# Single-cell mutational burden distributions in birth–death processes

## TL;DR

The paper introduces a new mathematical framework to understand how mutations accumulate in tumors by connecting different mutation statistics from bulk and single-cell data.

## Contribution

The study introduces dynamical matrices and recurrence relations to unify and analyze site frequency spectra, division distributions, and mutational burden distributions in tumor evolution.

## Key findings

- A new expression for the single-cell mutational burden distribution (MBD) was derived.
- The MBD is primarily influenced by stochasticity in cell division rather than mutation rates per division.
- The framework successfully connects bulk and single-cell mutation statistics using dynamic matrices.

## Abstract

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.

Somatic mutations accumulated in tissue growth and maintenance lead to genetic variation in tumours and healthy tissues. The patterns of those mutations have been used to reveal tumour history. Here, we developed a general framework to unite different statistical properties of mutation distributions between bulk sequencing data and single-cell data. The site frequency spectra from bulk data, division distributions and single-cell mutational burden distributions from single-cell data can be connected using dynamic matrices and recurrence relations. Counter-intuitively, the stochasticity in the number of mutations acquired in each cell division does not play a critical role in the single-cell mutation burden distribution.