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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 |
References:
[1] |
M. Aigner, Combinatorial Theory, reprint,, Springer, (1997).
doi: 10.1007/978-3-642-59101-3. |
[2] |
H. Amann, Gewöhnliche Differentialgleichungen,, 2nd ed., (1995). |
[3] |
E. Baake, Deterministic and stochastic aspects of single-crossover recombination,, in: Proceedings of the International Congress of Mathematicians, (2010), 3037.
|
[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. |
[5] |
M. Baake, Recombination semigroups on measure spaces,, Monatsh. Math., 146 (2005), 267.
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.
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.
|
[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. |
[9] |
M. Baake and R. Speicher, in, preparation., (). |
[10] |
J. H. Bennett, On the theory of random mating,, Ann. Human Gen., 18 (1954), 311.
|
[11] |
R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation,, Wiley, (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.
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.
doi: 10.1016/S0024-3795(01)00586-9. |
[14] |
R. Durrett, Probability Models for DNA Sequence Evolution,, 2nd ed., (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, (). |
[16] |
W. J. Ewens and G. Thomson, Properties of equilibria in multi-locus genetic systems,, Genetics, 87 (1977), 807.
|
[17] |
W. Feller, An Introduction to Probability Theory and Its Applications,, Vol. I, (1986).
doi: 10.1063/1.3062516. |
[18] |
H. Geiringer, On the probability theory of linkage in Mendelian heredity,, Ann. Math. Stat., 15 (1944), 25.
doi: 10.1214/aoms/1177731313. |
[19] |
Y. Lyubich, Mathematical Structures in Population Genetics,, Springer, (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.
doi: 10.1007/s002850050143. |
[21] |
J. R. Norris, Markov Chains,, Cambridge University Press, (1998).
|
[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. |
[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. |
[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. |
[25] |
E. Spiegel and C. J. O'Donnell, Incidence Algebras,, Marcel Dekker, (1997).
|
[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. |
show all references
References:
[1] |
M. Aigner, Combinatorial Theory, reprint,, Springer, (1997).
doi: 10.1007/978-3-642-59101-3. |
[2] |
H. Amann, Gewöhnliche Differentialgleichungen,, 2nd ed., (1995). |
[3] |
E. Baake, Deterministic and stochastic aspects of single-crossover recombination,, in: Proceedings of the International Congress of Mathematicians, (2010), 3037.
|
[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. |
[5] |
M. Baake, Recombination semigroups on measure spaces,, Monatsh. Math., 146 (2005), 267.
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.
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.
|
[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. |
[9] |
M. Baake and R. Speicher, in, preparation., (). |
[10] |
J. H. Bennett, On the theory of random mating,, Ann. Human Gen., 18 (1954), 311.
|
[11] |
R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation,, Wiley, (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.
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.
doi: 10.1016/S0024-3795(01)00586-9. |
[14] |
R. Durrett, Probability Models for DNA Sequence Evolution,, 2nd ed., (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, (). |
[16] |
W. J. Ewens and G. Thomson, Properties of equilibria in multi-locus genetic systems,, Genetics, 87 (1977), 807.
|
[17] |
W. Feller, An Introduction to Probability Theory and Its Applications,, Vol. I, (1986).
doi: 10.1063/1.3062516. |
[18] |
H. Geiringer, On the probability theory of linkage in Mendelian heredity,, Ann. Math. Stat., 15 (1944), 25.
doi: 10.1214/aoms/1177731313. |
[19] |
Y. Lyubich, Mathematical Structures in Population Genetics,, Springer, (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.
doi: 10.1007/s002850050143. |
[21] |
J. R. Norris, Markov Chains,, Cambridge University Press, (1998).
|
[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. |
[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. |
[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. |
[25] |
E. Spiegel and C. J. O'Donnell, Incidence Algebras,, Marcel Dekker, (1997).
|
[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. |
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