# Filling a Protein Scaffold with a Reference

## Abstract

In mass spectrometry-based de novo protein sequencing, it is hard to complete the sequence of the whole protein. Motivated by this we study the (one-sided) problem of filling a protein *scaffold* \(\mathcal{S}\) with some missing amino acids, given a sequence of contigs none of which is allowed to be altered, with respect to a complete reference protein \(\mathcal{P}\) of length *n*, such that the BLOSUM62 score between \(\mathcal{P}\) and the filled sequence \(\mathcal{S}'\) is maximized. We show that this problem is polynomial-time solvable in \(O(n^{26})\) time. We also consider the case when the contigs are not of high quality and they are concatenated into an (incomplete) sequence \(\mathcal{I}\), where the missing amino acids can be inserted anywhere in \(\mathcal{I}\) to obtain \(\mathcal{I}'\), such that the BLOSUM62 score between \(\mathcal{P}\) and \(\mathcal{I}'\) is maximized. We show that this problem is polynomial-time solvable in \(O(n^{22})\) time. Due to the high running time, both of these algorithms are impractical, we hence present several algorithms based on greedy and local search, trying to solve the problems practically. The empirical results show that the algorithms can fill protein scaffolds almost perfectly, provided that a good pair of scaffold and reference are given.

## Keywords

Matched Pair Protein Scaffold Tandem Mass Spectrum Local Search Method Practical Algorithm## Notes

### Acknowledgments

This research is partially supported by NSF of China under grant 60928006 and by the Opening Fund of Top Key Discipline of Computer Software and Theory in Zhejiang Provincial Colleges at Zhejiang Normal University. We also thank anonymous reviewers for several useful comments.

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