A probabilistic view on the deterministic mutation–selection equation: dynamics, equilibria, and ancestry via individual lines of descent

Abstract

We reconsider the deterministic haploid mutation–selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation’s solution (and, in particular, of its equilibria) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.

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Acknowledgements

It is our pleasure to thank Anton Wakolbinger and Ute Lenz for stimulating and fruitful discussions. Furthermore, we are grateful to Anton Wakolbinger, Jay Taylor, and an unknown referee for helpful comments on the manuscript. This project received financial support from Deutsche Forschungsgemeinschaft (Priority Programme SPP 1590 Probabilistic Structures in Evolution, Grant No. BA 2469/5-1, and CRC 1283 Taming Uncertainty, Project C1).

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Correspondence to Ellen Baake.

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Baake, E., Cordero, F. & Hummel, S. A probabilistic view on the deterministic mutation–selection equation: dynamics, equilibria, and ancestry via individual lines of descent. J. Math. Biol. 77, 795–820 (2018). https://doi.org/10.1007/s00285-018-1228-8

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Keywords

  • Mutation–selection equation
  • Pruned lookdown ancestral selection graph
  • Killed ancestral selection graph
  • Error threshold

Mathematics Subject Classification

  • 92D15
  • 60J28
  • 60J75
  • 05C80