Comparison between Mean Forces and Swarms-of-Trajectories String Methods

Finite Temperature String with Order Parameter as Collective Variables for Molecular Crystal: A Case of Polymorphic Transformation of TNT under External Electric Field

An external electric field is an effective tool to induce the polymorphic transformation of molecular crystals, which is important practically in the chemical, material, and energy storage industries. However, the understanding of this mechanism is poor at the molecular level. In this work, two types of order parameters (OPs) were constructed for the molecular crystal based on the intermolecular distance, bond orientation, and molecular orientation. Using the K-means clustering algorithm for the sampling of OPs based on the Euclidean distance and density weight, the polymorphic transformation of TNT was investigated using a finite temperature string (FTS) under external electric fields. The potential of mean force (PMF) was obtained, and the essence of the polymorphic transformation between o-TNT and m-TNT was revealed, which verified the effectiveness of the FTS method based on K-means clustering to OPs. The differences in PMFs between the o-TNT and transition state were decreased under external electric fields in comparison with those in no field. The fields parallel to the c-axis obviously affected the difference in PMF, and the relationship between the changes in PMFs and field strengths was found. Although the external electric field did not promote the convergence, the time of the polymorphic transformation was reduced under the external electric field in comparison to its absence. Moreover, under the external electric field, the polymorphic transformation from o-TNT to m-TNT occurred while that from m-TNT to o-TNT was prevented, which was explained by the dipole moment of molecule, relative permittivity, chemical potential difference, nucleation work and nucleation rate. This confirmed that the polymorphic transformation orientation of the molecular crystal could be controlled by the external electric field. This work provides an effective way to explore the polymorphic transformation of the molecular crystals at a molecular level, and it is useful to control the production process and improve the performance of energetic materials by using the external electric fields.

A weak coupling mechanism for the early steps of the recovery stroke of myosin VI: A free energy simulation and string method analysis

Myosin motors use the energy of ATP to produce force and directed movement on actin by a swing of the lever-arm. ATP is hydrolysed during the off-actin re-priming transition termed recovery stroke. To provide an understanding of chemo-mechanical transduction by myosin, it is critical to determine how the reverse swing of the lever-arm and ATP hydrolysis are coupled. Previous studies concluded that the recovery stroke of myosin II is initiated by closure of the Switch II loop in the nucleotide-binding site. Recently, we proposed that the recovery stroke of myosin VI starts with the spontaneous re-priming of the converter domain to a putative pre-transition state (PTS) intermediate that precedes Switch II closing and ATPase activation. Here, we investigate the transition from the pre-recovery, post-rigor (PR) state to PTS in myosin VI using geometric free energy simulations and the string method. First, our calculations rediscover the PTS state agnostically and show that it is accessible from PR via a low free energy transition path. Second, separate path calculations using the string method illuminate the mechanism of the PR to PTS transition with atomic resolution. In this mechanism, the initiating event is a large movement of the converter/lever-arm region that triggers rearrangements in the Relay-SH1 region and the formation of the kink in the Relay helix with no coupling to the active site. Analysis of the free-energy barriers along the path suggests that the converter-initiated mechanism is much faster than the one initiated by Switch II closure, which supports the biological relevance of PTS as a major on-pathway intermediate of the recovery stroke in myosin VI. Our analysis suggests that lever-arm re-priming and ATP hydrolysis are only weakly coupled, so that the myosin recovery stroke is initiated by thermal fluctuations and stabilised by nucleotide consumption via a ratchet-like mechanism.

Experimental and Theoretical Investigation of External Electric-Field-Induced Crystallization of TKX-50 from Solution by Finite-Temperature String with Order Parameters as Collective Variables for Ionic Crystals

External electric fields are an effective tool to induce phase transformations. The crystallization of ionic crystals from solution is a common phase transformation. However, understanding of mechanisms is poor at the molecular level. In this work, we carried out an experimental and theoretical investigation of the external electric-field-induced crystallization of TKX-50 from saturated formic acid solution by finite-temperature string (FTS) with order parameters (OPs) as collective variables for ionic crystals. The minimum-free-energy path was sketched by the string method in collective variables. The results show that the K-means clustering algorithm based on Euclidean distance and density weights can be used for enhanced sampling of the OPs in external electric-field-induced crystallization of ionic crystal from solution, which improves the conventional FTS. The crystallization from solution is a process of surface-mediated nucleation. The external electric field can accelerate the evolution of the string and decrease the difference in the potential of mean forces between the crystal and the transition state. Due to the significant change in OPs induced by the external electric field in nucleation, the crystalline quality was enhanced, which explains the experimental results that the external electric field enhanced the density, detonation velocity, and detonation pressure of TKX-50. This work provides an effective way to explore the crystallization of ionic crystals from solution at the molecular level, and it is useful for improving the properties of ionic crystal explosives by using external electric fields.

Ligand Lipophilicity Determines Molecular Mechanisms of Nanoparticle Adsorption to Lipid Bilayers

The interactions of ligand-functionalized nanoparticles with the cell membrane affect cellular uptake, cytotoxicity, and related behaviors, but relating these interactions to ligand properties remains challenging. In this work, we perform coarse-grained molecular dynamics simulations to study how the adsorption of ligand-functionalized cationic gold nanoparticles (NPs) to a single-component lipid bilayer (as a model cell membrane) is influenced by ligand end group lipophilicity. A set of 2 nm diameter NPs, each coated with a monolayer of organic ligands that differ only in their end groups, was simulated to mimic NPs recently studied experimentally. Metadynamics calculations were performed to determine key features of the free energy landscape for adsorption as a function of the distance of the NP from the bilayer and the number of NP-lipid contacts. These simulations revealed that NP adsorption is thermodynamically favorable for all NPs due to the extraction of lipids from the bilayer and into the NP monolayer. To resolve ligand-dependent differences in adsorption behavior, string method calculations were performed to compute minimum free energy pathways for adsorption. These calculations revealed a surprising nonmonotonic dependence of the free energy barrier for adsorption on ligand end group lipophilicity. Large free energy barriers are predicted for the least lipophilic end groups because favorable NP-lipid contacts are initiated only through the unfavorable protrusion of lipid tail groups out of the bilayer. The smallest free energy barriers are predicted for end groups of intermediate lipophilicity which promote NP-lipid contacts by intercalating within the bilayer. Unexpectedly, large free energy barriers are also predicted for the most lipophilic end groups which remain sequestered within the ligand monolayer rather than intercalating within the bilayer. These trends are broadly in agreement with past experimental measurements and reveal how subtle variations in ligand lipophilicity dictate adsorption mechanisms and associated kinetics by influencing the interplay of lipid-ligand interactions.

Finite temperature string by K-means clustering sampling with order parameters as collective variables for molecular crystals: application to polymorphic transformation between β-CL-20 and ε-CL-20

Polymorphic transformation of molecular crystals is a fundamental phase transition process, and it is important practically in the chemical, material, biopharmaceutical, and energy storage industries. However, understanding of the transformation mechanism at the molecular level is poor due to the extreme simulating challenges in enhanced sampling and formulating order parameters (OPs) as the collective variables that can distinguish polymorphs with quite similar and complicated structures so as to describe the reaction coordinate. In this work, two kinds of OPs for CL-20 were constructed by the bond distances, bond orientations and relative orientations. A K-means clustering algorithm based on the Euclidean distance and sample weight was used to smooth the initial finite temperature string (FTS), and the minimum free energy path connecting β-CL-20 and ε-CL-20 was sketched by the string method in collective variables, and the free energy profile along the path and the nucleation kinetics were obtained by Markovian milestoning with Voronoi tessellations. In comparison with the average-based sampling, the K-means clustering algorithm provided an improved convergence rate of FTS. The simulation of transformation was independent of OP types but was affected greatly by finite-size effects. A surface-mediated local nucleation mechanism was confirmed and the configuration located at the shoulder of potential of mean force, rather than overall maximum, was confirmed to be the critical nucleus formed by the cooperative effect of the intermolecular interactions. This work provides an effective way to explore the polymorphic transformation of caged molecular crystals at the molecular level.

Computing committors via Mahalanobis diffusion maps with enhanced sampling data

The study of phenomena such as protein folding and conformational changes in molecules is a central theme in chemical physics. Molecular dynamics (MD) simulation is the primary tool for the study of transition processes in biomolecules, but it is hampered by a huge timescale gap between the processes of interest and atomic vibrations that dictate the time step size. Therefore, it is imperative to combine MD simulations with other techniques in order to quantify the transition processes taking place on large timescales. In this work, the diffusion map with Mahalanobis kernel, a meshless approach for approximating the Backward Kolmogorov Operator (BKO) in collective variables, is upgraded to incorporate standard enhanced sampling techniques, such as metadynamics. The resulting algorithm, which we call the target measure Mahalanobis diffusion map (tm-mmap), is suitable for a moderate number of collective variables in which one can approximate the diffusion tensor and free energy. Imposing appropriate boundary conditions allows use of the approximated BKO to solve for the committor function and utilization of transition path theory to find the reactive current delineating the transition channels and the transition rate. The proposed algorithm, tm-mmap, is tested on the two-dimensional Moro-Cardin two-well system with position-dependent diffusion coefficient and on alanine dipeptide in two collective variables where the committor, the reactive current, and the transition rate are compared to those computed by the finite element method (FEM). Finally, tm-mmap is applied to alanine dipeptide in four collective variables where the use of finite elements is infeasible.

Finite temperature string method with umbrella sampling using path collective variables: application to secondary structure change in a protein

The transition of an α-helix to a β-sheet in proteins is among the most complex conformational changes seen in biomolecular systems. Due to long time scales involved in the transition, it is challenging to study such protein conformational changes using direct molecular dynamics simulations. This limitation is typically overcome using an indirect approach wherein one computes the free energy landscape associated with the transition. Computation of free energy landscapes, however, requires a suitable set of collective variables that describe the transition. In this work, we demonstrate the use of path collective variables [D. Branduardi, F. L. Gervasio and M. Parrinello, J. Chem. Phys., 2007, 126, 054103] and combine it with the finite temperature string (FTS) method [E. Weinan, W. Ren and E. Vanden-Eijnden, J. Phys. Chem. B, 2005, 109, 6688-6693] to determine the molecular mechanisms involved during the structural transition of the mini G-protein from an α-helix to a β-hairpin. The transition from the α-helix proceeds via unfolding of the terminal residues, giving rise to a β-turn unfolded intermediate to eventually form the β-hairpin. Our proposed algorithm uses umbrella sampling simulations to simulate images along the string and the weighted histogram analysis to compute the free energy along the computed transition path. This work demonstrates that the string method in combination with path collective variables can be exploited to study complex protein conformational changes such as a complete change in the secondary structure.

Pre-exascale HPC approaches for molecular dynamics simulations. Covid-19 research: A use case

Exascale computing has been a dream for ages and is close to becoming a reality that will impact how molecular simulations are being performed, as well as the quantity and quality of the information derived for them. We review how the biomolecular simulations field is anticipating these new architectures, making emphasis on recent work from groups in the BioExcel Center of Excellence for High Performance Computing. We exemplified the power of these simulation strategies with the work done by the HPC simulation community to fight Covid-19 pandemics. This article is categorized under:Data Science > Computer Algorithms and ProgrammingData Science > Databases and Expert SystemsMolecular and Statistical Mechanics > Molecular Dynamics and Monte-Carlo Methods.

A Companion Guide to the String Method with Swarms of Trajectories: Characterization, Performance, and Pitfalls

The string method with swarms of trajectories (SMwST) is an algorithm that identifies a physically meaningful transition pathway─a one-dimensional curve, embedded within a high-dimensional space of selected collective variables. The SMwST algorithm leans on a series of short, unbiased molecular dynamics simulations spawned at different locations of the discretized path, from whence an average dynamic drift is determined to evolve the string toward an optimal pathway. However conceptually simple in both its theoretical formulation and practical implementation, the SMwST algorithm is computationally intensive and requires a careful choice of parameters for optimal cost-effectiveness in applications to challenging problems in chemistry and biology. In this contribution, the SMwST algorithm is presented in a self-contained manner, discussing with a critical eye its theoretical underpinnings, applicability, inherent limitations, and use in the context of path-following free-energy calculations and their possible extension to kinetics modeling. Through multiple simulations of a prototypical polypeptide, combining the search of the transition pathway and the computation of the potential of mean force along it, several practical aspects of the methodology are examined with the objective of optimizing the computational effort, yet without sacrificing accuracy. In light of the results reported here, we propose some general guidelines aimed at improving the efficiency and reliability of the computed pathways and free-energy profiles underlying the conformational transitions at hand.

Computational methods and theory for ion channel research

Ion channels are fundamental biological devices that act as gates in order to ensure selective ion transport across cellular membranes; their operation constitutes the molecular mechanism through which basic biological functions, such as nerve signal transmission and muscle contraction, are carried out. Here, we review recent results in the field of computational research on ion channels, covering theoretical advances, state-of-the-art simulation approaches, and frontline modeling techniques. We also report on few selected applications of continuum and atomistic methods to characterize the mechanisms of permeation, selectivity, and gating in biological and model channels.

Role of salt-bridging interactions in recognition of viral RNA by arginine-rich peptides

Interactions between RNA molecules and proteins are critical to many cellular processes and are implicated in various diseases. The RNA-peptide complexes are good model systems to probe the recognition mechanism of RNA by proteins. In this work, we report studies on the binding-unbinding process of a helical peptide from a viral RNA element using nonequilibrium molecular dynamics simulations. We explored the existence of various dissociation pathways with distinct free-energy profiles that reveal metastable states and distinct barriers to peptide dissociation. We also report the free-energy differences for each of the four pathways to be 96.47 ± 12.63, 96.1 ± 10.95, 91.83 ± 9.81, and 92 ± 11.32 kcal/mol. Based on the free-energy analysis, we further propose the preferred pathway and the mechanism of peptide dissociation. The preferred pathway is characterized by the formation of sequential hydrogen-bonding and salt-bridging interactions between several key arginine amino acids and the viral RNA nucleotides. Specifically, we identified one arginine amino acid (R8) of the peptide to play a significant role in the recognition mechanism of the peptide by the viral RNA molecule.

String Method with Swarms-of-Trajectories, Mean Drifts, Lag Time, and Committor

The kinetics of a dynamical system comprising two metastable states is formulated in terms of a finite-time propagator in phase space (position and velocity) adapted to the underdamped Langevin equation. Dimensionality reduction to a subspace of collective variables yields familiar expressions for the propagator, committor, and steady-state flux. A quadratic expression for the steady-state flux between the two metastable states can serve as a robust variational principle to determine an optimal approximate committor expressed in terms of a set of collective variables. The theoretical formulation is exploited to clarify the foundation of the string method with swarms-of-trajectories, which relies on the mean drift of short trajectories to determine the optimal transition pathway. It is argued that the conditions for Markovity within a subspace of collective variables may not be satisfied with an arbitrary short time-step and that proper kinetic behaviors appear only when considering the effective propagator for longer lag times. The effective propagator with finite lag time is amenable to an eigenvalue-eigenvector spectral analysis, as elaborated previously in the context of position-based Markov models. The time-correlation functions calculated by swarms-of-trajectories along the string pathway constitutes a natural extension of these developments. The present formulation provides a powerful theoretical framework to characterize the optimal pathway between two metastable states of a system.

Large-Scale Conformational Changes and Protein Function: Breaking the in silico Barrier

Large-scale conformational changes are essential to link protein structures with their function at the cell and organism scale, but have been elusive both experimentally and computationally. Over the past few years developments in cryo-electron microscopy and crystallography techniques have started to reveal multiple snapshots of increasingly large and flexible systems, deemed impossible only short time ago. As structural information accumulates, theoretical methods become central to understand how different conformers interconvert to mediate biological function. Here we briefly survey current in silico methods to tackle large conformational changes, reviewing recent examples of cross-validation of experiments and computational predictions, which show how the integration of different scale simulations with biological information is already starting to break the barriers between the in silico, in vitro, and in vivo worlds, shedding new light onto complex biological problems inaccessible so far.

String Method for Protein-Protein Binding Free-Energy Calculations

A powerful computational strategy to determine the equilibrium association constant of two macromolecules with explicit-solvent molecular dynamics (MD) simulations is the "geometric route", which considers the reversible physical separation of the bound complex in solution. Nonetheless, multiple challenges remain to render this type of methodology reliable and computationally efficient in practice. In particular, in one, formulation of the geometric route relies on the potential of mean force (PMF) for physically separating the two binding partners restrained along a straight axis, which must be selected prior to the calculation. However, practical applications indicate that the calculation of the separation PMF along the predefined rectilinear pathway may be suboptimal and slowly convergent. Recognizing that a rectilinear straight separation pathway is generally not representative of how the protein complex physically separates in solution, we put forth a novel theoretical framework for binding free-energy calculations, leaning on the optimal curvilinear minimum free-energy path (MFEP) determined from the string method. The proposed formalism is validated by comparing the results obtained using both rectilinear and curvilinear pathways for a prototypical host-guest complex formed by cucurbit[7]uril (CB[7]) binding benzene, and for the barnase-barstar protein complex. On the basis of multi-microsecond MD calculations, we find that the calculations following the traditional rectilinear pathway and the string-based curvilinear pathway agree quantitatively, but convergence is faster with the latter.

Dissociation Process of a MDM2/p53 Complex Investigated by Parallel Cascade Selection Molecular Dynamics and the Markov State Model

Recently, we efficiently generated dissociation pathways of a protein-ligand complex without applying force bias with parallel cascade selection molecular dynamics (PaCS-MD) and showed that PaCS-MD in combination with the Markov state model (MSM) yielded a binding free energy comparable to experimental values. In this work, we applied the same procedure to a complex of MDM2 protein and the transactivation domain of p53 protein (TAD-p53), the latter of which is known to be very flexible in the unbound state. Using 30 independent MD simulations in PaCS-MD, we successfully generated 25 dissociation pathways of the complex, which showed complete or partial unfolding of the helical region of TAD-p53 during the dissociation process within an average simulation time of 154.8 ± 46.4 ns. The standard binding free energy obtained in combination with one-dimensional-, three-dimensional (3D)- or C_α-MSM was in good agreement with those determined experimentally. Using 3D-MSM based on the center of mass position of TAD-p53 relative to MDM2, the dissociation rate constant was calculated, which was comparable to those measured experimentally. C_α-MSM based on all C_α coordinates of TAD-p53 reproduced the experimentally measured standard binding free energy, and dissociation and association rate constants. We conclude that the combination of PaCS-MD and MSM offers an efficient computational procedure to calculate binding free energies and kinetic rates.

Simulation of spontaneous G protein activation reveals a new intermediate driving GDP unbinding

Activation of heterotrimeric G proteins is a key step in many signaling cascades. However, a complete mechanism for this process, which requires allosteric communication between binding sites that are ~30 Å apart, remains elusive. We construct an atomically detailed model of G protein activation by combining three powerful computational methods: metadynamics, Markov state models (MSMs), and CARDS analysis of correlated motions. We uncover a mechanism that is consistent with a wide variety of structural and biochemical data. Surprisingly, the rate-limiting step for GDP release correlates with tilting rather than translation of the GPCR-binding helix 5. β-Strands 1 - 3 and helix 1 emerge as hubs in the allosteric network that links conformational changes in the GPCR-binding site to disordering of the distal nucleotide-binding site and consequent GDP release. Our approach and insights provide foundations for understanding disease-implicated G protein mutants, illuminating slow events in allosteric networks, and examining unbinding processes with slow off-rates.

A minimization principle for transition paths of maximum flux for collective variables

Considered is the construction of transition paths of conformational changes for proteins and other macromolecules, using methods that do not require the generation of dynamics trajectories. Special attention is given to the use of a reduced set of collective variables for describing such paths. A favored way to define transition paths is to seek channels through the transition state having cross sections with a high reactive flux (density of last hitting points of reactive trajectories). Given here is a formula for reactive flux that is independent of the parameterization of "collective variable space." This formula is needed for the principal curve of the reactive flux (as in the revised finite temperature string method) and for the maximum flux transition (MaxFlux) path. Additionally, a resistance functional is derived for narrow tubes, which when minimized yields a MaxFlux path. A strategy for minimization is outlined in the spirit of the string method. Finally, alternative approaches based on determining trajectories of high probability are considered, and it is observed that they yield paths that depend on the parameterization of collective variable space, except in the case of zero temperature, where such a path coincides with a MaxFlux path.

Computational membrane biophysics: From ion channel interactions with drugs to cellular function

The rapid development of experimental and computational techniques has changed fundamentally our understanding of cellular-membrane transport. The advent of powerful computers and refined force-fields for proteins, ions, and lipids has expanded the applicability of Molecular Dynamics (MD) simulations. A myriad of cellular responses is modulated through the binding of endogenous and exogenous ligands (e.g. neurotransmitters and drugs, respectively) to ion channels. Deciphering the thermodynamics and kinetics of the ligand binding processes to these membrane proteins is at the heart of modern drug development. The ever-increasing computational power has already provided insightful data on the thermodynamics and kinetics of drug-target interactions, free energies of solvation, and partitioning into lipid bilayers for drugs. This review aims to provide a brief summary about modeling approaches to map out crucial binding pathways with intermediate conformations and free-energy surfaces for drug-ion channel binding mechanisms that are responsible for multiple effects on cellular functions. We will discuss post-processing analysis of simulation-generated data, which are then transformed to kinetic models to better understand the molecular underpinning of the experimental observables under the influence of drugs or mutations in ion channels. This review highlights crucial mathematical frameworks and perspectives on bridging different well-established computational techniques to connect the dynamics and timescales from all-atom MD and free energy simulations of ion channels to the physiology of action potentials in cellular models. This article is part of a Special Issue entitled: Biophysics in Canada, edited by Lewis Kay, John Baenziger, Albert Berghuis and Peter Tieleman.

Ab initio sampling of transition paths by conditioned Langevin dynamics

We propose a novel stochastic method to generate Brownian paths conditioned to start at an initial point and end at a given final point during a fixed time t_f under a given potential U(x). These paths are sampled with a probability given by the overdamped Langevin dynamics. We show that these paths can be exactly generated by a local stochastic partial differential equation. This equation cannot be solved in general but we present several approximations that are valid either in the low temperature regime or in the presence of barrier crossing. We show that this method warrants the generation of statistically independent transition paths. It is computationally very efficient. We illustrate the method first on two simple potentials, the two-dimensional Mueller potential and the Mexican hat potential, and then on the multi-dimensional problem of conformational transitions in proteins using the "Mixed Elastic Network Model" as a benchmark.

GENESIS 1.1: A hybrid-parallel molecular dynamics simulator with enhanced sampling algorithms on multiple computational platforms

GENeralized-Ensemble SImulation System (GENESIS) is a software package for molecular dynamics (MD) simulation of biological systems. It is designed to extend limitations in system size and accessible time scale by adopting highly parallelized schemes and enhanced conformational sampling algorithms. In this new version, GENESIS 1.1, new functions and advanced algorithms have been added. The all-atom and coarse-grained potential energy functions used in AMBER and GROMACS packages now become available in addition to CHARMM energy functions. The performance of MD simulations has been greatly improved by further optimization, multiple time-step integration, and hybrid (CPU + GPU) computing. The string method and replica-exchange umbrella sampling with flexible collective variable choice are used for finding the minimum free-energy pathway and obtaining free-energy profiles for conformational changes of a macromolecule. These new features increase the usefulness and power of GENESIS for modeling and simulation in biological research. © 2017 Wiley Periodicals, Inc.

The Activation of c-Src Tyrosine Kinase: Conformational Transition Pathway and Free Energy Landscape

Tyrosine kinases are important cellular signaling allosteric enzymes that regulate cell growth, proliferation, metabolism, differentiation, and migration. Their activity must be tightly controlled, and malfunction can lead to a variety of diseases, particularly cancer. The nonreceptor tyrosine kinase c-Src, a prototypical model system and a representative member of the Src-family, functions as complex multidomain allosteric molecular switches comprising SH2 and SH3 domains modulating the activity of the catalytic domain. The broad picture of self-inhibition of c-Src via the SH2 and SH3 regulatory domains is well characterized from a structural point of view, but a detailed molecular mechanism understanding is nonetheless still lacking. Here, we use advanced computational methods based on all-atom molecular dynamics simulations with explicit solvent to advance our understanding of kinase activation. To elucidate the mechanism of regulation and self-inhibition, we have computed the pathway and the free energy landscapes for the "inactive-to-active" conformational transition of c-Src for different configurations of the SH2 and SH3 domains. Using the isolated c-Src catalytic domain as a baseline for comparison, it is observed that the SH2 and SH3 domains, depending upon their bound orientation, promote either the inactive or active state of the catalytic domain. The regulatory structural information from the SH2-SH3 tandem is allosterically transmitted via the N-terminal linker of the catalytic domain. Analysis of the conformational transition pathways also illustrates the importance of the conserved tryptophan 260 in activating c-Src, and reveals a series of concerted events during the activation process.

Propagation of the Allosteric Modulation Induced by Sodium in the δ-Opioid Receptor

Allosteric sodium in the helix bundle of a G protein-coupled receptor (GPCR) can modulate the receptor activation on the intracellular side. This phenomenon has confounded the GPCR community for decades. In this work, we present a theoretical model that reveals the mechanism of the allosteric modulation induced by sodium in the δ-opioid receptor. We found that the allosteric sodium ion exploits a distinct conformation of the key residue Trp274^6.48 to propagate the modulation to helices 5 and 6, which further transmits along the helices and regulates their positions on the intracellular side. This mechanism is supported by subsequent functional assays. Remarkably, our results highlight the contrast between the allosteric effects towards two GPCR partners, the G protein and β-arrestin, as indicated by the fact that the allosteric modulation initiated by the sodium ion significantly affects the β-arrestin recruitment, while it alters the G protein signaling only moderately. We believe that the mechanism revealed in this work can be used to explain allosteric effects initiated by sodium in other GPCRs since the allosteric sodium is highly conserved across GPCRs.

Minimum action transition paths connecting minima on an energy surface

Dynamics is essential to the biological functions of many bio-molecules, yet our knowledge of dynamics remains fragmented. Experimental techniques for studying bio-molecules either provide high resolution information on static conformations of the molecule or provide low-resolution, ensemble information that does not shed light on single molecule dynamics. In parallel, bio-molecular dynamics occur at time scale that are not yet attainable through detailed simulation methods. These limitations are especially noticeable when studying transition paths. To address this issue, we report in this paper two methods that derive meaningful trajectories for proteins between two of their conformations. The first method, MinActionPath, uses approximations of the potential energy surface for the molecule to derive an analytical solution of the equations of motion related to the concept of minimum action path. The second method, RelaxPath, follows the same principle of minimum action path but implements a more sophisticated potential, including a mixed elastic potential and a collision term to alleviate steric clashes. Using this new potential, the equations of motion cannot be solved analytically. We have introduced a relaxation method for solving those equations. We describe both the theories behind the two methods and their implementations, focusing on the specific techniques we have used that make those implementations amenable to study large molecular systems. We have illustrated the performance of RelaxPath on simple 2D systems. We have also compared MinActionPath and RelaxPath to other methods for generating transition paths on a well suited test set of large proteins, for which the end points of the trajectories as well as an intermediate conformation between those end points are known. We have shown that RelaxPath outperforms those other methods, including MinActionPath, in its ability to generate trajectories that get close to the known intermediates. We have also shown that the structures along the RelaxPath trajectories remain protein-like. Open source versions of the two programs MinActionPath and RelaxPath are available by request.

Effective Riemannian Diffusion Model for Conformational Dynamics of Biomolecular Systems

We present a Riemannian formalism for effective diffusion of biomolecules in collective variable spaces that provides a robust framework for conformational free energy calculation methods. Unlike their Euclidean counterparts, the Riemannian potential of mean force (PMF) and minimum free energy path (MFEP) are invariant under coordinate transformations. The presented formalism can be readily employed to modify the collective variable based enhanced sampling techniques, such as umbrella sampling (US) commonly used in biomolecular simulations, to take into account the role of intrinsic geometry of collective variable space. Although our model is mathematically equivalent to a Euclidean diffusion with a position-dependent diffusion tensor, the Riemannian formulation provides a more convenient framework for free energy calculation methods and path-finding algorithms aimed at characterizing the effective conformational dynamics of biomolecules. A simple three-dimensional toy model and a pentapeptide (met-enkephalin) simulated in an explicit solvent environment are used to illustrate the workings of the formalism and its implementation.

The allosteric switching mechanism in bacteriophage MS2

We use all-atom simulations to elucidate the mechanisms underlying conformational switching and allostery within the coat protein of the bacteriophage MS2. Assembly of most icosahedral virus capsids requires that the capsid protein adopts different conformations at precise locations within the capsid. It has been shown that a 19 nucleotide stem loop (TR) from the MS2 genome acts as an allosteric effector, guiding conformational switching of the coat protein during capsid assembly. Since the principal conformational changes occur far from the TR binding site, it is important to understand the molecular mechanism underlying this allosteric communication. To this end, we use all-atom simulations with explicit water combined with a path sampling technique to sample the MS2 coat protein conformational transition, in the presence and absence of TR-binding. The calculations find that TR binding strongly alters the transition free energy profile, leading to a switch in the favored conformation. We discuss changes in molecular interactions responsible for this shift. We then identify networks of amino acids with correlated motions to reveal the mechanism by which effects of TR binding span the protein. We find that TR binding strongly affects residues located at the 5-fold and quasi-sixfold interfaces in the assembled capsid, suggesting a mechanism by which the TR binding could direct formation of the native capsid geometry. The analysis predicts amino acids whose substitution by mutagenesis could alter populations of the conformational substates or their transition rates.

QM/MM free energy simulations: recent progress and challenges

Due to the higher computational cost relative to pure molecular mechanical (MM) simulations, hybrid quantum mechanical/molecular mechanical (QM/MM) free energy simulations particularly require a careful consideration of balancing computational cost and accuracy. Here we review several recent developments in free energy methods most relevant to QM/MM simulations and discuss several topics motivated by these developments using simple but informative examples that involve processes in water. For chemical reactions, we highlight the value of invoking enhanced sampling technique (e.g., replica-exchange) in umbrella sampling calculations and the value of including collective environmental variables (e.g., hydration level) in metadynamics simulations; we also illustrate the sensitivity of string calculations, especially free energy along the path, to various parameters in the computation. Alchemical free energy simulations with a specific thermodynamic cycle are used to probe the effect of including the first solvation shell into the QM region when computing solvation free energies. For cases where high-level QM/MM potential functions are needed, we analyze two different approaches: the QM/MM-MFEP method of Yang and co-workers and perturbative correction to low-level QM/MM free energy results. For the examples analyzed here, both approaches seem productive although care needs to be exercised when analyzing the perturbative corrections.

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.

Principles and Overview of Sampling Methods for Modeling Macromolecular Structure and Dynamics

Investigation of macromolecular structure and dynamics is fundamental to understanding how macromolecules carry out their functions in the cell. Significant advances have been made toward this end in silico, with a growing number of computational methods proposed yearly to study and simulate various aspects of macromolecular structure and dynamics. This review aims to provide an overview of recent advances, focusing primarily on methods proposed for exploring the structure space of macromolecules in isolation and in assemblies for the purpose of characterizing equilibrium structure and dynamics. In addition to surveying recent applications that showcase current capabilities of computational methods, this review highlights state-of-the-art algorithmic techniques proposed to overcome challenges posed in silico by the disparate spatial and time scales accessed by dynamic macromolecules. This review is not meant to be exhaustive, as such an endeavor is impossible, but rather aims to balance breadth and depth of strategies for modeling macromolecular structure and dynamics for a broad audience of novices and experts.

Extended Phase-Space Methods for Enhanced Sampling in Molecular Simulations: A Review

Molecular Dynamics simulations are a powerful approach to study biomolecular conformational changes or protein-ligand, protein-protein, and protein-DNA/RNA interactions. Straightforward applications, however, are often hampered by incomplete sampling, since in a typical simulated trajectory the system will spend most of its time trapped by high energy barriers in restricted regions of the configuration space. Over the years, several techniques have been designed to overcome this problem and enhance space sampling. Here, we review a class of methods that rely on the idea of extending the set of dynamical variables of the system by adding extra ones associated to functions describing the process under study. In particular, we illustrate the Temperature Accelerated Molecular Dynamics (TAMD), Logarithmic Mean Force Dynamics (LogMFD), and Multiscale Enhanced Sampling (MSES) algorithms. We also discuss combinations with techniques for searching reaction paths. We show the advantages presented by this approach and how it allows to quickly sample important regions of the free-energy landscape via automatic exploration.

Technical advances in molecular simulation since the 1980s

This review describes how the theory and practice of molecular simulation have evolved since the beginning of the 1980s when the author started his career in this field. The account is of necessity brief and subjective and highlights the changes that the author considers have had significant impact on his research and mode of working.

Enzymatic minimum free energy path calculations using swarms of trajectories

The development of approaches for simulating rare events in complex molecular systems is a central concern in chemical physics. In recent work, Roux and co-workers proposed a novel, swarms of trajectories (SoT) method for determining the transition paths of such events. It consists of the dynamical refinement on the system's free energy surface of a putative transition path that is parametrized in terms of a set of collective variables (CVs) that are identified as being important for the transition. In this work, we have implemented the SoT method and used it to investigate the catalytic mechanisms of two enzymatic reactions using hybrid QM/MM potentials. Our aim has been to test the performance of SoT for enzyme systems and to devise robust simulation protocols that can be employed in future studies of this type. We identify the conditions under which converged results can be obtained using inertial and Brownian dynamical evolutions of the CVs, show that the inclusion of several CVs can give significant additional insight into the mechanisms of the reactions, and show that the use of minimum energy paths as starting guesses can greatly accelerate path refinement.