A Polynomial Time Solution for Protein Chain Pair Simplification under the Discrete Fréchet Distance
The comparison and simplification of polygonal chains is an important and active topic in many areas of research. In the study of protein structure alignment and comparison, a lot of work has been done using RMSD as the distance measure. This method has certain drawbacks, and thus recently, the discrete Fréchet distance was applied to the problem of protein (backbone) structure alignment and comparison with promising results. Another important area within protein structure research is visualization, due to the number of nodes along each backbone. Protein chain backbones can have as many as 500~600 α-carbon atoms which constitute the vertices in the comparison. Even with an excellent alignment, the similarity of two polygonal chains can be very difficult to see visually unless the two chains are nearly identical. To address this issue, 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. It is unknown whether CPS-3F is NP-complete, and so heuristic methods have been developed. Here, we first define a version of CPS-3F, denoted CPS-3F + , and prove that it is polynomially solvable by presenting a dynamic programming solution. Then we compare the CPS-3F + solutions with previous empirical results, and further demonstrate some of the benefits of the simplified comparison. Finally, we discuss future work and implications along with a web-based software implementation, named FPACT (The Fréchet-based Protein Alignment & Comparison Tool), allowing users to align, simplify, and compare protein backbone chains using methods based on the discrete Fréchet distance.
KeywordsStructure Alignment Protein Backbone Polygonal Chain Protein Structure Comparison Polynomial Time Solution
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