Dynamics Simulations Of Folding Research Articles

Data-driven detection of critical points of phase transitions in complex systems

Detecting the critical points of phase transitions and their driver factors in complex systems from data is a very challenging task. In these regards, the dynamic network biomarker/marker (DNB) method derived from the bifurcation theory is currently very popular, but a unified criterion to pick the most appropriate DNBs is lacking. Here, we propose a giant-component-based DNB (GDNB) method inspired by the percolation theory, that directly selects the largest DNB as the transition core to reflect the progress of the transition. We test the effectiveness of this scheme to detect transitions on three distinct systems, differing in terms of interactions and transitions: Monte Carlo simulations of the 2D Ising model, molecular dynamics simulations of protein folding, and measured gene expression time course in mouse muscle regeneration. These results suggest that the GDNB method inherits all the advantages of the DNB method, while it improves the interpretability at a reduced computational complexity.

A Study of a Protein-Folding Machine: Transient Rotation of the Polypeptide Backbone Facilitates Rapid Folding of Protein Domains in All-Atom Molecular Dynamics Simulations.

Molecular dynamics simulations of protein folding typically consider the polypeptide chain at equilibrium and in isolation from the cellular components. We argue that in order to understand protein folding as it occurs in vivo, it should be modeled as an active, energy-dependent process, in which the cellular protein-folding machine directly manipulates the polypeptide. We conducted all-atom molecular dynamics simulations of four protein domains, whose folding from the extended state was augmented by the application of rotational force to the C-terminal amino acid, while the movement of the N-terminal amino acid was restrained. We have shown earlier that such a simple manipulation of peptide backbone facilitated the formation of native structures in diverse α-helical peptides. In this study, the simulation protocol was modified, to apply the backbone rotation and movement restriction only for a short time at the start of simulation. This transient application of a mechanical force to the peptide is sufficient to accelerate, by at least an order of magnitude, the folding of four protein domains from different structural classes to their native or native-like conformations. Our in silico experiments show that a compact stable fold may be attained more readily when the motions of the polypeptide are biased by external forces and constraints.

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.

Caliber Corrected Markov Modeling (C2M2): Correcting Equilibrium Markov Models.

Rate processes are often modeled using Markov State Models (MSMs). Suppose you know a prior MSM and then learn that your prediction of some particular observable rate is wrong. What is the best way to correct the whole MSM? For example, molecular dynamics simulations of protein folding may sample many microstates, possibly giving correct pathways through them while also giving the wrong overall folding rate when compared to experiment. Here, we describe Caliber Corrected Markov Modeling (C2M2), an approach based on the principle of maximum entropy for updating a Markov model by imposing state- and trajectory-based constraints. We show that such corrections are equivalent to asserting position-dependent diffusion coefficients in continuous-time continuous-space Markov processes modeled by a Smoluchowski equation. We derive the functional form of the diffusion coefficient explicitly in terms of the trajectory-based constraints. We illustrate with examples of 2D particle diffusion and an overdamped harmonic oscillator.

Accelerated Molecular Dynamics Simulations of Protein Folding

Accelerated Molecular Dynamics Simulations of Protein Folding

Accelerated molecular dynamics simulations of protein folding.

Folding of four fast-folding proteins, including chignolin, Trp-cage, villin headpiece and WW domain, was simulated via accelerated molecular dynamics (aMD). In comparison with hundred-of-microsecond timescale conventional molecular dynamics (cMD) simulations performed on the Anton supercomputer, aMD captured complete folding of the four proteins in significantly shorter simulation time. The folded protein conformations were found within 0.2-2.1 Å of the native NMR or X-ray crystal structures. Free energy profiles calculated through improved reweighting of the aMD simulations using cumulant expansion to the second-order are in good agreement with those obtained from cMD simulations. This allows us to identify distinct conformational states (e.g., unfolded and intermediate) other than the native structure and the protein folding energy barriers. Detailed analysis of protein secondary structures and local key residue interactions provided important insights into the protein folding pathways. Furthermore, the selections of force fields and aMD simulation parameters are discussed in detail. Our work shows usefulness and accuracy of aMD in studying protein folding, providing basic references in using aMD in future protein-folding studies.

Non-Native Structure Appears in Microseconds during the Folding of E. coli RNase H

The folding pathway of Escherichia coli RNase H is one of the best experimentally characterized for any protein. In spite of this, spectroscopic studies have never captured the earliest events. Using continuous-flow microfluidic mixing, we have now observed the first several milliseconds of folding by monitoring the tryptophan fluorescence lifetime (60μs dead time). Two folding intermediates are observed, the second of which is the previously characterized Icore millisecond intermediate. The new earlier intermediate is likely on-pathway and appears to have long-range non-native structure, providing a rare example of such non-native structure formation in a folding pathway. The tryptophan fluorescence lifetimes also suggest a deviation from native packing in the second intermediate, Icore. Similar results from a fragment of RNase H demonstrate that only half of the protein is significantly involved in this early structure formation. These studies give us a view of the formation of tertiary structure on the folding pathway, which complements previous hydrogen-exchange studies that monitored only secondary structure and observed sequential native structure formation. Our results provide detailed folding information on both a timescale and a size-scale accessible to all-atom molecular dynamics simulations of protein folding.

Bayesian nonparametric analysis of reversible Markov chains

We introduce a three-parameter random walk with reinforcement, called the $(\theta,\alpha,\beta)$ scheme, which generalizes the linearly edge reinforced random walk to uncountable spaces. The parameter $\beta$ smoothly tunes the $(\theta,\alpha,\beta)$ scheme between this edge reinforced random walk and the classical exchangeable two-parameter Hoppe urn scheme, while the parameters $\alpha$ and $\theta$ modulate how many states are typically visited. Resorting to de Finetti’s theorem for Markov chains, we use the $(\theta,\alpha,\beta)$ scheme to define a nonparametric prior for Bayesian analysis of reversible Markov chains. The prior is applied in Bayesian nonparametric inference for species sampling problems with data generated from a reversible Markov chain with an unknown transition kernel. As a real example, we analyze data from molecular dynamics simulations of protein folding.

Multimolecule test-tube simulations of protein unfolding and aggregation

Molecular dynamics simulations of protein folding or unfolding, unlike most in vitro experimental methods, are performed on a single molecule. The effects of neighboring molecules on the unfolding/folding pathway are largely ignored experimentally and simply not modeled computationally. Here, we present two all-atom, explicit solvent molecular dynamics simulations of 32 copies of the Engrailed homeodomain (EnHD), an ultrafast-folding and -unfolding protein for which the folding/unfolding pathway is well-characterized. These multimolecule simulations, in comparison with single-molecule simulations and experimental data, show that intermolecular interactions have little effect on the folding/unfolding pathway. EnHD unfolded by the same mechanism whether it was simulated in only water or also in the presence of other EnHD molecules. It populated the same native state, transition state, and folding intermediate in both simulation systems, and was in good agreement with experimental data available for each of the three states. Unfolding was slowed slightly by interactions with neighboring proteins, which were mostly hydrophobic in nature and ultimately caused the proteins to aggregate. Protein-water hydrogen bonds were also replaced with protein-protein hydrogen bonds, additionally contributing to aggregation. Despite the increase in protein-protein interactions, the protein aggregates formed in simulation did not do so at the total exclusion of water. These simulations support the use of single-molecule techniques to study protein unfolding and also provide insight into the types of interactions that occur as proteins aggregate at high temperature at an atomic level.

Common Structural Transitions in Explicit-Solvent Simulations of Villin Headpiece Folding

Molecular dynamics simulations of protein folding can provide very high-resolution data on the folding process; however, due to computational challenges most studies of protein folding have been limited to small peptides, or made use of approximations such as Gō potentials or implicit solvent models. We have performed a set of molecular dynamics simulations totaling >50 μs on the villin headpiece subdomain, one of the most stable and fastest-folding naturally occurring proteins, in explicit solvent. We find that the wild-type villin headpiece reliably folds to a native conformation on timescales similar to experimentally observed folding, but that a fast folding double-norleucine mutant shows significantly more heterogeneous behavior. Along with other recent simulation studies, we note the occurrence of nonnative structures intermediates, which may yield a nativelike signal in the fluorescence measurements typically used to study villin folding. Based on the wild-type simulations, we propose alternative approaches to measure the formation of the native state.

Force Field Bias in Protein Folding Simulations

Long timescale (>1 μs) 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.

An Independent Method for the Analysis of Protein Folding Kinetics from All-atom Molecular Dynamics Simulations

We propose a method for extracting useful kinetic information from all-atom molecular dynamics simulations of protein folding. By calculating the time correlation functions between the evolution of different structural properties during the course of the simulation we can determine the endpoint of the reaction and the mechanism by which it occurs. As a test of our method we use thermal denaturation simulations on a 76 residue protein, ubiquitin. The method we present should be used in combination with current techniques for analyzing molecular dynamics trajectories.

Computer simulations of protein folding: Classical trajectories by optimization of action

The vast range of time scales (from nanoseconds to seconds) during protein folding is a challenge for experiments and computations. To make concrete predictions of folding mechanisms, atomically detailed simulations of protein folding using potentials derived from chemical physics principles, are desired. However, due to their computational complexity, straightforward molecular dynamics simulations of protein folding are impossible using today's technology. An alternative algorithm for atomically detailed simulations is discussed that makes it possible to compute approximate atomically detailed long time trajectories (SDEL—the Stochastic Difference Equation in Length). This algorithm is used to compute folding trajectories of a helical peptide, protein A, and the protein cytochrome c. A brief account of the methodology, the results, and comparison to experiments are provided.

Kinetics of cytochrome C folding: atomically detailed simulations.

The vast range of time scales (from nanoseconds to seconds) during protein folding is a challenge for experiments and computations. To make concrete predictions on folding mechanisms, atomically detailed simulations of protein folding, using potentials derived from chemical physics principles, are desired. However, due to their computational complexity, straightforward molecular dynamics simulations of protein folding are impossible today. An alternative algorithm is used that makes it possible to compute approximate atomically detailed long time trajectories (the Stochastic Difference Equation in Length). This algorithm is used to compute 26 atomically detailed folding trajectories of cytochrome c (a millisecond process). The early collapse of the protein chain (with marginal formation of secondary structure), and the earlier formation of the N and C helices (compare to the 60's helix) are consistent with the experiment. The existence of an energy barrier upon entry to the molten globule is examined as well. In addition to (favorable) comparison to experiments, we show that non-native contacts drive the formation of the molten globule. In contrast to popular folding models, the non-native contacts do not form off-pathway kinetic traps in cytochrome c.

Molecular dynamics simulations of protein folding from the transition state.

Putative transition-state ensemble (TSE) conformations of src SH3 were identified by monitoring the deviation from the experimental phi values along molecular dynamics (MD) simulations of unfolding. Sixty MD trajectories (for a total of about 7 micros) were then started from the putative TSE. About one-half of the 60 runs reached the folded state while unfolding was observed in the remaining half of the runs. This result validates phi-value analysis as an approach to obtain structural information on the transition state. It also demonstrates that an atomic resolution description of the TSE can be extracted from MD simulations. All conformations in the TSE have the central three-stranded beta-sheet formed in agreement with experimental data. An elongation of strand beta 2 as well as non-native side-chain interactions between the diverging turn and the distal hairpin are observed. The simulation results indicate that the tight packing of the side chains between the diverging turn and the distal hairpin is a necessary condition for rapid folding. Contacts between residues in the most structured element of the TSE, the central beta-sheet, are kinetically more important than those between the N- and C-terminal strands.

Computational protein folding: From lattice to all-atom

Understanding the mechanism of protein folding is often referred to as the second half of genetics. Computational approaches have been instrumental in the efforts. Simplified models have been applied to understand the physical principles governing the folding processes and will continue to play important roles in the endeavor. Encouraging results have been obtained from all-atom molecular dynamics simulations of protein folding. A recent microsecond-length molecular dynamics simulation on a small protein, villin headpiece subdomain, with an explict atomic-level representation of both protein and solvent, has marked the beginning of direct and realistic simulations of the folding processes. With growing computer power and increasingly accurate representations together with the advancement of experimental methods, such approaches will help us to achieve a detailed understanding of protein folding mechanisms.

On the Temperature and Pressure Dependence of a Range of Properties of a Type of Water Model Commonly Used in High-Temperature Protein Unfolding Simulations

Molecular dynamics simulations of protein folding and unfolding are often carried out at temperatures (400–600 K) that are much higher than physiological or room temperature to speed up the (un)folding process. Use of such high temperatures changes both the protein and solvent properties considerably, compared to physiological or room temperature. Water models designed for use in conjunction with biomolecules, such as the simple point charge (SPC) model, have generally been calibrated at room temperature and pressure. To determine the distortive effect of high simulation temperatures on the behavior of such “room temperature” water models, the structural, dynamic, and thermodynamic properties of the much-used SPC water model are investigated in the temperature range from 300 to 500 K. Both constant pressure and constant volume conditions, as used in protein simulations, were analyzed. We found that all properties analyzed change markedly with increasing temperature, but no phase transition in this temperature range was observed.

Brownian dynamics simulations of protein folding: access to milliseconds time scale and beyond.

Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion-collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion-collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments.

Brownian dynamics simulations of protein folding

The torsional angle space macromolecular conformational dynamics treatment presented in the preceding paper is used to study the mechanism and kinetics of protein folding by using continuum rigid chain molecules. The main purpose is to test the treatment using simple macromolecular systems. It is found that the torsional angle space approach is much faster and more reliable than similar approaches in atomic coordinate space. The simulation results also suggest that the short-ranged Lennard-Jones binary interactions alone are not sufficient to fold the chain molecules, and that hydrophobic collapse is essential for the folding processes. In our simplified protein folding model, the hydrophobic collapse is achieved by introducing global dipole interactions. The collapse of the chain molecule induced by dipole interactions significantly reduces the folding time. The chain collapse processes effectively bring the atoms into the (short) range of Lennard-Jones attractions, which then, in turn, are able to play their role in the folding processes; without such collapse the folding processes are highly frustrated.

Molecular Dynamics Simulations of Folding in Cyclic Alkanes

Molecular Dynamics Simulations of Folding in Cyclic Alkanes