Tandem mass spectrometry is widely used in proteomic studies because of its ability to identify large numbers of peptides from complex mixtures. In a typical LC-MS/MS experiment, thousands of tandem mass spectra will be collected and peptide identification algorithms are of great importance to translate them into peptide sequences. Though these spectra contain both m/z and intensity values, most popular protein identification algorithms primarily use predicted fragment m/z values to assign peptide sequences to fragmentation spectra. The intensity information is often undervalued, because it is not as easy to predict and incorporate into algorithms. Nevertheless, the use of intensity to assist peptide identification is an attractive prospect and can potentially improve the confidence of matches and generate more identifications. In this dissertation, an unsupervised statistical method, K-means clustering, was used to study peptide fragmentation patterns for both CID and ETD data, and many unique fragmentation features were discovered. For instance, strong c(n-1) ions were observed in ETD, indicating that the fragmentation site in ETD is highly related to the amino acid residue location. Based on the fragmentation patterns observed through data mining, a peptide identification algorithm that makes use of these patterns was developed. The program is named SQID and it is the first algorithm in our bioinformatics project. Our testing results using multiple public datasets indicated an improvement in the number of identified peptides compared with popular proteomics algorithms such as Sequest or X!Tandem. SQID was further extended to improve cross-linked peptide identification (SQID-XLink) as well as blind modification identification (SQID-Mod), and both of them showed significant improvement compared with existing methods. In this dissertation the SQID algorithm was also successfully applied to a mosquito proteomics project. We are incorporating new features and new algorithms to our software, such as more fragmentation methods, more accurate spectra prediction and more user-friendly interface. We hope the SQID project can continually benefit researchers and help to improve the data analysis of proteomics community.

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Part I discusses the problem of determining the set of intersection points, with corresponding multiplicities, of two algebraic plane curves. We derive an algorithm based on the Euclidean Algorithm for polynomials and show how to use it to find the intersection points of two given curves. We also show that an easy proof of Bézout’s Theorem follows. We then discuss how, for curves with rational coefficients, this algorithm can bemodified to find the intersection points with coordinates expressed in terms of algebraic extensions of the rational numbers. Part II deals with the problem of determining the (monic) integer transfinite diameter of a given real interval. We show how this problem relates to the problem of determining the structure of the spectrum of normalised leading coefficients of polynomials with integer coefficients and all roots in the given interval. We then find dense regions of this spectrum for a number of intervals and discuss algorithms for finding discrete subsets of the spectrum for the interval [0,1]. This leads to an improvement in the known upper bound for the integer transfinite diameter. Finally, we discuss the connection between the infimum of the spectrum and the monic integer transfinite diameter.