Practical Protein Chemistry - A. Darbre 1989

Prediction of Peptide and Protein Conformation
The Arsenal of Modern Theoretical Methods
Monte Carlo Method

Since the primary interest lies in The equilibrium state rather than the kinetic mechanism of reaching equilibrium, the kinetic component of energy is frequently neglected, with the exception of the part present as kТ.

Neglecting kinetic energy allows us to treat the system's nuclei as inertialess, thereby enabling an arbitrary choice of Conformations when calculating the potential energy surface. Another approach, resembling The Structure/8.html">Molecular Dynamics method, is known as the Monte Carlo method. The underlying objective, as well as the Organization of data storage and analysis, are identical for both Methods. In the Monte Carlo method, navigating the potential energy surface relies on a technique based on random number generation. At the same time, a fundamental principle is observed: The sequence of the conformations being studied is genuinely random in nature, and calculations continue until a sufficient degree of accuracy is achieved. As a result of these computations, any physical characteristic of interest to the researcher, such as Free energy, can be obtained.

Modern computers feature software tools for generating random numbers, which are used to obtain new conformations—each formed through minor random perturbations of the preceding conformation. The generated random numbers are actually "pseudorandom," and the algorithm for producing such numbers relies on some starting value (seed). The resulting random number, in turn, serves as an argument for generating the next random number. Thus, unless the algorithm is specifically tied to the computer clock or values present in random-access memory, the same starting number will reproduce identical results every time. This is sometimes even desirable, as it provides The ability to repeat computations. However, this does not imply any correlation between the generated numbers other than the fact that they are all random variables. Nevertheless, there is always a risk that the random number algorithm will at some point reproduce the initial seed, causing the entire sequence of numbers to repeat. Therefore, the algorithm and the starting value must be chosen so that the cycle length is sufficiently large. Even so, if this risk is minimized in the software used, a subtle underlying correlation among the resulting values may still persist.

Despite the desire to eliminate any regularities in the sample under study, it must be acknowledged that the Monte Carlo method requires certain conditions to be met. Naturally, the accuracy and reliability of the results should not depend on these requirements, and their impact is compensated for in some way. The justification for introducing such restrictions is the impracticality of considering an excessively large number of high-energy conformations, since the studied sample must be rich in low-energy conformations. The way these restrictions are implemented forms The Essence of the Monte Carlo method; without them, the approach would not only be trivial but also devoid of practical meaning. Depending on the specific techniques employed, various modifications of the Monte Carlo method are known, such as polymer chain constraints and Metropolis sampling, among others. The approach proposed by Metropolis is so widespread that it is frequently mistaken for the Monte Carlo method itself, and this section will examine it in greater detail. The "polymer chain constraints" technique is most popular when studying The properties of random-coil Polypeptides [52, 53]. This application is associated with generating conformations that correspond to fully extended, denatured polypeptides.

The essence of the Metropolis algorithm [38] is as follows. The positions of the system's particles are altered within designated limits relative to their original coordinates. The potential energy of the new conformation is calculated, and the difference in potential energy between the new and old conformations is denoted as ∆Е. It should be kept in mind that this refers not to the conformation of a single molecule, but to the configuration of an entire system of particles consisting of peptide and Water molecules. If ∆Е<0, the new conformation is accepted outright. However, if ∆Е>0, a random number between 0 and 1 is generated and compared with the value e^-∆E/kT. If this threshold is exceeded, the new conformation is accepted; otherwise, it is rejected, and the previous conformation is reinstated as the starting point.

An important feature of the Metropolis approach lies in the method used to calculate the average Properties of the object (i.e., averaged over all conformations). To achieve this, it is sufficient to divide the sum over all generated conformers by the total number of conformers. It should be noted that only "successfully generated" conformations participate in the averaging Procedure. Attempted but rejected conformers are not counted, while conformers to which the system returns after an unsuccessful step are counted again. This method is unconventional for Statistical Mechanics (see below) and is valid solely for the Metropolis procedure, constituting an integral part of it. The difference is that standard calculations typically compute a weighted average, using exponential factors of the form e^-∆Е/kТ as weighting coefficients. In the Metropolis method, the frequency of successfully generated conformers turns out to be proportional to these exponential weighting factors, thereby ensuring the correctness of the resulting averaged property. The primary value of the Metropolis method lies in the fact that statistical properties are imparted to the algorithm and its ergodicity is ensured—meaning the existence of all possible conformers is legitimately accounted for.

The described method has proven particularly successful in investigating aqueous solutions of Peptides and Proteins. It is important to note that the number of conformational parameters is limited primarily by computation time rather than The complexity of the system under study. Therefore, if the protein molecule does not undergo conformational changes, accounting for its effect on the surrounding environment does not significantly complicate computer simulations.

A suitable test system is a hydrated crystal, since the locations of certain water molecules can be compared with crystallographic data. Of particular interest in this regard are studies on Lysozyme crystals and hydrated cyclic dipeptide crystals [16, 21]. These Examples illustrate how to bypass the difficulties associated with edge effects at the system boundaries. In these calculations, it is possible to account for the presence of 350–400 water molecules, yet even this quantity is insufficient to form a surface protein film in a vacuum. This surface cannot be accommodated in calculations of an isolated unit Cell of a hydrated crystal contacting analogous neighboring Cells, which would otherwise fill all space. However, the properties of water molecules in each unit cell are mirrored by the properties of equivalent water molecules in all other cells, and any molecule leaving The Cell at its lower left boundary is replaced by a new one at the upper right boundary. Thus, "periodic boundary conditions" are introduced.

Although the periodic boundary conditions discussed above are particularly intuitive in the case of single crystal unit cells, they can also be successfully applied to solutions. In this case, it is necessary to determine the size of the unit cell and the cutoff distance up to which the energy is calculated and METABOLISM/18.html">The Influence of long-range forces is taken into account. Hagler et al. [22] used the Monte Carlo method to study The behavior of N-acetylalanine N'-methylamide in an aqueous solution. This molecule is of interest as a classical model of a polypeptide possessing chemical characteristics. It possesses a sufficient number of degrees of freedom (5) to exhibit various conformations. In solution, it turns out to be feasible to consider only the influence of water on the conformational preference of the peptide while neglecting The Effect of the dissolved peptide on the water structure. As a result of the performed calculations, the following data regarding the behavior of N-acetylalanine N'-methylamide in solution were obtained:

1) as expected, water molecules are attached to the peptide groups via Hydrogen Bonds and apparently exert an influence on the conformational capabilities of the peptide molecule;

2) the peptide's influence on the behavior of water is restricted to the space occupied by the first Hydration shell;

3) polar Solvents tend to stabilize the conformation of the peptide molecule that is characterized by the highest dipole moment. In the presence of polar solvents, the stability of conformations featuring an intramolecular Hydrogen bond yet characterized by a low dipole moment is explained by steric factors. Interestingly, these results support well-known models of solvent behavior, namely: (a) the supermolecular model, (b) the solvation shell model, and (c) the reaction field model. Each of these models reflects aspects of the solvation process from a single perspective. In this case, "exact" computations clearly demonstrate the effectiveness of approximate calculation methods. Of course, computational methods are not yet capable of overcoming all arising difficulties, as free energy calculations for a peptide-solvent system, for example, require tens of hours of computer time.