# American Institute of Mathematical Sciences

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.
 [1] Jianjun Tian, Bai-Lian Li. Coalgebraic Structure of Genetic Inheritance. Mathematical Biosciences & Engineering, 2004, 1 (2) : 243-266. doi: 10.3934/mbe.2004.1.243 [2] Doston Jumaniyozov, Ivan Kaygorodov, Abror Khudoyberdiyev. The algebraic classification of nilpotent commutative algebras. Electronic Research Archive, 2021, 29 (6) : 3909-3993. doi: 10.3934/era.2021068 [3] Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167 [4] Jesse Berwald, Marian Gidea. Critical transitions in a model of a genetic regulatory system. Mathematical Biosciences & Engineering, 2014, 11 (4) : 723-740. doi: 10.3934/mbe.2014.11.723 [5] Feng Rong. Non-algebraic attractors on $\mathbf{P}^k$. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 977-989. doi: 10.3934/dcds.2012.32.977 [6] JinHyon Kim, HyonHui Ju, WiJong An. Inheritance of ${\mathscr F}-$chaos and ${\mathscr F}-$sensitivities under an iteration for non-autonomous discrete systems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022053 [7] Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755 [8] Antonio Algaba, Natalia Fuentes, Cristóbal García, Manuel Reyes. Non-formally integrable centers admitting an algebraic inverse integrating factor. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 967-988. doi: 10.3934/dcds.2018041 [9] Pieter C. Allaart. An algebraic approach to entropy plateaus in non-integer base expansions. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6507-6522. doi: 10.3934/dcds.2019282 [10] Ping-Chen Lin. Portfolio optimization and risk measurement based on non-dominated sorting genetic algorithm. Journal of Industrial and Management Optimization, 2012, 8 (3) : 549-564. doi: 10.3934/jimo.2012.8.549 [11] Jiao-Yan Li, Xiao Hu, Zhong Wan. An integrated bi-objective optimization model and improved genetic algorithm for vehicle routing problems with temporal and spatial constraints. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1203-1220. doi: 10.3934/jimo.2018200 [12] Boris Khots, Dmitriy Khots. P-groups applications in genetics. Conference Publications, 2001, 2001 (Special) : 224-228. doi: 10.3934/proc.2001.2001.224 [13] Liu Liu, Weinian Zhang. Genetics of iterative roots for PM functions. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2391-2409. doi: 10.3934/dcds.2020369 [14] Zhilan Feng, Carlos Castillo-Chavez. The influence of infectious diseases on population genetics. Mathematical Biosciences & Engineering, 2006, 3 (3) : 467-483. doi: 10.3934/mbe.2006.3.467 [15] Daniele Bartoli, Leo Storme. On the functional codes arising from the intersections of algebraic hypersurfaces of small degree with a non-singular quadric. Advances in Mathematics of Communications, 2014, 8 (3) : 271-280. doi: 10.3934/amc.2014.8.271 [16] Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144 [17] A. A. Kirillov. Family algebras. Electronic Research Announcements, 2000, 6: 7-20. [18] Reinhard Bürger. A survey of migration-selection models in population genetics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 883-959. doi: 10.3934/dcdsb.2014.19.883 [19] Peng Zhou, Jiang Yu, Dongmei Xiao. A nonlinear diffusion problem arising in population genetics. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 821-841. doi: 10.3934/dcds.2014.34.821 [20] Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169

2021 Impact Factor: 1.865