In the PGMC method, conformations of a polypeptide are generated by rotating backbone () and/or sidechain () dihedral angles for individual amino acids. These dihedral angles are shown in Figure for arginine, which has the largest number of freely-rotating dihedrals. The conformations are not chosen randomly, but are selected from probability grids calculated from a selected subset of proteins from the Brookhave Protein Database. Each grid is an -dimensional matrix, where is the number of dihedrals involved. For instance, backbone sampling involves two-dimensional grids, and each point on the grid is the probability of chosing a particular pair. The grids have spacing, where = 5, 10, 15, 30, or 60. Therefore, grids have points, where . The probabilities were derived by partitioning every pair in a set of high-resolution protein crystal structures into -degree bins. The probabilities () are normalized so that
Sidechain probability grids have varying dimensionality, depending upon the number of dihedrals needed to specify the conformation. This ranges from = 1 for small sidechains like serine and threonine, to = 5 for arginine. For alanine and glycine, = 0.
Our standard approach for doing Monte Carlo simulations uses these probability grids to generate trial conformations and assesses these conformations using the Metropolis criterion. The protein is assigned an initial conformation, usually by rotating all backbone and sidechain dihedrals to the highest-probability grid values. A new conformation is generated by modifying one amino acid, which is chosen at random. Depending on the nature of the simulation, either a new main-chain conformation is chosen from the probability grid, or a new side-chain conformation is chosen from the probability grid. The potential energy of the new conformation () is calculated, using a standard forcefield such as DREIDING or AMBER and is compared to the energy of the previous conformation (). Conformation is either accepted or rejected according to the Metropolis criterion. If the new structure is lower in energy (, where ), it is accepted. If it is higher in energy, the probability of acceptance, , is defined by:
where is the Boltzmann constant and is the simulation temperature. A random number is generated between 0.0 and 1.0. If , the new conformation is accepted. Otherwise, the structure is rejected and the previous structure is restored. This is the Metropolis criterion for accepting or rejecting a new structure, designed to ensure a Boltzmann distribution of conformations. This criterion means, for example, that there is a 50%chance of accepting a new structure if . At 300 K, this value is 0.413 kcal/mol. At higher temperatures, the probability of accepting a bad structure increases.
We are generally interested in finding the lowest-energy conformation of a given peptide. This can be done, theoretically, by the simulated annealing method: starting at a high simulation temperature, and slowly cooling the system in a process called simulated annealing. However, it is usually not possible to know beforehand exactly what cooling rate is necessary to achieve the global minimum. Considerable work has been done in optimizing the heating and cooling process in Monte Carlo simulations of peptides. We have employed both constant temperature and simulated annealing in our studies of protein conformations.