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The computer-aided investigation of protein folding has greatly benefited from coarse-grained models, that is, simplified representations at a resolution level lower than atomistic, providing access to qualitative and quantitative details of the folding process that would be hardly attainable, via all-atom descriptions, for medium to long molecules. Nonetheless, the effectiveness of low-resolution models is itself hampered by the presence, in a small but significant number of proteins, of nontrivial topological self-entanglements. Features such as native state knots or slipknots introduce conformational bottlenecks, affecting the probability to fold into the correct conformation; this limitation is particularly severe in the context of coarse-grained models. In this work, we tackle the relationship between folding probability, protein folding pathway, and protein topology in a set of proteins with a nontrivial degree of topological complexity. To avoid or mitigate the risk of incurring in kinetic traps, we make use of the elastic folder model, a coarse-grained model based on angular potentials optimized toward successful folding via a genetic procedure. This light-weight representation allows us to estimate in silico folding probabilities, which we find to anti-correlate with a measure of topological complexity as well as to correlate remarkably well with experimental measurements of the folding rate. These results strengthen the hypothesis that the topological complexity of the native state decreases the folding probability and that the force-field optimization mimics the evolutionary process these proteins have undergone to avoid kinetic traps.

In this paper, we present a fast and adaptive correlation guided enhanced sampling method (CORE-MD II). The CORE-MD II technique relies, in part, on partitioning of the entire pathway into short trajectories that we refer to as instances. The sampling within each instance is accelerated by adaptive path-dependent metadynamics simulations. The second part of this approach involves kinetic Monte Carlo (kMC) sampling between the different states that have been accessed during each instance. Through the combination of the partition of the total simulation into short non-equilibrium simulations and the kMC sampling, the CORE-MD II method is capable of sampling protein folding without any a priori definitions of reaction pathways and additional parameters. In the validation simulations, we applied the CORE-MD II on the dialanine peptide and the folding of two peptides: TrpCage and TrpZip2. In a comparison with long time equilibrium Molecular Dynamics (MD), 1 µs replica exchange MD (REMD), and CORE-MD I simulations, we find that the level of convergence of the CORE-MD II method is improved by a factor of 8.8, while the CORE-MD II method reaches acceleration factors of ∼120. In the CORE-MD II simulation of TrpZip2, we observe the formation of the native state in contrast to the REMD and the CORE-MD I simulations. The method is broadly applicable for MD simulations and is not restricted to simulations of protein folding or even biomolecules but also applicable to simulations of protein aggregation, protein signaling, or even materials science simulations.

Made available in DSpace on 2020-12-10T19:36:37Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-09-21 Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) National Science Foundation (NSF) National Science Foundation CAREER Award The stochastic drift-diffusion (DrDiff) theory is an approach used to characterize the dynamical properties of simulation data. With new features in transition times analyses, the framework characterized the thermodynamic free-energy profile [F(Q)], the folding time (tau(f)), and transition path time (tau(TP)) by determining the coordinate-dependent drift-velocity [v(Q)] and diffusion [D(Q)] coefficients from trajectory time traces. In order to explore the DrDiff approach and to tune it with two other methods (Bayesian analysis and fep1D algorithm), a numerical integration of the Langevin equation with known D(Q) and F(Q) was performed and the inputted coefficients were recovered with success by the diffusion models. DrDiff was also applied to investigate the prion protein (PrP) kinetics and thermodynamics by analyzing folding/unfolding simulations. The protein structure-based model, the well-known Go over bar -model, was employed in a coarse-grained C-alpha level to generate long constant-temperature time series. PrP was chosen due to recent experimental single-molecule studies in D and tau(TP) that stressed the importance and the difficulty of probing these quantities and the rare transition state events related to prion misfolding and aggregation. The PrP thermodynamic double-well F(Q) profile, the X shape of tau(f)(T), and the linear shape of tau(TP)(T) were predicted with v(Q) and D(Q) obtained by the DrDiff algorithm. With the advance of single-molecule techniques, the DrDiff framework might be a useful ally for determining kinetic and thermodynamic properties by analyzing time observables of biomolecular systems. The code is freely available at . Univ Fed Triangulo Mineiro, Inst Ciencias Exatas Nat & Educ, Dept Fis, Lab Biofis Teor, Uberaba, MG, Brazil Univ Fed ABC, Lab Biol Computac & Bioinformat, Santo Andre, SP, Brazil Rice Univ, Ctr Theoret Biol Phys, Houston, TX 77005 USA Univ Estadual Paulista, Dept Fis, Sao Jose Do Rio Preto, SP, Brazil Brazilian Ctr Res Energy & Mat CNPEM, Brazilian Biorenewables Natl Lab LNBR, Campinas, SP, Brazil Northeastern Univ, Dept Phys, Boston, MA 02115 USA Univ Estadual Paulista, Dept Fis, Sao Jose Do Rio Preto, SP, Brazil FAPEMIG: APQ-00941-14 FAPEMIG: CEX-RED-00010-14 CNPq: 438316/2018-5 FAPESP: 2016/13998-8 FAPESP: 2017/09662-7 National Science Foundation (NSF): PHY-1427654 National Science Foundation CAREER Award: MCB-1350312

Adaptive sampling methods, often used in combination with Markov state models, are becoming increasingly popular for speeding up rare events in simulation such as molecular dynamics (MD) without biasing the system dynamics. Several adaptive sampling strategies have been proposed, but it is not clear which methods perform better for different physical systems. In this work, we present a systematic evaluation of selected adaptive sampling strategies on a wide selection of fast folding proteins. The adaptive sampling strategies were emulated using models constructed on already existing MD trajectories. We provide theoretical limits for the sampling speed-up and compare the performance of different strategies with and without using some a priori knowledge of the system. The results show that for different goals, different adaptive sampling strategies are optimal. In order to sample slow dynamical processes such as protein folding without a priori knowledge of the system, a strategy based on the identification of a set of metastable regions is consistently the most efficient, while a strategy based on the identification of microstates performs better if the goal is to explore newer regions of the conformational space. Interestingly, the maximum speed-up achievable for the adaptive sampling of slow processes increases for proteins with longer folding times, encouraging the application of these methods for the characterization of slower processes, beyond the fast-folding proteins considered here.

As molecular dynamics simulations access increasingly longer time scales, complementary advances in the analysis of biomolecular time-series data are necessary. Markov state models offer a powerful framework for this analysis by describing a system’s states and the transitions between them. A recently established variational theorem for Markov state models now enables modelers to systematically determine the best way to describe a system’s dynamics. In the context of the variational theorem, we analyze ultra-long folding simulations for a canonical set of twelve proteins [K. Lindorff-Larsen et al., Science 334, 517 (2011)] by creating and evaluating many types of Markov state models. We present a set of guidelines for constructing Markov state models of protein folding; namely, we recommend the use of cross-validation and a kinetically motivated dimensionality reduction step for improved descriptions of folding dynamics. We also warn that precise kinetics predictions rely on the features chosen to describe the system and pose the description of kinetic uncertainty across ensembles of models as an open issue.

In the last years, it has become evident that computer simulations can assume a relevant role in modelling protein dynamical motions for their ability to provide a full atomistic image of the processes under investigation. The ability of the current protein force-fields in reproducing the correct thermodynamics and kinetics systems behaviour is thus an essential ingredient to improve our understanding of many relevant biological functionalities. In this work, employing the last developments of the metadynamics framework, we compare the ability of state-of-the-art all-atom empirical functions and water models to consistently reproduce the folding and unfolding of a helix turn motif in a model peptide. This theoretical study puts in evidence that the choice of the water models can influence the thermodynamic and the kinetics of the system under investigation, and for this reason cannot be considered trivial.

Biomolecular folding, at least in simple systems, can be described as a two state transition in a free energy landscape with two deep wells separated by a high barrier. Transition paths are the short part of the trajectories that cross the barrier. Average transition path times and, recently, their full probability distribution have been measured for several biomolecular systems, e.g. in the folding of nucleic acids or proteins. Motivated by these experiments, we have calculated the full transition path time distribution for a single stochastic particle crossing a parabolic barrier, focusing on the underdamped regime. Our analysis thus includes inertial terms, which were neglected in previous studies. These terms influence the short time scale dynamics of a stochastic system, and can be of experimental relevance in view of the short duration of transition paths. We derive the full transition path time distribution in the underdamped case and discuss the similarities and differences with the high friction (overdamped) limit., Comment: 16 pages, 5 figures

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.

In this paper, we present an improved method for obtaining unbiased estimates of the free energy difference between two thermodynamic states using the work distribution measured in nonequilibrium driven experiments connecting these states. The method is based on the assumption that any observed work distribution is given by a mixture of Gaussian distributions, whose normal components are identical in either direction of the nonequilibrium process, with weights regulated by the Crooks theorem. Using the prototypical example for the driven unfolding/folding of deca-alanine, we show that the predicted behavior of the forward and reverse work distributions, assuming a combination of only two Gaussian components with Crooks derived weights, explains surprisingly well the striking asymmetry in the observed distributions at fast pulling speeds. The proposed methodology opens the way for a perfectly parallel implementation of Jarzynski-based free energy calculations in complex systems.

Biased simulations have great potential for the study of slow processes, including protein folding. Atomic motions in molecules are nonlinear, which suggests that simulations with enhanced sampling of collective motions traced by nonlinear dimensionality reduction methods may perform better than linear ones. In this study, we compare an unbiased folding simulation of the Trp-cage miniprotein with metadynamics simulations using both linear (principle component analysis) and nonlinear (Isomap) low dimensional embeddings as collective variables. Folding of the mini-protein was successfully simulated in 200 ns simulation with linear biasing and non-linear motion biasing. The folded state was correctly predicted as the free energy minimum in both simulations. We found that the advantage of linear motion biasing is that it can sample a larger conformational space, whereas the advantage of nonlinear motion biasing lies in slightly better resolution of the resulting free energy surface. In terms of sampling efficiency, both methods are comparable.

A method is introduced to construct a better approximation for the reaction coordinate for protein folding from known order parameters. The folding of a two-state off-lattice alpha helical Go-type protein is studied using molecular dynamics simulations. Folding times are computed directly from simulation, as well as theoretically using an equation derived by considering Brownian-type dynamics for the putative reaction coordinate. Theoretical estimates of the folding time using the number of native contacts (Qn) as a reaction coordinate were seen to differ quite significantly from the true folding time of the protein. By considering the properties of the bimodal free energy surface of this protein as a function of Qn and another relevant coordinate for folding Q (the total number of contacts), we show that by introducing a rotation in the phase space of the order parameters Q and Qn, we can construct a new reaction coordinate q that leads to a fivefold improvement in the estimate of the folding rate. This new coordinate q, resulting from the rotation, lies along the line connecting the unfolded and folded ensemble minima of the free energy map plotted as a function of the original order parameters Q and Qn. Possible reasons for the remaining discrepancy between the folding time computed theoretically and from folding simulations are discussed.

Extended state methods are powerful tools for studying the conformational equilibria of proteins. This study focuses on three aspects of their implementation. First, existing approaches for determining importance weights (namely, recursion, random walk, and transition probability schemes) are compared in the context of their use with the method of expanded ensembles (EXE). Second, a combined scheme (REXE) involving EXE and replica exchange (REX) updates is developed for simulating a small number of replicas within a much larger macrostate space. Finally, variants of the extended state methods are considered for accelerating folding, either through special-purpose ensembles which target specific force-field parameters, or through biased sampling of extended macrostates that favor structural fluctuations. All methods are applied to a three-dimensional lattice protein model. Overall, it is found that transition probability approaches employing multiple system replicas perform naturally better than methods that intrinsically require macrostate equilibration by a single replica; the transition probability approaches need about an order of magnitude fewer steps to reach the same degree of convergence in the importance weights. The specific REXE protocol implemented is observed to have an efficiency intermediate to that of EXE and REX schemes at high temperatures, but to outperform them at more glassy conditions. Finally, special-purpose and locally enhanced tempering ensembles are shown to promote faster folding than conventional tempering.

For the small peptide met-enkephalin in implicit aqueous solvent a connected database of potential energy minima and transition states is constructed and refined based on kinetic criteria, using the discrete path sampling method. A comparison of this technique with parallel tempering Monte Carlo simulations shows that it produces a satisfactory sampling of the configuration space occupied at 298 K. The peptide is predicted to show a folding transition, and time scales for this folding are obtained, along with a description of significant folding intermediates.

A master equation approach is developed to find the rate-limiting steps in biopolymer folding, where the folding kinetics is described as a linear combination of basic kinetic modes determined from the eigenvalues and eigenvectors of the rate matrix. Because the passage of a rate-limiting step is intrinsically related to the folding speed, it is possible to probe and to identify the rate-limiting steps through the folding from different unfolded initial conformations. In a master equation approach, slow and fast folding speeds are directly correlated to the large and small contributions of the (rate-limiting) slow kinetic modes. Because the contributions from the slow modes can be computed from the corresponding eigenvectors, the rate-limiting steps can be identified from the eigenvectors of the slow modes. Our rate-limiting searching method has been tested for a simplified hairpin folding kinetics model, and it may provide a general transition state searching method for biopolymer folding.

The sequential importance sampling method and its various modifications have been developed intensively and used effectively in diverse research areas ranging from polymer simulation to signal processing and statistical inference. We propose a new variant of the method, sequential importance sampling with pilot-exploration resampling (SISPER), and demonstrate its successful application in folding polypeptide chains described by a two-dimensional hydrophobic-hydrophilic (HP) lattice model. We show by numerical results that SISPER outperformed several existing approaches, e.g., a genetic algorithm, the pruned-enriched Rosenbluth method, and the evolutionary Monte Carlo, in finding the ground folding states of 2D square-lattice HP sequences. In a few difficult cases, the new method can find the ground states without using any prior structural information on the chain. We also discuss the potential applications of SISPER in more general problems.

This work addresses the question of whether it is possible to define simple pair-wise interaction terms to approximate free energies of proteins or polymers. Rather than ask how reliable a potential of mean force is, one can ask how reliable it could possibly be. In a two-dimensional, infinite lattice model system one can calculate exact free energies by exhaustive enumeration. A series of approximations were fitted to exact results to assess the feasibility and utility of pair-wise free energy terms. Approximating the true free energy with pair-wise interactions gives a poor fit with little transferability between systems of different size. Adding extra artificial terms to the approximation yields better fits, but does not improve the ability to generalise from one system size to another. Further, one cannot distinguish folding from non-folding sequences via the approximated free energies. Most usefully, the methodology shows how one can assess the utility of various terms in lattice protein/polymer models., HTML/GIF-Format, incl. 5 figures, 5 tables - to be published in the The Journal of Chemical Physics

We explore the similarities and differences between the energy landscapes of proteins that have been selected by nature and those of some proteins designed by humans. Natural proteins have evolved to function as well as fold, and this is a source of energetic frustration. The sequence of Top7, on the other hand, was designed with architecture alone in mind using only native state stability as the optimization criterion. Its topology had not previously been observed in nature. Experimental studies show that the folding kinetics of Top7 is more complex than the kinetics of folding of otherwise comparable naturally occurring proteins. In this paper, we use structure prediction tools, frustration analysis, and free energy profiles to illustrate the folding landscapes of Top7 and two other proteins designed by Takada. We use both perfectly funneled (structure-based) and predictive (transferable) models to gain insight into the role of topological versus energetic frustration in these systems and show how they differ from those found for natural proteins. We also study how robust the folding of these designs would be to the simplification of the sequences using fewer amino acid types. Simplification using a five amino acid type code results in comparable quality of structure prediction to the full sequence in some cases, while the two-letter simplification scheme dramatically reduces the quality of structure prediction.

We study the dynamics on a biased globally connected energy landscape with correlations, using the generalized random energy model. Though the kinetic model has many applications, we discuss the model in the context of protein folding kinetics. Restricting the hierarchy to just two tiers, we find that the time scale of the overall folding coincides with that predicted by the random energy model at temperatures above the transition to partial freezing. At intermediate temperatures below the partial freezing transition, the search time is reduced due to the correlation effects. The dispersion of the relaxation time from sample to sample is also discussed.

The full characterization of protein folding is a remarkable long-standing challenge both for experiment and simulation. Working towards a complete understanding of this process, one needs to cover the full diversity of existing folds and identify the general principles driving the process. Here, we want to understand and quantify the diversity in folding routes for a large and representative set of protein topologies covering the full range from all alpha helical topologies towards beta barrels guided by the key question: Does the majority of the observed routes contribute to the folding process or only a particular route? We identified a set of two-state folders among non-homologous proteins with a sequence length of 40-120 residues. For each of these proteins, we ran native-structure based simulations both with homogeneous and heterogeneous contact potentials. For each protein, we simulated dozens of folding transitions in continuous uninterrupted simulations and constructed a large database of kinetic parameters. We investigate folding routes by tracking the formation of tertiary structure interfaces and discuss whether a single specific route exists for a topology or if all routes are equiprobable. These results permit us to characterize the complete folding space for small proteins in terms of folding barrier ΔG(‡), number of routes, and the route specificity RT.

The folding kinetics of Rd-apocytochrome b562 is two-state, but native-state hydrogen exchange experiments show that there are discrete partially unfolded (PUF) structures in equilibrium with the native state. These PUF structures are called hidden intermediates because they are not detected in kinetic experiments and they exist after the rate-limiting step. Structures of the mimics of hidden intermediates of Rd-apocytochrome b562 are resolved by NMR. Based upon their relative stability and structural features, the folding mechanism was proposed to follow a specific pathway (unfolded → rate-limiting transition state → PUF1 → PUF2 → native). Investigating the roles of equilibrium PUF structures in folding kinetics and their interrelationship not only deepens our understanding of the details of folding mechanism but also provides guides in protein design and prevention of misfolding. We performed molecular dynamics simulations starting from a hidden intermediate and the native state of Rd-apocytochrome b562 in explicit solvent, for a total of 37.18 μs mainly with Anton. We validated our simulations by detailed comparison with experimental data and other computations. We have verified that we sampled the post rate-limiting transition state region only. Markov state model was used to analyze the simulation results. We replace the specific pathway model with a network model. Transition-path theory was employed to calculate the net effective flux from the most unfolded state towards the most folded state in the network. The proposed sequential folding pathway via PUF1 then more stable, more native-like PUF2 is one of the routes in our network, but it is not dominant. The dominant path visits PUF2 without going through PUF1. There is also a route from PUF1 directly to the most folded state in the network without visiting PUF2. Our results indicate that the PUF states are not necessarily sequential in the folding. The major routes predicted in our network are testable by future experiments such as single molecule experiment.

We propose a new type of transition network for modeling of protein dynamics. The nodes of the network correspond to the conformations taken from random sampling of equilibrium ensemble available, e.g., by Monte Carlo simulations. Although this approach does not provide absolute values of folding/unfolding rates, it allows identification of reaction pathways, transition state ensemble, and, eventually, intermediates. The new method is verified by a comparison with direct molecular dynamic simulations performed for a coarse-grained Gō-like model of proteins. As an illustrative example, we analyze kinetics of formation of a small β-hairpin (Trp zipper 1) in the all-atom representation.

We propose a multiscale simulation method combining the efficiency of a coarse-grained model (CGM) and the accuracy of an all-atom model (AAM) for free-energy landscape calculation of protein systems. A protein’s conformation space is quickly searched first using CGM. Then the obtained information is incorporated into AAM simulations. The free-energy landscape is subsequently obtained from AAM simulations. This method was tested on chignolin folding. The results demonstrated that the computational time was reduced by as much as 90%.

The identification of metastable states of a molecule plays an important role in the interpretation of molecular simulation data because the free-energy surface, the relative populations in this landscape, and ultimately also the dynamics of the molecule under study can be described in terms of these states. We compare the results of three different geometric cluster algorithms (neighbor algorithm, K-medoids algorithm, and common-nearest-neighbor algorithm) among each other and to the results of a kinetic cluster algorithm. First, we demonstrate the characteristics of each of the geometric cluster algorithms using five two-dimensional data sets. Second, we analyze the molecular dynamics data of a beta-heptapeptide in methanol--a molecule that exhibits a distinct folded state, a structurally diverse unfolded state, and a fast folding/unfolding equilibrium--using both geometric and kinetic cluster algorithms. We find that geometric clustering strongly depends on the algorithm used and that the density based common-nearest-neighbor algorithm is the most robust of the three geometric cluster algorithms with respect to variations in the input parameters and the distance metric. When comparing the geometric cluster results to the metastable states of the beta-heptapeptide as identified by kinetic clustering, we find that in most cases the folded state is identified correctly but the overlap of geometric clusters with further metastable states is often at best approximate.

Topology-based potentials (also known as Gō-type models) have been widely used in the study of the protein folding problem. When a topology-based potential is applied, the structure of the native state of the protein considered has to be known in advance. This fact gives to these models a semiempirical character, and therefore the quality of the simulation results obtained for the folding transition relies, among other factors, on the accuracy of the experimental structural data employed. In this work, we use a topology-based potential to carry out folding simulations of a protein whose structure has been determined both with NMR spectroscopy and x-ray crystallography. This way, we have been able to establish to which extent the differences in the topologies of the two experimental structures, easily ignored in a standard structural analysis for this protein, affect the thermodynamic characteristics of the folding transition defined in the simulations.

Master equations are widely used for modeling protein folding. Here an approximate solution to such master equations is presented. The approach used may be viewed as a discrete variational transition-state theory. The folding rate constant kf is approximated by the outgoing reaction flux J, when the unfolded set of macrostates assumes an equilibrium distribution. Correspondingly the unfolding rate constant ku is calculated as Jpu(1-pu), where pu is the equilibrium fraction of the unfolded state. The dividing surface between the unfolded and folded states is chosen to minimize the reaction flux J. This minimum-reaction-flux surface plays the role of the transition-state ensemble and identifies rate-limiting steps. Test against exact results of master-equation models of Zwanzig [Proc. Natl. Acad. Sci. USA 92, 9801 (1995)] and Munoz et al. [Proc. Natl. Acad. Sci. USA 95, 5872 (1998)] shows that the minimum-reaction-flux solution works well. Macrostates separated by the minimum-reaction-flux surface show a gap in p(fold) values. The approach presented here significantly simplifies the solution of master-equation models and, at the same time, directly yields insight into folding mechanisms.

Exhaustive enumeration of all compact self‐avoiding conformations of a chain of 27 monomers on the 3*3*3 fragment of a simple cubic lattice is given. Total number of conformations unrelated by symmetry is 103 346. This number is relatively small which makes it possible to make a numerically exact calculation of all thermodynamic functions this chain. Heteropolymers with random sequence of links were considered, and the freezing transition at finite temperature was observed. This transition is analogous to folding transition in proteins where unique structure is formed. The numeric results demonstrate the equivalence between random 3‐dimensional heteropolymers and the random energy model found previously in analytical investigations. The possible application of these results to some problems of combinational optimization is discussed.

A new approach is developed for identifying suitable reaction coordinates to describe the progression of rare events in complex systems. The method is based on the forward flux sampling (FFS) technique and standard least-square estimation (LSE) and it is denoted as FFS-LSE. The FFS algorithm generates trajectories for the transition between stable states as chains of partially connected paths, which can then be used to obtain "on-the-fly" estimates for the committor probability to the final region, p(B). These p(B) data are then used to screen a set of candidate collective properties for an optimal order parameter (i.e., reaction coordinate) that depends on a few relevant variables. LSE is used to find the coefficients of the proposed reaction coordinate model and an analysis of variance is used to determine the significant terms in the model. The method is demonstrated for several test systems, including the folding of a lattice protein. It is shown that a simple approximation to p(B) via a model linear on energy and number of native contacts is sufficient to describe the intrinsic dynamics of the protein system and to ensure an efficient sampling of pathways. In addition, since the p(B) surface found from the FFS-LSE approach leads to the identification of the transition state ensemble, mechanistic details of the dynamics of the system can be readily obtained during a single FFS-type simulation without the need to perform additional committor simulations.

A novel scheme for fast conformational search has been developed by combining the replica exchange method (REM) with the generalized effective potential concept. The new method, referred to Q-REM [S. Jang et al. Phys. Rev. Lett. 91, 058305 (2003)], is expected to provide a useful alternative to the conventional REM for effective conformational sampling of complex systems. The authors have performed folding simulations of the Trp-cage miniprotein using Q-REM. All atom level simulations with generalized Born solvent access-area solvation model show that successful folding can be observed with much smaller number of replicas in Q-REM compared to the conventional REM. It can be concluded that the new method has potential to significantly improve sampling efficiency, allowing simulations of more challenging systems.

The Bacteriophage T4 Lysozyme (T4L) is a prototype modular protein comprised of an N-terminal and a C-domain domain, which was extensively studied to understand the folding/unfolding mechanism of modular proteins. To offer detailed structural and dynamic insights to the folded-state stability and the mechanical unfolding behaviors of T4L, we have performed extensive equilibrium and steered molecular dynamics simulations of both the wild-type (WT) and a circular permutation (CP) variant of T4L using all-atom and coarse-grained force fields. Our all-atom and coarse-grained simulations of the folded state have consistently found greater stability of the C-domain than the N-domain in isolation, which is in agreement with past thermostatic studies of T4L. While the all-atom simulation cannot fully explain the mechanical unfolding behaviors of the WT and the CP variant observed in an optical tweezers study, the coarse-grained simulations based on the Go model or a modified elastic network model (mENM) are in qualitative agreement with the experimental finding of greater unfolding cooperativity in the WT than the CP variant. Interestingly, the two coarse-grained models predict different structural mechanisms for the observed change in cooperativity between the WT and the CP variant--while the Go model predicts minor modification of the unfolding pathways by circular permutation (i.e., preserving the general order that the N-domain unfolds before the C-domain), the mENM predicts a dramatic change in unfolding pathways (e.g., different order of N/C-domain unfolding in the WT and the CP variant). Based on our simulations, we have analyzed the limitations of and the key differences between these models and offered testable predictions for future experiments to resolve the structural mechanism for cooperative folding/unfolding of T4L.

We propose an equi-energy (EE) sampling approach to study protein folding in the two-dimensional hydrophobic-hydrophilic (HP) lattice model. This approach enables efficient exploration of the global energy landscape and provides accurate estimates of the density of states, which then allows us to conduct a detailed study of the thermodynamics of HP protein folding, in particular, on the temperature dependence of the transition from folding to unfolding and on how sequence composition affects this phenomenon. With no extra cost, this approach also provides estimates on global energy minima and ground states. Without using any prior structural information of the protein the EE sampler is able to find the ground states that match the best known results in most benchmark cases. The numerical results demonstrate it as a powerful method to study lattice protein folding models.

A variation of the line integral method of Elber with self-avoiding walk has been implemented using a state of the art nonlinear constrained optimization procedure. The new implementation appears to be robust in finding approximate reaction paths for small and large systems. Exact transition states and intermediates for the resulting paths can easily be pinpointed with subsequent application of the conjugate peak refinement method [S. Fischer and M. Karplus, Chem. Phys. Lett. 194, 252 (1992)] and unconstrained minimization, respectively. Unlike previous implementations utilizing a penalty function approach, the present implementation generates an exact solution of the underlying problem. Most importantly, this formulation does not require an initial guess for the path, which makes it particularly useful for studying complex molecular rearrangements. The method has been applied to conformational rearrangements of the alanine dipeptide in the gas phase and in water, and folding of the beta hairpin of protein G in water. In the latter case a procedure was developed to systematically sample the potential energy surface underlying folding and reconstruct folding pathways within the nearest-neighbor hopping approximation.

In this paper, we introduce a parallel continuous simulated tempering (PCST) method for enhanced sampling in studying large complex systems. It mainly inherits the continuous simulated tempering (CST) method in our previous studies [C. Zhang and J. Ma, J. Chem. Phys. 130, 194112 (2009); C. Zhang and J. Ma, J. Chem. Phys. 132, 244101 (2010)], while adopts the spirit of parallel tempering (PT), or replica exchange method, by employing multiple copies with different temperature distributions. Differing from conventional PT methods, despite the large stride of total temperature range, the PCST method requires very few copies of simulations, typically 2-3 copies, yet it is still capable of maintaining a high rate of exchange between neighboring copies. Furthermore, in PCST method, the size of the system does not dramatically affect the number of copy needed because the exchange rate is independent of total potential energy, thus providing an enormous advantage over conventional PT methods in studying very large systems. The sampling efficiency of PCST was tested in two-dimensional Ising model, Lennard-Jones liquid and all-atom folding simulation of a small globular protein trp-cage in explicit solvent. The results demonstrate that the PCST method significantly improves sampling efficiency compared with other methods and it is particularly effective in simulating systems with long relaxation time or correlation time. We expect the PCST method to be a good alternative to parallel tempering methods in simulating large systems such as phase transition and dynamics of macromolecules in explicit solvent.

Thermal unfolding of proteins is compared to folding and mechanical stretching in a simple topology-based dynamical model. We define the unfolding time and demonstrate its low-temperature divergence. Below a characteristic temperature, contacts break at separate time scales and unfolding proceeds approximately in a way reverse to folding. Features in these scenarios agree with experiments and atomic simulations on titin., 3 figures, a text in latex

Combinatorial protein libraries provide a promising route to investigate the determinants and features of protein folding and to identify novel folding amino acid sequences. A library of sequences based on a pool of different monomer types are screened for folding molecules, consistent with a particular foldability criterion. The number of sequences grows exponentially with the length of the polymer, making both experimental and computational tabulations of sequences infeasible. Herein a statistical theory is extended to specify the properties of sequences having particular values of global energetic quantities that specify their energy landscape. The theory yields the site-specific monomer probabilities. A foldability criterion is derived that characterizes the properties of sequences by quantifying the energetic separation of the target state from low-energy states in the unfolded ensemble and the fluctuations of the energies in the unfolded state ensemble. For a simple lattice model of proteins, excellent agreement is observed between the theory and the results of exact enumeration. The theory may be used to provide a quantitative framework for the design and interpretation of combinatorial experiments.

We apply several methods to probe the ensemble kinetic and structural properties of a model system of poly-phenylacetylene (pPA) oligomer folding trajectories. The kinetic methods employed included a brute force accounting of conformations, a Markovian state matrix method, and a nonlinear least squares fit to a minimalist kinetic model used to extract the folding time. Each method gave similar measures for the folding time of the 12-mer chain, calculated to be on the order of 7 ns for the complete folding of the chain from an extended conformation. Utilizing both a linear and a nonlinear scaling relationship between the viscosity and the folding time to correct for a low simulation viscosity, we obtain an upper and a lower bound for the approximate folding time within the range 70 ns

The theoretical concept of folding probability, p(fold), has proven to be a useful means to characterize the kinetics of protein folding. Here, we illustrate the practical importance of p(fold) and demonstrate how it can be determined theoretically. We derive a general analytical expression for p(fold) and show how it can be estimated from simulations for systems where the transition rates between the relevant microstates are not known. By analyzing the Ising model we are able to determine the scaling behavior of the numerical error in the p(fold) estimate as function of the number of analyzed Monte Carlo runs. We apply our method to a simple, newly developed protein folding model for the formation of alpha helices. It is demonstrated that our technique highly parallelizes the calculation of p(fold) and that it is orders of magnitude more efficient than conventional approaches.

The discrete path sampling technique is used to calculate folding pathways of the 16-amino acid beta hairpin-forming sequence from residues 41-56 of the B1 domain of protein G. The folding time is obtained using master equation dynamics and kinetic Monte Carlo simulations, and the time evolution of different order parameters and occupation probabilities of groups of minima are calculated and used to characterize intermediates on the folding pathway.

We propose an efficient method for the prediction of protein folding rate constants and mechanisms. We use molecular dynamics simulation data to build Markovian state models (MSMs), discrete representations of the pathways sampled. Using these MSMs, we can quickly calculate the folding probability (P(fold)) and mean first passage time of all the sampled points. In addition, we provide techniques for evaluating these values under perturbed conditions without expensive recomputations. To demonstrate this method on a challenging system, we apply these techniques to a two-dimensional model energy landscape and the folding of a tryptophan zipper beta hairpin.

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 {\mu}s 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., Comment: 7 pages, 7 figures

We study two-state protein folding in the framework of a toy model of protein dynamics. This model has an important advantage: it allows for an analytical solution for the sum of folding and unfolding rate constants [A. M. Berezhkovskii, F. Tofoleanu, and N.-V. Buchete, J. Chem. Theory Comput. 7, 2370 (2011)10.1021/ct200281d] and hence for the reactive flux at equilibrium. We use the model to test the Kramers-type formula for the reactive flux, which was derived assuming that the protein dynamics is described by a Markov random walk on a network of complex connectivity [A. Berezhkovskii, G. Hummer, and A. Szabo, J. Chem. Phys. 130, 205102 (2009)10.1063/1.3139063]. It is shown that the Kramers-type formula leads to the same result for the reactive flux as the sum of the rate constants. AMS

Temperature-based replica exchange (T-ReX) enhances sampling of molecular dynamics simulations by autonomously heating and cooling simulation clients via a Metropolis exchange criterion. A pathological case for T-ReX can occur when a change in state (e.g., folding to unfolding of a protein) has a large energetic difference over a short temperature interval leading to insufficient exchanges amongst replica clients near the transition temperature. One solution is to allow the temperature set to dynamically adapt in the temperature space, thereby enriching the population of clients near the transition temperature. In this work, we evaluated two approaches for adapting the temperature set: a method that equalizes exchange rates over all neighbor temperature pairs and a method that attempts to induce clients to visit all temperatures (dubbed "current maximization") by positioning many clients at or near the transition temperature. As a test case, we simulated the 57-residue SH3 domain of alpha-spectrin. Exchange rate equalization yielded the same unfolding-folding transition temperature as fixed-temperature ReX with much smoother convergence of this value. Surprisingly, the current maximization method yielded a significantly lower transition temperature, in close agreement with experimental observation, likely due to more extensive sampling of the transition state.

The study of the mechanism which is at the basis of the phenomenon of protein folding requires the knowledge of multiple folding trajectories under biological conditions. Using a biasing molecular-dynamics algorithm based on the physics of the ratchet-and-pawl system, we carry out all-atom, explicit solvent simulations of the sequence of folding events which proteins G, CI2, and ACBP undergo in evolving from the denatured to the folded state. Starting from highly disordered conformations, the algorithm allows the proteins to reach, at the price of a modest computational effort, nativelike conformations, within a root mean square deviation (RMSD) of approximately 1 Å. A scheme is developed to extract, from the myriad of events, information concerning the sequence of native contact formation and of their eventual correlation. Such an analysis indicates that all the studied proteins fold hierarchically, through pathways which, although not deterministic, are well-defined with respect to the order of contact formation. The algorithm also allows one to study unfolding, a process which looks, to a large extent, like the reverse of the major folding pathway. This is also true in situations in which many pathways contribute to the folding process, like in the case of protein G.

We introduce a reweighting scheme for the path ensembles in the transition interface sampling framework. The reweighting allows for the analysis of free energy landscapes and committor projections in any collective variable space. We illustrate the reweighting scheme on a two dimensional potential with a nonlinear reaction coordinate and on a more realistic simulation of the Trp-cage folding process. We suggest that the reweighted path ensemble can be used to optimize possible nonlinear reaction coordinates.

We derive simple analytical expressions for the error and computational efficiency of replica exchange molecular dynamics (REMD) simulations (and by analogy replica exchange Monte Carlo simulations). The theory applies to the important case of systems whose dynamics at long times is dominated by the slow interconversion between two metastable states. As a specific example, we consider the folding and unfolding of a protein. The efficiency is defined as the rate with which the error in an estimated equilibrium property, as measured by the variance of the estimator over repeated simulations, decreases with simulation time. For two-state systems, this rate is in general independent of the particular property. Our main result is that, with comparable computational resources used, the relative efficiency of REMD and molecular dynamics (MD) simulations is given by the ratio of the number of transitions between the two states averaged over all replicas at the different temperatures, and the number of transitions at the single temperature of the MD run. This formula applies if replica exchange is frequent, as compared to the transition times. High efficiency of REMD is thus achieved by including replica temperatures in which the frequency of transitions is higher than that at the temperature of interest. In tests of the expressions for the error in the estimator, computational efficiency, and the rate of equilibration we find quantitative agreement with the results both from kinetic models of REMD and from actual all-atom simulations of the folding of a peptide in water.

An empirical sampling method for molecular simulation based on "temperature intervals with global exchange of replicas" (TIGER2) has been developed to reduce the high demand for computational resources and the low computational efficiency of the conventional replica-exchange molecular dynamics (REMD) method. This new method overcomes the limitation of its previous version, called TIGER, which requires the assumption of constant heat capacity during quenching of replicas from elevated temperatures to the baseline temperature. The robustness of the TIGER2 method is examined by comparing it against a Metropolis Monte Carlo simulation for sampling the conformational distribution of a single butane molecule in vacuum, a REMD simulation for sampling the behavior of alanine dipeptide in explicit solvent, and REMD simulations for sampling the folding behavior of two peptides, (AAQAA)(3) and chignolin, in implicit solvent. The agreement between the results from these conventional sampling methods and the TIGER2 simulations indicates that the TIGER2 algorithm is able to closely approximate a Boltzmann-weighted ensemble of states for these systems but without the limiting assumptions that were required for the original TIGER algorithm. TIGER2 is an efficient replica-exchange sampling method that enables the number of replicas that are used for a replica-exchange simulation to be substantially reduced compared to the conventional REMD method.

Theory of multi-dimensional representation of free energy surface of protein folding is developed by adopting structural order parameters of multiple regions in protein as multiple coordinates. Various scenarios of folding are classified in terms of cooperativity within individual regions and interactions among multiple regions and thus obtained classification is used to analyze the folding process of several example proteins. Ribosomal protein S6, src-SH3 domain, CheY, barnase, and BBL domain are analyzed with the two-dimensional representation by using a structure-based Hamiltonian model. Extension to the higher dimensional representation leads to the finer description of the folding process. Barnase, NtrC, and an ankyrin repeat protein are examined with the three-dimensional representation. The multi-dimensional representation allows us to directly address questions on folding pathways, intermediates, and transition states., Comment: to appear in J. Chem. Phys

While a variety of double-ended transition state search methods have been developed, their relative performance in characterizing complex multistep pathways between structurally disparate molecular conformations remains unclear. Three such methods (doubly-nudged elastic band, a string method, and a growing string method) are compared for a series of benchmarks ranging from permutational isomerizations of the seven-atom Lennard-Jones cluster (LJ(7)) to highly cooperative LJ(38) and LJ(75) rearrangements, and the folding pathways of two peptides. A database of short paths between LJ(13) local minima is used to explore the effects of parameters and suggest reasonable default values. Each double-ended method was employed within the framework of a missing connection network flow algorithm to construct more complicated multistep pathways. We find that in our implementation none of the three methods definitively outperforms the others, and that their relative effectiveness is strongly system and parameter dependent.

The Crooks equation [Eq. (10) in J. Stat. Phys. 90, 1481 (1998)], originally derived for microscopically reversible Markovian systems, relates the work done on a system during an irreversible transformation to the free energy difference between the final and the initial state of the transformation. In the present work we provide a theoretical proof of the Crooks equation in the context of constant volume, constant temperature steered molecular dynamics simulations of systems thermostated by means of the Nosé-Hoover method (and its variant using a chain of thermostats). As a numerical test we use the folding and unfolding processes of decaalanine in vacuo at finite temperature. We show that the distribution of the irreversible work for the folding process is markedly non-Gaussian thereby implying, according to Crooks equation, that also the work distribution of the unfolding process must be inherently non-Gaussian. The clearly asymmetric behavior of the forward and backward irreversible work distributions is a signature of a non-Markovian regime for the folding/unfolding of decaalanine.

Many different protein folding potentials have been developed in the last decades, based upon knowledge of experimentally determined protein structures. Decoy-based techniques are frequently used to assess these force fields, but other methods can explore different features in the performance of the interaction schemes, thus helping in their evaluation. Here, we propose an evolutionary strategy to efficiently assess folding potentials. We apply it to three potentials with different characteristics, taken from the bibliography. A search for minimum energy protein topologies, treated as arrangements of rigid protein fragments, is performed. The method, applied to a set of helix bundle proteins, shows the different behavior of the studied potentials, providing a reasonably fast tool to evaluate their advantages and limitations.

We report on the use of large-scale distributed computing simulation and novel analysis techniques for examining the dynamics of a small protein. Matters addressed include folding rate, very long time scale kinetics, ensemble properties, and interaction with water. The target system for the study, the villin headpiece, has been of great interest to experimentalists and theorists both. Sampling totaled nearly 500 mus-the most extensive published to date for a system of villin's size in explicit solvent with all atom detail-and was in the form of tens of thousands of independent molecular dynamics trajectories, each several tens of nanoseconds in length. We report on kinetics sensitivity analyses that, using a set of short simulations, probed the role of water in villin's folding and sensitivity to the simulation's electrostatics treatment. By constructing Markovian state models (MSMs) from the collected data, we were able to propagate dynamics to times far beyond those directly simulated and to rapidly compute mean first passage times, long time kinetics (tens of microseconds), and evolution of ensemble property distributions over long times, otherwise currently impossible. We also tested our MSM by using it to predict the structure of villin de novo.