Closed-form stochastic solutions for non-equilibrium dynamics and inheritance of cellular components over many cell divisions

TL;DR

This paper develops a mathematical framework with closed-form solutions to model stochastic inheritance of cellular components over multiple cell divisions, applicable to processes like mitochondrial DNA dynamics.

Contribution

It introduces a novel formalism for describing stochastic inheritance in dividing cells, providing exact solutions for various partitioning and change mechanisms.

Findings

Closed-form generating functions derived for diverse inheritance scenarios.

Solutions validated against stochastic simulations.

Applicable to mitochondrial DNA dynamics during development.

Abstract

Stochastic dynamics govern many important processes in cellular biology, and an underlying theoretical approach describing these dynamics is desirable to address a wealth of questions in biology and medicine. Mathematical tools exist for treating several important examples of these stochastic processes, most notably gene expression, and random partitioning at single cell divisions or after a steady state has been reached. Comparatively little work exists exploring different and specific ways that repeated cell divisions can lead to stochastic inheritance of unequilibrated cellular populations. Here we introduce a mathematical formalism to describe cellular agents that are subject to random creation, replication, and/or degradation, and are inherited according to a range of random dynamics at cell divisions. We obtain closed-form generating functions describing systems at any time after any number of cell divisions for binomial partitioning and divisions provoking a deterministic or random, subtractive or additive change in copy number, and show that these solutions agree exactly with stochastic simulation. We apply this general formalism to several example problems involving the dynamics of mitochondrial DNA (mtDNA) during development and organismal lifetimes.