A strategy for reducing gross errors in the generalized Born models of implicit solvation

The "canonical" generalized Born (GB) formula [C. Still, A. Tempczyk, R. C. Hawley, and T. Hendrickson, J. Am. Chem. Soc. 112, 6127 (1990)] is known to provide accurate estimates for total electrostatic solvation energies ΔG(el) of biomolecules if the corresponding effective Born radii are accurate. Here we show that even if the effective Born radii are perfectly accurate, the canonical formula still exhibits significant number of gross errors (errors larger than 2k(B)T relative to numerical Poisson equation reference) in pairwise interactions between individual atomic charges. Analysis of exact analytical solutions of the Poisson equation (PE) for several idealized nonspherical geometries reveals two distinct spatial modes of the PE solution; these modes are also found in realistic biomolecular shapes. The canonical GB Green function misses one of two modes seen in the exact PE solution, which explains the observed gross errors. To address the problem and reduce gross errors of the GB formalism, we have used exact PE solutions for idealized nonspherical geometries to suggest an alternative analytical Green function to replace the canonical GB formula. The proposed functional form is mathematically nearly as simple as the original, but depends not only on the effective Born radii but also on their gradients, which allows for better representation of details of nonspherical molecular shapes. In particular, the proposed functional form captures both modes of the PE solution seen in nonspherical geometries. Tests on realistic biomolecular structures ranging from small peptides to medium size proteins show that the proposed functional form reduces gross pairwise errors in all cases, with the amount of reduction varying from more than an order of magnitude for small structures to a factor of 2 for the largest ones.     

Efficient evaluation of the effective dielectric function of a macromolecule in aqueous solution

We propose an analytical approach to calculate the effective dielectric function of proteins in aqueous solution. The screening effect if quantified by a measure of enclosure which is based on the distribution of solute atomic volumes around a pair of charges in a macromolecule. For protein conformations that vary significantly in size and shape, a comparison with finite difference Poisson calculations shows that pair interaction energies, their sums and solvation energies are well reproduced. The approach rivals the speed of simple distance dependent dielectric functions and the accuracy of the generalized Born model.     

Solvation model based on order parameters and a fast sampling method for the calculation of the solvation free energies of peptides

An analytical solvation model is proposed as a function of an order parameter, which represents the local arrangement of water molecules in the first solvation shell of peptide atoms. The model is combined with a fast sampling method, rotational isomeric state Monte Carlo, to sample efficiently the torsional degrees of freedom on a peptide backbone. This order parameter solvation model is shown to reproduce without ad hoc fitting parameters the solvation free energies of single amino acids and tripeptides with slightly better accuracy than the generalized Born model but with several orders of magnitude improvement in efficiency. This method is a potential candidate for efficiently and accurately tackling some important issues in biophysical chemistry that are related to solvation, for example, protein folding, ligand binding, etc. Our results also present fundamental new insights into solvation. Specifically, the local water geometry, represented in this work by a properly defined order parameter, carries the majority, if not all, of the energetic information of solvation, including solute-solvent interactions and solvent reorganization in the presence of the solute.     

The importance of excluded solvent volume effects in computing hydration free energies

Continuum dielectric methods such as the Born equation have been widely used to compute the electrostatic component of the solvation free energy, DeltaG(solv)(elec), because they do not need to include solvent molecules explicitly and are thus far less costly compared to molecular simulations. All of these methods can be derived from Gauss Law of Maxwell's equations, which yields an analytical solution for the solvation free energy, DeltaG(Born), when the solute is spherical. However, in Maxwell's equations, the solvent is assumed to be a structureless continuum, whereas in reality, the near-solute solvent molecules are highly structured unlike far-solute bulk solvent. Since we have recently reformulated Gauss Law of Maxwell's equations to incorporate the near-solute solvent structure by considering excluded solvent volume effects, we have used it in this work to derive an analytical solution for the hydration free energy of an ion. In contrast to continuum solvent models, which assume that the normalized induced solvent electric dipole density P(n) is constant, P(n) mimics that observed from simulations. The analytical formula for the ionic hydration free energy shows that the Born radius, which has been used as an adjustable parameter to fit experimental hydration free energies, is no longer ill defined but is related to the radius and polarizability of the water molecule, the hydration number, and the first peak position of the solute-solvent radial distribution function. The resulting DeltaG(solv)(elec) values are shown to be close to the respective experimental numbers.     

Balancing an accurate representation of the molecular surface in generalized born formalisms with integrator stability in molecular dynamics simulations

Different integrator time steps in NVT and NVE simulations of protein and nucleic acid systems are tested with the GBMV (Generalized Born using Molecular Volume) and GBSW (Generalized Born with simple SWitching) methods. The simulation stability and energy conservation is investigated in relation to the agreement with the Poisson theory. It is found that very close agreement between generalized Born methods and the Poisson theory based on the commonly used sharp molecular surface definition results in energy drift and simulation artifacts in molecular dynamics simulation protocols with standard 2-fs time steps. New parameters are proposed for the GBMV method, which maintains very good agreement with the Poisson theory while providing energy conservation and stable simulations at time steps of 1 to 1.5 fs.     

Effective Born radii in the generalized Born approximation: the importance of being perfect

Generalized Born (GB) models provide, for many applications, an accurate and computationally facile estimate of the electrostatic contribution to aqueous solvation. The GB models involve two main types of approximations relative to the Poisson equation (PE) theory on which they are based. First, the self-energy contributions of individual atoms are estimated and expressed as "effective Born radii." Next, the atom-pair contributions are estimated by an analytical function f(GB) that depends upon the effective Born radii and interatomic distance of the atom pairs. Here, the relative impacts of these approximations are investigated by calculating "perfect" effective Born radii from PE theory, and enquiring as to how well the atom-pairwise energy terms from a GB model using these perfect radii in the standard f(GB) function duplicate the equivalent terms from PE theory. In tests on several biological macromolecules, the use of these perfect radii greatly increases the accuracy of the atom-pair terms; that is, the standard form of f(GB) performs quite well. The remaining small error has a systematic and a random component. The latter cannot be removed without significantly increasing the complexity of the GB model, but an alternative choice of f(GB) can reduce the systematic part. A molecular dynamics simulation using a perfect-radii GB model compares favorably with simulations using conventional GB, even though the radii remain fixed in the former. These results quantify, for the GB field, the importance of getting the effective Born radii right; indeed, with perfect radii, the GB model gives a very good approximation to the underlying PE theory for a variety of biomacromolecular types and conformations.     

On the hydrogen atom beyond the Born–Oppenheimer approximation

Recently a new formulation of quantum mechanics has been suggested which is based on the concept of signed particles, that is, classical objects provided with a position, a momentum and a sign simultaneously. In this article, we comment on the plausibility of simulating atomic systems beyond the Born–Oppenheimer approximation by means of the signed particle formulation of quantum mechanics. First, to show the new perspective offered by this new formalism, we provide an example studying quantum tunnelling through a simple Gaussian barrier in terms of the signed particle formulation. Then, we perform a direct simulation of the hydrogen atom as a full quantum two‐body system, showing that the formalism can be a very promising tool for first‐principle‐only quantum chemistry.     

A simple procedure for deriving an electrical double layer equation from the Born—Green—Yvon theory

A simple procedure for deriving electrical double layer equations from electrolyte theories is applied to the Born—Green—Yvon equation. It is shown that when the auto-field effects are not considered in the wall-ion potential, the equation obtained with this procedure leads to gross inconsistencies.     

Distance and exposure dependent effective dielectric function

In an effort to develop a dielectric screening function for molecular dynamics simulations of biomolecules in implicit solvent, effective dielectric constants (D(eff)) for a large number of atom pairs in a typical globular protein are calculated by continuum electrostatics. Plots of D(eff) versus the intercharge distance are in general sigmoidal with the characteristics of the curve depending on the distance of the two charges from the dielectric boundary and, secondarily, on the extent to which the area surrounding each charge is occupied by solvent (the "exposure"). The D(eff) values were fitted to an empirical, analytical function of these parameters that reproduces the data reasonably well, although considerable scatter exists in the range of D(eff) from 30 to 80. In the system used for parameterization, the mean square deviation of electrostatic interaction energies with this function is 0.48 kcal/mol, compared to 1.45 for an analytical Generalized Born model and 1.52 for the linear distance-dependent dielectric model. When tested in other proteins of varying size and compactness, the present function is superior to both of the above models, except for a fully unfolded polypeptide chain, where the Generalized Born model is superior.     

Some remarks on the quantitative expression of the selectivity of an analytical procedure

The selectivity of an analytical method may be defined as expressing the degree to which a component can be determined in the presence of other but similarly behaving components without interference. The method itself will usually serve for determination of the other components, under slightly different conditions. The selectivity is closely related to the resolution of the method and also to the resolution of the instrument used for obtaining the signals. A simple formula is suggested for expressing the percentual degree of selectivity of an analytical procedure, on the basis of signal overlap caused by the interfering components, and can be used quite generally. A distinction is made between the terms analytical selectivity and selectivity of an analytical procedure, and between selectivity and specificity.     

On the calculation of the electron capture within the strong potential born approximation

A new method to perform the calculation the limits of the electron capture cross-section in the Strong Potential Born Approximation is presented. Within this approach, the high velocity limit is obtained by a simple algebraic procedure as an alternative derivation to the complex variable analysis used previously.     

AGBNP: an analytic implicit solvent model suitable for molecular dynamics simulations and high-resolution modeling

We have developed an implicit solvent effective potential (AGBNP) that is suitable for molecular dynamics simulations and high-resolution modeling. It is based on a novel implementation of the pairwise descreening Generalized Born model for the electrostatic component and a new nonpolar hydration free energy estimator. The nonpolar term consists of an estimator for the solute-solvent van der Waals dispersion energy designed to mimic the continuum solvent solute-solvent van der Waals interaction energy, in addition to a surface area term corresponding to the work of cavity formation. AGBNP makes use of a new parameter-free algorithm to calculate the scaling coefficients used in the pairwise descreening scheme to take into account atomic overlaps. The same algorithm is also used to calculate atomic surface areas. We show that excellent agreement is achieved for the GB self-energies and surface areas in comparison to accurate, but much more expensive, numerical evaluations. The parameter-free approach used in AGBNP and the sensitivity of the AGBNP model with respect to large and small conformational changes makes the model suitable for high-resolution modeling of protein loops and receptor sites as well as high-resolution prediction of the structure and thermodynamics of protein-ligand complexes. We present illustrative results for these kinds of benchmarks. The model is fully analytical with first derivatives and is computationally efficient. It has been incorporated into the IMPACT molecular simulation program.     

Electrostatics of ligand binding: parametrization of the generalized Born model and comparison with the Poisson-Boltzmann approach

An accurate and fast evaluation of the electrostatics in ligand-protein interactions is crucial for computer-aided drug design. The pairwise generalized Born (GB) model, a fast analytical method originally developed for studying the solvation of organic molecules, has been widely applied to macromolecular systems, including ligand-protein complexes. However, this model involves several empirical scaling parameters, which have been optimized for the solvation of organic molecules, peptides, and nucleic acids but not for energetics of ligand binding. Studies have shown that a good solvation energy does not guarantee a correct model of solvent-mediated interactions. Thus, in this study, we have used the Poisson-Boltzmann (PB) approach as a reference to optimize the GB model for studies of ligand-protein interactions. Specifically, we have employed the pairwise descreening approximation proposed by Hawkins et al.(1) for GB calculations and DelPhi for PB calculations. The AMBER all-atom force field parameters have been used in this work. Seventeen protein-ligand complexes have been used as a training database, and a set of atomic descreening parameters has been selected with which the pairwise GB model and the PB model yield comparable results on atomic Born radii, the electrostatic component of free energies of ligand binding, and desolvation energies of the ligands and proteins. The energetics of the 15 test complexes calculated with the GB model using this set of parameters also agrees well with the energetics calculated with the PB method. This is the first time that the GB model has been parametrized and thoroughly compared with the PB model for the electrostatics of ligand binding.     

A remark on the Born–Oppenheimer approximation

The method of calculating wave functions for an electron-nucleon system by a variational method originally suggested by Born and Oppenheimer [1] is rigorously investigated. As an application we sketch the calculation of a simple nonadiabatic wave function for the system.     

Ionisation dynamics at intermediate momentum transfer: an (e, 2e) investigation on argon

Relative triple differential cross section for the coplanar asymmetric (e, 2e) reaction in argon have been measured at 1.5 KeV incident energy and 40 eV ejected electron energy in several kinematics. Depending on the scattering angle, ? _a , the chosen kinematics select either ionising collisions belonging to the Bethe ridge (? _a =9.2°) or processes in the intermediate region between the pure dipolar and binary regimes. The more relevant finding is the presence of a minimum in the recoil lobe, almost opposite to the direction of the momentum transfer. This feature is qualitatively explained by a first Born model, which describes the ejected electron by a Coulomb wave-function. This result suggests that in the investigated kinematics the interaction of the slow ejected electron with the residual ion is the dominant effect beyond the first order electron-electron interactions.     

The Gaussian Generalized Born model: application to small molecules

This work presents a Generalized Born model for the computation of the electrostatic component of solvation energies which is based on volume integration. An analytic masking function is introduced to remove Coulombic singularities. This approach leads to analytic formulae for the computation of Born radii, which are differentiable to arbitrary order, and computationally straightforward to implement.     

Kinetic-spectrometric three-dimensional chemiluminescence as an effective analytical tool. Application to the determination of benzo(a)pyrene

Kinetic and spectroscopic methods were used in combination in this work to develop a new analytical tool for use in chemiluminescence detection processes. Specifically, time-resolved chemiluminescence was used jointly with a stopped-flow assembly in order to monitor the chemiluminescence produced in the oxidation of bis(2,4-dinitrophenyl)oxalate (DNPO) by hydrogen peroxide in the presence of a polycyclic aromatic hydrocarbon. Recording of successive two-dimensional spectra during the emission process and treating the acquired spectral data with dedicated software allows the obtainment of three-dimensional chemiluminescence spectra, a result of the joint use of two analytical techniques. Thus, using a flow cell specifically designed for direct coupling to the charge-coupled device (CCD) detector increases the emission intensity without the need for fibre optics. Also, using dedicated software to process the acquired two-dimensional spectra affords a comprehensive kinetic and spectroscopic characterization of the chemiluminescence signal via the three-dimensional spectrum previously obtained. The analytical potential of this new tool was assessed by application to the chemiluminescent reaction between a peroxyoxalate and an oxidant (hydrogen peroxide); the reaction is induced by benzo(a)pyrene, which was used to determine this polycyclic aromatic hydrocarbon in an organic solvent. A linear calibration graph was obtained between 0.5 and 20 mg L(-1). The limit of detection found to be 3.97 μg L(-1) and a relative standard error of 0.64% and a relative standard deviation of 1.87% were obtained. The results reached testify to the usefulness of the proposed analytical tool for simple determinations and its potential for the resolution of complex mixtures or determinations in complex matrices.     

A variational treatment of some excited e ⊗ ε jahn-teller states

Using a simple variational treatment for the first-order Jahn-Teller coupling, E ? ε, all the pairs of states for which i=(v+$\text{1}\text{2}$), v=0,1,2 … are to first order vertically displaced in energy with an energy difference of one vibrational quantum. An analytical expression is given for the reduced matrix element of Child and Longuet-Higgins.