Continuum electrostatic methods applied to pH-dependent properties of antibody-antigen association

Protein association events are a critical component of the functioning of biological systems. Antibody/antigen association, which involves extraordinarily specific interactions, has been a paradigm for the study of structural factors and intermolecular forces controlling protein-protein association. As new experimental approaches to the study of antibody/antigen affinity have become routine, and as more structures of complexes of antibodies and their antigens have become available, it has become possible to use computational approaches to study these interactions. Electrostatic interactions are known to play an important role in protein complex formation. In this review, we focus on the use of continuum electrostatic methods to compute pH-dependent properties of proteins and discuss the use of these methods in the study of antibody/antigen complexes.     

Van der Waals forces involving thin rods

The Susceptibility theory of van der Waals dispersion forces is used to calculate the non-retarded dispersion free energy between two parallel anisotropic elliptical rods, between a rod and an ellipsoid, and between a rod and a wall, when the dimensions of the bodies (semiaxes) are small compared to their separation. The results show that the dispersion energy is orientation dependent, so that in general the bodies will experience, in addition to a force, a torque which will tend to align them relative to each other so as to minimize the free energy. The results should prove useful in calculating the dispersion interactions in biological systems involving chainlike molecules.     

Diffusing colloidal probes of protein and synthetic macromolecule interactions

A new approach is described for measuring kT and nanometer scale protein-protein and protein-synthetic macromolecule interactions. The utility of this method is demonstrated by measuring interactions of bovine serum albumin (BSA) and copolymers with exposed polyethyleneoxide (PEO) moieties adsorbed to hydrophobically modified colloids and surfaces. Total internal reflection and video microscopy are used to track three-dimensional trajectories of many single diffusing colloids that are analyzed to yield interaction potentials, mean-square displacements, and colloid-surface association lifetimes. A criterion is developed to identify colloids as being levitated, associated, or deposited based on energetic, spatial, statistical, and temporal information. Whereas levitation and deposition occur for strongly repulsive or attractive potentials, association is exponentially sensitive to weak interactions influenced by adsorbed layer architectures and surface heterogeneity. Systematic experiments reveal how BSA orientation and PEO molecular weight produce adsorbed layers that either conceal or expose substrate heterogeneities to generate a continuum of colloid-surface association lifetimes. These measurements provide simultaneous access to a broad range of information that consistently indicates purely repulsive BSA and PEO interactions and a role for surface heterogeneity in colloid-surface association. The demonstrated capability to measure nonspecific protein interactions provides a basis for future measurements of specific protein interactions.     

Intermolecular interactions between protein and other molecules including hydration effects

The structural aspects of protein functions, e.g., molecular recognition such as enzyme-substrate and antibody-antigen interactions, are elucidated in terms of dehydration and atomic interactions. When a protein interacts with some target molecule, water molecules at the interacting regions of both molecules are removed, with loss of the hydration free energy, but gaining atomic interactions between atoms of the contact sites in both molecules. The free energies of association originating from the dehydration and interactions between the atoms can be computed from changes in the accessible surface areas of the atoms involved. The free energy due to interactions between atomic groups at the contact sites is estimated as the sum of those estimated from the changes in the accessible surface area of 7 atomic groups, assuming that the interactions are proportional to the change of the area. The chain enthalpies and entropies evaluated from experimental thermodynamic properties and hydration quantities at the standard temperature for 10 proteins were available to determine the proportional constants for the atomic groups. This method was applied to the evaluation of association constants for the dimerization of proteins and the formation of proteolytic enzyme-inhibitor complexes, and the computed constants were in agreement with the experimental ones. However, the method is not accurate enough to account quantitatively for the change in the thermal stability of mutants of T4 lysozyme. Nevertheless, this method provides a way to elucidate the interactions between molecules in solution.     

Van der Waals forces. Special characteristics in lipid-water systems and a general method of calculation based on the Lifshitz theory

A practical method for examining and calculating van der Waals forces is derived from Lifshitz'' theory. Rather than treat the total van der Waals energy as a sum of pairwise interactions between atoms, the Lifshitz theory treats component materials as continua in which there are electromagnetic fluctuations at all frequencies over the entire body. It is necessary in principle to use total macroscopic dielectric data from component substances to analyze the permitted fluctuations; in practice it is possible to use only partial information to perform satisfactory calculations. The biologically interesting case of lipid-water systems is considered in detail for illustration. The method gives good agreement with measured van der Waals energy of interaction across a lipid film. It appears that fluctuations at infrared frequencies and microwave frequencies are very important although these are usually ignored in preference to UV contributions. “Retardation effects” are such as to damp out high frequency fluctuation contributions; if interaction specificity is due to UV spectra, this will be revealed only at interactions across <200 angstrom (A). Dependence of van der Waals forces on material electric properties is discussed in terms of illustrative numerical calculations.     

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.     

Free energetics of rigid body association of ubiquitin binding domains: A biochemical model for binding mediated by hydrophobic interaction

Weak intermolecular interactions, such as hydrophobic associations, underlie numerous biomolecular recognition processes. Ubiquitin is a small protein that represents a biochemical model for exploring thermodynamic signatures of hydrophobic association as it is widely held that a major component of ubiquitin's binding to numerous partners is mediated by hydrophobic regions on both partners. Here, we use atomistic molecular dynamics simulations in conjunction with the Adaptive Biasing Force sampling method to compute potentials of mean force (the reversible work, or free energy, associated with the binding process) to investigate the thermodynamic signature of complexation in this well‐studied biochemical model of hydrophobic association. We observe that much like in the case of a purely hydrophobic solute (i.e., graphene, carbon nanotubes), association is favored by entropic contributions from release of water from the interprotein regions. Moreover, association is disfavored by loss of enthalpic interactions, but unlike in the case of purely hydrophobic solutes, in this case protein‐water interactions are lost and not compensated for by additional water‐water interactions generated upon release of interprotein and moreso, hydration, water. We further find that relative orientations of the proteins that mutually present hydrophobic regions of each protein to its partner are favored over those that do not. In fact, the free energy minimum as predicted by a force field based method recapitulates the experimental NMR solution structure of the complex. Proteins 2014; 82:1453–1468. © 2014 Wiley Periodicals, Inc.     

Molecular dynamics simulations of a calmodulin-peptide complex in solution

We have performed an 4-ns MD simulation of calmodulin complexed with a target peptide in explicit water, under realistic conditions of constant temperature and pressure, in the presence of a physiological concentration of counterions and using Ewald summation to avoid truncation of long-range electrostatic forces. During the simulation the system tended to perform small fluctuations around a structure similar to, but somewhat looser than the starting crystal structure. The calmodulin-peptide complex was quite rigid and did not exhibit any large amplitude domain motions such as previously seen in apo- and calcium-bound calmodulin. We analyzed the calmodulin-peptide interactions by calculating buried surface areas, CHARMM interaction energies and continuum model interaction free energies. In the trajectory, the protein surface area buried by contact with the peptide is 1373 A(2) approximately evenly divided between the calmodulin N-terminal, C-terminal and central linker regions. A majority of this buried surface, 803 A(2), comes from nonpolar residues, in contrast to the protein as a whole, for which the surface is made up of mostly polar and charged groups. Our continuum calculations indicate that the largest favorable contribution to peptide binding comes from burial of molecular surface upon complex formation. Electrostatic contributions are favorable but smaller in the trajectory structures, and actually unfavorable for binding in the crystal structure. Since nonpolar groups make up most of buried surface of the protein, our calculations suggest that the hydrophobic effect is the main driving force for binding the helical peptide to calmodulin, consistent with thermodynamic analysis of experimental data. Besides the burial of nonpolar surface area, secondary contributions to peptide binding come from burial of polar surface and electrostatic interactions. In the nonpolar interactions a crucial role is played by the nine methionines of calmodulin. In the electrostatic interactions the negatively charged protein residues and positively charged peptide residues play a dominant role.     

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.     

Calculation of the total electrostatic energy of a macromolecular system: solvation energies, binding energies, and conformational analysis

In this report we describe an accurate numerical method for calculating the total electrostatic energy of molecules of arbitrary shape and charge distribution, accounting for both Coulombic and solvent polarization terms. In addition to the solvation energies of individual molecules, the method can be used to calculate the electrostatic energy associated with conformational changes in proteins as well as changes in solvation energy that accompany the binding of charged substrates. The validity of the method is examined by calculating the hydration energies of acetate, methyl ammonium, ammonium, and methanol. The method is then used to study the relationship between the depth of a charge within a protein and its interaction with the solvent. Calculations of the relative electrostatic energies of crystal and misfolded conformations of Themiste dyscritum hemerythrin and the VL domain of an antibody are also presented. The results indicate that electrostatic charge-solvent interactions strongly favor the crystal structures. More generally, it is found that charge-solvent interactions, which are frequently neglected in protein structure analysis, can make large contributions to the total energy of a macromolecular system.     

Electrostatic Exclusion of Neutral Solutes from Condensed DNA and Other Charged Phases

Motivated by experiments on condensed DNA phases in binary mixtures of water and a low-dielectric solute, we develop a theory for the electrostatic contribution to solute exclusion from a highly charged phase, within the continuum approximation of the medium. Because the electric field is maximum at the surface of each ion, the electrostatic energy is dominated by the Born energy; interactions between charges are of secondary importance. Neglecting interactions and considering only the competition between the Born energy and the free energy of mixing, we predict that low dielectric solutes are excluded from condensed DNA phases in water-cosolvent mixtures. This suggests that the traditional continuum electrostatic approach of modeling binary mixtures with a uniform dielectric constant needs to be modified. The linking of solute exclusion to solute dielectric properties also suggests a mechanism for predicting the electrostatic contribution to preferential hydration of polar and charged surfaces.     

Effect of the external electrostatic field on the conformation of a macromolecule containing charged groups

A method for calculating the free energy of a macromolecule containing charged groups in electrostatic field in aqueous solution was proposed. The non-electrostatic component of free energy was calculated with consideration of van der Waals interactions between uncharged parts of the macromolecule. The electrostatic component of free energy was calculated with regard for the interactions of charged groups of the macromolecule with each other and with water molecules. It was found that, depending on the strength of external electric field, the free energy of the system passes through a minimum, whereas the internal energy passes through a maximum. By minimizing the free energy, relative changes in the mean radius 'r' and the distance between the termini of the macromolecule 'h' were calculated. It was found that, at some values of field strength, both 'r' and 'h' decrease. An increase in strength led to an increase in 'r' and 'h'. The regularities observed depend on the charge of the macromolecule and the spatial redistribution of macromolecules and counterions.     

Calculation of the free energy of association for protein complexes.

We have developed a method for calculating the association energy of quaternary complexes starting from their atomic coordinates. The association energy is described as the sum of two solvation terms and an energy term to account for the loss of translational and rotational entropy. The calculated solvation energy, using atomic solvation parameters and the solvent accessible surface areas, has a correlation of 96% with experimentally determined values. We have applied this methodology to examine intermediates in viral assembly and to assess the contribution isomerization makes to the association energy of molecular complexes. In addition, we have shown that the calculated association can be used as a predictive tool for analyzing modeled molecular complexes.     

Solvation thermodynamics of biopolymers. I. Separation of the volume and surface interactions with estimates for proteins

The present paper is a systematic first approach to the problem of solvation thermodynamics of biomolecules. Most previous approaches have been only crude estimates of solvent contributions, and have simply assessed solvation free energy as proportional to surface areas. Here we estimate the various contributions and divide them into (a) hard-core interactions dependent upon the entire volume of solute and (b) the remainder of interactions manifested through surfaces, such as van der Waals, charge-charge, or hydrogen bonds. We have estimated the work to create a cavity with scaled-particle theory (SPT), the van der Waals interactions on the surface, and hydrogen bonds between the surface and the solvent. The conclusion here is that this latter term is the largest component of the solvation free energy of proteins. From estimates on nine diverse proteins, it is clear that the larger the protein, the more dominant is the hydrogen-bond term. In the next paper, we indicate that correlations between hydrogen-bonding groups on the surfaces could increase the magnitude of the hydrogen-bond contribution.     

On Calculation of the Electrostatic Potential of a Phosphatidylinositol Phosphate-Containing Phosphatidylcholine Lipid Membrane Accounting for Membrane Dynamics

Many signaling events require the binding of cytoplasmic proteins to cell membranes by recognition of specific charged lipids, such as phosphoinositol-phosphates. As a model for a protein-membrane binding site, we consider one charged phosphoinositol phosphate (PtdIns(3)P) embedded in a phosphatidylcholine bilayer. As the protein-membrane binding is driven by electrostatic interactions, continuum solvent models require an accurate representation of the electrostatic potential of the phosphoinositol phosphate-containing membrane. We computed and analyzed the electrostatic potentials of snapshots taken at regular intervals from molecular dynamics simulations of the bilayer. We observe considerable variation in the electrostatic potential of the bilayer both along a single simulation and between simulations performed with the GAFF or CHARMM c36 force fields. However, we find that the choice of GAFF or CHARMM c36 parameters has little effect on the electrostatic potential of a given configuration of the bilayer with a PtdIns(3)P embedded in it. From our results, we propose a remedian averaging method for calculating the electrostatic potential of a membrane system that is suitable for simulations of protein-membrane binding with a continuum solvent model.     

Explicit and implicit water simulations of a beta-hairpin peptide.

The conformational properties of a beta-hairpin peptide (YITNSDGTWT) were studied by using both explicit and implicit water simulations. The conformational space of the peptide was scanned by using a restricted hydrogen-bonding search method. The search method used generated the conformational space with enough diversity and good representation of beta-hairpin structures. By using a total surface area-based treatment of hydrophobic interactions, implicit water simulations failed to discriminate between experimental beta-hairpin structures from the rest of the conformers present in the authors' conformation library. However, with inclusion of vibrational free energy and accounting separately for polar and nonpolar surface areas, the nuclear magnetic resonance structure was ranked successfully as the most stable conformation. There is a loose correlation between the conformational energies by the continuum model and the conformational energies by explicit water simulation for conformers with similar structures. However, in terms of solvation energy, both approaches have a much better correlation. By using proper treatment of surface effect (partition of the surface area into polar and nonpolar areas) and including vibrational free-energy contribution, the continuum models should be reliable. Furthermore, the authors found that, for this peptide, beta-hairpin structures have large vibrational entropy that contributes decisively to the stability of folded beta-hairpin structures. Proteins 1999;37:73-87.     

Molecular Dynamics Simulations of a Calmodulin-peptide Complex in Solution

Abstract

We have performed an 4-ns MD simulation of calmodulin complexed with a target peptide in explicit water, under realistic conditions of constant temperature and pressure, in the presence of a physiological concentration of counterions and using Ewald summation to avoid truncation of long-range electrostatic forces. During the simulation the system tended to perform small fluctuations around a structure similar to, but somewhat looser than the starting crystal structure. The calmodulin-peptide complex was quite rigid and did not exhibit any large amplitude domain motions such as previously seen in apo- and calcium-bound calmodulin. We analyzed the calmodulin-peptide interactions by calculating buried surface areas, CHARMM interaction energies and continuum model interaction free energies. In the trajectory, the protein surface area buried by contact with the peptide is 1373 Ų, approximately evenly divided between the calmodulin N-terminal, C-terminal and central linker regions. A majority of this buried surface, 803 ·A², comes from nonpolar residues, in contrast to the protein as a whole, for which the surface is made up of mostly polar and charged groups. Our continuum calculations indicate that the largest favorable contribution to pep- tide binding comes from burial of molecular surface upon complex formation. Electrostatic contributions are favorable but smaller in the trajectory structures, and actually unfavorable for binding in the crystal structure. Since nonpolar groups make up most of buried surface of the protein, our calculations suggest that the hydrophobic effect is the main driving force for binding the helical peptide to calmodulin, consistent with thermodynamic analysis of experimental data. Besides the burial of nonpolar surface area, secondary contributions to peptide binding come from burial of polar surface and electrostatic interactions. In the nonpolar interactions a crucial role is played by the nine methionines of calmodulin. In the electrostatic interactions the negatively charged protein residues and positively charged peptide residues play a dominant role.     

Protein folding and association: insights from the interfacial and thermodynamic properties of hydrocarbons.

We demonstrate in this work that the surface tension, water-organic solvent, transfer-free energies and the thermodynamics of melting of linear alkanes provide fundamental insights into the nonpolar driving forces for protein folding and protein binding reactions. We first develop a model for the curvature dependence of the hydrophobic effect and find that the macroscopic concept of interfacial free energy is applicable at the molecular level. Application of a well-known relationship involving surface tension and adhesion energies reveals that dispersion forces play little or no net role in hydrophobic interactions; rather, the standard model of disruption of water structure (entropically driven at 25 degrees C) is correct. The hydrophobic interaction is found, in agreement with the classical picture, to provide a major driving force for protein folding. Analysis of the melting behavior of hydrocarbons reveals that close packing of the protein interior makes only a small free energy contribution to folding because the enthalpic gain resulting from increased dispersion interactions (relative to the liquid) is countered by the freezing of side chain motion. The identical effect should occur in association reactions, which may provide an enormous simplification in the evaluation of binding energies. Protein binding reactions, even between nearly planar or concave/convex interfaces, are found to have effective hydrophobicities considerably smaller than the prediction based on macroscopic surface tension. This is due to the formation of a concave collar region that usually accompanies complex formation. This effect may preclude the formation of complexes between convex surfaces.