Notes

¹Strictly speaking, simulated annealing is guaranteed to find the global optimum of the target function only in the case of the so-called Boltzmann annealing, for which the temperature T(k) at each step k of the simulation is given by T(k) = T₀/log(k), where T₀ is the starting temperature (Ingber, 1993). Only with this annealing schedule and with T₀ `sufficiently high' is the algorithm guaranteed to find the global optimum of the target function. In this respect, the linear slow-cooling protocol discussed in §3.1 of this paper is more accurately described by the term `simulated quenching' than the conventionally used term `simulated annealing'.

²The word `grid' is used here metaphorically. For all practical purposes, the values of , x, y and z returned by the random-number generator are continuous [if, for example, the generator returns values in the range 0-2<sup>31</sup> - 1 and max() = , then the `grid size' on is less than 9 × 10^-8 °].

³All references to physical time measurements of the program's speed of execution refer to a UNIX workstation which in single-user mode gave the following SPEC95 benchmark results: SPECint95 = 16.6, SPECint_rate95 = 149, SPECfp95 = 21.9 and SPECfp_rate95 = 197 (Standard Performance Evaluation Corporation, 10754 Ambassador Drive, Suite 201, Manassas, VA 21109, USA; http://www.specbench.org/ ). UNIX is a registered trademark of UNIX System Laboratories, Inc.

⁴This search was conducted as follows: one polyalanine helix was fixed in orientation and position by combining the best 99 orientations from its cross-rotation function with the top 20 peaks from each of the corresponding translation functions, giving a total of 1980 starting models for the first helix. For each of these models, we calculated the translation functions corresponding to each and every of the 99 best orientations for a second copy of the model, giving a grand total of 1980 × 99 = 196 020 translation functions or a list of 3 920 400 correlation coefficients. This search resulted in a more or less uniform distribution of the linear correlation coefficients from the translation functions, with the best solutions being clearly wrong as judged by packing considerations.

⁵It should be mentioned, however, that even this simple proposition, i.e. to use the cross-rotation function as an orientational probability distribution, is still an oversimplification for problems with more than one molecule per asymmetric unit. The reason is that for such problems the probability distribution for the orientation of one molecule ought to be treated as conditional on the orientation of the other search models. One way that this could be achieved is through the active use of the self-rotation function as a means to calculate, based on the probability distribution for the orientation of one of the search models, the orientational probability distributions for the rest of the molecules (which is a generalization of the principle behind the locked rotation function; Tong & Rossmann, 1990). It goes without saying that the computational cost for performing such a calculation (which would involve updating the orientational probability distributions at each and every step) would be prohibitive with present-day computing capabilities.