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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

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

show all references

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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