ORIGINAL PAPERS

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

Expand
  • 1. School of Mathematics, University of Minnesota, Twin Cities, MN, 55455, USA;
    2. Department of Industrial and Systems Engineering, University of Minnesota, Twin Cities, MN, 55455, USA;
    3. Departments of Statistics and Mathematics, The Ohio State University, Columbus, OH, 43210, USA

Received date: 2022-09-12

  Revised date: 2023-02-28

  Online published: 2024-04-16

Supported by

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.

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.

Cite this article

Jasmine Foo, Einar Bjarki Gunnarsson, Kevin Leder, David Sivakoff . Dynamics of Advantageous Mutant Spread in Spatial Death-Birth and Birth-Death Moran Models[J]. Communications on Applied Mathematics and Computation, 2024 , 6(1) : 576 -604 . DOI: 10.1007/s42967-023-00278-6

References

[1] Allen, B., Sample, C., Jencks, R., Withers, J., Steinhagen, P., Brizuela, L., Kolodny, J., Parke, D., Lippner, G., Dementieva, Y.A.:Transient amplifiers of selection and reducers of fixation for death-birth updating on graphs. PLoS Comput. Biol. 16(1), e1007529 (2020)
[2] Antal, T., Redner, S., Sood, V.:Evolutionary dynamics on degree-heterogeneous graphs. Phys. Rev. Lett. 96(18), 188104 (2006)
[3] Armitage, P., Doll, R.:The age distribution of cancer and a multi-stage theory of carcinogenesis. Br. J. Cancer 8(1), 1-12 (1954)
[4] Armitage, P., Doll, R.:A two-stage theory of carcinogenesis in relation to the age distribution of human cancer. Br. J. Cancer 11(2), 161-169 (1957)
[5] Bozic, I., Antal, T., Ohtsuki, H., Carter, H., Kim, D., Chen, S., Karchin, R., Kinzler, K.W., Vogelstein, B., Nowak, M.A.:Accumulation of driver and passenger mutations during tumor progression. Proc. Natl. Acad. Sci. 107(43), 18545-18550 (2010)
[6] Bramson, M., Griffeath, D.:On the Williams-Bjerknes tumour growth model. II. Math. Proc. Camb. Philos. Soc. 88(2), 339-357 (1980)
[7] Bramson, M., Griffeath, D.:On the Williams-Bjerknes tumor growth model I. Ann. Probab. 9, 173-185 (1981)
[8] Brock, C.K., Wallin, S.T., Ruiz, O.E., Samms, K.M., Mandal, A., Sumner, E.A.:Stem cell proliferation is induced by apoptotic bodies from dying cells during epithelial tissue maintenance. Nat. Commun. 10(1), 1-11 (2019)
[9] Broom, M., Rychtář, J.:An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc. R. Soc. A Math. Phys. Eng. Sci. 464(2098), 2609-2627 (2008)
[10] Durrett, R.:Lecture Notes on Particle Systems and Percolation. Brooks/Cole Pub Co, Pacific Grove (1988)
[11] Durrett, R.:Ten lectures on particle systems. In:Lectures on Probability Theory (Saint-Flour, 1993), Lecture Notes in Math., vol. 1608, pp. 97-201. Springer, Berlin (1995)
[12] Durrett, R.:Probability Models for DNA Sequence Evolution. Springer Science & Business Media, New York (2008)
[13] Durrett, R.:Probability-Theory and Examples, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 49, 5th edn. Cambridge University Press, Cambridge (2019)
[14] Durrett, R., Foo, J., Leder, K.:Spatial Moran models, II:cancer initiation in spatially structured tissue. J. Math. Biol. 72(5), 1369-1400 (2016)
[15] Durrett, R., Griffeath, D.:Contact processes in several dimensions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 59, 535-552 (1982)
[16] Durrett, R., Moseley, S.:Spatial Moran models I. Stochastic tunneling in the neutral case. Ann. Appl. Probab. 25(1), 104-115 (2015)
[17] Foo, J., Gunnarsson, E.B., Leder, K., Storey, K.:Spread of premalignant mutant clones and cancer initiation in multilayered tissue. Ann. Appl. Probab. 33(1), 299-343 (2023)
[18] Foo, J., Leder, K., Ryser, M.D.:Multifocality and recurrence risk:a quantitative model of field cancerization. J. Theoret. Biol. 355, 170-184 (2014)
[19] Foo, J., Leder, K., Schweinsberg, J.:Mutation timing in a spatial model of evolution. Stoch. Process. Appl. 130(10), 6388-6413 (2020)
[20] Fuchs, Y., Steller, H.:Live to die another way:modes of programmed cell death and the signals emanating from dying cells. Nat. Rev. Mol. Cell Biol. 16(6), 329-344 (2015)
[21] Gray, L.:Duality for general attractive spin systems with applications in one dimension. Ann. Probab. 14(2), 371-396 (1986)
[22] Hindersin, L., Traulsen, A.:Most undirected random graphs are amplifiers of selection for birth-death dynamics, but suppressors of selection for death-birth dynamics. PLoS Comput. Biol. 11(11), e1004437 (2015)
[23] Kaveh, K., Komarova, N.L., Kohandel, M.:The duality of spatial death-birth and birth-death processes and limitations of the isothermal theorem. R. Soc. Open Sci. 2(4), 140465 (2015)
[24] Kimura, M., Weiss, G.H.:The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49(4), 561 (1964)
[25] Knudson, A.G.:Mutation and cancer:statistical study of retinoblastoma. Proc. Natl. Acad. Sci. 68(4), 820-823 (1971)
[26] Knudson, A.:Two genetic hits (more or less) to cancer. Nat. Rev. Cancer 1(2), 157-161 (2001)
[27] Komarova, N.L.:Spatial stochastic models for cancer initiation and progression. Bull. Math. Biol. 68(7), 1573-1599 (2006)
[28] Lieberman, E., Hauert, C., Nowak, M.A.:Evolutionary dynamics on graphs. Nature 433(7023), 312-316 (2005)
[29] Loomis, L.H., Whitney, H.:An inequality related to the isoperimetric inequality. Bull. Am. Math. Soc. 55(10), 961-962 (1949)
[30] Maruyama, T.:On the fixation probability of mutant genes in a subdivided population. Genet. Res. 15(2), 221-225 (1970)
[31] Maruyama, T.:A simple proof that certain quantities are independent of the geographical structure of population. Theor. Popul. Biol. 5, 148-154 (1974)
[32] Masuda, N., Ohtsuki, H.:Evolutionary dynamics and fixation probabilities in directed networks. New J. Phys. 11(3), 033012 (2009)
[33] Moran, P.A.P.:Random processes in genetics. In:Mathematical Proceedings of the Cambridge Philosophical Society, vol. 54, pp. 60-71. Cambridge University Press, Cambridge (1958)
[34] NIH National Cancer Institute.:SEER training cancer classification. https://training.seer.cancer.gov/disease/categories/classification.html
[35] Richter, H.:Spectral analysis of transient amplifiers for death-birth updating constructed from regular graphs. J. Math. Biol. 82(7), 61 (2021)
[36] Sharma, N., Traulsen, A.:Suppressors of fixation can increase average fitness beyond amplifiers of selection. Proc. Natl. Acad. Sci. 119(37), e2205424119 (2022)
[37] Slatkin, M.:Fixation probabilities and fixation times in a subdivided population. Evolution 164(2), 477-488 (1981)
[38] Sood, V., Antal, T., Redner, S.:Voter models on heterogeneous networks. Phys. Rev. E 77(4), 041121 (2008)
[39] Tkadlec, J., Pavlogiannis, A., Chatterjee, K., Nowak, M.A.:Limits on amplifiers of natural selection under death-birth updating. PLoS Comput. Biol. 16(1), e1007494 (2020)
[40] Williams, T., Bjerknes, R.:Stochastic model for abnormal clone spread through epithelial basal layer. Nature 236(5340), 19-21 (1972)
Options
Outlines

/