Melding of Quantum Mechanics with Simpler Models
A daunting challenge for the future is to accurately model chemical reactions in phases and at the active site of enzymes. An ability to do so would be of great importance in designing new biological catalysts as well as fully understanding the chemica l mechanism of those that already exist. This would be of significant technological as well as scientific importance. One could imagine that many new molecules could be made and made much more efficiently by such catalysts.
Methods to do this in an approximate way have been available since 1976 (Warshel and Levitt, 1976) and have involved using the Schrödinger equation (Equation 1 or a variant or empiricized form of it) for the parts of the system where bonds are bein g made or broken and thus the electronic structure is changing, combined with representations such as Equation (10), which assume transferable electronic structure, for the remaining atoms of the system. Typically, the number of atoms for which Equation (1) must be solved is much smaller, of the order of 20 to 30, than the number in the whole system, which for the chemical reactions mentioned is typically more than a thousand.
In fact, some exciting results for simple reactions involving organic molecules in different solvents have been achieved (Blake and Jorgensen, 1991). In these cases, one has solved Equation (1) to high accuracy for a simple reactive pathway in vacuo and then, employing these energies, has used free energy calculation methods to evaluate the solvation free energy of different structures along the reactive pathway. This is in some sense a proof of concept for the combined application of Equations (1) and (10) because impressive agreement with experiment has been achieved in these simple, well-defined cases.
For more complex cases, such as enzyme reactions, the reaction pathway might involve many steps, and some of the reacting groups are chemically bonded to the protein, thus requiring some additional technical challenges in simulating the atoms at the jun ction between those that are participating in the chemical reactions and those that are not. In addition, one might have to consider many conformations of the enzyme and its substrate and accurately represent their relative energy by using the energy fun ction of Equation (10), all the while considering the electronic energy (Equation 1) and the relative total free energy of the system.
As noted above, progress on this problem has been made when employing much simplified representations of the electronic structure of the system, which enable the solution of equations such as Equation (1) for the few "quantum mechanical" atoms as rapidl y or more so than the classical molecular dynamical equations of motion, using Equation (10) as a potential energy (Field et al., 1990; Warshel, 1991).
These methods use semi-empirical or empirical valence bond approximations to solve Equation (1). Although these methods are not highly accurate, the use of non-empirical quantum mechanical methods for systems of 20 to 30 atoms (the traditional ab initi o approach) requires 104 to 106 more computations than the semi-empirical or empirical approaches and much more time than it takes to solve the classical equations of motion for the rest of the system (Singh and Kollman, 1986).
To elaborate, calculation of the free energy of a complex chemical system by using classical molecular dynamics (Kollman, 1993) requires one to calculate the energy of the system and its gradient with respect to all the 3N coordinates. This can be done for noncovalent processes (those using only Equation 10) quite efficiently because the energy function in Equation (10) is very simple and its derivatives are quick and easy to evaluate. When one adds quantum mechanical (bond making or breaking) effects via Equation (1), in order to make the calculation of the free energy tractable, one must be able to evaluate the quantum mechanical energy and its gradient for the few quantum mechanical atoms as rapidly as the classical molecular mechanical energy and gradient for the thousands of atoms in the remainder of the system. This can be done by using simpler empirical (Warshel, 1991) and, to a reasonable approximation, semi-empirical (Field et al., 1990) quantum mechanical methods, but not with the first pri nciple ab initio methods.
Density functional methods (Labanowski and Andzelm, 1991), particularly the divide-and-conquer strategy (Yang, 1991), show promise in leading to accurate and rapid solutions of Equation (9) for the electronic structure, but they are still a long way fro m being fully developed, so one cannot tell how efficient and useful they will be in this regard.
Thus, accurate simulation of chemical reactions at the active sites of macromolecules will likely require significant progress in the conformational search problem, even if one considers only the active site of the enzyme. As should be emphasized, the "conformational search problem" requires one not only to consider many conformations, but also to rank their relative free energy in solution. On top of this, one places the problem of accurate and very rapid electronic structure calculations. The above problems are very challenging conceptually, practically, and computationally.
References
Blake, J.F., and W. Jorgensen, 1991, Solvent effects on a Diels-Alder reaction from computer simulations, J. Am. Chem. Soc. 113:7430-7432.
Field, M.J., P.A. Bash, and M. Karplus, 1990, A combined quantum mechanical and molecular mechanical potential for molecular dynamics simulations, J. Comput. Chem. 11:700-733.
Kollman, P., 1993, Free energy Calculations--Applications to chemical and biochemical phenomena, Chem. Rev. 93:2395-2417.
Labanowski, J., and J.W. Andzelm, eds., 1991, Density Functional Methods in Chemistry, Springer Verlag, New York.
Singh, U.C., and P.A. Kollman, 1986, A combined ab initio QM/MM method for carrying out simulations on complex systems: Application to the CH3Cl + Cl- exchange reaction and gas phase protonation of polyethers, J. Comput. Chem. 7:718-730.
Warshel, A., 1991, Computer Modeling of Chemical Reactions in Enzymes and Solutions, John Wiley, New York.
Warshel, A., and M. Levitt, 1976, Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme, J. Mol. Biol. 103:227-249.
Yang, W., 1991, Direct calculation of electron density in density-functional theory-implementation for benzene and a tetrapeptide, Phys. Rev. A. 44:7823-7826.
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