Accurate Free Energy Profiles Research Articles

Efficient machine learning approach for accurate free-energy profiles and kinetic rates.

The computational exploration of reactive processes is challenging due to the requirement of thorough sampling across the free energy landscape using accurate ab initio methods. To address these constraints, machine learning potentials are employed, yet their training for this kind of problem is still a laborious and tedious task. In this study, we present an efficient approach to train these potentials by cleverly using a single batch of unbiased trajectories that avoid the pitfalls of trajectories artificially biased along a suboptimal collective variable. This strategy, when integrated with current enhanced sampling techniques, allows to obtain free energy profiles and kinetic rates of ab initio quality, yet dramatically reducing the computational cost.

OGRe: Optimal Grid Refinement Protocol for Accurate Free Energy Surfaces and Its Application in Proton Hopping in Zeolites and 2D COF Stacking.

While free energy surfaces are the crux of our understanding of many chemical and biological processes, their accuracy is generally unknown. Moreover, many developments to improve their accuracy are often complicated, limiting their general use. Luckily, several tools and guidelines are already in place to identify these shortcomings, but they are typically lacking in flexibility or fail to systematically determine how to improve the accuracy of the free energy calculation. To overcome these limitations, this work introduces OGRe, a Python package for optimal grid refinement in an arbitrary number of dimensions. OGRe is based on three metrics that gauge the confinement, consistency, and overlap of each simulation in a series of umbrella sampling (US) simulations, an enhanced sampling technique ubiquitously adopted to construct free energy surfaces for hindered processes. As these three metrics are fundamentally linked to the accuracy of the weighted histogram analysis method adopted to generate free energy surfaces from US simulations, they facilitate the systematic construction of accurate free energy profiles, where each metric is driven by a specific umbrella parameter. This allows for the derivation of a consistent and optimal collection of umbrellas for each simulation, largely independent of the initial values, thereby dramatically increasing the ease-of-use toward accurate free energy surfaces. As such, OGRe is particularly suited to determine complex free energy surfaces with large activation barriers and shallow minima, which underpin many physical and chemical transformations and hence to further our fundamental understanding of these processes.

Combining Machine Learning Approaches and Accurate Ab Initio Enhanced Sampling Methods for Prebiotic Chemical Reactions in Solution.

The study of the thermodynamics, kinetics, and microscopic mechanisms of chemical reactions in solution requires the use of advanced free-energy methods for predictions to be quantitative. This task is however a formidable one for atomistic simulation methods, as the cost of quantum-based ab initio approaches, to obtain statistically meaningful samplings of the relevant chemical spaces and networks, becomes exceedingly heavy. In this work, we critically assess the optimal structure and minimal size of an ab initio training set able to lead to accurate free-energy profiles sampled with neural network potentials. The results allow one to propose an ab initio protocol where the ad hoc inclusion of a machine-learning (ML)-based task can significantly increase the computational efficiency, while keeping the ab initio accuracy and, at the same time, avoiding some of the notorious extrapolation risks in typical atomistic ML approaches. We focus on two representative, and computationally challenging, reaction steps of the classic Strecker-cyanohydrin mechanism for glycine synthesis in water solution, where the main precursors are formaldehyde and hydrogen cyanide. We demonstrate that indistinguishable ab initio quality results are obtained, thanks to the ML subprotocol, at about 1 order of magnitude less of computational load.

Exploration of Free Energy Surfaces Across a Membrane Channel Using Metadynamics and Umbrella Sampling.

To reach their site of action, it is essential for antibiotic molecules to cross the bacterial outer membrane. The progress of enhanced sampling techniques in molecular dynamics simulations enables us to understand these translocations at an atomic level. To this end, calculations of free energy surfaces for these permeation processes are of key importance. Herein, we investigate the translocation of a variety of anionic solutes through the outer membrane pore OprO of the Gram-negative bacterium Pseudomonas aeruginosa using the metadynamics and umbrella sampling techniques at the all-atom level. Free energy calculations have been performed employing these two distinct methods in order to illustrate the difference in computed free energies, if any. The investigated solutes range from a single atomic chloride ion over a multiatomic monophosphate ion to a more bulky fosmidomycin antibiotic. The role of complexity of the permeating solutes in estimating accurate free energy profiles is demonstrated by performing extensive convergence analysis. For simple monatomic ions, good agreement between the well-tempered metadynamics and the umbrella sampling approaches is achieved, while for the permeation of the monophosphate ion differences start to appear. In the case of larger molecules such as fosmidomycin it is a tough challenge to achieve converged free energy profiles. This issue is mainly due to neglecting orthogonal degrees of freedom during the free energy calculations. Nevertheless, the freely driven metadynamics approach leads to clearly advantageous results. Additionally, atomistic insights of the translocation mechanisms of all three solutes are discussed.

Targeted Molecular Dynamics Calculations of Free Energy Profiles Using a Nonequilibrium Friction Correction.

As standard unbiased molecular dynamics (MD) simulations become impractical for sampling rare events, "targeted MD" employs a moving distance constraint to enforce rare transitions along some reaction coordinate x. To calculate free energy profiles from these nonequilibrium simulations via Δ G( x) = W( x) - Wdiss( x), apart from the (readily obtained) work W( x) performed on the system, the dissipated work Wdiss( x) is also required. By employing a second-order cumulant expansion of Jarzynski's equality combined with an analysis within Langevin theory, the dissipated work can be expressed via a nonequilibrium friction coefficient ΓNEQ( x) that may be calculated on-the-fly from constraint force fluctuations. Adopting the ion dissociation of NaCl in water as a test system, this friction correction is shown to result in accurate free energy profiles, even for a modest number of simulations and at high constraint velocities. As a bonus, the analysis of ΓNEQ( x) may yield valuable insight into the microscopic mechanism of friction.

Protocol for Identifying Accurate Collective Variables in Enhanced Molecular Dynamics Simulations for the Description of Structural Transformations in Flexible Metal-Organic Frameworks.

Various kinds of flexibility have been observed in metal–organic frameworks, which may originate from the topology of the material or the presence of flexible ligands. The construction of free energy profiles describing the full dynamical behavior along the phase transition path is challenging since it is not trivial to identify collective variables able to identify all metastable states along the reaction path. In this work, a systematic three-step protocol to uniquely identify the dominant order parameters for structural transformations in flexible metal–organic frameworks and subsequently construct accurate free energy profiles is presented. Methodologically, this protocol is rooted in the time-structure based independent component analysis (tICA), a well-established statistical modeling technique embedded in the Markov state model methodology and often employed to study protein folding, that allows for the identification of the slowest order parameters characterizing the structural transformation. To ensure an unbiased and systematic identification of these order parameters, the tICA decomposition is performed based on information from a prior replica exchange (RE) simulation, as this technique enhances the sampling along all degrees of freedom of the system simultaneously. From this simulation, the tICA procedure extracts the order parameters—often structural parameters—that characterize the slowest transformations in the material. Subsequently, these order parameters are adopted in traditional enhanced sampling methods such as umbrella sampling, thermodynamic integration, and variationally enhanced sampling to construct accurate free energy profiles capturing the flexibility in these nanoporous materials. In this work, the applicability of this tICA-RE protocol is demonstrated by determining the slowest order parameters in both MIL-53(Al) and CAU-13, which exhibit a strongly different type of flexibility. The obtained free energy profiles as a function of this extracted order parameter are furthermore compared to the profiles obtained when adopting less-suited collective variables, indicating the importance of systematically selecting the relevant order parameters to construct accurate free energy profiles for flexible metal–organic frameworks, which is in correspondence with experimental findings. The method succeeds in mapping the full free energy surface in terms of appropriate collective variables for MOFs exhibiting linker flexibility. For CAU-13, we show the decreased stability of the closed pore phase by systematically adding adsorbed xylene molecules in the framework.

Steered Molecular Dynamics Methods Applied to Enzyme Mechanism and Energetics.

One of the main goals of chemistry is to understand the underlying principles of chemical reactions, in terms of both its reaction mechanism and the thermodynamics that govern it. Using hybrid quantum mechanics/molecular mechanics (QM/MM)-based methods in combination with a biased sampling scheme, it is possible to simulate chemical reactions occurring inside complex environments such as an enzyme, or aqueous solution, and determining the corresponding free energy profile, which provides direct comparison with experimental determined kinetic and equilibrium parameters. Among the most promising biasing schemes is the multiple steered molecular dynamics method, which in combination with Jarzynski's Relationship (JR) allows obtaining the equilibrium free energy profile, from a finite set of nonequilibrium reactive trajectories by exponentially averaging the individual work profiles. However, obtaining statistically converged and accurate profiles is far from easy and may result in increased computational cost if the selected steering speed and number of trajectories are inappropriately chosen. In this small review, using the extensively studied chorismate to prephenate conversion reaction, we first present a systematic study of how key parameters such as pulling speed, number of trajectories, and reaction progress are related to the resulting work distributions and in turn the accuracy of the free energy obtained with JR. Second, and in the context of QM/MM strategies, we introduce the Hybrid Differential Relaxation Algorithm, and show how it allows obtaining more accurate free energy profiles using faster pulling speeds and smaller number of trajectories and thus smaller computational cost.

Improving Efficiency in SMD Simulations Through a Hybrid Differential Relaxation Algorithm.

The fundamental object for studying a (bio)chemical reaction obtained from simulations is the free energy profile, which can be directly related to experimentally determined properties. Although quite accurate hybrid quantum (DFT based)-classical methods are available, achieving statistically accurate and well converged results at a moderate computational cost is still an open challenge. Here, we present and thoroughly test a hybrid differential relaxation algorithm (HyDRA), which allows faster equilibration of the classical environment during the nonequilibrium steering of a (bio)chemical reaction. We show and discuss why (in the context of Jarzynski's Relationship) this method allows obtaining accurate free energy profiles with smaller number of independent trajectories and/or faster pulling speeds, thus reducing the overall computational cost. Moreover, due to the availability and straightforward implementation of the method, we expect that it will foster theoretical studies of key enzymatic processes.

Improved Reweighting of Accelerated Molecular Dynamics Simulations for Free Energy Calculation

Accelerated molecular dynamics (aMD) simulations greatly improve the efficiency of conventional molecular dynamics (cMD) for sampling biomolecular conformations, but they require proper reweighting for free energy calculation. In this work, we systematically compare the accuracy of different reweighting algorithms including the exponential average, Maclaurin series, and cumulant expansion on three model systems: alanine dipeptide, chignolin, and Trp-cage. Exponential average reweighting can recover the original free energy profiles easily only when the distribution of the boost potential is narrow (e.g., the range ≤20kBT) as found in dihedral-boost aMD simulation of alanine dipeptide. In dual-boost aMD simulations of the studied systems, exponential average generally leads to high energetic fluctuations, largely due to the fact that the Boltzmann reweighting factors are dominated by a very few high boost potential frames. In comparison, reweighting based on Maclaurin series expansion (equivalent to cumulant expansion on the first order) greatly suppresses the energetic noise but often gives incorrect energy minimum positions and significant errors at the energy barriers (∼2–3kBT). Finally, reweighting using cumulant expansion to the second order is able to recover the most accurate free energy profiles within statistical errors of ∼kBT, particularly when the distribution of the boost potential exhibits low anharmonicity (i.e., near-Gaussian distribution), and should be of wide applicability. A toolkit of Python scripts for aMD reweighting “PyReweighting” is distributed free of charge at http://mccammon.ucsd.edu/computing/amdReweighting/.

Molecular Simulation of the Adsorption of Amino Acid Sidechain Analogs to the TiO2 (100) Surface

Titanium dioxide (TiO2) in the form of nanoparticles (NPs) are found in many common food and health products. TiO2-NPs have been shown to attract biological material, such as proteins and lipids, which form a protective on the NP surface. This coating is believed to be a key element in determining the toxicity behavior of the nanoparticle, and its potential hazard to human health.Large-scale computer simulations promise to provide atomistically detailed information on the molecular basis of NP corona formation, but are limited by the absence of suitable interaction parameters for inorganic nanomaterials (such as TiO2) and biomolecules. We have developed forcefield parameters for simulations of TiO2--biomolecular interfaces to alleviate this situation. The new forcefield is based on high-quality experimental data and is directly compatible with the widely used AMBER parameters for biomolecules.Side chain analogs (SCA) are small molecules designed to mimic the naturally occuring amino acids, and serve as a model system for studies of NP and protein/peptide interactions. We have performed systematic molecular dynamics simulations with our new forcefield parameters to study the adsorption of SCAs to the TiO2 (100) surface. Advanced sampling techniques are employed to determine accurate free energy profiles of SCA binding to TiO2, which are used to map the affinities of individual peptide segments to TiO2. The impact for general protein-surface adsorption are discussed, also in light of recent experimental data.

Topologically Invariant Reaction Coordinates for Simulating Multistate Chemical Reactions

Evaluating free energy profiles of chemical reactions in complex environments such as solvents and enzymes requires extensive sampling, which is usually performed by potential of mean force (PMF) techniques. The reliability of the sampling depends not only on the applied PMF method but also the reaction coordinate space within the dynamics is biased. In contrast to simple geometrical collective variables that depend only on the positions of the atomic coordinates of the reactants, the E(gap) reaction coordinate (the energy difference obtained by evaluating a suitable force field using reactant and product state topologies) has the unique property that it is able to take environmental effects into account leading to better convergence, a more faithful description of the transition state ensemble and therefore more accurate free energy profiles. However, E(gap) requires predefined topologies and is therefore inapplicable for multistate reactions, in which the barrier between the chemically equivalent topologies is comparable to the reaction activation barrier, because undesired "side reactions" occur. In this article, we introduce a new energy-based collective variable by generalizing the E(gap) reaction coordinate such that it becomes invariant to equivalent topologies and show that it yields more well behaved free energy profiles than simpler geometrical reaction coordinates.

Equilibrium Distribution from Distributed Computing (Simulations of Protein Folding)

Multiple independent molecular dynamics (MD) simulations are often carried out starting from a single protein structure or a set of conformations that do not correspond to a thermodynamic ensemble. Therefore, a significant statistical bias is usually present in the Markov state model generated by simply combining the whole MD sampling into a network whose nodes and links are clusters of snapshots and transitions between them, respectively. Here, we introduce a depth-first search algorithm to extract from the whole conformation space network the largest ergodic component, i.e., the subset of nodes of the network whose transition matrix corresponds to an ergodic Markov chain. For multiple short MD simulations of a globular protein (as in distributed computing), the steady state, i.e., stationary distribution determined using the largest ergodic component, yields more accurate free energy profiles and mean first passage times than the original network or the ergodic network obtained by imposing detailed balance by means of symmetrization of the transition counts.

Minimum free energy pathways and free energy profiles for conformational transitions based on atomistic molecular dynamics simulations

An efficient method for the calculation of minimum free energy pathways and free energy profiles for conformational transitions is presented. Short restricted perturbation-targeted molecular dynamics trajectories are used to generate an approximate free energy surface. Approximate reaction pathways for the conformational change are constructed from one-dimensional line segments on this surface using a Monte Carlo optimization. Accurate free energy profiles are then determined along the pathways by means of one-dimensional adaptive umbrella sampling simulations. The method is illustrated by its application to the alanine "dipeptide." Due to the low computational cost and memory demands, the method is expected to be useful for the treatment of large biomolecular systems.

Free energy calculations for DNA base stacking by replica-exchange umbrella sampling

We have investigated the free energy change of the stacking process of DNA dimers using molecular dynamics simulations based on replica-exchange umbrella sampling. Pairs of replicas with different umbrella potentials are exchanged in this method, which allows the simulation to sample much wider conformational space and, therefore, to yield more accurate free energy profiles than by the conventional umbrella sampling. From the free energy profiles, we observed good stacking for all DNA dimers and sequence-dependent stacking stability. This sequence dependence of the stacking free energy is in accord with the experimental results.