Reaction coordinates and rates from transition paths

Can Local Probes Go Global? A Joint Experiment-Simulation Analysis of λ(6-85) Folding

The process of protein folding is known to involve global motions in a cooperative affair; the structure of most of the protein sequences is gained or lost over a narrow range of temperature, denaturant, or pressure perturbations. At the same time, recent simulations and experiments reveal a complex structural landscape with a rich set of local motions and conformational changes. We couple experimental kinetic and thermodynamic measurements with specifically tailored analysis of simulation data to isolate local versus global folding probes. We find that local probes exhibit lower melting temperatures, smaller surface area changes, and faster kinetics compared to global ones. We also see that certain local probes of folding match the global behavior more closely than others. Our work highlights the importance of using multiple probes to fully characterize protein folding dynamics by theory and experiment.

A Simple and Accurate Method To Calculate Free Energy Profiles and Reaction Rates from Restrained Molecular Simulations of Diffusive Processes

A method is developed to obtain simultaneously free energy profiles and diffusion constants from restrained molecular simulations in diffusive systems. The method is based on low-order expansions of the free energy and diffusivity as functions of the reaction coordinate. These expansions lead to simple analytical relationships between simulation statistics and model parameters. The method is tested on 1D and 2D model systems; its accuracy is found to be comparable to or better than that of the existing alternatives, which are briefly discussed. An important aspect of the method is that the free energy is constructed by integrating its derivatives, which can be computed without need for overlapping sampling windows. The implementation of the method in any molecular simulation program that supports external umbrella potentials (e.g., CHARMM) requires modification of only a few lines of code. As a demonstration of its applicability to realistic biomolecular systems, the method is applied to model the α-helix ↔ β-sheet transition in a 16-residue peptide in implicit solvent, with the reaction coordinate provided by the string method. Possible modifications of the method are briefly discussed; they include generalization to multidimensional reaction coordinates [in the spirit of the model of Ermak and McCammon (Ermak, D. L.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352-1360)], a higher-order expansion of the free energy surface, applicability in nonequilibrium systems, and a simple test for Markovianity. In view of the small overhead of the method relative to standard umbrella sampling, we suggest its routine application in the cases where umbrella potential simulations are appropriate.

Capturing transition paths and transition states for conformational rearrangements in the ribosome

To reveal the molecular determinants of biological function, one seeks to characterize the interactions that are formed in conformational and chemical transition states. In other words, what interactions govern the molecule's energy landscape? To accomplish this, it is necessary to determine which degrees of freedom can unambiguously identify each transition state. Here, we perform simulations of large-scale aminoacyl-transfer RNA (aa-tRNA) rearrangements during accommodation on the ribosome and project the dynamics along experimentally accessible atomic distances. From this analysis, we obtain evidence for which coordinates capture the correct number of barrier-crossing events and accurately indicate when the aa-tRNA is on a transition path. Although a commonly used coordinate in single-molecule experiments performs poorly, this study implicates alternative coordinates along which rearrangements are accurately described as diffusive movements across a one-dimensional free-energy profile. From this, we provide the theoretical foundation required for single-molecule techniques to uncover the energy landscape governing aa-tRNA selection by the ribosome.

Dimensionality of Collective Variables for Describing Conformational Changes of a Multi-Domain Protein

Collective variables (CVs) are often used in molecular dynamics simulations based on enhanced sampling algorithms to investigate large conformational changes of a protein. The choice of CVs in these simulations is essential because it affects simulation results and impacts the free-energy profile, the minimum free-energy pathway (MFEP), and the transition-state structure. Here we examine how many CVs are required to capture the correct transition-state structure during the open-to-close motion of adenylate kinase using a coarse-grained model in the mean forces string method to search the MFEP. Various numbers of large amplitude principal components are tested as CVs in the simulations. The incorporation of local coordinates into CVs, which is possible in higher dimensional CV spaces, is important for capturing a reliable MFEP. The Bayesian measure proposed by Best and Hummer is sensitive to the choice of CVs, showing sharp peaks when the transition-state structure is captured. We thus evaluate the required number of CVs needed in enhanced sampling simulations for describing protein conformational changes.

Microscopic interpretation of folding ϕ-values using the transition path ensemble

All-atom molecular dynamics simulations now allow us to create movies of proteins folding and unfolding. However, it is difficult to assess the accuracy of the folding mechanisms observed because experiments cannot yet directly resolve events occurring along the transition paths between unfolded and folded states. Protein folding ϕ-values provide residue-resolved information about folding mechanisms by comparing the effects of mutations on folding rates and stability, but determining ϕ-values by separately simulating mutant proteins would be computationally demanding and prone to large statistical errors. Here we use transition path theory to develop a method for computing ϕ-values directly from the transition path ensemble, without the need for additional simulations. This path-based approach uses the full transition path information available from equilibrium folding and unfolding trajectories, or from transition path sampling, and does not require identification of folding transition states. Applying our approach to a set of simulations of 10 small proteins by Shaw and coworkers [Lindorff-Larsen K, Piana S, Dror RO, Shaw DE (2011) Science 334(6055):517-520; Piana S, Lindorff-Larsen K, Shaw DE (2011) Biophys J100(9):L47-L49; and Piana S, Lindorff-Larsen K, Shaw DE (2013) Proc Natl Acad Sci USA 110(15):5915-5920], we find good agreement with experiments in most cases where data are available. We can further resolve the contributions to fractional ϕ-values coming from partial contact formation versus transition path heterogeneity. Although in some cases, there is substantial heterogeneity of folding mechanism, in others, such as Ubiquitin, the mechanism is strongly conserved.

Spectral gap optimization of order parameters for sampling complex molecular systems

In modern-day simulations of many-body systems, much of the computational complexity is shifted to the identification of slowly changing molecular order parameters called collective variables (CVs) or reaction coordinates. A vast array of enhanced-sampling methods are based on the identification and biasing of these low-dimensional order parameters, whose fluctuations are important in driving rare events of interest. Here, we describe a new algorithm for finding optimal low-dimensional CVs for use in enhanced-sampling biasing methods like umbrella sampling, metadynamics, and related methods, when limited prior static and dynamic information is known about the system, and a much larger set of candidate CVs is specified. The algorithm involves estimating the best combination of these candidate CVs, as quantified by a maximum path entropy estimate of the spectral gap for dynamics viewed as a function of that CV. The algorithm is called spectral gap optimization of order parameters (SGOOP). Through multiple practical examples, we show how this postprocessing procedure can lead to optimization of CV and several orders of magnitude improvement in the convergence of the free energy calculated through metadynamics, essentially giving the ability to extract useful information even from unsuccessful metadynamics runs.

Exploring the mechanochemical cycle of dynein motor proteins: structural evidence of crucial intermediates

Exploration of the biologically relevant pathways of dynein's mechanochemical cycle using structure based models.

S-shooting: a Bennett–Chandler-like method for the computation of rate constants from committor trajectories

Mechanisms of rare transitions between long-lived stable states are often analyzed in terms of commitment probabilities, determined from swarms of short molecular dynamics trajectories. Here, we present a computer simulation method to determine rate constants from such short trajectories combined with free energy calculations. The method, akin to the Bennett–Chandler approach for the calculation of reaction rate constants, requires the definition of a valid reaction coordinate and can be applied to both under- and overdamped dynamics. We verify the correctness of the algorithm using a one-dimensional random walker in a double-well potential and demonstrate its applicability to complex transitions in condensed systems by calculating cavitation rates for water at negative pressures.

High-resolution free energy landscape analysis of protein folding

The free energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. The profile together with the optimal coordinate allows one to directly determine such basic properties of folding dynamics as the configurations of the minima and transition states, the heights of the barriers, the value of the pre-exponential factor and its relation to the transition path times. In the present study, we review the framework, in particular, the approach to determine such an optimal coordinate, and its application to the analysis of simulated protein folding dynamics.

Binding of Disordered Peptides to Kelch: Insights from Enhanced Sampling Simulations

Keap1 protein plays an essential role in regulating cellular oxidative stress response and is a crucial binding hub for multiple proteins, several of which are intrinsically disordered proteins (IDP). Among Kelch's IDP binding partners, NRF2 and PTMA are the two most interesting cases. They share a highly similar binding motif; however, NRF2 binds to Kelch with a binding affinity of approximately 100-fold higher than that of PTMA. In this study, we perform an exhaustive sampling composed of 6 μs well-tempered metadynamics and 2 μs unbiased molecular dynamics (MD) simulations aiming at characterizing the binding mechanisms and structural properties of these two peptides. Our results agree with previous experimental observations that PTMA is remarkably more disordered than NRF2 in both the free and bound states. This explains PTMA's lower binding affinity. Our extensive sampling also provides valuable insights into the vast conformational ensembles of both NRF2 and PTMA, supports the hypothesis of coupled folding-binding, and confirms the essential role of linear motifs in IDP binding.

Accelerating the Conformational Sampling of Intrinsically Disordered Proteins

Intrinsically disordered proteins (IDPs) are a class of proteins lacking a well-defined secondary structure. Instead, they are able to attain multiple conformations, bind to multiple targets, and respond to changes in their surroundings. Functionally, IDPs have been associated with molecular recognition, cell regulation, and signal transduction. The dynamic conformational ensemble of IDPs is highly environmental and binding partner dependent, rendering the characterization of IDPs extremely challenging. Here, we compare the sampling efficiencies of conventional molecular dynamics (MD), well-tempered metadynamics (WT-META), and bias-exchange metadynamics (BE-META). The total simulation time was over 10 μs, and a 20-mer peptide derived from the Neh2 domain of the Nuclear factor erythroid 2-related factor 2 (Nrf2) protein was simulated. BE-META, with a neutral replica and seven biased replicas employing a set of seven relevant collective variables (CVs), provided the most reliable and efficient sampling. Finally, we propose a free-energy reconstruction method based on the probability distribution of the secondary structure contents. This postprocessing analysis confirms the presence of not only the β-hairpin conformation of the free Neh2 peptide but also its rare bound-state-like conformation, both of that have been experimentally observed. In addition, our simulations also predict other possible conformations to be verified with future experiments.

Transition-State Theory, Dynamics, and Narrow Time Scale Separation in the Rate-Promoting Vibrations Model of Enzyme Catalysis

The power of transition-state theory (TST) for understanding enzymes is evidenced by its recent use in the design and synthesis of highly active de novo enzymes. However, dynamics can influence reaction kinetics, and some studies of rate-promoting vibrations even claim that dynamical theories instead of TST are needed to understand enzymatic reaction mechanisms. For the rate-promoting vibration (RPV) model of enzyme catalysis [Antoniou et al., J. Chem. Phys. 2004, 121, 6442], a reactive flux correlation function analysis shows that dynamical effects do slow the kinetics. However, the RPV model also shows extremely long-lived correlations because the RPV and the bath are not directly coupled. Additionally, earlier studies of the RPV model show a narrow time scale separation due to a small 5kT barrier. Thus earlier findings based on the RPV model may have little bearing on the properties of real enzymes. The intrinsic reaction coordinate (IRC) reveals that the RPV is an important component of the reaction coordinate at early and late stages of the pathway, but the RPV is not an important component of the reaction coordinate direction at the transition state. The unstable eigenmode from harmonic TST (which coincides with the IRC at the saddle point) gives a larger transmission coefficient than the coordinate used in the correlation functions of Antoniou et al. Thus while TST cannot predict the transmission coefficient, the RPV model suggests that TST can provide mechanistic insights on elementary steps in enzyme catalysis. Finally, we propose a method for using the transition-state ensemble as predicted from harmonic TST to distinguish promoting vibrations from other more mundane bath variables.

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

Theoretical models have often modeled protein folding dynamics as diffusion on a low-dimensional free energy surface, a remarkable simplification. However, the accuracy of such an approximation and the number of dimensions required were not clear. For all-atom folding simulations of ten small proteins in explicit solvent we show that the folding dynamics can indeed be accurately described as diffusion on just a single coordinate, the fraction of native contacts (Q). The diffusion models reproduce both folding rates, and finer details such as transition-path durations and diffusive propagators. The Q-averaged diffusion coefficients decrease with chain length, as anticipated from energy landscape theory. Although the Q-diffusion model does not capture transition-path durations for the protein NuG2, we show that this can be accomplished by designing an improved coordinate Qopt. Overall, one-dimensional diffusion on a suitable coordinate turns out to be a remarkably faithful model for the dynamics of the proteins considered.

Reducing the cost of evaluating the committor by a fitting procedure

Correct identification of reaction coordinates in complex systems is essential for understanding the mechanisms of their reaction dynamics. Existing methods for identifying reaction coordinates typically require knowledge of the committor--the probability of a given configuration to reach the product basin. The high computational cost of evaluating committors has limited applications of methods for identifying reaction coordinates. We proposed a fitting procedure that can reduce the cost of evaluating committors by an order of magnitude or more. The method only requires evaluating the committors of a few configurations in a transition path by the standard and costly shooting procedure. The committors of the other configurations are then estimated with great accuracy by a sigmoid function derived from fitting the few numerically evaluated committors. The method has been systematically tested on a model system of a Brownian particle moving in a one-dimensional double-well potential, and a small biomolecular system--the isomerization of alanine dipeptide in vacuum and in explicit water.

Diffusion maps, clustering and fuzzy Markov modeling in peptide folding transitions

Using the helix-coil transitions of alanine pentapeptide as an illustrative example, we demonstrate the use of diffusion maps in the analysis of molecular dynamics simulation trajectories. Diffusion maps and other nonlinear data-mining techniques provide powerful tools to visualize the distribution of structures in conformation space. The resulting low-dimensional representations help in partitioning conformation space, and in constructing Markov state models that capture the conformational dynamics. In an initial step, we use diffusion maps to reduce the dimensionality of the conformational dynamics of Ala5. The resulting pretreated data are then used in a clustering step. The identified clusters show excellent overlap with clusters obtained previously by using the backbone dihedral angles as input, with small--but nontrivial--differences reflecting torsional degrees of freedom ignored in the earlier approach. We then construct a Markov state model describing the conformational dynamics in terms of a discrete-time random walk between the clusters. We show that by combining fuzzy C-means clustering with a transition-based assignment of states, we can construct robust Markov state models. This state-assignment procedure suppresses short-time memory effects that result from the non-Markovianity of the dynamics projected onto the space of clusters. In a comparison with previous work, we demonstrate how manifold learning techniques may complement and enhance informed intuition commonly used to construct reduced descriptions of the dynamics in molecular conformation space.

Structural origin of slow diffusion in protein folding

Experimental, theoretical, and computational studies of small proteins suggest that interresidue contacts not present in the folded structure play little or no role in the self-assembly mechanism. Non-native contacts can, however, influence folding kinetics by introducing additional local minima that slow diffusion over the global free-energy barrier between folded and unfolded states. Here, we combine single-molecule fluorescence with all-atom molecular dynamics simulations to discover the structural origin for the slow diffusion that markedly decreases the folding rate for a designed α-helical protein. Our experimental determination of transition path times and our analysis of the simulations point to non-native salt bridges between helices as the source, which provides a quantitative glimpse of how specific intramolecular interactions influence protein folding rates by altering dynamics and not activation free energies.

Investigating rare events with nonequilibrium work measurements. II. Transition and reaction rates

We present a formalism for investigating transition pathways and transition probabilities for rare events in biomolecular systems. The formalism is based on combining Transition Path Theory with the results of nonequilibrium work relations, and shows that the equilibrium and nonequilibrium transition rates are in fact related. Aside from its fundamental importance, this allows for the calculation of relative equilibrium reaction rates with driven nonequilibrium simulations such as Steered Molecular Dynamics. The workings of the formalism are illustrated with a few typical numerical examples.

Hairpins participating in folding of human telomeric sequence quadruplexes studied by standard and T-REMD simulations

DNA G-hairpins are potential key structures participating in folding of human telomeric guanine quadruplexes (GQ). We examined their properties by standard MD simulations starting from the folded state and long T-REMD starting from the unfolded state, accumulating ∼130 μs of atomistic simulations. Antiparallel G-hairpins should spontaneously form in all stages of the folding to support lateral and diagonal loops, with sub-μs scale rearrangements between them. We found no clear predisposition for direct folding into specific GQ topologies with specific syn/anti patterns. Our key prediction stemming from the T-REMD is that an ideal unfolded ensemble of the full GQ sequence populates all 4096 syn/anti combinations of its four G-stretches. The simulations can propose idealized folding pathways but we explain that such few-state pathways may be misleading. In the context of the available experimental data, the simulations strongly suggest that the GQ folding could be best understood by the kinetic partitioning mechanism with a set of deep competing minima on the folding landscape, with only a small fraction of molecules directly folding to the native fold. The landscape should further include non-specific collapse processes where the molecules move via diffusion and consecutive random rare transitions, which could, e.g. structure the propeller loops.

Role of water and steric constraints in the kinetics of cavity-ligand unbinding

A key factor influencing a drug's efficacy is its residence time in the binding pocket of the host protein. Using atomistic computer simulation to predict this residence time and the associated dissociation process is a desirable but extremely difficult task due to the long timescales involved. This gets further complicated by the presence of biophysical factors such as steric and solvation effects. In this work, we perform molecular dynamics (MD) simulations of the unbinding of a popular prototypical hydrophobic cavity-ligand system using a metadynamics-based approach that allows direct assessment of kinetic pathways and parameters. When constrained to move in an axial manner, the unbinding time is found to be on the order of 4,000 s. In accordance with previous studies, we find that the cavity must pass through a region of sharp wetting transition manifested by sudden and high fluctuations in solvent density. When we remove the steric constraints on ligand, the unbinding happens predominantly by an alternate pathway, where the unbinding becomes 20 times faster, and the sharp wetting transition instead becomes continuous. We validate the unbinding timescales from metadynamics through a Poisson analysis, and by comparison through detailed balance to binding timescale estimates from unbiased MD. This work demonstrates that enhanced sampling can be used to perform explicit solvent MD studies at timescales previously unattainable, to our knowledge, obtaining direct and reliable pictures of the underlying physiochemical factors including free energies and rate constants.

Prediction of the optimal set of contacts to fold the smallest knotted protein

Knotted protein chains represent a new motif in protein folds. They have been linked to various diseases, and recent extensive analysis of the Protein Data Bank shows that they constitute 1.5% of all deposited protein structures. Despite thorough theoretical and experimental investigations, the role of knots in proteins still remains elusive. Nonetheless, it is believed that knots play an important role in mechanical and thermal stability of proteins. Here, we perform a comprehensive analysis of native, shadow-specific and non-native interactions which describe free energy landscape of the smallest knotted protein (PDB id 2efv). We show that the addition of shadow-specific contacts in the loop region greatly enhances folding kinetics, while the addition of shadow-specific contacts along the C-terminal region (H3 or H4) results in a new folding route with slower kinetics. By means of direct coupling analysis (DCA) we predict non-native contacts which also can accelerate kinetics. Next, we show that the length of the C-terminal knot tail is responsible for the shape of the free energy barrier, while the influence of the elongation of the N-terminus is not significant. Finally, we develop a concept of a minimal contact map sufficient for 2efv protein to fold and analyze properties of this protein using this map.

Extended molecular dynamics of a c-kit promoter quadruplex

The 22-mer c-kit promoter sequence folds into a parallel-stranded quadruplex with a unique structure, which has been elucidated by crystallographic and NMR methods and shows a high degree of structural conservation. We have carried out a series of extended (up to 10 μs long, ∼50 μs in total) molecular dynamics simulations to explore conformational stability and loop dynamics of this quadruplex. Unfolding no-salt simulations are consistent with a multi-pathway model of quadruplex folding and identify the single-nucleotide propeller loops as the most fragile part of the quadruplex. Thus, formation of propeller loops represents a peculiar atomistic aspect of quadruplex folding. Unbiased simulations reveal μs-scale transitions in the loops, which emphasizes the need for extended simulations in studies of quadruplex loops. We identify ion binding in the loops which may contribute to quadruplex stability. The long lateral-propeller loop is internally very stable but extensively fluctuates as a rigid entity. It creates a size-adaptable cleft between the loop and the stem, which can facilitate ligand binding. The stability gain by forming the internal network of GA base pairs and stacks of this loop may be dictating which of the many possible quadruplex topologies is observed in the ground state by this promoter quadruplex.

Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories

Structural self-assembly in biopolymers, such as proteins and nucleic acids, involves a diffusive search for the minimum-energy state in a conformational free-energy landscape. The likelihood of folding proceeding to completion, as a function of the reaction coordinate used to monitor the transition, can be described by the splitting probability, p(fold)(x). P(fold) encodes information about the underlying energy landscape, and it is often used to judge the quality of the reaction coordinate. Here, we show how p(fold) can be used to reconstruct energy landscapes from single-molecule folding trajectories, using force spectroscopy measurements of single DNA hairpins. Calculating p(fold)(x) directly from trajectories of the molecular extension measured for hairpins fluctuating in equilibrium between folded and unfolded states, we inverted the result expected from diffusion over a 1D energy landscape to obtain the implied landscape profile. The results agreed well with the landscapes reconstructed by established methods, but, remarkably, without the need to deconvolve instrumental effects on the landscape, such as tether compliance. The same approach was also applied to hairpins with multistate folding pathways. The relative insensitivity of the method to the instrumental compliance was confirmed by simulations of folding measured with different tether stiffnesses. This work confirms that the molecular extension is a good reaction coordinate for these measurements, and validates a powerful yet simple method for reconstructing landscapes from single-molecule trajectories.

Exploring the balance between folding and functional dynamics in proteins and RNA

As our understanding of biological dynamics continues to be refined, it is becoming clear that biomolecules can undergo transitions between ordered and disordered states as they execute functional processes. From a computational perspective, studying disorder events poses a challenge, as they typically occur on long timescales, and the associated molecules are often large (i.e., hundreds of residues). These size and time requirements make it advantageous to use computationally inexpensive models to characterize large-scale dynamics, where more highly detailed models can provide information about individual sub-steps associated with function. To reduce computational demand, one often uses a coarse-grained representation of the molecule or a simplified description of the energetics. In order to use simpler models to identify transient disorder in RNA and proteins, it is imperative that these models can accurately capture structural fluctuations about folded configurations, as well as the overall stability of each molecule. Here, we explore a class of simplified model for which all non-hydrogen atoms are explicitly represented. We find that this model can provide a consistent description of protein folding and native-basin dynamics for several representative biomolecules. We additionally show that the native-basin fluctuations of tRNA and the ribosome are robust to variations in the model. Finally, the extended variable loop in tRNAIle is predicted to be very dynamic, which may facilitate biologically-relevant rearrangements. Together, this study provides a foundation that will aid in the application of simplified models to study disorder during function in ribonucleoprotein (RNP) assemblies.

Transition-Path Probability as a Test of Reaction-Coordinate Quality Reveals DNA Hairpin Folding Is a One-Dimensional Diffusive Process

Chemical reactions are typically described in terms of progress along a reaction coordinate. However, the quality of reaction coordinates for describing reaction dynamics is seldom tested experimentally. We applied a framework for gauging reaction-coordinate quality based on transition-path analysis to experimental data for the first time, looking at folding trajectories of single DNA hairpin molecules measured under tension applied by optical tweezers. The conditional probability for being on a reactive transition path was compared with the probability expected for ideal diffusion over a 1D energy landscape based on the committor function. Analyzing measurements and simulations of hairpin folding where end-to-end extension is the reaction coordinate, after accounting for instrumental effects on the analysis, we found good agreement between transition-path and committor analyses for model two-state hairpins, demonstrating that folding is well-described by 1D diffusion. This work establishes transition-path analysis as a powerful new tool for testing experimental reaction-coordinate quality.

What protein folding teaches us about biological function and molecular machines

Protein folding was the first area of molecular biology for which a systematic statistical-mechanical analysis of dynamics was developed. As a result, folding is described as a process by which a disordered protein chain diffuses across a high-dimensional energy landscape and finally reaches the folded ensemble. Folding studies have produced countless theoretical concepts that are generalizable to other biomolecular processes, such as the functional dynamics of molecular assemblies. Common themes in folding and function include the dominant role of excluded volume, that a balance between energetic roughness and geometrical effects guides dynamics, and that folding/functional landscapes are relatively smooth. Here, we discuss how insights into protein folding have been applied to investigate the functional dynamics of biomolecular assemblies.

Simulations and experiments in protein folding

The interplay between simulations and experiments of protein folding has largely contributed to the elucidation of many important aspects of the phenomenon. In this chapter, I briefly describe the experiments which provide information on the kinetics of the protein folding process, and help to characterize the folding transition state. Then, I show how to probe the kinetics of protein folding using molecular dynamics simulations, how to compare the simulations with the experiments and how to help and rationalize the latter, ultimately offering a molecular picture of the process. After the production of suitable molecular dynamics simulation data in the form of trajectories, the procedure involves sequentially the identification of the stable states of the protein, the identification of the transition pathways connecting the stable states, the identification of the transition state conformations, comparison with experimental results, and finally, the identification of the molecular determinants or reaction coordinates of the folding process, that is, the features that clearly help distinguishing the transition state from the stable states.

Multiple-Replica Strategies for Free-Energy Calculations in NAMD: Multiple-Walker Adaptive Biasing Force and Walker Selection Rules

From the most powerful supercomputers to multicore desktops and laptops, parallel computing architectures have been in the mainstream for some time. However, numerical schemes for calculating free energies in molecular systems that directly leverage this hardware paradigm, usually taking the form of multiple-replica strategies, are just now on the cusp of becoming standard practice. Here, we present a modification of the popular molecular dynamics program NAMD that is envisioned to facilitate the use of powerful multiple-replica strategies to improve ergodic sampling for a specific class of free-energy methods known as adaptive biasing force. We describe the software implementation in a so-called multiple-walker context, alongside the interface that makes the proposed approach accessible to the end users. We further evaluate the performance of the adaptive biasing force multiple-walker strategy for a model system, namely, the reversible folding of a short peptide, and show, in particular, in regions of the transition coordinate where convergence of the free-energy calculation is encumbered by hidden barriers, that the multiple-walker strategy can yield far more reliable results in appreciably less real time on parallel architectures, relative to standard, single-replica calculations.

Coarse-Grained Langevin Equation for Protein Dynamics: Global Anisotropy and a Mode Approach to Local Complexity

We utilize a multiscale approach where molecular dynamic simulations are performed to obtain quantitative structural averages used as input to a coarse-grained Langevin equation for protein dynamics, which can be solved analytically. The approach describes proteins as fundamentally semiflexible objects collapsed into the free energy well representing the folded state. The normal-mode analytical solution to this Langevin equation naturally separates into global modes describing the fully anisotropic tumbling of the macromolecule as a whole and internal modes which describe local fluctuations about the folded structure. Complexity in the configurational free-energy landscape of the macromolecule leads to a renormalization of the internal modes, while the global modes provide a basis set in which the dipolar orientation and global anisotropy can be accounted for when comparing to experiments. This simple approach predicts the dynamics of both global rotational diffusion and internal motion from the picosecond to the nanosecond regime and is quantitative when compared to time correlation functions calculated from molecular dynamic simulations and in good agreement with nuclear magnetic resonance relaxation experiments. Fundamental to this approach is the inclusion of internal dissipation, which is absent in any rigid-body hydrodynamical modeling scheme.

Automatic Selection of Order Parameters in the Analysis of Large Scale Molecular Dynamics Simulations

Given the large number of crystal structures and NMR ensembles that have been solved to date, classical molecular dynamics (MD) simulations have become powerful tools in the atomistic study of the kinetics and thermodynamics of biomolecular systems on ever increasing time scales. By virtue of the high-dimensional conformational state space that is explored, the interpretation of large-scale simulations faces difficulties not unlike those in the big data community. We address this challenge by introducing a method called clustering based feature selection (CB-FS) that employs a posterior analysis approach. It combines supervised machine learning (SML) and feature selection with Markov state models to automatically identify the relevant degrees of freedom that separate conformational states. We highlight the utility of the method in the evaluation of large-scale simulations and show that it can be used for the rapid and automated identification of relevant order parameters involved in the functional transitions of two exemplary cell-signaling proteins central to human disease states.

The adaptive biasing force method: everything you always wanted to know but were afraid to ask

In the host of numerical schemes devised to calculate free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an average force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy estimates. Much of the appeal of the algorithm to the practitioner is in its physically intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth mathematical analysis of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chemistry and biology. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a critical eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calculations of prototypical molecular systems, many methodological aspects are examined, from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a number of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.

Effect of confinement on DNA, solvent and counterion dynamics in a model biological nanopore

The application of recent advances in nanopore technology to high-throughput DNA sequencing requires a more detailed understanding of solvent, ion and DNA interactions occurring within these pores. Here we present a combination of atomistic and coarse-grained modeling studies of the dynamics of short single-stranded DNA (ssDNA) homopolymers within the alpha-hemolysin pore, for the two single-stranded homopolymers poly(dA)40 and poly(dC)40. Analysis of atomistic simulations along with the per-residue decomposition of protein-DNA interactions in these simulations gives new insight into the very complex issues that have yet to be fully addressed with detailed MD simulations. We discuss a modification of the solvent properties and ion distribution around DNA within nanopore confinement and put it into the general framework of counterion condensation theory. There is a reasonable agreement in computed properties from our all-atom simulations and the resulting predictions from analytical theories with experimental data, and our equilibrium results here support the conclusions from our previous non-equilibrium Brownian dynamics studies with a recently developed BROMOC protocol that cations are the primary charge carriers through alpha-hemolysin nanopores under an applied voltage in the presence of ssDNA. Clustering analysis led to an identification of distinct conformational states of captured polymer and depth of the current blockade. Therefore, our data suggest that confined polymer may act as a flickering gate, thus contributing to excess noise phenomena. We also discuss the extent of water structuring due to nanopore confinement and the relationship between the conformational dynamics of a captured polymer and the distribution of blocked current.

Testing for memory-free spectroscopic coordinates by 3D IR exchange spectroscopy

Using 3D infrared (IR) exchange spectroscopy, the ultrafast hydrogen-bond forming and breaking (i.e., complexation) kinetics of phenol to benzene in a benzene/CCl4 mixture is investigated. By introducing a third time point at which the hydrogen-bonding state of phenol is measured (in comparison with 2D IR exchange spectroscopy), the spectroscopic method can serve as a critical test of whether the spectroscopic coordinate used to observe the exchange process is a memory-free, or Markovian, coordinate. For the system under investigation, the 3D IR results suggest that this is not the case. This conclusion is reconfirmed by accompanying molecular dynamics simulations, which furthermore reveal that the non-Markovian kinetics is caused by the heterogeneous structure of the mixed solvent.