Malcolm E. Davis

A Comparison of Particle-Particle, Particle-Mesh and Ewald Methods for Calculating Electrostatic Interactions in Periodic Molecular Systems

Abstract We compare the Particle-Particle Particle-Mesh (PPPM) and Ewald methods for calculating electrostatic interactions in periodic molecular systems. A brief comparison of the theories shows that the methods are very similar differing mainly in the technique which is used to perform the “k-space” or mesh calculation. Because the PPPM utilizes the highly efficient numerical Fast Fourier Transform (FFT) method it requires significantly less computational effort than the Ewald method and scales almost linearly with system size.

Finite difference Poisson-Boltzmann electrostatic calculations: Increased accuracy achieved by harmonic dielectric smoothing and charge antialiasing

A common problem in the calculation of electrostatic potentials with the Poisson‐Boltzmann equation using finite difference methods is the effect of molecular position relative to the grid. Previously a uniform charging method was shown to reduce the grid dependence substantially over the point charge model used in commercially available codes. In this article we demonstrate that smoothing the charge and dielectric values on the grid can improve the grid independence, as measured by the spread of calculated values, by another order of magnitude. Calculations of Born ion solvation energies, small molecule solvation energies, the electrostatic field of superoxide dismutase, and protein‐protein binding energies are used to demonstrate that this method yields the same results as the point charge model while reducing the positional errors by several orders of magnitude.

Poisson-Boltzmann analysis of the lambda repressor-operator interaction.

A theoretical study of the ion atmosphere contribution to the binding free energy of the lambda repressor-operator complex is presented. The finite-difference form of the Poisson-Boltzmann equation was solved to calculate the electrostatic interaction energy of the amino-terminal domain of the lambda repressor with a 9 or 45 base pair oligonucleotide. Calculations were performed at various distances between repressor and operator as well as at different salt concentrations to determine ion atmosphere contributions to the total electrostatic interaction. Details in the distribution of charges on DNA and protein atoms had a strong influence on the calculated total interaction energies. In contrast, the calculated salt contributions are relatively insensitive to changes in the details of the charge distribution. The results indicate that the ion atmosphere contribution favors association at all protein-DNA distances studied. The theoretical number of ions released upon repressor-operator binding appears to be in reasonable agreement with experimental data.

Diffusion-controlled enzymatic reactions

Publisher Summary The rate of diffusional encounter among reactant molecules in solution sets the ultimate limit on the speed of enzymatic and other reactions. If the reactant molecules are such that subsequent events develop very rapidly when the reactants come into contact, the net rate of the reaction will be equal to the rate of diffusional encounter. The reaction is then said to be diffusion-controlled. This chapter describes the way computer simulations may be used to highlight the nature of diffusion-controlled reactions. Such simulations can, in principle, aid in the detailed interpretation of experimental results or in the design of molecules with prescribed kinetic properties. The chapter also describes methods for calculating electrostatic interactions among solute molecules in solution. The long range of such interactions makes them particularly important in the consideration of diffusional encounters. Diffusional encounter involves the interaction of molecules at large separations. Electrostatic interactions are, therefore, particularly important because of the long-range nature of the Coulombic potential. At small separations, the effects of other interactions can often be represented by suitable boundary conditions. As a result, most attention has been paid to the calculation of electrostatic forces. The methods described in the chapter can also be used to study the role of electrostatics in determining equilibrium properties, such as the stability of molecular folding and association.

Implementation of a six-dimensional search using the AMoRe translation function for difficult molecular-replacement problems

A six-dimensional molecular-replacement procedure has been implemented using Perl scripts and the CCP4 version of the translation function of the molecular-replacement program AMoRe. These tools have allowed the structure determination of CTLA-4 monomer from NMR-derived coordinates, and of a potential complex of MurD with a substrate. In both cases other molecular-replacement programs, X-PLOR and AMoRe, when used in a conventional manner, either failed or did not yield an obvious solution.

The inducible multipole solvation model: A new model for solvation effects on solute electrostatics

A new approach to modeling the electrostatics of molecules in solution is presented, which is comparable in accuracy to finite‐difference continuum models, yet is comparable in speed to polarizable in vacuo models. This is achieved by modeling the polarization charge resulting from solvation as inducible multipoles at the solute’s atom centers. Pilot studies of small molecule solvation energies, and of pKa shifts and molecular fields in proteins are presented. The results compare favorably with the finite‐difference results while requiring significantly shorter execution time. Bearing in mind that no parameter optimization was performed in this study and that nonelectrostatic energy contributions were not considered, the results also compare reasonably with experimental results where those are available.

DRUG DISCOVERY INTERFACE: Functional Group Dependence of Solute Partitioning to Various Locations within a DOPC Bilayer: A Comparison of Molecular Dynamics Simulations with Experiment

Atomic-level molecular dynamics simulations of 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) bilayers containing small, amphiphilic, drug-like molecules were carried out to examine the influence of polar functionality on membrane partitioning and transport. Three related molecules (tyramine, phenethylamine, and 4-ethylphenol) were chosen to allow a detailed study of the isolated effects of the amine and hydroxyl functionalities on the preferred solute location, free energies of transfer, and the effect of combining both functional groups in a same molecule. Transfer free energy profiles (from water) generated from molecular dynamics (MD) simulations as a function of bilayer depth compared favorably to comparable experimental results. The simulations allowed the determination of the location of the barrier domain for permeability where the transfer free energy is highest and the preferred binding region at which the free energy is a minimum for each of the three solutes. Comparisons of the free energy profiles reveal that the hydrocarbon chain interior is the region most selective to chemical structure of different solutes because the free energies of transfer in that region vary to a significantly greater extent than in other regions of the bilayer. The contributions of the hydroxyl and amino groups to the free energies of solute transfer from water to the interfacial region were close to zero in both the MD simulations and experimental measurements. This suggests that the free energy decrease observed for solute transfer into the head group region occurs with minimal loss in solvation by hydrogen bonding to polar functional groups on the solute and is largely driven by hydrophobicity. Overall, the joint experimental and simulation studies suggest that the assumption of additivity of free energy contributions from multiple polar functional groups on the same molecule may hold for predictions of passive bilayer permeability coefficients providing that the groups are well isolated. However, this assumption does not hold for predictions of relative liposome-binding affinities.

An atomic and molecular view of the depth dependence of the free energies of solute transfer from water into lipid bilayers.

Molecular interactions and orientations responsible for differences in 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) bilayer partitioning of three structurally related drug-like molecules (4-ethylphenol, phenethylamine, and tyramine) were investigated. This work is based on previously reported molecular dynamics (MD) simulations that determined their transverse free energy profiles across the bilayer. Previously, the location where the transfer free energy of the three solutes is highest, which defines the barrier domain for permeability, was found to be the bilayer center, while the interfacial region was found to be the preferred binding region. Contributions of the amino (NH2) and hydroxyl (OH) functional groups to the transfer free energies from water to the interfacial region were found to be very small both experimentally and by MD simulation, suggesting that the interfacial binding of these solutes is hydrophobically driven and occurs with minimal loss of hydrogen-bonding interactions of the polar functional groups which can occur with either water or phospholipid head groups. Therefore, interfacial binding is relatively insensitive to the number or type of polar functional groups on the solute. In contrast, the relative solute free energy in the barrier domain is highly sensitive to the number of polar functional groups on the molecule. The number and types of hydrogen bonds formed between the three solutes and polar phospholipid atoms or with water molecules were determined as a function of solute position in the bilayer. Minima were observed in the number of hydrogen bonds formed by each solute at the center of the bilayer, coinciding with a decrease in the number of water molecules in DOPC as a function of distance away from the interfacial region. In all regions, hydrogen bonds with water molecules account for the majority of hydrogen-bonding interactions observed for each solute. Significant orientational preferences for the solutes are evident in certain regions of the bilayer (e.g., within the ordered chain region and near the interfacial region 20-25 Å from the bilayer center). The preferred orientations are those that preserve favorable molecular interactions for each solute, which vary with the solute structure.

Conserved core substructures in the overlay of protein-ligand complexes

The method of conserved core substructure matching (CSM) for the overlay of protein-ligand complexes is described. The method relies upon distance geometry to align structurally similar substructures without regard to sequence similarity onto substructures from a reference protein empirically selected to include key determinants of binding site location and geometry. The error in ligand position is reduced in reoriented ensembles generated with CSM when compared to other overlay methods. Since CSM can only succeed when the selected core substructure is geometrically conserved, misalignments only rarely occur. The method may be applied to reliably overlay large numbers of protein-ligand complexes in a way that optimizes ligand position at a specific binding site or subsite or to align structures from large and diverse protein families where the conserved binding site is localized to only a small portion of either protein. Core substructures may be complex and must be chosen with care. We have created a database of empirically selected core substructures to demonstrate the utility of CSM alignment of ligand binding sites in important drug targets. A Web-based interface can be used to apply CSM to align large collections of protein-ligand complexes for use in drug design using these substructures or to evaluate the use of alternative core substructures that may then be shared with the larger user community. Examples show the benefit of CSM in the practice of structure-based drug design.

Electric-field distribution inside the bacterial photosynthetic reaction center of Rhodopseudomonas viridis

Abstract The electric-field distribution inside the bacterial photosynthetic reaction center of Rhodopseudomonas viridis was obtained by solving the linearized Poisson—Boltzmann equation with a grid size of (2 A)3. The order of potentials induced by the protein medium for the four heme groups in the cytochrome part is Hm1: 310 mV; Hm2: 370mV; Hm4: 380 mV; Hm3: 130 mV. The potential profile is similar along both the L and the M branches, a result of the C2 symmetry related environment. In both the L and the M subunits, bacteriochlorophyll has the lowest potential. It is shown that the Poisson—Boltzmann method can also be used to analyze the variation of local fields inside proteins in response to applied fields. For the reaction center, the dielectric response to an applied field is anistropic. There are significant induced x and y components of the internal field for an applied field along the z direction (the C2 axis). Thus the effective dielectric-constant tensor of the protein medium has non-zero off-diagonal elements. Analysis of how the applied field and ionic strength influence the internal field indicates that there is relatively small screening due to free solvent in the complex. The difference between the potentials at various cofactors is due to the sum of small contributions from the protein environment, rather than a few charged residues.