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Phase-Type Distribution Approximations of the Waiting Time Until Coordinated Mutations Get Fixed in a Population

  • Ola Hössjer
  • Günter Bechly
  • Ann Gauger
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 271)

Abstract

In this paper we study the waiting time until a number of coordinated mutations occur in a population that reproduces according to a continuous time Markov process of Moran type. It is assumed that any individual can have one of \(m+1\) different types, numbered as \(0,1,\ldots ,m\), where initially all individuals have the same type 0. The waiting time is the time until all individuals in the population have acquired type m, under different scenarios for the rates at which forward mutations \(i\rightarrow i+1\) and backward mutations \(i\rightarrow i-1\) occur, and the selective fitness of the mutations. Although this waiting time is the time until the Markov process reaches its absorbing state, the state space of this process is huge for all but very small population sizes. The problem can be simplified though if all mutation rates are smaller than the inverse population size. The population then switches abruptly between different fixed states, where one type at a time dominates. Based on this, we show that phase-type distributions can be used to find closed form approximations for the waiting time law. Our results generalize work by Schweinsberg [60] and Durrett et al. [20], and they have numerous applications. This includes onset and growth of cancer for a cell population within a tissue, with type representing the severity of the cancer. Another application is temporal changes of gene expression among the individuals in a species, with type representing different binding sites that appear in regulatory sequences of DNA.

Keywords

Coordinated mutations Fixed state population Moran model Phase-type distribution Waiting time 

Notes

Acknowledgements

The authors wish to thank an anonymous reviewer for several helpful suggestions that improved the clarity and presentation of the paper.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden
  2. 2.Biologic InstituteRedmondUSA

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