Simulated annealing is a general procedure for finding a solution to an optimization problem (1, 2). Nuclear magnetic resonance (NMR) experiments can be used to produce constraints on the possible secondary structures and tertiary structures of a biological macromolecule. The experimental observations usually include nuclear Overhauser effects (NOEs), which provide information about the distances between pairs of nuclei of the molecule, and three-bond coupling constants, which can provide estimates of dihedral angles. Constraints may also be produced by experiments that define the orientation of chemical shift tensors or dipolar interactions within the molecular framework. Determination of the three-dimensional structure consists of finding a conformation that is consistent with all constraints. This is a kind of optimization problem; simulated annealing thus provides a method for finding tertiary structures that are consistent with known constraints.
The use of simulated annealing for determination of tertiary structure is based on the type of empirical molecular force field that is usually used in molecular modeling. Such a force field is a collection of mathematical expressions that attempt to account for the energies associated with the deformation of chemical bonds, nonbonded van der Waals interactions between atoms of a molecule, and electrostatic interactions between charged groups of the structure. Molecular modeling programs usually contain a facility for conformational energy minimization. This feature uses an algorithm to adjust the positions of the atoms of a molecule so that the total energy, computed using the force field of the program, is at a minimum.
The simplest approach to finding a molecular conformation consistent with a set of NMR constraints is to include a term in the molecular force field that adds nothing to the total energy of the molecule when a given constraint is satisfied, but adds a large, unfavorable energy contribution when that constraint is violated. With such terms present, the process of energy minimization will tend to drive the molecule toward a conformation in which all constraints are satisfied. The defect in this procedure is that, although all energy minimization algorithms are capable of finding a minimum energy, no way exists to guarantee that it will be the conformation of lowest energy. Significant energy maxima (barriers) may separate local energy minima from the global minimum, and these must be overcome before the conformation that satisfies all constraints represented in the force field is identified. Simulated annealing is useful at this stage of optimization problem.
If the mathematical expressions in a force field can be differentiated, it is possible to calculate the force experienced by any atom of the molecule for any specific conformation. By employing Newton’s second law (F = ma), one can calculate the acceleration of each atom and, therefore, the change of position of an atom with time. Using a molecular force field in conjunction with the solution of a family of equations that embody the second law is the basis for molecular dynamics simulations.
The method of simulated annealing is based conceptually on the behavior of materials as they undergo the transition from the liquid state to the solid. If a sample starts at high temperature and is cooled rapidly, a polycrystalline or amorphous material may be obtained. This disordered sample represents a material that is not is its lowest free energy state. If the liquid is cooled slowly (annealed), a single crystal can often be formed. In the crystal, molecules are ordered in a way that represents the conformation of lowest possible energy, that is, the global minimum energy.
A direct proportionality exists between the kinetic energy of a system of moving particles and the temperature of the system. To find the global energy minimum for a system that includes NMR constraints, a molecular dynamics simulation is run in which the system is subjected to a schedule of temperature changes. The system is held at a high temperature for a period of time and then cooled slowly. It is hoped that during the high temperature phase, there will be sufficient motions of the atoms of the structure that barriers separating local minima from the global minimum are overcome. The cooling phase of the calculation potentially "anneals" the system into the conformation representing the global energy minimum—presumably the conformation in which experimentally derived constraints are best satisfied. In practice, several cycles of heating and cooling (annealing) are used, and the best schedule for these must often be worked out by trial-and-error. (See also Distance Geometry.)
