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

Algebraic model of non-Mendelian inheritance

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.
Citation: Jianjun Paul Tian. Algebraic model of non-Mendelian inheritance. Discrete and 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.

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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).

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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).

[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.

[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.

[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.

[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.

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