Sampling from T cell receptor repertoires

Modern single-cell sequencing techniques allow the unique TCR signature of each of a sample of hundreds of T cells to be read. The mathematical challenge is to extrapolate from the properties of a sample to those of the whole repertoire of an individual, made up of many millions of T cells. We consider the distribution of the number of repeats of any TCR in a sample, the mean number of samples needed to find a repeat with probability one half, and the relationship between the true distribution of clonal sizes and that experimentally observed in the sample. We consider two special cases, where the distribution of clonal sizes is geometric, and where a subset of clones in the repertoire is expanded.