American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An Age-Structured Model of HIV Infection that Allows for Variations in the Production Rate of Viral Particles and the Death Rate of Productively Infected Cells
  • MBE Home
  • This Issue
  • Next Article
    Dynamic Multidrug Therapies for HIV: Optimal and STI Control Approaches
2004, 1(2): 243-266. doi: 10.3934/mbe.2004.1.243

Coalgebraic Structure of Genetic Inheritance

Jianjun Tian 1, and  Bai-Lian Li 2,

1. 

Department of Mathematics, University of California, Riverside, CA 92521-0135, United States
2. 

Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA 92521-0124, United States

Received  December 2003 Revised  May 2004 Published  July 2004

Although in the broadly defined genetic algebra, multiplication suggests a forward direction of from parents to progeny, when looking from the reverse direction, it also suggests to us a new algebraic structure --- coalgebraic structure, which we call genetic coalgebras. It is not the dual coalgebraic structure and can be used in the construction of phylogenetic trees. Mathematically, to construct phylogenetic trees means we need to solve equations x[n] =a , or x(n)=b. It is generally impossible to solve these equations in algebras. However, we can solve them in coalgebras in the sense of tracing back for their ancestors. A thorough exploration of coalgebraic structure in genetics is apparently necessary. Here, we develop a theoretical framework of the coalgebraic structure of genetics. From biological viewpoint, we defined various fundamental concepts and examined their elementary properties that contain genetic significance. Mathematically, by genetic coalgebra, we mean any coalgebra that occurs in genetics. They are generally noncoassociative and without counit; and in the case of non-sex-linked inheritance, they are cocommutative. Each coalgebra with genetic realization has a baric property. We have also discussed the methods to construct new genetic coalgebras, including cocommutative duplication, the tensor product, linear combinations and the skew linear map, which allow us to describe complex genetic traits. We also put forward certain theorems that state the relationship between gametic coalgebra and gametic algebra. By Brower's theorem in topology, we prove the existence of equilibrium state for the in-evolution operator.
Keywords: general genetic algebras, baric coalgebras, general genetic coalgebras, in-evolution operators., conilpotent coalgebras.
Mathematics Subject Classification: 16W9.
Citation: 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
[1]

Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure & Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109
[2]

Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016
[3]

Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061
[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]

Johannes Giannoulis, Alexander Mielke. Dispersive evolution of pulses in oscillator chains with general interaction potentials. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 493-523. doi: 10.3934/dcdsb.2006.6.493
[6]

Thaís Jordão, Xingping Sun. General types of spherical mean operators and $K$-functionals of fractional orders. Communications on Pure & Applied Analysis, 2015, 14 (3) : 743-757. doi: 10.3934/cpaa.2015.14.743
[7]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Enrico Obrecht, Silvia Romanelli. Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2475-2487. doi: 10.3934/cpaa.2016045
[8]

Bertrand Lods, Mustapha Mokhtar-Kharroubi, Mohammed Sbihi. Spectral properties of general advection operators and weighted translation semigroups. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1469-1492. doi: 10.3934/cpaa.2009.8.1469
[9]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[10]

Yaw Chang, Lin Chen. Solve the vehicle routing problem with time windows via a genetic algorithm. Conference Publications, 2007, 2007 (Special) : 240-249. doi: 10.3934/proc.2007.2007.240
[11]

Massimiliano Zanin, Ernestina Menasalvas, Pedro A. C. Sousa, Stefano Boccaletti. Preprocessing and analyzing genetic data with complex networks: An application to Obstructive Nephropathy. Networks & Heterogeneous Media, 2012, 7 (3) : 473-481. doi: 10.3934/nhm.2012.7.473
[12]

Didem Cinar, José António Oliveira, Y. Ilker Topcu, Panos M. Pardalos. A priority-based genetic algorithm for a flexible job shop scheduling problem. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1391-1415. doi: 10.3934/jimo.2016.12.1391
[13]

Yuanyuan Huang, Yiping Hao, Min Wang, Wen Zhou, Zhijun Wu. Optimality and stability of symmetric evolutionary games with applications in genetic selection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 503-523. doi: 10.3934/mbe.2015.12.503
[14]

William Chad Young, Adrian E. Raftery, Ka Yee Yeung. A posterior probability approach for gene regulatory network inference in genetic perturbation data. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1241-1251. doi: 10.3934/mbe.2016041
[15]

Abdel-Rahman Hedar, Alaa Fahim. Filter-based genetic algorithm for mixed variable programming. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 99-116. doi: 10.3934/naco.2011.1.99
[16]

Francesca Verrilli, Hamed Kebriaei, Luigi Glielmo, Martin Corless, Carmen Del Vecchio. Effects of selection and mutation on epidemiology of X-linked genetic diseases. Mathematical Biosciences & Engineering, 2017, 14 (3) : 755-775. doi: 10.3934/mbe.2017042
[17]

María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637
[18]

Bernd Ammann, Robert Lauter and Victor Nistor. Algebras of pseudodifferential operators on complete manifolds. Electronic Research Announcements, 2003, 9: 80-87.

[19]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Dirichlet - transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 999-1018. doi: 10.3934/dcdsb.2011.15.999
[20]

David Damanik, Serguei Tcheremchantsev. A general description of quantum dynamical spreading over an orthonormal basis and applications to Schrödinger operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1381-1412. doi: 10.3934/dcds.2010.28.1381

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences