Arieh Warshel

Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme

Abstract A general method for detailed study of enzymic reactions is presented. The method considers the complete enzyme-substrate complex together with the surrounding solvent and evaluates all the different quantum mechanical and classical energy factors that can affect the reaction pathway. These factors include the quantum mechanical energies associated with bond cleavage and charge redistribution of the substrate and the classical energies of steric and electrostatic interactions between the substrate and the enzyme. The electrostatic polarization of the enzyme atoms and the orientation of the dipoles of the surrounding water molecules is simulated by a microscopic dielectric model. The solvation energy resulting from this polarization is considerable and must be included in any realistic calculation of chemical reactions involving anything more than an isolated molecule in vacuo . Without it, acidic groups can never become ionized and the charge distribution on the substrate will not be reasonable. The same dielectric model can also be used to study the reaction of the substrate in solution. In this way the reaction in solution can be compared with the enzymic reaction. In this paper we study the stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme. It is found that electrostatic stabilization is an important factor in increasing the rate of the reaction step that leads to the formation of the carbonium ion intermediate. Steric factors, such as the strain of the substrate on binding to lysozyme, do not seem to contribute significantly.

Calculations of electrostatic interactions in biological systems and in solutions

Correlating the structure and action of biological molecules requires knowledge of the corresponding relation between structure and energy. Probably the most important factors in such a structure– energy correlation are associated with electrostatic interactions. Thus the key requirement for quantative understanding of the action of biological molecules is the ability to correlate electrostatic interactions with structural information. To appreciate this point it is useful to compare the electrostatic energy of a charged amino acid in a polar solvent to the corresponding van der Waals energy. The electrostatic free energy, Δ G el , can be approximated (as will be shown in Section II) by the Born formula (Δ G el = –(166Q 2 /ā) (I – I/ E )). Where Δ G el is given in kcal/mol, Q is the charge of the given group, in units of electron charge, ā is the effective radius of the group, and E is the dielectric constant of the solvent. With an effective radius of charged amino acids of approximately 2 A, Borns formula gives about – 80 kcal/mol for their energy in polar solvents where E is larger than 10. This energy is two orders of magnitude larger than the van der Waals interaction of such groups and their surroundings.

Computer simulation of protein folding

A new and very simple representation of protein conformations has been used together with energy minimisation and thermalisation to simulate protein folding. Under certain conditions, the method succeeds in ‘renaturing’ bovine pancreatic trypsin inhibitor from an open-chain conformation into a folded conformation close to that of the native molecule.

What are the dielectric "constants" of proteins and how to validate electrostatic models?

Implicit models for evaluation of electrostatic energies in proteins include dielectric constants that represent effect of the protein environment. Unfortunately, the results obtained by such models are very sensitive to the value used for the dielectric constant. Furthermore, the factors that determine the optimal value of these constants are far from being obvious. This review considers the meaning of the protein dielectric constants and the ways to determine their optimal values. It is pointed out that typical benchmarks for validation of electrostatic models cannot discriminate between consistent and inconsistent models. In particular, the observed pKa values of surface groups can be reproduced correctly by models with entirely incorrect physical features. Thus, we introduce a discriminative benchmark that only includes residues whose pKa values are shifted significantly from their values in water. We also use the semimacroscopic version of the protein dipole Langevin dipole (PDLD/S) formulation to generate a series of models that move gradually from microscopic to fully macroscopic models. These include the linear response version of the PDLD/S models, Poisson Boltzmann (PB)‐type models, and Tanford Kirkwwod (TK)‐type models. Using our different models and the discriminative benchmark, we show that the protein dielectric constant, εp, is not a universal constant but simply a parameter that depends on the model used. It is also shown in agreement with our previous works that εp represents the factors that are not considered explicitly. The use of a discriminative benchmark appears to help not only in identifying nonphysical models but also in analyzing effects that are not reproduced in an accurate way by consistent models. These include the effect of water penetration and the effect of the protein reorganization. Finally, we show that the optimal dielectric constant for self‐energies is not the optimal constant for charge‐charge interactions. Proteins 2001;44:400–417.

Consistent Force Field for Calculations of Conformations, Vibrational Spectra, and Enthalpies of Cycloalkane and n‐Alkane Molecules

An inductive method for a systematic selection of energy functions of interatomic interactions in large families of molecules is suggested and is applied to the family of cycloalkane and n‐alkane molecules. Equilibrium conformations, vibrational frequencies, and excess enthalpies, including strain energies and vibrational enthalpies, are all derived from the same set of energy functions. The energy‐function parameters are optimized by a least‐squares algorithm to give the best possible agreement with a large amount and variety of observed data. Analytical derivatives of the various calculated quantities with respect to the energy parameters help to facilitate the computational procedures. The resulting agreement with experiment is used as a measure of success of the energy functions with optimized parameters, referred to as “consistent force field” (CFF). Different CFFs are compared and selected according to their relative success. Energy functions commonly used in conformational analysis are examined in...

Q-Chem 2.0: A High-Performance Ab Initio Electronic Structure Program Package

Q‐Chem 2.0 is a new release of an electronic structure program package, capable of performing first principles calculations on the ground and excited states of molecules using both density functional theory and wave function‐based methods. A review of the technical features contained within Q‐Chem 2.0 is presented. This article contains brief descriptive discussions of the key physical features of all new algorithms and theoretical models, together with sample calculations that illustrate their performance.

ELECTROSTATIC ORIGIN OF THE CATALYTIC POWER OF ENZYMES AND THE ROLE OF PREORGANIZED ACTIVE SITES

Enzymatic reactions are involved in most biological processes. Thus, there is a major practical and fundamental interest in finding out what makes enzymes so efficient. Many crucial pieces of this puzzle were provided by biochemical and structural studies (1). Yet, as will be shown below, the actual reason for the catalytic power of enzymes is not widely understood. It is clearly not explained by the statement that “the enzyme binds the transition state stronger than the ground state” because the real question is how the differential binding can be accomplished. Similarly, it is not true that “evolution can use any factor to accelerate reactions.” This review uses energy considerations and the results of computational studies to clarify open questions about enzyme catalysis.

A surface constrained all-atom solvent model for effective simulations of polar solutions

A consistent simulation of ionic or strongly polar solutes in polar solvents presents a major challenge from both fundamental and practical aspects. The frequently used method of periodic boundary conditions (PBC) does not correctly take into account the symmetry of the solute field. Instead of using PBC, it is natural to model this type of system as a sphere (with the solute at the origin), but the boundary conditions to be used in such a model are not obvious. Early calculations performed with our surface constrained soft sphere dipoles (SCSSD) model indicated that the dipoles near the surface of the sphere will show unusual orientational preferences (they will overpolarize) unless a corrective force is included in the model, and thus we implemented polarization constraints in this spherical model of polar solutions. More recent approaches that treated the surface with stochastic dynamics, but did not take into account the surface polarization effects, were also found to exhibit these nonphysical orient...

Microscopic and semimicroscopic calculations of electrostatic energies in proteins by the POLARIS and ENZYMIX programs

Different microscopic and semimicroscopic approaches for calculations of electrostatic energies in macromolecules are examined. This includes the Protein Dipoles Langevin Dipoles (PDLD) method, the semimicroscopic PDLD (PDLD/S) method, and a free energy perturbation (FEP) method. The incorporation of these approaches in the POLARIS and ENZYMIX modules of the MOLARIS package is described in detail. The PDLD electrostatic calculations are augmented by estimates of the relevant hydrophobic and steric contributions, as well as the effects of the ionic strength and external pH. Determination of the hydrophobic energy involves an approach that considers the modification of the effective surface area of the solute by local field effects. The steric contributions are analyzed in terms of the corresponding reorganization energies. Ionic strength effects are studied by modeling the ionic environment around the given system using a grid of residual charges and evaluating the relevant interaction using Coulombs law with the dielectric constant of water. The performance of the FEP calculations is significantly enhanced by using special boundary conditions and evaluating the long‐range electrostatic contributions using the Local Reaction Field (LRF) model. A diverse set of electrostatic effects are examined, including the solvation energies of charges in proteins and solutions, energetics of ion pairs in proteins and solutions, interaction between surface charges in proteins, and effect of ionic strength on such interactions, as well as electrostatic contributions to binding and catalysis in solvated proteins. Encouraging results are obtained by the microscopic and semimicroscopic approaches and the problems associated with some macroscopic models are illustrated. The PDLD and PDLD/S methods appear to be much faster than the FEP approach and still give reasonable results. In particular, the speed and simplicity of the PDLD/S method make it an effective strategy for calculations of electrostatic free energies in interactive docking studies. Nevertheless, comparing the results of the three approaches can provide a useful estimate of the accuracy of the calculated energies.

Investigation of the free energy functions for electron transfer reactions

The free energy functions for electron transfer reactions in solution are explored using a previously developed microscopic simulation approach that provides a clear definition of these functions and the variable (the reaction coordinate) used as their argument. The issue of whether the curvatures of the two functions (which correspond to states with nonpolar and polar solutes) are different is given special attention. It is found, in contrast to some previous suggestions, that the curvatures of the two functions are quite similar, even when one would expect differences due to dielectric saturation effects, and that Marcus’ approximation (and, in fact, the linear response theory inherent in this approximation) provides a valid description of the solvent’s role in electron transfer reactions over a wide range of conditions. The present study demonstrates that direct simulations of the reactant and product states do not provide the data needed for determination of the free energy functions in high energy re...