Accurate Multiple Time Step in Biased Molecular Simulations

Understanding Glucagon Aggregation: In Silico Insights and Experimental Validation

Peptide aggregation poses a significant challenge in biopharmaceutical development and neurodegenerative diseases. This study combines computational simulations and experimental validation to uncover the underlying mechanisms and countermeasures for the aggregation of glucagon, a peptide with a high tendency to aggregate. In silico simulations demonstrate that lactose and 2-hydroxypropyl-β-cyclodextrin (2-HPβCD) influence glucagon aggregation differently: lactose stabilizes glucagon by increasing the α-helical content, while 2-HPβCD disrupts protein-protein interactions. According to the simulations, 2-HPβCD is particularly effective at preserving the monomeric form of glucagon. Experimental validation with microfluidic modulation spectroscopy (MMS) confirms these findings, showing that glucagon in the presence of 2-HPβCD remains structurally stable, supporting the antiaggregation effect of this excipient. This research provides essential insights into glucagon aggregation obtained through a new powerful tool for monitoring the critical properties of peptide aggregation, suggesting new strategies for addressing this challenge in therapeutic peptide development.

Accurate model and ensemble refinement using cryo-electron microscopy maps and Bayesian inference

Converting cryo-electron microscopy (cryo-EM) data into high-quality structural models is a challenging problem of outstanding importance. Current refinement methods often generate unbalanced models in which physico-chemical quality is sacrificed for excellent fit to the data. Furthermore, these techniques struggle to represent the conformational heterogeneity averaged out in low-resolution regions of density maps. Here we introduce EMMIVox, a Bayesian inference approach to determine single-structure models as well as structural ensembles from cryo-EM maps. EMMIVox automatically balances experimental information with accurate physico-chemical models of the system and the surrounding environment, including waters, lipids, and ions. Explicit treatment of data correlation and noise as well as inference of accurate B-factors enable determination of structural models and ensembles with both excellent fit to the data and high stereochemical quality, thus outperforming state-of-the-art refinement techniques. EMMIVox represents a flexible approach to determine high-quality structural models that will contribute to advancing our understanding of the molecular mechanisms underlying biological functions.

Collective Variable-Based Enhanced Sampling: From Human Learning to Machine Learning

Enhanced-sampling algorithms relying on collective variables (CVs) are extensively employed to study complex (bio)chemical processes that are not amenable to brute-force molecular simulations. The selection of appropriate CVs characterizing the slow movement modes is of paramount importance for reliable and efficient enhanced-sampling simulations. In this Perspective, we first review the application and limitations of CVs obtained from chemical and geometrical intuition. We also introduce path-sampling algorithms, which can identify path-like CVs in a high-dimensional free-energy space. Machine-learning algorithms offer a viable approach to finding suitable CVs by analyzing trajectories from preliminary simulations. We discuss both the performance of machine-learning-derived CVs in enhanced-sampling simulations of experimental models and the challenges involved in applying these CVs to realistic, complex molecular assemblies. Moreover, we provide a prospective view of the potential advancements of machine-learning algorithms for the development of CVs in the field of enhanced-sampling simulations.

Vendi sampling for molecular simulations: Diversity as a force for faster convergence and better exploration

Molecular dynamics (MD) is the method of choice for understanding the structure, function, and interactions of molecules. However, MD simulations are limited by the strong metastability of many molecules, which traps them in a single conformation basin for an extended amount of time. Enhanced sampling techniques, such as metadynamics and replica exchange, have been developed to overcome this limitation and accelerate the exploration of complex free energy landscapes. In this paper, we propose Vendi Sampling, a replica-based algorithm for increasing the efficiency and efficacy of the exploration of molecular conformation spaces. In Vendi sampling, replicas are simulated in parallel and coupled via a global statistical measure, the Vendi Score, to enhance diversity. Vendi sampling allows for the recovery of unbiased sampling statistics and dramatically improves sampling efficiency. We demonstrate the effectiveness of Vendi sampling in improving molecular dynamics simulations by showing significant improvements in coverage and mixing between metastable states and convergence of free energy estimates for four common benchmarks, including Alanine Dipeptide and Chignolin.

Reaction Coordinates for Conformational Transitions Using Linear Discriminant Analysis on Positions

In this work, we demonstrate that Linear Discriminant Analysis (LDA) applied to atomic positions in two different states of a biomolecule produces a good reaction coordinate between those two states. Atomic coordinates of a macromolecule are a direct representation of a macromolecular configuration, and yet, they are not used in enhanced sampling studies due to a lack of rotational and translational invariance. We resolve this issue using the technique of our prior work, whereby a molecular configuration is considered a member of an equivalence class in size-and-shape space, which is the set of all configurations that can be translated and rotated to a single point within a reference multivariate Gaussian distribution characterizing a single molecular state. The reaction coordinates produced by LDA applied to positions are shown to be good reaction coordinates both in terms of characterizing the transition between two states of a system within a long molecular dynamics (MD) simulation and also ones that allow us to readily produce free energy estimates along that reaction coordinate using enhanced sampling MD techniques.

Gentle and fast all-atom model refinement to cryo-EM densities via a maximum likelihood approach

Better detectors and automated data collection have generated a flood of high-resolution cryo-EM maps, which in turn has renewed interest in improving methods for determining structure models corresponding to these maps. However, automatically fitting atoms to densities becomes difficult as their resolution increases and the refinement potential has a vast number of local minima. In practice, the problem becomes even more complex when one also wants to achieve a balance between a good fit of atom positions to the map, while also establishing good stereochemistry or allowing protein secondary structure to change during fitting. Here, we present a solution to this challenge using a maximum likelihood approach by formulating the problem as identifying the structure most likely to have produced the observed density map. This allows us to derive new types of smooth refinement potential-based on relative entropy-in combination with a novel adaptive force scaling algorithm to allow balancing of force-field and density-based potentials. In a low-noise scenario, as expected from modern cryo-EM data, the relative-entropy based refinement potential outperforms alternatives, and the adaptive force scaling appears to aid all existing refinement potentials. The method is available as a component in the GROMACS molecular simulation toolkit.

Improving Speed and Affordability without Compromising Accuracy: Standard Binding Free-Energy Calculations Using an Enhanced Sampling Algorithm, Multiple-Time Stepping, and Hydrogen Mass Repartitioning

Accurate evaluation of protein-ligand binding free energies in silico is of paramount importance for understanding the mechanisms of biological regulation and providing a theoretical basis for drug design and discovery. Based on a series of atomistic molecular dynamics simulations in an explicit solvent, using well-tempered metadynamics extended adaptive biasing force (WTM-eABF) as an enhanced sampling algorithm, the so-called "geometrical route" offers a rigorous theoretical framework for binding affinity calculations that match experimental values. However, although robust, this strategy remains expensive, requiring substantial computational time to achieve convergence of the simulations. Improving the efficiency of the geometrical route, while preserving its reliability through improved ergodic sampling, is, therefore, highly desirable. In this contribution, having identified the computational bottleneck of the geometrical route, to accelerate the calculations we combine (i) a longer time step for the integration of the equations of motion with hydrogen-mass repartitioning (HMR), and (ii) multiple time-stepping (MTS) for collective-variable and biasing-force evaluation. Altogether, we performed 50 independent WTM-eABF simulations in triplicate for the "physical" separation of the Abl kinase-SH3 domain:p41 complex, following different HMR and MTS schemes, while tuning, in distinct protocols, the parameters of the enhanced-sampling algorithm. To demonstrate the consistency and reliability of the results obtained with the best-performing setups, we carried out quintuple simulations. Furthermore, we demonstrated the transferability of our method to other complexes by triplicating a 200 ns separation simulation of nine chosen protocols for the MDM2-p53:NVP-CGM097 complex. [Holzer et al. J. Med. Chem. 2015, 58, 6348-6358.] Our results, based on an aggregate simulation time of 14.4 μs, allowed an optimal set of parameters to be identified, able to accelerate convergence by a factor of three without any noticeable loss of accuracy.

Extending and validating bubble nucleation rate predictions in a Lennard-Jones fluid with enhanced sampling methods and transition state theory

We calculate bubble nucleation rates in a Lennard-Jones fluid through explicit molecular dynamics simulations. Our approach—based on a recent free energy method (dubbed reweighted Jarzynski sampling), transition state theory, and a simple recrossing correction—allows us to probe a fairly wide range of rates in several superheated and cavitation regimes in a consistent manner. Rate predictions from this approach bridge disparate independent literature studies on the same model system. As such, we find that rate predictions based on classical nucleation theory, direct brute force molecular dynamics simulations, and seeding are consistent with our approach and one another. Published rates derived from forward flux sampling simulations are, however, found to be outliers. This study serves two purposes: First, we validate the reliability of common modeling techniques and extrapolation approaches on a paradigmatic problem in materials science and chemical physics. Second, we further test our highly generic recipe for rate calculations, and establish its applicability to nucleation processes.

Using Constrained Density Functional Theory to Track Proton Transfers and to Sample Their Associated Free Energy Surface

Ab initio molecular dynamics (AIMD) and quantum mechanics/molecular mechanics (QM/MM) methods are powerful tools for studying proton solvation, transfer, and transport processes in various environments. However, due to the high computational cost of such methods, achieving sufficient sampling of rare events involving excess proton motion-especially when Grotthuss proton shuttling is involved-usually requires enhanced free energy sampling methods to obtain informative results. Moreover, an appropriate collective variable (CV) that describes the effective position of the net positive charge defect associated with an excess proton is essential both for tracking the trajectory of the defect and for the free energy sampling of the processes associated with the resulting proton transfer and transport. In this work, such a CV is derived from first principles using constrained density functional theory (CDFT). This CV is applicable to a broad array of proton transport and transfer processes as studied via AIMD and QM/MM simulations.

Reweighting of molecular simulations with explicit-solvent SAXS restraints elucidates ion-dependent RNA ensembles

Small-angle X-ray scattering (SAXS) experiments are increasingly used to probe RNA structure. A number of forward models that relate measured SAXS intensities and structural features, and that are suitable to model either explicit-solvent effects or solute dynamics, have been proposed in the past years. Here, we introduce an approach that integrates atomistic molecular dynamics simulations and SAXS experiments to reconstruct RNA structural ensembles while simultaneously accounting for both RNA conformational dynamics and explicit-solvent effects. Our protocol exploits SAXS pure-solute forward models and enhanced sampling methods to sample an heterogenous ensemble of structures, with no information towards the experiments provided on-the-fly. The generated structural ensemble is then reweighted through the maximum entropy principle so as to match reference SAXS experimental data at multiple ionic conditions. Importantly, accurate explicit-solvent forward models are used at this reweighting stage. We apply this framework to the GTPase-associated center, a relevant RNA molecule involved in protein translation, in order to elucidate its ion-dependent conformational ensembles. We show that (a) both solvent and dynamics are crucial to reproduce experimental SAXS data and (b) the resulting dynamical ensembles contain an ion-dependent fraction of extended structures.

Determination of Protein Structural Ensembles by Hybrid-Resolution SAXS Restrained Molecular Dynamics

Small-angle X-ray scattering (SAXS) experiments provide low-resolution but valuable information about the dynamics of biomolecular systems, which could be ideally integrated into molecular dynamics (MD) simulations to accurately determine conformational ensembles of flexible proteins. The applicability of this strategy is hampered by the high computational cost required to calculate scattering intensities from three-dimensional structures. We previously presented a hybrid resolution method that makes atomistic SAXS-restrained MD simulation feasible by adopting a coarse-grained approach to efficiently back-calculate scattering intensities; here, we extend this technique, applying it in the framework of metainference with the aim to investigate the dynamical behavior of flexible biomolecules. The efficacy of the method is assessed on the K63-diubiquitin, showing that the inclusion of SAXS restraints is effective in generating a reliable conformational ensemble, improving the agreement with independent experimental data.

Anncolvar: Approximation of Complex Collective Variables by Artificial Neural Networks for Analysis and Biasing of Molecular Simulations

The state of a molecular system can be described in terms of collective variables. These low-dimensional descriptors of molecular structure can be used to monitor the state of the simulation, to calculate free energy profiles or to accelerate rare events by a bias potential or a bias force. Frequent calculation of some complex collective variables may slow down the simulation or analysis of trajectories. Moreover, many collective variables cannot be explicitly calculated for newly sampled structures. In order to address this problem, we developed a new package called anncolvar. This package makes it possible to build and train an artificial neural network model that approximates a collective variable. It can be used to generate an input for the open-source enhanced sampling simulation PLUMED package, so the collective variable can be monitored and biased by methods available in this program. The computational efficiency and the accuracy of anncolvar are demonstrated on selected molecular systems (cyclooctane derivative, Trp-cage miniprotein) and selected collective variables (Isomap, molecular surface area).

Simultaneous Determination of Protein Structure and Dynamics Using Cryo-Electron Microscopy

Cryo-electron microscopy is rapidly emerging as a powerful technique to determine the structures of complex macromolecular systems elusive to other techniques. Because many of these systems are highly dynamical, characterizing their movements is also a crucial step to unravel their biological functions. To achieve this goal, we report an integrative modeling approach to simultaneously determine structure and dynamics of macromolecular systems from cryo-electron microscopy density maps. By quantifying the level of noise in the data and dealing with their ensemble-averaged nature, this approach enables the integration of multiple sources of information to model ensembles of structures and infer their populations. We illustrate the method by characterizing structure and dynamics of the integral membrane receptor STRA6, thus providing insights into the mechanisms by which it interacts with retinol binding protein and translocates retinol across the membrane.

Martini bead form factors for nucleic acids and their application in the refinement of protein–nucleic acid complexes against SAXS data

The use of small-angle X-ray scattering (SAXS) in combination with molecular dynamics simulation is hampered by its heavy computational cost. The calculation of SAXS from atomic structures can be speeded up by using a coarse-grain representation of the structure. Following the work of Niebling, Björling & Westenhoff [J. Appl. Cryst. (2014), 47, 1190–1198], the Martini bead form factors for nucleic acids have been derived and then implemented, together with those previously determined for proteins, in the publicly available PLUMED library. A hybrid multi-resolution strategy has also been implemented to perform SAXS restrained simulations at atomic resolution by calculating the virtual positions of the Martini beads on the fly and using them for the calculation of SAXS. The accuracy and efficiency of the method are demonstrated by refining the structure of two protein–nucleic acid complexes. Instrumental for this result is the use of metainference, which allows the consideration and alleviation of the approximations at play in the present SAXS calculations.

Analyzing and Biasing Simulations with PLUMED

This chapter discusses how the PLUMED plugin for molecular dynamics can be used to analyze and bias molecular dynamics trajectories. The chapter begins by introducing the notion of a collective variable and by then explaining how the free energy can be computed as a function of one or more collective variables. A number of practical issues mostly around periodic boundary conditions that arise when these types of calculations are performed using PLUMED are then discussed. Later parts of the chapter discuss how PLUMED can be used to perform enhanced sampling simulations that introduce simulation biases or multiple replicas of the system and Monte Carlo exchanges between these replicas. This section is then followed by a discussion on how free-energy surfaces and associated error bars can be extracted from such simulations by using weighted histogram and block averaging techniques.

Conformational ensembles of RNA oligonucleotides from integrating NMR and molecular simulations

RNA molecules are key players in numerous cellular processes and are characterized by a complex relationship between structure, dynamics, and function. Despite their apparent simplicity, RNA oligonucleotides are very flexible molecules, and understanding their internal dynamics is particularly challenging using experimental data alone. We show how to reconstruct the conformational ensemble of four RNA tetranucleotides by combining atomistic molecular dynamics simulations with nuclear magnetic resonance spectroscopy data. The goal is achieved by reweighting simulations using a maximum entropy/Bayesian approach. In this way, we overcome problems of current simulation methods, as well as in interpreting ensemble- and time-averaged experimental data. We determine the populations of different conformational states by considering several nuclear magnetic resonance parameters and point toward properties that are not captured by state-of-the-art molecular force fields. Although our approach is applied on a set of model systems, it is fully general and may be used to study the conformational dynamics of flexible biomolecules and to detect inaccuracies in molecular dynamics force fields.

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.

Coarse-Grained Directed Simulation

Many free-energy sampling and quantum mechanics/molecular mechanics (QM/MM) computations on protein complexes have been performed where, by necessity, a single component is studied isolated in solution while its overall configuration is kept in the complex-like state by either rigid restraints or harmonic constraints. A drawback in these studies is that the system's native fluctuations are lost, both due to the change of environment and the imposition of the extra potential. Yet, we know that having both accurate structure and fluctuations is likely crucial to achieving correct simulation estimates for the subsystem within its native larger protein complex context. In this work, we provide a new approach to this problem by drawing on ideas developed to incorporate experimental information into a molecular simulation by relative entropy minimization to a target system. We show that by using linear biases on coarse-grained (CG) observables (such as distances or angles between large subdomains within a protein), we can maintain the protein in a particular target conformation while also preserving the correct equilibrium fluctuations of the subsystem within its larger biomolecular complex. As an application, we demonstrate this algorithm by training a bias that causes an actin monomer (and trimer) in solution to sample the same average structure and fluctuations as if it were embedded within a much larger actin filament. Additionally, we have developed a number of algorithmic improvements that accelerate convergence of the on-the-fly relative entropy minimization algorithms for this type of application. Finally, we have contributed these methods to the PLUMED open source free energy sampling software library.

Exploring Valleys without Climbing Every Peak: More Efficient and Forgiving Metabasin Metadynamics via Robust On-the-Fly Bias Domain Restriction

Metadynamics is an enhanced sampling method designed to flatten free energy surfaces uniformly. However, the highest-energy regions are often irrelevant to study and dangerous to explore because systems often change irreversibly in unforeseen ways in response to driving forces in these regions, spoiling the sampling. Introducing an on-the-fly domain restriction allows metadynamics to flatten only up to a specified energy level and no further, improving efficiency and safety while decreasing the pressure on practitioners to design collective variables that are robust to otherwise irrelevant high energy driving. This paper describes a new method that achieves this using sequential on-the-fly estimation of energy wells and redefinition of the metadynamics hill shape, termed metabasin metadynamics. The energy level may be defined a priori or relative to unknown barrier energies estimated on-the-fly. Altering only the hill ensures that the method is compatible with many other advances in metadynamics methodology. The hill shape has a natural interpretation in terms of multiscale dynamics, and the computational overhead in simulation is minimal when studying systems of any reasonable size, for instance proteins or other macromolecules. Three example applications show that the formula is accurate and robust to complex dynamics, making metadynamics significantly more forgiving with respect to CV quality and thus more feasible to apply to the most challenging biomolecular systems.

ATP dependent NS3 helicase interaction with RNA: insights from molecular simulations

Non-structural protein 3 (NS3) helicase from hepatitis C virus is an enzyme that unwinds and translocates along nucleic acids with an ATP-dependent mechanism and has a key role in the replication of the viral RNA. An inchworm-like mechanism for translocation has been proposed based on crystal structures and single molecule experiments. We here perform atomistic molecular dynamics in explicit solvent on the microsecond time scale of the available experimental structures. We also construct and simulate putative intermediates for the translocation process, and we perform non-equilibrium targeted simulations to estimate their relative stability. For each of the simulated structures we carefully characterize the available conformational space, the ligand binding pocket, and the RNA binding cleft. The analysis of the hydrogen bond network and of the non-equilibrium trajectories indicates an ATP-dependent stabilization of one of the protein conformers. Additionally, enthalpy calculations suggest that entropic effects might be crucial for the stabilization of the experimentally observed structures.

Exploring LacI-DNA dynamics by multiscale simulations using the SIRAH force field

The lac repressor protein (LacI) together with its target regulatory sequence are a common model for studying DNA looping and its implications on transcriptional control in bacteria. Owing to the molecular size of this system, standard all-atom (AA) simulations are prohibitive for achieving relevant biological time scales. As an alternative, multiscale models, which combine AA descriptions at particular regions with coarse-grained (CG) representations of the remaining components, were used to address this computational challenge while preserving the relevant details of the system. In this work, we implement a new multiscale approach based on the SIRAH force field to gain deeper insights into the dynamics of the LacI-DNA system. Our methodology allows for a dual resolution treatment of the solute and solvent, explicitly representing the protein, DNA, and solvent environment without compromising the AA region. Starting from the P1 loop configuration in an undertwisted conformation, we were able to observe the transition to the more stable overtwisted state. Additionally, a detailed characterization of the conformational space sampled by the DNA loop was done. In agreement with experimental and theoretical evidence, we observed the transient formation of kinks at the loop, which were stabilized by the presence of counterions at the minor groove. We also show that the loop's intrinsic flexibility can account for reported FRET measurements and bent conformations required to bind the CAP transcription factor.

Enhanced Conformational Sampling Using Replica Exchange with Collective-Variable Tempering

The computational study of conformational transitions in RNA and proteins with atomistic molecular dynamics often requires suitable enhanced sampling techniques. We here introduce a novel method where concurrent metadynamics are integrated in a Hamiltonian replica-exchange scheme. The ladder of replicas is built with different strengths of the bias potential exploiting the tunability of well-tempered metadynamics. Using this method, free-energy barriers of individual collective variables are significantly reduced compared with simple force-field scaling. The introduced methodology is flexible and allows adaptive bias potentials to be self-consistently constructed for a large number of simple collective variables, such as distances and dihedral angles. The method is tested on alanine dipeptide and applied to the difficult problem of conformational sampling in a tetranucleotide.