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) from bulk or single-cell data, as well as single-cell division distributions (DD) and mutational burden distributions (MBD). An integrated understanding of these distributions obtained from different sequencing information is important to illuminate the ecological and evolutionary dynamics of tumours, and requires novel mathematical and computational tools. We introduce dynamical matrices to analyse and unite the SFS, DD and MBD based on a birth-death process. Using the Markov nature of the model, we derive recurrence relations for the expectations of these three distributions. While recovering classic exact results in pure-birth cases for the SFS and the DD through our new framework, we also derive a new expression for the MBD as well as approximations for all three distributions when death is introduced, confirming our results with stochastic simulations. Moreover, we demonstrate a natural link between the SFS and the single-cell MBD, and show that the MBD can be regenerated through the DD. Surprisingly, 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.