DelPhiForce,a tool for electrostatic force calculations: Applications to macromolecular binding

Long‐range electrostatic forces play an important role in molecular biology, particularly in macromolecular interactions. However, calculating the electrostatic forces for irregularly shaped molecules immersed in water is a difficult task. Here, we report a new tool, DelPhiForce, which is a tool in the DelPhi package that calculates and visualizes the electrostatic forces in biomolecular systems. In parallel, the DelPhi algorithm for modeling electrostatic potential at user‐defined positions has been enhanced to include triquadratic and tricubic interpolation methods. The tricubic interpolation method has been tested against analytical solutions and it has been demonstrated that the corresponding errors are negligibly small at resolution 4 grids/Å. The DelPhiForce is further applied in the study of forces acting between partners of three protein–protein complexes. It has been demonstrated that electrostatic forces play a dual role by steering binding partners (so that the partners recognize their native interfaces) and exerting an electrostatic torque (if the mutual orientations of the partners are not native‐like). The output of DelPhiForce is in a format that VMD can read and visualize, and provides additional options for analysis of protein–protein binding. DelPhiForce is available for download from the DelPhi webpage at http://compbio.clemson.edu/downloadDir/delphiforce.tar.gz © 2017 Wiley Periodicals, Inc.     

A rapid finite difference algorithm,utilizing successive over-relaxation to solve the Poisson–Boltzmann equation

An efficient algorithm is presented for the numerical solution of the Poisson–Boltzmann equation by the finite difference method of successive over-relaxation. Improvements include the rapid estimation of the optimum relaxation parameter, reduction in number of operations per iteration, and vector-oriented array mapping. The algorithm has been incorporated into the electrostatic program DelPhi, reducing the required computing time by between one and two orders of magnitude. As a result the estimation of electrostatic effects such as solvent screening, ion distributions, and solvation energies of small solutes and biological macromolecules in solution, can be performed rapidly, and with minimal computing facilities.     

Modeling the electrostatic potential of asymmetric lipopolysaccharide membranes: The MEMPOT algorithm implemented in DelPhi

Four chemotypes of the rough lipopolysaccharides (LPS) membrane from Pseudomonas aeruginosa were investigated by a combined approach of explicit water molecular dynamics (MD) simulations and Poisson–Boltzmann continuum electrostatics with the goal to deliver the distribution of the electrostatic potential across the membrane. For the purpose of this investigation, a new tool for modeling the electrostatic potential profile along the axis normal to the membrane, MEMbrane POTential (MEMPOT), was developed and implemented in DelPhi. Applying MEMPOT on the snapshots obtained by MD simulations, two observations were made: (a) the average electrostatic potential has a complex profile but is mostly positive inside the membrane due to the presence of Ca²⁺ ions, which overcompensate for the negative potential created by lipid phosphate groups; and (b) correct modeling of the electrostatic potential profile across the membrane requires taking into account the water phase, while neglecting it (vacuum calculations) results in dramatic changes including a reversal of the sign of the potential inside the membrane. Furthermore, using DelPhi to assign different dielectric constants for different regions of the LPS membranes, it was investigated whether a single frame structure before MD simulations with appropriate dielectric constants for the lipid tails, inner, and the external leaflet regions, can deliver the same average electrostatic potential distribution as obtained from the MD‐generated ensemble of structures. Indeed, this can be attained by using smaller dielectric constant for the tail and inner leaflet regions (mostly hydrophobic) than for the external leaflet region (hydrophilic) and the optimal dielectric constant values are chemotype‐specific. © 2014 Wiley Periodicals, Inc.     

Continuous development of schemes for parallel computing of the electrostatics in biological systems: Implementation in DelPhi

Due to the enormous importance of electrostatics in molecular biology, calculating the electrostatic potential and corresponding energies has become a standard computational approach for the study of biomolecules and nano‐objects immersed in water and salt phase or other media. However, the electrostatics of large macromolecules and macromolecular complexes, including nano‐objects, may not be obtainable via explicit methods and even the standard continuum electrostatics methods may not be applicable due to high computational time and memory requirements. Here, we report further development of the parallelization scheme reported in our previous work (Li, et al., J. Comput. Chem. 2012, 33, 1960) to include parallelization of the molecular surface and energy calculations components of the algorithm. The parallelization scheme utilizes different approaches such as space domain parallelization, algorithmic parallelization, multithreading, and task scheduling, depending on the quantity being calculated. This allows for efficient use of the computing resources of the corresponding computer cluster. The parallelization scheme is implemented in the popular software DelPhi and results in speedup of several folds. As a demonstration of the efficiency and capability of this methodology, the electrostatic potential, and electric field distributions are calculated for the bovine mitochondrial supercomplex illustrating their complex topology, which cannot be obtained by modeling the supercomplex components alone. © 2013 Wiley Periodicals, Inc.     

Rapid grid-based construction of the molecular surface and the use of induced surface charge to calculate reaction field energies: applications to the molecular systems and geometric objects

This article describes a number of algorithms that are designed to improve both the efficiency and accuracy of finite difference solutions to the Poisson-Boltzmann equation (the FDPB method) and to extend its range of application. The algorithms are incorporated in the DelPhi program. The first algorithm involves an efficient and accurate semianalytical method to map the molecular surface of a molecule onto a three-dimensional lattice. This method constitutes a significant improvement over existing methods in terms of its combination of speed and accuracy. The DelPhi program has also been expanded to allow the definition of geometrical objects such as spheres, cylinders, cones, and parallelepipeds, which can be used to describe a system that may also include a standard atomic level depiction of molecules. Each object can have a different dielectric constant and a different surface or volume charge distribution. The improved definition of the surface leads to increased precision in the numerical solutions of the PB equation that are obtained. A further improvement in the precision of solvation energy calculations is obtained from a procedure that calculates induced surface charges from the FDPB solutions and then uses these charges in the calculation of reaction field energies. The program allows for finite difference grids of large dimension; currently a maximum of 571(3) can be used on molecules containing several thousand atoms and charges. As described elsewhere, DelPhi can also treat mixed salt systems containing mono- and divalent ions and provide electrostatic free energies as defined by the nonlinear PB equation.     

Comparison of methods to estimate the free energy of solvation:Importance in the modulation of the affinity of 3-benzazepines for the D1receptor

We computed the free energy of solvation for a series of ions and neutral molecules using two different continuum approaches. First, we used the AM1–SM1 technique, where the AM1 Fock matrix is modified to include a generalized Born contribution. Second, we applied the DelPhi approach, where the electrostatic component of the free energy of solvation is evaluated by resolving the Poisson–Boltzman equation by a finite difference method. Both methods appear equally reliable for ionic systems. For neutral compounds, AM1–SM1 performs better than DelPhi; however, the differences become less pronounced for compounds with larger free energies of solvation. In parallel, both methods were applied to study the influence of the solvation process in the overall drug receptor interaction for a series of closely related ligands for the D₁ dopamine receptor. An inverse linear relationship was found between the free energy of solvation and the logarithm of the affinity of the ligands; nevertheless, electrostatic properties are likely to modulate affinity as well. © 1993 John Wiley & Sons, Inc.     

Extended implementation of canonical transformation theory: parallelization and a new level-shifted condition

The canonical transformation (CT) theory has been developed as a multireference electronic structure method to compute high-level dynamic correlation on top of a large active space reference treated with the ab initio density matrix renormalization group method. This article describes a parallelized algorithm and implementation of the CT theory to handle large computational demands of the CT calculation, which has the same scaling as the coupled cluster singles and doubles theory. To stabilize the iterative solution of the CT method, a modification to the CT amplitude equation is introduced with the inclusion of a level shift parameter. The level-shifted condition has been found to effectively remove a type of intruder state that arises in the linear equations of CT and to address the discontinuity problems in the potential energy curves observed in the previous CT studies.     

Statistical mechanics of energy transfer in gas–surface scattering: A dynamical Lie algebraic approach

The dynamical Lie algebraic (DLA) method is used to describe statistical mechanics of energy transfer in rotationally inelastic molecule–surface scattering. Statistical average values of an observable for the scattering system are calculated in terms of density operator formalism in statistical mechanics. Employing a cubic expansion procedure of molecule–surface interaction potential leads to generation of a dynamical Lie algebra. Thus these statistical average values as a function of the group parameters can be obtained analytically in this formulation. The group parameters can be found from solving a set of coupled nonlinear differential equations. The DLA method, which has no need for determination of transition probabilities in advance as made routinely in the calculation, offers an efficient alternative to the method for computing the statistical average values. This method is much less computationally intensive because most of calculations can be analytically carried out. The average final rotational energies and their dependence on the main dynamic variables and the average interaction potential are presented for the rotationally inelastic scattering of NO molecules from a flat, static Ag(111) surface. Direct comparison is made between the predictions of this model calculation and experiment. The model reproduces well the degree of rotational excitation and correlation between the average final translational and the average rotational energies. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Achieving linear-scaling computational cost for the polarizable continuum model of solvation

This work describes a new and low-scaling implementation of the polarizable continuum model (PCM) for computing the self-consistent solvent reaction field. The PCM approach is both general and accurate. It is applicable in the framework of both quantum and classical calculations, and also to hybrid quantum/classical methods. In order to further extend the range of applicability of PCM we addressed the problem of its computational cost. The generation of the finite-elements molecular cavity has been reviewed and reimplemented, achieving linear scaling for systems containing up to 500 atoms. Linear scaling behavior has been achieved also for the iterative solution of the PCM equations, by exploiting the fast multipole method (FMM) for computing electrostatic interactions. Numerical results for large (both linear and globular) chemical systems are discussed.     

Multiple solutions of coupled-cluster equations for PPP model of [10]annulene

Multiple (real) solutions of the coupled-cluster (CC) equations (corresponding to the CCD, ACP and ACPQ methods) are studied for the Pariser–Parr–Pople (PPP) model of [10]annulene, C₁₀H₁₀. The long-range electrostatic interactions are represented either by the Mataga–Nishimoto potential, or Pople’s R⁻¹ potential. The multiple solutions are obtained in a quasi-random manner, by generating a pool of starting amplitudes and applying a standard CC iterative procedure combined with Pulay’s DIIS method. Several unexpected features of these solutions are uncovered, including the switching between two CCD solutions when moving between the weakly and strongly correlated regime of the PPP model with Pople’s potential.     

Efficient treatment of solvation shells in 3D molecular theory of solvation

We developed a technique to decrease memory requirements when solving the integral equations of three‐dimensional (3D) molecular theory of solvation, a.k.a. 3D reference interaction site model (3D‐RISM), using the modified direct inversion in the iterative subspace (MDIIS) numerical method of generalized minimal residual type. The latter provides robust convergence, in particular, for charged systems and electrolyte solutions with strong associative effects for which damped iterations do not converge. The MDIIS solver (typically, with 2 × 10 iterative vectors of argument and residual for fast convergence) treats the solute excluded volume (core), while handling the solvation shells in the 3D box with two vectors coupled with MDIIS iteratively and incorporating the electrostatic asymptotics outside the box analytically. For solvated systems from small to large macromolecules and solid–liquid interfaces, this results in 6‐ to 16‐fold memory reduction and corresponding CPU load decrease in MDIIS. We illustrated the new technique on solvated systems of chemical and biomolecular relevance with different dimensionality, both in ambient water and aqueous electrolyte solution, by solving the 3D‐RISM equations with the Kovalenko–Hirata (KH) closure, and the hypernetted chain (HNC) closure where convergent. This core–shell‐asymptotics technique coupling MDIIS for the excluded volume core with iteration of the solvation shells converges as efficiently as MDIIS for the whole 3D box and yields the solvation structure and thermodynamics without loss of accuracy. Although being of benefit for solutes of any size, this memory reduction becomes critical in 3D‐RISM calculations for large solvated systems, such as macromolecules in solution with ions, ligands, and other cofactors. © 2012 Wiley Periodicals, Inc.     

A reduced-restricted-quasi-Newton–Raphson method for locating and optimizing energy crossing points between two potential energy surfaces

We present a method for the location and optimization of an intersection energy point between two potential energy surfaces. The procedure directly optimizes the excited state energy using a quasi-Newton–Raphson method coupled with a restricted step algorithm. A linear transformation is also used for the solution of the quasi-Newton–Raphson equations. The efficiency of the algorithm is analyzed and demonstrated in some examples. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :992–1003, 1997     

Efficient parallelization of the energy,surface, and derivative calculations for internal coordinate mechanics

An efficient algorithm for parallelization of a molecular mechanics program operating in the space of internal coordinates such as dihedral angles, bond angles, and bond lengths is described. The iterative procedure to calculate analytical energy derivatives with respect to the internal coordinates was modified to allow parallelization. Computationally intensive modules that calculate energy and its derivatives, solvent-accessible surface, electrostatic polarization energy and that update lists of interactions were parallelized with nearly 100% efficiency. The proposed strategy for the shared-memory computer architecture is easily scalable and requires minimum changes in a program code. The overall speedup for a realistic calculation minimizing the energy of a myoglobin reaches a factor of 3 for 4 processors. © 1994 by John Wiley & Sons, Inc.     

Double Layer at the Pt(111)–Aqueous Electrolyte Interface: Potential of Zero Charge and Anomalous Gouy–Chapman Screening

We report, for the first time, the observation of a Gouy–Chapman capacitance minimum at the potential of zero charge of the Pt(111)‐aqueous perchlorate electrolyte interface. The potential of zero charge of 0.3 V vs. NHE agrees very well with earlier values obtained by different methods. The observation of the potential of zero charge of this interface requires a specific pH (pH 4) and anomalously low electrolyte concentrations (<10^?3 m ). By comparison to gold and mercury double‐layer data, we conclude that the diffuse double layer structure at the Pt(111)‐electrolyte interface deviates significantly from the Gouy–Chapman theory in the sense that the electrostatic screening is much better than predicted by purely electrostatic mean‐field Poisson–Boltzmann theory.     

A near linear-scaling smooth local coupled cluster algorithm for electronic structure

We demonstrate near linear scaling of a new algorithm for computing smooth local coupled-cluster singles-doubles (LCCSD) correlation energies of quantum mechanical systems. The theory behind our approach has been described previously, [J. Subotnik and M. Head-Gordon, J. Chem. Phys. 123, 064108 (2005)], and requires appropriately multiplying standard iterative amplitude equations by a bump function, creating local amplitude equations (which are smooth according to the implicit function theorem). Here, we provide an example that this theory works in practice: we show that our algorithm leads to smooth potential energy surfaces and yields large computational savings. As an example, we apply our LCCSD approach to measure the post-MP2 correction to the energetic gap between two different alanine tetrapeptide conformations.     

Fast evaluation of polarizable forces

Polarizability is considered to be the single most significant development in the next generation of force fields for biomolecular simulations. However, the self-consistent computation of induced atomic dipoles in a polarizable force field is expensive due to the cost of solving a large dense linear system at each step of a simulation. This article introduces methods that reduce the cost of computing the electrostatic energy and force of a polarizable model from about 7.5 times the cost of computing those of a nonpolarizable model to less than twice the cost. This is probably sufficient for the routine use of polarizable forces in biomolecular simulations. The reduction in computing time is achieved by an efficient implementation of the particle-mesh Ewald method, an accurate and robust predictor based on least-squares fitting, and non-stationary iterative methods whose fast convergence is accelerated by a simple preconditioner. Furthermore, with these methods, the self-consistent approach with a larger timestep is shown to be faster than the extended Lagrangian approach. The use of dipole moments from previous timesteps to calculate an accurate initial guess for iterative methods leads to an energy drift, which can be made acceptably small. The use of a zero initial guess does not lead to perceptible energy drift if a reasonably strict convergence criterion for the iteration is imposed.     

Determination and quantification of astragalosides in Radix Astragali and its medicinal products using LC–MS

In the present study, a liquid chromatography–tandem mass spectrometry method was developed for the separation and simultaneous quantification of astragalosides I–IV in samples of Radix Astragali and a medicinal product thereof (Jinqi Jiangtang tablets). Chromatographic separation was achieved on an Agilent Eclipse XDB (ODS)‐C18 column with a mobile phase consisting of acetonitrile and 0.05% formic acid aqueous solution by use of an efficient 17‐min program. A triple quadrupole mass spectrometer was operated in positive ionization mode with multiple reaction monitoring for the detection of four astragalosides. The saponin ginsenoside Rg1 (similar structure to astralagosides) was used as an internal standard. All calibration curves showed excellent linear regressions (r² ? 0.9912) within the range of tested concentrations. The intra‐ and inter‐day variations were below 4.57% in terms of RSD. The recoveries were 94.38–103.53% with RSD of 1.39–3.58% for spiked Radix Astragali samples. The method was successfully used for the analysis of samples of Radix Astragali and Jinqi Jiangtang tablets. In conclusion, we have developed a rapid, efficient, and accurate LC–MS/MS method for the detection of astragalosides, which can be applied for quality control of Radix Astragali and related medicinal products.     

An assessment of the accuracy of the RRIGS hydration potential: Comparison to solutions of the Poisson–Boltzmann equation

A rapid, pairwise hydration potential, the reduced radius independent Gaussian sphere (RRIGS) approximation, has been presented recently. Because experimental values of the conformational dependence of the hydration free energy are unavailable, this hydration potential is testable by comparison to a presumably more accurate (yet more computationally intensive) model. One such method is the electrostatic hydration approach, which models the protein as a collection of point charges in a low-dielectric medium and the solvent as a high-dielectric continuum. The electrostatic free energy can be determined by solving the Poisson–Boltzmann equation, which is carried out with the program DelPhi. The total free energy of hydration is calculated by adding a free energy of cavity formation term to this electrostatic term. Comparison is made for many conformations of two proteins, bovine pancreatic trypsin inhibitor (BPTI) and the carboxy-terminal fragment of the L7/L12 ribosomal protein (CTF). Thirty-nine near-native structures of BPTI, previously generated by Ripoll and coworkers, and 150 conformations of CTF, generated by a threading algorithm to cover a wide range of conformational space, were used in these comparisons. It is shown that, for the neutral forms of these proteins, the RRIGS hydration potential correlates very well with the electrostatic model hydration free energy, although the correlation is better for the CTF conformations than for the near-native BPTI conformations. For charged forms, the correlation is much poorer. These results serve as evidence that solvent-exposure models of hydration, which leave out cooperative effects between different groups, may be appropriate for modeling neutral or slightly charged species, because these cooperative effects are likely to be small. However, for highly charged species where cooperative effects are surely large, such an approach will be less accurate. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :1072–1078, 1997