Lattice and Off-Lattice Side Chain Models of Protein Folding: Linear Time Structure Prediction Better than 86% of Optimal

This paper considers the protein energy minimization problem for lattice and off-lattice protein folding models that explicitly represent side chains. Lattice models of proteins have proven useful tools for reasoning about protein folding in unrestricted continuous space through analogy. This paper provides the first illustration of how rigorous algorithmic analyses of lattice models can lead to rigorous algorithmic analyses of off-lattice models. We consider two side chain models: a lattice model that generalizes the HP model (Dill, 1985) to explicitly represent side chains on the cubic lattice and a new off-lattice model, the HP Tangent Spheres Side Chain model (HP-TSSC), that generalizes this model further by representing the backbone and side chains of proteins with tangent spheres. We describe algorithms with mathematically guaranteed error bounds for both of these models. In particular, we describe a linear time performance guaranteed approximation algorithm for the HP side chain model that constructs conformations whose energy is better than 86% of optimal in a face-centered cubic lattice, and we demonstrate how this provides a better than 70% performance guarantee for the HP-TSSC model. Our analysis provides a mathematical methodology for transferring performance guarantees on lattices to off-lattice models. These results partially answer the open question of Ngo et al. (1994) concerning the complexity of protein folding models that include side chains.