Reservoir pH replica exchange

We present the reservoir pH replica exchange (R-pH-REM) method for constant pH simulations. The R-pH-REM method consists of a two-step procedure; the first step involves generation of one or more reservoirs of conformations. Each reservoir is obtained from a standard or enhanced molecular dynamics simulation with a constrained (fixed) protonation state. In the second step, fixed charge constraints are relaxed, as the structures from one or more reservoirs are periodically injected into a constant pH or a pH-replica exchange (pH-REM) simulation. The benefit of this two-step process is that the computationally intensive part of conformational search can be decoupled from constant pH simulations, and various techniques for enhanced conformational sampling can be applied without the need to integrate such techniques into the pH-REM framework. Simulations on blocked Lys, KK, and KAAE peptides were used to demonstrate an agreement between pH-REM and R-pH-REM simulations. While the reservoir simulations are not needed for these small test systems, the real need arises in cases when ionizable molecules can sample two or more conformations separated by a large energy barrier, such that adequate sampling is not achieved on a time scale of standard constant pH simulations. Such problems might be encountered in protein systems that exploit conformational transitions for function. A hypothetical case is studied, a small molecule with a large torsional barrier; while results of pH-REM simulations depend on the starting structure, R-pH-REM calculations on this model system are in excellent agreement with a theoretical model.

Absolute and relative pKa predictions via a DFT approach applied to the SAMPL6 blind challenge

In this work, quantum mechanical methods were used to predict the microscopic and macroscopic pK_a values for a set of 24 molecules as a part of the SAMPL6 blind challenge. The SMD solvation model was employed with M06-2X and different basis sets to evaluate three pK_a calculation schemes (direct, vertical, and adiabatic). The adiabatic scheme is the most accurate approach (RMSE = 1.40 pK_a units) and has high correlation (R² = 0.93), with respect to experiment. This approach can be improved by applying a linear correction to yield an RMSE of 0.73 pK_a units. Additionally, we consider including explicit solvent representation and multiple lower-energy conformations to improve the predictions for outliers. Adding three water molecules explicitly can reduce the error by 2-4 pK_a units, with respect to experiment, whereas including multiple local minima conformations does not necessarily improve the pK_a prediction.

Prediction of CB[8] host-guest binding free energies in SAMPL6 using the double-decoupling method

This study reports the results of binding free energy calculations for CB[8] host-guest systems in the SAMPL6 blind challenge (receipt ID 3z83m). Force-field parameters were developed specific for each of host and guest molecules to improve configurational sampling. We used quantum mechanical (QM) implicit solvent calculations and QM force matching to determine non-bonded (partial atomic charges) and bonded terms, respectively. Free energy calculations were carried out using the double-decoupling method (DDM) combined with Hamiltonian replica exchange method (HREM) and Bennett acceptance ratio (BAR) method. The root mean square error (RMSE) of the predicted values using DDM with respect to the experimental results was 4.32 kcal/mol. The coefficient of determination (R²) and Kendall rank coefficient (τ) were 0.49 and 0.52, respectively, highest of all submissions. In addition, these were compared to the results obtained by umbrella sampling (US) and weighted histogram analysis method (WHAM). Overall, DDM achieved a higher prediction accuracy than the US method. Results are discussed in terms of parameterization and free energy simulations.

Mapping the Drude polarizable force field onto a multipole and induced dipole model

The induced dipole and the classical Drude oscillator represent two major approaches for the explicit inclusion of electronic polarizability into force field-based molecular modeling and simulations. In this work, we explore the equivalency of these two models by comparing condensed phase properties computed using the Drude force field and a multipole and induced dipole (MPID) model. Presented is an approach to map the electrostatic model optimized in the context of the Drude force field onto the MPID model. Condensed phase simulations on water and 15 small model compounds show that without any reparametrization, the MPID model yields properties similar to the Drude force field with both models yielding satisfactory reproduction of a range of experimental values and quantum mechanical data. Our results illustrate that the Drude oscillator model and the point induced dipole model are different representations of essentially the same physical model. However, results indicate the presence of small differences between the use of atomic multipoles and off-center charge sites. Additionally, results on the use of dispersion particle mesh Ewald further support its utility for treating long-range Lennard Jones dispersion contributions in the context of polarizable force fields. The main motivation in demonstrating the transferability of parameters between the Drude and MPID models is that the more than 15 years of development of the Drude polarizable force field can now be used with MPID formalism without the need for dual-thermostat integrators nor self-consistent iterations. This opens up a wide range of new methodological opportunities for polarizable models.

Hydronium Ions Accompanying Buried Acidic Residues Lead to High Apparent Dielectric Constants in the Interior of Proteins

Internal ionizable groups are known to play important roles in protein functions. A mystery that has attracted decades of extensive experimental and theoretical studies is the apparent dielectric constants experienced by buried ionizable groups, which are much higher than values expected for protein interiors. Many interpretations have been proposed, such as water penetration, conformational relaxation, local unfolding, protein intrinsic backbone fluctuations, etc. However, these interpretations conflict with many experimental observations. The virtual mixture of multiple states (VMMS) simulation method developed in our lab provides a direct approach for studying the equilibrium of multiple chemical states and can monitor p K_a values along simulation trajectories. Through VMMS simulations of staphylococcal nuclease (SNase) variants with internal Asp or Glu residues, we discovered that cations were attracted to buried deprotonated acidic groups and the presence of the nearby cations were essential to reproduce experimentally measured p K_a values. This finding, combined with structural analysis and validation simulations, suggests that the proton released from a deprotonation process stays near the deprotonated group inside proteins, possibly in the form of a hydronium ion. The existence of a proton near a buried charge has many implications in our understanding of protein functions.

On the convergence of multi-scale free energy simulations

In this work we employ simple model systems to evaluate the relative performance of two of the most important free energy methods: The Zwanzig equation (also known as "Free energy perturbation") and Bennett's acceptance ratio method (BAR). Although our examples should be transferable to other kinds of free energy simulations, we focus on applications of multi-scale free energy simulations. Such calculations are especially complex, since they connect two different levels of theory with very different requirements in terms of speed, accuracy, sampling and parallelizability. We try to reconcile all those different factors by developing some simple criteria to guide the early stages of the development of a free energy protocol. This is accomplished by quantifying how many λ intermediate steps and how many potential energy evaluations are necessary in order to reach a certain level of convergence.

Structural Modulation of Human Amylin Protofilaments by Naturally Occurring Mutations

Human islet amyloid polypeptide (hIAPP), also known as amylin, is a 37-amino-acid peptide, co-secreted with insulin, and widely found in fibril form in type-2 diabetes patients. By using all-atom molecular dynamics simulations, we study hIAPP fibril segments (i.e., fibrillar oligomers) formed with sequences of naturally occurring variants from cat, rat, and pig, presenting different aggregation propensities. We characterize the effect of mutations on the structural dynamics of solution-formed hIAPP fibril models built from solid-state NMR data. Results from this study are in agreement with experimental observations regarding their respective relative aggregation propensities. We analyze in detail the specific structural characteristics and infer mechanisms that modulate the conformational stability of amylin fibrils. Results provide a platform for further studies and the design of new drugs that could interfere with amylin aggregation and its cytotoxicity. One particular mutation, N31K, has fibril-destabilizing properties, and could potentially improve the solubility of therapeutic amylin analogs.

Comparison of Additive and Polarizable Models with Explicit Treatment of Long-Range Lennard-Jones Interactions Using Alkane Simulations

Long-range Lennard-Jones (LJ) interactions have a significant impact on the structural and thermodynamic properties of nonpolar systems. While several methods have been introduced for the treatment of long-range LJ interactions in molecular dynamics (MD) simulations, increased accuracy and extended applicability is required for anisotropic systems such as lipid bilayers. The recently refined Lennard-Jones particle-mesh Ewald (LJ-PME) method extends the particle-mesh Ewald (PME) method to long-range LJ interactions and is suitable for use with anisotropic systems. Implementation of LJ-PME with the CHARMM36 (C36) additive and CHARMM Drude polarizable force fields improves agreement with experiment for density, isothermal compressibility, surface tension, viscosity, translational diffusion, and ¹³C T₁ relaxation times of pure alkanes. Trends in the temperature dependence of the density and isothermal compressibility of hexadecane are also improved. While the C36 additive force field with LJ-PME remains a useful model for liquid alkanes, the Drude polarizable force field with LJ-PME is more accurate for nearly all quantities considered. LJ-PME is also preferable to the isotropic long-range correction for hexadecane because the molecular order extends to nearly 20 Å, well beyond the usual 10-12 Å cutoffs used in most simulations.

Global organization of a binding site network gives insight into evolution and structure-function relationships of proteins

The global organization of protein binding sites is analyzed by constructing a weighted network of binding sites based on their structural similarities and detecting communities of structurally similar binding sites based on the minimum description length principle. The analysis reveals that there are two central binding site communities that play the roles of the network hubs of smaller peripheral communities. The sizes of communities follow a power-law distribution, which indicates that the binding sites included in larger communities may be older and have been evolutionary structural scaffolds of more recent ones. Structurally similar binding sites in the same community bind to diverse ligands promiscuously and they are also embedded in diverse domain structures. Understanding the general principles of binding site interplay will pave the way for improved drug design and protein design.

Machine Learning Force Field Parameters from Ab Initio Data

Machine learning (ML) techniques with the genetic algorithm (GA) have been applied to determine a polarizable force field parameters using only ab initio data from quantum mechanics (QM) calculations of molecular clusters at the MP2/6-31G(d,p), DFMP2(fc)/jul-cc-pVDZ, and DFMP2(fc)/jul-cc-pVTZ levels to predict experimental condensed phase properties (i.e., density and heat of vaporization). The performance of this ML/GA approach is demonstrated on 4943 dimer electrostatic potentials and 1250 cluster interaction energies for methanol. Excellent agreement between the training data set from QM calculations and the optimized force field model was achieved. The results were further improved by introducing an offset factor during the machine learning process to compensate for the discrepancy between the QM calculated energy and the energy reproduced by optimized force field, while maintaining the local "shape" of the QM energy surface. Throughout the machine learning process, experimental observables were not involved in the objective function, but were only used for model validation. The best model, optimized from the QM data at the DFMP2(fc)/jul-cc-pVTZ level, appears to perform even better than the original AMOEBA force field (amoeba09.prm), which was optimized empirically to match liquid properties. The present effort shows the possibility of using machine learning techniques to develop descriptive polarizable force field using only QM data. The ML/GA strategy to optimize force fields parameters described here could easily be extended to other molecular systems.

OpenMM 7: Rapid development of high performance algorithms for molecular dynamics

OpenMM is a molecular dynamics simulation toolkit with a unique focus on extensibility. It allows users to easily add new features, including forces with novel functional forms, new integration algorithms, and new simulation protocols. Those features automatically work on all supported hardware types (including both CPUs and GPUs) and perform well on all of them. In many cases they require minimal coding, just a mathematical description of the desired function. They also require no modification to OpenMM itself and can be distributed independently of OpenMM. This makes it an ideal tool for researchers developing new simulation methods, and also allows those new methods to be immediately available to the larger community.

Finding multiple reaction pathways via global optimization of action

Global searching for reaction pathways is a long-standing challenge in computational chemistry and biology. Most existing approaches perform only local searches due to computational complexity. Here we present a computational approach, Action-CSA, to find multiple diverse reaction pathways connecting fixed initial and final states through global optimization of the Onsager-Machlup action using the conformational space annealing (CSA) method. Action-CSA successfully overcomes large energy barriers via crossovers and mutations of pathways and finds all possible pathways of small systems without initial guesses on pathways. The rank order and the transition time distribution of multiple pathways are in good agreement with those of long Langevin dynamics simulations. The lowest action folding pathway of FSD-1 is consistent with recent experiments. The results show that Action-CSA is an efficient and robust computational approach to study the multiple pathways of complex reactions and large-scale conformational changes.

Isotropic periodic sum for multipole interactions and a vector relation for calculation of the Cartesian multipole tensor

Isotropic periodic sum (IPS) is a method to calculate long-range interactions based on the homogeneity of simulation systems. By using the isotropic periodic images of a local region to represent remote structures, long-range interactions become a function of the local conformation. This function is called the IPS potential; it folds long-ranged interactions into a short-ranged potential and can be calculated as efficiently as a cutoff method. It has been demonstrated that the IPS method produces consistent simulation results, including free energies, as the particle mesh Ewald (PME) method. By introducing the multipole homogeneous background approximation, this work derives multipole IPS potentials, abbreviated as IPSMm, with m being the maximum order of multipole interactions. To efficiently calculate the multipole interactions in Cartesian space, we propose a vector relation that calculates a multipole tensor as a dot product of a radial potential vector and a directional vector. Using model systems with charges, dipoles, and/or quadrupoles, with and without polarizability, we demonstrate that multipole interactions of order m can be described accurately with the multipole IPS potential of order 2 or m - 1, whichever is higher. Through simulations with the multipole IPS potentials, we examined energetic, structural, and dynamic properties of the model systems and demonstrated that the multipole IPS potentials produce very similar results as PME with a local region radius (cutoff distance) as small as 6 Å.

An empirical extrapolation scheme for efficient treatment of induced dipoles

Many cutting edge force fields include polarization, to enhance their accuracy and range of applicability. In this work, we develop efficient strategies for the induced dipole polarization method. By fitting various orders of perturbation theory (PT) dipoles to a diverse training set, we arrive at a family of fully analytic methods - whose nth order is referred to OPTn - that span the full spectrum of polarization methods from the fast zeroth-order approach that neglects mutual dipole coupling, approaching the fully variational approach at high order. Our training set contains many difficult cases where the PT series diverges, and we demonstrate that our OPTn methods still deliver excellent results in these cases. Our tests show that the OPTn methods exhibit rapid convergence towards the exact answer with each increasing PT order. The fourth order OPT4 method, whose costs are commensurate with three iterations of the leading conjugate gradient method, is a particularly promising candidate to be used as a drop-in replacement for existing solvers without further parameterization.

An Estimation of Hybrid Quantum Mechanical Molecular Mechanical Polarization Energies for Small Molecules Using Polarizable Force-Field Approaches

In this work, we report two polarizable molecular mechanics (polMM) force field models for estimating the polarization energy in hybrid quantum mechanical molecular mechanical (QM/MM) calculations. These two models, named the potential of atomic charges (PAC) and potential of atomic dipoles (PAD), are formulated from the ab initio quantum mechanical (QM) response kernels for the prediction of the QM density response to an external molecular mechanical (MM) environment (as described by external point charges). The PAC model is similar to fluctuating charge (FQ) models because the energy depends on external electrostatic potential values at QM atomic sites; the PAD energy depends on external electrostatic field values at QM atomic sites, resembling induced dipole (ID) models. To demonstrate their uses, we apply the PAC and PAD models to 12 small molecules, which are solvated by TIP3P water. The PAC model reproduces the QM/MM polarization energy with a R² value of 0.71 for aniline (in 10,000 TIP3P water configurations) and 0.87 or higher for other 11 solute molecules, while the PAD model has a much better performance with R² values of 0.98 or higher. The PAC model reproduces reference QM/MM hydration free energies for 12 solute molecules with a RMSD of 0.59 kcal/mol. The PAD model is even more accurate, with a much smaller RMSD of 0.12 kcal/mol, with respect to the reference. This suggests that polarization effects, including both local charge distortion and intramolecular charge transfer, can be well captured by induced dipole type models with proper parametrization.

Absolute binding free energies for octa-acids and guests in SAMPL5 : Evaluating binding free energies for octa-acid and guest complexes in the SAMPL5 blind challenge

As part of the SAMPL5 blind prediction challenge, we calculate the absolute binding free energies of six guest molecules to an octa-acid (OAH) and to a methylated octa-acid (OAMe). We use the double decoupling method via thermodynamic integration (TI) or Hamiltonian replica exchange in connection with the Bennett acceptance ratio (HREM-BAR). We produce the binding poses either through manual docking or by using GalaxyDock-HG, a docking software developed specifically for this study. The root mean square deviations for our most accurate predictions are 1.4 kcal mol-1 for OAH with TI and 1.9 kcal mol-1 for OAMe with HREM-BAR. Our best results for OAMe were obtained for systems with ionic concentrations corresponding to the ionic strength of the experimental solution. The most problematic system contains a halogenated guest. Our attempt to model the σ-hole of the bromine using a constrained off-site point charge, does not improve results. We use results from molecular dynamics simulations to argue that the distinct binding affinities of this guest to OAH and OAMe are due to a difference in the flexibility of the host. We believe that the results of this extensive analysis of host-guest complexes will help improve the protocol used in predicting binding affinities for larger systems, such as protein-substrate compounds.

Structural Characterization of Arginine Fingers: Identification of an Arginine Finger for the Pyrophosphatase dUTPases

Arginine finger is a highly conserved and essential residue in many GTPase and AAA+ ATPase enzymes that completes the active site from a distinct protomer, forming contacts with the γ-phosphate of the nucleotide. To date, no pyrophosphatase has been identified that employs an arginine finger fulfilling all of the above properties; all essential arginine fingers are used to catalyze the cleavage of the γ-phosphate. Here, we identify and unveil the role of a conserved arginine residue in trimeric dUTPases that meets all the criteria established for arginine fingers. We found that the conserved arginine adjacent to the P-loop-like motif enables structural organization of the active site for efficient catalysis via its nucleotide coordination, while its direct electrostatic role in transition state stabilization is secondary. An exhaustive structure-based comparison of analogous, conserved arginines from nucleotide hydrolases and transferases revealed a consensus amino acid location and orientation for contacting the γ-phosphate of the substrate nucleotide. Despite the structurally equivalent position, functional differences between arginine fingers of dUTPases and NTPases are explained on the basis of the unique chemistry performed by the pyrophosphatase dUTPases.

Origin of pKa Shifts of Internal Lysine Residues in SNase Studied Via Equal-Molar VMMS Simulations in Explicit Water

Protein internal ionizable groups can exhibit large shifts in pK_a values. Although the environment and interaction changes have been extensively studied both experimentally and computationally, direct calculation of pK_a values of these internal ionizable groups in explicit water is challenging due to energy barriers in solvent interaction and in conformational transition. The virtual mixture of multiple states (VMMS) method is a new approach designed to study chemical state equilibrium. This method constructs a virtual mixture of multiple chemical states in order to sample the conformational space of all states simultaneously and to avoid crossing energy barriers related to state transition. By applying VMMS to 25 variants of staphylococcal nuclease with lysine residues at internal positions, we obtained the pK_a values of these lysine residues and investigated the physics underlining the pK_a shifts. Our calculation results agree reasonably well with experimental measurements, validating the VMMS method for pK_a calculation and providing molecular details of the protonation equilibrium for protein internal ionizable groups. Based on our analyses of protein conformation relaxation, lysine side chain flexibility, water penetration, and the microenvironment, we conclude that the hydrophobicity of the microenvironment around the lysine side chain (which affects water penetration differently for different protonation states) plays an important role in the pK_a shifts.

Partition coefficients for the SAMPL5 challenge using transfer free energies

SAMPL challenges (Mobley et al. in J Comput Aided Mol Des 28:135-150, 2014; Skillman in J Comput Aided Mol Des 26:473-474, 2012; Geballe in J Comput Aided Mol Des 24:259-279, 2010; Guthrie in J Phys Chem B 113:4501-4507, 2009) provide excellent opportunities to assess theoretical approaches on new data sets with a goal of gaining greater insight towards protein and ligand modeling. In the SAMPL5 experiment, cyclohexane-water partition coefficients were determined using a vertical solvation scheme in conjunction with the SMD continuum solvent model. Several DFT functionals partnered with correlation consistent basis sets were evaluated for the prediction of the partition coefficients. The approach chosen for the competition, a B3PW91 vertical solvation scheme, yields a mean absolute deviation of 1.9 logP units and performs well at estimating the correct hydrophilicity and hydrophobicity for the full SAMPL5 molecule set.

Blind prediction of distribution in the SAMPL5 challenge with QM based protomer and pK a corrections

The computation of distribution coefficients between polar and apolar phases requires both an accurate characterization of transfer free energies between phases and proper accounting of ionization and protomerization. We present a protocol for accurately predicting partition coefficients between two immiscible phases, and then apply it to 53 drug-like molecules in the SAMPL5 blind prediction challenge. Our results combine implicit solvent QM calculations with classical MD simulations using the non-Boltzmann Bennett free energy estimator. The OLYP/DZP/SMD method yields predictions that have a small deviation from experiment (RMSD = 2.3 [Formula: see text] D units), relative to other participants in the challenge. Our free energy corrections based on QM protomer and [Formula: see text] calculations increase the correlation between predicted and experimental distribution coefficients, for all methods used. Unfortunately, these corrections are overly hydrophilic, and fail to account for additional effects such as aggregation, water dragging and the presence of polar impurities in the apolar phase. We show that, although expensive, QM-NBB free energy calculations offer an accurate and robust method that is superior to standard MM and QM techniques alone.

Calculating distribution coefficients based on multi-scale free energy simulations: an evaluation of MM and QM/MM explicit solvent simulations of water-cyclohexane transfer in the SAMPL5 challenge

One of the central aspects of biomolecular recognition is the hydrophobic effect, which is experimentally evaluated by measuring the distribution coefficients of compounds between polar and apolar phases. We use our predictions of the distribution coefficients between water and cyclohexane from the SAMPL5 challenge to estimate the hydrophobicity of different explicit solvent simulation techniques. Based on molecular dynamics trajectories with the CHARMM General Force Field, we compare pure molecular mechanics (MM) with quantum-mechanical (QM) calculations based on QM/MM schemes that treat the solvent at the MM level. We perform QM/MM with both density functional theory (BLYP) and semi-empirical methods (OM1, OM2, OM3, PM3). The calculations also serve to test the sensitivity of partition coefficients to solute polarizability as well as the interplay of the quantum-mechanical region with the fixed-charge molecular mechanics environment. Our results indicate that QM/MM with both BLYP and OM2 outperforms pure MM. However, this observation is limited to a subset of cases where convergence of the free energy can be achieved.