Intraprotein electrostatics derived from first principles: divide-and-conquer approaches for QM/MM calculations

Two divide-and-conquer (DAQ) approaches for building multipole-based molecular electrostatic potentials of proteins are presented and evaluated for use in QM/MM calculations. One approach is a further development of the neutralization method of Bellido and Rullmann (J Comput Chem 1989, 10, 479-487) while the other is based on removing part of the electron density before performing the multipole expansion. Both methods create systems with integer charges without using charge renormalization. To determine their performance in terms of location of cuts and distance to QM region, the new DAQ approaches are tested in calculations of the proton affinity of N(zeta) of Lys55 in the inhibitor turkey ovomucoid third domain. Finally, the two methods are used to build a variety of MM regions, applied to calculations of the pK(a) of Lys55, and compared to other computational methodologies in which force field charges are employed.     

Role of short-range electrostatics in torsional potentials

A force field needs to decide if it should contain a torsional potential or not. A helpful guide to this decision should come from a quantum mechanical energy partitioning. Here we analyze the energy profiles of eight simple molecules (ethane, hydrogen peroxide, hydrazine, methanol, acetaldehyde, formamide, acetamide and N-methylacetamide) subject to rotation around a torsion angle. Coulomb interaction energies between all atom pairs in a molecule are monitored during the rotation. Atoms are defined as finite electron density fragments by quantum chemical topology, a method that enables well-defined short-range interactions (1-2, 1-3 and 1-4). Energy profiles of Coulomb interaction energies mostly counteract the ab initio energy profiles. This and future work strives to settle ambiguities in current force field design.     

Boundary conditions in local electrostatics algorithms

We study the simulation of charged systems in the presence of general boundary conditions in a local Monte Carlo algorithm based on a constrained electric field. We first show how to implement constant-potential, Dirichlet boundary conditions by introducing extra Monte Carlo moves to the algorithm. Second, we show the interest of the algorithm for studying systems which require anisotropic electrostatic boundary conditions for simulating planar geometries such as membranes.     

Limiting assumptions in molecular modeling: electrostatics

Molecular mechanics attempts to represent intermolecular interactions in terms of classical physics. Initial efforts assumed a point charge located at the atom center and coulombic interactions. It is been recognized over multiple decades that simply representing electrostatics with a charge on each atom failed to reproduce the electrostatic potential surrounding a molecule as estimated by quantum mechanics. Molecular orbitals are not spherically symmetrical, an implicit assumption of monopole electrostatics. This perspective reviews recent evidence that requires use of multipole electrostatics and polarizability in molecular modeling.     

Comment on the validation of continuum electrostatics models

A validation based on solvation energies (vacuum to water transfer) is not sufficient to justify the use of approximated models of electrostatics to rank ligand/protein complexes. A full validation should be based on energies in solution, i.e., solvation plus vacuum Coulomb energies, because of the anticorrelation between solvation and vacuum energies. The energy in solution is the relevant quantity in simulations of biological macromolecules and complexes. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1533–1536, 1999     

Multipole electrostatics in hydration free energy calculations

Hydration free energy (HFE) is generally used for evaluating molecular solubility, which is an important property for pharmaceutical and chemical engineering processes. Accurately predicting HFE is also recognized as one fundamental capability of molecular mechanics force field. Here, we present a systematic investigation on HFE calculations with AMOEBA polarizable force field at various parameterization and simulation conditions. The HFEs of seven small organic molecules have been obtained alchemically using the Bennett Acceptance Ratio method. We have compared two approaches to derive the atomic multipoles from quantum mechanical calculations: one directly from the new distributed multipole analysis and the other involving fitting to the electrostatic potential around the molecules. Wave functions solved at the MP2 level with four basis sets (6-311G*, 6-311++G(2d,2p), cc-pVTZ, and aug-cc-pVTZ) are used to derive the atomic multipoles. HFEs from all four basis sets show a reasonable agreement with experimental data (root mean square error 0.63 kcal/mol for aug-cc-pVTZ). We conclude that aug-cc-pVTZ gives the best performance when used with AMOEBA, and 6-311++G(2d,2p) is comparable but more efficient for larger systems. The results suggest that the inclusion of diffuse basis functions is important for capturing intermolecular interactions. The effect of long-range correction to van der Waals interaction on the hydration free energies is about 0.1 kcal/mol when the cutoff is 12?, and increases linearly with the number of atoms in the solute/ligand. In addition, we also discussed the results from a hybrid approach that combines polarizable solute with fixed-charge water in the HFE calculation.     

Redox reactions in lipid bilayers and membrane bioenergetics

This paper begins with a consideration of electronic processes in ultrathin artificial bilayer lipid membranes (BLM) and biomembranes of chloroplasts and mitochondria. Methods and materials used in the study of BLMs are described next. The results of recent experiments, as obtained by voltammetric techniques, are summarized. The remaining sections of the paper deal with the application of basic electrochemistry to membrane research, in particular, the Eyring equation, the Butler-Volmer equation and the Tafel equation, and the origin of electronic processes in membranes in terms of electrostenolysis. The paper concludes with a discussion on the determination of standard potentials (U_o′) of redox protein components in the absence of the lipid bilayer, and finally a suggestion of the future experiments in relation to electron-transfer and bioredox reactions in bilayer lipid membranes.     

Effective multipoles and Yukawa electrostatics in dressed molecule theory

In this paper we derive the multipolar expansion of the screened Coulomb potential in electrolyte solutions with molecular solvent. The solute and solvent molecules can have arbitrary sizes, shapes, and internal charge distributions. We use the exact statistical mechanical definition of renormalized charge distributions coming from "dressed molecule theory" to determine the effective multipoles of a molecule immersed in an electrolyte. The effects of many-body correlations are fully included in our formally exact theory. We restrict ourselves to sufficiently dilute solutions so the screened Coulomb potential decays for large distances like a Yukawa function, exp(-kappa r)/r, where r is the distance and 1/kappa is the decay length (it is normally different from the Debye length). The resulting "Yukawa electrostatics" differ in many respects from ordinary, unscreened electrostatics. The "Yukawa charge" of a molecule (the lowest order moment in the multipolar expansion) is in general not equal to its Coulombic charge and it is not the integral of the renormalized charge distribution of the molecule. Moreover, as shown in this paper, the multipolar expansion of the Yukawa potential does not correspond, contrary to the case of the Coulomb potential, to its asymptotic expansion for large r. As a consequence, the charge term in the multipolar expansion is not the leading term in the asymptotic expansion. Instead, for large r values, multipoles of all orders contribute to the leading asymptotic term. Thus, the electrostatic potential from, for example, an electroneutral solvent molecule in an electrolyte solution has generally the same range as that from an ion. The proper asymptotic expansion for electrostatic interactions in electrolytes is derived. It is briefly shown how the multipole expansion formalism can also be applied in the Poisson-Boltzmann approximation for primitive model electrolytes.     

Solid-state energetics and electrostatics: Madelung constants and Madelung energies

The Madelung constants of ionic solids relate to their geometry and electrostatic interactions. Furthermore, because of issues in their evaluation, they are also of considerable mathematical interest. The corresponding Madelung (electrostatic, coulomb) energy is the principal contributor to the lattice energies of ionic systems, and these energies largely influence many of their physical properties. The Madelung constants are here defined and their properties considered. A difficulty with their application is that they may be defined relative to various lattice distances, and with various conventions for inclusion of the charges, leading to possible confusion in their use. Instead, the unambiguous Madelung energy, E(M), is to be preferred in chemistry. An extensive list of Madelung energies is presented. From this data set, it is observed that there is a strong linear correlation between the lattice energies of ionic solids, U(POT), and their Madelung energies: U(POT)/kJ mol(-1) = 0.8519E(M) + 293.9. This correlation establishes that the lattice energy, U(POT), for ionic solids is about 15% smaller than the attractive Madelung energy, the difference arising from the repulsions unaccounted for by the solely coulombic Madelung energy calculation. Correlations of U(POT) against E(M) for alkali metal hydrides and transition metal compounds, each having considerable covalency, show much reduced Madelung contributions to the lattice energy. These correlations permit ready estimation of lattice energies, and are the first to be based on actual data rather than a broad analysis. The independent volume-based thermodynamic (VBT) method, which relies on a separate correlation with the formula unit volume of the ionic material, complements these correlations.     

Multiple protonation equilibria in electrostatics of protein-protein binding

All proteins contain groups capable of exchanging protons with their environment. We present here an approach, based on a rigorous thermodynamic cycle and the partition functions for energy levels characterizing protonation states of the associating proteins and their complex, to compute the electrostatic pH-dependent contribution to the free energy of protein-protein binding. The computed electrostatic binding free energies include the pH of the solution as the variable of state, mutual "polarization" of associating proteins reflected as changes in the distribution of their protonation states upon binding and fluctuations between available protonation states. The only fixed property of both proteins is the conformation; the structure of the monomers is kept in the same conformation as they have in the complex structure. As a reference, we use the electrostatic binding free energies obtained from the traditional Poisson-Boltzmann model, computed for a single macromolecular conformation fixed in a given protonation state, appropriate for given solution conditions. The new approach was tested for 12 protein-protein complexes. It is shown that explicit inclusion of protonation degrees of freedom might lead to a substantially different estimation of the electrostatic contribution to the binding free energy than that based on the traditional Poisson-Boltzmann model. This has important implications for the balancing of different contributions to the energetics of protein-protein binding and other related problems, for example, the choice of protein models for Brownian dynamics simulations of their association. Our procedure can be generalized to include conformational degrees of freedom by combining it with molecular dynamics simulations at constant pH. Unfortunately, in practice, a prohibitive factor is an enormous requirement for computer time and power. However, there may be some hope for solving this problem by combining existing constant pH molecular dynamics algorithms with so-called accelerated molecular dynamics algorithms.     

Local Monte Carlo for electrostatics in anisotropic and nonperiodic geometries

We present an implementation of a local Monte Carlo algorithm for simulating charged particles in anisotropic and nonperiodic geometries. Specifically, we consider a quasi-two-dimensional periodic slab geometry with an either infinite or finite third dimension. For the infinite case, we show that the method generates accurate electrostatics equivalent to standard two-dimensional Ewald formulas. We then implement constant charge or constant potential (Dirichlet) boundary conditions, which frequently occur in experimental studies of charged complex fluids or polyelectrolytes. As a demonstration of the versatility of the approach, we compute ion density profiles in front of oppositely charged surfaces (the electric double layer) and find excellent agreement with theory in known analytic limits.     

Combined electrostatics and hydrogen bonding determine intermolecular interactions between polyphosphoinositides

Membrane lipids are active contributors to cell function as key mediators in signaling pathways controlling cell functions including inflammation, apoptosis, migration, and proliferation. Recent work on multimolecular lipid structures suggests a critical role for lipid organization in regulating the function of both lipids and proteins. Of particular interest in this context are the polyphosphoinositides (PPI's), especially phosphatidylinositol (4,5) bisphosphate (PIP 2). The cellular functions of PIP 2 are numerous but the organization of PIP 2 in the inner leaflet of the plasma membrane, as well as the factors controlling targeting of PIP 2 to specific proteins, remains poorly understood. To analyze the organization of PIP 2 in a simplified planar system, we used Langmuir monolayers to study the effects of subphase conditions on monolayers of purified naturally derived PIP 2 and other anionic or zwitterionic phospholipids. We report a significant molecular area expanding effect of subphase monovalent salts on PIP 2 at biologically relevant surface densities. This effect is shown to be specific to PIP 2 and independent of subphase pH. Chaotropic agents (e.g., salts, trehalose, urea, temperature) that disrupt water structure and the ability of water to mediate intermolecular hydrogen bonding also specifically expanded PIP 2 monolayers. These results suggest a combination of water-mediated hydrogen bonding and headgroup repulsion in determining the organization of PIP 2, and may contribute to an explanation for the unique functionality of PIP 2 compared to other anionic phospholipids.     

A sphere-based model for the electrostatics of globular proteins

We propose a model for the electrostatics of globular proteins in which the low dielectric region is replaced by concentric spheres of the appropriate size. The method uses analytical formulas for the dielectric sphere and allows an efficient and accurate treatment of bulk charges. For surface charges, we propose a numerical determination of the sphere radius based on the solvent exposure of the individual atoms. The present implementation of the sphere model yields a good approximation of finite-difference Poisson solvation and interaction energies for a test set of 12 proteins.