Adaptive-Partitioning Redistributed Charge and Dipole Schemes for QM/MM Dynamics Simulations: On-the-fly Relocation of Boundaries that Pass through Covalent Bonds

PyDFT-QMMM: A modular, extensible software framework for DFT-based QM/MM molecular dynamics

PyDFT-QMMM is a Python-based package for performing hybrid quantum mechanics/molecular mechanics (QM/MM) simulations at the density functional level of theory. The program is designed to treat short-range and long-range interactions through user-specified combinations of electrostatic and mechanical embedding procedures within periodic simulation domains, providing necessary interfaces to external quantum chemistry and molecular dynamics software. To enable direct embedding of long-range electrostatics in periodic systems, we have derived and implemented force terms for our previously described QM/MM/PME approach [Pederson and McDaniel, J. Chem. Phys. 156, 174105 (2022)]. Communication with external software packages Psi4 and OpenMM is facilitated through Python application programming interfaces (APIs). The core library contains basic utilities for running QM/MM molecular dynamics simulations, and plug-in entry-points are provided for users to implement custom energy/force calculation and integration routines, within an extensible architecture. The user interacts with PyDFT-QMMM primarily through its Python API, allowing for complex workflow development with Python scripting, for example, interfacing with PLUMED for free energy simulations. We provide benchmarks of forces and energy conservation for the QM/MM/PME and alternative QM/MM electrostatic embedding approaches. We further demonstrate a simple example use case for water solute in a water solvent system, for which radial distribution functions are computed from 100 ps QM/MM simulations; in this example, we highlight how the solvation structure is sensitive to different basis-set choices due to under- or over-polarization of the QM water molecule's electron density.

Reshaping the QM Region On-the-Fly: Adaptive-Shape QM/MM Dynamic Simulations of a Hydrated Proton in Bulk Water

Adaptive quantum mechanics/molecular mechanics (QM/MM) reclassifies on-the-fly a molecule or molecular fragment as QM or MM during dynamics simulations without abrupt changes in the energy or forces. Notably, the permuted adaptive-partitioning (PAP) algorithms have been applied to simulate a hydrated proton, with a mobile QM zone anchored at a pseudoatom called a proton indicator. The position of the proton indicator approximates the location of the delocalized excess proton, yielding a smooth trajectory of the proton diffusing via the Grotthuss mechanism in aqueous solutions. The mobile QM zone, which has been taken to be a sphere with a preset radius, follows the proton wherever it goes. Although the simulations are successful, the use of a spherical QM zone has one disadvantage: A large preset radius must be utilized to minimize the chance of missing water molecules that are important to proton translocation. A large radius leads to a large QM zone, which is computationally expensive. In this work, we report a new way to set up the QM zone, where one includes only the water molecules important to proton transfer. The importance of a given water molecule is quantified by its "weight" that depends on its relation to the reaction path of proton transfer. The weight varies smoothly, ensuring that a water molecule gradually appears in or disappears from the QM zone without abrupt changes, as required by the PAP method. Consequently, the shape of the QM zone evolves on-the-fly, keeping the QM zone as small as possible and as large as necessary. Test simulations demonstrate that the new algorithm significantly improves the computation efficiency while maintaining the proper descriptions of proton transfer in bulk water.

Adaptive-Partitioning Multilayer Dynamics Simulations: 2. Implementations of the Permuted and Interpolated Adaptive-Partitioning Gradients

Recently, an adaptive-partitioning multilayer Q1/Q2/MM method was proposed, where Q1 and Q2 denote, respectively, two distinct quantum-mechanical levels of theory and MM, the molecular-mechanical force fields. Such a multilayer model resembles the ONIOM (our own N-layered integrated molecular orbital and molecular mechanics) model by Morokuma and co-workers, but it is distinguished by on-the-fly reclassifying atoms to be Q1, Q2, or MM in dynamics simulations. To smoothly blend the levels of descriptions of the atoms, buffer zones are introduced between adjacent layers, and the energy is smoothly interpolated. In particular, the Q1/Q2 interaction energy was expressed in two different formalisms: permuted and interpolated adaptive-partitioning (PAP and IAP), respectively. While the PAP energy is based on a weighted many-body expansion, the IAP energy is derived via alchemical quantum calculations with interpolated Fock and overlap matrices. In this article, we examine in-depth the irregularities in the IAP energy near the boundary between the buffer and Q2 zones, which were found prominent in some calculations. These irregularities are due to basis-set linear dependencies, which can be effectively suppressed using a cutoff for the weighted atomic orbital coefficients. Furthermore, we derived and implemented the gradients for both PAP and IAP. Test calculations on a series of water cluster models show perfectly smooth gradients in PAP, while a minor discontinuity occurs in IAP gradients at the buffer/Q2 boundary. The energy and gradient discontinuities in IAP become smaller when moving the buffer/Q2 boundary further away from the Q1 center and when increasing the size of the basis sets used. Overall, those discontinuities are controllable, and possible ways to further diminish them are discussed.

Tracking the Delocalized Proton in Concerted Proton Transfer in Bulk Water

A solvated proton in water is often characterized as a charge or structural defect, and it is important to track its evolution on-the-fly in certain dynamics simulations. Previously, we introduced the proton indicator, a pseudo-atom, whose position approximates the location of the excess proton modeled as a structural defect. The proton indicator generally yields a smooth trajectory of a hydrated proton diffusing in aqueous solutions, including in the events of stepwise proton transfer via the Grotthuss mechanism; however, the proton indicator did not perform well in the events of concerted proton transfer, for which it occasionally yielded large position displacements between two successive time steps. To overcome this hurdle, we develop a new algorithm of a proton indicator with greatly enhanced performance for concerted proton transfer in bulk water. A protocol is proposed to exhaustively explore the hydrogen-bonding network of the water wires over which the excess proton is delocalized and to properly account for the contributions of the water molecules in this network as the geometry evolves. The new proton indicator (called Indicator 2.0) is assessed in dynamics simulations of an excess proton in bulk water and in specially constructed model systems of more complex architectures. The results demonstrate that the new indicator yields a smooth trajectory in both stepwise and concerted proton transfers.

Adaptive partitioning molecular dynamics using an extended Hamiltonian approach

A recently proposed extended Hamiltonian approach to switching interaction potentials is generalized to enable adaptive partitioning molecular dynamics simulations. Switching is performed along a fictitious classical degree of freedom whose value determines the mixing ratio of the two potentials on a time scale determined by its associated mass. We propose to choose this associated fictitious mass adaptively so as to ensure a constant time scale for all switching processes. For different model systems, including a harmonic oscillator and a Lennard-Jones fluid, we investigate the window of switching time scales that guarantees the conservation of the extended Hamiltonian for a large number of switching events. The methodology is first applied in the microcanonical ensemble and then generalized to the canonical ensemble using a Nosé-Hoover chain thermostat. It is shown that the method is stable for thousands of consecutive switching events during a single simulation, with constant temperature and a conserved extended Hamiltonian. A slight modification of the original Hamiltonian is introduced to avoid accumulation of small numerical errors incurred after each switching process.

Hybrid Quantum Mechanics/Molecular Mechanics (QM/MM) Simulation: A Tool for Structure-based Drug Design and Discovery

Quantum mechanics (QM) is physics based theory which explains the physical properties of nature at the level of atoms and sub-atoms. Molecular mechanics (MM) construct molecular systems through the use of classical mechanics. So, hybrid quantum mechanics and molecular mechanics (QM/MM) when combined together can act as computer-based methods which can be used to calculate structure and property data of molecular structures. Hybrid QM/MM combines the strengths of QM with accuracy and MM with speed. QM/MM simulation can also be applied for the study of chemical process in solutions as well as in the proteins, and has a great scope in structure-based drug design (CADD) and discovery. Hybrid QM/MM also applied to HTS, to derive QSAR models and due to availability of many protein crystal structures; it has a great role in computational chemistry, especially in structure- and fragment-based drug design. Fused QM/MM simulations have been developed as a widespread method to explore chemical reactions in condensed phases. In QM/MM simulations, the quantum chemistry theory is used to treat the space in which the chemical reactions occur; however the rest is defined through molecular mechanics force field (MMFF). In this review, we have extensively reviewed recent literature pertaining to the use and applications of hybrid QM/MM simulations for ligand and structure-based computational methods for the design and discovery of therapeutic agents.

Adaptive-Partitioning Multilayer Dynamics Simulations: 1. On-the-Fly Switch between Two Quantum Levels of Theory

We propose to generalize the previously developed two-layer permuted adaptive-partitioning quantum-mechanics/molecular-mechanics (QM/MM), which reclassifies atoms as QM or MM on-the-fly in dynamics simulations, to multilayer adaptive-partitioning algorithms that enable multiple levels of theory. In this work, we formulate two new algorithms that smoothly interpolate the energy between two QM (Q1 and Q2) levels of theory. The first "permuted adaptive-partitioning" scheme is based on the weighted many-body expansion of the potential, as in the adaptive-partitioning QM/MM. Unconventional and potentially more efficient, the second "interpolated adaptive-partitioning" method employs alchemical QM calculations with Q1/Q2-mixed basis sets, Fock matrices, and overlap matrices. To our knowledge, this is the first time that such alchemical calculations are performed in QM, although they are routinely done in MM. Test calculations on water-cluster models show that both new algorithms indeed yield smooth energy curves when water molecules shift between Q1 and Q2.

General Many-Body Framework for Data-Driven Potentials with Arbitrary Quantum Mechanical Accuracy: Water as a Case Study

We present a general framework for the development of data-driven many-body (MB) potential energy functions (MB-QM PEFs) that represent the interactions between small molecules at an arbitrary quantum-mechanical (QM) level of theory. As a demonstration, a family of MB-QM PEFs for water is rigorously derived from density functionals belonging to different rungs across Jacob's ladder of approximations within density functional theory (MB-DFT) and from Møller-Plesset perturbation theory (MB-MP2). Through a systematic analysis of individual MB contributions to the interaction energies of water clusters, we demonstrate that all MB-QM PEFs preserve the same accuracy as the corresponding ab initio calculations, with the exception of those derived from density functionals within the generalized gradient approximation (GGA). The differences between the DFT and MB-DFT results are traced back to density-driven errors that prevent GGA functionals from accurately representing the underlying molecular interactions for different cluster sizes and hydrogen-bonding arrangements. We show that this shortcoming may be overcome, within the MB formalism, by using density-corrected functionals (DC-DFT) that provide a more consistent representation of each individual MB contribution. This is demonstrated through the development of a MB-DFT PEF derived from DC-PBE-D3 data, which more accurately reproduce the corresponding ab initio results.

Proton transfer in bulk water using the full adaptive QM/MM method: integration of solute- and solvent-adaptive approaches

The quantum mechanical/molecular mechanical (QM/MM) method is a hybrid molecular simulation technique that increases the accessibility of local electronic structures of large systems. The technique combines the benefit of accuracy found in the QM method and that of cost efficiency found in the MM method. However, it is difficult to directly apply the QM/MM method to the dynamics of solution systems, particularly for proton transfer. As explained in the Grotthuss mechanism, proton transfer is a structural interconversion between hydronium ions and solvent water molecules. Hence, when the QM/MM method is applied, an adaptive treatment, namely on-the-fly revisions on molecular definitions, is required for both the solute and solvent. Although several solvent-adaptive methods have been proposed, a full adaptive framework, which is an approach that also considers adaptation for solutes, remains untapped. In this paper, we propose a new numerical expression for the coordinates of the excess proton and its control algorithm. Furthermore, we confirm that this method can stably and accurately simulate proton transfer dynamics in bulk water.

Protonation Dynamics in the K-Channel of Cytochrome c Oxidase Estimated from Molecular Dynamics Simulations

Proton transfer reactions are one of the most fundamental processes in biochemistry. We present a simplistic approach for estimating proton transfer probabilities in a membrane protein, cytochrome c oxidase. We combine short molecular dynamics simulations at discrete protonation states with a Monte Carlo approach to exchange between those states. Requesting for a proton transfer the existence of a hydrogen-bonded connection between the two source and target residues of the exchange, restricts the acceptance of transfers to only those in which a proton-relay is possible. Together with an analysis of the hydrogen-bonded connectivity in one of the proton-conducting channels of cytochrome c oxidase, this approach gives insight into the protonation dynamics of the hydrogen-bonded networks. The connectivity and directionality of the networks are coupled to the conformation of an important protein residue in the channel, K362, rendering proton transfer in the entire channel feasible in only one of the two major conformations. Proton transport in the channel can thus be regulated by K362 not only through its possible role as a proton carrier itself, but also by allowing or preventing proton transport via water residues.

A Many-Body, Fully Polarizable Approach to QM/MM Simulations

We present a new development in quantum mechanics/molecular mechanics (QM/MM) methods by replacing conventional MM models with data-driven many-body (MB) representations rigorously derived from high-level QM calculations. The new QM/MM approach builds on top of mutually polarizable QM/MM schemes developed for polarizable force fields with inducible dipoles and uses permutationally invariant polynomials to effectively account for quantum-mechanical contributions (e.g., exchange-repulsion and charge transfer and penetration) that are difficult to describe by classical expressions adopted by conventional MM models. Using the many-body MB-pol and MB-DFT potential energy functions for water, which include explicit two-body and three-body terms fitted to reproduce the corresponding CCSD(T) and PBE0 two-body and three-body energies for water, we demonstrate a smooth energetic transition as molecules are transferred between QM and MM regions, without the need of a transition layer. By effectively elevating the accuracy of both the MM region and the QM/MM interface to that of the QM region, the new QM/MB-MM approach achieves an accuracy comparable to that obtained with a fully QM treatment of the entire system.

Extending scaled-interaction adaptive-partitioning QM/MM to covalently bonded systems

Quantum mechanics/molecular mechanics (QM/MM) is the method of choice for atomistic simulations of large systems that can be partitioned into active and environmental regions. Adaptive-partitioning (AP) methods extend the applicability of QM/MM, allowing active regions to change during the simulation. AP methods achieve continuous potential energy surface (PES) by introducing buffer regions in which atoms have both QM and MM characters. Most of the existing AP-QM/MM methods require multiple QM calculations per time step, which can be expensive for systems with many atoms in buffer regions. Although one can lower the computational cost by grouping atoms into fragments, this may not be possible for all systems, especially for applications in covalent solids. The SISPA method [Field, J. Chem. Theory Comput., 2017, 13, 2342] differs from other AP-QM/MM methods by only requiring one QM calculation per time step, but it has the flaw that the QM charge density and wavefunction near the buffer/MM boundary tend to those of isolated atoms/fragments. Besides, regular QM/MM methods for treating covalent bonds cut by the QM/MM boundary are incompatible with SISPA. Due to these flaws, SISPA in its original form cannot treat covalently bonded systems properly. In this work, I show that a simple modification to the SISPA method improves the treatment of covalently bonded systems. I also study the effect of correcting the charge density in SISPA by developing a density-corrected pre-scaled algorithm. I demonstrate the methods with simple molecules and bulk solids.

Adaptive QM/MM for Molecular Dynamics Simulations: 5. On the Energy-Conserved Permuted Adaptive-Partitioning Schemes

In combined quantum-mechanical/molecular-mechanical (QM/MM) dynamics simulations, the adaptive-partitioning (AP) schemes reclassify atoms on-the-fly as QM or MM in a smooth manner. This yields a mobile QM subsystem with contents that are continuously updated as needed. Here, we tailor the Hamiltonian adaptive many-body correction (HAMBC) proposed by Boreboom et al. [J. Chem. Theory Comput.2016, 12, 3441] to the permuted AP (PAP) scheme. The treatments lead to the HAMBC-PAP method (HPAP), which both conserves energy and produces accurate solvation structures in the test of "water-in-water" model system.

Restrained Proton Indicator in Combined Quantum-Mechanics/Molecular-Mechanics Dynamics Simulations of Proton Transfer through a Carbon Nanotube

Recently, a collective variable "proton indicator" was purposed for tracking an excess proton solvated in bulk water in molecular dynamics simulations. In this work, we demonstrate the feasibility of utilizing the position of this proton indicator as a reaction coordinate to model an excess proton migrating through a hydrophobic carbon nanotube in combined quantum-mechanics/molecular-mechanics simulations. Our results indicate that applying a harmonic restraint to the proton indicator in the bulk solvent near the nanotube pore entrance leads to the recruitment of water molecules into the pore. This is consistent with an earlier study that employed a multistate empirical valence bond potential and a different representation (center of excess charge) of the proton. We attribute this water recruitment to the delocalized nature of the solvated proton, which prefers to be in high-dielectric bulk solvent. While water recruitment into the pore is considered an artifact in the present simulations (because of the artificially imposed restraint on the proton), if the proton were naturally restrained, it could assist in building water wires prior to proton transfer through the pore. The potential of mean force for a proton translocation through the water-filled pore was computed by umbrella sampling, where the bias potentials were applied to the proton indicator. The free energy curve and barrier heights agree reasonably with those in the literature. The results suggest that the proton indicator can be used as a reaction coordinate in simulations of proton transport in confined environments.

Adaptive switching of interaction potentials in the time domain: an extended Lagrangian approach tailored to transmute force field to QM/MM simulations and back

An extended Lagrangian formalism that allows for a smooth transition between two different descriptions of interactions during a molecular dynamics simulation is presented. This time-adaptive method is particularly useful in the context of multiscale simulation as it provides a sound recipe to switch on demand between different hierarchical levels of theory, for instance between ab initio ("QM") and force field ("MM") descriptions of a given (sub)system in the course of a molecular dynamics simulation. The equations of motion can be integrated straightforwardly using the usual propagators, such as the Verlet algorithm. First test cases include a bath of harmonic oscillators, of which a subset is switched to a different force constant and/or equilibrium position, as well as an all-MM to QM/MM transition in a hydrogen-bonded water dimer. The method is then applied to a smectic 8AB8 liquid crystal and is shown to be able to switch dynamically a preselected 8AB8 molecule from an all-MM to a QM/MM description which involves partition boundaries through covalent bonds. These examples show that the extended Lagrangian approach is not only easy to implement into existing code but that it is also efficient and robust. The technique moreover provides easy access to a conserved energy quantity, also in cases when Nosé-Hoover chain thermostatting is used throughout dynamical switching. A simple quadratic driving potential proves to be sufficient to guarantee a smooth transition whose time scale can be easily tuned by varying the fictitious mass parameter associated with the auxiliary variable used to extend the Lagrangian. The method is general and can be applied to time-adaptive switching on demand between two different levels of theory within the framework of hybrid scale-bridging simulations.

Mathematical and computational modeling in biology at multiple scales

A variety of topics are reviewed in the area of mathematical and computational modeling in biology, covering the range of scales from populations of organisms to electrons in atoms. The use of maximum entropy as an inference tool in the fields of biology and drug discovery is discussed. Mathematical and computational methods and models in the areas of epidemiology, cell physiology and cancer are surveyed. The technique of molecular dynamics is covered, with special attention to force fields for protein simulations and methods for the calculation of solvation free energies. The utility of quantum mechanical methods in biophysical and biochemical modeling is explored. The field of computational enzymology is examined.

Recent advances in QM/MM free energy calculations using reference potentials

Background

Recent years have seen enormous progress in the development of methods for modeling (bio)molecular systems. This has allowed for the simulation of ever larger and more complex systems. However, as such complexity increases, the requirements needed for these models to be accurate and physically meaningful become more and more difficult to fulfill. The use of simplified models to describe complex biological systems has long been shown to be an effective way to overcome some of the limitations associated with this computational cost in a rational way.

Scope of review

Hybrid QM/MM approaches have rapidly become one of the most popular computational tools for studying chemical reactivity in biomolecular systems. However, the high cost involved in performing high-level QM calculations has limited the applicability of these approaches when calculating free energies of chemical processes. In this review, we present some of the advances in using reference potentials and mean field approximations to accelerate high-level QM/MM calculations. We present illustrative applications of these approaches and discuss challenges and future perspectives for the field.

Major conclusions

The use of physically-based simplifications has shown to effectively reduce the cost of high-level QM/MM calculations. In particular, lower-level reference potentials enable one to reduce the cost of expensive free energy calculations, thus expanding the scope of problems that can be addressed.

General significance

As was already demonstrated 40 years ago, the usage of simplified models still allows one to obtain cutting edge results with substantially reduced computational cost. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

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We present some of the advances to accelerate high-level QM/MM calculations.

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Quantitative limitations of low-level methods can be overcome by these approaches.

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Reference potentials make free energy simulations feasible for large systems.

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Automated fitting reduces the need of expensive sampling of high-level approaches.

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Application of reference potentials can be extended to a wide range of processes.

Computational electrochemistry: prediction of liquid-phase reduction potentials

This article reviews recent developments and applications in the area of computational electrochemistry. Our focus is on predicting the reduction potentials of electron transfer and other electrochemical reactions and half-reactions in both aqueous and nonaqueous solutions. Topics covered include various computational protocols that combine quantum mechanical electronic structure methods (such as density functional theory) with implicit-solvent models, explicit-solvent protocols that employ Monte Carlo or molecular dynamics simulations (for example, Car-Parrinello molecular dynamics using the grand canonical ensemble formalism), and the Marcus theory of electronic charge transfer. We also review computational approaches based on empirical relationships between molecular and electronic structure and electron transfer reactivity. The scope of the implicit-solvent protocols is emphasized, and the present status of the theory and future directions are outlined.

Multiscale Modeling of Chemistry in Water: Are We There Yet?

This paper critically evaluates the state of the art in combined quantum mechanical/molecular mechanical (QM/MM) approaches to the computational description of chemistry in water and supplies guidelines for the setup of customized multiscale simulations of aqueous processes. We differentiate between structural and dynamic performance, since some tasks, e.g., the reproduction of NMR or UV-vis spectra, require only structural accuracy, while others, i.e., reaction mechanisms, require accurate dynamic data as well. As a model system for aqueous solutions in general, the approaches were tested on a QM water cluster in an environment of MM water molecules. The key difficulty is the description of the possible diffusion of QM molecules into the MM region and vice versa. The flexible inner region ensemble separator (FIRES) approach constrains QM solvent molecules within an active (QM) region. Sorted adaptive partitioning (SAP), difference-based adaptive solvation (DAS), and buffered-force (BF) are all adaptive approaches that use a buffer zone in which solvent molecules gradually adapt from QM to MM (or vice versa). The costs of SAP and DAS are relatively high, while BF is fast but sacrifices conservation of both energy and momentum. Simulations in the limit of an infinitely small buffer zone, where DAS and SAP become equivalent, are discussed as well and referred to as ABRUPT. The best structural accuracy is obtained with DAS, BF, and ABRUPT, all three of similar quality. FIRES performs very well for dynamic properties localized deep within the QM region. By means of elimination DAS emerges as the best overall compromise between structural and dynamic performance. Eliminating the buffer zone (ABRUPT) improves efficiency and still leads to surprisingly good results. While none of the many new flavors are perfect, all together this new field already allows accurate description of a wide range of structural and dynamic properties of aqueous solutions.

Combined quantum mechanics/molecular mechanics (QM/MM) methods in computational enzymology

Computational enzymology is a rapidly maturing field that is increasingly integral to understanding mechanisms of enzyme-catalyzed reactions and their practical applications. Combined quantum mechanics/molecular mechanics (QM/MM) methods are important in this field. By treating the reacting species with a quantum mechanical method (i.e., a method that calculates the electronic structure of the active site) and including the enzyme environment with simpler molecular mechanical methods, enzyme reactions can be modeled. Here, we review QM/MM methods and their application to enzyme-catalyzed reactions to investigate fundamental and practical problems in enzymology. A range of QM/MM methods is available, from cheaper and more approximate methods, which can be used for molecular dynamics simulations, to highly accurate electronic structure methods. We discuss how modeling of reactions using such methods can provide detailed insight into enzyme mechanisms and illustrate this by reviewing some recent applications. We outline some practical considerations for such simulations. Further, we highlight applications that show how QM/MM methods can contribute to the practical development and application of enzymology, e.g., in the interpretation and prediction of the effects of mutagenesis and in drug and catalyst design.