# Error-induced extinction in a multi-type critical birth–death process

## TL;DR

This paper studies how high mutation rates can lead to extinction in growing populations of cells, using a mathematical model.

## Contribution

The paper introduces a multi-type critical birth–death process to model error-induced extinction and derives new asymptotic results for cell type distributions.

## Key findings

- The mass function of cell types has algebraic tails with exponents $\chi_k = 2^{1-k}$.
- The survival probability decays as a power law with exponents $\xi_k$.
- The model shows how mutation rates can lead to certain extinction in biological populations.

## Abstract

Extreme mutation rates in microbes and cancer cells can result in error-induced extinction (EEX), where every descendant cell eventually acquires a lethal mutation. In this work, we investigate critical birth–death processes with n distinct types as a birth–death model of EEX in a growing population. Each type-i cell divides independently \documentclass[12pt]{minimal}
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\begin{document}$$(i)\rightarrow (i)+(i)$$\end{document}(i)→(i)+(i) or mutates \documentclass[12pt]{minimal}
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\begin{document}$$(i)\rightarrow (i+1)$$\end{document}(i)→(i+1) at the same rate. The total number of cells grows exponentially as a Yule process until a cell of type-n appears, which cell type can only divide or die at rate one. This makes the whole process critical and hence after the exponentially growing phase eventually all cells die with probability one. We present large-time asymptotic results for the general n-type critical birth–death process. We find that the mass function of the number of cells of type-k has algebraic and stationary tail \documentclass[12pt]{minimal}
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\begin{document}$$(\text {size})^{-1-\chi _k}$$\end{document}(size)-1-χk, with \documentclass[12pt]{minimal}
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\begin{document}$$\chi _k=2^{1-k}$$\end{document}χk=21-k, for \documentclass[12pt]{minimal}
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\begin{document}$$k=2,\dots ,n$$\end{document}k=2,⋯,n, in sharp contrast to the exponential tail of the first type. The same exponents describe the tail of the asymptotic survival probability \documentclass[12pt]{minimal}
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\begin{document}$$(\text {time})^{-\xi _k}$$\end{document}(time)-ξk. We present applications of the results for studying extinction due to intolerable mutation rates in biological populations.