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A protein's amino acid sequence determines both its chemical and its physical structures, and together these two structures determine its function. Protein designers seek new amino acid sequences with chemical and physical structures capable of performing some function. The vast size of sequence space frustrates efforts to find useful sequences. Protein designers model proteins on computers and search through amino acid sequence space computationally. They represent the three-dimensional structures for the sequences they examine, specifying the location of each atom, and evaluate the stability of these structures. Good structures are tightly packed but are free of collisions. Designers seek a sequence with a stable structure that meets the geometric and chemical requirements to function as desired; they frame their search as an optimization problem. In this dissertation, I present a graphical model of the central optimization problem in protein design, the side-chain-placement problem. This model allows the formulation of a dynamic programming solution, thus connecting side-chain placement with the class of NP-complete problems for which certain instances admit polynomial time solutions. Moreover, the graphical model suggests a natural data structure for storing the energies used in design. With this data structure, I have created an extensible framework for the representation of energies during side-chain-placement optimization and have incorporated this framework into the Rosetta molecular modeling program. I present one extension that incorporates a new degree of structural variability into the optimization process. I present another extension that includes a non-pairwise decomposable energy function, the first of its kind in protein design, laying the ground-work to capture aspects of protein stability that could not previously be incorporated into the optimization of side-chain placement.