Protein chain pair simplification under the discrete Fréchet distance

For protein structure alignment and comparison, a lot of work has been done using RMSD as the distance measure, which has drawbacks under certain circumstances. Thus, the discrete Fréchet distance was recently applied to the problem of protein (backbone) structure alignment and comparison with promising results. For this problem, visualization is also important because protein chain backbones can have as many as 500-600 $(\alpha)$-carbon atoms, which constitute the vertices in the comparison. Even with an excellent alignment, the similarity of two polygonal chains can be difficult to visualize unless the chains are nearly identical. Thus, the chain pair simplification problem (CPS-3F) was proposed in 2008 to simultaneously simplify both chains with respect to each other under the discrete Fréchet distance. The complexity of CPS-3F is unknown, so heuristic methods have been developed. Here, we define a variation of CPS-3F, called the constrained CPS-3F problem ($({\rm CPS\hbox{-}3F}^+)$), and prove that it is polynomially solvable by presenting a dynamic programming solution, which we then prove is a factor-2 approximation for CPS-3F. We then compare the $({\rm CPS\hbox{-}3F}^+)$ solutions with previous empirical results, and further demonstrate some of the benefits of the simplified comparisons. Chain pair simplification based on the Hausdorff distance (CPS-2H) is known to be NP-complete, and here we prove that the constrained version ($(\rm CPS\hbox{-}2H^+)$) is also NP-complete. Finally, we discuss future work and implications along with a software library implementation, named the Fréchet-based Protein Alignment & Comparison Toolkit (FPACT).