This notation simplifies the description of the pairs criterion below. Singles elimination criterion If a particular rotamer of sidechain cannot possibly give a better energy than another rotamer of the same sidechain, then rotamer A can be eliminated from further consideration, which reduces the search space. Mathematically, this condition is expressed by the inequality where is the minimum (best) energy possible between rotamer of sidechain and any rotamer X of side chain . Similarly, is the maximum (worst) energy possible between rotamer of sidechain and any rotamer X of side chain . Pairs elimination criterion The pairs criterion is more difficult to describe and to implement, but it adds significant eliminating power. For brevity, we define the shorthand variable that is the intrinsic energy of a pair of rotamers and at positions and , respectively A given pair of rotamers and at positions and , respectively, cannot both be in the final solution (although one or the other may be) if there is another pair and that always gives a better energy. Expressed mathematically, where , and . Energy matrices For large , the matrices of precomputed energies can become costly to store. Let be the number of amino acid positions, as above, and let be the number of rotamers at each position (this is usually, but not necessarily, constant over all positions). Each self-energy matrix for a given position requires entries, so the total number of self-energies to store is . Each pair energy matrix between two positions and , for discrete rotamers at each position, requires a matrix. This makes the total number of entries in an unreduced pair matrix . This can be trimmed somewhat, at the cost of additional complexity in implementation, because pair energies are symmetrical and the pair energy between a rotamer and itself is zero. Implementation and efficiency The above two criteria are normally applied iteratively until convergence, defined as the point at which no more rotamers or pairs can be eliminated. Since this is normally a reduction in the sample space by many orders of magnitude, simple enumeration will suffice to determine the minimum within this pared-down set. Given this model, it is clear that the DEE algorithm is guaranteed to find the optimal solution; that is, it is a global optimization process. The single-rotamer search scales quadratically in time with total number of rotamers. The pair search scales cubically and is