Applications of cancer biology to statistics and combinatorics

I. Single cell microscopy enables individual cells to be tracked as they divide, giving detailed information on the lifetimes of single cells, and the identity if their daughter cells. This gives detailed information that we can use to describe the growth of a colony of cells. We use this information to parameterize a model of cell homeostasis in epithelial tissue, and describe how these models may be perturbed in pre-cancerous lesions. This work has led to an elegant unification of two old statistical problems. The Kolmogorov Master equation describes branching processes and has seen many applications ranging from queues to chemical equations. The McKendrick equation describes age-dependent populations and has seen many applications in demography. We describe a hierarchy of equations that incorporates both systems.

II. Genomic rearrangements are a major source of variation in cancer genomes. We firstly examine next generation sequencing data and describe some of the problems associated with (i) describing how DNA segments combine into chromosomes, and (ii) the evolutionary steps that have taken place to produce the observed data. We secondly describe some combinatorial patterns that emerge from these processes, which tell us that even with perfect sequence data, we still cannot necessarily describe how the genomes have evolved.