[Paper Review] Single-cell mutational burden distributions in birth-death processes

Christo Morison, Dudley Stark+1 more|arXiv (Cornell University)|Sep 12, 2023

Mathematical Biology Tumor Growth4 citations

TL;DR

This paper introduces a novel dynamical matrix framework to unify the analysis of site frequency spectra (SFS), division distributions (DD), and single-cell mutational burden distributions (MBD) in birth-death processes. By deriving recurrence relations for expectations under neutral evolution, it provides exact expressions for MBD in pure-birth models and approximations when death is included, revealing that MBD is primarily driven by stochasticity in cell division history rather than mutation count variability at division.

ABSTRACT

Genetic mutations are footprints of tumour growth. While mutation data in bulk samples has been used to infer evolutionary parameters hard to measure in vivo, the advent of single-cell data has led to strong interest in the mutational burden distribution (MBD) among tumour cells. We introduce dynamical matrices and recurrence relations to integrate this single-cell MBD with known statistics, and derive new analytical expressions. Surprisingly, we find that the shape of the MBD is driven by cell lineage-level stochasticity rather than by the distribution of mutations in each cell division.

Motivation & Objective

• To develop a unified mathematical framework for analyzing key mutational distributions—SFS, DD, and MBD—in the context of tumour evolution.
• To address the lack of analytical expressions for the single-cell mutational burden distribution (MBD) under neutral evolution with cell death.
• To establish a mechanistic link between MBD and DD, showing that MBD can be regenerated from DD via a binning and rescaling procedure.
• To quantify the relative contributions of division stochasticity versus mutation count stochasticity to MBD shape.
• To provide approximations for all three distributions (SFS, DD, MBD) under general birth-death processes, validated through stochastic simulations.

Proposed method

• The authors employ dynamical matrices to model the time evolution of cell lineage trees under a birth-death process, leveraging the Markov property to derive recurrence relations.
• Key equations include recurrence relations for the expected values of SFS, DD, and MBD, derived using generating functions and moment-generating techniques.
• For the MBD, the paper derives an exact expression in pure-birth (Yule) processes and approximates it under general birth-death dynamics using asymptotic expansions.
• The framework enables conversion between DD and MBD via a histogram-based binning and rescaling procedure, assuming constant mutation rate per division.
• Stochastic simulations are used to validate analytical approximations, particularly for non-zero death rates.
• The method incorporates the infinite sites approximation and assumes neutral mutations, allowing focus on stochastic dynamics of lineage and mutation accumulation.

Figure 1 : (a) Discrete-time Markov chain description of the population size. (b) Growing binary tree representation of an example realisation of the birth-death process with mutations described in the main text, with birth probability $\beta=2/3$ , death probability $\delta=1/3$ , mutational mean $

Experimental results

Research questions

• RQ1 How can the single-cell mutational burden distribution (MBD) be analytically derived under a birth-death process with neutral mutations?
• RQ2 What is the quantitative relationship between the division distribution (DD) and the MBD in clonal tumour evolution?
• RQ3 To what extent is the shape of the MBD determined by stochasticity in cell division history versus stochasticity in mutation counts per division?
• RQ4 Can the MBD be reconstructed from the DD, and under what conditions does this conversion hold?
• RQ5 How do birth and death rates affect the expected MBD, and what approximations are valid for non-zero death rates?

Key findings

• The paper derives an exact analytical expression for the single-cell mutational burden distribution (MBD) in the pure-birth (Yule) process, confirming known results for SFS and DD.
• For general birth-death processes with non-zero death rates, the authors provide a first-order approximation for the MBD, showing that it scales linearly with the expected number of divisions.
• The MBD is found to be primarily driven by stochasticity in the division distribution (DD), not by variation in the number of mutations per division, as confirmed by simulations with non-Poisson mutational distributions.
• A conversion procedure from DD to MBD is established: by binning the MBD into intervals of width μ (mean mutations per division) and summing over μ adjacent bins, the DD can be recovered via rescaling.
• The expected total number of mutational occurrences in cells with i divisions is approximated as 𝔼[C_i] ≈ 2μβ ∑_{i'=1}^{i+1} 1/𝔼[N_{i'}], where β is the birth rate and N_i is the expected population size at time i.
• Simulations confirm that the MBD conversion from DD holds under various mutational distributions (e.g., uniform, geometric) with mean μ, but fails when mutation counts have small or degenerate support (e.g., delta distribution).

Figure 2 : The matrix framework described in the main text, where $i$ refers to the Markov step count (in this example, $i=12$ ). (a) The matrix $Y_{i}$ corresponding to the example realisation of the birth-death process depicted in Figure 1 b, where entry $(n,m)$ is $1$ if cell $n$ possesses mutati