Rapid boundary element solvation electrostatics calculations in folding simulations: successful folding of a 23-residue peptide.

Solvation effects play a profound role in the energetics of protein folding. While a continuum dielectric model of solvation may provide a sufficiently accurate estimate of the solvation effects, until now this model was too computationally expensive and unstable for folding simulations. Here we proposed a fast yet accurate and robust implementation of the boundary element solution of the Poisson equation, the REBEL algorithm. Using our earlier double-energy scheme, we included for the first time the mathematically rigorous continuous REBEL solvation term in our Biased Probability Monte Carlo (BPMC) simulations of the peptide folding. The free energy of a 23-residue beta beta alpha-peptide was then globally optimized with and without the solvation electrostatics contribution. An ensemble of beta beta alpha conformations was found at and near the global minimum of the energy function with the REBEL electrostatic solvation term. Much poorer correspondence to the native solution structure was found in the "control" simulations with a traditional method to account for solvation via a distance-dependent dielectric constant. Each simulation took less than 40 h (21 h without electrostatic solvation calculation) on a single Alpha 677 MHz CPU and involved more than 40,000 solvation energy evaluations. This work demonstrates for the first time that such a simulation can be performed in a realistic time frame. The proposed procedure may eliminate the energy evaluation accuracy bottleneck in folding simulations.     

Solvation energy density occlusion approximation for evaluation of desolvation penalties in biomolecular interactions

In calculations involving many displacements of an interacting pair of biomolecules, such as brownian dynamics, the docking of a substrate/ligand to an enzyme/receptor, or the screening of a large number of ligands as prospective inhibitors for a particular receptor site, there is a need for rapid evaluation of the desolvation penalties of the interacting pair. Although continuum electrostatic treatments with distinct dielectric constants for solute and solvent provide an account of the electrostatics of solvation and desolvation, it is necessary to re-solve the Poisson equation, at considerable computational cost, for each displacement of the interacting pair. We present a new method that uses a formulation of continuum electrostatic solvation in terms of the solvation energy density and approximates desolvation in terms of the occlusion of this density. We call it the SEDO approximation. It avoids the need to re-solve the Poisson equation, as desolvation is now estimated by an integral over the occluded volume. Test calculations are presented for some simple model systems and for some real systems that have previously been studied using the Poisson equation approach: MHC class I protein-peptide complexes and a congeneric series of human immunodeficiency virus type 1 (HIV-1) protease--ligand complexes. For most of the systems considered, the trends and magnitudes of the desolvation component of interaction energies obtained using the SEDO approximation are in reasonable correlation with those obtained by re-solving the Poisson equation. In most cases, the error introduced by the SEDO approximation is much less than that of the often-used test-charge approximation for the charge-charge components of intermolecular interactions. Proteins 2001;43:12-27.     

Coupling between conformation and proton binding in proteins

Interest centers here on whether the use of a fixed charge distribution of a protein solute, or a treatment that considers proton-binding equilibria by solving the Poisson equation, is a better approach to discriminate native from non-native conformations of proteins. In this analysis of the charge distribution of 7 proteins, we estimate the solvation free energy contribution to the total free energy by exploring the 2(zeta) possible ionization states of the whole molecule, with zeta being the number of ionizable groups in the amino acid sequence, for every conformation in the ensembles of 7 proteins. As an additional consideration of the role of electrostatic interactions in determining the charge distribution of native folds, we carried out a comparison of alternative charge assignment models for the ionizable residues in a set of 21 native-like proteins. The results of this work indicate that (1) for 6 out of 7 proteins, estimation of solvent polarization based on the Generalized Born model with a fixed charge distribution provides the optimal trade-off between accuracy, with respect to the Poisson equation, and speed when compared to the accessible surface area model; for the seventh protein, consideration of all possible ionization states of the whole molecule appears to be crucial to discriminate the native from non-native conformations; (2) significant differences in the degree of ionization and hence the charge distribution for native folds are found between the different charge models examined; (3) the stability of the native state is determined by a delicate balance of all the energy components, and (4) conformational entropy, and hence the dynamics of folding, may play a crucial role for a successful ab initio protein folding prediction.     

Prediction of the aqueous solvation free energy of organic compounds by using autocorrelation of molecular electrostatic potential surface properties combined with response surface analysis

Several quantitative structure-property relationship (QSPR) approaches have been explored for the prediction of aqueous solubility or aqueous solvation free energies, DeltaG(sol), as crucial parameter affecting the pharmacokinetic profile and toxicity of chemical compounds. It is mostly accepted that aqueous solvation free energies can be expressed quantitatively in terms of properties of the molecular surface electrostatic potentials of the solutes. In the present study we have introduced autocorrelation molecular electrostatic potential (autoMEP) vectors in combination with nonlinear response surface analysis (RSA) as alternative 3D-QSPR strategy to evaluate the aqueous solvation free energy of organic compounds. A robust QSPR model (r(cv)=0.93) has been obtained by using a collection of 248 organic chemicals. An external test set based on 23 molecules confirmed the good predictivity of the autoMEP/RSA model suggesting its further applicability in the in silico prediction of water solubility of large organic compound libraries.     

pH-dependent stability of sperm whale myoglobin in water-guanidine hydrochloride solutions

An experimental-theoretical approach for the elucidation of protein stability is proposed. The theoretical prediction of pH-dependent protein stability is based on the macroscopic electrostatic model for calculation of the pH-dependent electrostatic free energy of proteins. As a test of the method we have considered the pH-dependent stability of sperm whale metmyoglobin. Two theoretical methods for evaluation of the electrostatic free energy and p K values are applied: the finite-difference Poisson-Boltzmann method and the semiempirical approach based on the modified Tanford-Kirkwood theory. The theoretical results for electrostatic free energy of unfolding are compared with the experimental data for guanidine hydrochloride unfolding under equilibrium conditions over a wide pH range. Using the optical parameters of the Soret absorbance to monitor conformational equilibrium and Tanford's method to estimate the resulting data, it was found that the conformational free energy of unfolding of metmyoglobin is 16.3 kcal mol(-1) at neutral pH values. The total unfolding free energies were calculated on the basis of the theoretically predicted electrostatic unfolding free energies and the experimentally measured midpoints (pH(1/2)) of acidic and alkaline denaturation transitions. Experimental data for alkaline denaturation were used for the first time in theoretical analysis of the pH-dependent unfolding of myoglobin. The present results demonstrate that the simultaneous application of appropriate theoretical and experimental methods permits a more complete analysis of the pH-dependent and pH-independent properties and stability of globular proteins.     

Modeling loop reorganization free energies of acetylcholinesterase: a comparison of explicit and implicit solvent models

The treatment of hydration effects in protein dynamics simulations varies in model complexity and spans the range from the computationally intensive microscopic evaluation to simple dielectric screening of charge-charge interactions. This paper compares different solvent models applied to the problem of estimating the free-energy difference between two loop conformations in acetylcholinesterase. Molecular dynamics (MD) simulations were used to sample potential energy surfaces of the two basins with solvent treated by means of explicit and implicit methods. Implicit solvent methods studied include the generalized Born (GB) model, atomic solvation potential (ASP), and the distance-dependent dieletric constant. By using the linear response approximation (LRA), the explicit solvent calculations determined a free-energy difference that is in excellent agreement with the experimental estimate, while rescoring the protein conformations with GB or the Poisson equation showed inconsistent and inferior results. While the approach of rescoring conformations from explicit water simulations with implicit solvent models is popular among many applications, it perturbs the energy landscape by changing the solvent contribution to microstates without conformational relaxation, thus leading to non-optimal solvation free energies. Calculations applying MD with a GB solvent model produced results of comparable accuracy as observed with LRA, yet the electrostatic free-energy terms were significantly different due to optimization on a potential energy surface favored by an implicit solvent reaction field. The simpler methods of ASP and the distance-dependent scaling of the dielectric constant both produced considerable distortions in the protein internal free-energy terms and are consequently unreliable.     

A CFF 91-based Continuum Solvation Model: Solvation Free Energies of Small Organic Molecules and Conformations of the Alanine Dipeptide in Solution

Abstract

For molecular mechanics simulations of solvated molecules, it is important to use a consistent approach for calculating both the force field energy and the solvation free energy. A continuum solvation model based upon the atomic charges provided with the CFF91 force field is derived. The electrostatic component of the solvation free energy is described by the Poisson-Bolzmann equation while the nonpolar comonent of the solvation energy is assumed to be proportional to the solvent accessible surface area of the solute. Solute atomic radii used to describe the interface between the solute and solvent are fitted to reproduce the energies of small organic molecules. Data for 140 compounds are presented and compared to experiment and to the results from the well-characterized quantum mechanical solvation model AM1-SM2. In particular, accurate results are obtained for amino acid neutral analogues (mean unsigned error of 0.3 kcal/mol). The conformational energetics of the solvated alanine dipeptide is discussed.     

Energy transfer in photosynthesis

A theoretical model is presented to account for the physical mechanism of energy transfer from antenna molecules to the reaction centers in photosynthesis. The energy transfer is described by a generalized transport equation or “master equation”. The solution of this equation for the proposed model gives a relationship between the antennae interaction energy and the transfer rate. The results are shown to be in agreement with inter-antenna transfer rates calculated from experimental fluorescence lifetimes. Previous theories were based either on the Förster mechanism, which is valid for very small interaction energies, or an exciton model valid for very large interactions, but experimental results seemed to indicate that the actual situation was intermediate between these two. The Förster theory and the exciton model are limiting cases of the master equation.     

Effective energy function for proteins in lipid membranes

A simple extension of the EEF1 energy function to heterogeneous membrane-aqueous media is proposed. The extension consists of (a) development of solvation parameters for a nonpolar phase using experimental data for the transfer of amino acid side-chains from water to cyclohexane, (b) introduction of a heterogeneous membrane-aqueous system by making the reference solvation free energy of each atom dependent on the vertical coordinate, (c) a modification of the distance-dependent dielectric model to account for reduced screening of electrostatic interactions in the membrane, and (d) an adjustment of the EEF1 aqueous model in light of recent calculations of the potential of mean force between amino acid side-chains in water. The electrostatic model is adjusted to match experimental observations for polyalanine, polyleucine, and the glycophorin A dimer. The resulting energy function (IMM1) reproduces the preference of Trp and Tyr for the membrane interface, gives reasonable energies of insertion into or adsorption onto a membrane, and allows stable 1-ns MD simulations of the glycophorin A dimer. We find that the lowest-energy orientation of melittin in bilayers varies, depending on the thickness of the hydrocarbon layer.     

Optimized atomic radii for protein continuum electrostatics solvation forces

Recently, we presented a Green's function approach for the calculation of analytic continuum electrostatic solvation forces based on numerical solutions of the finite-difference Poisson-Botzmann (FDPB) equation [Im et al., Comp. Phys. Comm. 111 (1998) 59]. In this treatment the analytic forces were explicitly defined as the first derivative of the FDPB continuum electrostatic free energy with respect to the coordinates of the solute atoms. A smooth intermediate region for the solute-solvent dielectric boundary needed to be introduced to avoid abrupt discontinuous variations in the solvation free energy and forces as a function of the atomic positions. In the present paper we extend the set of optimized radii, which was previously parametrized from molecular dynamics free energy simulations of the 20 standard amino acids with explicit solvent molecules [Nina et al., J. Phys. Chem. 101 (1997) 5239], to yield accurate solvation free energy by taking the influence of the smoothed dielectric region into account.     

Free energy simulations: The meaning of the individual contributions from a component analysis

A theoretical analysis is made of the decomposition into contributions from individual interactions of the free energy calculated by thermodynamic integration. It is demonstrated that such a decomposition, often referred to as “component analysis,” is meaningful, even though it is a function of the integration path. Moreover, it is shown that the path dependence can be used to determine the relation of the contribution of a given interaction to the state of the system. To illustrate these conclusions, a simple transformation(Cl^? to Br^? in aqueous solution) is analyzed by use of the Reference Interaction Site Model-Hypernetted Chain Closure integral equation approach; it avoids the calculational difficulties of macromolecular simulation while retaining their conceptual complexity. The difference in the solvation free energy between chloride and bromide is calculated, and the contributions of the Lennard-Jones and electrostatic terms in the potential function are analyzed by the use of suitably chosen integration paths. The model is also used to examine the path dependence of individual contributions to the double free energy differences (ΔΔG or ΔΔA) that are often employed in free energy simulations of biological systems. The alchemical path, as contrasted with the experimental path, is shown to be appropriate for interpreting the effects of mutations on ligand binding and protein stability. The formulation is used to obtain a better understanding of the success of the Poisson-Boltzmann continuum approach for determining the solvation properties of polar and ionic systems. © 1994 Wiley-Liss, Inc.     

Prediction of the binding energy for small molecules,peptides and proteins

A fast and reliable evaluation of the binding energy from a single conformation of a molecular complex is an important practical task. Knowledge‐based scoring schemes may not be sufficiently general and transferable, while molecular dynamics or Monte Carlo calculations with explicit solvent are too computationally expensive for many applications. Recently, several empirical schemes using finite difference Poisson–Boltzmann electrostatics to predict energies for particular types of complexes were proposed. Here, an improved empirical binding energy function has been derived and validated on three different types of complexes: protein–small ligand, protein–peptide and protein–protein. The function uses the boundary element algorithm to evaluate the electrostatic solvation energy. We show that a single set of parameters can predict the relative binding energies of the heterogeneous validation set of complexes with 2.5 kcal/mol accuracy. We also demonstrate that global optimization of the ligand and of the flexible side‐chains of the receptor improves the accuracy of the evaluation. Copyright © 1999 John Wiley & Sons, Ltd.     

Electrostatic Oscillations Along Cylindrical Micelles

Theoretical analysis of ionic electrostatic oscillations of a charged cylindrical micelle is presented within the framework of the fluid theory for surface counterions of the cell, in conjunction with the Poisson equation. General expression of dispersion relation is obtained for electrostatic excitations of system.     

Prediction of Protein-Protein Interaction Sites Using Electrostatic Desolvation Profiles

Protein-protein complex formation involves removal of water from the interface region. Surface regions with a small free energy penalty for water removal or desolvation may correspond to preferred interaction sites. A method to calculate the electrostatic free energy of placing a neutral low-dielectric probe at various protein surface positions has been designed and applied to characterize putative interaction sites. Based on solutions of the finite-difference Poisson equation, this method also includes long-range electrostatic contributions and the protein solvent boundary shape in contrast to accessible-surface-area-based solvation energies. Calculations on a large set of proteins indicate that in many cases (>90%), the known binding site overlaps with one of the six regions of lowest electrostatic desolvation penalty (overlap with the lowest desolvation region for 48% of proteins). Since the onset of electrostatic desolvation occurs even before direct protein-protein contact formation, it may help guide proteins toward the binding region in the final stage of complex formation. It is interesting that the probe desolvation properties associated with residue types were found to depend to some degree on whether the residue was outside of or part of a binding site. The probe desolvation penalty was on average smaller if the residue was part of a binding site compared to other surface locations. Applications to several antigen-antibody complexes demonstrated that the approach might be useful not only to predict protein interaction sites in general but to map potential antigenic epitopes on protein surfaces.     

Application of the Poisson Boltzmann polyelectrolyte model for analysis of thermal denaturation of DNA in the presence of Na+ and polyamine cations

The Poisson Boltzmann (PB) cell model of polyelectrolyte solution has been used for calculation of the electrostatic free energy difference, Delta G(el), between double- and single-stranded DNA. The calculations have been performed for conditions relevant to describe the DNA helix-coil transition in NaCl solution in the presence of the natural polyamines putrescine(2+), spermidine(3+), spermine(4+) and their synthetic homologs with different spacing between the charged amino groups, for which experimental values of the DNA 'melting' transition temperature (T(m)) are available. Using the PB theory and the polyamine ion radius as an adjusting parameter provides quantitative agreement between experimental and theoretical T(m)--salt concentration dependencies only by using physically unreasonable radii for the polyamine. Thus, modeling the linear and flexible polyamines as charged spheres within the PB cell model is an implausible oversimplification. We propose another explanation for the experimental observations, still within the frame of the 'primitive' PB polyelectrolyte theory. This explanation is based on an analysis of the Delta G(el) dependence on the stoichiometry of polyamine-polyanion binding to double- and single-stranded DNA.     

Comparison of end-point continuum-solvation methods for the calculation of protein-ligand binding free energies

We have compared the predictions of ligand‐binding affinities from several methods based on end‐point molecular dynamics simulations and continuum solvation, that is, methods related to MM/PBSA (molecular mechanics combined with Poisson–Boltzmann and surface area solvation). Two continuum‐solvation models were considered, viz., the Poisson–Boltzmann (PB) and generalised Born (GB) approaches. The nonelectrostatic energies were also obtained in two different ways, viz., either from the sum of the bonded, van der Waals, nonpolar solvation energies, and entropy terms (as in MM/PBSA), or from the scaled protein–ligand van der Waals interaction energy (as in the linear interaction energy approach, LIE). Three different approaches to calculate electrostatic energies were tested, viz., the sum of electrostatic interaction energies and polar solvation energies, obtained either from a single simulation of the complex or from three independent simulations of the complex, the free protein, and the free ligand, or the linear‐response approximation (LRA). Moreover, we investigated the effect of scaling the electrostatic interactions by an effective internal dielectric constant of the protein (?_int). All these methods were tested on the binding of seven biotin analogues to avidin and nine 3‐amidinobenzyl‐1H‐indole‐2‐carboxamide inhibitors to factor Xa. For avidin, the best results were obtained with a combination of the LIE nonelectrostatic energies with the MM+GB electrostatic energies from a single simulation, using ?_int = 4. For fXa, standard MM/GBSA, based on one simulation and using ?_int = 4–10 gave the best result. The optimum internal dielectric constant seems to be slightly higher with PB than with GB solvation. © Proteins 2012; © 2012 Wiley Periodicals, Inc.     

Integral equation theories for predicting water structure around molecules

Water plays a crucial role in the structure and function of proteins and other biological macromolecules; thus, theories of aqueous solvation for these molecules are of great importance. However, water is a complex solvent whose properties are still not completely understood. Statistical mechanical integral equation theories predict the density distribution of water molecules around a solute so that all particles are fully represented and thus potentially both molecular and macroscopic properties are included. Here we discuss how several theoretical tools we have developed have been integrated into an integral equation theory designed for globular macromolecular solutes such as proteins. Our approach predicts the three-dimensional spatial and orientational distribution of water molecules around a solute. Beginning with a three-dimensional Ornstein-Zernike equation, a separation is made between a reference part dependent only on the spatial distribution of solvent and a perturbation part dependent also on the orientational distribution of solvent. The spatial part is treated at a molecular level by a modified hypernetted chain closure whereas the orientational part is treated as a Boltzmann prefactor using a quasi-continuum theory we developed for solvation of simple ions. A potential energy function for water molecules is also needed and the sticky dipole models of water, such as our recently developed soft-sticky dipole (SSD) model, are ideal for the proposed separation. Moreover, SSD water is as good as or better than three point models typically used for simulations of biological macromolecules in structural, dielectric and dynamics properties and yet is seven times faster in Monte Carlo and four times faster in molecular dynamics simulations. Since our integral equation theory accurately predicts results from Monte Carlo simulations for solvation of a variety of test cases from a single water or ion to ice-like clusters and ion pairs, the application of this theory to biological macromolecules is promising.     

Effective water model for Monte Carlo simulations of proteins

We present an effective theory for water. Our goal is to formulate on accurate model for the effects of solvation on protein dynamics, without incurring the huge computational cost and the slow temporal evolution typical of molecular dynamics simulations of liquids. We replace the individual water molecules in an all-atom potential with a local dielectric density field, with self interactions given by the Landau-Ginzburg free energy and external interactions by Lennard-Jones forces at the surface of the protein atoms. We explore conformational space with finite temperature Monte Carlo dynamics, using parallel Langevin and Fourier acceleration algorithms well suited to data-parallel computer architectures such as the Connection Machine. To establish the validity of our approximations, we compare our electrostatic contribution to the solvalion energy with the results of Lim, Bashford, and Karplus using a conventional static continuum dielectric cavity model, and the non electrostatic contributions with estimates of hydrophohic surface free energy. Our model can also accommodate ionic charges and temperature fluctuations, We propose future investigations extending our effective theory of solvation to include explicit orientational entropy and hydroxen-bonding terms. © 1995 John Wiley & Sons, Inc.