Practical Protein Chemistry - A. Darbre 1989

Prediction of Peptide and Protein Conformation
The Arsenal of Modern Theoretical Methods
Minimization

Energy minimization can be described as an evaluative Procedure designed to find the most probable conformation. Such a search is intrinsically tied to energy minimization, i.e., locating the most probable conformation characterized by the lowest energy.

Strictly speaking, finding the most favorable conformation requires calculating the Free energy of the system. In practice, however, an assumption is made that a minimum on the potential energy surface corresponds to a minimum on the free energy surface. Furthermore, it is generally assumed that the deepest minimum of potential energy also corresponds to the deepest minimum of free energy. If there is insufficient certainty regarding this latter assumption, free energy values at all local minima are calculated and compared. Various free energy calculation Methods clearly indicate that a broad, shallow potential energy minimum may correspond to a lower free energy minimum. Naturally, the difference in potential energy between the two minima must not be excessively large. When calculating the free energy of a system, the local Distribution Function Z (see above) is found by integration in the vicinity of the minimum. Free energy calculations also utilize second derivatives and vibrational frequencies (see, e.g., [20]). In all cases, evaluating free energy involves applying Statistical Mechanics concepts to analyze minima on the potential energy surface. Nevertheless, this approach is relatively new in peptide conformational analysis and is still rarely used. It is worth noting that it is theoretically possible to calculate free energy near any point on the potential surface, rather than just near a minimum, using a specialized computer program that implements free energy minimization.

Energy minimization as a function of conformation is widely used. This procedure forms the basis for computer simulation of protein molecule self-assembly and the energetic refinement of protein structures determined by X-ray crystallography. A large number of software packages are available for performing minimization calculations; most of these allow users to define their own energy calculation Functions based on conformational parameters. This software diversity reflects the wide range of mathematical techniques employed to locate function minima. Overall, however, these techniques fall into two categories.

The first category consists of gradient methods. Although the primary condition is finding a point where the first derivatives with respect to conformational variables are zero and the second derivatives are greater than zero, some methods do not require the explicit calculation of derivatives. These methods can be viewed as a family of Procedures that differ in how they transform E_i+1 − E_i (where i and i+1 correspond to adjacent points in conformational space) to compute derivatives at point i and select the conformation X_i+1. The steepest descent method is one of the simplest yet most popular in conformational calculations due to its reliable convergence. Its main drawback is relatively slow speed. The Newton-Raphson method is more complex and requires the computation of second derivatives; nevertheless, such methods are now quite common. The Fletcher-Reeves method avoids calculating second derivatives, obtaining information about the curvature of the energy surface using quadratic approximation forms. The Davidon [9] and Fletcher-Powell [11] methods combine the advantages of both the Newton-Raphson and Fletcher-Reeves procedures. While these methods are quite efficient, they share a common limitation: the search terminates at any local minimum. Indeed, escaping a minimum is impossible due to the fundamental characteristics of this Class of algorithms.

The second type of minimization algorithms is represented by valley-seeking (or ravine) methods, which can be well illustrated by the simplex procedure. It works effectively when minimizing energy as a function of torsion angles rather than x, y, and z coordinates. A simplex—a polyhedron with n+1 vertices in n-dimensional space—analyzes the function values at all vertices and moves the vertex with the highest energy to a point with a lower energy. To achieve this, the simplex procedure employs operations of reflection, expansion, or contraction of the polyhedron, as well as movement of the trial point relative to the centroid of the remaining points. Gradient search methods can arguably be compared to a ball rolling down a hillside, whereas the simplex method resembles a blind octopus feeling its way through a coral reef. Naturally, the simplex method also has limitations, but overall it should be considered highly robust. It functions even in the presence of discontinuities in the derivatives or the function itself; it is well-suited for incorporating various additional constraints imposed on conformational variables, energy values, or Other functions of conformational variables. Moreover, this algorithm can overcome shallow minima if the trial point enters a region of lower energy. The probability of such a "jump" depends on the depth and profile of the encountered minimum, the width of the energy barrier, and the control parameters of the simplex procedure. Provided that a new vertex is generated at a significant distance from the polyhedron centroid, the probability of encountering a new minimum increases noticeably. Depending on specific circumstances, the simplex procedure can be modified for greater efficiency in solving particular problems (e.g., the complex method). One drawback of the simplex method is the high computational cost required to obtain reliable results.

On the one hand, gradient methods rapidly locate local minima, but they struggle when The behavior of the function is insufficiently simple (e.g., containing discontinuities). On the other hand, the simplex method can handle most functions and does not get trapped in local minima, which increases the likelihood of finding the global minimum. The desirability of combining both methods in studies of Peptide and Protein Structure has been repeatedly emphasized. For instance, Robson and Osguthorpe [62] employed a dual minimization scheme, switching from a gradient method to a simplex whenever convergence stalled in a local minimum or a difficult point was encountered, and vice versa, returning to the gradient method once a new minimum was found. This switching occurs automatically via diagnostic variables calculated within each algorithm based on current convergence values.

This approach eliminates the need to introduce special algorithms for escaping local minima (see, for example, the thermalization method [33]).