Continuum Electrostatic Model Research Articles

Molecular electrostatics and pKa shifts calculations with the Generalized Born model. A tutorial through examples with Bluues2

Biomolecular electrostatics is of key importance for biological function and recognition. The continuum electrostatic model based on the Poisson-Boltzmann (PB) equation has been widely used to study biomolecular electrostatics. The solution of the PB equation gives the electrostatic potential over the space enclosing the molecule(s) of interest, typically obtained at points of a suitable grid.The Generalized Born (GB) model has been used to provide a useful approximation to the solution obtained by solving numerically the PB equation. The main advantage of the GB approach is to express the electrostatic energy of the system as the sum of pairwise interactions. The latter depend on geometric parameters (the GB radii) which depend on all atoms, and their radii, of the system. In this work we present a tutorial through examples for a more efficient and general version of the program Bluues which is able to compute:1) the generalized Born radius of each atom;2) the electrostatic potential at the surface of the molecule mapped to solvent accessible atoms;3) the solvent accessible surface in a PDB formatted file;4) the electrostatic potential in the volume surrounding the molecule;5) the electrostatic free energy and different contributions to it;6) the pH-dependent properties of proteins (total charge and pH-dependent free energy of folding) in the pH range −4 to 18;7) the pKa shifts due to molecular structure of all ionizable groups. Program summaryProgram Title: bluues2CPC Library link to program files:https://doi.org/10.17632/bx98gcrzbg.1Developer's repository link:https://github.com/federico-fogolari/bluues2Code Ocean capsule:https://codeocean.com/capsule/9338710Licensing provisions: GPLv3Programming language: CNature of problem: Biomolecular electrostatics is of key importance for biological function and recognition. Computation of electrostatic effects and their dissection in atomic contributions is a general problem in molecular modeling.Solution method: The GBR6 version of the Generalized Born (GB) model is implemented through surface integrals and all common molecular electrostatic analyses are performed by a single command line program.Additional comments including restrictions and unusual features: The program requires, as input, files in pqr format, which may be generated by independent software (e.g. babel, pdb2pqr). For the optional generation of the solvent excluded surface, the software msms must be installed and present in the path.

Solvation Thermodynamics of Solutes in Water and Ionic Liquids Using the Multiscale Solvation-Layer Interface Condition Continuum Model.

Molecular assembly processes are generally driven by thermodynamic properties in solutions. Atomistic modeling can be very helpful in designing and understanding complex systems, except that bulk solvent is very inefficient to treat explicitly as discrete molecules. In this work, we develop and assess two multiscale solvation models for computing solvation thermodynamic properties. The new SLIC/CDC model combines continuum solvent electrostatics based on the solvent layer interface condition (SLIC) with new statistical thermodynamic models for hydrogen bonding and nonpolar modes: cavity formation, dispersion interactions, combinatorial mixing (CDC). Given the structures of 500 solutes, the SLIC/CDC model predicts Gibbs energies of solvation in water with an average accuracy better than 1 kcal/mol, when compared to experimental measurements, and better than 0.8 kcal/mol, when compared to explicit-solvent molecular dynamics simulations. The individual SLIC/CDC energy mode values agree quantitatively with those computed from explicit-solvent molecular dynamics. The previously published SLIC/SASA multiscale model combines the SLIC continuum electrostatic model with the solvent-accessible surface area (SASA) nonpolar energy mode. With our new, improved parametrization method, the SLIC/SASA model now predicts Gibbs energies of solvation with better than 1.4 kcal/mol average accuracy in aqueous systems, compared to experimental and explicit-solvent molecular dynamics, and better than 1.6 kcal/mol average accuracy in ionic liquids, compared to explicit-solvent molecular dynamics. Both models predict solvation entropies, and are the first implicit-solvation models capable of predicting solvation heat capacities.

Interplay of Interfacial Viscosity, Specific-Ion, and Impurity Adsorption Determines Zeta Potentials of Phospholipid Membranes.

Ion-specific induced changes of the ζ-potential of phospholipid vesicles are commonly used to quantify the affinity of different ions to the lipid interface. The negative ζ-potential of zwitterionic net-neutral phospholipid vesicles in neat water, which changes sign and increases in solutions of NaCl or KCl, is a phenomenon consistently observed in experiments but not fully understood theoretically. Using atomistic molecular dynamics simulations in the presence of applied electric fields which drive electroosmotic flows, in combination with an electrostatic continuum model based on the modified Poisson-Boltzmann and Helmholtz-Smoluchowski equations, we study the electrokinetic and electrostatic properties as well as the specific ion affinities to the phospholipid-water interface, in order to resolve these puzzling observations. Our modified continuum equations account for the dielectric profile at the lipid-water interface, ion-specific interactions between ions and the lipid-water interface, and the interfacial viscosity profile, which are all extracted from our atomistic simulations and rather accurately predict ion-density and electrostatic-potential distributions as well as ζ-potentials in comparison with our atomistic simulations. Our continuum model can explain experimental ζ-potentials only when we assume minute amounts of surface-active anionic impurities in the aqueous solution. In fact, the amount of impurities needed to explain the experimental data increases linearly with the salt concentration, suggesting that surface-active species, which might be already present in the lab water or lipid samples, could further be introduced through the added salt.

Implicit Solvents for the Polarizable Atomic Multipole AMOEBA Force Field.

Computational protein design, ab initio protein/RNA folding, and protein-ligand screening can be too computationally demanding for explicit treatment of solvent. For these applications, implicit solvent offers a compelling alternative, which we describe here for the polarizable atomic multipole AMOEBA force field based on three treatments of continuum electrostatics: numerical solutions to the nonlinear and linearized versions of the Poisson-Boltzmann equation (PBE), the domain-decomposition conductor-like screening model (ddCOSMO) approximation to the PBE, and the analytic generalized Kirkwood (GK) approximation. The continuum electrostatics models are combined with a nonpolar estimator based on novel cavitation and dispersion terms. Electrostatic model parameters are numerically optimized using a least-squares style target function based on a library of 103 small-molecule solvation free energy differences. Mean signed errors for the adaptive Poisson-Boltzmann solver (APBS), ddCOSMO, and GK models are 0.05, 0.00, and 0.00 kcal/mol, respectively, while the mean unsigned errors are 0.70, 0.63, and 0.58 kcal/mol, respectively. Validation of the electrostatic response of the resulting implicit solvents, which are available in the Tinker (or Tinker-HP), OpenMM, and Force Field X software packages, is based on comparisons to explicit solvent simulations for a series of proteins and nucleic acids. Overall, the emergence of performative implicit solvent models for polarizable force fields opens the door to their use for folding and design applications.

Experimental Evidence of Solvent-Separated Ion Pairs as Metastable States in Electrostatic Interactions of Biological Macromolecules.

Electrostatic interactions via ion pairs are vital for biological macromolecules. Regarding the free energy of each ion pair as a function of the interionic distance, continuum electrostatic models predict a single energy minimum corresponding to the contact ion-pair (CIP) state, whereas atomically detailed theoretical hydration studies predict multiple energy minima corresponding to the CIP and solvent-separated ion-pair (SIP) states. Through a statistical analysis of high-resolution crystal structures, we present experimental evidence of the SIP as a metastable state. The histogram of interionic distances between protein side-chain NH3+ and DNA phosphate groups clearly shows two major peaks corresponding to the CIP and SIP states. The statistical data are consistent with the probability distribution of the CIP-SIP equilibria previously obtained with molecular dynamics simulations. Spatial distributions of NH3+ ions and water molecules around phosphates reveal preferential sites for CIP and SIP formations and show how the ions compete with water molecules.

Ion Pairs Buried in Hydrophobic Environments within Proteins: Electrostatic Crosstalk between Buried Groups

Internal ion pairs buried in the hydrophobic interior of proteins are essential for processes such as H+ transport, e- transfer, and catalysis. Despite their importance, the properties of buried ion pairs remain poorly understood. It has been suggested that medium- or long-range Coulomb interactions could play a role in tuning biological function, but this would only be possible if proteins behaved as materials with low dielectric constant, as assumed in many continuum electrostatics models. Buried ion pairs are of interest then not only for their functional roles, but also because they can be used to examine the balance between Coulomb and hydration energies in the dry, hydrophobic interior of proteins. As the distance between a pair of internal ionizable groups increases and the energy of their Coulomb interaction decreases, the balance of these energies will favor burial of the groups in the neutral state, eliminating any direct electrostatic interaction between them. To examine the magnitude of electrostatic crosstalk between buried ionizable groups experimentally, many variants were engineered on a highly stable form of staph nuclease. All variants have His66 in the hydrophobic core and either Asp or Glu residues buried at another internal position. Many His-Asp/Glu pairs were studied, and in no case were medium- or long-range Coulomb interactions observed. When the groups were within H-bond distance, strong, favorable coupling energies were measured, but only when the side chains of the buried pair could achieve a geometry favorable for an H-bond interaction. These data suggest that at short range the buried pair behaved as a strong hydrogen bond, but no evidence was found that they ever behave as titratable point charges interacting at a distance through Coulomb forces enhanced by the low dielectric protein medium.

A Systematic Approach to Understanding Macromolecular Crowding Effects on Biomolecular Interactions through Computational “Toy” Models

The cellular environment is crowded, with macromolecules occupying up to 40% of the cell's volume. These “crowders” have been shown to have significant impacts on biomolecular recognition in the cell. Thus, it is important to realistically account for these effects when studying biological complexes. Rather than approaching an understanding of crowding effects through a specific system, this study uses theoretical “toy” models to investigate the impact of macromolecular crowding on binding energetics of a complex as a function of the intrinsic properties of the binding partners and the crowders, focusing on electrostatic interactions and solvation properties. The use of toy model molecules allows for controlled, systematic variation of their physical properties, such as charge distribution, size, and shape. Using a continuum electrostatics model, binding partners and randomly-placed crowders are represented as low-dielectric regions. By studying the trends of crowding effects on both binding affinity and specificity through the lens of the intrinsic, physical properties of the binding partners and the crowders, we hope to achieve a unified, general framework for understanding and predicting how macromolecular crowding affects biomolecular interactions that will help guide future crowding studies, both computational and experimental.

Robustness and Efficiency of Poisson-Boltzmann Modeling on Graphics Processing Units.

Poisson-Boltzmann equation (PBE) based continuum electrostatics models have been widely used in modeling electrostatic interactions in biochemical processes, particularly in estimating protein-ligand binding affinities. Fast convergence of PBE solvers is crucial in binding affinity computations as numerous snapshots need to be processed. Efforts have been reported to develop PBE solvers on graphics processing units (GPUs) for efficient modeling of biomolecules, though only relatively simple successive over-relaxation and conjugate gradient methods were implemented. However, neither convergence nor scaling properties of the two methods are optimal for large biomolecules. On the other hand, geometric multigrid (MG) has been shown to be an optimal solver on CPUs, though no MG have been reported for biomolecular applications on GPUs. This is not a surprise as it is a more complex method and depends on simpler but limited iterative methods such as Gauss-Seidel in its core relaxation procedure. The robustness and efficiency of MG on GPUs are also unclear. Here we present an implementation and a thorough analysis of MG on GPUs. Our analysis shows that robustness is a more pronounced issue than efficiency for both MG and other tested solvers when the single precision is used for complex biomolecules. We further show how to balance robustness and efficiency utilizing MG's overall efficiency and conjugate gradient's robustness, pointing to a hybrid GPU solver with a good balance of efficiency and accuracy. The new PBE solver will significantly improve the computational throughput for a range of biomolecular applications on the GPU platforms.

Dielectric Properties of a Protein Probed by Reversal of a Buried Ion Pair

Thirty years ago, Hwang and Warshel suggested that a microenvironment preorganized to stabilize an ion pair would be incapable of reorganizing to stabilize the reverse ion pair. The implications were that (1) proteins have a limited capacity to reorganize, even under the influence of strong interactions, such as those present when ionizable groups are buried in the hydrophobic interior of a protein, and (2) the inability of proteins to tolerate the reversal of buried ion pairs demonstrates the limitations inherent to continuum electrostatic models of proteins. Previously we showed that when buried individually in the interior of staphylococcal nuclease, Glu23 and Lys36 have p Ka values near pH 7, but when buried simultaneously, they establish a strong interaction of ∼5 kcal/mol and have p Ka values shifted toward more normal values. Here, using equilibrium thermodynamic measurements, crystal structures, and NMR spectroscopy experiments, we show that although the reversed, individual substitutions-Lys23 and Glu36-also have p Ka values near 7, when buried together, they neither establish a strong interaction nor promote reorganization of their microenvironment. These experiments both confirm Warshel's original hypothesis and expand it by showing that it applies to reorganization, as demonstrated by our artificial ion pairs, as well as to preorganization as is commonly argued for motifs that stabilize naturally occurring ion pairs in polar microenvironments. These data constitute a challenging benchmark useful to test the ability of structure-based algorithms to reproduce the compensation between self-energy, Coulomb and polar interactions in hydrophobic environments of proteins.

Extending the Solvation-Layer Interface Condition Continum Electrostatic Model to a Linearized Poisson-Boltzmann Solvent.

We extend the linearized Poisson-Boltzmann (LPB) continuum electrostatic model for molecular solvation to address charge-hydration asymmetry. Our new solvation-layer interface condition (SLIC)/LPB corrects for first-shell response by perturbing the traditional continuum-theory interface conditions at the protein-solvent and the Stern-layer interfaces. We also present a GPU-accelerated treecode implementation capable of simulating large proteins, and our results demonstrate that the new model exhibits significant accuracy improvements over traditional LPB models, while reducing the number of fitting parameters from dozens (atomic radii) to just five parameters, which have physical meanings related to first-shell water behavior at an uncharged interface. In particular, atom radii in the SLIC model are not optimized but uniformly scaled from their Lennard-Jones radii. Compared to explicit-solvent free-energy calculations of individual atoms in small molecules, SLIC/LPB is significantly more accurate than standard parametrizations (RMS error 0.55 kcal/mol for SLIC, compared to RMS error of 3.05 kcal/mol for standard LPB). On parametrizing the electrostatic model with a simple nonpolar component for total molecular solvation free energies, our model predicts octanol/water transfer free energies with an RMS error 1.07 kcal/mol. A more detailed assessment illustrates that standard continuum electrostatic models reproduce total charging free energies via a compensation of significant errors in atomic self-energies; this finding offers a window into improving the accuracy of Generalized-Born theories and other coarse-grained models. Most remarkably, the SLIC model also reproduces positive charging free energies for atoms in hydrophobic groups, whereas standard PB models are unable to generate positive charging free energies regardless of the parametrized radii. The GPU-accelerated solver is freely available online, as is a MATLAB implementation.

A Hybrid Molecular Dynamics/Multiconformer Continuum Electrostatics (MD/MCCE) Approach for the Determination of Surface Charge of Nanomaterials

The surface charge of nanomaterials determines their stability in solution and interaction with other molecules and surfaces, yet experimental determination of surface charge of complex nanomaterials is not straightforward. We propose a hybrid approach that iteratively integrates explicit solvent molecular dynamics simulations and a multiconformer continuum electrostatic model (MCCE) to efficiently sample the configurational and titration spaces of surface ligands of nanomaterials. Test calculations of model systems indicate that the iterative approach converges rapidly even for systems that contain hundreds of titratable sites, making the approach complementary to more elaborate methods such as explicit solvent-based constant-pH molecular dynamics. The hybrid method is applied to analyze the pKa distribution of alkylamines attached to a carbon-based nanoparticle as a function of ligand density, nanoparticle surface curvature, and ligand heterogeneity. The results indicate that functionalization strategie...

Effects of the dielectric properties of the ceramic–solvent interface on the binding of proteins to oxide ceramics: a non-local electrostatic approach

The rapid development of nanoscience and nanotechnology has raised many fundamental questions that significantly impede progress in these fields. In particular, understanding the physicochemical processes at the interface in aqueous solvents requires the development and application of efficient and accurate methods. In the present work we evaluate the electrostatic contribution to the energy of model protein–ceramic complex formation in an aqueous solvent. We apply a non-local (NL) electrostatic approach that accounts for the effects of the short-range structure of the solvent on the electrostatic interactions of the interfacial systems. In this approach the aqueous solvent is considered as a non-ionic liquid, with the rigid and strongly correlated dipoles of the water molecules. We have found that an ordered interfacial aqueous solvent layer at the protein– and ceramic–solvent interfaces reduces the charging energy of both the ceramic and the protein in the solvent, and significantly increases the electrostatic contribution to their association into a complex. This contribution in the presented NL approach was found to be significantly shifted with respect to the classical model at any dielectric constant value of the ceramics. This implies a significant increase of the adsorption energy in the protein–ceramic complex formation for any ceramic material. We show that for several biocompatible ceramics (for example HfO2, ZrO2, and Ta2O5) the above effect predicts electrostatically induced protein–ceramic complex formation. However, in the framework of the classical continuum electrostatic model (the aqueous solvent as a uniform dielectric medium with a high dielectric constant ∼80) the above ceramics cannot be considered as suitable for electrostatically induced complex formation. Our results also show that the protein–ceramic electrostatic interactions can be strong enough to compensate for the unfavorable desolvation effect in the process of protein–ceramic complex formation.

Generalising the mean spherical approximation as a multiscale, nonlinear boundary condition at the solute–solvent interface

ABSTRACTIn this paper, we extend the familiar continuum electrostatic model to incorporate finite-size effects in the solvation layer, by perturbing the usual macroscopic interface condition. The perturbation is based on the mean spherical approximation (MSA), to derive a multiscale solvation-layer interface condition (SLIC/MSA). We show that SLIC/MSA reproduces MSA predictions for Born ions in a variety of polar solvents, including water as well as other protic and aprotic solvents. Importantly, the SLIC/MSA model predicts not only solvation free energies accurately but also solvation entropies, which standard continuum electrostatic models fail to predict. The SLIC/MSA model depends only on the normal electric field at the dielectric boundary, similar to our recent development of a SLIC model for charge-sign hydration asymmetry, and the reformulation of the MSA as an effective boundary condition enables its straightforward application to complex molecules such as proteins, whereas traditionally it is primarily a bulk theory. This work also opens the possibility for other electrolyte models to be incorporated into fast implicit-solvent models of biomolecular electrostatics.

Computational Tools for Interpreting Ion Channel pH-Dependence.

Activity in many biological systems is mediated by pH, involving proton titratable groups with pKas in the relevant pH range. Experimental analysis of pH-dependence in proteins focusses on particular sidechains, often with mutagenesis of histidine, due to its pKa near to neutral pH. The key question for algorithms that predict pKas is whether they are sufficiently accurate to effectively narrow the search for molecular determinants of pH-dependence. Through analysis of inwardly rectifying potassium (Kir) channels and acid-sensing ion channels (ASICs), mutational effects on pH-dependence are probed, distinguishing between groups described as pH-coupled or pH-sensor. Whereas mutation can lead to a shift in transition pH between open and closed forms for either type of group, only for pH-sensor groups does mutation modulate the amplitude of the transition. It is shown that a hybrid Finite Difference Poisson-Boltzmann (FDPB) – Debye-Hückel continuum electrostatic model can filter mutation candidates, providing enrichment for key pH-coupled and pH-sensor residues in both ASICs and Kir channels, in comparison with application of FDPB alone.

Self-consistent field model for strong electrostatic correlations and inhomogeneous dielectric media.

Electrostatic correlations and variable permittivity of electrolytes are essential for exploring many chemical and physical properties of interfaces in aqueous solutions. We propose a continuum electrostatic model for the treatment of these effects in the framework of the self-consistent field theory. The model incorporates a space- or field-dependent dielectric permittivity and an excluded ion-size effect for the correlation energy. This results in a self-energy modified Poisson-Nernst-Planck or Poisson-Boltzmann equation together with state equations for the self energy and the dielectric function. We show that the ionic size is of significant importance in predicting a finite self energy for an ion in an inhomogeneous medium. Asymptotic approximation is proposed for the solution of a generalized Debye-Hückel equation, which has been shown to capture the ionic correlation and dielectric self energy. Through simulating ionic distribution surrounding a macroion, the modified self-consistent field model is shown to agree with particle-based Monte Carlo simulations. Numerical results for symmetric and asymmetric electrolytes demonstrate that the model is able to predict the charge inversion at high correlation regime in the presence of multivalent interfacial ions which is beyond the mean-field theory and also show strong effect to double layer structure due to the space- or field-dependent dielectric permittivity.

Domain and Interdomain Energetics Underlying Gating in Shaker-Type KV Channels

To understand gating events with a time-base many orders-of-magnitude slower than that of atomic motion in voltage-gated ion channels such as the Shaker-type KV channels, a multiscale physical model is constructed from the experimentally well-characterized voltage-sensor (VS) domains coupled to a hydrophobic gate. The four VS domains are described by a continuum electrostatic model under voltage-clamp conditions, the control of ion flow by the gate domain is described by a vapor-lock mechanism, and the simple coupling principle is informed by known experimental results and trial-and-error. The configurational energy computed for each element is used to produce a total Hamiltonian that is a function of applied voltage, VS positions, and gate radius. We compute statistical-mechanical expectation values of macroscopic laboratory observables. This approach stands in contrast with molecular-dynamic models which are challenged by increasing scale, and kinetic models which assume a probability distribution rather than derive it from the underlying physics. This generic model predicts well the Shaker charge/voltage and conductance/voltage relations; the tight constraints underlying these results allow us to quantitatively assess the underlying physical mechanisms. The total electrical work picked up by the VS domains is an order-of-magnitude larger than the work required to actuate the gate itself, suggesting an energetic basis for the evolutionary flexibility of the voltage-gating mechanism. The cooperative slide-and-interlock behavior of the VS domains described by the VS-gate coupling relation leads to the experimentally observed bistable gating. This engineering approach should prove useful in the investigation of various elements underlying gating characteristics and degraded behavior due to mutation.

Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding

We analyze and suggest improvements to a recently developed approximate continuum-electrostatic model for proteins. The model, called BIBEE/I (boundary-integral based electrostatics estimation with interpolation), was able to estimate electrostatic solvation free energies to within a mean unsigned error of 4% on a test set of more than 600 proteins-a significant improvement over previous BIBEE models. In this work, we tested the BIBEE/I model for its capability to predict residue-by-residue interactions in protein-protein binding, using the widely studied model system of trypsin and bovine pancreatic trypsin inhibitor (BPTI). Finding that the BIBEE/I model performs surprisingly less well in this task than simpler BIBEE models, we seek to explain this behavior in terms of the models' differing spectral approximations of the exact boundary-integral operator. Calculations of analytically solvable systems (spheres and tri-axial ellipsoids) suggest two possibilities for improvement. The first is a modified BIBEE/I approach that captures the asymptotic eigenvalue limit correctly, and the second involves the dipole and quadrupole modes for ellipsoidal approximations of protein geometries. Our analysis suggests that fast, rigorous approximate models derived from reduced-basis approximation of boundary-integral equations might reach unprecedented accuracy, if the dipole and quadrupole modes can be captured quickly for general shapes.

Electrostatic Effects on Proton Coupled Electron Transfer in Oxomanganese Complexes Inspired by the Oxygen-Evolving Complex of Photosystem II

The influence of electrostatic interactions on the free energy of proton coupled electron transfer in biomimetic oxomanganese complexes inspired by the oxygen-evolving complex (OEC) of photosystem II (PSII) are investigated. The reported study introduces an enhanced multiconformer continuum electrostatics (MCCE) model, parametrized at the density functional theory (DFT) level with a classical valence model for the oxomanganese core. The calculated pKa's and oxidation midpoint potentials (E(m)'s) match experimental values for eight complexes, indicating that purely electrostatic contributions account for most of the observed couplings between deprotonation and oxidation state transitions. We focus on pKa's of terminal water ligands in [Mn(II/III)(H2O)6](2+/3+) (1), [Mn(III)(P)(H2O)2](3-) (2, P = 5,10,15,20-tetrakis(2,6-dichloro-3-sulfonatophenyl)porphyrinato), [Mn2(IV,IV)(μ-O)2(terpy)2(H2O)2](4+) (3, terpy = 2,2':6',2″-terpyridine), and [Mn3(IV,IV,IV)(μ-O)4(phen)4(H2O)2](4+) (4, phen = 1,10-phenanthroline) and the pKa's of μ-oxo bridges and Mn E(m)'s in [Mn2(μ-O)2(bpy)4] (5, bpy = 2,2'-bipyridyl), [Mn2(μ-O)2(salpn)2] (6, salpn = N,N'-bis(salicylidene)-1,3-propanediamine), [Mn2(μ-O)2(3,5-di(Cl)-salpn)2] (7), and [Mn2(μ-O)2(3,5-di(NO2)-salpn)2] (8). The analysis of complexes 6-8 highlights the strong coupling between electron and proton transfers, with any Mn oxidation lowering the pKa of an oxo bridge by 10.5 ± 0.9 pH units. The model also accounts for changes in the E(m)'s by ligand substituents, such as found in complexes 6-8, due to the electron withdrawing Cl (7) and NO2 (8). The reported study provides the foundation for analysis of electrostatic effects in other oxomanganese complexes and metalloenzymes, where proton coupled electron transfer plays a fundamental role in redox-leveling mechanisms.

A Modified Nonlocal Continuum Electrostatic Model for Protein in Water and Its Analytical Solutions for Ionic Born Models

AbstractA nonlocal continuum electrostatic model, defined as integro-differential equations, can significantly improve the classic Poisson dielectric model, but is too costly to be applied to large protein simulations. To sharply reduce the model’s complexity, a modified nonlocal continuum electrostatic model is presented in this paper for a protein immersed in water solvent, and then transformed equivalently as a system of partial differential equations. By using this new differential equation system, analytical solutions are derived for three different nonlocal ionic Born models, where a monoatomic ion is treated as a dielectric continuum ball with point charge either in the center or uniformly distributed on the surface of the ball. These solutions are analytically verified to satisfy the original integro-differential equations, thereby, validating the new differential equation system.

Thermodynamics of Transport Through the Ammonium Transporter Amt-1 Investigated with Free Energy Calculations

Amt-1 from Archaeoglobus fulgidus (AfAmt-1) belongs to the Amt/Rh family of ammonium/ammonia transporting membrane proteins. The transport mode and the precise microscopic permeation mechanism utilized by these proteins are intensely debated. Open questions concern the identity of the transported substrate (ammonia and/or ammonium) and whether the transport is passive or active. To address these questions, we studied the overall thermodynamics of the different transport modes as a function of the environmental conditions. Then, we investigated the thermodynamics of the underlying microscopic transport mechanisms with free energy calculations within a continuum electrostatics model. The formalism developed for this purpose is of general utility in the calculation of binding free energies for ligands with multiple protonation forms or other binding forms. The results of our calculations are compared to the available experimental and theoretical data on Amt/Rh proteins and discussed in light of the current knowledge on the physiological conditions experienced by microorganisms and plants. We found that microscopic models of electroneutral and electrogenic transport modes are in principle thermodynamically viable. However, only the electrogenic variants have a net thermodynamic driving force under the physiological conditions experienced by microorganisms and plants. Thus, the transport mechanism of AfAmt-1 is most likely electrogenic.