Calculation Of Electrostatic Free Energy Research Articles

Calculation of electrostatic free energy for the nonlinear Poisson-Boltzmann model based on the dimensionless potential

The Poisson-Boltzmann (PB) equation governing the electrostatic potential with a unit is often transformed to a normalized form for a dimensionless potential in numerical studies. To calculate the electrostatic free energy (EFE) of biological interests, a unit conversion has to be conducted, because the existing PB energy functionals are all described in terms of the original potential. To bypass this conversion, this paper proposes energy functionals in terms of the dimensionless potential for the first time in the literature, so that the normalized PB equation can be directly derived by using the Euler-Lagrange variational analysis. Moreover, alternative energy forms have been rigorously derived to avoid approximating the gradient of singular functions in the electrostatic stress term. A systematic study has been carried out to examine the surface integrals involved in alternative energy forms and their dependence on finite domain size and mesh step size, which leads to a recommendation on the EFE forms for efficient computation of protein systems. The calculation of the EFE in the regularization formulation, which is an analytical approach for treating singular charge sources of the PB equation, has also been studied. The proposed energy forms have been validated by considering smooth dielectric settings, such as diffuse interface and super-Gaussian, for which the EFE of the nonlinear PB model is found to be significantly different from that of the linearized PB model. All proposed energy functionals and EFE forms are designed such that the dimensionless potential can be simply plugged in to compute the EFE in the unit of kcal/mol, and they can also be applied in the classical sharp interface PB model.

Interplay between Ionic Strength, Association Rates and Electrostatic Interaction in the C3d:CR2 Complex

The complement system, consisting of several plasma proteins and cell-bound receptors, is an important part of innate immunity. Within the complement system, the interaction between complement fragment C3d and complement receptor 2 (CR2) acts as a link between innate and adaptive immunity. Mutations in the C3d:CR2 interaction are implicated in the pathogenesis of autoimmune diseases. Electrostatic interactions have been suggested to amplify complement response by driving conformational changes, and, in the case of C3d:CR2 complex, electrostatic interactions have been suggested as the basis for association. In our study, we utilize the more recent structure of C3d:CR2 (PDB Code: 3OED), which has been established to be physiologically relevant, to evaluate the effect of ionic strength on the association rate of CR2 to C3d using Brownian dynamics simulations and electrostatic free energy calculations. We also utilize mutants from experimental studies to investigate the contributions of specific residues to the association of the C3d:CR2 complex. We demonstrate that the association rate constant is ionic strength-dependent and that the rate constant decreases with increasing ionic strength. Mutations disrupting favorable or unfavorable interactions in the computational mutagenesis study result in slower or higher association rate constants, respectively. Our results are in line with the presence of electrostatic steering in accelerating the interaction between C3d and CR2. The mutagenesis results demonstrate the applicability of this method in optimizing the design of protein-protein interaction interfaces.

An evaluation of Poisson–Boltzmann electrostatic free energy calculations through comparison with experimental mutagenesis data

For systems involving highly and oppositely charged proteins, electrostatic forces dominate association and contribute to biomolecular complex stability. Using experimental or theoretical alanine-scanning mutagenesis, it is possible to elucidate the contribution of individual ionizable amino acids to protein association. We evaluated our electrostatic free energy calculations by comparing calculated and experimental data for alanine mutants of five protein complexes. We calculated Poisson-Boltzmann electrostatic free energies based on a thermodynamic cycle, which incorporates association in a reference (Coulombic) and solvated (solution) state, as well as solvation effects. We observe that Coulombic and solvation free energy values correlate with experimental data in highly and oppositely charged systems, but not in systems comprised of similarly charged proteins. We also observe that correlation between solution and experimental free energies is dependent on dielectric coefficient selection for the protein interior. Free energy correlations improve as protein dielectric coefficient increases, suggesting that the protein interior experiences moderate dielectric screening, despite being shielded from solvent. We propose that higher dielectric coefficients may be necessary to more accurately predict protein-protein association. Additionally, our data suggest that Coulombic potential calculations alone may be sufficient to predict relative binding of protein mutants.

Molecular orientations at interfaces by extended polarizable continuum model

This study presents an investigation into orientation of molecular solutes at the interface of liquid water and other media. The calculation of electrostatic free energy of molecular solute is based on an extension of the polarizable continuum model (PCM) to interfacial system. The extended PCM computational scheme is incorporated with the self-consistent field procedure which is necessary to obtain more accurate electrostatic free energy and charge density distribution. The computation of non-electrostatic energy for interfacial system is also realized. Applying the numerical procedure to molecular systems, N,N′-diethyl-p-nitroaniline (DEPNA) at air/water interface and p-nitrophenol (PNP) at cyclohexane/water interface, the average orientational angles are in reasonable agreement with the experimental results. Taking both the electrostatic and the non-electrostatic energies into account, the analysis on the energy profiles shows that the electrostatic solvation energy is the dominant factor in determining the orientation angle for PNP, whereas for DEPNA, the orientation angle mainly depends on the cavitation energy. This suggests that, in addition to the electrostatic energy, taking the cavitation energy into account may provide a more complete view when we survey the molecular orientation at interface. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

A structural model for K2P potassium channels based on 23 pairs of interacting sites and continuum electrostatics.

K2PØ, the two-pore domain potassium background channel that determines cardiac rhythm in Drosophila melanogaster, and its homologues that establish excitable membrane activity in mammals are of unknown structure. K2P subunits have two pore domains flanked by transmembrane (TM) spans: TM1-P1-TM2-TM3-P2-TM4. To establish spatial relationships in K2PØ, we identified pairs of sites that display electrostatic compensation. Channels silenced by the addition of a charge in pore loop 1 (P1) or P2 were restored to function by countercharges at specific second sites. A three-dimensional homology model was determined using the crystal structure of KV1.2, effects of K2PØ mutations to establish alignment, and compensatory charge–charge pairs. The model was refined and validated by continuum electrostatic free energy calculations and covalent linkage of introduced cysteines. K2P channels use two subunits arranged so that the P1 and P2 loops contribute to one pore, identical P loops face each other diagonally across the pore, and the channel complex has bilateral symmetry with a fourfold symmetric selectivity filter.

Electrostatic free energy calculations using the generalized solvent boundary potential method

Free energy perturbation (FEP) calculations using all-atom molecular dynamics simulations with a large number of explicit solvent molecules are a powerful approach to study ligand–macromolecule association processes at the atomic level. One strategy to carry out FEP calculations efficiently and reduce computational time is to consider the explicit dynamics of only a small number of atoms in a localized region around the ligand. Such an approximation is motivated by the observation that the factors governing binding specificity are dominated by interactions in the vicinity of the ligand. However, a straightforward truncation of the system may yield inaccurate results as the influence exerted by the remote regions of the macromolecule and the surrounding solvent through long-range electrostatic effects may be significant. To obtain meaningful results, it is important to incorporate the influence of the remote regions of the ligand–macromolecule complex implicitly using some effective potential. The generalized solvent boundary potential (GSBP) that was developed recently [W. Im, S. Bernèche, and B. Roux, J. Chem. Phys. 114, 2924 (2001)] is an efficient computational method to represent the long-range electrostatic interactions arising from remote (outer) regions in simulations of a localized (inner) region with a small number of explicit atoms. In the present work, FEP calculations combined with GSBP are used to illustrate the importance of these long-range electrostatic factors in estimation of the charging free energy of an aspartate ligand bound to the aspartyl-tRNA synthetase. Calculations with explicit spherical simulation inner regions of different radii are used to test the accuracy of the GSBP method and also illustrate the importance of explicit protein and solvent dynamics in the free energy estimation. The influence of the represented outer region is tested using separate simulations in which the reaction field and/or the protein static field are excluded. Both components are shown to be essential to obtain quantitatively meaningful results. The ability of implicitly treating the influence of protein fluctuations in the outer region using a protein dielectric constant is examined. It is shown that accurate charging free energy calculations can be performed for this system with a spherical region of 15 to 20 Å radius, which roughly corresponds to 1500–3500 moving atoms. The results indicate that GSBP in combination with FEP calculations is a precise and efficient approach to include long-range electrostatic effects in the study of ligand binding to large macromolecules.

Theoretical descriptors for the quantitative rationalisation of plastocyanin mutant functional propertiess.

A quantitative rationalisation of the effect of specific amino acids on the recognition process and redox characteristics of plastocyanin towards cytochrome f, as determined by point mutation experiments, has been attempted in this study. To achieve this goal we derived theoretical descriptors directly from the three-dimensional structure of the plastocyanin mutants, in the same manner as it is usually done for small drug-like molecules. The protein descriptors computed can be related to: (a) the electrostatic and dipole-dipole interactions, effective at long distance; (b) polar interactions whose features are encoded by charged partial surface area descriptors; (c) the propensity of the surface residues to form hydrogen bonding interactions; and (d) dispersion and repulsive interactions. Moreover, an estimation of mutation-dependent variation of redox potential observed has been obtained by electrostatic free energy calculations. The quantitative structure-activity relationship (QSAR) models offer structural interpretation of the point mutation experiment responses and can be of help in the design of new protein engineering experiments.

Titration calculations of foot-and-mouth disease virus capsids and their stabilities as a function of pH

Foot-and-mouth disease virus (FMDV), a non-enveloped picornavirus, is sensitive to acidic conditions. At pH values below 7 the icosahedral virus capsid, formed from 60 copies of a protomer containing four polypeptides (VP1 to 4), dissociates into 12 pentamers, releasing the viral RNA. Evidence suggests that this acid lability may assist FMDV cell entry via an endosomal pathway. Calculations of titration curves and pH-stability profiles are presented for three different strains of FMDV, O1BFS, A1061 and A22 Iraq, and compared with experimental data for complete virions and empty capsids (which lack RNA). The finite difference Poisson-Boltzmann method was used for the calculation of electrostatic free energies with the solvent treated as a dielectric continuum. The inter-pentamer interface in the virus is formed by two protomers related by 2-fold icosahedral symmetry. As a simple model for inter-pentamer interactions, a dimer and two separate protomers were compared. The association free energy was computed by integrating the difference between the titration curves of the two species. The calculations reproduced the observed decrease in capsid stability at acidic pH but not the difference in pH sensitivities of the two type A viruses. It is shown that only residues within 15 Å of the interface play a significant role in determining acid lability. For the experimentally studied pH range (5 to 7.6), histidine residues were found to dominate the pH-dependence of the stability. Two histidine residues in VP3, H142 and H145, are shown to have the greatest effect by virtue of their interactions with many polar residues across the inter-pentamer interface; the interaction of H142 with an α-helix in the opposite pentamer contributes only a small proportion of the destabilization energy.