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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Under a Creative Commons license

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 and Continuous Dynamical Systems, 2016, 36 (1) : 63-95. doi: 10.3934/dcds.2016.36.63

References:

[1]	M. Aigner, Combinatorial Theory, reprint, Springer, Berlin, 1997. doi: 10.1007/978-3-642-59101-3.
[2]	H. Amann, Gewöhnliche Differentialgleichungen, 2nd ed., de Gryuter, Berlin, 1995.
[3]	E. Baake, Deterministic and stochastic aspects of single-crossover recombination, in: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, ed. J. Bhatia, Hindustan Book Agency, New Delhi (2010), 3037-3053.
[4]	E. Baake and I. Herms, Single-crossover dynamics: Finite versus infinite populations, Bull. Math. Biol., 70 (2008), 603-624; arXiv:q-bio/0612024. doi: 10.1007/s11538-007-9270-5.
[5]	M. Baake, Recombination semigroups on measure spaces, Monatsh. Math., 146 (2005), 267-278 and 150 (2007), 83-84 (Addendum); arXiv:math.CA/0506099. doi: 10.1007/s00605-005-0326-z.
[6]	M. Baake and E. Baake, An exactly solved model for mutation, recombination and selection, Can. J. Math., 55 (2003), 3-41 and 60 (2008), 264-265 (Erratum); arXiv:math.CA/0210422. doi: 10.4153/CJM-2003-001-0.
[7]	E. Baake and T. Hustedt, Moment closure in a Moran model with recombination, Markov Proc. Rel. Fields, 17 (2011), 429-446; arXiv:1105.0793.
[8]	E. Baake and U. von Wangenheim, Single-crossover recombination and ancestral recombination trees, J. Math. Biol., 68 (2014), 1371-1402; arXiv:1206.0950. doi: 10.1007/s00285-013-0662-x.
[9]	M. Baake and R. Speicher, in preparation.
[10]	J. H. Bennett, On the theory of random mating, Ann. Human Gen., 18 (1954), 311-317.
[11]	R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, Wiley, Chichester, 2000.
[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), 1-20. doi: 10.1006/tpbi.2000.1471.
[13]	K. J. Dawson, The evolution of a population under recombination: How to linearise the dynamics, Lin. Alg. Appl., 348 (2002), 115-137. doi: 10.1016/S0024-3795(01)00586-9.
[14]	R. Durrett, Probability Models for DNA Sequence Evolution, 2nd ed., Springer, New York, 2008. doi: 10.1007/978-0-387-78168-6.
[15]	M. Esser, S. Probst and E. Baake, Partitioning, duality, and linkage disequilibria in the Moran model with recombination, submitted, preprint arXiv:1502.05194.
[16]	W. J. Ewens and G. Thomson, Properties of equilibria in multi-locus genetic systems, Genetics, 87 (1977), 807-819.
[17]	W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, New York, 1986. doi: 10.1063/1.3062516.
[18]	H. Geiringer, On the probability theory of linkage in Mendelian heredity, Ann. Math. Stat., 15 (1944), 25-57. doi: 10.1214/aoms/1177731313.
[19]	Y. Lyubich, Mathematical Structures in Population Genetics, Springer, Berlin, 1992. doi: 10.1007/978-3-642-76211-6.
[20]	T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak selection, J. Math. Biol., 38 (1999), 103-133. doi: 10.1007/s002850050143.
[21]	J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998, reprint, 2005.
[22]	O. Redner and M. Baake, Unequal crossover dynamics in discrete and continuous time, J. Math. Biol., 49 (2004), 201-226; arXiv:math.DS/0402351. doi: 10.1007/s00285-004-0273-7.
[23]	N. J. A. Sloane, The On-line encyclopedia of integer sequences, Lecture Notes in Computer Science, 4573 (2007), p 130, https://oeis.org/ doi: 10.1007/978-3-540-73086-6_12.
[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), 1028-1047; arXiv:math.DS/0002113. doi: 10.1109/9.935056.
[25]	E. Spiegel and C. J. O'Donnell, Incidence Algebras, Marcel Dekker, New York, 1997.
[26]	U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in discrete time, J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678. doi: 10.1007/s00285-009-0277-4.

show all references

References:

[1]	M. Aigner, Combinatorial Theory, reprint, Springer, Berlin, 1997. doi: 10.1007/978-3-642-59101-3.
[2]	H. Amann, Gewöhnliche Differentialgleichungen, 2nd ed., de Gryuter, Berlin, 1995.
[3]	E. Baake, Deterministic and stochastic aspects of single-crossover recombination, in: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010, Vol. VI, ed. J. Bhatia, Hindustan Book Agency, New Delhi (2010), 3037-3053.
[4]	E. Baake and I. Herms, Single-crossover dynamics: Finite versus infinite populations, Bull. Math. Biol., 70 (2008), 603-624; arXiv:q-bio/0612024. doi: 10.1007/s11538-007-9270-5.
[5]	M. Baake, Recombination semigroups on measure spaces, Monatsh. Math., 146 (2005), 267-278 and 150 (2007), 83-84 (Addendum); arXiv:math.CA/0506099. doi: 10.1007/s00605-005-0326-z.
[6]	M. Baake and E. Baake, An exactly solved model for mutation, recombination and selection, Can. J. Math., 55 (2003), 3-41 and 60 (2008), 264-265 (Erratum); arXiv:math.CA/0210422. doi: 10.4153/CJM-2003-001-0.
[7]	E. Baake and T. Hustedt, Moment closure in a Moran model with recombination, Markov Proc. Rel. Fields, 17 (2011), 429-446; arXiv:1105.0793.
[8]	E. Baake and U. von Wangenheim, Single-crossover recombination and ancestral recombination trees, J. Math. Biol., 68 (2014), 1371-1402; arXiv:1206.0950. doi: 10.1007/s00285-013-0662-x.
[9]	M. Baake and R. Speicher, in preparation.
[10]	J. H. Bennett, On the theory of random mating, Ann. Human Gen., 18 (1954), 311-317.
[11]	R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, Wiley, Chichester, 2000.
[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), 1-20. doi: 10.1006/tpbi.2000.1471.
[13]	K. J. Dawson, The evolution of a population under recombination: How to linearise the dynamics, Lin. Alg. Appl., 348 (2002), 115-137. doi: 10.1016/S0024-3795(01)00586-9.
[14]	R. Durrett, Probability Models for DNA Sequence Evolution, 2nd ed., Springer, New York, 2008. doi: 10.1007/978-0-387-78168-6.
[15]	M. Esser, S. Probst and E. Baake, Partitioning, duality, and linkage disequilibria in the Moran model with recombination, submitted, preprint arXiv:1502.05194.
[16]	W. J. Ewens and G. Thomson, Properties of equilibria in multi-locus genetic systems, Genetics, 87 (1977), 807-819.
[17]	W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, New York, 1986. doi: 10.1063/1.3062516.
[18]	H. Geiringer, On the probability theory of linkage in Mendelian heredity, Ann. Math. Stat., 15 (1944), 25-57. doi: 10.1214/aoms/1177731313.
[19]	Y. Lyubich, Mathematical Structures in Population Genetics, Springer, Berlin, 1992. doi: 10.1007/978-3-642-76211-6.
[20]	T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak selection, J. Math. Biol., 38 (1999), 103-133. doi: 10.1007/s002850050143.
[21]	J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998, reprint, 2005.
[22]	O. Redner and M. Baake, Unequal crossover dynamics in discrete and continuous time, J. Math. Biol., 49 (2004), 201-226; arXiv:math.DS/0402351. doi: 10.1007/s00285-004-0273-7.
[23]	N. J. A. Sloane, The On-line encyclopedia of integer sequences, Lecture Notes in Computer Science, 4573 (2007), p 130, https://oeis.org/ doi: 10.1007/978-3-540-73086-6_12.
[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), 1028-1047; arXiv:math.DS/0002113. doi: 10.1109/9.935056.
[25]	E. Spiegel and C. J. O'Donnell, Incidence Algebras, Marcel Dekker, New York, 1997.
[26]	U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in discrete time, J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678. doi: 10.1007/s00285-009-0277-4.

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