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December  2011, 4(6): 1577-1586. doi: 10.3934/dcdss.2011.4.1577

Algebraic model of non-Mendelian inheritance

Jianjun Paul Tian 1,

1. 

Mathematics Department, College of William and Mary, Williamsburg, VA 23187, United States

Received  April 2009 Revised  October 2009 Published  December 2010

Evolution algebra theory is used to study non-Mendelian inheritance, particularly organelle heredity and population genetics of Phytophthora infectans. We not only can explain a puzzling feature of establishment of homoplasmy from heteroplasmic cell population and the coexistence of mitochondrial triplasmy, but also can predict all mechanisms to form the homoplasmy of cell populations, which are hypothetical mechanisms in current mitochondrial disease research. The algebras also provide a way to easily find different genetically dynamic patterns from the complexity of the progenies of Phytophthora infectans which cause the late blight of potatoes and tomatoes. Certain suggestions to pathologists are made as well.
Keywords: non-Mendelian inheritance, Genetics, genetic algebras., algebraic model.
Mathematics Subject Classification: Primary: 17D99, 17D92; Secondary: 92D1.
Citation: Jianjun Paul Tian. Algebraic model of non-Mendelian inheritance. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1577-1586. doi: 10.3934/dcdss.2011.4.1577
References:
[1]

C. W. Jr. Birky, The inheritance of genes in mitochondria and chloroplasts: Laws, mechanisms, and models, Annu. Rev. Genet., 35 (2001), 125-148. doi: 10.1146/annurev.genet.35.102401.090231.  Google Scholar

[2]

C. W. Jr. Birky, "Inheritance of Mitochondrial Mutations," Mitochondrial DNA Mutations and Aging, Disease and Cancer, K.K. Singh, edit, Spring, 1998. Google Scholar

[3]

Jianjun Paul Tian, "Evolution Algebras and their Applications," Lecture Note in Mathematics, 1921, Springer, Berlin, 2008.  Google Scholar

[4]

G. Mendel, "Experiments in Plant-Hybridization," Classic Papers in Genetics, 1-20, J. A. Peter editor, Prentice-Hall Inc. 1959. Google Scholar

[5]

Y. I. Lyubich, "Mathematical Structures in Population Genetics," Springer-Verlag, New York, 1992. Google Scholar

[6]

A. Worz-Busekros, "Algebras in Genetics," Lecture Notes in Biomath. 36, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[7]

M. L. Reed, Algebraic structure of genetic inheritance, Bull. of AMS, 34 (1997), 107-130. doi: 10.1090/S0273-0979-97-00712-X.  Google Scholar

[8]

N. W. Gillham, "Organelle Genes and Genomes," Oxford University Press, 1994. Google Scholar

[9]

C. F. Emmerson, G. K. Brown and J. Poulton, Synthesis of mitochondrial DNA in permeabilised human cultured cells, Nucleic Acids Res., 29 (2001), e1-e1(1). Google Scholar

[10]

F. Ling and T. Shibata, Mhr1p-dependent concatemeric mitochondrial DNA formation for generating yeast mitochondrial homoplasmic cells, Mol. Biol. Cell, 15 (2004), 310-322. doi: 10.1091/mbc.E03-07-0508.  Google Scholar

[11]

Y. Tang, G. Manfredi, M. Hirano and E. A. Schon, Maintenance of human rearranged mitochondrial DNAs in long-term transmitochondrial cell lines, Mol. Biol. Cell, 11 (2000), 2349-2358. Google Scholar

[12]

F. M. A. Samen, G. A. Secor and N. C. Gudmestad, Variability in virulence among asexual progenies of Phytophthora infestans, Phytopathology, 93 (2003), 293-304. doi: 10.1094/PHYTO.2003.93.3.293.  Google Scholar

[13]

W. E. Fry, and S. B. Goodwin, Re-emergence of potato and tomato late blight in the United States, Plant Disease, 81 (1997), 1349-1357. doi: 10.1094/PDIS.1997.81.12.1349.  Google Scholar

show all references

References:
[1]

C. W. Jr. Birky, The inheritance of genes in mitochondria and chloroplasts: Laws, mechanisms, and models, Annu. Rev. Genet., 35 (2001), 125-148. doi: 10.1146/annurev.genet.35.102401.090231.  Google Scholar

[2]

C. W. Jr. Birky, "Inheritance of Mitochondrial Mutations," Mitochondrial DNA Mutations and Aging, Disease and Cancer, K.K. Singh, edit, Spring, 1998. Google Scholar

[3]

Jianjun Paul Tian, "Evolution Algebras and their Applications," Lecture Note in Mathematics, 1921, Springer, Berlin, 2008.  Google Scholar

[4]

G. Mendel, "Experiments in Plant-Hybridization," Classic Papers in Genetics, 1-20, J. A. Peter editor, Prentice-Hall Inc. 1959. Google Scholar

[5]

Y. I. Lyubich, "Mathematical Structures in Population Genetics," Springer-Verlag, New York, 1992. Google Scholar

[6]

A. Worz-Busekros, "Algebras in Genetics," Lecture Notes in Biomath. 36, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[7]

M. L. Reed, Algebraic structure of genetic inheritance, Bull. of AMS, 34 (1997), 107-130. doi: 10.1090/S0273-0979-97-00712-X.  Google Scholar

[8]

N. W. Gillham, "Organelle Genes and Genomes," Oxford University Press, 1994. Google Scholar

[9]

C. F. Emmerson, G. K. Brown and J. Poulton, Synthesis of mitochondrial DNA in permeabilised human cultured cells, Nucleic Acids Res., 29 (2001), e1-e1(1). Google Scholar

[10]

F. Ling and T. Shibata, Mhr1p-dependent concatemeric mitochondrial DNA formation for generating yeast mitochondrial homoplasmic cells, Mol. Biol. Cell, 15 (2004), 310-322. doi: 10.1091/mbc.E03-07-0508.  Google Scholar

[11]

Y. Tang, G. Manfredi, M. Hirano and E. A. Schon, Maintenance of human rearranged mitochondrial DNAs in long-term transmitochondrial cell lines, Mol. Biol. Cell, 11 (2000), 2349-2358. Google Scholar

[12]

F. M. A. Samen, G. A. Secor and N. C. Gudmestad, Variability in virulence among asexual progenies of Phytophthora infestans, Phytopathology, 93 (2003), 293-304. doi: 10.1094/PHYTO.2003.93.3.293.  Google Scholar

[13]

W. E. Fry, and S. B. Goodwin, Re-emergence of potato and tomato late blight in the United States, Plant Disease, 81 (1997), 1349-1357. doi: 10.1094/PDIS.1997.81.12.1349.  Google Scholar

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