American Institute of Mathematical Sciences

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

The general recombination equation in continuous time and its solution

 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

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.
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.

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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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