Appendix of “Computational Protein Design Using AND/OR Branch and Bound Search”

Suppose we try to find the global minimum value of the energy function E(r), in which r ∈ R and R is the conformational space of the rotamers. The BnB algorithm executes two steps recursively. The first step is called branching, in which we split the conformational space R into two or more smaller spaces, i.e., R1, R2, . . . , Rm, where R1 ∪ R2 ∪ · · · ∪ Rm = R. If we are able to find r̂i = argminr∈Ri E(r) for all i ∈ {1, 2, . . . ,m}, we can compute the minimum energy conformation in the conformational space R by identifying one of r̂i that has the lowest energy. The second step of BnB is called bounding. Suppose the current lowest energy conformation is ri. For any sub-space Rj, if we can ensure that the lower bound of the energy of all conformations in Rj is greater than E(ri), we do not need to further search this sub-space, that is, we can prune the whole sub-space Rj safely. The lower bound of the energy of the conformations in a given space usually can be computed based on some heuristic functions. The BnB algorithm generally performs the branching step recursively until the current conformational space contains only one single conformation. The space generated from the branching step can form a BnB search tree, in which the union of sub-spaces represented by the children of a node covers the whole conformational space of this node. In the protein design problem, we can split a conformational space by assigning a particular rotamer in a residue. For each node in the search tree, the bounding step is applied to prune some branches and thus shorten the search time. Fig. A1 shows an example of the traditional branch-and-bound (BnB) search tree. In order to traverse the BnB search tree, a queue Q is often used to store the nodes to be expanded. Initially, Q only contains the node representing the whole conformational space. Also, we maintain a global variable u to store the current lowest energy value. Initially, u can be initialized to the energy of any conformation. In practice, we often use a stochastic local search algorithm to generate a local optimal conformation so that it can be used to prune more nodes at the beginning of the search process. BnB can be executed by looping the following steps until Q becomes empty: