Accurate,robust, and reliable calculations of Poisson–Boltzmann binding energies

Poisson–Boltzmann (PB) model is one of the most popular implicit solvent models in biophysical modeling and computation. The ability of providing accurate and reliable PB estimation of electrostatic solvation free energy, , and binding free energy, , is important to computational biophysics and biochemistry. In this work, we investigate the grid dependence of our PB solver (MIBPB) with solvent excluded surfaces for estimating both electrostatic solvation free energies and electrostatic binding free energies. It is found that the relative absolute error of obtained at the grid spacing of 1.0 Å compared to at 0.2 Å averaged over 153 molecules is less than 0.2%. Our results indicate that the use of grid spacing 0.6 Å ensures accuracy and reliability in calculation. In fact, the grid spacing of 1.1 Å appears to deliver adequate accuracy for high throughput screening. © 2017 Wiley Periodicals, Inc.     

On the accurate estimation of free energies using the jarzynski equality

The Jarzynski equality is one of the most widely celebrated and scrutinized nonequilibrium work theorems, relating free energy to the external work performed in nonequilibrium transitions. In practice, the required ensemble average of the Boltzmann weights of infinite nonequilibrium transitions is estimated as a finite sample average, resulting in the so-called Jarzynski estimator, . Alternatively, the second-order approximation of the Jarzynski equality, though seldom invoked, is exact for Gaussian distributions and gives rise to the Fluctuation-Dissipation estimator . Here we derive the parametric maximum-likelihood estimator (MLE) of the free energy considering unidirectional work distributions belonging to Gaussian or Gamma families, and compare this estimator to . We further consider bidirectional work distributions belonging to the same families, and compare the corresponding bidirectional to the Bennett acceptance ratio () estimator. We show that, for Gaussian unidirectional work distributions, is in fact the parametric MLE of the free energy, and as such, the most efficient estimator for this statistical family. We observe that and perform better than and , for unidirectional and bidirectional distributions, respectively. These results illustrate that the characterization of the underlying work distribution permits an optimal use of the Jarzynski equality. © 2018 Wiley Periodicals, Inc.     

Using the fast fourier transform in binding free energy calculations

According to implicit ligand theory, the standard binding free energy is an exponential average of the binding potential of mean force (BPMF), an exponential average of the interaction energy between the unbound ligand ensemble and a rigid receptor. Here, we use the fast Fourier transform (FFT) to efficiently evaluate BPMFs by calculating interaction energies when rigid ligand configurations from the unbound ensemble are discretely translated across rigid receptor conformations. Results for standard binding free energies between T4 lysozyme and 141 small organic molecules are in good agreement with previous alchemical calculations based on (1) a flexible complex ( for 24 systems) and (2) flexible ligand with multiple rigid receptor configurations ( for 141 systems). While the FFT is routinely used for molecular docking, to our knowledge this is the first time that the algorithm has been used for rigorous binding free energy calculations. © 2017 Wiley Periodicals, Inc.     

Aspartate: An interesting model for analyzing dipole-ion and ion pair interactions through its oppositely charged amine and acid groups

Anionic species of aspartic acid, Asp⁻, having a zwitterionic backbone and a deprotonated side chain, appears to be a good example for analyzing dipole-ion and ion pair interactions. Density functional theory calculations were herein performed to investigate the low energy conformers of Asp⁻ embedded in a dielectric continuum modeling an aqueous environment, through a scan of the potential energy as a function of the side chain (χ₁, χ₂) torsion angles. The most energetically favorable conformers having g⁺g⁻ and g⁻g⁺ side chain orientations are found to be stabilized by charge-enhanced intramolecular H-bonding involving the positively charged () and the two negatively charged (COO⁻) groups. These conformers were further used to analyze Asp⁻ + nW clusters (W: water, n = 1 or 3), and Asp⁻/Asp⁻ pair formation. COO⁻ groups were found to be the most attractive sites for hosting a water molecule (binding energy: −6.0 ± 1.5 kcal/mol), compared to groups (binding energy: −4.7 ± 1.1 kcal/mol). Energy separation between g⁺g⁻ and g⁻g⁺ conformers increases upon explicit hydration. Asp⁻/Asp⁻ ion pairs, stabilized by the interaction between the group of a partner and the COO⁻ group of the other, shows a quite constant binding energy (−8.1 ± 0.2 kcal/mol), whatever the pair type, and the relative orientation of the two interacting partners. This study suggests a first step to achieve a more realistic image of intermolecular interactions in aqueous environment, especially upon increasing concentration. It can also be considered as a preliminary attempt to assess the interactions of the Lys⁺…Asp⁻/Glu⁻ ion pairs stabilizing intra- and interchain interactions in proteins.     

Fast and accurate determination of the relative binding affinities of small compounds to HIV‐1 protease using non‐equilibrium work

The fast pulling ligand (FPL) out of binding cavity using non‐equilibrium molecular dynamics (MD) simulations was demonstrated to be a rapid, accurate and low CPU demand method for the determination of the relative binding affinities of a large number of HIV‐1 protease (PR) inhibitors. In this approach, the ligand is pulled out of the binding cavity of the protein using external harmonic forces, and the work of pulling force corresponds to the relative binding affinity of HIV‐1 PR inhibitor. The correlation coefficient between the pulling work and the experimental binding free energy of shows that FPL results are in good agreement with experiment. It is thus easier to rank the binding affinities of HIV‐1 PR inhibitors, that have similar binding affinities because the mean error bar of pulling work amounts to . The nature of binding is discovered using the FPL approach. © 2016 Wiley Periodicals, Inc.     

Free energies of solvation in the context of protein folding: Implications for implicit and explicit solvent models

Implicit solvent models for biomolecular simulations have been developed to use in place of more expensive explicit models; however, these models make many assumptions and approximations that are likely to affect accuracy. Here, the changes in free energies of solvation upon folding of several fast folding proteins are calculated from previously run μs–ms simulations with a number of implicit solvent models and compared to the values needed to be consistent with the explicit solvent model used in the simulations. In the majority of cases, there is a significant and substantial difference between the values calculated from the two approaches that is robust to the details of the calculations. These differences could only be remedied by selecting values for the model parameters—the internal dielectric constant for the polar term and the surface tension coefficient for the nonpolar term—that were system‐specific or physically unrealistic. We discuss the potential implications of our findings for both implicit and explicit solvent simulations. © 2015 Wiley Periodicals, Inc.     

Ab initio structure and vibration-rotation dynamics of germylene,GeH2

Accurate structure and potential energy surface of germylene, GeH₂, in its ground electronic state ¹A₁ were determined from ab initio calculations using the coupled-cluster approach in conjunction with the correlation-consistent basis sets up to sextuple-zeta quality. The Born-Oppenheimer equilibrium structural parameters for the ¹A₁ state are estimated to be r_e(GeH) = 1.5793 Å and ∠_e(HGeH) = 91.19^∘. The term value T_e for the lowest excited electronic state ã³B₁ of GeH₂ is predicted to be 9140 cm^–1. The vibration-rotation energy levels for the ¹A₁ state of the ⁷⁴GeH₂, ⁷⁴GeD₂, ⁷²GeH₂, and ⁷⁰GeH₂ isotopologues were determined using a variational approach and compared with the experimental data. The role of the core-electron correlation, higher-order valence-electron correlation, scalar relativistic, spin-orbit, and adiabatic effects for prediction of the structure and vibration-rotation dynamics of the GeH₂ molecule is discussed. © 2019 Wiley Periodicals, Inc.     

Energy decomposition analysis of cation–π, metal ion–lone pair,hydrogen bonded,charge‐assisted hydrogen bonded,and π–π interactions

This study probes the nature of noncovalent interactions, such as cation–π, metal ion–lone pair (M–LP), hydrogen bonding (HB), charge‐assisted hydrogen bonding (CAHB), and π–π interactions, using energy decomposition schemes—density functional theory (DFT)–symmetry‐adapted perturbation theory and reduced variational space. Among cation–π complexes, the polarization and electrostatic components are the major contributors to the interaction energy (IE) for metal ion–π complexes, while for onium ion–π complexes ( , , , and ) the dispersion component is prominent. For M–LP complexes, the electrostatic component contributes more to the IE except the dicationic metal ion complexes with H₂S and PH₃ where the polarization component dominates. Although electrostatic component dominates for the HB and CAHB complexes, dispersion is predominant in π–π complexes.Copyright © 2015 Wiley Periodicals, Inc.     

Efficient singular-value decomposition of the coupled-cluster triple excitation amplitudes

We demonstrate a novel technique to obtain singular-value decomposition (SVD) of the coupled-cluster triple excitations amplitudes, . The presented method is based on the Golub-Kahan bidiagonalization strategy and does not require to be stored. The computational cost of the method is comparable to several coupled cluster singles and doubles (CCSD) iterations. Moreover, the number of singular vectors to be found can be predetermined by the user and only those singular vectors which correspond to the largest singular values are obtained at convergence. We show how the subspace of the most important singular vectors obtained from an approximate triple amplitudes tensor can be used to solve equations of the CC3 method. The new method is tested for a set of small and medium-sized molecular systems in basis sets ranging in quality from double- to quintuple-zeta. It is found that to reach the chemical accuracy (≈1 kJ/mol) in the total CC3 energies as little as 5 − 15% of SVD vectors are required. This corresponds to the compression of the amplitudes by a factor of about 0.0001 − 0.005 . Significant savings are obtained also in calculation of interaction energies or rotational barriers, as well as in bond-breaking processes. © 2019 Wiley Periodicals, Inc.     

Fluxional bonds in quasi-planar and half-sandwich (M = K,Rb, and Cs)

Detailed molecular orbital and bonding analyses reveal the existence of both fluxional σ- and π-bonds in the global minima C_s ( 1 ) and C_s MB₁₈ ( 3 ) and transition states C_s ( 2 ) and C_s ( 4 ) of dianion and monoanions (M = K, Rb, and Cs). It is the fluxional bonds that facilitate the fluxional behaviors of the quasi-planar and half-sandwich which possess energy barriers smaller than the difference of the corresponding zero-point corrections. © 2019 Wiley Periodicals, Inc.     

Low‐energy structures of benzene clusters with a novel accurate potential surface

The benzene‐benzene (Bz‐Bz) interaction is present in several chemical systems and it is known to be crucial in understanding the specificity of important biological phenomena. In this work, we propose a novel Bz‐Bz analytical potential energy surface which is fine‐tuned on accurate ab initio calculations in order to improve its reliability. Once the Bz‐Bz interaction is modeled, an analytical function for the energy of the clusters may be obtained by summing up over all pair potentials. We apply an evolutionary algorithm (EA) to discover the lowest‐energy structures of clusters (for ), and the results are compared with previous global optimization studies where different potential functions were employed. Besides the global minimum, the EA also gives the structures of other low‐lying isomers ranked by the corresponding energy. Additional ab initio calculations are carried out for the low‐lying isomers of and clusters, and the global minimum is confirmed as the most stable structure for both sizes. Finally, a detailed analysis of the low‐energy isomers of the n = 13 and 19 magic‐number clusters is performed. The two lowest‐energy isomers show S₆ and C₃ symmetry, respectively, which is compatible with the experimental results available in the literature. The structures reported here are all non‐symmetric, showing two central Bz molecules surrounded by 12 nearest‐neighbor monomers in the case of the five lowest‐energy structures. © 2015 Wiley Periodicals, Inc.     

Ab initio potential energy surface and vibration–rotation energy levels of germanium dicarbide,GeC2

The accurate ground‐state potential energy surface of germanium dicarbide, GeC₂, has been determined from ab initio calculations using the coupled‐cluster approach. The core–electron correlation, higher‐order valence‐electron correlation, and scalar relativistic effects were taken into account. The potential energy surface of GeC₂ was shown to be extraordinarily flat near the T‐shaped equilibrium configuration. The potential energy barrier to the linear CCGe configuration was predicted to be 1218 cm⁻¹. The vibration–rotation energy levels of some GeC₂ isotopologues were calculated using a variational method. The vibrational bending mode ν₃ was found to be highly anharmonic, with the fundamental wavenumber being only 58 cm⁻¹. Vibrational progressions due to this mode were predicted for the , , and states of GeC₂. © 2018 Wiley Periodicals, Inc.     

A Density Functional Theory Study on Nonlinear Optical Properties of Double Cage Excess Electron Compounds: Theoretically Design M[Cu(Ag)@(NH3)n](M = Be,Mg and Ca; n = 1–3)

In this work, we investigated the nonlinear optical (NLO) properties of excess electron electride molecules of M[Cu(Ag)@(NH₃)_n](M = Be, Mg and Ca; n = 1–3) using density functional theory (DFT). This electride molecules consist of an alkaline-earth (Be, Mg and Ca) together with transition metal (Cu and Ag) doped in NH₃ cluster. The natural population analysis of charge and their highest occupied molecular orbital suggests that the M[Cu(Ag)@(NH₃)_n] compound has excess electron like alkaline-earth metal form double cage electrides molecules, which exhibit a large static first hyperpolarizability () (electron contribution part) and one of which owns a peak value of 216,938 (a.u.) for Be[Ag@(NH₃)₂] and vibrational harmonic first hyperpolarizability () (nuclear contribution part) values and the ratio of /, namely, η values from 0.02 for Be[Ag@(NH₃)] to 0.757 for Mg[Ag@(NH₃)₃]. The electron density contribution in different regions on values mainly come from alkaline-earth and transition metal atoms by first hyperpolarizability density analysis, and also explains the reason why values are positive and negative. Moreover, the frequency-dependent values β(−2ω,ω,ω) are also estimated to make a comparison with experimental measures. © 2018 Wiley Periodicals, Inc.     

Accurate double many‐body expansion potential energy surface by extrapolation to the complete basis set limit and dynamics calculations for ground state of NH2

An accurate single‐sheeted double many‐body expansion potential energy surface is reported for the title system. A switching function formalism has been used to warrant the correct behavior at the and dissociation channels involving nitrogen in the ground and first excited states. The topographical features of the novel global potential energy surface are examined in detail, and found to be in good agreement with those calculated directly from the raw ab initio energies, as well as previous calculations available in the literature. The novel surface can be using to treat well the Renner–Teller degeneracy of the and states of . Such a work can both be recommended for dynamics studies of the reaction and as building blocks for constructing the double many‐body expansion potential energy surface of larger nitrogen/hydrogen‐containing systems. In turn, a test theoretical study of the reaction has been carried out with the method of quantum wave packet on the new potential energy surface. Reaction probabilities, integral cross sections, and differential cross sections have been calculated. Threshold exists because of the energy barrier (68.5 meV) along the minimum energy path. On the curve of reaction probability for total angular momentum J = 0, there are two sharp peaks just above threshold. The value of integral cross section increases quickly from zero to maximum with the increase of collision energy, and then stays stable with small oscillations. The differential cross section result shows that the reaction is a typical forward and backward scatter in agreement with experimental measurement result. © 2013 Wiley Periodicals, Inc.     

Aqueous acidities of primary benzenesulfonamides: Quantum chemical predictions based on density functional theory and SMD

Aqueous of selected primary benzenesulfonamides are predicted in a systematic manner using density functional theory methods and the SMD solvent model together with direct and proton exchange thermodynamic cycles. Some test calculations were also performed using high‐level composite CBS‐QB3 approach. The direct scheme generally does not yield a satisfactory agreement between calculated and measured acidities due to a severe overestimation of the Gibbs free energy changes of the gas‐phase deprotonation reaction by the used exchange‐correlation functionals. The relative values calculated using proton exchange method compare to experimental data very well in both qualitative and quantitative terms, with a mean absolute error of about 0.4 units. To achieve this accuracy, we find it mandatory to perform geometry optimization of the neutral and anionic species in the gas and solution phases separately, because different conformations are stabilized in these two cases. We have attempted to evaluate the effect of the conformer‐averaged free energies in the predictions, and the general conclusion is that this procedure is highly too costly as compared with the very small improvement we have gained. © 2015 Wiley Periodicals, Inc.     

Fast search algorithms for computational protein design

One of the main challenges in computational protein design (CPD) is the huge size of the protein sequence and conformational space that has to be computationally explored. Recently, we showed that state‐of‐the‐art combinatorial optimization technologies based on Cost Function Network (CFN) processing allow speeding up provable rigid backbone protein design methods by several orders of magnitudes. Building up on this, we improved and injected CFN technology into the well‐established CPD package Osprey to allow all Osprey CPD algorithms to benefit from associated speedups. Because Osprey fundamentally relies on the ability of to produce conformations in increasing order of energy, we defined new strategies combining CFN lower bounds, with new side‐chain positioning‐based branching scheme. Beyond the speedups obtained in the new ‐CFN combination, this novel branching scheme enables a much faster enumeration of suboptimal sequences, far beyond what is reachable without it. Together with the immediate and important speedups provided by CFN technology, these developments directly benefit to all the algorithms that previously relied on the DEE/ combination inside Osprey* and make it possible to solve larger CPD problems with provable algorithms. © 2016 Wiley Periodicals, Inc.     

FEREBUS: Highly parallelized engine for kriging training

FFLUX is a novel force field based on quantum topological atoms, combining multipolar electrostatics with IQA intraatomic and interatomic energy terms. The program FEREBUS calculates the hyperparameters of models produced by the machine learning method kriging. Calculation of kriging hyperparameters ( θ and p ) requires the optimization of the concentrated log‐likelihood . FEREBUS uses Particle Swarm Optimization (PSO) and Differential Evolution (DE) algorithms to find the maximum of . PSO and DE are two heuristic algorithms that each use a set of particles or vectors to explore the space in which is defined, searching for the maximum. The log‐likelihood is a computationally expensive function, which needs to be calculated several times during each optimization iteration. The cost scales quickly with the problem dimension and speed becomes critical in model generation. We present the strategy used to parallelize FEREBUS, and the optimization of through PSO and DE. The code is parallelized in two ways. MPI parallelization distributes the particles or vectors among the different processes, whereas the OpenMP implementation takes care of the calculation of , which involves the calculation and inversion of a particular matrix, whose size increases quickly with the dimension of the problem. The run time shows a speed‐up of 61 times going from single core to 90 cores with a saving, in one case, of ～98% of the single core time. In fact, the parallelization scheme presented reduces computational time from 2871 s for a single core calculation, to 41 s for 90 cores calculation. © 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.