The general recombination equation in continuous time and its solution

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January 2016, 36(1): 63-95. doi: 10.3934/dcds.2016.36.63

The general recombination equation in continuous time and its solution

Ellen Baake ^1, , Michael Baake ^2, and Majid Salamat ^1,

1.	Technische Fakultät, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany, Germany
2.	Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld

Received August 2014 Revised March 2015 Published June 2015

Abstract
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The process of recombination in population genetics, in its deterministic limit, leads to a nonlinear ODE in the Banach space of finite measures on a locally compact product space. It has an embedding into a larger family of nonlinear ODEs that permits a systematic analysis with lattice-theoretic methods for general partitions of finite sets. We discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and establish a connection with an ancestral partitioning process, backward in time. We solve the finite-dimensional problem recursively for generic sets of parameters and briefly discuss the singular cases, and how to extend the solution to this situation.

Keywords: population genetics, lattice of partitions, Nonlinear recombination equation, measure-valued ODE, recursive solution..

Mathematics Subject Classification: 34G20, 06B23, 92D1.

Citation: Ellen Baake, Michael Baake, Majid Salamat. The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 63-95. doi: 10.3934/dcds.2016.36.63

References:

[1]	M. Aigner, Combinatorial Theory, reprint,, Springer, (1997). doi: 10.1007/978-3-642-59101-3. Google Scholar
[2]	H. Amann, Gewöhnliche Differentialgleichungen,, 2nd ed., (1995). Google Scholar
[3]	E. Baake, Deterministic and stochastic aspects of single-crossover recombination,, in: Proceedings of the International Congress of Mathematicians, (2010), 3037. Google Scholar
[4]	E. Baake and I. Herms, Single-crossover dynamics: Finite versus infinite populations,, Bull. Math. Biol., 70 (2008), 603. doi: 10.1007/s11538-007-9270-5. Google Scholar
[5]	M. Baake, Recombination semigroups on measure spaces,, Monatsh. Math., 146 (2005), 267. doi: 10.1007/s00605-005-0326-z. Google Scholar
[6]	M. Baake and E. Baake, An exactly solved model for mutation, recombination and selection,, Can. J. Math., 55 (2003), 3. doi: 10.4153/CJM-2003-001-0. Google Scholar
[7]	E. Baake and T. Hustedt, Moment closure in a Moran model with recombination,, Markov Proc. Rel. Fields, 17 (2011), 429. Google Scholar
[8]	E. Baake and U. von Wangenheim, Single-crossover recombination and ancestral recombination trees,, J. Math. Biol., 68 (2014), 1371. doi: 10.1007/s00285-013-0662-x. Google Scholar
[9]	M. Baake and R. Speicher, in, preparation., (). Google Scholar
[10]	J. H. Bennett, On the theory of random mating,, Ann. Human Gen., 18 (1954), 311. Google Scholar
[11]	R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation,, Wiley, (2000). Google Scholar
[12]	K. J. Dawson, The decay of linkage disequilibrium under random union of gametes: How to calculate Bennett's principal components,, Theor. Popul. Biol., 58 (2000), 1. doi: 10.1006/tpbi.2000.1471. Google Scholar
[13]	K. J. Dawson, The evolution of a population under recombination: How to linearise the dynamics,, Lin. Alg. Appl., 348 (2002), 115. doi: 10.1016/S0024-3795(01)00586-9. Google Scholar
[14]	R. Durrett, Probability Models for DNA Sequence Evolution,, 2nd ed., (2008). doi: 10.1007/978-0-387-78168-6. Google Scholar
[15]	M. Esser, S. Probst and E. Baake, Partitioning, duality, and linkage disequilibria in the Moran model with recombination,, submitted, (). Google Scholar
[16]	W. J. Ewens and G. Thomson, Properties of equilibria in multi-locus genetic systems,, Genetics, 87 (1977), 807. Google Scholar
[17]	W. Feller, An Introduction to Probability Theory and Its Applications,, Vol. I, (1986). doi: 10.1063/1.3062516. Google Scholar
[18]	H. Geiringer, On the probability theory of linkage in Mendelian heredity,, Ann. Math. Stat., 15 (1944), 25. doi: 10.1214/aoms/1177731313. Google Scholar
[19]	Y. Lyubich, Mathematical Structures in Population Genetics,, Springer, (1992). doi: 10.1007/978-3-642-76211-6. Google Scholar
[20]	T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak selection,, J. Math. Biol., 38 (1999), 103. doi: 10.1007/s002850050143. Google Scholar
[21]	J. R. Norris, Markov Chains,, Cambridge University Press, (1998). Google Scholar
[22]	O. Redner and M. Baake, Unequal crossover dynamics in discrete and continuous time,, J. Math. Biol., 49 (2004), 201. doi: 10.1007/s00285-004-0273-7. Google Scholar
[23]	N. J. A. Sloane, The On-line encyclopedia of integer sequences,, Lecture Notes in Computer Science, 4573* (2007). doi: 10.1007/978-3-540-73086-6_12. Google Scholar*
[24]	E. D. Sontag, Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction,, IEEE Trans. Automatic Control, 46 (2001), 1028. doi: 10.1109/9.935056. Google Scholar
[25]	E. Spiegel and C. J. O'Donnell, Incidence Algebras,, Marcel Dekker, (1997). Google Scholar
[26]	U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in discrete time,, J. Math. Biol., 60 (2010), 727. doi: 10.1007/s00285-009-0277-4. Google Scholar

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References:

[1]	M. Aigner, Combinatorial Theory, reprint,, Springer, (1997). doi: 10.1007/978-3-642-59101-3. Google Scholar
[2]	H. Amann, Gewöhnliche Differentialgleichungen,, 2nd ed., (1995). Google Scholar
[3]	E. Baake, Deterministic and stochastic aspects of single-crossover recombination,, in: Proceedings of the International Congress of Mathematicians, (2010), 3037. Google Scholar
[4]	E. Baake and I. Herms, Single-crossover dynamics: Finite versus infinite populations,, Bull. Math. Biol., 70 (2008), 603. doi: 10.1007/s11538-007-9270-5. Google Scholar
[5]	M. Baake, Recombination semigroups on measure spaces,, Monatsh. Math., 146 (2005), 267. doi: 10.1007/s00605-005-0326-z. Google Scholar
[6]	M. Baake and E. Baake, An exactly solved model for mutation, recombination and selection,, Can. J. Math., 55 (2003), 3. doi: 10.4153/CJM-2003-001-0. Google Scholar
[7]	E. Baake and T. Hustedt, Moment closure in a Moran model with recombination,, Markov Proc. Rel. Fields, 17 (2011), 429. Google Scholar
[8]	E. Baake and U. von Wangenheim, Single-crossover recombination and ancestral recombination trees,, J. Math. Biol., 68 (2014), 1371. doi: 10.1007/s00285-013-0662-x. Google Scholar
[9]	M. Baake and R. Speicher, in, preparation., (). Google Scholar
[10]	J. H. Bennett, On the theory of random mating,, Ann. Human Gen., 18 (1954), 311. Google Scholar
[11]	R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation,, Wiley, (2000). Google Scholar
[12]	K. J. Dawson, The decay of linkage disequilibrium under random union of gametes: How to calculate Bennett's principal components,, Theor. Popul. Biol., 58 (2000), 1. doi: 10.1006/tpbi.2000.1471. Google Scholar
[13]	K. J. Dawson, The evolution of a population under recombination: How to linearise the dynamics,, Lin. Alg. Appl., 348 (2002), 115. doi: 10.1016/S0024-3795(01)00586-9. Google Scholar
[14]	R. Durrett, Probability Models for DNA Sequence Evolution,, 2nd ed., (2008). doi: 10.1007/978-0-387-78168-6. Google Scholar
[15]	M. Esser, S. Probst and E. Baake, Partitioning, duality, and linkage disequilibria in the Moran model with recombination,, submitted, (). Google Scholar
[16]	W. J. Ewens and G. Thomson, Properties of equilibria in multi-locus genetic systems,, Genetics, 87 (1977), 807. Google Scholar
[17]	W. Feller, An Introduction to Probability Theory and Its Applications,, Vol. I, (1986). doi: 10.1063/1.3062516. Google Scholar
[18]	H. Geiringer, On the probability theory of linkage in Mendelian heredity,, Ann. Math. Stat., 15 (1944), 25. doi: 10.1214/aoms/1177731313. Google Scholar
[19]	Y. Lyubich, Mathematical Structures in Population Genetics,, Springer, (1992). doi: 10.1007/978-3-642-76211-6. Google Scholar
[20]	T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak selection,, J. Math. Biol., 38 (1999), 103. doi: 10.1007/s002850050143. Google Scholar
[21]	J. R. Norris, Markov Chains,, Cambridge University Press, (1998). Google Scholar
[22]	O. Redner and M. Baake, Unequal crossover dynamics in discrete and continuous time,, J. Math. Biol., 49 (2004), 201. doi: 10.1007/s00285-004-0273-7. Google Scholar
[23]	N. J. A. Sloane, The On-line encyclopedia of integer sequences,, Lecture Notes in Computer Science, 4573* (2007). doi: 10.1007/978-3-540-73086-6_12. Google Scholar*
[24]	E. D. Sontag, Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction,, IEEE Trans. Automatic Control, 46 (2001), 1028. doi: 10.1109/9.935056. Google Scholar
[25]	E. Spiegel and C. J. O'Donnell, Incidence Algebras,, Marcel Dekker, (1997). Google Scholar
[26]	U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in discrete time,, J. Math. Biol., 60 (2010), 727. doi: 10.1007/s00285-009-0277-4. Google Scholar

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