Coalgebraic Structure of Genetic Inheritance
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1.
Department of Mathematics, University of California, Riverside, CA 92521-0135
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2.
Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA 92521-0124
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Received:
01 December 2003
Accepted:
29 June 2018
Published:
01 July 2004
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MSC :
16W99.
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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.
Citation: Jianjun Tian, Bai-Lian Li. Coalgebraic Structure of Genetic Inheritance[J]. Mathematical Biosciences and Engineering, 2004, 1(2): 243-266. doi: 10.3934/mbe.2004.1.243
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Abstract
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.
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