Electrostatic Free Energy Research Articles

A self-consistent phase-field approach to implicit solvation of charged molecules with Poisson-Boltzmann electrostatics.

Dielectric boundary based implicit-solvent models provide efficient descriptions of coarse-grained effects, particularly the electrostatic effect, of aqueous solvent. Recent years have seen the initial success of a new such model, variational implicit-solvent model (VISM) [Dzubiella, Swanson, and McCammon Phys. Rev. Lett. 96, 087802 (2006) and J. Chem. Phys. 124, 084905 (2006)], in capturing multiple dry and wet hydration states, describing the subtle electrostatic effect in hydrophobic interactions, and providing qualitatively good estimates of solvation free energies. Here, we develop a phase-field VISM to the solvation of charged molecules in aqueous solvent to include more flexibility. In this approach, a stable equilibrium molecular system is described by a phase field that takes one constant value in the solute region and a different constant value in the solvent region, and smoothly changes its value on a thin transition layer representing a smeared solute-solvent interface or dielectric boundary. Such a phase field minimizes an effective solvation free-energy functional that consists of the solute-solvent interfacial energy, solute-solvent van der Waals interaction energy, and electrostatic free energy described by the Poisson-Boltzmann theory. We apply our model and methods to the solvation of single ions, two parallel plates, and protein complexes BphC and p53/MDM2 to demonstrate the capability and efficiency of our approach at different levels. With a diffuse dielectric boundary, our new approach can describe the dielectric asymmetry in the solute-solvent interfacial region. Our theory is developed based on rigorous mathematical studies and is also connected to the Lum-Chandler-Weeks theory (1999). We discuss these connections and possible extensions of our theory and methods.

Protein-Ligand Electrostatic Binding Free Energies from Explicit and Implicit Solvation.

Accurate yet efficient computational models of solvent environment are central for most calculations that rely on atomistic modeling, such as prediction of protein-ligand binding affinities. In this study, we evaluate the accuracy of a recently developed generalized Born implicit solvent model, GBNSR6 (Aguilar et al. J. Chem. Theory Comput. 2010, 6, 3613-3639), in estimating the electrostatic solvation free energies (ΔG(pol)) and binding free energies (ΔΔG(pol)) for small protein-ligand complexes. We also compare estimates based on three different explicit solvent models (TIP3P, TIP4PEw, and OPC). The two main findings are as follows. First, the deviation (RMSD = 7.04 kcal/mol) of GBNSR6 binding affinities from commonly used TIP3P reference values is comparable to the deviations between explicit models themselves, e.g. TIP4PEw vs TIP3P (RMSD = 5.30 kcal/mol). A simple uniform adjustment of the atomic radii by a single scaling factor reduces the RMS deviation of GBNSR6 from TIP3P to within the above "error margin" - differences between ΔΔG(pol) estimated by different common explicit solvent models. The simple radii scaling virtually eliminates the systematic deviation (ΔΔG(pol)) between GBNSR6 and two out of the three explicit water models and significantly reduces the deviation from the third explicit model. Second, the differences between electrostatic binding energy estimates from different explicit models is disturbingly large; for example, the deviation between TIP4PEw and TIP3P estimates of ΔΔG(pol) values can be up to ∼50% or ∼9 kcal/mol, which is significantly larger than the "chemical accuracy" goal of ∼1 kcal/mol. The absolute ΔG(pol) calculated with different explicit models could differ by tens of kcal/mol. These discrepancies point to unacceptably high sensitivity of binding affinity estimates to the choice of common explicit water models. The absence of a clear "gold standard" among these models strengthens the case for the use of accurate implicit solvation models for binding energetics, which may be orders of magnitude faster.

A structural model for the in vivo human cornea including collagen-swelling interaction.

A structural model of the in vivo cornea, which accounts for tissue swelling behaviour, for the three-dimensional organization of stromal fibres and for collagen-swelling interaction, is proposed. Modelled as a binary electrolyte gel in thermodynamic equilibrium, the stromal electrostatic free energy is based on the mean-field approximation. To account for active endothelial ionic transport in the in vivo cornea, which modulates osmotic pressure and hydration, stromal mobile ions are shown to satisfy a modified Boltzmann distribution. The elasticity of the stromal collagen network is modelled based on three-dimensional collagen orientation probability distributions for every point in the stroma obtained by synthesizing X-ray diffraction data for azimuthal angle distributions and second harmonic-generated image processing for inclination angle distributions. The model is implemented in a finite-element framework and employed to predict free and confined swelling of stroma in an ionic bath. For the in vivo cornea, the model is used to predict corneal swelling due to increasing intraocular pressure (IOP) and is adapted to model swelling in Fuchs' corneal dystrophy. The biomechanical response of the in vivo cornea to a typical LASIK surgery for myopia is analysed, including tissue fluid pressure and swelling responses. The model provides a new interpretation of the corneal active hydration control (pump-leak) mechanism based on osmotic pressure modulation. The results also illustrate the structural necessity of fibre inclination in stabilizing the corneal refractive surface with respect to changes in tissue hydration and IOP.

Finite electric boundary-layer solutions of a generalized Poisson–Boltzmann equation

We solve the nonlinear Poisson–Boltzmann (P–B) equation of statistical thermodynamics for the external electrostatic potential of a uniformly charged flat plate immersed in an unbounded strong aqueous electrolyte. Our rather general variational formulation yields new solutions for the external potential derived from both the classical Boltzmann distribution and its heuristic Eigen–Wicke modification for concentrated symmetric electrolytes. Electrostatic potentials of these mean-field solutions satisfy a homogeneous condition at a free boundary plane parallel to the electrically conducting plate. The preferred position of this plane, characterizing the outer limit of the charged electrolyte, is determined by minimizing electrostatic free energy of the electrolyte. For a given uniform density of surface charge exceeding a well-defined and experimentally accessible threshold, we show that the generalized nonlinear P–B equation predicts a unique sharp interface separating a charged boundary layer or double layer from electroneutral bulk electrolyte. Sharp electric boundary layers are shown to be an essentially nonlinear phenomenon. In the super-threshold regime, the diffuse Gouy–Chapman solution is inapplicable and thus the Derjaguin–Landau–Verwey–Overbeek analysis, predicting electrostatic repulsion between two sufficiently separated and identically charged parallel plates must be rejected. Similar limitations restrict the applicability of the Grahame equation relating surface charge density to surface potential.

Liquid-vapor equilibria of ionic liquids from a SAFT equation of state with explicit electrostatic free energy contributions.

Statistically associating fluid theory (SAFT) provides closed-form free energies by perturbation methods. We propose here a SAFT equation of state for ionic liquids that models the contribution from Coulomb forces after that of the Restricted Primitive Model (RPM) in the Mean Spherical Approximation (MSA). The resulting SAFT-MSA equation, fitted to simulated orthobaric curves of imidazolium based ionic liquids, predicts experimental density data with errors ≈1% and the characteristic decrease of all critical coordinates with increasing cation size. The SAFT-MSA equation can be applied to calculate thermodynamic coefficients, the speed of sound and surface tension (among other properties) of pure ionic liquids and can be generalized straightforwardly to mixtures.

Continuum electrostatic approach for evaluating positions and interactions of proteins in a bilayer membrane

Orientations of proteins in the membranes are crucial to their function and stability. Unfortunately the exact positions of these proteins in the lipid bilayer are mostly undetermined. Here, the spatial orientation of membrane proteins within the lipid membrane was evaluated using a Poisson–Boltzmann solvent continuum approach to calculate the electrostatic free energy of the protein solvation at various orientations in an implicit bilayer. The solvation energy was obtained by computing the difference in electrostatic energies of the protein in water and in lipid/water environments, treating each as an implicit solvent model. The optimal position of transmembrane proteins (TMP) in a lipid bilayer is identified by the minimum in the “downhill” pathway of the solvation energy landscape. The energy landscape pattern was considerably conserved in various TMP classes. Evaluation of the position of 1060 membrane proteins from the orientations of proteins in membranes (OPM) database revealed that most of the polytopic and β-barrel proteins were in good agreement with those of the OPM database. The study provides a useful scheme for estimating the membrane solvation energy made by lipid-exposed amino acids in membrane proteins. In addition, our results tested with the bacterial potassium channel model demonstrated the potential usefulness of the approach in assessing the quality of membrane protein models. The present approach should be applicable for constructing transmembrane proteins–lipid configuration suitable for membrane protein simulations and will have utility for the structural modeling of membrane proteins.

TBI server: a web server for predicting ion effects in RNA folding.

BackgroundMetal ions play a critical role in the stabilization of RNA structures. Therefore, accurate prediction of the ion effects in RNA folding can have a far-reaching impact on our understanding of RNA structure and function. Multivalent ions, especially Mg2+, are essential for RNA tertiary structure formation. These ions can possibly become strongly correlated in the close vicinity of RNA surface. Most of the currently available software packages, which have widespread success in predicting ion effects in biomolecular systems, however, do not explicitly account for the ion correlation effect. Therefore, it is important to develop a software package/web server for the prediction of ion electrostatics in RNA folding by including ion correlation effects.ResultsThe TBI web server http://rna.physics.missouri.edu/tbi_index.html provides predictions for the total electrostatic free energy, the different free energy components, and the mean number and the most probable distributions of the bound ions. A novel feature of the TBI server is its ability to account for ion correlation and ion distribution fluctuation effects.ConclusionsBy accounting for the ion correlation and fluctuation effects, the TBI server is a unique online tool for computing ion-mediated electrostatic properties for given RNA structures. The results can provide important data for in-depth analysis for ion effects in RNA folding including the ion-dependence of folding stability, ion uptake in the folding process, and the interplay between the different energetic components.

Extended Debye-Hückel theory for studying the electrostatic solvation energy.

The electrostatic part of the solvation energy has been studied by using extended Debye-Hückel (DH) theories. Specifically, our molecular Debye-Hückel theory [J. Chem. Phys. 2011, 135, 104104] and its simplified version, an energy-scaled Debye-Hückel theory, were applied to electrolytes with strong electrostatic coupling. Our theories provide a practical methodology for calculating the electrostatic solvation free energies, and the accuracy was verified for atomic and diatomic charged solutes.

Interplay between Electrostatics and Cation-PI Interactions Governs the Specific Membrane Binding of Phosphatidylinositol-Specific Phospholipase-C

Bacillus thuringiensis phosphatidylinositol-specific phospholipase C (BtPI-PLC) is an amphitropic enzyme which cleaves GPI-anchored proteins off the outer surface of eukaryotic plasma membranes. Amphitropic proteins bind specifically and transiently to the surface of cell membranes, and their functions are regulated upon binding. It is commonly acknowledged that non-specific electrostatic forces are responsible for their long-range interactions with membranes. Using continuum electrostatics calculations we show how, despite having an overall negative charge (−7e), the charge distribution of BtPI-PLC leads to favorable electrostatic interactions with anionic membranes. However, the resulting electrostatic binding free energy, which is essential for membrane binding, is quite low. Mutation of a single, key basic residue to alanine diminishes this long range electrostatic contribution making it difficult for BtPI-PLC to associate with membranes. Once close to the membrane surface, short range non-specific hydrophobic interactions and specific cation-pi interactions with the N(Me)3 groups of phosphatidylcholine (PC) lipids of the membrane likely come into play for BtPI-PLC binding to the membrane surface. 500ns-long all-atom molecular dynamics simulations of BtPI-PLC docked to mixed bilayers with varying ratio of zwitterionic lipids indeed confirm this. Finally, we see that the interplay between long range electrostatics and short range, PC specific cation-pi interactions governs the specificity of BtPI-PLC for PC rich membrane. Moreover, our results show that BtPI-PLC can achieve favorable electrostatics interactions with lipid bilayers without having surface-exposed basic clusters suggesting that such clusters are not always necessary for the regulation of amphitropic enzyme binding.

DIFFUSED SOLUTE-SOLVENT INTERFACE WITH POISSON-BOLTZMANN ELECTROSTATICS: FREE-ENERGY VARIATION AND SHARP-INTERFACE LIMIT.

A phase-field free-energy functional for the solvation of charged molecules (e.g., proteins) in aqueous solvent (i.e., water or salted water) is constructed. The functional consists of the solute volumetric and solute-solvent interfacial energies, the solute-solvent van der Waals interaction energy, and the continuum electrostatic free energy described by the Poisson-Boltzmann theory. All these are expressed in terms of phase fields that, for low free-energy conformations, are close to one value in the solute phase and another in the solvent phase. A key property of the model is that the phase-field interpolation of dielectric coefficient has the vanishing derivative at both solute and solvent phases. The first variation of such an effective free-energy functional is derived. Matched asymptotic analysis is carried out for the resulting relaxation dynamics of the diffused solute-solvent interface. It is shown that the sharp-interface limit is exactly the variational implicit-solvent model that has successfully captured capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states of underlying biomolecular systems as found in experiment and molecular dynamics simulations. Our phase-field approach and analysis can be used to possibly couple the description of interfacial fluctuations for efficient numerical computations of biomolecular interactions.

Alumina surfaces with nanoscale topography reduce attachment and biofilm formation by Escherichia coli and Listeria spp.

This work reports on a simple, robust and scientifically sound method to develop surfaces able to reduce microbial attachment and biofilm development, with possible applications in medicine, dentistry, food processing, or water treatment. Anodic surfaces with cylindrical nanopores 15 to 100 nm in diameter were manufactured and incubated with Escherichia coli ATCC 25922 and Listeria innocua. Surfaces with 15 and 25 nm pore diameters significantly repressed attachment and biofilm formation. Surface–bacteria interaction forces calculated using the extended Derjaguin Landau Verwey-Overbeek (XDLVO) theory indicate that reduction in attachment and biofilm formation is due to a synergy between electrostatic repulsion and surface effective free energy. An attachment study using E. coli K12 strains unable to express appendages also suggests that the small-pore surfaces may inhibit flagella-dependent attachment. These results can have immediate, far-reaching implications and commercial applications, with substantial benefits for human health and life.

Ionic strength independence of charge distributions in solvation of biomolecules

Electrostatic forces enormously impact the structure, interactions, and function of biomolecules. We perform all-atom molecular dynamics simulations for 5 proteins and 5 RNAs to determine the dependence on ionic strength of the ion and water charge distributions surrounding the biomolecules, as well as the contributions of ions to the electrostatic free energy of interaction between the biomolecule and the surrounding salt solution (for a total of 40 different biomolecule/solvent combinations). Although water provides the dominant contribution to the charge density distribution and to the electrostatic potential even in 1M NaCl solutions, the contributions of water molecules and of ions to the total electrostatic interaction free energy with the solvated biomolecule are comparable. The electrostatic biomolecule/solvent interaction energies and the total charge distribution exhibit a remarkable insensitivity to salt concentrations over a huge range of salt concentrations (20 mM to 1M NaCl). The electrostatic potentials near the biomolecule's surface obtained from the MD simulations differ markedly, as expected, from the potentials predicted by continuum dielectric models, even though the total electrostatic interaction free energies are within 11% of each other.

Lamellar cationic lipid-DNA complexes from lipids with a strong preference for planar geometry: A Minimal Electrostatic Model.

We formulate and analyze a minimal model, based on condensation theory, of the lamellar cationic lipid (CL)-DNA complex of alternately charged lipid bilayers and DNA monolayers in a salt solution. Each lipid bilayer, composed by a random mixture of cationic and neutral lipids, is assumed to be a rigid uniformly charged plane. Each DNA monolayer, located between two lipid bilayers, is formed by the same number of parallel DNAs with a uniform separation distance. For the electrostatic calculation, the model lipoplex is collapsed to a single plane with charge density equal to the net lipid and DNA charge. The free energy difference between the lamellar lipoplex and a reference state of the same number of free lipid bilayers and free DNAs, is calculated as a function of the fraction of CLs, of the ratio of the number of CL charges to the number of negative charges of the DNA phosphates, and of the total number of planes. At the isoelectric point the free energy difference is minimal. The complex formation, already favoured by the decrease of the electrostatic charging free energy, is driven further by the free energy gain due to the release of counterions from the DNAs and from the lipid bilayers, if strongly charged. This minimal model compares well with experiment for lipids having a strong preference for planar geometry and with major features of more detailed models of the lipoplex.

Charge and force on a conductive sphere between two parallel electrodes: A Stokesian dynamics approach

We present an accurate and efficient method to compute the electrostatic charge and force on a conductive sphere between two parallel electrodes. The method relies on a Stokesian dynamics-like approach, in which the capacitance tensor is divided into two contributions: (1) a far field contribution that captures the long range, many body interactions between the sphere and the two electrodes and (2) a near field contribution that captures the pairwise interactions between nearly contacting surfaces. The accuracy of this approach is confirmed by comparison to “exact” numerical results obtained by finite element modeling. From the capacitance tensor, we derive the charge and dipole moment on the sphere, the electrostatic free energy of the system, and the electrostatic force on the sphere. These quantities are used to describe the dynamics of micron-scale particles oscillating within a viscous dielectric liquid between two parallel electrodes subject to constant voltage. Simulated particle trajectories agree quantitatively with those captured experimentally by high speed imaging.

Pressure response of amide one-bond J-couplings in model peptides and proteins.

The pressure dependence of the one-bond indirect spin-spin coupling constants (1)J(N-H) was studied in the protected tetrapeptides Ac-Gly-Gly-Xxx-Ala-NH2 (with Xxx being one of the 20 proteinogenic amino acids). The response of the (1)J(N-H) coupling constants is amino acid type specific, with an average increase of its magnitude by 0.6 Hz at 200 MPa. The variance of the pressure response is rather large, the largest pressure effect is observed for asparagine where the coupling constant becomes more negative by -2.9 Hz at 200 MPa. The size of the J-coupling constant at high pressure is positively correlated with its low pressure value and the β-propensity, and negatively correlated with the amide proton shift and the first order nitrogen pressure coefficient and the electrostatic solvation free energy.

A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent

In this paper, a novel solution decomposition of the Poisson dielectric model is proposed to modify a traditional electrostatic free energy minimization problem into one that is well defined for the case of protein in ionic solvent. The target function of this modified problem is shown to be strictly convex, weak sequentially lower semicontinuous, and twice continuously Fréchet differentiable. Its first and second Gâteaux derivatives are then found. Moreover, it is proved that this modified electrostatic free energy minimization problem has a unique solution, and its solution existence and uniqueness is equivalent to that of the Poisson–Boltzmann equation, a widely-used implicit solvent model for computing the electrostatics of biomolecules.

Combined kinetic studies and computational analysis on kojic acid analogous as tyrosinase inhibitors.

Tyrosinase is a key enzyme in melanin synthesis and widely distributed in plants and animals tissues. In mammals, this enzyme is related to pigment production, involved in wound healing, primary immune response and it can also contribute to catecholamines synthesis in the brain. Consequently, tyrosinase enzyme represents an attractive and selective target in the field of the medicine, cosmetics and bio-insecticides. In this paper, experimental kinetics and computational analysis were used to study the inhibition of tyrosinase by analogs of Kojic acid. The main interactions occurring between inhibitors-tyrosinase complexes and the influence of divalent cation (Cu2+) in enzymatic inhibition were investigated by using molecular docking, molecular dynamic simulations and electrostatic binding free energy by using the Linear Interaction Energy (LIE) method. The results showed that the electrostatic binding free energy are correlated with values of constant inhibition (r2 = 0.97).Thus, the model obtained here could contribute to future studies of this important system and, therefore, eventually facilitate development of tyrosinase inhibitors.

An Overview of Electrostatic Free Energy Computations for Solutions and Proteins.

Free energy simulations for electrostatic and charging processes in complex molecular systems encounter specific difficulties owing to the long-range, 1/r Coulomb interaction. To calculate the solvation free energy of a simple ion, it is essential to take into account the polarization of nearby solvent but also the electrostatic potential drop across the liquid-gas boundary, however distant. The latter does not exist in a simulation model based on periodic boundary conditions because there is no physical boundary to the system. An important consequence is that the reference value of the electrostatic potential is not an ion in a vacuum. Also, in an infinite system, the electrostatic potential felt by a perturbing charge is conditionally convergent and dependent on the choice of computational conventions. Furthermore, with Ewald lattice summation and tinfoil conducting boundary conditions, the charges experience a spurious shift in the potential that depends on the details of the simulation system such as the volume fraction occupied by the solvent. All these issues can be handled with established computational protocols, as reviewed here and illustrated for several small ions and three solvated proteins.

Recent progress in adapting Poisson–Boltzmann methods to molecular simulations

Electrostatic solvation modeling based upon the Poisson–Boltzmann equation is widely used in studies of biomolecular structures and functions. This manuscript provides a thorough review of published efforts to adapt the numerical Poisson–Boltzmann methods to molecular simulations so that these methods can be extended to biomolecular studies involving conformational fluctuation and/or dynamics. We first review the fundamental works on how to define the electrostatic free energy and the Maxwell stress tensor. These topics are followed by three different strategies in developing algorithms to compute electrostatic forces and how to improve their numerical performance. Finally procedures are also presented in detail on how to discretize these algorithms for numerical calculations. Given the pioneer works reviewed here, further developmental efforts will be on how to balance efficiency and accuracy in these theoretical sound approaches — two important issues in applying any numerical algorithms for routine biomolecular applications. Even if not reviewed here, more advanced numerical solvers are certainly necessary to achieve higher accuracy than the widely used classical methods to improve the overall performance of the numerical Poisson–Boltzmann methods.

Introducing Charge Hydration Asymmetry into the Generalized Born Model.

The effect of charge hydration asymmetry (CHA)—non-invariance of solvation free energy upon solute charge inversion—is missing from the standard linear response continuum electrostatics. The proposed charge hydration asymmetric–generalized Born (CHA–GB) approximation introduces this effect into the popular generalized Born (GB) model. The CHA is added to the GB equation via an analytical correction that quantifies the specific propensity of CHA of a given water model; the latter is determined by the charge distribution within the water model. Significant variations in CHA seen in explicit water (TIP3P, TIP4P-Ew, and TIP5P-E) free energy calculations on charge-inverted “molecular bracelets” are closely reproduced by CHA–GB, with the accuracy similar to models such as SEA and 3D-RISM that go beyond the linear response. Compared against reference explicit (TIP3P) electrostatic solvation free energies, CHA–GB shows about a 40% improvement in accuracy over the canonical GB, tested on a diverse set of 248 rigid small neutral molecules (root mean square error, rmse = 0.88 kcal/mol for CHA–GB vs 1.24 kcal/mol for GB) and 48 conformations of amino acid analogs (rmse = 0.81 kcal/mol vs 1.26 kcal/mol). CHA–GB employs a novel definition of the dielectric boundary that does not subsume the CHA effects into the intrinsic atomic radii. The strategy leads to finding a new set of intrinsic atomic radii optimized for CHA–GB; these radii show physically meaningful variation with the atom type, in contrast to the radii set optimized for GB. Compared to several popular radii sets used with the original GB model, the new radii set shows better transferability between different classes of molecules.