The parameterization and validation of generalized born models using the pairwise descreening approximation

Generalized Born Surface Area (GBSA) models for water using the Pairwise Descreening Approximation (PDA) have been parameterized by two different methods. The first method, similar to that used in previously reported parameterizations, optimizes all parameters against the experimental free energies of hydration of organic molecules. The second method optimizes the PDA parameters to compensate only for systematic errors of the PDA. The best models are compared to Poisson-Boltzmann calculations and applied to the computation of potentials of mean force (PMFs) for the association of various molecules. PMFs present a more rigorous test of the ability of a solvation model to correctly reproduce the screening of intermolecular interactions by the solvent, than its accuracy at predicting free energies of hydration of small molecules. Models derived with the first method are sometimes shown to fail to compute accurate potentials of mean force because of large errors in the computation of Born radii, while no such difficulties are observed with the second method. Furthermore, accurate computation of the Born radii appears to be more important than good agreement with experimental free energies of solvation. We discuss the source of errors in the potentials of mean force and suggest means to reduce them. Our findings suggest that Generalized Born models that use the Pairwise Descreening Approximation and that are derived solely by unconstrained optimization of parameters against free energies of hydration should be applied to the modeling of intermolecular interactions with caution.     

Is Poisson-Boltzmann theory insufficient for protein folding simulations?

The Poisson-Boltzmann theory has been widely used in the studies of energetics and conformations of biological macromolecules. Recently, introduction of the efficient generalized Born approximation has greatly extended its applicability to areas such as protein folding simulations where highly efficient computation is crucial. However, limitations have been found in the folding simulations of a well-studied beta hairpin with several generalized Born implementations and different force fields. These studies have raised the question whether the underlining Poisson-Boltzmann theory, on which the generalized Born model is calibrated, is adequate in the treatment of polar interactions for the challenging protein folding simulations. To address the question whether the Poisson-Boltzmann theory in the current formalism might be insufficient, we directly tested our efficient numerical Poisson-Boltzmann implementation in the beta-hairpin folding simulation. Good agreement between simulation and experiment was found for the beta-hairpin equilibrium structures when the numerical Poisson-Boltzmann solvent and a recently improved generalized Born solvent were used. In addition simulated thermodynamic properties also agree well with experiment in both solvents. Finally, an overall agreement on the beta-hairpin folding mechanism was found between the current and previous studies. Thus, our simulations indicate that previously observed limitations are most likely due to imperfect calibration in previous generalized Born models but not due to the limitation of the Poisson-Boltzmann theory.     

Dispersion forces at arbitrary distances

The formalism of Boehm and Yaris is used to evaluate explicitly the leading term of the London dispersion force between closed-shell atoms. Instead of using the usual multipole expansion, which breaks down at intermediate internuclear distances, an analytic representation of the Born amplitude together with a general angular momentum analysis is used. As a result expressions are obtained which reduce to the usual dispersion forces at large distances and are finite at all distances.     

Adjustment of Born‐Oppenheimer electronic wave functions to simplify close coupling calculations

Technical problems connected with use of the Born‐Oppenheimer clamped‐nuclei approximation to generate electronic wave functions, potential energy surfaces (PES), and associated properties are discussed. A computational procedure for adjusting the phases of the wave functions, as well as their order when potential crossings occur, is presented which is based on the calculation of overlaps between sets of molecular orbitals and configuration interaction eigenfunctions obtained at neighboring nuclear conformations. This approach has significant advantages for theoretical treatments describing atomic collisions and photo‐dissociation processes by means of ab initio PES, electronic transition moments, and nonadiabatic radial and rotational coupling matrix elements. It ensures that the electronic wave functions are continuous over the entire range of nuclear conformations considered, thereby greatly simplifying the process of obtaining the above quantities from the results of single‐point Born‐Oppenheimer calculations. The overlap results are also used to define a diabatic transformation of the wave functions obtained for conical intersections that greatly simplifies the computation of off‐diagonal matrix elements by eliminating the need for complex phase factors. © 2013 Wiley Periodicals, Inc.     

Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. I. Formalism

An efficient and general method for the analytic computation of the nonandiabatic coupling vector at the multireference configuration interaction (MR-CI) level is presented. This method is based on a previously developed formalism for analytic MR-CI gradients adapted to the use for the computation of nonadiabatic coupling terms. As was the case for the analytic energy gradients, very general, separate choices of invariant orbital subspaces at the multiconfiguration self-consistent field and MR-CI levels are possible, allowing flexible selections of MR-CI wave functions. The computational cost for the calculation of the nonadiabatic coupling vector at the MR-CI level is far below the cost for the energy calculation. In this paper the formalism of the method is presented and in the following paper [Dallos et al., J. Chem. Phys. 120, 7330 (2004)] applications concerning the optimization of minima on the crossing seam are described.     

Computation of derivatives for parameter optimization in least-squares fitting of linear combinations of Slater-type orbitals by Gaussians

The approximation of a linear combination of Slater-type orbitals in terms of Gaussian functions is a many-parameter optimization problem. Formulas for computation of the gradient of the overlap in parameter space are reported. An alternative method of computing the gradient is described, which is of general applicability. This technique permits the exact evaluation of a derivative, without derivation and programming of its analytic expression.     

Rapid estimation of relative protein-ligand binding affinities using a high-throughput version of MM-PBSA

By employing a modified protocol of the Molecular Mechanics with Poisson-Boltzmann Surface Area (MM-PBSA) methodology we substantially decrease the required computation time for calculating relative estimates of protein-ligand binding affinities. The modified method uses a generalized Born implicit solvation model during molecular dynamics to enhance conformational sampling as well as a very efficient Poisson-Boltzmann solver and a computational design based on a distributed-computing paradigm. This construction allows for reduction of the computational cost of the calculations by roughly 2 orders of magnitude compared to the traditional formulation of MM-PBSA. With this high-throughput version of MM-PBSA we show that one can produce efficient physics-based estimates of relative binding free energies with reasonable correlation to experimental data and a total computation time that is sufficiently low such that an industrially relevant throughput can be realized given currently accessible computing resources. We demonstrate this approach by performing a comparison of different MM-PBSA implementations on a set of 18 ligands for the protein target urokinase.     

New analytic approximation to the standard molecular volume definition and its application to generalized Born calculations

In a recent article (Lee, M. S.; Salsbury, F. R. Jr.; Brooks, C. L., III. J Chem Phys 2002, 116, 10606), we demonstrated that generalized Born (GB) theory provides a good approximation to Poisson electrostatic solvation energy calculations if one uses the same definitions of molecular volume for each. In this work, we present a new and improved analytic method for reproducing the Lee-Richards molecular volume, which is the most common volume definition for Poisson calculations. Overall, 1% errors are achieved for absolute solvation energies of a large set of proteins and relative solvation energies of protein conformations. We also introduce an accurate SASA approximation that uses the same machinery employed by our GB method and requires a small addition of computational cost. The combined methodology is shown to yield an efficient and accurate implicit solvent representation for simulations of biopolymers.     

An analytic model for atomic clusters

The paper presents an analytic model which is based on approximation of the catchment basins in cluster potential energy surface by structures of ash-tray type. In order to see how the quantities to be specified relate to those obtained by means of numerical computation, predictions for the Ar₁₃ caloric curve are made and they are compared with computer simulation data.     

Accuracy assessment of the linear Poisson–Boltzmann equation and reparametrization of the OBC generalized Born model for nucleic acids and nucleic acid–protein complexes

The generalized Born model in the Onufriev, Bashford, and Case (Onufriev et al., Proteins: Struct Funct Genet 2004, 55, 383) implementation has emerged as one of the best compromises between accuracy and speed of computation. For simulations of nucleic acids, however, a number of issues should be addressed: (1) the generalized Born model is based on a linear model and the linearization of the reference Poisson–Boltmann equation may be questioned for highly charged systems as nucleic acids; (2) although much attention has been given to potentials, solvation forces could be much less sensitive to linearization than the potentials; and (3) the accuracy of the Onufriev–Bashford–Case (OBC) model for nucleic acids depends on fine tuning of parameters. Here, we show that the linearization of the Poisson Boltzmann equation has mild effects on computed forces, and that with optimal choice of the OBC model parameters, solvation forces, essential for molecular dynamics simulations, agree well with those computed using the reference Poisson–Boltzmann model. © 2015 Wiley Periodicals, Inc.     

Implicit solvation based on generalized Born theory in different dielectric environments

In this paper we are investigating the effect of the dielectric environment on atomic Born radii used in generalized Born (GB) methods. Motivated by the Kirkwood expression for the reaction field of a single off-center charge in a spherical cavity, we are proposing extended formalisms for the calculation of Born radii as a function of external and internal dielectric constants. We demonstrate that reaction field energies calculated from environmentally dependent Born radii lead to much improved agreement with Poisson-Boltzmann solutions for low dielectric external environments, such as biological membranes or organic solvent, compared to previous methods where the calculation of Born radii does not depend on the environment. We also examine how this new approach can be applied for the calculation of transfer free energies from vacuum to a given external dielectric for a system with an internal dielectric larger than one. This has not been possible with standard GB theory but is relevant when scoring minimized or average structures with implicit solvent.     

Molecular structure calculations without clamping the nuclei

A number of recent papers have considered ways in which molecular structure may be calculated when both the electrons and the nuclei are treated from the outset as quantum particles. This is in contrast to the conventional approach in which the nuclei initially have their positions fixed and so merely provide a potential for electronic motion. The usual approach is generally assumed to be justified by the 1927 work of Born and Oppenheimer. In this paper we discuss what precisely might be anticipated in the way of molecular structure from a mathematical consideration of the spectral properties of the full Coulomb Hamiltonian, to what extent the very idea of molecular structure might be dependent upon treating the nuclei simply as providing a potential and the extent to which the work of Born and Oppenheimer can be used to support this position.     

Density functional theory of born couplings: Consequences for electron flow in Jahn–Teller molecules and superconductors

The dynamics of Jahn–Teller systems has recently been discussed in terms of generalized electronic charge and current densities in nuclear-coordinate space. The introduction of the electronic phase as a function of both electronic and nuclear coordinates, in addition to the electronic density, was a crucial component of this formulation. Here, a densitybased treatment of Born couplings is derived from first-principles quantum mechanics beyond the Born–Oppenheimer approximation. Because of the degenerate electronic configuration of a Jahn–Teller molecule, there are an infinite number of ways in which the charge distribution can be oriented for the same energy, leading to a vanishing bond hardness for the molecule in the symmetric nuclear configuration. Further, the moving nuclear framework serves as the perturbation necessary to define the orientation of the charge density, leading to unhindered rotation of the charge cloud. This leads to the dynamical Jahn–Teller problem, namely, the coupling of electronic and nuclear motions through the Born coupling terms. Applications to superconductivity theory are discussed. © 1995 John Wiley & Sons, Inc.     

Optical oscillator strengths for lithium

A simple two-parameter analytic potential adjusted so as to reproduce the experimental energy levels is used to generate wave functions for the ground and excited states of the lithium atom. Using these wave functions in conjunction with the Born approximation and the Russell–Saunders LS-coupling scheme, we calculate the optical oscillator strengths for various excitations from the 1s²2s(²S_1/2) ground state. The results are compared to experiment and other calculations.     

New Born radii deriving method for Generalized Born model

Here we report a method to calculate Born radii, an important parameter used in a Generalized Born model. Traditional methods to derive Born radii are mostly based on a complicated formula, while our method is easier and more direct. Atoms are classified according to their atom type, and the Born radii of each type are obtained by fitting to experimental solvation free energy. The SMARTS language is used for the exact definition of atoms types, and Ullmann's subgraph isomorphism algorithm is used to deduce the environment. A generic algorithm is used for the parameter fitting because of its efficiency in searching a huge phase space, and its results are then optimized by using the conjugate gradient method. The final parameter set is fitting from a training set containing 357 molecules and is tested using a test set of 44 small organic molecules, and the average error is 0.58 kcal/mol for 36 neutral molecules and is 1.67 kcal/mol for 8 ions. The model is further tested under organic molecules, biopolymers, and a protein-inhibitor complex and yields reliable results in all these cases. This method can be used to accelerate molecular docking calculations.     

Transport properties of normal liquid helium: comparison of various methodologies

We revisit the problem of self-diffusion in normal liquid helium above the lambda transition. Several different methods are applied to compute the velocity autocorrelation function. Since it is still impossible to determine the exact result for the velocity autocorrelation function from simulation, we appeal to the computation of short-time moments to determine the accuracy of the different approaches at short times. The main conclusion reached from our study is that both the quantum mode-coupling theory and the numerical analytic continuation approach must be regarded as a viable and competitive methods for the computation of dynamical properties of quantum systems.     

An improved Born approximation for the electron impact ionization of atomic hydrogen

A two-electron continuum wave function satisfying exact asymptotic boundary conditions (see [1]) is simplified in the high energy limit for asymmetric kinematics. The long range correlation between the fast escaping electron and the target atom in a low continuum state is properly taken into account, and corrects therefore the major shortcoming of the ordinary Born approximation. This improved Born approximation has been employed to calculate triply differential ionization cross sections at intermediate energies.     

A high‐performance parallel‐generalized born implementation enabled by tabulated interaction rescaling

Implicit solvent representations, in general, and generalized Born models, in particular, provide an attractive way to reduce the number of interactions and degrees of freedom in a system. The instantaneous relaxation of the dielectric shielding provided by an implicit solvent model can be extremely efficient for high‐throughput and Monte Carlo studies, and a reduced system size can also remove a lot of statistical noise. Despite these advantages, it has been difficult for generalized Born implementations to significantly outperform optimized explicit‐water simulations due to more complex functional forms and the two extra interaction stages necessary to calculate Born radii and the derivative chain rule terms contributing to the force. Here, we present a method that uses a rescaling transformation to make the standard generalized Born expression a function of a single variable, which enables an efficient tabulated implementation on any modern CPU hardware. The total performance is within a factor 2 of simulations in vacuo. The algorithm has been implemented in Gromacs, including single‐instruction multiple‐data acceleration, for three different Born radius models and corresponding chain rule terms. We have also adapted the model to work with the virtual interaction sites commonly used for hydrogens to enable long‐time steps, which makes it possible to achieve a simulation performance of 0.86 μs/day for BBA5 with 1‐nm cutoff on a single quad‐core desktop processor. Finally, we have also implemented a set of streaming kernels without neighborlists to accelerate the non‐cutoff setup occasionally used for implicit solvent simulations of small systems. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010