Yury N. Vorobjev

Discrimination between native and intentionally misfolded conformations of proteins: ES/IS, a new method for calculating conformational free energy that uses both dynamics simulations with an explicit solvent and an implicit solvent continuum model†

A new method for calculating the total conformational free energy of proteins in water solvent is presented. The method consists of a relatively brief simulation by molecular dynamics with explicit solvent (ES) molecules to produce a set of microstates of the macroscopic conformation. Conformational energy and entropy are obtained from the simulation, the latter in the quasi‐harmonic approximation by analysis of the covariance matrix. The implicit solvent (IS) dielectric continuum model is used to calculate the average solvation free energy as the sum of the free energies of creating the solute‐size hydrophobic cavity, of the van der Waals solute‐solvent interactions, and of the polarization of water solvent by the solutes charges. The reliability of the solvation free energy depends on a number of factors: the details of arrangement of the proteins charges, especially those near the surface; the definition of the molecular surface; and the method chosen for solving the Poisson equation. Molecular dynamics simulation in explicit solvent relaxes the proteins conformation and allows polar surface groups to assume conformations compatible with interaction with solvent, while averaging of internal energy and solvation free energy tend to enhance the precision. Two recently developed methods—SIMS, for calculation of a smooth invariant molecular surface, and FAMBE, for solution of the Poisson equation via a fast adaptive multigrid boundary element—have been employed. The SIMS and FAMBE programs scale linearly with the number of atoms. SIMS is superior to Connollys MS (molecular surface) program: it is faster, more accurate, and more stable, and it smooths singularities of the molecular surface. Solvation free energies calculated with these two programs do not depend on molecular position or orientation and are stable along a molecular dynamics trajectory. We have applied this method to calculate the conformational free energy of native and intentionally misfolded globular conformations of proteins (the EMBL set of deliberately misfolded proteins) and have obtained good discrimination in favor of the native conformations in all instances. Proteins 32:399–413, 1998.

A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent

A fast multigrid boundary element (MBE) method for solving the Poisson equation for macromolecular electrostatic calculations in a solvent is developed. To convert the integral equation of the BE method into a numerical linear equation of low dimensions, the MBE method uses an adaptive tesselation of the molecular surface by BEs with nonregular size. The size of the BEs increases in three successive levels as the uniformity of the electrostatic field on the molecular surface increases. The MBE method provides a high degree of consistency, good accuracy, and stability when the sizes of the BEs are varied. The computational complexity of the unrestricted MBE method scales as O(Nat), where Nat is the number of atoms in the macromolecule. The MBE method is ideally suited for parallel computations and for an integrated algorithm for calculations of solvation free energy and free energy of ionization, which are coupled with the conformation of a solute molecule. The current version of the 3‐level MBE method is used to calculate the free energy of transfer from a vacuum to an aqueous solution and the free energy of the equilibrium state of ionization of a 17‐residue peptide in a given conformation at a given pH in ∼ 400 s of CPU time on one node of the IBM SP2 supercomputer.

SIMS: computation of a smooth invariant molecular surface

SIMS, a new method of calculating a smooth invariant molecular dot surface, is presented. The SIMS method generates the smooth molecular surface by rolling two probe spheres. A solvent probe sphere is rolled over the molecule and produces a Richards-Connolly molecular surface (MS), which envelops the solvent-excluded volume of the molecule. In deep crevices, Connollys method of calculating the MS has two deficiencies. First, it produces self-intersecting parts of the molecular surface, which must be removed to obtain the correct MS. Second, the correct MS is not smooth, i.e., the direction of the normal vector of the MS is not continuous, and some points of the MS are singular. We present an exact method for removing self-intersecting parts and smoothing the singular regions of the MS. The singular MS is smoothed by rolling a smoothing probe sphere over the inward side of the singular MS. The MS in the vicinity of singularities is replaced with the reentrant surface of the smoothing probe sphere. The smoothing method does not disturb the topology of a singular MS, and the smooth MS is a better approximation of the dielectric border between high dielectric solvent and the low dielectric molecular interior. The SIMS method generates a smooth molecular dot surface, which has a quasi-uniform dot distribution in two orthogonal directions on the molecular surface, which is invariant with molecular rotation and stable under changes in the molecular conformation, and which can be used in a variety of implicit methods of modeling solvent effects. The SIMS program is faster than the Connolly MS program, and in a matter of seconds generates a smooth dot MS of a 200-residue protein. The program is available from the authors on request (see http:@femto.med.unc.edu/SIMS).

ES/IS: Estimation of conformational free energy by combining dynamics simulations with explicit solvent with an implicit solvent continuum model

This paper reviews a recently developed method for calculating the total conformational free energy of a solute macromolecule in water solvent. The method consists of a relatively short simulation by molecular dynamics with explicit solvent molecules (ES) to produce a set of microstates of the macroscopic conformation. Conformational internal solute energy and entropy are obtained from the simulation, the latter in the quasi-harmonic approximation by analysis of the covariance matrix. The implicit solvent (IS) surface energy-dielectric continuum model is used to calculate the average solvation free energy as the sum of the free energies of creating the solute-size hydrophobic cavity, of the van der Waals solute-solvent interactions and of the polarization of water solvent by the solutes charges. We have earlier applied this method to calculate the conformational free energy of native and intentionally misfolded globular conformations of proteins (the EMBL set of deliberately misfolded proteins), and have obtained good discrimination in favor of the native conformations in all instances. These results are summarized and further analyzed to show that, on average, three major component terms of the free energy all contribute in favor of discrimination. We discuss possible improvements of the ES/IS method. It is shown how the force field can be made self-consistent by adapting the parameters for calculation of surface and polarization free energies closely to the molecular mechanics force field used in the dynamics simulation, using established simulation methods to compute free energies for cavity formation and a charging process with the molecular mechanics force field to provide a set of (quasi-experimental) reference data that can be used to refine the parameters of the continuum models. The molecular surface area together with a microscopic surface free energy near 70 cal/(mol A2) is found to be a consistent descriptor of the cavity free energy. Preliminary results indicate that a linear-response approximation for the polarization of water solvent reaction near typical polar and charged protein groups is accurate to within approximately 90%.

Free energies of protein decoys provide insight into determinants of protein stability

We have calculated the stability of decoy structures of several proteins (from the CASP3 models and the Park and Levitt decoy set) relative to the native structures. The calculations were performed with the force field-consistent ES/IS method, in which an implicit solvent (IS) model is used to calculate the average solvation free energy for snapshots from explicit simulations (ESs). The conformational free energy is obtained by adding the internal energy of the solute from the ESs and an entropic term estimated from the covariance positional fluctuation matrix. The set of atomic Born radii and the cavity-surface free energy coefficient used in the implicit model has been optimized to be consistent with the all-atom force field used in the ESs (cedar/gromos with simple point charge (SPC) water model). The decoys are found to have a consistently higher free energy than that of the native structure; the gap between the native structure and the best decoy varies between 10 and 15 kcal/mole, on the order of the free energy difference that typically separates the native state of a protein from the unfolded state. The correlation between the free energy and the extent to which the decoy structures differ from the native (as root mean square deviation) is very weak; hence, the free energy is not an accurate measure for ranking the structurally most native-like structures from among a set of models. Analysis of the energy components shows that stability is attained as a result of three major driving forces: (1) minimum size of the protein-water surface interface; (2) minimum total electrostatic energy, which includes solvent polarization; and (3) minimum protein packing energy. The detailed fit required to optimize the last term may underlie difficulties encountered in recovering the native fold from an approximate decoy or model structure.

Assessing the fractions of tautomeric forms of the imidazole ring of histidine in proteins as a function of pH

A method is proposed to determine the fraction of the tautomeric forms of the imidazole ring of histidine in proteins as a function of pH, provided that the observed and chemical shifts and the protein structure, or the fraction of H+ form, are known. This method is based on the use of quantum chemical methods to compute the 13C NMR shieldings of all the imidazole ring carbons (13Cγ, , and ) for each of the two tautomers, Nδ1-H and Nϵ2-H, and the protonated form, H+, of histidine. This methodology enabled us (i) to determine the fraction of all the tautomeric forms of histidine for eight proteins for which the and chemical shifts had been determined in solution in the pH range of 3.2 to 7.5 and (ii) to estimate the fraction of tautomeric forms of eight histidine-containing dipeptide crystals for which the chemical shifts had been determined by solid-state 13C NMR. Our results for proteins indicate that the protonated form is the most populated one, whereas the distribution of the tautomeric forms for the imidazole ring varies significantly among different histidines in the same protein, reflecting the importance of the environment of the histidines in determining the tautomeric forms. In addition, for ∼70% of the neutral histidine-containing dipeptides, the method leads to fairly good agreement between the calculated and the experimental tautomeric form. Coexistence of different tautomeric forms in the same crystal structure may explain the remaining 30% of disagreement.

Role of Hydrophobicity and Solvent-Mediated Charge-Charge Interactions in Stabilizing α-Helices

A theoretical study to identify the conformational preferences of lysine-based oligopeptides has been carried out. The solvation free energy and free energy of ionization of the oligopeptides have been calculated by using a fast multigrid boundary element method that considers the coupling between the conformation of the molecule and the ionization equilibria explicitly, at a given pH value. It has been found experimentally that isolated alanine and lysine residues have somewhat small intrinsic helix-forming tendencies; however, results from these simulations indicate that conformations containing right-handed alpha-helical turns are energetically favorable at low values of pH for lysine-based oligopeptides. Also, unusual patterns of interactions among lysine side chains with large hydrophobic contacts and close proximity (5-6 A) between charged NH3+ groups are observed. Similar arrangements of charged groups have been seen for lysine and arginine residues in experimentally determined structures of proteins available from the Protein Data Bank. The lowest-free-energy conformation of the sequence Ac-(LYS)6-NMe from these simulations showed large pKalpha shifts for some of the NH3+ groups of the lysine residues. Such large effects are not observed in the lowest-energy conformations of oligopeptide sequences with two, three, or four lysine residues. Calculations on the sequence Ac-LYS-(ALA)4-LYS-NMe also reveal low-energy alpha-helical conformations with interactions of one of the LYS side chains with the helix backbone in an arrangement quite similar to the one described recently by (Proc. Natl. Acad. Sci. U.S.A. 93:4025-4029). The results of this study provide a sound basis with which to discuss the nature of the interactions, such as hydrophobicity, charge-charge interaction, and solvent polarization effects, that stabilize right-handed alpha-helical conformations.

Advances in Implicit Models of Water Solvent to Compute Conformational Free Energy and Molecular Dynamics of Proteins at Constant pH

Modern implicit solvent models for macromolecular simulations in water-proton bath are considered. The fundamental quantity that implicit models approximate is the solute potential of mean force, which is obtained by averaging over solvent degrees of freedom. The implicit solvent models suggest practical ways to calculate free energies of macromolecular conformations taking into account equilibrium interactions with water solvent and proton bath, while the explicit solvent approach is unable to do that due to the need to account for a large number of solvent degrees of freedom. The most advanced realizations of the implicit continuum models by different research groups are discussed, their accuracy are examined, and some applications of the implicit solvent models to macromolecular modeling, such as free energy calculations, protein folding, and constant pH molecular dynamics are highlighted.

FAMBE-pH: A Fast and Accurate Method to Compute the Total Solvation Free Energies of Proteins

A fast and accurate method to compute the total solvation free energies of proteins as a function of pH is presented. The method makes use of a combination of approaches, some of which have already appeared in the literature; (i) the Poisson equation is solved with an optimized fast adaptive multigrid boundary element (FAMBE) method; (ii) the electrostatic free energies of the ionizable sites are calculated for their neutral and charged states by using a detailed model of atomic charges; (iii) a set of optimal atomic radii is used to define a precise dielectric surface interface; (iv) a multilevel adaptive tessellation of this dielectric surface interface is achieved by using multisized boundary elements; and (v) 1:1 salt effects are included. The equilibrium proton binding/release is calculated with the Tanford-Schellman integral if the proteins contain more than approximately 20-25 ionizable groups; for a smaller number of ionizable groups, the ionization partition function is calculated directly. The FAMBE method is tested as a function of pH (FAMBE-pH) with three proteins, namely, bovine pancreatic trypsin inhibitor (BPTI), hen egg white lysozyme (HEWL), and bovine pancreatic ribonuclease A (RNaseA). The results are (a) the FAMBE-pH method reproduces the observed pK as of the ionizable groups of these proteins within an average absolute value of 0.4 p K units and a maximum error of 1.2 p K units and (b) comparison of the calculated total pH-dependent solvation free energy for BPTI, between the exact calculation of the ionization partition function and the Tanford-Schellman integral method, shows agreement within 1.2 kcal/mol. These results indicate that calculation of total solvation free energies with the FAMBE-pH method can provide an accurate prediction of protein conformational stability at a given fixed pH and, if coupled with molecular mechanics or molecular dynamics methods, can also be used for more realistic studies of protein folding, unfolding, and dynamics, as a function of pH.

MDTRA: a molecular dynamics trajectory analyzer with a graphical user interface.

Most of existing software for analysis of molecular dynamics (MD) simulation results is based on command‐line, script‐guided processes that require the researchers to have an idea about programming language constructions used, often applied to the one and only product. Here, we describe an open‐source cross‐platform program, MD Trajectory Reader and Analyzer (MDTRA), that performs a large number of MD analysis tasks assisted with a graphical user interface. The program has been developed to facilitate the process of search and visualization of results. MDTRA can handle trajectories as sets of protein data bank files and presents tools and guidelines to convert some other trajectory formats into such sets. The parameters analyzed by MDTRA include interatomic distances, angles, dihedral angles, angles between planes, one‐dimensional and two‐dimensional root‐mean‐square deviation, solvent‐accessible area, and so on. As an example of using the program, we describe the application of MDTRA to analyze the MD of formamidopyrimidine‐DNA glycosylase, a DNA repair enzyme from Escherichia coli.