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The mutational catalog of a cancer genome, defined over an alphabet of mutation types Ξ, can be mathematically expressed as, m g, a mapping from a genome g and finite alphabet of mutation types Ξ to a specific nonnegative K-tuple.
The mutational catalog of a cancer genome g defined over the alphabet of mutation types Ξ is represented by m g : Ξ → N 0 K.
NMF can estimate the contribution of each mutational signature to the mutational catalog of each breast cancer.
For example, the mutational catalog of a lung cancer in a tobacco smoker will carry the signature of ∼60 chemicals that bind and mutate DNA (Pleasance et al., 2010).
The mutational catalog of the cancer genome formed by these three processes, however, does not have any notable or specific features and does not obviously resemble any of the mutational signatures that generated it.
In our model, the mutational catalog of a cancer genome is examined as a linear superposition of the signatures of the mutational processes operative in this genome and their respective exposures.
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I curated the majority of publicly available data and compiled a data set encompassing ~5 million somatic mutations from the mutational catalogs of 7042 primary cancers of 30 different classes.
We recently described five mutational signatures derived from the 96 possible mutated trinucleotides within the mutational catalogs of 21 whole breast cancer genomes, named Signatures A E (Nik-Zainal et al., 2012a).
Simulating the mutational catalogs of 50 cancer genomes with different average numbers of mutations indicates that two or three signatures can be effectively extracted from catalogs with very few mutations, whereas extracting seven or more signatures requires an average of at least 1,000 mutations per catalog.
Mathematically, a set of mutational catalogs of cancer genomes could be examined as a linear mixture of unknown numbers of mutational signatures.
The mutational catalogs of these cancer genomes are known from DNA sequencing, and the aim is to identify the patterns of the mutational signatures as well as the number of mutations attributed to each signature in each sample.
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