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The dead-end elimination theorems are used to rule out certain rotamer assignments. One keeps a list of all the rotamers at all molten residues, and then removes from this list those rotamers that can be proven to not belong in the optimal configuration. The optimal solution remains when all but one rotamer from each molten residue has been removed from this list. The dead-end elimination theorems provide the means by which rotamers may be scratched off the list; they have the general form “If rotamers is not as goodas rotamers′_{, then sis not a member of the GMEC and may be removed from}

the list” where the “not as good”portion of the theorem is an inequality.

Desmetet al.originally introduced the dead-end elimination theorem in the context of the side-chain-placement problem in 1992. Their theorem applies to a single residue

v, on which a pair of rotamers s and s′ _{have been chosen. It relies on the idea of a}

conformational background. A conformational background for residue v is a rotamer assignment to the rest of the residues in the protein not including v. The theorem says that if the conformational background that favors rotamer s′ _{the least favors s}′

more than the conformational background that favors s the most favors s, then every conformational background favors s′ _{over s and therefore s is not a member of the}

be expressed by the following inequality: Eself(s) + X u6=v min t∈S(u)Epair(t, s)>Eself(s ′_{) +}X u6=v max t∈S(u)Epair(t, s ′_{) (2.8)}

where each rotamer set _S(u) contains only those rotamers for residue u that have not yet been eliminated. Through iterative applications of the theorem, the conformational backgrounds favoring or penalizing various rotamers change. A rotamer that could not be eliminated in the first round of eliminations might be eliminatable after some other rotamers have been eliminated. An iterative method based on this theorem would repeatedly attempt to eliminate each rotamer until it could eliminate none; at this point either a single rotamer for each residue remains, and the problem is solved, or an irreducible sub problem remains. If this sub-problem is small enough, then it can be treated with brute-force enumeration.

The time it takes to find the conformational background that penalizes s′ _{the most,}

as well as the time it takes to find the conformational background that favors s the most, scales linearly with the number of rotamers being considered at all other residues in the protein; P

u inV,u6=v|Su|. Notice that the time does not scale exponentially with the number of other rotamers at all other residues; there are after all Q

u inV,u6=v|Su| conformational backgrounds available for consideration. The key to the speed is the pairwise decomposability of the energy function; the rotamer assigned to each residue in the conformational background that penalizess′ _{the most can be computed indepen-}

dently of the rest of the rotamer assignments to the other residues. The same is true, of course, for the rotamers in the conformational background that favors s the most. Pairwise decomposability is required of any energy function to be solved by iterative applications of the dead-end elimination theorems.

Desmetet al.’s theorem was expanded to include rotamer pairs (Lasters and Desmet, 1993). The idea is that, not only does one maintain a list of rotamers that have not yet been eliminated, one also maintains a list of rotamer pairs that have not yet been eliminated. As one applies the theorem to a rotamer pair and finds it is not a member of the GMEC, one removes that rotamer pair from the list. The new theorem, called the fuzzy ended theorem, considers the rotamer pair s1 and s2 on residues u and v in

comparison with an alternate rotamer pairs′

(s1, s2) is not a member of the GMEC if the following inequality holds: E(s1) + E(s2) +E(s1, s2) > E(s′1) + E(s′2) +E(s′1, s′2) + P w6=u,vmint∈S(w)E(t, s1) + P w6=u,vmaxt∈S(w)E(t, s′1) + P w6=u,vmint∈S(w)E(t, s2) + P w6=u,vmaxt∈S(w)E(t, s′2) (2.9)

where both the left and right half of the inequality exclude any rotamertthat has been eliminated, and where the left half of the inequality may exclude any pair (t, s1) or

(t, s2) from the min operations if that pair has already been eliminated; the right half

of the inequality may not exclude eliminated pairs. This discrepancy between the left and the right hand sides earns this theorem its fuzzy-ended name. The rotamer pairs eliminated by this theorem can also be excluded from the left-hand side of Inequality 2.8 for the elimination of single rotamers.

Goldstein formulated a more lenient elimination criteria for rotamer singles (Gold- stein, 1994). He noted that the alternate rotamers′ _{had only to prove more favorable}

than s in each individual conformational background. Instead of searching for two conformational backgrounds, the one favorings the most and the one penalizing s′ _the

most, he searches instead for the single conformational background that favors s′ _over

s the least. If in this background, s′ _{is still preferred, then s may be eliminated from}

consideration. Specifically, s′ _{eliminates s if}

E(s′)_{− E}(s) +X u6=v max t∈S(u)(E(t, s ′₎ − E(t, s))<0 (2.10) where again the rotamer set S(u) contains only un-eliminated rotamers, and any ro- tamert may be ignored if the pair (t, s) has been eliminated; whether or nott may be ignored is independent of the elimination status of pair (t, s′_{). Goldstein’s theorem is}

weaker in the sense that any rotamer pair that satisfies Inequality 2.8 also satisfies his inequality; it is more powerful in the sense that it can eliminate rotamers even after Inequality 2.8 fails. His inequality ends up more expensive to include in an iterative method for rotamer elimination than Inequality 2.8, not because once s and s′ _have

been chosen it takes more time (both Inequalities 2.8 and 2.10 take Θ(P

u inV,u6=v|Su|) time to compute), but rather because the conformational backgrounds for s and s′

may not be computed independently. If s is chosen, then the iterative method for ro- tamer elimination chooses from the rest of _Sv for alternate rotamers s′, and for each combination of s and s′_{, check if s and s}′ _{satisfy Inequality 2.10.}

The elimination of state singles and state pairs lead to the generalized dead-end elimination theorem (Looger and Hellinga, 2001) which eliminates arbitrarily large rotamer clusters (rotamer triples, rotamer quadruples, etc.) if it can find a winning replacement cluster. Of course, the complexity of eliminating larger and larger rotamer clusters increases to the complexity of the original problem. Loogeret al. (Looger and Hellinga, 2001) have reported that the generalized DEE algorithm has succeeded on problems with state spaces as large as 101044_{. In their numbers, however, Loogeret al.}

include rotamers that collide with the background, which are pruned from consideration before DEE begins; that is, these rotamers are never proven not to be in the GMEC by DEE, so it is somewhat misleading to say that DEE solves problems of this size. It is likely that DEE could have eliminated such rotamers. What is most misleading is the idea that Dezymer ever computes and stores as many rotamer-pair energies for as many rotamers as are implied to exist in a problem that large