Adaptively biased molecular dynamics for free energy calculations

Comparing Nonbonded Metal Ion Models in the Divalent Cation Binding Protein PsaA

Divalent metal cations are essential for many biological processes; however, accurately modeling divalent metal ions has proved a significant challenge for molecular dynamics force fields. Here we show that the choice of ion model influences the observed dynamics in PsaA, a metal binding protein from Streptococcus pneumoniae. We conduct extensive unbiased simulations and free energy calculations of PsaA bound to its cognate ligand Mn²⁺ and inhibitory ligand Zn²⁺ using three nonbonded ion models: a 12-6 model, a 12-6-4 model, and a multisite model. The observed coordination geometries and metal binding dynamics are sensitive to the choice of ion model, with the most dramatic differences observed in free energy calculations of ion release. We show that the conformational ensemble of Mn-bound PsaA is more similar to the crystallographic metal bound open state. This work extends the current model of PsaA metal binding and provides a framework for the rationalization of experimentally determined metal binding behavior. Our findings support the use of the 12-6-4 ion model for further simulations of divalent cation binding proteins.

FSATOOL: A useful tool to do the conformational sampling and trajectory analysis work for biomolecules

Reliable conformational sampling and trajectory analysis are always important to the study of the folding or binding mechanisms of biomolecules. Generally, one has to prepare many complicated parameters and follow a lot of steps to obtain the final data. The whole process is too complicated to new users. In this article, we provide a convenient and user-friendly tool that is compatible to AMBER, called fast sampling and analysis tool (FSATOOL). FSATOOL has some useful features. First and the most important, the whole work is extremely simplified into two steps, one is the fast sampling procedure and the other is the trajectory analysis procedure. Second, it contains several powerful sampling methods for the simulation on graphics process unit, including our previous mixing replica exchange molecular dynamics method. The method combines the advantages of the biased and unbiased simulations. Finally, it extracts the dominant transition pathways automatically from the folding network by Markov state model. Users do not need to do the tedious intermediate steps by hand. To illustrate the usage of FSATOOL in practice, we perform one simulation for a RNA hairpin in explicit solvent. All the results are presented. © 2019 Wiley Periodicals, Inc.

Calculating the absolute binding free energy of the insulin dimer in an explicit solvent

Insulin is a significant hormone in the regulation of glucose level in the blood. Its monomers bind to each other to form dimers or hexamers through a complex process. To study the binding of the insulin dimer, we first calculate its absolute binding free energy by the steered molecular dynamics method and the confinement method based on a fictitious thermodynamic cycle. After considering some special correction terms, the final calculated binding free energy at 298 K is −8.97 ± 1.41 kcal mol⁻¹, which is close to the experimental value of −7.2 ± 0.8 kcal mol⁻¹. Furthermore, we discuss the important residue–residue interactions between the insulin monomers, including hydrophobic interactions, π–π interactions and hydrogen bond interactions. The analysis reveals five key residues, Vla^B12, Tyr^B16, Phe^B24, Phe^B25, and Tyr^B26, for the dimerization of the insulin. We also perform MM-PBSA calculations for the wild-type dimer and some mutants and study the roles of the key residues by the change of the binding energy of the insulin dimer.

In this paper, we calculate the absolute binding free energy of an insulin dimer by steered MD method. The result of −8.97 kcal mol⁻¹ is close to the experimental value −7.2 kcal mol⁻¹. We also analyze the residue–residue interactions.

Free energy profile for unwrapping outer superhelical turn of CENP-A nucleosome

Eukaryotic genome is packaged in a nucleus in the form of chromatin. The fundamental structural unit of the chromatin is the protein-DNA complex, nucleosome, where DNA of about 150 bp is wrapped around a histone core almost twice. In cellular processes such as gene expression, DNA repair and duplication, the nucleosomal DNA has to be unwrapped. Histone proteins have their variants, indicating there are a variety of constitutions of nucleosomes. These different constitutions are observed in different cellular processes. To investigate differences among nucleosomes, we calculated free energy profiles for unwrapping the outer superhelical turn of CENP-A nucleosome and compared them with those of the canonical nucleosome. The free energy profiles for CENP-A nucleosome suggest that CENP-A nucleosome is the most stable when 16 to 22 bps are unwrapped in total whereas the canonical nucleosome is the most stable when it is fully wrapped. This indicates that the flexible conformation of CENP-A nucleosome is ready to provide binding sites for the structural integrity of the centromere.

Combining the biased and unbiased sampling strategy into one convenient free energy calculation method

Constructing a free energy landscape for a large molecule is difficult. One has to use either a high temperature or a strong driving force to enhance the sampling on the free energy barriers. In this work, we propose a mixed method that combines these two kinds of acceleration strategies into one simulation. First, it applies an adaptive biasing potential to some replicas of the molecule. These replicas are particularly accelerated in a collective variable space. Second, it places some unbiased and exchangeable replicas at various temperature levels. These replicas generate unbiased sampling data in the canonical ensemble. To improve the sampling efficiency, biased replicas transfer their state variables to the unbiased replicas after equilibrium by Monte Carlo trial moves. In comparison to previous integrated methods, it is more convenient for users. It does not need an initial reference biasing potential to guide the sampling of the molecule. And it is also unnecessary to insert many replicas for the requirement of passing the free energy barriers. The free energy calculation is accomplished in a single stage. It samples the data as fast as a biased simulation and it processes the data as simple as an unbiased simulation. The method provides a minimalist approach to the construction of the free energy landscape. © 2019 Wiley Periodicals, Inc.

Insights into the Stereospecificity in Papain-Mediated Chemoenzymatic Polymerization from Quantum Mechanics/Molecular Mechanics Simulations

Chemoenzymatic peptide synthesis is an efficient and clean method to generate polypeptides for new applications in the fields of biomedical and functional materials. However, this enzyme-mediated synthesis is dependent on the reaction rate of the protease biocatalyst, which is essentially determined by the natural substrate specificity of the enzyme. Papain, one of the most studied cysteine proteases, is extensively used for the chemoenzymatic synthesis of new polypeptides. Similar to most proteases, papain displays high stereospecificity toward l-amino acids, with limited reactivity for the d-stereoisomer counterparts. However, the incorporation of d-amino acids into peptides is a promising approach to increase their biostability by conferring intrinsic resistance to proteolysis. Herein, we determined the stereospecific-limiting step of the papain-mediated polymerization reaction with the chiral substrates l/d-alanine ethyl ester (Ala-OEt). Afterward, we used Quantum Mechanics/Molecular Mechanics (QM/MM) simulations to study the catalytic mechanism at atomic level of detail and investigate the origin of its stereospecificity. The experimental and computational results show that papain is able to attack both l- and d-stereoisomers of Ala-OEt, forming an enzyme-substrate intermediate, and that the two reactions display a similar activation barrier. Moreover, we found that the reduced catalytic activity of papain in the polymerization of d-amino acids arises from the aminolysis step of the reaction, in which l-Ala-OEt displays a significantly lower free-energy barrier (12 kcal/mol) than d-Ala-OEt (30 kcal/mol). Further simulations suggest that the main factor affecting the polymerization of d-amino acids is the configuration of the d-acyl-intermediate enzyme, and in particular the orientation of its methyl group, which hinders the nucleophilic attack by other monomers and thus the formation of polypeptides.

Alchemical Free-Energy Calculations by Multiple-Replica λ-Dynamics: The Conveyor Belt Thermodynamic Integration Scheme

A new method is proposed to calculate alchemical free-energy differences based on molecular dynamics (MD) simulations, called the conveyor belt thermodynamic integration (CBTI) scheme. As in thermodynamic integration (TI), K replicas of the system are simulated at different values of the alchemical coupling parameter λ. The number K is taken to be even, and the replicas are equally spaced on a forward-turn-backward-turn path, akin to a conveyor belt (CB) between the two physical end-states; and as in λ-dynamics (λD), the λ-values associated with the individual systems evolve in time along the simulation. However, they do so in a concerted fashion, determined by the evolution of a single dynamical variable Λ of period 2π controlling the advance of the entire CB. Thus, a change of Λ is always associated with K/2 equispaced replicas moving forward and K/2 equispaced replicas moving backward along λ. As a result, the effective free-energy profile of the replica system along Λ is periodic of period 2 πK^-1, and the magnitude of its variations decreases rapidly upon increasing K, at least as K^-1 in the limit of large K. When a sufficient number of replicas is used, these variations become small, which enables a complete and quasi-homogeneous coverage of the λ-range by the replica system, without application of any biasing potential. If desired, a memory-based biasing potential can still be added to further homogenize the sampling, the preoptimization of which is computationally inexpensive. The final free-energy profile along λ is calculated similarly to TI, by binning of the Hamiltonian λ-derivative as a function of λ considering all replicas simultaneously, followed by quadrature integration. The associated quadrature error can be kept very low owing to the continuous and quasi-homogeneous λ-sampling. The CBTI scheme can be viewed as a continuous/deterministic/dynamical analog of the Hamiltonian replica-exchange/permutation (HRE/HRP) schemes or as a correlated multiple-replica analog of the λD or λ-local elevation umbrella sampling (λ-LEUS) schemes. Compared to TI, it shares the advantage of the latter schemes in terms of enhanced orthogonal sampling, i.e. the availability of variable-λ paths to circumvent conformational barriers present at specific λ-values. Compared to HRE/HRP, it permits a deterministic and continuous sampling of the λ-range, is expected to be less sensitive to possible artifacts of the thermo- and barostating schemes, and bypasses the need to carefully preselect a λ-ladder and a swapping-attempt frequency. Compared to λ-LEUS, it eliminates (or drastically reduces) the dead time associated with the preoptimization of a biasing potential. The goal of this article is to provide the mathematical/physical formulation of the proposed CBTI scheme, along with an initial application of the method to the calculation of the hydration free energy of methanol.

The Adaptive Path Collective Variable: A Versatile Biasing Approach to Compute the Average Transition Path and Free Energy of Molecular Transitions

In the past decade, great progress has been made in the development of enhanced sampling methods, aimed at overcoming the time-scale limitations of molecular dynamics (MD) simulations. Many sampling schemes rely on adding an external bias to favor the sampling of transitions and to estimate the underlying free energy landscape. Nevertheless, sampling molecular processes described by many order parameters, or collective variables (CVs), such as complex biomolecular transitions, remains often very challenging. The computational cost has a prohibitive scaling with the dimensionality of the CV-space. Inspiration can be taken from methods that focus on localizing transition pathways: the CV-space can be projected onto a path-CV that connects two stable states, and a bias can be exerted onto a one-dimensional parameter that captures the progress of the transition along the path-CV. In principle, such a sampling scheme can handle an arbitrarily large number of CVs. A standard enhanced sampling technique combined with an adaptive path-CV can then locate the mean transition pathway and obtain the free energy profile along the path. In this chapter, we discuss the adaptive path-CV formalism and its numerical implementation. We apply the path-CV with several enhanced sampling methods-steered MD, metadynamics, and umbrella sampling-to a biologically relevant process: the Watson-Crick to Hoogsteen base-pairing transition in double-stranded DNA. A practical guide is provided on how to recognize and circumvent possible pitfalls during the calculation of a free energy landscape that contains multiple pathways. Examples are presented on how to perform enhanced sampling simulations using PLUMED, a versatile plugin that can work with many popular MD engines.

In Silico Study Reveals How E64 Approaches, Binds to, and Inhibits Falcipain-2 of Plasmodium falciparum that Causes Malaria in Humans

Plasmodium falciparum malaria, which degrades haemoglobin through falcipain-2 (FP2), is a serious disease killing 445 thousand people annually. Since the P. falciparum's survival in humans depends on its ability to degrade human's haemoglobin, stoppage or hindrance of FP2 has antimalarial effects. Therefore, we studied the atomic details of how E64 approaches, binds to, and inhibits FP2. We found that E64 (1) gradually approaches FP2 by first interacting with FP2's D170 and Q171 or N81, N77, and K76; (2) binds FP2 tightly (ΔG_binding = -12.2 ± 1.1 kJ/mol); and (3) persistently blocks access to FP2's catalytic residues regardless of whether or not E64 has already been able to form a covalent bond with FP2's C42. Furthermore, the results suggest that S41, D234, D170, N38, N173, and L172 (which are located in or near the FP2's catalytic site's binding pocket) contribute the most towards the favourable binding of E64 to FP2. Their in silico mutations adversely affect E64-FP2 binding affinity with D234L/A, N173L/A, W43F/A, D234L/A, H174F/A, and N38L/A having the most significant adverse effects on E64-FP2 binding and interactions. The findings presented in this article, which has antimalarial implications, suggest that hydrogen bonding and electrostatic interactions play important roles in E64-FP2 binding, and that a potential FP2-blocking E64-based/E64-like antimalarial drug should be capable of being both hydrogen-bond donor and acceptor, and/or have the ability to favourably interact with polar amino acids (such as S41, S149, N38, N173, N77, Q171) and with charged amino acids (such as D234, D170, H174) of FP2. The abilities to favourably interact with ASN, ASP, and SER appears to be important characteristics that such potential drug should have.

Adaptive ensemble simulations of biomolecules

Recent advances in both theory and computational power have created opportunities to simulate biomolecular processes more efficiently using adaptive ensemble simulations. Ensemble simulations are now widely used to compute a number of individual simulation trajectories and analyze statistics across them. Adaptive ensemble simulations offer a further level of sophistication and flexibility by enabling high-level algorithms to control simulations-based on intermediate results. We review some of the adaptive ensemble algorithms and software infrastructure currently in use and outline where the complexities of implementing adaptive simulation have limited algorithmic innovation to date. We describe an adaptive ensemble API to overcome some of these barriers and more flexibly and simply express adaptive simulation algorithms to help realize the power of this type of simulation.

Zooming across the Free-Energy Landscape: Shaving Barriers, and Flooding Valleys

A robust importance-sampling algorithm for mapping free-energy surfaces over geometrical variables, coined meta-eABF, is introduced. This algorithm shaves the free-energy barriers and floods valleys by incorporating a history-dependent potential term in the extended adaptive biasing force (eABF) framework. Numerical applications on both toy models and nontrivial examples indicate that meta-eABF explores the free-energy surface significantly faster than either eABF or metadynamics (MtD) alone, without the need to stratify the reaction pathway. In some favorable cases, meta-eABF can be as much as five times faster than other importance-sampling algorithms. Many of the shortcomings inherent to eABF and MtD, like kinetic trapping in regions of configurational space already adequately sampled, the requirement of prior knowledge of the free-energy landscape to set up the simulation, are readily eliminated in meta-eABF. Meta-eABF, therefore, represents an appealing solution for a broad range of applications, especially when both eABF and MtD fail to achieve the desired result.

The Impairment of TorsinA's Binding to and Interactions With Its Activator: An Atomistic Molecular Dynamics Study of Primary Dystonia

Primary dystonia's prolonged muscle contractions and the associated abnormal postures and twisting movements remain incurable. Genetic mutation/deletion of GAG from TorsonA's gene resulting in ΔE303 (which weakens the binding between TorsinA and its activator, such as LULL1) primarily cause this neurodegenerative disorder. We studied TorsinA-LULL1 (or TorsinAΔE303-LULL1) bindings and interactions. For the first time, we show the atomic details of TorsinA-LULL1 dynamic interactions and TorsinAΔE303-LULL1 dynamic interactions and their binding affinities. Our results show extensive effects of ΔE303 on TorsinAΔE303-LULL1 interactions, and suggest that the differences between TorsinA-LULL1 interactions and TorsinAΔE303-LULL1 interactions are non-subtle. ΔE303 significantly weakens TorsinAΔE303-LULL1's binding affinity. We present pieces of evidence proving that the effects of ΔE303 (on the differences between TorsinA-LULL1 interactions and TorsinAΔE303-LULL1 interactions) are more pronounced than previously suggested, and that the nanobody used for achieving the X-ray crystallization in the previous study attenuated the differences between TorsinA-LULL1 and TorsinAΔE303-LULL1 interactions. Our accounts of the dynamic interactions between “TorsinA and LULL1” and between “TorsinAΔE303 and LULL1” and the detailed effects of ΔE303 on TorsinA-/TorsinAΔE303-LULL1 build on previous findings and offer new insights for a better understanding of the molecular basis of Primary Dystonia. Our results have long-term potentials of guiding the development of medications for the disease.

Adaptive enhanced sampling with a path-variable for the simulation of protein folding and aggregation

In this article, we present a novel adaptive enhanced sampling molecular dynamics (MD) method for the accelerated simulation of protein folding and aggregation. We introduce a path-variable L based on the un-biased momenta p and displacements dq for the definition of the bias s applied to the system and derive 3 algorithms: general adaptive bias MD, adaptive path-sampling, and a hybrid method which combines the first 2 methodologies. Through the analysis of the correlations between the bias and the un-biased gradient in the system, we find that the hybrid methodology leads to an improved force correlation and acceleration in the sampling of the phase space. We apply our method on SPC/E water, where we find a conservation of the average water structure. We then use our method to sample dialanine and the folding of TrpCage, where we find a good agreement with simulation data reported in the literature. Finally, we apply our methodologies on the initial stages of aggregation of a hexamer of Alzheimer's amyloid β fragment 25-35 (Aβ 25-35) and find that transitions within the hexameric aggregate are dominated by entropic barriers, while we speculate that especially the conformation entropy plays a major role in the formation of the fibril as a rate limiting factor.

Molecular dynamics simulations of lipid nanodiscs

A lipid nanodisc is a discoidal lipid bilayer stabilized by proteins, peptides, or polymers on its edge. Nanodiscs have two important connections to structural biology. The first is associated with high-density lipoprotein (HDL), a particle with a variety of functionalities including lipid transport. Nascent HDL (nHDL) is a nanodisc stabilized by Apolipoprotein A-I (APOA1). Determining the structure of APOA1 and its mimetic peptides in nanodiscs is crucial to understanding pathologies related to HDL maturation and designing effective therapies. Secondly, nanodiscs offer non-detergent membrane-mimicking environments and greatly facilitate structural studies of membrane proteins. Although seemingly similar, natural and synthetic nanodiscs are different in that nHDL is heterogeneous in size, due to APOA1 elasticity, and gradually matures to become spherical. Synthetic nanodiscs, in contrast, should be homogenous, stable, and size-tunable. This report reviews previous molecular dynamics (MD) simulation studies of nanodiscs and illustrates convergence and accuracy issues using results from new multi-microsecond atomistic MD simulations. These new simulations reveal that APOA1 helices take 10-20 μs to rearrange on the nanodisc, while peptides take 2 μs to migrate from the disc surfaces to the edge. These systems can also become kinetically trapped depending on the initial conditions. For example, APOA1 was trapped in a biologically irrelevant conformation for the duration of a 10 μs trajectory; the peptides were similarly trapped for 5 μs. It therefore remains essential to validate MD simulations of these systems with experiments due to convergence and accuracy issues. This article is part of a Special Issue entitled: Emergence of Complex Behavior in Biomembranes edited by Marjorie Longo.

RNA Structural Dynamics As Captured by Molecular Simulations: A Comprehensive Overview

With both catalytic and genetic functions, ribonucleic acid (RNA) is perhaps the most pluripotent chemical species in molecular biology, and its functions are intimately linked to its structure and dynamics. Computer simulations, and in particular atomistic molecular dynamics (MD), allow structural dynamics of biomolecular systems to be investigated with unprecedented temporal and spatial resolution. We here provide a comprehensive overview of the fast-developing field of MD simulations of RNA molecules. We begin with an in-depth, evaluatory coverage of the most fundamental methodological challenges that set the basis for the future development of the field, in particular, the current developments and inherent physical limitations of the atomistic force fields and the recent advances in a broad spectrum of enhanced sampling methods. We also survey the closely related field of coarse-grained modeling of RNA systems. After dealing with the methodological aspects, we provide an exhaustive overview of the available RNA simulation literature, ranging from studies of the smallest RNA oligonucleotides to investigations of the entire ribosome. Our review encompasses tetranucleotides, tetraloops, a number of small RNA motifs, A-helix RNA, kissing-loop complexes, the TAR RNA element, the decoding center and other important regions of the ribosome, as well as assorted others systems. Extended sections are devoted to RNA-ion interactions, ribozymes, riboswitches, and protein/RNA complexes. Our overview is written for as broad of an audience as possible, aiming to provide a much-needed interdisciplinary bridge between computation and experiment, together with a perspective on the future of the field.

Structure and Dynamics of DNA and RNA Double Helices Obtained from the CCG and GGC Trinucleotide Repeats

Expansions of both GGC and CCG sequences lead to a number of expandable, trinucleotide repeat (TR) neurodegenerative diseases. Understanding of these diseases involves, among other things, the structural characterization of the atypical DNA and RNA secondary structures. We have performed molecular dynamics simulations of (GCC) _n and (GGC) _n homoduplexes in order to characterize their conformations, stability, and dynamics. Each TR has two reading frames, which results in eight nonequivalent RNA/DNA homoduplexes, characterized by CpG or GpC steps between the Watson-Crick base pairs. Free energy maps for the eight homoduplexes indicate that the C-mismatches prefer anti-anti conformations, while G-mismatches prefer anti-syn conformations. Comparison between three modifications of the DNA AMBER force field shows good agreement for the mismatch free energy maps. The mismatches in DNA-GCC (but not CCG) are extrahelical, forming an extended e-motif. The mismatched duplexes exhibit characteristic sequence-dependent step twist, with strong variations in the G-rich sequences and the e-motif. The distribution of Na⁺ is highly localized around the mismatches, especially G-mismatches. In the e-motif, there is strong Na⁺ binding by two G(N7) atoms belonging to the pseudo GpC step created when cytosines are extruded and by extrahelical cytosines. Finally, we used a novel technique based on fast melting by means of an infrared laser pulse to classify the relative stability of the different DNA-CCG and -GGC homoduplexes.

Free energy profiles for unwrapping the outer superhelical turn of nucleosomal DNA

The eukaryotic genome is packaged into a nucleus in the form of chromatin. The fundamental structural unit of chromatin is a protein-DNA complex, the nucleosome, where 146 or 147 base pairs of DNA wrap 1.75 times around a histone core. To function in cellular processes, however, nucleosomal DNA must be unwrapped. Although this unwrapping has been experimentally investigated, details of the process at an atomic level are not yet well understood. Here, we used molecular dynamics simulation with an enhanced sampling method to calculate the free energy profiles for unwrapping the outer superhelical turn of nucleosomal DNA. A free energy change of about 11.5 kcal/mol for the unwrapping agrees well with values obtained in single molecule experiments. This simulation revealed a variety of conformational states, indicating there are many potential paths to outer superhelicdal turn unwrapping, but the dominant path is likely asymmetric. At one end of the DNA, the first five bps unwrap, after which a second five bps unwrap at the same end with no increase in free energy. The unwrapping then starts at the other end of the DNA, where 10 bps are unwrapped. During further unwrapping of 15 bps, the unwrapping advances at one of the ends, after which the other end of the DNA unwraps to complete the unwrapping of the outer superhelical turn. These results provide insight into the construction, disruption, and repositioning of nucleosomes, which are continuously ongoing during cellular processes.

Efficient Construction of Free Energy Profiles of Breathing Metal-Organic Frameworks Using Advanced Molecular Dynamics Simulations

In order to reliably predict and understand the breathing behavior of highly flexible metal–organic frameworks from thermodynamic considerations, an accurate estimation of the free energy difference between their different metastable states is a prerequisite. Herein, a variety of free energy estimation methods are thoroughly tested for their ability to construct the free energy profile as a function of the unit cell volume of MIL-53(Al). The methods comprise free energy perturbation, thermodynamic integration, umbrella sampling, metadynamics, and variationally enhanced sampling. A series of molecular dynamics simulations have been performed in the frame of each of the five methods to describe structural transformations in flexible materials with the volume as the collective variable, which offers a unique opportunity to assess their computational efficiency. Subsequently, the most efficient method, umbrella sampling, is used to construct an accurate free energy profile at different temperatures for MIL-53(Al) from first principles at the PBE+D3(BJ) level of theory. This study yields insight into the importance of the different aspects such as entropy contributions and anharmonic contributions on the resulting free energy profile. As such, this thorough study provides unparalleled insight in the thermodynamics of the large structural deformations of flexible materials.

H4 Tails Potentially Produce the Diversity in the Orientation of Two Nucleosomes

Histone tails play an important role in internucleosomal interaction and chromatin compaction. To understand how the H4 tails are involved in the internucleosomal interaction, an adaptively biased molecular dynamics simulation of 63 models of two stacked nucleosomes, each with the H4 tails in different locations, was carried out. This simulation generated a variety of orientations of the separated nucleosomes depending on the formation of the H4 tail bridge between the H4 tails and the DNA of the neighboring nucleosomes. For the models that showed distinctive orientations of the two nucleosomes, the free energies of the separation of the nucleosomes were further investigated using umbrella sampling simulations. The attractive force between the nucleosomes was estimated from the free energies; the force when two H4 tail bridges formed varied from 36 to 63 pN, depending on the formation of the H4 tail-bridge and the interfacial interaction, whereas the force reduced to 15-18 pN after either one of the H4 tail bridges had broken, regardless of the conformation of the H4 tail. Additional simulations of the nucleosomes show that when the H4 tail was truncated, the force between the nucleosomes became repulsive (from-3 to -7 pN). We concluded that the H4 tails potentially produce the diversity in the orientation of the two nucleosomes, which would contribute to the polymorphism of the chromatin structure.

Constructing a multidimensional free energy surface like a spider weaving a web

Complete free energy surface in the collective variable space provides important information of the reaction mechanisms of the molecules. But, sufficient sampling in the collective variable space is not easy. The space expands quickly with the number of the collective variables. To solve the problem, many methods utilize artificial biasing potentials to flatten out the original free energy surface of the molecule in the simulation. Their performances are sensitive to the definitions of the biasing potentials. Fast-growing biasing potential accelerates the sampling speed but decreases the accuracy of the free energy result. Slow-growing biasing potential gives an optimized result but needs more simulation time. In this article, we propose an alternative method. It adds the biasing potential to a representative point of the molecule in the collective variable space to improve the conformational sampling. And the free energy surface is calculated from the free energy gradient in the constrained simulation, not given by the negative of the biasing potential as previous methods. So the presented method does not require the biasing potential to remove all the barriers and basins on the free energy surface exactly. Practical applications show that the method in this work is able to produce the accurate free energy surfaces for different molecules in a short time period. The free energy errors are small in the cases of various biasing potentials. © 2017 Wiley Periodicals, Inc.

Structure and Dynamics of DNA and RNA Double Helices of CAG and GAC Trinucleotide Repeats

CAG trinucleotide repeats are known to cause 10 late-onset progressive neurodegenerative disorders as the repeats expand beyond a threshold, whereas GAC repeats are associated with skeletal dysplasias and expand from the normal five to a maximum of seven repeats. The TR secondary structure is believed to play a role in CAG expansions. We have carried out free energy and molecular dynamics studies to determine the preferred conformations of the A-A noncanonical pairs in (CAG)_n and (GAC)_n trinucleotide repeats (n = 1, 4) and the consequent changes in the overall structure of the RNA and DNA duplexes. We find that the global free energy minimum corresponds to A-A pairs stacked inside the core of the helix with anti-anti conformations in RNA and (high-anti)-(high-anti) conformations in DNA. The next minimum corresponds to anti-syn conformations, whereas syn-syn conformations are higher in energy. Transition rates of the A-A conformations are higher for RNA than DNA. Mechanisms for these various transitions are identified. Additional structural and dynamical aspects of the helical conformations are explored, with a focus on contrasting CAG and GAC duplexes. The neutralizing ion distribution around the noncanonical pairs is described.

Fast exploration of an optimal path on the multidimensional free energy surface

In a reaction, determination of an optimal path with a high reaction rate (or a low free energy barrier) is important for the study of the reaction mechanism. This is a complicated problem that involves lots of degrees of freedom. For simple models, one can build an initial path in the collective variable space by the interpolation method first and then update the whole path constantly in the optimization. However, such interpolation method could be risky in the high dimensional space for large molecules. On the path, steric clashes between neighboring atoms could cause extremely high energy barriers and thus fail the optimization. Moreover, performing simulations for all the snapshots on the path is also time-consuming. In this paper, we build and optimize the path by a growing method on the free energy surface. The method grows a path from the reactant and extends its length in the collective variable space step by step. The growing direction is determined by both the free energy gradient at the end of the path and the direction vector pointing at the product. With fewer snapshots on the path, this strategy can let the path avoid the high energy states in the growing process and save the precious simulation time at each iteration step. Applications show that the presented method is efficient enough to produce optimal paths on either the two-dimensional or the twelve-dimensional free energy surfaces of different small molecules.

Imaging Metastable States and Transitions in Proteins by Trajectory Map

It has been a long-standing and intriguing issue to develop robust methods to identify metastable states and interstate transitions from simulations or experimental data to understand the functional conformational changes of proteins. It is usually hard to define the complicated boundaries of the states in the conformational space using most of the existing methods, and they often lead to parameter-sensitive results. Here, we present a new approach, visualized Trajectory Map (vTM), to identify the metastable states and the rare interstate transitions, by considering both the conformational similarity and the temporal successiveness of conformations. The vTM is able to give a nonambiguous description of slow dynamics. The case study of a β-hairpin peptide shows that the vTM can reveal the states and transitions from all-atom MD trajectory data even when a single observable (i.e, one-dimensional reaction coordinate) is used. We also use the vTM to refine the folding/unfolding mechanism of HP35 in explicit water by analyzing a 125 μs all-atom MD trajectory and obtain folding/unfolding rates of about 1/μs, which are in good agreement with the experimental values.

Efficient Conformational Search Based on Structural Dissimilarity Sampling: Applications for Reproducing Structural Transitions of Proteins

Structural Dissimilarity Sampling (SDS) is proposed as an efficient conformational search method to promote structural transitions essential for the biological functions of proteins. In SDS, initial structures are selected based on structural dissimilarity, and conformational resampling is repeated. Conformational resampling is performed as follows: (I) arrangement of initial structures for a diverse distribution at the edge of a conformational subspace and (II) promotion of the structural transitions with multiple short-time molecular dynamics (MD) simulations restarting from the diversely distributed initial structures. Cycles of (I) and (II) are repeated to intensively promote structural transitions because conformational resampling from the initial structures would quickly expand conformational distributions toward unvisited conformational subspaces. As a demonstration, SDS was first applied to maltodextrin binding protein (MBP) in explicit water to reproduce structural transitions between the open and closed states of MBP. Structural transitions of MBP were successfully reproduced with SDS in nanosecond-order simulation times. Starting from both the open and closed forms, SDS successfully reproduced the structural transitions within 25 cycles (a total of 250 ns of simulation time). For reference, a conventional long-time (500 ns) MD simulation under NPT (300 K and 1 bar) starting from the open form failed to reproduce the structural transition. In addition to the open-closed motions of MBP, SDS was applied to folding processes of the fast-folding proteins (chignolin, Trp-cage, and villin) and successfully sampled their native states. To confirm how the selections of initial structures affected conformational sampling efficiency, numbers of base sets for characterizing structural dissimilarity of initial structures were addressed in distinct trials of SDS. The parameter searches showed that the conformational sampling efficiency was relatively insensitive with respect to the numbers of base sets, indicating the robustness of SDS for actual applications.

A fast, open source implementation of adaptive biasing potentials uncovers a ligand design strategy for the chromatin regulator BRD4

In this communication we introduce an efficient implementation of adaptive biasing that greatly improves the speed of free energy computation in molecular dynamics simulations. We investigated the use of accelerated simulations to inform on compound design using a recently reported and clinically relevant inhibitor of the chromatin regulator BRD4 (bromodomain-containing protein 4). Benchmarking on our local compute cluster, our implementation achieves up to 2.5 times more force calls per day than plumed2. Results of five 1 μs-long simulations are presented, which reveal a conformational switch in the BRD4 inhibitor between a binding competent and incompetent state. Stabilization of the switch led to a -3 kcal/mol improvement of absolute binding free energy. These studies suggest an unexplored ligand design principle and offer new actionable hypotheses for medicinal chemistry efforts against this druggable epigenetic target class.

Quantum free energy landscapes from ab initio path integral metadynamics: Double proton transfer in the formic acid dimer is concerted but not correlated

With the goal of computing quantum free energy landscapes of reactive (bio)chemical systems in multi-dimensional space, we combine the metadynamics technique for sampling potential energy surfaces with the ab initio path integral approach to treating nuclear quantum motion. This unified method is applied to the double proton transfer process in the formic acid dimer (FAD), in order to study the nuclear quantum effects at finite temperatures without imposing a one-dimensional reaction coordinate or reducing the dimensionality. Importantly, the ab initio path integral metadynamics technique allows one to treat the hydrogen bonds and concomitant proton transfers in FAD strictly independently and thus provides direct access to the much discussed issue of whether the double proton transfer proceeds via a stepwise or concerted mechanism. The quantum free energy landscape we compute for this H-bonded molecular complex reveals that the two protons move in a concerted fashion from initial to product state, yet world-line analysis of the quantum correlations demonstrates that the protons are as quantum-uncorrelated at the transition state as they are when close to the equilibrium structure.

Multistate λ-local-elevation umbrella-sampling (MS-λ-LEUS): method and application to the complexation of cations by crown ethers

An extension of the λ-local-elevation umbrella-sampling (λ-LEUS) scheme [ Bieler et al. J. Chem. Theory Comput. 2014 , 10 , 3006 ] is proposed to handle the multistate (MS) situation, i.e. the calculation of the relative free energies of multiple physical states based on a single simulation. The key element of the MS-λ-LEUS approach is to use a single coupling variable Λ controlling successive pairwise mutations between the states of interest in a cyclic fashion. The Λ variable is propagated dynamically as an extended-system variable, using a coordinate transformation with plateaus and a memory-based biasing potential as in λ-LEUS. Compared to other available MS schemes (one-step perturbation, enveloping distribution sampling and conventional λ-dynamics) the proposed method presents a number of important advantages, namely: (i) the physical states are visited explicitly and over finite time periods; (ii) the extent of unphysical space required to ensure transitions is kept minimal and, in particular, one-dimensional; (iii) the setup protocol solely requires the topologies of the physical states; and (iv) the method only requires limited modifications in a simulation code capable of handling two-state mutations. As an initial application, the absolute binding free energies of five alkali cations to three crown ethers in three different solvents are calculated. The results are found to reproduce qualitatively the main experimental trends and, in particular, the experimental selectivity of 18C6 for K(+) in water and methanol, which is interpreted in terms of opposing trends along the cation series between the solvation free energy of the cation and the direct electrostatic interactions within the complex.

Calculation of the Local Free Energy Landscape in the Restricted Region by the Modified Tomographic Method

The free energy landscape is the most important information in the study of the reaction mechanisms of the molecules. However, it is difficult to calculate. In a large collective variable space, a molecule must take a long time to obtain the sufficient sampling during the simulation. To save the calculation quantity, decreasing the sampling region and constructing the local free energy landscape is required in practice. However, the restricted region in the collective variable space may have an irregular shape. Simply restricting one or more collective variables of the molecule cannot satisfy the requirement. In this paper, we propose a modified tomographic method to perform the simulation. First, it divides the restricted region by some hyperplanes and connects the centers of hyperplanes together by a curve. Second, it forces the molecule to sample on the curve and the hyperplanes in the simulation and calculates the free energy data on them. Finally, all the free energy data are combined together to form the local free energy landscape. Without consideration of the area outside the restricted region, this free energy calculation can be more efficient. By this method, one can further optimize the path quickly in the collective variable space.

Semiempirical QM/MM calculations reveal a step-wise proton transfer and an unusual thiolate pocket in the mechanism of the unique arylpropionate racemase AMDase G74C

Semiempirical calculations on the mechanism of the arylpropionate racemase AMDase G74C reveal a step-wise mechanism involving a planar-enedionate intermediate.

Investigating rare events with nonequilibrium work measurements. I. Nonequilibrium transition path probabilities

We have developed a formalism for investigating transition pathways and transition probabilities for rare events in biomolecular systems. In this paper, we set the theoretical framework for employing nonequilibrium work relations to estimate the relative reaction rates associated with different classes of transition pathways. Particularly, we derive an extension of Crook's transient fluctuation theorem, which relates the relative transition rates of driven systems in the forward and reverse directions, and allows for the calculation of these relative rates using work measurements (e.g., in Steered Molecular Dynamics). The formalism presented here can be combined with Transition Path Theory to relate the equilibrium and driven transition rates. The usefulness of this framework is illustrated by means of a Gaussian model and a driven proline dimer.

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.

Multiscale methods for computational RNA enzymology

RNA catalysis is of fundamental importance to biology and yet remains ill-understood due to its complex nature. The multidimensional "problem space" of RNA catalysis includes both local and global conformational rearrangements, changes in the ion atmosphere around nucleic acids and metal ion binding, dependence on potentially correlated protonation states of key residues, and bond breaking/forming in the chemical steps of the reaction. The goal of this chapter is to summarize and apply multiscale modeling methods in an effort to target the different parts of the RNA catalysis problem space while also addressing the limitations and pitfalls of these methods. Classical molecular dynamics simulations, reference interaction site model calculations, constant pH molecular dynamics (CpHMD) simulations, Hamiltonian replica exchange molecular dynamics, and quantum mechanical/molecular mechanical simulations will be discussed in the context of the study of RNA backbone cleavage transesterification. This reaction is catalyzed by both RNA and protein enzymes, and here we examine the different mechanistic strategies taken by the hepatitis delta virus ribozyme and RNase A.

Enhanced sampling techniques in biomolecular simulations

Biomolecular simulations are routinely used in biochemistry and molecular biology research; however, they often fail to match expectations of their impact on pharmaceutical and biotech industry. This is caused by the fact that a vast amount of computer time is required to simulate short episodes from the life of biomolecules. Several approaches have been developed to overcome this obstacle, including application of massively parallel and special purpose computers or non-conventional hardware. Methodological approaches are represented by coarse-grained models and enhanced sampling techniques. These techniques can show how the studied system behaves in long time-scales on the basis of relatively short simulations. This review presents an overview of new simulation approaches, the theory behind enhanced sampling methods and success stories of their applications with a direct impact on biotechnology or drug design.

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.

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.

Computational Recipe for Efficient Description of Large-Scale Conformational Changes in Biomolecular Systems

Characterizing large-scale structural transitions in biomolecular systems poses major technical challenges to both experimental and computational approaches. On the computational side, efficient sampling of the configuration space along the transition pathway remains the most daunting challenge. Recognizing this issue, we introduce a knowledge-based computational approach toward describing large-scale conformational transitions using (i) nonequilibrium, driven simulations combined with work measurements and (ii) free energy calculations using empirically optimized biasing protocols. The first part is based on designing mechanistically relevant, system-specific reaction coordinates whose usefulness and applicability in inducing the transition of interest are examined using knowledge-based, qualitative assessments along with nonequilirbrium work measurements which provide an empirical framework for optimizing the biasing protocol. The second part employs the optimized biasing protocol resulting from the first part to initiate free energy calculations and characterize the transition quantitatively. Using a biasing protocol fine-tuned to a particular transition not only improves the accuracy of the resulting free energies but also speeds up the convergence. The efficiency of the sampling will be assessed by employing dimensionality reduction techniques to help detect possible flaws and provide potential improvements in the design of the biasing protocol. Structural transition of a membrane transporter will be used as an example to illustrate the workings of the proposed approach.

Evaluation of enhanced sampling provided by accelerated molecular dynamics with Hamiltonian replica exchange methods

Many problems studied via molecular dynamics require accurate estimates of various thermodynamic properties, such as the free energies of different states of a system, which in turn requires well-converged sampling of the ensemble of possible structures. Enhanced sampling techniques are often applied to provide faster convergence than is possible with traditional molecular dynamics simulations. Hamiltonian replica exchange molecular dynamics (H-REMD) is a particularly attractive method, as it allows the incorporation of a variety of enhanced sampling techniques through modifications to the various Hamiltonians. In this work, we study the enhanced sampling of the RNA tetranucleotide r(GACC) provided by H-REMD combined with accelerated molecular dynamics (aMD), where a boosting potential is applied to torsions, and compare this to the enhanced sampling provided by H-REMD in which torsion potential barrier heights are scaled down to lower force constants. We show that H-REMD and multidimensional REMD (M-REMD) combined with aMD does indeed enhance sampling for r(GACC), and that the addition of the temperature dimension in the M-REMD simulations is necessary to efficiently sample rare conformations. Interestingly, we find that the rate of convergence can be improved in a single H-REMD dimension by simply increasing the number of replicas from 8 to 24 without increasing the maximum level of bias. The results also indicate that factors beyond replica spacing, such as round trip times and time spent at each replica, must be considered in order to achieve optimal sampling efficiency.