an approximate variational scheme which can be used to compute reaction pathways in conformational transitions undergone by large biomolecules (up to ~10(3) amino-acids) using realistic all-atom force fields. We first review the status of development of this method. Next, we discuss its validation against the results of plain MD protein folding simulations performed by the DE-Shaw group using the Anton supercomputer. Finally, we review a few representative applications of the DRP approach to study reactions which are far too complex and rare to be investigated by plain MD, even on the Anton machine.

Comparing Fast Pressure Jump and Temperature Jump Protein Folding Experiments and Simulations

The unimolecular folding reaction of small proteins is now amenable to a very direct mechanistic comparison between experiment and simulation. We present such a comparison of microsecond pressure and temperature jump refolding kinetics of the engineered WW domain FiP35, a model system for β-sheet folding. Both perturbations produce experimentally a faster and a slower kinetic phase, and the "slow" microsecond phase is activated. The fast phase shows differences between perturbation methods and is closer to the downhill limit by temperature jump, but closer to the transiently populated intermediate limit by pressure jump. These observations make more demands on simulations of the folding process than just a rough comparison of time scales. To complement experiments, we carried out several pressure jump and temperature jump all-atom molecular dynamics trajectories in explicit solvent, where FiP35 folded in five of the six simulations. We analyzed our pressure jump simulations by kinetic modeling and found that the pressure jump experiments and MD simulations are most consistent with a 4-state kinetic mechanism. Together, our experimental and computational data highlight FiP35's position at the boundary where activated intermediates and downhill folding meet, and we show that this model protein is an excellent candidate for further pressure jump molecular dynamics studies to compare experiment and modeling at the folding mechanism level.

Variational scheme to compute protein reaction pathways using atomistic force fields with explicit solvent

We introduce a variational approximation to the microscopic dynamics of rare conformational transitions of macromolecules. Within this framework it is possible to simulate on a small computer cluster reactions as complex as protein folding, using state of the art all-atom force fields in explicit solvent. We test this method against MD simulations of the folding of an α and a β protein performed with the same all-atom force field on the Anton supercomputer. We find that our approach yields results consistent with those of MD simulations, at a computational cost orders of magnitude smaller.

Folding kinetics of WW domains with the united residue force field for bridging microscopic motions and experimental measurements

To demonstrate the utility of the coarse-grained united-residue (UNRES) force field to compare experimental and computed kinetic data for folding proteins, we have performed long-time millisecond-timescale canonical Langevin molecular dynamics simulations of the triple β-strand from the Formin binding protein 28 WW domain and six nonnatural variants, using UNRES. The results have been compared with available experimental data in both a qualitative and a quantitative manner. Complexities of the folding pathways, which cannot be determined experimentally, were revealed. The folding mechanisms obtained from the simulated folding kinetics are in agreement with experimental results, with a few discrepancies for which we have accounted. The origins of single- and double-exponential kinetics and their correlations with two- and three-state folding scenarios are shown to be related to the relative barrier heights between the various states. The rate constants obtained from time profiles of the fractions of the native, intermediate, and unfolded structures, and the kinetic equations fitted to them, correlate with the experimental values; however, they are about three orders of magnitude larger than the experimental ones for most of the systems. These differences are in agreement with the timescale extension derived by scaling down the friction of water and averaging out the fast degrees of freedom when passing from all-atom to a coarse-grained representation. Our results indicate that the UNRES force field can provide accurate predictions of folding kinetics of these WW domains, often used as models for the study of the mechanisms of proein folding.

Serpin latency transition at atomic resolution

Protease inhibition by serpins requires a large conformational transition from an active, metastable state to an inactive, stable state. Similar reactions can also occur in the absence of proteases, and these latency transitions take hours, making their time scales many orders of magnitude larger than are currently accessible using conventional molecular dynamics simulations. Using a variational path sampling algorithm, we simulated the entire serpin active-to-latent transition in all-atom detail with a physically realistic force field using a standard computing cluster. These simulations provide a unifying picture explaining existing experimental data for the latency transition of the serpin plasminogen activator inhibitor-1 (PAI-1). They predict a long-lived intermediate that resembles a previously proposed, partially loop-inserted, prelatent state; correctly predict the effects of PAI-1 mutations on the kinetics; and provide a potential means to identify ligands able to accelerate the latency transition. Interestingly, although all of the simulated PAI-1 variants readily access the prelatent intermediate, this conformation is not populated in the active-to-latent transition of another serpin, α1-antitrypsin, which does not readily go latent. Thus, these simulations also help elucidate why some inhibitory serpin families are more conformationally labile than others.

High-resolution visualisation of the states and pathways sampled in molecular dynamics simulations

We have recently developed a scalable algorithm for ordering the instantaneous observations of a dynamical system evolving continuously in time. Here, we apply the method to long molecular dynamics trajectories. The procedure requires only a pairwise, geometrical distance as input. Suitable annotations of both structural and kinetic nature reveal the free energy basins visited by biomolecules. The profile is supplemented by a trace of the temporal evolution of the system highlighting the sequence of events. We demonstrate that the resultant SAPPHIRE (States And Pathways Projected with HIgh REsolution) plots provide a comprehensive picture of the thermodynamics and kinetics of complex, molecular systems exhibiting dynamics covering a range of time and length scales. Information on pathways connecting states and the level of recurrence are quickly inferred from the visualisation. The considerable advantages of our approach are speed and resolution: the SAPPHIRE plot is scalable to very large data sets and represents every single snapshot. This minimizes the risk of missing states because of overlap or prior coarse-graining of the data.

Tracking conformational dynamics of polypeptides by nonlinear electronic spectroscopy of aromatic residues: a first-principles simulation study

The ability of nonlinear electronic spectroscopy to track folding/unfolding processes of proteins in solution by monitoring aromatic interactions is investigated by first-principles simulations of two-dimensional (2D) electronic spectra of a model peptide. A dominant reaction pathway approach is employed to determine the unfolding pathway of a tetrapeptide, which connects the initial folded configuration with stacked aromatic side chains and the final unfolded state with distant noninteracting aromatic residues. The π-stacking and excitonic coupling effects are included through ab initio simulations based on multiconfigurational methods within a hybrid quantum mechanics/molecular mechanics scheme. It is shown that linear absorption spectroscopy in the ultraviolet (UV) region is unable to resolve the unstacking dynamics characterized by the three-step process: T-shaped→twisted offset stacking→unstacking. Conversely, pump-probe spectroscopy can be used to resolve aromatic interactions by probing in the visible region, the excited-state absorptions (ESAs) that involve charge-transfer states. 2D UV spectroscopy offers the highest sensitivity to the unfolding process, by providing the disentanglement of ESA signals belonging to different aromatic chromophores and high correlation between the conformational dynamics and the quartic splitting.

Towards a true protein movie: a perspective on the potential impact of the ensemble-based structure determination using exact NOEs

Confined by the Boltzmann distribution of the energies of the states, a multitude of structural states are inherent to biomolecules. For a detailed understanding of a protein's function, its entire structural landscape at atomic resolution and insight into the interconversion between all the structural states (i.e. dynamics) are required. Whereas dedicated trickery with NMR relaxation provides aspects of local dynamics, and 3D structure determination by NMR is well established, only recently have several attempts been made to formulate a more comprehensive description of the dynamics and the structural landscape of a protein. Here, a perspective is given on the use of exact NOEs (eNOEs) for the elucidation of structural ensembles of a protein describing the covered conformational space.

Consensus for the Fip35 folding mechanism?

Recent advances in computational power and simulation programs finally delivered the first examples of reversible folding for small proteins with an all-atom description. But having at hand the atomistic details of the process did not lead to a straightforward interpretation of the mechanism. For the case of the Fip35 WW-domain where multiple long trajectories of 100 μs are available from D. E. Shaw Research, different interpretations emerged. Some of those are in clear contradiction with each other while others are in qualitative agreement. Here, we present a network-based analysis of the same data by looking at the local fluctuations of conventional order parameters for folding. We found that folding occurs through two major pathways, one almost four times more populated than the other. Each pathway involves the formation of an intermediate with one of the two hairpins in a native configuration. The quantitative agreement of our results with a state-of-the-art reaction coordinate optimization procedure as well as qualitative agreement with other Markov-state-models and different simulation schemes provides strong evidence for a multiple folding pathways scenario with the presence of intermediates.

Dynamics of an ultrafast folding subdomain in the context of a larger protein fold

Small fast folding subdomains with low contact order have been postulated to facilitate the folding of larger proteins. We have tested this idea by determining how the fastest folding linear β-hairpin, CLN025, which folds on the nanosecond time scale, folds within the context of a two-hairpin WW domain system, which folds on the microsecond time scale. The folding of the wild type FBP28 WW domain was compared to constructs in which each of the loops was replaced by CLN025. A combination of FTIR spectroscopy and laser-induced temperature-jump coupled with infrared spectroscopy was used to probe changes in the peptide backbone. The relaxation dynamics of the β-sheets and β-turn were measured independently by probing the corresponding bands assigned in the amide I region. The folding rate of the CLN025 β-hairpin is unchanged within the larger protein. Insertion of the β-hairpin into the second loop results in an overall stabilization of the WW domain and a relaxation lifetime five times faster than the parent WW domain. In both mutants, folding is initiated in the turns and the β-sheets form last. These results demonstrate that fast folding subdomains can be used to speed the folding of more complex proteins, and that the folding dynamics of the subdomain is unchanged within the context of the larger protein.

The role of non-native interactions in the folding of knotted proteins: insights from molecular dynamics simulations

For several decades, the presence of knots in naturally-occurring proteins was largely ruled out a priori for its supposed incompatibility with the efficiency and robustness of folding processes. For this very same reason, the later discovery of several unrelated families of knotted proteins motivated researchers to look into the physico-chemical mechanisms governing the concerted sequence of folding steps leading to the consistent formation of the same knot type in the same protein location. Besides experiments, computational studies are providing considerable insight into these mechanisms. Here, we revisit a number of such recent investigations within a common conceptual and methodological framework. By considering studies employing protein models with different structural resolution (coarse-grained or atomistic) and various force fields (from pure native-centric to realistic atomistic ones), we focus on the role of native and non-native interactions. For various unrelated instances of knotted proteins, non-native interactions are shown to be very important for favoring the emergence of conformations primed for successful self-knotting events.

Unfolding thermodynamics of cysteine-rich proteins and molecular thermal-adaptation of marine ciliates

Euplotes nobilii and Euplotes raikovi are phylogenetically closely allied species of marine ciliates, living in polar and temperate waters, respectively. Their evolutional relation and the sharply different temperatures of their natural environments make them ideal organisms to investigate thermal-adaptation. We perform a comparative study of the thermal unfolding of disulfide-rich protein pheromones produced by these ciliates. Recent circular dichroism (CD) measurements have shown that the two psychrophilic (E. nobilii) and mesophilic (E. raikovi) protein families are characterized by very different melting temperatures, despite their close structural homology. The enhanced thermal stability of the E. raikovi pheromones is realized notwithstanding the fact that these proteins form, as a rule, a smaller number of disulfide bonds. We perform Monte Carlo (MC) simulations in a structure-based coarse-grained (CG) model to show that the higher stability of the E. raikovi pheromones is due to the lower locality of the disulfide bonds, which yields a lower entropy increase in the unfolding process. Our study suggests that the higher stability of the mesophilic E. raikovi phermones is not mainly due to the presence of a strongly hydrophobic core, as it was proposed in the literature. In addition, we argue that the molecular adaptation of these ciliates may have occurred from cold to warm, and not from warm to cold. To provide a testable prediction, we identify a point-mutation of an E. nobilii pheromone that should lead to an unfolding temperature typical of that of E. raikovi pheromones.

Analyzing milestoning networks for molecular kinetics: definitions, algorithms, and examples

Network representations are becoming increasingly popular for analyzing kinetic data from techniques like Milestoning, Markov State Models, and Transition Path Theory. Mapping continuous phase space trajectories into a relatively small number of discrete states helps in visualization of the data and in dissecting complex dynamics to concrete mechanisms. However, not only are molecular networks derived from molecular dynamics simulations growing in number, they are also getting increasingly complex, owing partly to the growth in computer power that allows us to generate longer and better converged trajectories. The increased complexity of the networks makes simple interpretation and qualitative insight of the molecular systems more difficult to achieve. In this paper, we focus on various network representations of kinetic data and algorithms to identify important edges and pathways in these networks. The kinetic data can be local and partial (such as the value of rate coefficients between states) or an exact solution to kinetic equations for the entire system (such as the stationary flux between vertices). In particular, we focus on the Milestoning method that provides fluxes as the main output. We proposed Global Maximum Weight Pathways as a useful tool for analyzing molecular mechanism in Milestoning networks. A closely related definition was made in the context of Transition Path Theory. We consider three algorithms to find Global Maximum Weight Pathways: Recursive Dijkstra's, Edge-Elimination, and Edge-List Bisection. The asymptotic efficiency of the algorithms is analyzed and numerical tests on finite networks show that Edge-List Bisection and Recursive Dijkstra's algorithms are most efficient for sparse and dense networks, respectively. Pathways are illustrated for two examples: helix unfolding and membrane permeation. Finally, we illustrate that networks based on local kinetic information can lead to incorrect interpretation of molecular mechanisms.