Dead-end elimination

The dead-end elimination algorithm (DEE) is a method for minimizing a function over a discrete set of independent variables. The basic idea is to identify "dead ends", i.e., combinations of variables that are not necessary to define a global minimum because there is always a way of replacing such combination by a better or equivalent one. Then we can refrain from searching such combinations further. Hence, dead-end elimination is a mirror image of dynamic programming, in which "good" combinations are identified and explored further. Although the method itself is general, it has been developed and applied mainly to the problems of predicting and designing the structures of proteins. It closely related to the notion of dominance in optimization also known as substitutability in a Constraint Satisfaction Problem. The original description and proof of the dead-end elimination theorem can be found in .
Basic requirements.
An effective DEE implementation requires four pieces of information:
Note that the criteria can easily be reversed to identify the maximum of a given function as well.
Applications to protein structure prediction.
Dead-end elimination has been used effectively to predict the structure of side chains on a given protein backbone structure by minimizing an energy function formula_1. The dihedral angle search space of the side chains is restricted to a discrete set of rotamers for each amino acid position in the protein (which is, obviously, of fixed length). The original DEE description included criteria for the elimination of single rotamers and of rotamer pairs, although this can be expanded.
In the following discussion, let formula_2 be the length of the protein and let formula_3 represent the rotamer of the formula_4 side chain. Since atoms in proteins are assumed to interact only by two-body potentials, the energy may be written
Where formula_6 represents the "self-energy" of a particular rotamer formula_3, and formula_8 represents the "pair energy" of the rotamers formula_9.
Also note that formula_10 (that is, the pair energy between a rotamer and itself) is taken to be zero, and thus does not affect the summations. This notation simplifies the description of the pairs criterion below.
Singles elimination criterion.
If a particular rotamer formula_11 of sidechain formula_12 cannot possibly give a better energy than another rotamer formula_13 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 formula_15 is the minimum (best) energy possible between rotamer formula_11 of sidechain formula_12 and "any" rotamer X of side chain formula_18. Similarly, formula_19 is the maximum (worst) energy possible between rotamer formula_13 of sidechain formula_12 and "any" rotamer X of side chain formula_18.
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 formula_23 that is the "intrinsic" energy of a pair of rotamers formula_24 and formula_25 at positions formula_12 and formula_18, respectively
A given pair of rotamers formula_24 and formula_25 at positions formula_12 and formula_18, respectively, cannot "both" be in the final solution (although one or the other may be) if there is another pair formula_33 and formula_34 that always gives a better energy. Expressed mathematically,
where formula_36, formula_37 and formula_38.
Energy matrices.
For large formula_2, the matrices of precomputed energies can become costly to store. Let formula_2 be the number of amino acid positions, as above, and let formula_41 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 formula_41 entries, so the total number of self-energies to store is formula_43. "Each" pair energy matrix between two positions formula_3 and formula_45, for formula_41 discrete rotamers at each position, requires a formula_47 matrix. This makes the total number of entries in an unreduced pair matrix formula_48. 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 the slowest part of the algorithm (aside from energy calculations). This is a dramatic improvement over the brute-force enumeration which scales as formula_49.
A large-scale benchmark of DEE compared with alternative methods of protein structure prediction and design finds that DEE reliably converges to the optimal solution for protein lengths for which it runs in a reasonable amount of time. It significantly outperforms the alternatives under consideration, which involved techniques derived from mean field theory, genetic algorithms, and the Monte Carlo method. However, the other algorithms are appreciably faster than DEE and thus can be applied to larger and more complex problems; their relative accuracy can be extrapolated from a comparison to the DEE solution within the scope of problems accessible to DEE.
Protein design.
The preceding discussion implicitly assumed that the rotamers formula_3 are all different orientations of the same amino acid side chain. That is, the sequence of the protein was assumed to be fixed. It is also possible to allow multiple side chains to "compete" over a position formula_12 by including both types of side chains in the set of rotamers for that position. This allows a novel sequence to be designed onto a given protein backbone. A short zinc finger protein fold has been redesigned this way. However, this greatly increases the number of rotamers per position and still requires a fixed protein length.
Generalizations.
More powerful and more general criteria have been introduced that improve both the efficiency and the eliminating power of the method for both prediction and design applications. One example is a refinement of the singles elimination criterion known as the Goldstein criterion, which arises from fairly straightforward algebraic manipulation before applying the minimization:
Thus rotamer formula_11 can be eliminated if any alternative rotamer from the set at formula_3 contributes less to the total energy than formula_11. This is an improvement over the original criterion, which requires comparison of the best possible (that is, the smallest) energy contribution from formula_11 with the "worst" possible contribution from an alternative rotamer.
An extended discussion of elaborate DEE criteria and a benchmark of their relative performance can be found in .