The SCC-DFTB method and its application to biological systems

Charge-dependent model for many-body polarization, exchange, and dispersion interactions in hybrid quantum mechanical/molecular mechanical calculations

This work explores a new charge-dependent energy model consisting of van der Waals and polarization interactions between the quantum mechanical (QM) and molecular mechanical (MM) regions in a combined QMMM calculation. van der Waals interactions are commonly treated using empirical Lennard-Jones potentials, whose parameters are often chosen based on the QM atom type (e.g., based on hybridization or specific covalent bonding environment). This strategy for determination of QMMM nonbonding interactions becomes tedious to parametrize and lacks robust transferability. Problems occur in the study of chemical reactions where the "atom type" is a complex function of the reaction coordinate. This is particularly problematic for reactions, where atoms or localized functional groups undergo changes in charge state and hybridization. In the present work we propose a new model for nonelectrostatic nonbonded interactions in QMMM calculations that overcomes many of these problems. The model is based on a scaled overlap model for repulsive exchange and attractive dispersion interactions that is a function of atomic charge. The model is chemically significant since it properly correlates atomic size, softness, polarizability, and dispersion terms with minimal one-body parameters that are functions of the atomic charge. Tests of the model are examined for rare-gas interactions with neutral and charged atoms in order to demonstrate improved transferability. The present work provides a new framework for modeling QMMM interactions with improved accuracy and transferability.

Description of phosphate hydrolysis reactions with the Self-Consistent-Charge Density-Functional-Tight-Binding (SCC-DFTB) theory. 1. Parameterization

Phosphate chemistry is involved in many key biological processes yet the underlying mechanism often remains unclear. For theoretical analysis to effectively complement experimental mechanistic analysis, it is essential to develop computational methods that can capture the complexity of the underlying potential energy surface and allow for sufficient sampling of the configurational space. To this end, we report the parameterization of an approximate density functional theory, Self-Consistent-Charge Density-Functional Tight-Binding (SCC-DFTB) method for systems containing phosphorus. Compared to high-level density functional theory and ab initio (MP2 and G3B3) results, the standard second-order parameterization is shown to give reliable structures for a diverse set of phosphate compounds but inaccurate energetics. With the on-site third-order terms included, referred to as SCC-DFTBPA, calculated proton affinities of phosphate compounds are substantially improved, although it remains difficult to obtain reliable proton affinity for both phosphates and compounds that do not contain phosphorus, indicating that further improvement in the formulation of SCC-DFTB is still a challenge to meet. To make SCC-DFTB applicable to phosphate reactions in the current (on-site-third-order-only) formulation, a "reaction-specific" parameterization, referred to as SCC-DFTBPR, is developed based on hydrolysis reactions of model phosphate species. Benchmark calculations in both the gas-phase and solution-phase indicate that SCC-DFTBPR gives reliable structural properties and semi-quantitative energetics for phosphate hydrolysis reactions. Since the number of reaction-specific parameters is small, it is likely that SCC-DFTBPR is applicable to a broad set of phosphate species. Indeed, for 56 reaction exothermicities and 47 energy barriers related to RNA catalysis model reactions collected from the QCRNA database, which involve molecules rather different from those used to parameterize SCC-DFTBPR, the corresponding root-mean-square difference between SCC-DFTBPR and high-level DFT results is only 5.3 kcal/mol. We hope that the parameterized SCC-DFTB models will complement NDDO based reaction-specific models (e.g., AM1-d/PhoT) and high-level ab initio QM/MM methods in better understanding the mechanism of phosphate chemistry in condensed phase, particularly biological systems.

Extension of the Self-Consistent-Charge Density-Functional Tight-Binding Method: Third-Order Expansion of the Density Functional Theory Total Energy and Introduction of a Modified Effective Coulomb Interaction

The standard self-consistent-charge density-functional-tight-binding (SCC-DFTB) method (Phys. Rev. B 1998, 58, 7260) is derived by a second-order expansion of the density functional theory total energy expression, followed by an approximation of the charge density fluctuations by charge monopoles and an effective damped Coulomb interaction between the atomic net charges. The central assumptions behind this effective charge-charge interaction are the inverse relation of atomic size and chemical hardness and the use of a fixed chemical hardness parameter independent of the atomic charge state. While these approximations seem to be unproblematic for many covalently bound systems, they are quantitatively insufficient for hydrogen-bonding interactions and (anionic) molecules with localized net charges. Here, we present an extension of the SCC-DFTB method to incorporate third-order terms in the charge density fluctuations, leading to chemical hardness parameters that are dependent on the atomic charge state and a modification of the Coulomb scaling to improve the electrostatic treatment within the second-order terms. These modifications lead to a significant improvement in the description of hydrogen-bonding interactions and proton affinities of biologically relevant molecules.

Benchmark RI-MP2 database of nucleic acid base trimers: performance of different density functional models for prediction of structures and binding energies

A new database of nucleic acid base trimers has been developed that includes 141 geometries and stabilization energies obtained at the RI-MP2 level of theory with the TZVPP basis set. Compared to previously compiled biologically oriented databases, this new construct includes considerably more complicated structures; the various intermolecular interactions in the trimers are quite heterogeneous and in particular include simultaneous hydrogen bonding and stacking interactions, which is similar to the situation in actual biopolymers. Validation against these benchmark data is therefore a more demanding task for approximate models, since correct descriptions of all energy terms are unlikely to be accomplished by fortuitous cancellations of systematic errors. The density functionals TPSS (both with and without an empirical dispersion term), PWB6K, M05-2X, and BH&H, and the self-consistent charge density functional tight binding method augmented with an empirical dispersion term (SCC-DFTB-D) were assessed for their abilities accurately to compute structures and energies. The best reproduction of the BSSE corrected RI-MP2 stabilization energies was achieved by the TPSS functional (TZVPP basis set) combined with empirical dispersion; removal of the dispersion correction leads to significantly degraded performance. The M05-2X and PWB6K functionals performed very well in reproducing the RI-MP2 geometries, but showed a systematic moderate underestimation of the magnitude of base stacking interactions. The SCC-DFTB-D method predicts geometries in fair agreement with RI-MP2; given its computational efficiency it represents a good option for initial scanning of analogous biopolymeric potential energy surfaces. BH&H gives geometries of comparable quality to the other functionals but significantly overestimates interaction energies other than stacking.

Implementation of the SCC-DFTB method for hybrid QM/MM simulations within the amber molecular dynamics package

Self-consistent charge density functional tight-binding (SCC-DFTB) is a semiempirical method based on density functional theory and has in many cases been shown to provide relative energies and geometries comparable in accuracy to full DFT or ab initio MP2 calculations using large basis sets. This article shows an implementation of the SCC-DFTB method as part of the new QM/MM support in the AMBER 9 molecular dynamics program suite. Details of the implementation and examples of applications are shown.

Simulating water with the self-consistent-charge density functional tight binding method: from molecular clusters to the liquid state

The recently developed self-consistent-charge density functional tight binding (SCCDFTB) method provides an accurate and inexpensive quantum mechanical solution to many molecular systems of interests. To examine the performance of the SCCDFTB method on (liquid) water, the most fundamental yet indispensable molecule in biological systems, we report here the simulation results of water in sizes ranging from molecular clusters to the liquid state. The latter simulation was achieved through the use of the linear scaling divide-and-conquer approach. The results of liquid water simulation indicate that the SCCDFTB method can describe the structural and energetics of liquid water in qualitative agreement with experiments, and the results for water clusters suggest potential future improvements of the SCCDFTB method.

SCC-DFTB: What Is the Proper Degree of Self-Consistency?

The approximate SCC-DFTB method (Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, Th.; Suhai, S.; Seifert, G. Phys. Rev. B 1998, 58, 7260) is derived from DFT by a second-order expansion of the total energy expression. In this article, basic approximations and assumptions underlying the DFTB method are discussed in detail, and further extensions to include third-order terms are proposed. Further, the SCC-DFTB and semiempirical NDDO formalisms are compared to elucidate similarities and differences.

Application of the Computationally Efficient Self-Consistent-Charge Density-Functional Tight-Binding Method to Magnesium-Containing Molecules

The geometric properties, ionization potentials, heats of formation, incremental binding energies, and protonation energies for up to 75 magnesium-containing compounds have been studied using the self-consistent-charge density-functional tight-binding method (SCC-DFTB), the complete-basis set (CBS-QB3) method, traditional B3LYP density-functional theory, and a number of modern semiempirical methods such as Austin Model 1 (AM1), modified neglect of diatomic overlap without and with inclusion of d functions (MNDO, MNDO/d), and the Parametric Method 3 (PM3) and its modification (PM5). The test set contains some widely varying chemical motifs including ionic or covalent, closed-shell or radical compounds, and many biologically relevant complexes. Geometric data are compared to experiment, if available, and otherwise to previous high-level ab initio calculations or the present B3LYP results. SCC-DFTB is found to predict bond lengths to high accuracy, with the root-mean-square (RMS) error being less than half that found for the other semiempirical methods. However, SCC-DFTB performs very poorly for absolute heats of formation, giving an RMS error of 29 kcal mol(-1), but for this property B3LYP and the other semiempirical methods also yield poor but useful results with errors of 12-22 kcal mol(-1). Nevertheless, SCC-DFTB does provide useful results for biologically relevant chemical-process energies such as protonation energies (RMS error 10 kcal mol(-1), with the range 6-19 kcal mol(-1) found for the other semiempirical methods) and ligation energies (RMS error 9 kcal mol(-1), less than the errors of 12-23 kcal mol(-1) found for the other semiempirical methods). SCC-DFTB is shown to provide a computationally expedient means of calculating properties of magnesium compounds, providing results with at most double the inaccuracy of the high-quality but dramatically more-expensive B3LYP method.

pKa analysis for the zinc-bound water in human carbonic anhydrase II: Benchmark for "multiscale" QM/MM simulations and mechanistic implications

To quantitatively explore the applicability of the generalized solvent boundary potential (GSBP) based QM/MM approach as a "multiscale" framework for studying chemical reactions in biomolecules, the structural and energetic properties of the Human Carbonic Anhydrase II (CAII) are analyzed and compared to those from periodic boundary condition (PBC) simulations and available experimental data. Although the atomic fluctuations from GSBP based simulations are consistently lower compared to those from PBC simulations or crystallographic data, the fluctuations and internal coordinate distributions for residues in the proximity of the active site as well as diffusion constants of active-site water molecules are fairly well described by GSBP simulations. The pKa of the zinc-bound water, calculated with a SCC-DFTB/MM-GSBP based thermodynamic integration approach, agrees well with experiments for the wild type CAII. For the E106Q mutant, however, a 9 pKa unit downward shift relative to the wild type is found in contrast with previous experiments that found little change. This dramatic discrepancy signals a possible change in the mechanism for the interconversion between CO2/HCO3- in the E106Q mutant, which may be similar to the bicarbonate mediated mechanism proposed for the Co2+ substituted CAII (J. Am. Chem. Soc. 2001, 123, 5861).1 The study highlights pKa analyses as a valuable approach for quantitatively validating the computational model for complex biomolecules as well as for revealing energetic properties intimately related to the chemical process of interest.

Sampling enhancement for the quantum mechanical potential based molecular dynamics simulations: a general algorithm and its extension for free energy calculation on rugged energy surface

An approach is developed in the replica exchange framework to enhance conformational sampling for the quantum mechanical (QM) potential based molecular dynamics simulations. Importantly, with our enhanced sampling treatment, a decent convergence for electronic structure self-consistent-field calculation is robustly guaranteed, which is made possible in our replica exchange design by avoiding direct structure exchanges between the QM-related replicas and the activated (scaled by low scaling parameters or treated with high "effective temperatures") molecular mechanical (MM) replicas. Although the present approach represents one of the early efforts in the enhanced sampling developments specifically for quantum mechanical potentials, the QM-based simulations treated with the present technique can possess the similar sampling efficiency to the MM based simulations treated with the Hamiltonian replica exchange method (HREM). In the present paper, by combining this sampling method with one of our recent developments (the dual-topology alchemical HREM approach), we also introduce a method for the sampling enhanced QM-based free energy calculations.

Looking at Self-Consistent-Charge Density Functional Tight Binding from a Semiempirical Perspective

The self-consistent-charge density functional tight binding (SCC-DFTB) method is compared with other semiempirical methods (MNDO, AM1, PM3, OM1, OM2, OM3). Despite the differences in the underlying philosophy and derivation, these methods share many common features. Systematic evaluations of their performance are reported for standard test sets that are in common use. The overall accuracy of SCC-DFTB and the other methods is in the same range, with the overall tendency AM1<SCC-DFTB<OM2, but any such ranking depends on the properties and compound classes considered. SCC-DFTB is excellent for geometries and performs well for biological systems. It seems less suitable for the energetics of radicals and electronically excited states, and suffers from occasional outliers (e.g., for compounds with NO bonds). In an overall assessment, SCC-DFTB is a viable semiempirical method with specific strengths and weaknesses which may be an attractive choice especially for biomolecular applications.

"Proton holes" in long-range proton transfer reactions in solution and enzymes: A theoretical analysis

Proton transfers are fundamental to chemical processes in solution and biological systems. Often, the well-known Grotthuss mechanism is assumed where a series of sequential "proton hops" initiates from the donor and combines to produce the net transfer of a positive charge over a long distance. Although direct experimental evidence for the sequential proton hopping has been obtained recently, alternative mechanisms may be possible in complex molecular systems. To understand these events, all accessible protonation states of the mediating groups should be considered. This is exemplified by transfers through water where the individual water molecules can exist in three protonation states (water, hydronium, and hydroxide); as a result, an alternative to the Grotthuss mechanism for a proton transfer through water is to generate a hydroxide by first protonating the acceptor and then transfer the hydroxide toward the donor through water. The latter mechanism can be most generally described as the transfer of a "proton hole" from the acceptor to the donor where the "hole" characterizes the deprotonated state of any mediating molecule. This pathway is distinct and is rarely considered in the discussion of proton-transfer processes. Using a calibrated quantum mechanical/molecular mechanical (QM/MM) model and an effective sampling technique, we study proton transfers in two solution systems and in Carbonic Anhydrase II. Although the relative weight of the "proton hole" and Grotthuss mechanisms in a specific system is difficult to determine precisely using any computational approach, the current study establishes an energetics motivated framework that hinges on the donor/acceptor pKa values and electrostatics due to the environment to argue that the "proton hole" transfer is likely as important as the classical Grotthuss mechanism for proton transport in many complex molecular systems.

QM/MM studies of enzymes

Combined quantum-mechanics/molecular-mechanics (QM/MM) methods are making rapid progress both methodologically and with respect to their range of application. Mechanistic studies on enzymes, including contributions towards the understanding of enzyme catalysis, continue to be a major target. They are joined by calculations of pK(a) values, redox properties, ground- and excited-state spectroscopic parameters, and excited-state dynamics. Methodological advances include improved QM/MM schemes, in particular new approaches for an effective treatment of the QM-MM electrostatic interactions, and the incorporation of new efficient and accurate QM methods in QM/MM schemes.

Diels−Alder Ribozyme Catalysis: A Computational Approach

A computational comparison of the Diels-Alder reaction of a maleimide and an anthracene in water and the active site of the ribozyme Diels-Alderase is reported. During the course of the catalyzed reaction, the maleimide is held in the hydrophobic pocket while the anthracene approaches to the maleimide through the back passage of the active site. The active site is so narrow that the anthracene has to adopt a tilted approach angle toward maleimide. The conformation of the active site changes marginally at different states of the reaction. Active site dynamics contribution to catalysis has been ruled out. The active site stabilizes the product more than the transition state (TS). The reaction coordinates of the ribozyme reaction in TS, RC1-CD1 and RC4-CD2, are 2.35 and 2.33 A, respectively, compared to 2.37 and 2.36 A in water. The approach angle of anthracene toward maleimide is twisted by 18 degrees in the TS structure of ribozyme reaction while no twisted angle is found in TS of the reaction in water. The free energy barriers for reactions in both ribozyme and water were obtained by umbrella sampling combined with SCCDFTB/MM. The calculated free energy barriers for the ribozyme and water reactions are in good agreement with the experimental values. As expected, Mulliken charges of the atoms involved in the ribozyme reaction change in a similar manner as that of the reaction in water. The proficiency of the Diels-Alder ribozyme reaction originates from the active site holding the two reactants in reactive conformations, in which the reacting atoms are brought together in van der Waals distances and reactants approach to each other at an appropriate angle.