Recovering kinetics from a simplified protein folding model using replica exchange simulations: a kinetic network and effective stochastic dynamics

Replica Exchange Molecular Dynamics of Diphenylalanine Amyloid Peptides in Electric Fields

The self-assembling propensity of amyloid peptides such as diphenylalanine (FF) allows them to form ordered, nanoscale structures, with biocompatible properties important for biomedical applications. Moreover, piezoelectric properties allow FF molecules and their aggregates (e.g., FF nanotubes) to be aligned in a controlled way by the application of external electric fields. However, while the behavior of FF nanostructures emerges from the biophysical properties of the monomers, the detailed responses of individual peptides to both temperature and electric fields are not fully understood. Here, we study the temperature-dependent conformational dynamics of FF peptides solvated in explicit water molecules, an environment relevant to biomedical applications, by using an enhanced sampling method, replica exchange molecular dynamics (REMD), in conjunction with applied electric fields. Our simulations highlight and overcome possible artifacts that may occur during the setup of REMD simulations of explicitly solvated peptides in the presence of external electric fields, a problem particularly important in the case of short peptides such as FF. The presence of the external fields could overstabilize certain conformational states in one or more REMD replicas, leading to distortions of the underlying potential energy distributions observed at each temperature. This can be overcome by correcting the REMD initial conditions to include the lower-energy conformations induced by the external field. We show that the converged REMD data can be analyzed using a Markovian description of conformational states and show that a rather complex, 3-state, temperature-dependent conformational dynamics in the absence of electric fields collapses to only one of these states in the presence of the electric fields. These details on the temperature- and electric-field-dependent thermodynamic and kinetic properties of small FF amyloid peptides can be useful in understanding and devising new methods to control their aggregation-prone biophysical properties and, possibly, the structural and biophysical properties of FF molecular nanostructures.

The Role of Calcium in Regulating the Conformational Dynamics of d-Galactose/d-Glucose-Binding Protein Revealed by Markov State Model Analysis

The d-glucose/d-galactose-binding protein (GGBP) from Escherichia coli is a substrate-binding protein (SBP) associated with sugar transport and chemotaxis. It is also a calcium-binding protein, which makes it unique in the SBP family. However, the functional importance of Ca²⁺ binding is not fully understood. Here, the calcium-dependent properties of GGBP were explored by all-atom molecular dynamics simulations and Markov state model (MSM) analysis as well as single-molecule Förster resonance energy transfer (smFRET) measurements. In agreement with previous experimental studies, we observed the structure stabilization effect of Ca²⁺ binding on the C-terminal domain of GGBP, especially the Ca²⁺-binding site. Interestingly, the MSMs of calcium-depleted GGBP and calcium-bound GGBP (GGBP/Ca²⁺) demonstrate that Ca²⁺ greatly stabilizes the open conformation, and smFRET measurements confirmed this result. Further analysis reveals that Ca²⁺ binding disturbs the local hydrogen bonding interactions and the conformational dynamics of the hinge region, thereby weakening the long-range interdomain correlations to favor the open conformation. These results suggest an active regulatory role of Ca²⁺ binding in GGBP, which finely tunes the conformational distribution. The work sheds new light on the study of calcium-binding proteins in prokaryotes.

Glycerol transport through the aquaglyceroporin GlpF: bridging dynamics and kinetics with atomic simulation

The aquaglyceroporin GlpF is a member of the aquaporin family. It selectively conducts small molecules, such as glycerol, across the cell membrane under a concentration gradient of the substrate. Atomistic molecular dynamics (MD) simulation would provide great insight into the substrate transport mechanism of GlpF and membrane channels alike. Ideally, non-equilibrium simulations under various concentration gradients of glycerol are desired to emulate the transportation in cells, but this kind of simulation is difficult due to a complicated system setup and high computational cost. Here, we present a new strategy to extract non-equilibrium kinetic information from equilibrium MD simulation. We first performed long-time (totally 22.5 μs) multi-copy equilibrium MD simulations of glycerol conduction through GlpF. Tens of times the spontaneous permeation of glycerol through GlpF was observed, allowing us to elucidate the detailed mechanism of the stereoselectivity for glycerol. Then we employed Markov state model (MSM) analysis of the MD trajectories to identify the intermediate states during glycerol transport and calculate the inter-state transition rate constants. Based on the results of MSM analysis, we built the kinetic models of glycerol transport and calculated the glycerol fluxes under various concentration gradients by solving the master equations. The results agree well with the experimental measurement at a certain glycerol concentration, and provide holistic information on the glycerol conduction capacity of GlpF. Our work demonstrates that long-time atomistic MD simulations can now bridge the microscopic dynamics and the kinetic description of substance transport through membrane channels, hopefully facilitating the engineering of new selective channels for various molecules.

On the removal of initial state bias from simulation data

Classical atomistic simulations of biomolecules play an increasingly important role in molecular life science. The structure of current computing architectures favors methods that run multiple trajectories at once without requiring extensive communication between them. Many advanced sampling strategies in the field fit this mold. These approaches often rely on an adaptive logic and create ensembles of comparatively short trajectories whose starting points are not distributed according to the correct Boltzmann weights. This type of bias is notoriously difficult to remove, and Markov state models (MSMs) are one of the few strategies available for recovering the correct kinetics and thermodynamics from these ensembles of trajectories. In this contribution, we analyze the performance of MSMs in the thermodynamic reweighting task for a hierarchical set of systems. We show that MSMs can be rigorous tools to recover the correct equilibrium distribution for systems of sufficiently low dimensionality. This is conditional upon not tampering with local flux imbalances found in the data. For a real-world application, we find that a pure likelihood-based inference of the transition matrix produces the best results. The removal of the bias is incomplete, however, and for this system, all tested MSMs are outperformed by an alternative albeit less general approach rooted in the ideas of statistical resampling. We conclude by formulating some recommendations for how to address the reweighting issue in practice.

Conformational analysis of replica exchange MD: Temperature-dependent Markov networks for FF amyloid peptides

Recent molecular modeling methods using Markovian descriptions of conformational states of biomolecular systems have led to powerful analysis frameworks that can accurately describe their complex dynamical behavior. In conjunction with enhanced sampling methods, such as replica exchange molecular dynamics (REMD), these frameworks allow the systematic and accurate extraction of transition probabilities between the corresponding states, in the case of Markov state models, and of statistically-optimized transition rates, in the case of the corresponding coarse master equations. However, applying automatically such methods to large molecular dynamics (MD) simulations, with explicit water molecules, remains limited both by the initial ability to identify good candidates for the underlying Markovian states and by the necessity to do so using good collective variables as reaction coordinates that allow the correct counting of inter-state transitions at various lag times. Here, we show that, in cases when representative molecular conformations can be identified for the corresponding Markovian states, and thus their corresponding collective evolution of atomic positions can be calculated along MD trajectories, one can use them to build a new type of simple collective variable, which can be particularly useful in both the correct state assignment and in the subsequent accurate counting of inter-state transition probabilities. In the case of the ubiquitously used root-mean-square deviation (RMSD) of atomic positions, we introduce the relative RMSD (RelRMSD) measure as a good reaction coordinate candidate. We apply this method to the analysis of REMD trajectories of amyloid-forming diphenylalanine (FF) peptides-a system with important nanotechnology and biomedical applications due to its self-assembling and piezoelectric properties-illustrating the use of RelRMSD in extracting its temperature-dependent intrinsic kinetics, without a priori assumptions on the functional form (e.g., Arrhenius or not) of the underlying conformational transition rates. The RelRMSD analysis enables as well a more objective assessment of the convergence of the REMD simulations. This type of collective variable may be generalized to other observables that could accurately capture conformational differences between the underlying Markov states (e.g., distance RMSD, the fraction of native contacts, etc.).

Peptide dimerization-dissociation rates from replica exchange molecular dynamics

We show how accurate rates of formation and dissociation of peptide dimers can be calculated using direct transition counting (DTC) from replica-exchange molecular dynamics (REMD) simulations. First, continuous trajectories corresponding to system replicas evolving at different temperatures are used to assign conformational states. Second, we analyze the entire REMD data to calculate the corresponding rates at each temperature directly from the number of transition counts. Finally, we compare the kinetics extracted directly, using the DTC method, with indirect estimations based on trajectory likelihood maximization using short-time propagators and on decay rates of state autocorrelation functions. For systems with relatively low-dimensional intrinsic conformational dynamics, the DTC method is simple to implement and leads to accurate temperature-dependent rates. We apply the DTC rate-extraction method to all-atom REMD simulations of dimerization of amyloid-forming NNQQ tetrapetides in explicit water. In an assessment of the REMD sampling efficiency with respect to standard MD, we find a gain of more than a factor of two at the lowest temperature.

Implications of short time scale dynamics on long time processes

This review provides a comprehensive overview of the structural dynamics in topical gas- and condensed-phase systems on multiple length and time scales. Starting from vibrationally induced dissociation of small molecules in the gas phase, the question of vibrational and internal energy redistribution through conformational dynamics is further developed by considering coupled electron/proton transfer in a model peptide over many orders of magnitude. The influence of the surrounding solvent is probed for electron transfer to the solvent in hydrated I-. Next, the dynamics of a modified PDZ domain over many time scales is analyzed following activation of a photoswitch. The hydration dynamics around halogenated amino acid side chains and their structural dynamics in proteins are relevant for iodinated TyrB26 insulin. Binding of nitric oxide to myoglobin is a process for which experimental and computational analyses have converged to a common view which connects rebinding time scales and the underlying dynamics. Finally, rhodopsin is a paradigmatic system for multiple length- and time-scale processes for which experimental and computational methods provide valuable insights into the functional dynamics. The systems discussed here highlight that for a comprehensive understanding of how structure, flexibility, energetics, and dynamics contribute to functional dynamics, experimental studies in multiple wavelength regions and computational studies including quantum, classical, and more coarse grained levels are required.

Coarse Master Equations for Binding Kinetics of Amyloid Peptide Dimers

We characterize the kinetics of dimer formation of the short amyloid microcrystal-forming tetrapeptides NNQQ by constructing coarse master equations for the conformational dynamics of the system, using temperature replica-exchange molecular dynamics (REMD) simulations. We minimize the effects of Kramers-type recrossings by assigning conformational states based on their sequential time evolution. Transition rates are further estimated from short-time state propagators by maximizing the likelihood that the extracted rates agree with the observed atomistic trajectories without any a priori assumptions about their temperature dependence. Here, we evaluate the rates for both continuous replica trajectories that visit different temperatures and for discontinuous data corresponding to each REMD temperature. While the binding-unbinding kinetic process is clearly Markovian, the conformational dynamics of the bound NNQQ dimer has a complex character. Our kinetic analysis allows us to discriminate between short-lived encounter pairs and strongly bound conformational states. The conformational dynamics of NNQQ dimers supports a kinetically driven aggregation mechanism, in agreement with the polymorphic character reported for amyloid aggregates such as microcrystals and fibrils.

Simulating Replica Exchange: Markov State Models, Proposal Schemes, and the Infinite Swapping Limit

Replica exchange molecular dynamics is a multicanonical simulation technique commonly used to enhance the sampling of solvated biomolecules on rugged free energy landscapes. While replica exchange is relatively easy to implement, there are many unanswered questions about how to use this technique most efficiently, especially because it is frequently the case in practice that replica exchange simulations are not fully converged. A replica exchange cycle consists of a series of molecular dynamics steps of a set of replicas moving under different Hamiltonians or at different thermodynamic states followed by one or more replica exchange attempts to swap replicas among the different states. How the replica exchange cycle is constructed affects how rapidly the system equilibrates. We have constructed a Markov state model of replica exchange (MSMRE) using long molecular dynamics simulations of a host-guest binding system as an example, in order to study how different implementations of the replica exchange cycle can affect the sampling efficiency. We analyze how the number of replica exchange attempts per cycle, the number of MD steps per cycle, and the interaction between the two parameters affects the largest implied time scale of the MSMRE simulation. The infinite swapping limit is an important concept in replica exchange. We show how to estimate the infinite swapping limit from the diagonal elements of the exchange transition matrix constructed from MSMRE "simulations of simulations" as well as from relatively short runs of the actual replica exchange simulations.

Estimating first-passage time distributions from weighted ensemble simulations and non-Markovian analyses

First-passage times (FPTs) are widely used to characterize stochastic processes such as chemical reactions, protein folding, diffusion processes or triggering a stock option. In previous work (Suarez et al., JCTC 2014;10:2658-2667), we demonstrated a non-Markovian analysis approach that, with a sufficient subset of history information, yields unbiased mean first-passage times from weighted-ensemble (WE) simulations. The estimation of the distribution of the first-passage times is, however, a more ambitious goal since it cannot be obtained by direct observation in WE trajectories. Likewise, a large number of events would be required to make a good estimation of the distribution from a regular "brute force" simulation. Here, we show how the previously developed non-Markovian analysis can generate approximate, but highly accurate, FPT distributions from WE data. The analysis can also be applied to any other unbiased trajectories, such as from standard molecular dynamics simulations. The present study employs a range of systems with independent verification of the distributions to demonstrate the success and limitations of the approach. By comparison to a standard Markov analysis, the non-Markovian approach is less sensitive to the user-defined discretization of configuration space.

Mapping the Protein Fold Universe Using the CamTube Force Field in Molecular Dynamics Simulations

It has been recently shown that the coarse-graining of the structures of polypeptide chains as self-avoiding tubes can provide an effective representation of the conformational space of proteins. In order to fully exploit the opportunities offered by such a ‘tube model’ approach, we present here a strategy to combine it with molecular dynamics simulations. This strategy is based on the incorporation of the ‘CamTube’ force field into the Gromacs molecular dynamics package. By considering the case of a 60-residue polyvaline chain, we show that CamTube molecular dynamics simulations can comprehensively explore the conformational space of proteins. We obtain this result by a 20 μs metadynamics simulation of the polyvaline chain that recapitulates the currently known protein fold universe. We further show that, if residue-specific interaction potentials are added to the CamTube force field, it is possible to fold a protein into a topology close to that of its native state. These results illustrate how the CamTube force field can be used to explore efficiently the universe of protein folds with good accuracy and very limited computational cost.

Modelling protein behaviour using computer simulations has progressively emerged in the last 50 years as a powerful strategy in structural and molecular biology. Over this period there has been a continuing interest in pushing the boundaries of this approach in terms of the size of the systems and the timescale of the processes that can be studied. Coarse-grained models offer in principle great opportunities in this context, but it has been extremely challenging to obtain force fields of accuracy comparable to that typical of fully atomistic models. We show here that the representation of protein molecules as self-avoiding tubes within the CamTube model enables the comprehensive, accurate and very fast exploration of the conformational space of proteins in molecular dynamics simulations. We illustrate in particular how the comprehensive mapping of the protein fold universe obtained using the CamTube model offers the possibility of analysing the behaviour of proteins in a wide range of non-native states.

A comparative analysis of clustering algorithms: O2 migration in truncated hemoglobin I from transition networks

The ligand migration network for O2-diffusion in truncated Hemoglobin N is analyzed based on three different clustering schemes. For coordinate-based clustering, the conventional k-means and the kinetics-based Markov Clustering (MCL) methods are employed, whereas the locally scaled diffusion map (LSDMap) method is a collective-variable-based approach. It is found that all three methods agree well in their geometrical definition of the most important docking site, and all experimentally known docking sites are recovered by all three methods. Also, for most of the states, their population coincides quite favourably, whereas the kinetics of and between the states differs. One of the major differences between k-means and MCL clustering on the one hand and LSDMap on the other is that the latter finds one large primary cluster containing the Xe1a, IS1, and ENT states. This is related to the fact that the motion within the state occurs on similar time scales, whereas structurally the state is found to be quite diverse. In agreement with previous explicit atomistic simulations, the Xe3 pocket is found to be a highly dynamical site which points to its potential role as a hub in the network. This is also highlighted in the fact that LSDMap cannot identify this state. First passage time distributions from MCL clusterings using a one- (ligand-position) and two-dimensional (ligand-position and protein-structure) descriptor suggest that ligand- and protein-motions are coupled. The benefits and drawbacks of the three methods are discussed in a comparative fashion and highlight that depending on the questions at hand the best-performing method for a particular data set may differ.

Extracting intrinsic dynamic parameters of biomolecular folding from single-molecule force spectroscopy experiments

Single-molecule studies in which a mechanical force is transmitted to the molecule of interest and the molecular extension or position is monitored as a function of time are versatile tools for probing the dynamics of protein folding, stepping of molecular motors, and other biomolecular processes involving activated barrier crossing. One complication in interpreting such studies, however, is the fact that the typical size of a force probe (e.g., a dielectric bead in optical tweezers or the atomic force microscope tip/cantilever assembly) is much larger than the molecule itself, and so the observed molecular motion is affected by the hydrodynamic drag on the probe. This presents the experimenter with a nontrivial task of deconvolving the intrinsic molecular parameters, such as the intrinsic free energy barrier and the effective diffusion coefficient exhibited while crossing the barrier from the experimental signal. Here we focus on the dynamical aspect of this task and show how the intrinsic diffusion coefficient along the molecular reaction coordinate can be inferred from single-molecule measurements of the rates of biomolecular folding and unfolding. We show that the feasibility of accomplishing this task is strongly dependent on the relationship between the intrinsic molecular elasticity and that of the linker connecting the molecule to the force probe and identify the optimal range of instrumental parameters allowing determination of instrument-free molecular dynamics.

Fibril elongation by Aβ(17-42): kinetic network analysis of hybrid-resolution molecular dynamics simulations

A critical step of β-amyloid fibril formation is fibril elongation in which amyloid-β monomers undergo structural transitions to fibrillar structures upon their binding to fibril tips. The atomic detail of the structural transitions remains poorly understood. Computational characterization of the structural transitions is limited so far to short Aβ segments (5–10 aa) owing to the long time scale of Aβ fibril elongation. To overcome the computational time scale limit, we combined a hybrid-resolution model with umbrella sampling and replica exchange molecular dynamics and performed altogether ∼1.3 ms of molecular dynamics simulations of fibril elongation for Aβ_17–42. Kinetic network analysis of biased simulations resulted in a kinetic model that encompasses all Aβ segments essential for fibril formation. The model not only reproduces key properties of fibril elongation measured in experiments, including Aβ binding affinity, activation enthalpy of Aβ structural transitions and a large time scale gap (τ_lock/τ_dock = 10³–10⁴) between Aβ binding and its structural transitions, but also reveals detailed pathways involving structural transitions not seen before, namely, fibril formation both in hydrophobic regions L17-A21 and G37-A42 preceding fibril formation in hydrophilic region E22-A30. Moreover, the model identifies as important kinetic intermediates strand–loop–strand (SLS) structures of Aβ monomers, long suspected to be related to fibril elongation. The kinetic model suggests further that fibril elongation arises faster at the fibril tip with exposed L17-A21, rather than at the other tip, explaining thereby unidirectional fibril growth observed previously in experiments.

Isolation and characterization of new secondary metabolites from Asphodelus microcarpus

Phytochemical study of the ethanolic extract of Asphodelus microcarpus Salzm. et Viv. (Asphodelaceae) resulted in the isolation of two new compounds, methyl-1,4,5-trihydroxy-7-methyl-9,10-dioxo-9,10-dihydroanthracene-2-carboxylate (1), and (1R) 3,10-dimethoxy-5-methyl-1H-1,4-epoxybenzo[h]isochromene (2) as well as three known compounds; 3,4-dihydroxy-methyl benzoate (3), 3,4-dihydroxybenzoic acid (4), and 6-methoxychrysophanol (5). Compound 1 showed a potent activity against methicillin-resistant Staphylococcus aureus (MRSA) and Staphylococcus aureus with IC50 values of 1.5 and 1.2 µg/mL, respectively. Compound 3 showed antileishmanial activity with an IC50 value of 33.2 µg/mL. Compound 2 is the first isochromene possessing a highly strained 1,4-epoxy moiety. The structure elucidation of isolated metabolites was carried out using spectroscopic data, the absolute configuration of 2 based on optical rotation and electronic circular dichroism experiments and calculations.

How long does it take to equilibrate the unfolded state of a protein?

How long does it take to equilibrate the unfolded state of a protein? The answer to this question has important implications for our understanding of why many small proteins fold with two state kinetics. When the equilibration within the unfolded state U is much faster than the folding, the folding kinetics will be two state even if there are many folding pathways with different barriers. Yet the mean first passage times (MFPTs) between different regions of the unfolded state can be much longer than the folding time. This seems to imply that the equilibration within U is much slower than the folding. In this communication we resolve this paradox. We present a formula for estimating the time to equilibrate the unfolded state of a protein. We also present a formula for the MFPT to any state within U, which is proportional to the average lifetime of that state divided by the state population. This relation is valid when the equilibration within U is very fast as compared with folding as it often is for small proteins. To illustrate the concepts, we apply the formulas to estimate the time to equilibrate the unfolded state of Trp-cage and MFPTs within the unfolded state based on a Markov State Model using an ultra-long 208 microsecond trajectory of the miniprotein to parameterize the model. The time to equilibrate the unfolded state of Trp-cage is ∼100 ns while the typical MFPTs within U are tens of microseconds or longer.

NMR relaxation in proteins with fast internal motions and slow conformational exchange: model-free framework and Markov state simulations

Calculating NMR relaxation effects for proteins with dynamics on multiple time scales generally requires very long trajectories based on conventional molecular dynamics simulations. In this report, we have built Markov state models from multiple MD trajectories and used the resulting MSM to capture the very fast internal motions of the protein within a free energy basin on a time scale up to hundreds of picoseconds and the more than 3 orders of magnitude slower conformational exchange between macrostates. To interpret the relaxation data, we derive new equations using the model-free framework which includes two slowly exchanging macrostates, each of which also exhibits fast local motions. Using simulations of HIV-1 protease as an example, we show how the populations of slowly exchanging conformational states as well as order parameters for the different states can be determined from the NMR relaxation data.

Insights into the dynamics of HIV-1 protease: a kinetic network model constructed from atomistic simulations

The conformational dynamics in the flaps of HIV-1 protease plays a crucial role in the mechanism of substrate binding. We develop a kinetic network model, constructed from detailed atomistic simulations, to determine the kinetic mechanisms of the conformational transitions in HIV-1 PR. To overcome the time scale limitation of conventional molecular dynamics (MD) simulations, our method combines replica exchange MD with transition path theory (TPT) to study the diversity and temperature dependence of the pathways connecting functionally important states of the protease. At low temperatures the large-scale flap opening is dominated by a small number of paths; at elevated temperatures the transition occurs through many structurally heterogeneous routes. The expanded conformation in the crystal structure 1TW7 is found to closely mimic a key intermediate in the flap-opening pathways at low temperature. We investigated the different transition mechanisms between the semi-open and closed forms. The calculated relaxation times reveal fast semi-open ↔ closed transitions, and infrequently the flaps fully open. The ligand binding rate predicted from this kinetic model increases by 38-fold from 285 to 309 K, which is in general agreement with experiments. To our knowledge, this is the first application of a network model constructed from atomistic simulations together with TPT to analyze conformational changes between different functional states of a natively folded protein.

Analysis of the bacterial luciferase mobile loop by replica-exchange molecular dynamics

Bacterial luciferase contains an extended 29-residue mobile loop. Movements of this loop are governed by binding of either flavin mononucleotide (FMNH2) or polyvalent anions. To understand this process, loop dynamics were investigated using replica-exchange molecular dynamics that yielded conformational ensembles in either the presence or absence of FMNH2. The resulting data were analyzed using clustering and network analysis. We observed the closed conformations that are visited only in the simulations with the ligand. Yet the mobile loop is intrinsically flexible, and FMNH2 binding modifies the relative populations of conformations. This model provides unique information regarding the function of a crystallographically disordered segment of the loop near the binding site. Structures at or near the fringe of this network were compatible with flavin binding or release. Finally, we demonstrate that the crystallographically observed conformation of the mobile loop bound to oxidized flavin was influenced by crystal packing. Thus, our study has revealed what we believe are novel conformations of the mobile loop and additional context for experimentally determined structures.

Quantitative analysis of ligand migration from transition networks

In this work we use transition network analysis for the first time to investigate ligand migration in truncated hemoglobin (trHbN) and obtain kinetic information about the docking-site dynamics in the protein. A comparison with explicit water molecular dynamics simulations (100 ns in total) shows that the rate constants derived from the network analysis are realistic. The transition network analysis provides 1) The time-resolved connectivity network in the protein; 2) The half-lives of the docking sites; 3) The transition timescales between two given docking sites; and 4) The extent of population transfer among different docking sites of the protein as a function of lag time. We investigate the role of the Tyr33 and Gln58 residues in ligand migration by studying ligand migration in four mutants of trHbN. The mutation study suggests that residues Tyr33 and Gln58 stabilize the NO ligand in the Xe2 docking site of trHbN, thus facilitating the efficiency of the NO detoxification reaction.

Kinetic network study of the diversity and temperature dependence of Trp-Cage folding pathways: combining transition path theory with stochastic simulations

We present a new approach to study a multitude of folding pathways and different folding mechanisms for the 20-residue mini-protein Trp-Cage using the combined power of replica exchange molecular dynamics (REMD) simulations for conformational sampling, transition path theory (TPT) for constructing folding pathways, and stochastic simulations for sampling the pathways in a high dimensional structure space. REMD simulations of Trp-Cage with 16 replicas at temperatures between 270 and 566 K are carried out with an all-atom force field (OPLSAA) and an implicit solvent model (AGBNP). The conformations sampled from all temperatures are collected. They form a discretized state space that can be used to model the folding process. The equilibrium population for each state at a target temperature can be calculated using the weighted-histogram-analysis method (WHAM). By connecting states with similar structures and creating edges satisfying detailed balance conditions, we construct a kinetic network that preserves the equilibrium population distribution of the state space. After defining the folded and unfolded macrostates, committor probabilities (P(fold)) are calculated by solving a set of linear equations for each node in the network and pathways are extracted together with their fluxes using the TPT algorithm. By clustering the pathways into folding "tubes", a more physically meaningful picture of the diversity of folding routes emerges. Stochastic simulations are carried out on the network, and a procedure is developed to project sampled trajectories onto the folding tubes. The fluxes through the folding tubes calculated from the stochastic trajectories are in good agreement with the corresponding values obtained from the TPT analysis. The temperature dependence of the ensemble of Trp-Cage folding pathways is investigated. Above the folding temperature, a large number of diverse folding pathways with comparable fluxes flood the energy landscape. At low temperature, however, the folding transition is dominated by only a few localized pathways.

Modelling proteins: conformational sampling and reconstruction of folding kinetics

In the last decades biomolecular simulation has made tremendous inroads to help elucidate biomolecular processes in-silico. Despite enormous advances in molecular dynamics techniques and the available computational power, many problems involve long time scales and large-scale molecular rearrangements that are still difficult to sample adequately. In this review we therefore summarise recent efforts to fundamentally improve this situation by decoupling the sampling of the energy landscape from the description of the kinetics of the process. Recent years have seen the emergence of many advanced sampling techniques, which permit efficient characterisation of the relevant family of molecular conformations by dispensing with the details of the short-term kinetics of the process. Because these methods generate thermodynamic information at best, they must be complemented by techniques to reconstruct the kinetics of the process using the ensemble of relevant conformations. Here we review recent advances for both types of methods and discuss their perspectives to permit efficient and accurate modelling of large-scale conformational changes in biomolecules. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.