Electrostatic component of binding energy: Interpreting predictions from poisson–boltzmann equation and modeling protocols

Macromolecular interactions are essential for understanding numerous biological processes and are typically characterized by the binding free energy. Important component of the binding free energy is the electrostatics, which is frequently modeled via the solutions of the Poisson–Boltzmann Equations (PBE). However, numerous works have shown that the electrostatic component (ΔΔG_elec) of binding free energy is very sensitive to the parameters used and modeling protocol. This prompted some researchers to question the robustness of PBE in predicting ΔΔG_elec. We argue that the sensitivity of the absolute ΔΔG_elec calculated with PBE using different input parameters and definitions does not indicate PBE deficiency, rather this is what should be expected. We show how the apparent sensitivity should be interpreted in terms of the underlying changes in several numerous and physical parameters. We demonstrate that PBE approach is robust within each considered force field (CHARMM‐27, AMBER‐94, and OPLS‐AA) once the corresponding structures are energy minimized. This observation holds despite of using two different molecular surface definitions, pointing again that PBE delivers consistent results within particular force field. The fact that PBE delivered ΔΔG_elec values may differ if calculated with different modeling protocols is not a deficiency of PBE, but natural results of the differences of the force field parameters and potential functions for energy minimization. In addition, while the absolute ΔΔG_elec values calculated with different force field differ, their ordering remains practically the same allowing for consistent ranking despite of the force field used. © 2016 Wiley Periodicals, Inc.     

Calculation of relative free energy differences for the covalent hydration of organic compounds: A combined quantum mechanical and free energy perturbation study

The relative free energy difference (ΔΔG_hyd) for the reversible addition of water to two unsaturated molecules is accurately computed using a combination of ab initio quantum mechanical calculations and free energy perturbation methods. Initial attempts to calculate the absolute hydration free energy difference (ΔG_hyd) for formaldehyde and trichloroacetaldehyde gave values that differed substantially from experimental results even after inclusion of electron correlation energy contributions using third-order (MP3) and fourth-order (MP4) Møller-Plesset perturbation theory and QCISD(T) correlation methods at the 6-31G** basis set level. Inaccuracies in ΔG_hyd were attributed to errors in the calculation of both ΔG_gas and ΔΔG_sol. Gas phase quantum mechanical free energies (ΔG_gas) varied significantly (2–3 kcal/mol) depending on the level of theory. Errors in ΔΔG_sol were attributed to slow convergence of the calculations using the thermodynamic cycle perturbation (TCP) method with explicit solvent. These errors were minimized or canceled, however, when relative hydration free energy differences (ΔΔG_hyd) were calculated using a combination of ab initio quantum mechanical calculations and free energy perturbation methods. Calculated values for a variety of aldehydes and ketones were consistent with experimental data. © 1995 John Wiley & Sons, Inc.     

Free energy perturbation calculations on parallel computers: Demonstrations of scalable linear speedup

A coarse-grain parallel implementation of the free energy perturbation (FEP) module of the AMBER molecular dynamics program is described and then demonstrated using five different molecular systems. The difference in the free energy of (aqueous) solvation is calculated for two monovalent cations ΔΔG_aq(Li⁺ Δ Cs⁺), and for the zero-sum ethane-to-ethane′ perturbation ΔΔG_aq(CH₃? methyl? X → X? methyl? CH₃), where X is a ghost methyl. The difference in binding free energy for a docked HIV-1 protease inhibitor into its ethylene mimetic is examined by mutating its fifth peptide bond, ΔG(CO? NH → CH?CH). A potassium ion (K⁺) is driven outward from the center of mass of ionophore salinomycin (SAL^?) in a potential of mean force calculation ΔG_MeOH(SAL^? · K⁺) carried out in methanol solvent. Parallel speedup obtained is linearly proportional to the number of parallel processors applied. Finally, the difference in free energy of solvation of phenol versus benzene, ΔΔG_oct(phenol → benzene), is determined in water-saturated octanol and then expressed in terms of relative partition coefficients, Δ log(P_o/w). Because no interprocessor communication is required, this approach is scalable and applicable in general for any parallel architecture or network of machines. FEP calculations run on the nCUBE/2 using 50 or 100 parallel processors were completed in clock times equivalent to or twice as fast as a Cray Y-MP. The difficulty of ensuring adequate system equilibrium when agradual configurational reorientation follows the mutation of the Hamiltonian is discussed and analyzed. The results of a successful protocol for overcoming this equilibration problem are presented. The types of molecular perturbations for which this method is expected to perform most efficiently are described. © 1994 by John Wiley & Sons, Inc.     

Role of protein structure and the role of individual fingers in zinc finger protein–DNA recognition: a molecular dynamics simulation study and free energy calculations

Molecular dynamics and MM_GBSA energy calculations on various zinc finger proteins containing three and four fingers bound to their target DNA gave insights into the role of each finger in the DNA binding process as part of the protein structure. The wild type Zif 268 (PDB code: 1AAY) gave a ΔG value of ??76.1 (14) kcal/mol. Zinc fingers ZF1, ZF2 and ZF3 were mutated in one experiment and in another experiment one finger was cut and the rest of the protein was studied for binding. The ΔΔG values for the Zinc Finger protein with both ZF1 and ZF2 mutated was +?80 kcal/mol, while mutating only ZF1 the ΔΔG value was +?52 kcal/mol (relative to the wild type). Cutting ZF3 and studying the protein consisting only of ZF1 linked to ZF2 gave a ΔΔG value of +?68 kcal/mol. Upon cutting ZF1, the resulting ZF2 linked to ZF3 protein gave a ΔΔG value of +?41 kcal/mol. The above results shed light on the importance of each finger in the binding process, especially the role of ZF1 as the anchoring finger followed in importance by ZF2 and ZF3. The energy difference between the binding of the wild type protein Zif268 (1AAY) and that for individual finger binding to DNA according to the formula: ΔΔG_{linkers, otherstructuralfactors}?=?ΔG_zif268???(ΔG_F1+F2+F3) gave a value?=???44.5 kcal/mol. This stabilization can be attributed to the contribution of linkers and other structural factors in the intact protein in the DNA binding process. DNA binding energies of variant proteins of the wild type Zif268 which differ in their ZF1 amino acid sequence gave evidence of a good relationship between binding energy and recognition and specificity, this finding confirms the reported vital role of ZF1 in the ZF protein scanning and anchoring to the target DNA sequence. The role of hydrogen bonds in both specific and nonspecific amino acid-DNA contacts is discussed in relation to mutations. The binding energies of variant Zinc Finger proteins confirmed the role of ZF1 in the recognition, specificity and anchoring of the zinc finger protein to DNA.     

An effective model for calculations of the hydrophobic component of the Gibbs energy of solution of small- and medium-sized molecules

A model for calculations of the hydrophobic component of the Gibbs energy G of solution of small- and medium-sized molecules in water was suggested. The model uses the decomposition G = G ^s + G ^ns , where G ^s is the contribution of the cavity volume characterized by the effective radius R _v calculated from the volume of the cavity replaced by a sphere equal in volume and G ^ns is the contribution caused by the difference of the shape of the cavity from spherical. The most substantial contribution to G for small- and medium-sized molecules is G ^s , which can be calculated once for spheres over the radius range under consideration and then used in calculations of G for cavities of arbitrary shapes. At the same time, the Gibbs energy part G ^ns specific with respect to the cavity shape is comparatively small and can be calculated in terms of a simple model with the use of nonspherical indexes, which quantitatively characterize the degree of cavity shape deviation from spherical. The model was verified by Monte Carlo calculations for a set of cavities with different volumes and shapes in an ensemble of water molecules with periodic boundary conditions under usual conditions. The error in G calculated using the model with respect to Monte Carlo results was ∼1 kcal/mol, which was close to the error of the Monte Carlo method itself for the problem under consideration. The time required to calculate G in terms of the model was a few seconds, whereas calculations of G by the Monte Carlo method took from several hours to several days.     

Scalar Relativistic Density Functional Theoretical Investigation of Higher Complexation Ability of Substituted 1,10‐Phenanthroline over Bipyridine Towards Am3+/Eu3+ Ions

The better selectivity of Am³⁺ over Eu³⁺ ion with N‐based CyMe4‐BTPhen compared to CyMe4‐BTBP for experimentally observed [ML₂(NO₃)]²⁺ complexes was demonstrated using scalar relativistic DFT in conjunction with Born‐Haber thermodynamic cycle and COSMO solvation model. The calculated free energy of extraction, ΔG_ext reveals strong dependence on the hydration free energies of Am³⁺ and Eu³⁺ ions and week dependence to the difference in Gibbs free energy of solvation of the ligand or metal‐ligand complexes. Further, for the first time, we have computed the effect of co‐anion species ([M(NO₃)₅]^2–) on ΔG_ext of Am³⁺ and Eu³⁺ ions with CyMe4‐BTPhen and CyMe4‐BTBP, which adds a positive contribution and thus reduces the ΔG_ext. The calculated values of ΔΔΔG_ext (= ΔΔG_ext,L1 – ΔΔG_ext,L2, ΔΔG_ext = ΔG_ext,M1 – ΔG_ext,M2) can be used to avoid the very sensitive metal ion solvation energy, effect of co‐anionic species and thus provides a robust approach to determine the selectivity between two metal ions towards different competitive ligands. The natural population analysis (NPA), molecular orbital analysis, Mayer bond order analysis, and bond character analysis using Bader's quantum theory of atoms in molecules indicates slightly more covalency for complexes of Am³⁺ ion that are correlated to the experiental selectvity of Am³⁺ ion over Eu³⁺ ion and hence might be useful in the design and development of next generation extractants.     

Relative silver(I) ion binding energies of α-amino acids: A determination by means of the kinetic method

The relative silver(I) ion binding energies of 19 α-amino acids have been measured by means of the kinetic method. In general, they are similar to the relative copper(I) ion binding energies of corresponding amino acids although there are differences that can be accounted for by differences in silver(I) and copper(I) chemistry. The correlation with proton basicities is comparatively poorer. Again, the differences between silver(I) and proton binding can be attributed to differences in silver(I) and proton chemistry. The relative silver(I) binding energies measured are best described as relative basicities or ΔΔG _Ag ^° ’s. The observed internal consistency during construction of a silver(I) ion basicity ladder implies that ΔΔS _Ag ^° is approximately zero except when histidine and lysine are involved. For 16 α-amino acids, their relative silver(I) ion basicities ≈ relative silver(I) ion affinities or ΔΔG° _Ag ≈ ΔΔH _Ag ^° .     

DFT plus PCM calculation for pairing specificity of Watson–Crick‐type bases in aqueous solutions

This study is aimed at explaining the preference for AT and CG pairings and the possible insertion of other tautomeric DNA base pairs such as G_enolT, that respect energetic and steric requirements including at least two hydrogen bonds and 11 ± 0.5Å distance between the 9‐CH₃ of purine and 5‐CH₃ of pyrimidine. The calculated free energy of formation ΔΔG at the DFT B3LYP/6‐31G*‐PCM/BEM level pointed out the CG and AT pairs as the most favored, followed closely by G_enolT, in good agreement with Michaelis–Menten first order kinetics (CG ≈ AT > G_enolT). Unusual DNA base pairs complexes such as AG (BEM) and CT (PCM) resulted to be stable, but it is very difficult to assume that they are likely to be included in the double strand DNA. The calculated enthalpy and dipole moments of isolated DNA bases agree well with experiment. The free energy of hydration, ΔG_hyd, was found to depend on the electrostatic term, while cavitation‐dispersion components are almost constant. The stability of DNA complexes in water resulted from PCM calculations is markedly influenced by the free energy of hydration. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

The configurational dependence of binding free energies: A Poisson–Boltzmann study of Neuraminidase inhibitors

The linear finite difference Poisson-Boltzmann (FDPB) equation is applied to the calculation of the electrostatic binding free energies of a group of inhibitors to the Neuraminidase enzyme. An ensemble of enzyme-inhibitor complex conformations was generated using Monte Carlo simulations and the electrostatic binding free energies of subtly different configurations of the enzyme-inhibitor complexes were calculated. It was seen that the binding free energies calculated using FDPB depend strongly on the configuration of the complex taken from the ensemble. This configurational dependence was investigated in detail in the electrostatic hydration free energies of the inhibitors. Differences in hydration energies of up to 7 kcal mol^–1 were obtained for root mean square (RMS) structural deviations of only 0.5 Å. To verify the result, the grid size and parameter dependence of the calculated hydration free energies were systematically investigated. This showed that the absolute hydration free energies calculated using the FDPB equation were very sensitive to the values of key parameters, but that the configurational dependence of the free energies was independent of the parameters chosen. Thus just as molecular mechanics energies are very sensitive to configuration, and single-structure values are not typically used to score binding free energies, single FDPB energies should be treated with the same caution.     

Computation of relative binding free energy for an inhibitor and its analogs binding with Erk kinase using thermodynamic integration MD simulation

In the present study, we carried out thermodynamic integration molecular dynamics simulation for a pair of analogous inhibitors binding with Erk kinase to investigate how computation performs in reproducing the relative binding free energy. The computation with BCC-AM1 charges for ligands gave ?1.1?kcal/mol, deviated from experimental value of ?2.3?kcal/mol by 1.2?kcal/mol, in good agreement with experimental result. The error of computed value was estimated to be 0.5?kcal/mol. To obtain convergence, switching vdw interaction on and off required approximately 10 times more CPU time than switching charges. Residue-based contributions and hydrogen bonding were analyzed and discussed. Furthermore, subsequent simulation using RESP charge for ligand gave ΔΔG of ?1.6?kcal/mol. The computed results are better than the result of ?5.6?kcal/mol estimated using PBSA method in a previous study. Based on these results, we further carried out computations to predict ΔΔG for five new analogs, focusing on placing polar and nonpolar functional groups at the meta site of benzene ring shown in the Fig.?1, to see if these ligands have better binding affinity than the above ligands. The computations resulted that a ligand with polar –OH group has better binding affinity than the previous examined ligand by ~2.0?kcal/mol and two other ligands have better affinity by ~1.0?kcal/mol. The predicted better inhibitors of this kind should be of interest to experimentalist for future experimental enzyme and/or cell assays.     

DNA与金属锇配合物相互作用的表面电化学研究

采用自己建立的DNA表面电化学研究微量方法, 研究了单双链DNA与两种锇配合物(联吡啶锇和二氯菲咯啉锇)的相互作用. 研究发现, 两种锇配合物都是通过静电作用与DNA结合, 其作用方式不受溶液离子强度的影响. 并计算得到了联吡啶锇和二氯菲咯啉锇与dsDNA和ssDNA相互作用的多个热力学和动力学参数, 如结合常数K_3＋或K_2＋, 结合常数比K_3＋/K_2＋, 离子强度为零时的极限比 , 结合自由能ΔG_b, 解离速度常数k, 结合位点数s.     

Chiral discrimination of ibuprofen isomers in β-cyclodextrin inclusion complexes: experimental (NMR) and theoretical (MD, MM/GBSA) studies

In this paper, we analyze the energetic and conformational preferences involved in the chiral discrimination of ibuprofen (Ibu) isomers by beta-cyclodextrin (β-CD) when forming inclusion complexes in water. This study was performed by means of atomistic molecular mechanics simulations upon four different penetration modes of the guest, and a structural 2D NMR experiment. The trajectories of these simulations were treated with the MM/GBSA method in order to obtain the relative weights of the different free energy components. The resulting values of the free energy of binding and other geometrical features indicate that this chiral selectivity is influenced by a preferred penetration mode involving the S-(+)-Ibu isomer. The calculated ΔΔG of binding is in good agreement with published experiments.     

Calculations of the electrostatic free energy contributions to the binding free energy of sulfonamides to carbonic anhydrase

The interactions between biologically important enzymes and drugs are of great interest. In order to address some aspects of these interactions we have initiated a program to investigate enzymedrug interactions. Specifically, the interactions between one of the isozymes of carbonic anhydrase and a family of drugs known as sulfonamides have been studied using computational methods. In particular the electrostatic free energy of binding of carbonic anhydrase II with acetazolamide, methazolamide,p-chlorobenzenesulfonamide,p-aminobenzenesulfonamide and three new compounds (MK1, MK2, and MK3) has been computed using finite-difference Poisson-Boltzmann (FDPB) [1] method and the semimacroscopic version [2, 3] of the protein dipole Langevin dipole (PDLD) method [4]. Both methods, FDPB and PDLD, give similar results for the electrostatic free energy of binding even though different charges and different treatments were used for the protein. The calculated electrostatic binding free energies are in reasonable agreement with the experimental data. The potential and the limitation of electrostatic models for studies of binding energies are discussed.     

A method for the calculation of heats of solution (ΔH s ) for organic compounds from the number of constituent carbon atoms and the functional groups

The heat of solution (ΔH _s ) of several homologous series of linear and monofunctional organic compounds (n-alkanes,n-alcohols,n-aldehyds and esters) in polar stationary phase (Carbowax 1540) can be expressed by ΔH _s =a+b·n _R +c·C _G , wherea is a constant,n _R is the number of carbon atoms not belonging to the functional group in the molecule andC _G represents the contribution of each functional group to the ΔH _s value. The accuracy of the ΔH _s values calculated by this equation is sufficient for all the compounds tested (average relative error=1.92%).C _G can be calculated from the plots ΔH _s versusn _R .     

Nematic—isotropic solution thermodynamics in di(p-methoxyphenyl)-trans-cyclohexane-1,4-dicarboxylate

Solution thermodynamic parameters of selected non-polar solutes have been determined in the nematic and isotropic fluid states of di(p-methoxyphenyl)-trans-cyclohexane-1,4-dicarboxylate. These states have been reported to exert no measurable differential in kinetic medium effects on the Claisen rearrangement¹. Partial molar enthalpies Δ$\text{H}$^soln₂ and entropies Δ$\text{S}$^soln₂ of solution of a series of substituted benzenes in the nematogenic solvent, determined by the gas—liquid chromatographic method of Martire, ^2–4, are reported. Changes in solute excess Gibbs free energy Δ$\text{G}$^E₂ over the nematic—isotropic transition of the solvent, corresponding to the changes in free energy of solution ΔΔ$\text{G}$^soln₂, have been calculated for the series. The results show the nematic and isotropic states of the medium to exhibit distinctly different solvent characteristics and suggest that Claisen reaction kinetics in the nematogenic solvent reflect compensating medium effects on the reactant and its activated state.     

Accurate ab initio determination of the van der Waals interaction in the X 2Σ+ ground state of LiHe

Accurate quantum-chemical ab initio calculations have been performed at the SCF and CEPA (coupled electron pair approximation) levels for the van der Waals interaction in the X ² Σ ⁺ ground state of LiHe. An extended basis set has been used and the counterpoise correction for the basis set superposition error (BSSE) has been applied. The calculated potential energy curve has a very shallow minimum at 11.56 a ₀ with a well depth of only 1.49 cm^?1. This is too small to allow for a bound vibrational level. The analysis of the results shows that the interaction mainly consists of the Pauli repulsion between Li(1s ²2s) and He (1s ²), which is decaying exponentially, and the attractive London dispersion energy. Van der Waals coefficients C₆, C₈, and C₁₀ have been determined by a least squares fit to the long-range part of the calculated potential curve.     

Thermodynamical study on the formation of some mixed ligand complexes of non-transition metal ions in solution I. The formation and the stability of some Pb(II) mixed ligand complexes

The stability constants of the binary complexes of type PbB, PbB₂, and the mixed ligand complexes of type PbAB have been studied by potentiometric pH titration technique at ionic strength I = 0.10 (KNO₃) and at temperature 15, 25, 35 and 45°C respectively, where A = 2,2′-bipyridyl (bipy) or 1,10-phenanthroline (phen); B = malonate (mal), succinate (suc), or anthranilate (anth). The equalibrium constants ΔlogK,¹ ΔΔG, ΔΔH, and ΔΔS of the reaction PbA + PbB = PbAB + Pb²⁺ have been calculated. The results show that the discriminating effects² between the primary ligand (bipy or phen) and the secondary ligand (mal, suc or anth) in those non-transition metal mixed ligand complexes are also evident, and as a measurement of this effect, ΔΔH is more appropriate than ΔlogK. The possible reasons which lead to these results have been discussed.     

Pseudo-adiabatic calorimetry

A procedure is described for dealing with the error sources inherently present in any real calorimeter: work of powerP _s input from stirrer and possibly temperature sensor, and heat exchange at a rate ?G(T?T _e ) whereT andT _c are the temperatures of calorimeter and surroundings respectively. The constantsP _s andG are calculated from a period of thermal decay, and afterwards are used to correct the entire run. A calorimeter was designed with high thermal homogeneity and used in a test. The curve of calculated temperature exactly traces the heater energy, even after 5 h, with a standard deviation of about 1 mK. The relative error inC _p is less than 1/1000.     

Stark manifolds of barium Rydberg states

In a CW laser-atomic beam experiment Stark manifolds in barium originating from the Rydberg states 6s40f ¹ F ₃, 6s40g ^1,3 G ₄,³ G ₅ and 6s40h ^1,3 H ₅ have been studied. Accurate quantum defect values for higher orbital angular momentum states (l=6, 7) have been determined. The Stark manifolds are also calculated by diagonalization of the energy matrix in the presence of an external electric field. Good agreement between experiment and calculations is obtained.     

Classification of Simple Oxides: A Polarizability Approach

A simple oxide classification has been proposed on the basis of correlation between electronic polarizabilities of the ions and their binding energies determined by XPS. Three groups of oxides have been considered taking into account the values obtained on refractive-index- or energy-gap-based oxide ion polarizability, cation polarizability, optical basicity, O 1s binding energy, metal (or nonmetal) binding energy, and Yamashita-Kurosawa's interaction parameter of the oxides. The group of semicovalent predominantly acidic oxides includes BeO, B₂O₃, P₂O₅, SiO₂, Al₂O₃, GeO₂, and Ga₂O₃ with low oxide ion polarizability, high O 1s binding energy, low cation polarizability, high metal (or nonmetal) outermost binding energy, comparatively low optical basicity, and strong interionic interaction, leading to the formation of strong covalent bonds. Some main group oxides so-called ionic or basic such as CaO, In₂O₃, SnO₂, and TeO₂ and most transition metal oxides show relatively high oxide ion polarizability, O 1s binding energy in a very narrow medium range, high cation polarizability, and low metal (or nonmetal) binding energy. Their optical basicity varies in a narrow range and it is close to that of CaO. The group of very ionic or very basic oxides includes CdO, SrO, and BaO as well as PbO, Sb₂O₃, and Bi₂O₃, which possess very high oxide ion polarizability, low O 1s binding energy, very high cation polarizability, and very low metal (or nonmetal) binding energy. Their optical basicity is higher than that of CaO and the interionic interaction is very weak, giving rise to the formation of very ionic chemical bonds.