Practical Protein Chemistry - A. Darbre 1989

Prediction of Peptide and Protein Conformation
An Arsenal of Modern Theoretical Methods
Heuristic Methods

Usually, a heuristic approach is considered to be a technique that facilitates the search for a solution to a complex problem and reduces the time required to obtain a result. Similar techniques, which allow for saving computer time, are applied in protein conformation prediction. Unlike the Metropolis method or energy Minimization algorithms, heuristic Methods do not imply the Introduction of special assumptions. This section discusses approaches that do not alter the potential surface, but instead utilize a priori constraints within the conformational parameter space.

A fundamental difficulty in Structure/41.html">Cell/13.html">Protein Structure Prediction lies in the existence of a vast number of local minima; the application of heuristic techniques is specifically aimed at overcoming this multiple-minima problem. The introduction of constraints makes it possible to exclude from the search those Regions of the conformational space that a priori do not contain the global minimum, or, conversely, to restrict the search to regions containing the solution of interest to the researcher.

In the pioneering work by Levitt and Warshel [33] on modeling the self-Organization process of pancreatic Trypsin inhibitor, a simplified representation of the protein molecule was proposed. This was designed to smooth out non-essential details of the investigated potential surface, including local minima. Naturally, this approach can be viewed as the application of peculiar potential Functions, which is equivalent to adopting a deliberately crude further approximation. Such a technique is not fully heuristic, and transitioning to such a high level of simplification represents a very specific innovation in The Study of protein self-organization. It should also be borne in mind that the proposed primitive protein model was the only one possible in this case, whereas a more accurate representation currently defies calculation. However, a real danger arises that the altered potential surface of the protein may bear little resemblance to the actual energy surface.    

Study [63] proposed an alternative approach that is, in principle, applicable even with an exact representation of the protein molecule. It is well known that the internal variables determining protein conformation are not independent within a single residue. The value of one variable (typically a torsion angle) predetermines the allowable values of several other variables. Consequently, there is no need to test all possible combinations, as many of them are highly improbable. The main difficulty consists in appropriately choosing The values of the principal variables and avoiding the exploration of improbable Conformations.

The aforementioned Procedure can be formalized as follows. First, let us express the minimized Conformational Energy in the usual way as a function of the torsion angle variables v₁, v₂ ...

Class="center">E= E (v₁, v₂, ..., v_m)      (21.12)

Next, let us consider sets of variables grouped according to THE PRINCIPLE OF interdependence. For example, variables from v_i-m to v_i+m may constitute one such set. For each of the generated sets of variables, a binding function w_j is formulated:

w_j = f(v_i-m, ..., v₁, ..., v_i+m)      (21.13)

The binding function traces the trajectory in a certain region of the conformational space defined by the set of variables v_i-m, ..., v_i, ..., v_i+m. The form of the function is chosen such that the value of w_j corresponds to the distance near the search trajectory. Thus, the value of w_j determines THE POSITION OF the point belonging to the trajectory itself within the conformational space of the variables v_i-m, ..., v_i, ..., v_i+m. The expression for the vector P(v_i-m, ..., v_i, ..., v_i+m) specifying the point's position is as follows:

P(v_i-m, ..., v_i, ..., v_i+m) = f^-1(w_j) (21.14)

where f^-1 is the inverse of the function w_j in equation (21.13). Performing such operations for each set of variables makes it possible to utilize all intervariable couplings and calculate the energy using equation (21.12). The advantage of the described approach is that the energy is minimized as a function of a smaller number of variables:

E = E(w₁, w₂, ..., w_N)      (21.15)

The conformation of the system, defined in terms of the variables w, can, if necessary, be represented in terms of the variables v using an approximation procedure in which the conformation is varied as a function of the variables w.

A useful property of the function f^-1 is its uniqueness. How is the function f^-1 chosen in practice? To date, studies are known that have employed both very simple functions corresponding to straightforward trajectories and more cumbersome functions leading to rather complex trajectories. In work [62], the equation of an inclined ellipse was chosen for f^-1, describing the conformational space of two torsion angles Φ and Ψ for each residue in the polypeptide chain of the protein. The value of w corresponds to the angle D between two vectors originating at the center of the ellipse. One of the vectors is fixed in space, while the second points toward the peripheral point under consideration.

By adjusting the parameters, it was possible to achieve D = 0° and 360° for the fully extended pleated sheet conformation of the polypeptide chain, D = +90° for the classical right-handed α-Helix, D = -90° for the left-handed α-helix, and so forth. The time savings achieved through the application of the heuristic method considered here for the self-organization process of the pancreatic trypsin inhibitor proved quite substantial compared to the method of Levitt and Warshel [33]. It is also important to note that this computational acceleration was achieved with a more accurate representation of the protein chain details than in the work of Levitt and Warshel.

Naturally, all analogous methods involve the application of constraints and conditions, since the desired solution must conform to a specific trajectory within the studied conformational space. Special care must be taken to ensure that the search trajectory encompasses all crucial conformations (e.g., the α-helix) and passes sufficiently close to all possible solutions. In this regard, the choice of an elliptic trajectory appears somewhat controversial, although it does provide The ability to cover α-helix, β-sheet, and β-turn conformations in the search [62]. There is, of course, a danger that in a protein whose tertiary structure is being predicted, the conformation of a particular amino acid residue may lie outside the elliptic search trajectory and thus be omitted from consideration entirely. Nevertheless, the Current state of the method permits minor deviations in residue conformations as long as the overall course of the protein chain is preserved. Moreover, significant deviations in the conformation of a single residue are often compensated by positional changes of adjacent residues, so that the overall profile of the protein molecule is maintained. This was demonstrated by imposing elliptic constraints on the conformation of each amino acid residue in Proteins, followed by the minimization of the ROOT-mean-square deviation of atomic localization coordinates (typically Cα-atoms) from experimental values. For A number of proteins, a very low root-mean-square atomic position deviation of 1.1 Å was obtained, which compares very favorably with the 4–6 Å deviation typically obtained in protein structure predictions using standard energy minimization Procedures.

Even better agreement can be achieved by choosing more complex search trajectories through the Selection of the function f^-1. One such example is a trajectory approximated by a series of short straight segments, where the (I+1)-th segment near the search trajectory is defined by the coordinates of the endpoints Q_I and Q_I+1. Let D be the parameter defining the trajectory in a given region. In the case where D = 6.3, the search trajectory is approximated by the 0.3 fraction of the seventh straight segment connecting points Q₆ and Q₇. In the more general case, if D = I + X (where I is the integer part and X is the fractional part of the parameter D), the search trajectory is defined by the fraction X of the (I+1)-th straight segment between points Q_I and Q_I+1. For a conformation at the point P_D = (v_i-m, ..., v_i, ..., v_i+m) = f^-1(w_j), where w_j = D, the following relation holds:

P_D = Q_I + X(Q_I+1 - Q_I)      (21.16)

P_D, Q_N, and Q_N+1 are vectors, each containing as many elements as there are variables v in the given set; the vectors Q can be represented in a computer as a stack of vectors corresponding to points of a specific trajectory. In contrast, the vector P_D varies continuously and corresponds to a trial conformation P = (v_i-m, ..., v_i, ..., v_i+m), which matches the distance D along the search trajectory. In practice, a cyclic trajectory is utilized, such that Q_I replaces Q_I+1, and this replacement is performed taking into account the function modulus at the current value of I and the number of straight segments approximating the trajectory. Unlike the ellipse used in work [62], this latter approach can be extended to any number of interrelated variables. However, difficulties arise associated with discontinuities in the derivatives of the potential function with respect to the distance D. These can be overcome by minimization using the simplex method. This method is highly useful in protein structure prediction [62], and the presence of discontinuities in the derivatives does not present an insuperable obstacle for the simplex method. At the same time, other ways of representing the search trajectory are known in which the energy derivatives remain continuous, but these approaches are quite complex and are not considered here.