The Mathematical Theory Of Selection Recombination And Mutation Download UPDATED

The Mathematical Theory Of Selection Recombination And Mutation Download

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 meaty product infinite. Information technology has an embedding into a larger family of nonlinear ODEs that permits a systematic assay with lattice-theoretic methods for general partitions of finite sets. Nosotros discuss this type of system, reduce it to an equivalent finite-dimensional nonlinear problem, and establish a connectedness with an bequeathed sectionalisation process, astern in fourth dimension. 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.

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

References:

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[2]	H. Amann, Gewöhnliche Differentialgleichungen, 2nd ed., de Gryuter, Berlin, 1995. Google Scholar
[three]	E. Baake, Deterministic and stochastic aspects of single-crossover recombination, in: Proceedings of the International Congress of Mathematicians, Hyderabad, Bharat, 2010, Vol. Half dozen, ed. J. Bhatia, Hindustan Book Agency, New Delhi (2010), 3037-3053.  Google Scholar
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[vii]	E. Baake and T. Hustedt, Moment closure in a Moran model with recombination, Markov Proc. Rel. Fields, 17 (2011), 429-446; arXiv:1105.0793.  Google Scholar
[8]	E. Baake and U. von Wangenheim, Unmarried-crossover recombination and ancestral recombination copse, J. Math. Biol., 68 (2014), 1371-1402; arXiv:1206.0950. doi: ten.1007/s00285-013-0662-x.  Google Scholar
[9]	1000. Baake and R. Speicher, in, preparation., ().   Google Scholar
[10]	J. H. Bennett, On the theory of random mating, Ann. Human Gen., eighteen (1954), 311-317.  Google Scholar
[11]	R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, Wiley, Chichester, 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-xx. doi: 10.1006/tpbi.2000.1471.  Google Scholar
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[17]	W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, third ed., Wiley, New York, 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), 25-57. doi: 10.1214/aoms/1177731313.  Google Scholar
[19]	Y. Lyubich, Mathematical Structures in Population Genetics, Springer, Berlin, 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 pick, J. Math. Biol., 38 (1999), 103-133. doi: 10.1007/s002850050143.  Google Scholar
[21]	J. R. Norris, Markov Bondage, Cambridge University Press, Cambridge, 1998, reprint, 2005.  Google Scholar
[22]	O. Redner and M. Baake, Unequal crossover dynamics in detached and continuous time, J. Math. Biol., 49 (2004), 201-226; arXiv:math.DS/0402351. doi: 10.1007/s00285-004-0273-7.  Google Scholar
[23]	Northward. J. A. Sloane, The On-line encyclopedia of integer sequences, Lecture Notes in Computer science, 4573 (2007), p 130, https://oeis.org/ doi: ten.1007/978-iii-540-73086-6_12.  Google Scholar
[24]	E. D. Sontag, Structure and stability of sure chemical networks and applications to the kinetic proofreading model of T-prison cell receptor betoken transduction, IEEE Trans. Automated Control, 46 (2001), 1028-1047; arXiv:math.DS/0002113. doi: x.1109/9.935056.  Google Scholar
[25]	Due east. Spiegel and C. J. O'Donnell, Incidence Algebras, Marcel Dekker, New York, 1997.  Google Scholar
[26]	U. von Wangenheim, E. Baake and Chiliad. Baake, Single-crossover recombination in discrete fourth dimension, J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678. doi: 10.1007/s00285-009-0277-4.  Google Scholar

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References:

[1]	Thou. Aigner, Combinatorial Theory, reprint, Springer, Berlin, 1997. doi: x.1007/978-3-642-59101-3.  Google Scholar
[two]	H. Amann, Gewöhnliche Differentialgleichungen, 2nd ed., de Gryuter, Berlin, 1995. Google Scholar
[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 Bureau, New Delhi (2010), 3037-3053.  Google Scholar
[iv]	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-five.  Google Scholar
[v]	M. Baake, Recombination semigroups on mensurate spaces, Monatsh. Math., 146 (2005), 267-278 and 150 (2007), 83-84 (Addendum); arXiv:math.CA/0506099. doi: x.1007/s00605-005-0326-z.  Google Scholar
[6]	K. Baake and Eastward. Baake, An exactly solved model for mutation, recombination and selection, Can. J. Math., 55 (2003), three-41 and 60 (2008), 264-265 (Erratum); arXiv:math.CA/0210422. doi: 10.4153/CJM-2003-001-0.  Google Scholar
[7]	Eastward. Baake and T. Hustedt, Moment closure in a Moran model with recombination, Markov Proc. Rel. Fields, 17 (2011), 429-446; arXiv:1105.0793.  Google Scholar
[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.  Google Scholar
[9]	M. Baake and R. Speicher, in, preparation., ().   Google Scholar
[ten]	J. H. Bennett, On the theory of random mating, Ann. Human Gen., 18 (1954), 311-317.  Google Scholar
[11]	R. Bürger, The Mathematical Theory of Selection, Recombination and Mutation, Wiley, Chichester, 2000.  Google Scholar
[12]	K. J. Dawson, The disuse of linkage disequilibrium under random union of gametes: How to summate Bennett's main components, Theor. Popul. Biol., 58 (2000), i-20. doi: ten.1006/tpbi.2000.1471.  Google Scholar
[13]	Thou. 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.  Google Scholar
[14]	R. Durrett, Probability Models for Dna Sequence Evolution, 2nd ed., Springer, New York, 2008. doi: 10.1007/978-0-387-78168-half dozen.  Google Scholar
[xv]	M. Esser, S. Probst and Due east. Baake, Sectionalization, duality, and linkage disequilibria in the Moran model with recombination,, submitted, ().   Google Scholar
[sixteen]	W. J. Ewens and Thousand. Thomson, Properties of equilibria in multi-locus genetic systems, Genetics, 87 (1977), 807-819.  Google Scholar
[17]	W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, third ed., Wiley, New York, 1986. doi: 10.1063/1.3062516.  Google Scholar
[18]	H. Geiringer, On the probability theory of linkage in Mendelian heredity, Ann. Math. Stat., fifteen (1944), 25-57. doi: 10.1214/aoms/1177731313.  Google Scholar
[nineteen]	Y. Lyubich, Mathematical Structures in Population Genetics, Springer, Berlin, 1992. doi: ten.1007/978-3-642-76211-half dozen.  Google Scholar
[20]	T. Nagylaki, J. Hofbauer and P. Brunovski, Convergence of multilocus systems under weak epistasis or weak choice, J. Math. Biol., 38 (1999), 103-133. doi: 10.1007/s002850050143.  Google Scholar
[21]	J. R. Norris, Markov Bondage, Cambridge Academy Press, Cambridge, 1998, reprint, 2005.  Google Scholar
[22]	O. Redner and G. Baake, Unequal crossover dynamics in discrete and continuous fourth dimension, J. Math. Biol., 49 (2004), 201-226; arXiv:math.DS/0402351. doi: 10.1007/s00285-004-0273-7.  Google Scholar
[23]	N. J. A. Sloane, The On-line encyclopedia of integer sequences, Lecture Notes in Informatics, 4573 (2007), p 130, https://oeis.org/ doi: 10.1007/978-3-540-73086-6_12.  Google Scholar
[24]	E. D. Sontag, Construction and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor bespeak transduction, IEEE Trans. Automated Control, 46 (2001), 1028-1047; arXiv:math.DS/0002113. doi: 10.1109/9.935056.  Google Scholar
[25]	East. Spiegel and C. J. O'Donnell, Incidence Algebras, Marcel Dekker, New York, 1997.  Google Scholar
[26]	U. von Wangenheim, E. Baake and M. Baake, Single-crossover recombination in detached time, J. Math. Biol., 60 (2010), 727-760; arXiv:0906.1678. doi: 10.1007/s00285-009-0277-4.  Google Scholar

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