Continuum Electrostatics Research Articles

Dimensionality-Reduced Fermi Level Pinning in Coplanar 2D Heterojunctions.

Electronic transport through a metal|semiconductor (M|S) heterojunction is largely determined by its Schottky barrier. In 3D M|S junctions, the barrier height determines the turn-on voltage and is often pinned by the interface states, causing Fermi level pinning (FLP). The pinning strength in 3D depends on the ratio Ci/CM between the interface quantum capacitance Ci and the metal surface capacitance CM. In 2D, the interface dipole does not influence the band alignment, but still affects the Schottky barrier and transport. In light of the general interest in building 2D electronics, in this work we discover the relevant material parameters which dictate the behavior and strength of FLP in 2D M|S contacts. Using a multiscale model combining first-principles, continuum electrostatics, and transport calculations, we study a realistic Gr|MoS2 interface as an example with high interface state density (Ci/CM ≫ 1). Transport calculations show partial pinning with a strength P ∼ 0.6, while a 3D junction with similar heterogeneity gives full pinning with P = 1 as expected. We further show that in 2D M|S contacts the turn-on voltage and pinning strength are affected by a physical parameter l/λD, the ratio between the interface width l, and the thermal de Broglie wavelength λD. Pinning is absent for ideal line-contacts (l/λD = 0), but increases for realistic l/λD values.

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.

Dielectric continuum methods for quantum chemistry

AbstractThis review describes the theory and implementation of implicit solvation models based on continuum electrostatics. Within quantum chemistry this formalism is sometimes synonymous with the polarizable continuum model, a particular boundary‐element approach to the problem defined by the Poisson or Poisson–Boltzmann equation, but that moniker belies the diversity of available methods. This work reviews the current state‐of‐the art, with emphasis on theory and methods rather than applications. The basics of continuum electrostatics are described, including the nonequilibrium polarization response upon excitation or ionization of the solute. Nonelectrostatic interactions, which must be included in the model in order to obtain accurate solvation energies, are also described. Numerical techniques for implementing the equations are discussed, including linear‐scaling algorithms that can be used in classical or mixed quantum/classical biomolecular electrostatics calculations. Anisotropic models that can describe interfacial solvation are briefly described.This article is categorized under:Electronic Structure Theory > Ab Initio Electronic Structure MethodsMolecular and Statistical Mechanics > Free Energy Methods

Numerical Modeling of Solutions of the Applied Problems for the Poisson’s Equation

Many problems of continuum physics, applied physics, electrostatics, heat engineering, and structural mechanics are described by boundary value problems for the Poisson equation. The article is devoted to the construction of a numerical algorithm for solving plane boundary value problems for the Poisson equation. The algorithm is based on the transition to a polyharmonic equation, which is solved by the method of linear boundary elements. We consider two boundary value problems with different boundary conditions: Dirichlet boundary conditions and Neumann boundary conditions. To construct a numerical solution using the linear boundary element method, the boundary of the area is replaced by an inscribed polygon. The boundary conditions are satisfied at the middle (control) points of the elements. A polyharmonic equation is reduced to a system of linear integral equations, which can be reduced to a system of linear algebraic equations. Test numerical examples confirming the effectiveness of the proposed algorithm are given.

Mreža protein-protein interakcija i protein-ligand doking - trenutno stanje i perspektive

Traditional research means, such as in vitro and in vivo models, have consistently been used by scientists to test hypotheses in biochemistry. Computational (in silico) methods have been increasingly devised and applied to testing and hypothesis development in biochemistry over the last decade. The aim of in silico methods is to analyze the quantitative aspects of scientific (big) data, whether these are stored in databases for large data or generated with the use of sophisticated modeling and simulation tools; to gain a fundamental understanding of numerous biochemical processes related, in particular, to large biological macromolecules by applying computational means to big biological data sets, and by computing biological system behavior. Computational methods used in biochemistry studies include proteomics-based bioinformatics, genome-wide mapping of protein-DNA interaction, as well as high-throughput mapping of the protein-protein interaction networks. Some of the vastly used molecular modeling and simulation techniques are Monte Carlo and Langevin (stochastic, Brownian) dynamics, statistical thermodynamics, molecular dynamics, continuum electrostatics, protein-ligand docking, protein-ligand affinity calculations, protein modeling techniques, and the protein folding process and enzyme action computer simulation. This paper presents a short review of two important methods used in the studies of biochemistry - protein-ligand docking and the prediction of protein-protein interaction networks.

Poor Person's pH Simulation of Membrane Proteins.

pH conditions are central to the functioning of all biomolecules. However, implications of pH changes are nontrivial on a molecular scale. Though a rigorous microscopic definition of pH exists, its implementation in classical molecular dynamics (MD) simulations is cumbersome, and more so in large integral membrane systems. In this chapter, an integrative pipeline is described that combines Multi-Conformation Continuum Electrostatics (MCCE) computations with MD simulations to capture the effect of transient protonation states on the coupled conformational changes in transmembrane proteins. The core methodologies are explained, and all the software required to set up this pipeline are outlined with their key parameters. All associated analyses of structure and function are provided using two case studies, namely those of bioenergetic complexes: NADH dehydrogenase (complex I) and Vo domain of V-type ATPase. The hybrid MCCE-MD pipeline has allowed the discovery of hydrogen bond networks, ligand binding pathways, and disease-causing mutations.

Physics-Based Computational Protein Design: An Update.

We describe methods for physics-based protein design and some recent applications from our work. We present the physical interpretation of a MC simulation in sequence space and show that sequences and conformations form a well-defined statistical ensemble, explored with Monte Carlo and Boltzmann sampling. The folded state energy combines molecular mechanics for solutes with continuum electrostatics for solvent. We usually assume one or a few fixed protein backbone structures and discrete side chain rotamers. Methods based on molecular dynamics, which introduce additional backbone and side chain flexibility, are under development. The redesign of a PDZ domain and an aminoacyl-tRNA synthetase enzyme were successful. We describe a versatile, adaptive, Wang-Landau MC method that can be used to design for substrate affinity, catalytic rate, catalytic efficiency, or the specificity of these properties. The methods are transferable to all biomolecules, can be systematically improved, and give physical insights.

Identification of a Na+-Binding Site near the Oxygen-Evolving Complex of Spinach Photosystem II.

The oxygen-evolving complex (OEC) of photosystem II (PSII) is an oxomanganese cluster composed of four redox-active Mn ions and one redox-inactive Ca2+ ion, with two nearby bound Cl- ions. Sodium is a common counterion of both chloride and hydroxide anions, and a sodium-specific binding site has not been identified near the OEC. Here, we find that the oxygen-evolution activity of spinach PSII increases with Na+ concentration, particularly at high pH. A Na+-specific binding site next to the OEC, becomes available after deprotonation of the D1-H337 amino acid residue, is suggested by the analysis of two recently published PSII cryo-electron microscopy maps in combination with quantum mechanical calculations and multiconformation continuum electrostatics simulations.

Characterizing the water wire in the Gramicidin channel found by Monte Carlo sampling using continuum electrostatics and in molecular dynamics trajectories with conventional or polarizable force fields

Water molecules play a key role in all biochemical processes. They help define the shape of proteins, and they are reactant or product in many reactions and are released as ligands are bound. They facilitate the transfer of protons through transmembrane proton channel, pump and transporter proteins. Continuum electrostatics (CE) force fields used by program Multiconformation CE (MCCE) capture electrostatic interactions in biomolecules with an implicit solvent, which captures the averaged solvent water equilibrium properties. Hybrid CE methods can use explicit water molecules within the protein surrounded by implicit solvent. These hybrid methods permit the study of explicit hydrogen bond networks within the protein and allow analysis of processes such as proton transfer reactions. Yet hybrid CE methods have not been rigorously tested. Here, we present an explicit treatment of water molecules in the Gramicidin A (gA) channel using MCCE and compare the resulting distributions of water molecules and key hydration features against those obtained with explicit solvent Molecular Dynamics (MD) simulations with the nonpolarizable CHARMM36 and polarizable Drude force fields. CHARMM36 leads to an aligned water wire in the channel characterized by a large absolute net water dipole moment; the MCCE and Drude analysis lead to a small net dipole moment as the water molecules change orientation within the channel. The correct orientation is not as yet known, so these calculations identify an open question.

Characterizing the Water Wire in the Gramicidin Channel Found by Monte Carlo Sampling Using Continuum Electrostatics and in Molecular Dynamics Trajectories with Conventional or Polarizable Force Fields

Water molecules play a key role in all biochemical processes. They help define the shape of proteins, and they are reactant or product in many reactions and are released as ligands are bound. They facilitate the transfer of protons through transmembrane proton channel, pump and transporter proteins. Continuum electrostatics (CE) force fields used by program Multiconformation CE (MCCE) capture electrostatic interactions in biomolecules with an implicit solvent, which captures the averaged solvent water equilibrium properties. Hybrid CE methods can use explicit water molecules within the protein surrounded by implicit solvent. These hybrid methods permit the study of explicit hydrogen bond networks within the protein and allow analysis of processes such as proton transfer reactions. Yet hybrid CE methods have not been rigorously tested. Here, we present an explicit treatment of water molecules in the Gramicidin A (gA) channel using MCCE and compare the resulting distributions of water molecules and key hydration features against those obtained with explicit solvent Molecular Dynamics (MD) simulations with the nonpolarizable CHARMM36 and polarizable Drude force fields. CHARMM36 leads to an aligned water wire in the channel characterized by a large absolute net water dipole moment; the MCCE and Drude analysis lead to a small net dipole moment as the water molecules change orientation within the channel. The correct orientation is not as yet known, so these calculations identify an open question.

Identifying the proton loading site cluster in the ba3 cytochrome c oxidase that loads and traps protons

Cytochrome c Oxidase (CcO) is the terminal electron acceptor in aerobic respiratory chain, reducing O2 to water. The released free energy is stored by pumping protons through the protein, maintaining the transmembrane electrochemical gradient. Protons are held transiently in a proton loading site (PLS) that binds and releases protons driven by the electron transfer reaction cycle. Multi-Conformation Continuum Electrostatics (MCCE) was applied to crystal structures and Molecular Dynamics snapshots of the B-type Thermus thermophilus CcO. Six residues are identified as the PLS, binding and releasing protons as the charges on heme b and the binuclear center are changed: the heme a3 propionic acids, Asp287, Asp372, His376 and Glu126B. The unloaded state has one proton and the loaded state two protons on these six residues. Different input structures, modifying the PLS conformation, show different proton distributions and result in different proton pumping behaviors. One loaded and one unloaded protonation states have the loaded/unloaded states close in energy so the PLS binds and releases a proton through the reaction cycle. The alternative proton distributions have state energies too far apart to be shifted by the electron transfers so are locked in loaded or unloaded states. Here the protein can use active states to load and unload protons, but has nearby trapped states, which stabilize PLS protonation state, providing new ideas about the CcO proton pumping mechanism. The distance between the PLS residues Asp287 and His376 correlates with the energy difference between loaded and unloaded states.

Interfacial solvation can explain attraction between like-charged objects in aqueous solution.

Over the past few decades, the experimental literature has consistently reported observations of attraction between like-charged colloidal particles and macromolecules in aqueous solution. Examples include nucleic acids and colloidal particles in the bulk solution and under confinement, and biological liquid-liquid phase separation. This observation is at odds with the intuitive expectation of an interparticle repulsion that decays monotonically with distance. Although attraction between like-charged particles can be rationalized theoretically in the strong-coupling regime, e.g., in the presence of multivalent counterions, recurring accounts of long-range attraction in aqueous solution containing monovalent ions at low ionic strength have posed an open conundrum. Here, we show that the behavior of molecular water at an interface-traditionally disregarded in the continuum electrostatics picture-provides a mechanism to explain the attraction between like-charged objects in a broad spectrum of experiments. This basic principle will have important ramifications in the ongoing quest to better understand intermolecular interactions in solution.

Capturing the Effects of Explicit Waters in Implicit Electrostatics Modeling: Qualitative Justification of Gaussian-Based Dielectric Models in DelPhi.

Our group has implemented a smooth Gaussian-based dielectric function in DelPhi (J. Chem. Theory Comput. 2013, 9 (4), 2126-2136) which models the solute as an object with inhomogeneous dielectric permittivity and provides a smooth transition of dielectric permittivity from surface-bound water to bulk solvent. Although it is well-understood that the protein hydrophobic core is less polarizable than the hydrophilic protein surface, less attention is paid to the polarizability of water molecules inside the solute and on its surface. Here, we apply explicit water simulations to study the behavior of water molecules buried inside a protein and on the surface of that protein and contrast it with the behavior of the bulk water. We selected a protein that is experimentally shown to have five cavities, most of which are occupied by water molecules. We demonstrate through molecular dynamics (MD) simulations that the behavior of water in the cavity is drastically different from that in the bulk. These observations were made by comparing the mean residence times, dipole orientation relaxation times, and average dipole moment fluctuations. We also show that the bulk region has a nonuniform distribution of these tempo-spatial properties. From the perspective of continuum electrostatics, we argue that the dielectric "constant" in water-filled cavities of proteins and the space close to the molecular surface should differ from that assigned to the bulk water. This provides support for the Gaussian-based smooth dielectric model for solving electrostatics in the Poisson-Boltzmann equation framework. Furthermore, we demonstrate that using a well-parametrized Gaussian-based model with a single energy-minimized configuration of a protein can also reproduce its ensemble-averaged polar solvation energy. Thus, we argue that the Gaussian-based smooth dielectric model not only captures accurate physics but also provides an efficient way of computing ensemble-averaged quantities.

Deciphering the selectivity of inhibitor MKC9989 towards residue K907 in IRE1α; a multiscale in silico approach.

The selectivity of the ligand MKC9989, as inhibitor of the Inositol-Requiring Enzyme 1α (IRE1α) transmembrane kinase/ribonuclease protein, towards the residue K907 in the context of Schiff base formation, has been investigated by employing an array of in silico techniques including Multi-Conformation Continuum Electrostatics (MCCE) simulations, Quantum Mechanics/Molecular Mechanics (QM/MM) calculations, covalent docking, and Molecular Dynamics (MD) simulations. According to the MCCE results, K907 displays the lowest pKa value among all 23 lysine residues in IRE1α. The MMCE simulations also indicate a critical interaction between K907 and D885 within the hydrophobic pocket which increases significantly at low protein dielectric constants. The QM/MM calculations reveal a spontaneous proton transfer from K907 to D885, consistent with the low pKa value of K907. A Potential Energy Surface (PES) scan confirms the lack of energy barrier and transition state associated with this proton transfer reaction. Covalent docking and MD simulations verify that the protein pocket containing K907 can effectively stabilize the inhibitor by strong π–π and hydrogen bonding interactions. In addition, Radial Distribution Function (RDF) analysis shows that the imine group formed in the chemical reaction between MKC9989 and K907 is inaccessible to water molecules and thus the probability of imine hydrolysis is almost zero. The results of the current study explain the high selectivity of the MKC9989 inhibitor towards the K907 residue of IRE1α.

Computational nanoplasmonics in the quasistatic limit for biosensing applications.

The phenomenon of localized surface plasmon resonance (LSPR) provides high sensitivity in detecting biomolecules through shifts in resonance frequency when a target is present. Computational studies in this field have used the full Maxwell equations with simplified models of a sensor-analyte system, or they neglected the analyte altogether. In the long-wavelength limit, one can simplify the theory via an electrostatics approximation while adding geometrical detail in the sensor and analytes (at moderate computational cost). This work uses the latter approach, expanding the open-source PyGBe code to compute the extinction cross section of metallic nanoparticles in the presence of any target for sensing. The target molecule is represented by a surface mesh, based on its crystal structure. PyGBe is research software for continuum electrostatics, written in python with computationally expensive parts accelerated on GPU hardware, via PyCUDA. It is also accelerated algorithmically via a treecode that offers O(NlogN) computational complexity. These features allow PyGBe to handle problems with half a million boundary elements or more. In this work, we demonstrate the suitability of PyGBe, extended to compute LSPR response in the electrostatic limit, for biosensing applications. Using a model problem consisting of an isolated silver nanosphere in an electric field, our results show grid convergence as 1/N, and accurate computation of the extinction cross section as a function of wavelength (compared with an analytical solution). For a model of a sensor-analyte system, consisting of a spherical silver nanoparticle and a set of bovine serum albumin (BSA) proteins, our results again obtain grid convergence as 1/N (with respect to the Richardson extrapolated value). Computing the LSPR response as a function of wavelength in the presence of BSA proteins captures a redshift of 0.5 nm in the resonance frequency due to the presence of the analytes at 1-nm distance. The final result is a sensitivity study of the biosensor model, obtaining the shift in resonance frequency for various distances between the proteins and the nanoparticle. All results in this paper are fully reproducible, and we have deposited in archival data repositories all the materials needed to run the computations again and recreate the figures. PyGBe is open source under a permissive license and openly developed. Documentation is available at http://pygbe.github.io/pygbe/docs/.

Visualizing biomolecular electrostatics in virtual reality with UnityMol‐APBS

Virtual reality is a powerful tool with the ability to immerse a user within a completely external environment. This immersion is particularly useful when visualizing and analyzing interactions between small organic molecules, molecular inorganic complexes, and biomolecular systems such as redox proteins and enzymes. A common tool used in the biomedical community to analyze such interactions is the Adaptive Poisson-Boltzmann Solver (APBS) software, which was developed to solve the equations of continuum electrostatics for large biomolecular assemblages. Numerous applications exist for using APBS in the biomedical community including analysis of protein ligand interactions and APBS has enjoyed widespread adoption throughout the biomedical community. Currently, typical use of the full APBS toolset is completed via the command line followed by visualization using a variety of two-dimensional external molecular visualization software. This process has inherent limitations: visualization of three-dimensional objects using a two-dimensional interface masks important information within the depth component. Herein, we have developed a single application, UnityMol-APBS, that provides a dual experience where users can utilize the full range of the APBS toolset, without the use of a command line interface, by use of a simple graphical user interface (GUI) for either a standard desktop or immersive virtual reality experience.

Computationally Modeling Electrostatic Binding Energetics in a Crowded, Dynamic Environment: Physical Insights from a Peptide-DNA System.

The cell is a crowded place, and it may be crucial at times to account for the local environment when studying determinants of molecular recognition. In this work, we use continuum electrostatics calculations on snapshots extracted from molecular dynamics simulations to understand how various aspects of a crowded environment affect electrostatic binding energies between the antimicrobial peptide buforin II and DNA. By comparing multiple models for representing crowding, sequentially introducing layers of model complexity for maximum control, we explore how electrostatic binding energetics depend on crowder physical properties, the sampling of the binding partners and crowder molecules, and the treatment of bulk solvent. We show that physical characteristics can combine to create an interplay of competing effects in this highly charged system. For example, increased ionic strength screening due to crowding partially cancels out the reduced solvent screening due to water depletion. We also quantify the effect of crowders' charge distributions on binding energetics. While we focus on electrostatic effects of crowding on binding, we begin to consider nonpolar components as well, and we implement a thermodynamic cycle accounting for both bound and unbound states to show the necessity of adequate crowder sampling in future studies. The insights developed here provide a rich starting point for experiments to further explore these competing effects and, ultimately, to rationally modulate molecular recognition in the complex cellular environment.

Computational chemistry approaches in studies on industrial enzymes

We review major computational chemistry techniques applied in industrial enzyme studies, especially approaches intended for guiding enzyme engineering. These include molecular mechanics force field and molecular dynamics simulation, quantum mechanical and combined quantum mechanical/molecular mechanical approaches, electrostatic continuum models, molecular docking, etc. These approaches are essentially introduced from the following two angles for viewing: one is about the methods themselves, including the basic concepts, the primary computational results, and potential advantages and limitations; the other is about obtaining valuable information from the respective calculations to guide the design of mutants and mutant libraries.

Photosystem II oxygen-evolving complex photoassembly displays an inverse H/D solvent isotope effect under chloride-limiting conditions

Photosystem II (PSII) performs the solar-driven oxidation of water used to fuel oxygenic photosynthesis. The active site of water oxidation is the oxygen-evolving complex (OEC), a Mn4CaO5 cluster. PSII requires degradation of key subunits and reassembly of the OEC as frequently as every 20 to 40 min. The metals for the OEC are assembled within the PSII protein environment via a series of binding events and photochemically induced oxidation events, but the full mechanism is unknown. A role of proton release in this mechanism is suggested here by the observation that the yield of in vitro OEC photoassembly is higher in deuterated water, D2O, compared with H2O when chloride is limiting. In kinetic studies, OEC photoassembly shows a significant lag phase in H2O at limiting chloride concentrations with an apparent H/D solvent isotope effect of 0.14 ± 0.05. The growth phase of OEC photoassembly shows an H/D solvent isotope effect of 1.5 ± 0.2. We analyzed the protonation states of the OEC protein environment using classical Multiconformer Continuum Electrostatics. Combining experiments and simulations leads to a model in which protons are lost from amino acid that will serve as OEC ligands as metals are bound. Chloride and D2O increase the proton affinities of key amino acid residues. These residues tune the binding affinity of Mn2+/3+ and facilitate the deprotonation of water to form a proposed μ-hydroxo bridged Mn2+Mn3+ intermediate.

A Mechanism of Modulating the Direction of Flagellar Rotation in Bacteria by Fumarate and Fumarate Reductase

Fumarate, an electron acceptor in anaerobic respiration of Escherichia coli, has an additional function of assisting the flagellar motor to shift from counterclockwise to clockwise rotation, with a consequent modulation of the bacterial swimming behavior. Fumarate transmits its effect to the motor via the fumarate reductase complex (FrdABCD), shown to bind to FliG—one of the motor’s switch proteins. How binding of the FrdABCD respiratory enzyme to FliG enhances clockwise rotation and how fumarate is involved in this activity have remained puzzling. Here we show that the FrdA subunit in the presence of fumarate is sufficient for binding to FliG and for clockwise enhancement. We further demonstrate by in vitro binding assays and super-resolution microscopy in vivo that the mechanism by which fumarate-occupied FrdA enhances clockwise rotation involves its preferential binding to the clockwise state of FliG (FliGcw). Continuum electrostatics combined with docking analysis and conformational sampling endorsed the experimental conclusions and suggested that the FrdA–FliGcw interaction is driven by the positive electrostatic potential generated by FrdA and the negatively charged areas of FliG. They further demonstrated that fumarate changes FrdA’s conformation to one that can bind to FliGcw. These findings also show that the reason for the failure of the succinate dehydrogenase flavoprotein SdhA (an almost-identical analog of FrdA shown to bind to FliG equally well) to enhance clockwise rotation is that it has no binding preference for FliGcw. We suggest that this mechanism is physiologically important as it can modulate the magnitude of ΔG0 between the clockwise and counterclockwise states of the motor to tune the motor to the growth conditions of the bacteria.