# Simulated annealing

The potential energy landscape of a protein is complex and studded with many local minima, even in the presence of experimental restraints and when using a simplified target function. Because the temperature, i.e. kinetic energy, determines the maximal height of energy barriers that can be overcome in a molecular dynamics trajectory, the temperature schedule is important for the success and efficiency of a simulated annealing calculation. Elaborated protocols have been devised for structure calculations using molecular dynamics in Cartesian space (Brünger, 1992; Nilges et al., 1988). In addition to the temperature, other parameters such as force constants and repulsive core radii are varied in these schedules that may involve several stages of heating and cooling. The fast exploration of conformation space with torsion angle dynamics allows for simpler schedules.

### Standard CYANA simulated annealing schedule

The standard simulated annealing protocol in the program CYANA includes *N* torsion angle dynamics time steps. It starts from a conformation with all torsion angles treated as independent, uniformly distributed random variables and consists of five stages:

- Initial minimization. A short conjugate gradient minimization is applied to reduce high energy interactions that could otherwise disturb the torsion angle dynamics algorithm: 100 conjugate gradient minimization steps are performed, including only distance restraints between atoms up to 3 residues apart along the sequence, followed by a further 100 minimization steps including all restraints. For efficiency, all hydrogen atoms are excluded from the check for steric overlap, the repulsive core radii of heavy atoms without covalently bound hydrogen atoms are decreased by 0.2 Å with respect to their standard values, and the radii of heavy atoms with covalently bound hydrogens are decreased by 0.05 Å. The weighting factors in the target function are set to 1 for user-defined upper and lower distance bounds, and to 0.5 for steric lower distance bounds.
- First simulated annealing stage with reduced heavy atom radii. A torsion angle dynamics trajectory with (
*N*- 200)/3 time steps is generated. Typically, one fifth of these torsion angle dynamics steps are performed at a constant high reference temperature*T*_{high}of, typically, 10000 K, followed by slow cooling according to a fourth-power law to an intermediate reference temperature*T*_{med}=*T*_{high}/20. The time step is initialized to 2 fs. The list of van der Waals lower distance bounds is updated every 50 steps using a cutoff equal to twice the largest van der Waals radius plus 1 Å (= 4.2 Å for proteins) for the van der Waals pair list generation throughout all torsion angle dynamics phases. - Second simulated annealing stage with normal heavy atom radii and, later, normal hydrogen atom radii. The repulsive core radii of all heavy atoms are reset to their standard values, 50 conjugate gradient minimization steps are performed, and the torsion angle dynamics trajectory is continued for 2(
*N*- 200)/3 time step starting with an initial time step that is half as long as the last preceding time step. The reference temperature is decreased according to a fourth-power law from the intermediate temperature*T*_{med}to zero reference temperature. After two thirds of these time steps, the hydrogen atoms are included, with their standard radii, in the steric overlap check, and 50 conjugate gradient minimization steps are performed before continuing the trajectory, starting with a time step that is half as long as the last preceding time step. - Low temperature phase with increased weight for steric repulsion. The weighting factor for steric restraints is increased to 2, and 50 conjugate gradient minimization steps are performed, followed by 200 torsion angle dynamics steps at zero reference temperature, starting with a time step that is half as long as the last preceding time step.
- Final minimization. A final minimization with a maximum of, typically, 1000 conjugate gradient steps is applied.

The standard simulated annealing schedule is implemented in the CYANA command anneal.