Single-cell mutational burden distributions in birth-death processes

Genetic mutations are footprints of cancer evolution and reveal critical dynamic parameters of tumour growth, which otherwise are hard to measure in vivo. The mutation accumulation in tumour cell populations has been described by various statistics, such as site frequency spectra (SFS), single-cell division distributions (DD) and mutational burden distributions (MBD). While DD and SFS have been intensively studied in phylogenetics especially after the development of whole genome sequencing technology of bulk samples, MBD has drawn attention more recently with the single-cell sequencing data. Although those statistics all arise from the same somatic evolutionary process, an integrated understanding of these distributions is missing and requires novel mathematical tools to better inform the ecological and evolutionary dynamics of tumours. Here we introduce dynamical matrices to analyse and unite the SFS, DD and MBD and derive recurrence relations for the expectations of these three distributions. While we successfully recover classic exact results in pure-birth cases for the SFS and the DD through our new framework, we derive a new expression for the MBD and approximate all three distributions when death is introduced. We demonstrate a natural link between the SFS and the single-cell MBD, and show that the MBD can be regenerated through the DD. Counter-intuitively, the single-cell MBD is mainly driven by the stochasticity arising in the DD, rather than the extra stochasticity in the number of mutations at each cell division.