Practical Protein Chemistry - A. Darbre 1989

Prediction of Peptide and Protein Conformation
The Arsenal of Modern Theoretical Methods
Quantum Mechanical Methods

The application of quantum mechanical approaches requires strict adherence to physical laws. Somewhat surprisingly, calculations using the most rigorous quantum mechanical Methods yield values comparable to experimental data, although, unfortunately, such computations are currently performed only when the necessary amount of experimental data is lacking.

The starting point for the scheme in Fig. 21.1 is the well-known premise that all system properties can be calculated using the Hamilton equation:

Class="center">НΨ = ЕΨ      (21.7)

where Ψ is the wave function, Е is the total energy of the system, and Н is the Hamiltonian operator.

Solving the equation for the wave function Ψ provides the probability of finding a particle at position q at time t, as well as the observable energy of the system relative to the Hamiltonian operator. The function of this operator in the equation is quite straightforward: like any operator, it specifies how one function is transformed into another (another operator, for instance, the differential sign, denotes the operation of differentiation). Let us consider a classical principle. Suppose that in the expression for the system's energy, the particle momentum must be replaced by the operator (—ih/2п) ∙ (d/dq). The Hamiltonian operator contains precisely an expression of this type. Equation (21.7) has many solutions, but only certain ones possess physical meaning. Each of these corresponds to a state of the system with a defined probability. The greatest difficulty in solving the Hamilton equation arises from the necessity of accounting not only for METABOLISM/18.html">The Influence of a given particle on all other particles in the system, but also for The Effect of the latter on The behavior of that particle. To mathematically evaluate this influence, one must introduce a first-order approximation in the form of perturbation theory, which was previously used to account for the mutual gravitational influence of planetary orbits.

The next highly significant point for practical conformational analysis is associated with the Born-Oppenheimer approximation principle. The latter implies that it is appropriate to refer the energy of the system to a strictly fixed configuration of atomic nuclei without considering the spatial distribution of electrons. Thus, the set of nuclear positions determines the molecular conformation, to which a distinct value of Conformational Energy can be assigned. According to Fig. 21.1, the ab initio method is applicable for energy calculation, but in practice, however, such calculations are feasible only for very simple chemical molecules. In reality, obtaining even an approximate energy value via quantum mechanical methods requires going through several stages, each involving varying degrees of approximation and simplification. Overall, we remain at the same level of approximation, meaning all simplifications are quantum mechanical in nature or represent analytical approximations of cumbersome computations. In any case, we estimate the energy of a simple conformation resulting from the Born-Oppenheimer electron-nuclear Separation. We can now examine the side branches in Fig. 21.1.

To calculate The properties of more complex molecules, including simple Peptides, with sufficient accuracy, it is necessary to invoke two additional principles. One of these is known as the "linear combination of atomic orbitals." It is assumed that atomic orbitals alone poorly convey molecular properties; a more successful, albeit more complex, description can be achieved by summing atomic orbitals with varying weighting coefficients. According to the second, variational principle, all solutions are permissible, but the solution corresponding to the conformation with the lowest energy is the most probable. Therefore, to find the lowest-energy solution, both the coefficients and the shape of the atomic orbitals are varied.

The set of atomic orbitals takes the form of parameters for three-dimensional Gaussian Functions or similar Slater functions. To represent each atomic orbital, several additional functions are typically employed, and the greater their number, the easier it is to use a computer to find the solution corresponding to the lowest energy. The set of functions describing the orbitals forms a basis set (minimal or expanded). Expanding the basis set involves using a larger number of orbital functions, giving the computer greater flexibility to alter the state of the system and find a solution corresponding to a lower final energy. It is generally accepted that any calculation resulting in a lower energy also provides a more reliable result. With a sufficiently expanded basis set, the calculation results become entirely independent of the starting point, which aligns with the meaning of ab initio. In practice, however, this demands immense computational time. For example, calculating a single conformation of 20–30 atoms may require about an hour of computer time. It should also be noted that the choice of a basis set merely defines a certain level of approximation; the method under consideration can be considered parameter-free, as is appropriate for ab initio calculations.

The evaluation of certain integrals is a time-consuming step. At the same time, semi-empirical quantum mechanical calculations can significantly reduce computational effort by introducing a series of ad hoc auxiliary parameters. Such calculations run much faster but yield less reliable results. The semi-empirical methods used in this type of calculation bear names derived from the initials of their full titles: (EXT — Extended Hückel Theory, CNDO — Complete Neglect of Differential Overlap, PCILO — Perturbative Configuration Interaction using Localized Orbitals, etc.). The PCILO method is attractive because it preserves information on excited states and allows for the calculation of dispersion attraction energy contributions. Obtaining such data via the ab initio method would require excessively long computations. However, it should be noted that THE CONTRIBUTION OF these interactions to the conformational energy is very small (∼0.4 kJ/mol per interacting atom pair) and an order of magnitude smaller than the total calculation error.

Programs for such calculations are available as standard software packages. Typical programs for ab initio calculations include the GAUSSIAN70 and ATMOL/3 packages, while semi-empirical calculation programs are known under various names depending on The Nature of the applied techniques. Prior to performing calculations, Cartesian coordinates of the nuclei are usually specified, although some programs can compute initial coordinates based on a geometrical representation of the system via Bond Lengths, Bond Angles, and torsion angles. To achieve the lowest conformational energy, computations are typically performed for a series of Conformations, each slightly different from the preceding one. The obtained results can be presented in tabular or graphical form as a potential energy surface, thereby defining energy as a function of conformation. A number of programs feature automated execution of a series of computations to locate the lowest-energy point on the potential energy surface. In cases where the search for favorable conformations is conducted at fixed bond lengths and bond angles, the calculation is termed "geometry optimization." The ensemble of conformations is then defined by a set of torsion angles.

The next level of approximation when calculating potential energy surfaces is the replacement of quantum mechanical relations with analytical functions—that is, The Use of potential functions, as discussed in Section 21.2. The parameters of these functions are usually fitted against experimental data.