×

zbMATH — the first resource for mathematics

Single-crossover recombination and ancestral recombination trees. (English) Zbl 1284.92063
Summary: We consider the Wright-Fisher model for a population of \(N\) individuals, each identified with a sequence of a finite number of sites, and single-crossover recombinations between them. We trace back the ancestry of single individuals from the present population. In the \(N\to\infty\) limit without rescaling of parameters or time, this ancestral process is described by a random tree, whose branching events correspond to the splitting of the sequence due to recombination. With the help of a decomposition of the trees into subtrees, we calculate the probabilities of the topologies of the ancestral trees. At the same time, these probabilities lead to a semi-explicit solution of the deterministic single-crossover equation. The latter is a discrete-time dynamical system that emerges from the Wright-Fisher model via a law of large numbers and has been waiting for a solution for many decades.

MSC:
92D15 Problems related to evolution
92D10 Genetics and epigenetics
60J28 Applications of continuous-time Markov processes on discrete state spaces
60F15 Strong limit theorems
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Baake, E, Mutation and recombination with tight linkage, J Math Biol, 42, 455-488, (2001) · Zbl 1010.92031
[2] Baake M (2005) Recombination semigroups on measure spaces. Monatsh Math 146:267-278 [Addendum 150:83-84 (2007)] · Zbl 1011.92038
[3] Baake M, Baake E (2003) An exactly solved model for mutation, recombination and selection. Can J Math 55:3-41 [Erratum 60:264-265 (2008)] · Zbl 1056.92040
[4] Bennett, JH, On the theory of random mating, Ann Human Genet, 18, 311-317, (1954)
[5] Dawson, KJ, The decay of linkage disequilibria under random union of gametes: how to calculate bennett’s principal components, Theor Popul Biol, 58, 1-20, (2000) · Zbl 1011.92038
[6] Dawson, KJ, The evolution of a population under recombination: how to linearise the dynamics, Linear Algebra Appl, 348, 115-137, (2002) · Zbl 1003.92023
[7] Donnelly, KP, The probability that related individuals share some section of genome identical by descent, Theor Popul Biol, 23, 34-63, (1983) · Zbl 0521.92011
[8] Durrett R (2008) Probability models for DNA sequence evolution, 2nd edn. Springer, New York · Zbl 1311.92007
[9] Ethier SN, Kurtz TG (2005) Markov processes—characterization and convergence. Wiley, New York (reprint)
[10] Ewens W (2004) Mathematical population genetics, Vol. I: Theoretical introduction, 2nd edn. Springer, Berlin · Zbl 1060.92046
[11] Geiringer, H, On the probability theory of linkage in Mendelian heredity, Ann Math Stat, 15, 25-57, (1944) · Zbl 0063.01560
[12] Gill, J; Linusson, S; Moulton, V; Steel, M, A regular decomposition of the edge-product space of phylogenetic trees, Adv Appl Math, 41, 158-176, (2008) · Zbl 1149.05305
[13] Grimmett G, Stirzaker D (2001) Probability and random processes, 3rd edn. Oxford University Press, Oxford · Zbl 1015.60002
[14] Gross J, Yellen J (1999) Graph theory and its applications. CRC Press, Boca Raton · Zbl 0920.05001
[15] Gupta, DK, Generation of binary trees from \((0{-}1)\) codes, Int J Comput Math, 42, 157-162, (1992) · Zbl 0742.68024
[16] Hein J, Schierup MH, Wiuf C (2005) Gene genealogies, variation and evolution: a primer in coalescent theory. Oxford University Press, Oxford (corr. reprint 2006) · Zbl 1113.92048
[17] Kauppi, L; Alec, J; Jeffreys, AL; Keeney, S, Where the crossovers are: recombination distributions in mammals, Nat Rev Genet, 5, 413-424, (2004)
[18] Lyubich YI (1992) Mathematical structures in population genetics. Springer, Berlin · Zbl 0747.92019
[19] McHale, D; Ringwood, GA, Haldane linearisation of baric algebras, J Lond Math Soc, 28, 17-26, (1983) · Zbl 0515.17010
[20] McVean, GAT; Cardin, NJ, Approximating the coalescent with recombination, Philos Trans R Soc B, 360, 1387-1393, (2005)
[21] Proskurowski, A, On the generation of binary trees, J ACM, 27, 1-2, (1980)
[22] Ralph P, Coop G (2013) The geography of recent genetic ancestry across Europe. Submitted; arXiv:1207.3815
[23] Semple C, Steel M (2003) Phylogenetics. Oxford University Press, Oxford · Zbl 1043.92026
[24] Stanley RP (1999) Enumerative combinatorics, vol 2. Cambridge University Press, Cambridge · Zbl 0928.05001
[25] Wakeley J (2008) Coalescent theory: an introduction. Roberts & Company Publishers, Greenwood Village · Zbl 1366.92001
[26] Wiuf, C; Hein, J, On the number of ancestors to a DNA sequence, Genetics, 147, 1459-1468, (1997) · Zbl 0920.90109
[27] Wangenheim, U; Baake, E; Baake, M, Single-crossover recombination in discrete time, J Math Biol, 60, 727-760, (2010) · Zbl 1208.92050
[28] Zaks, S, Lexicographic generation of ordered trees, Theor Comput Sci, 10, 63-82, (1980) · Zbl 0422.05026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.