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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>)</mml:mo> <mml:mo>→</mml:mo> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> or mutates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>)</mml:mo> <mml:mo>→</mml:mo> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo></mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mo>(</mml:mo> <mml:mtext>size</mml:mtext> <mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msub><mml:mi>χ</mml:mi> <mml:mi>k</mml:mi></mml:msub> </mml:mrow> </mml:msup> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>χ</mml:mi> <mml:mi>k</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:msup><mml:mn>2</mml:mn> <mml:mrow><mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi></mml:mrow> </mml:msup> </mml:mrow> </mml:math> , for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi></mml:mrow> </mml:math> , in sharp contrast to the exponential tail of the first type. The same exponents describe the tail of the asymptotic survival probability <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mo>(</mml:mo> <mml:mtext>time</mml:mtext> <mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mo>-</mml:mo> <mml:msub><mml:mi>ξ</mml:mi> <mml:mi>k</mml:mi></mml:msub> </mml:mrow> </mml:msup> </mml:math> . We present applications of the results for studying extinction due to intolerable mutation rates in biological populations.