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Dynamics of Advantageous Mutant Spread in Spatial Death-Birth and Birth-Death Moran Models

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Abstract

The spread of an advantageous mutation through a population is of fundamental interest in population genetics. While the classical Moran model is formulated for a well-mixed population, it has long been recognized that in real-world applications, the population usually has an explicit spatial structure which can significantly influence the dynamics. In the context of cancer initiation in epithelial tissue, several recent works have analyzed the dynamics of advantageous mutant spread on integer lattices, using the biased voter model from particle systems theory. In this spatial version of the Moran model, individuals first reproduce according to their fitness and then replace a neighboring individual. From a biological standpoint, the opposite dynamics, where individuals first die and are then replaced by a neighboring individual according to its fitness, are equally relevant. Here, we investigate this death-birth analogue of the biased voter model. We construct the process mathematically, derive the associated dual process, establish bounds on the survival probability of a single mutant, and prove that the process has an asymptotic shape. We also briefly discuss alternative birth-death and death-birth dynamics, depending on how the mutant fitness advantage affects the dynamics. We show that birth-death and death-birth formulations of the biased voter model are equivalent when fitness affects the former event of each update of the model, whereas the birth-death model is fundamentally different from the death-birth model when fitness affects the latter event.

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Acknowledgements

This manuscript is dedicated to the memory of Ching-Shan Chou, our friend and colleague. We would like to acknowledge Rick Durrett for his insights on the dual process for the death-birth model during discussions at the Mathematical Biosciences Institute’s thematic program on cancer evolution (2014).

Funding

EBG and JF were supported in part by the NIH grant R01CA241134. EBG and KL were supported in part by the NSF grant CMMI-1552764. JF was supported in part by the NSF grants DMS-1349724 and DMS-2052465. DS was supported in part by the NSF grant CCF-1740761. JF and KL were supported in part by the U.S.-Norway Fulbright Foundation and the Research Council of Norway R &D Grant 309273. EBG was supported in part by the Norwegian Centennial Chair grant and the Doctoral Dissertation Fellowship from the University of Minnesota.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Twin Cities, MN, 55455, USA

    Jasmine Foo & Einar Bjarki Gunnarsson

  2. Department of Industrial and Systems Engineering, University of Minnesota, Twin Cities, MN, 55455, USA

    Einar Bjarki Gunnarsson & Kevin Leder

  3. Departments of Statistics and Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    David Sivakoff

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  1. Jasmine Foo
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  2. Einar Bjarki Gunnarsson
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Correspondence to Jasmine Foo or Einar Bjarki Gunnarsson.

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Foo, J., Gunnarsson, E.B., Leder, K. et al. Dynamics of Advantageous Mutant Spread in Spatial Death-Birth and Birth-Death Moran Models. Commun. Appl. Math. Comput. 6, 576–604 (2024). https://doi.org/10.1007/s42967-023-00278-6

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