PlanetPhysics/Category of Molecular Sets 4

Molecular sets as representations of chemical reactions

A uni-molecular chemical reaction is defined by the natural transformations ${\displaystyle \eta :h^{A}\longrightarrow h^{B},}$ specified in the following commutative diagram:

$\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d]_{h^A(t)} & h^B (A) = Hom(B,A)\ar[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}},$

with the states of molecular sets ${\displaystyle A_{u}=a_{1},\ldots ,a_{n}}$ and ${\displaystyle B_{u}=b_{1},\ldots b_{n}}$ being defined as the endomorphism sets ${\displaystyle Hom(A,A)}$ and ${\displaystyle Hom(B,B)}$, respectively. In general, molecular sets ${\displaystyle M_{S}}$ are defined as finite sets whose elements are molecules; the molecules are mathematically defined in terms of their molecular observables as specified next. In order to define molecular observables one needs to define first the concept of a molecular class variable or ${\displaystyle m.c.v}$.

A molecular class variables is defined as a family of molecular sets ${\displaystyle [M_{S}]_{i\in I}}$, with ${\displaystyle I}$ being either an indexing set, or a proper class, that defines the variation range of the ${\displaystyle m.c.v}$. Most physical, chemical or biochemical applications require that ${\displaystyle I}$ is restricted to a finite set, (that is, without any sub-classes). A morphism, or molecular mapping, ${\displaystyle M_{t}:M_{S}\to M_{S}}$ of molecular sets, with ${\displaystyle t\in T}$ being real time values, is defined as a time-dependent mapping or function ${\displaystyle M_{S}(t)}$ also called a molecular transformation, ${\displaystyle M_{t}}$.

An ${\displaystyle m.c.v.}$ observable of ${\displaystyle B}$, characterizing the products of chemical type "B" of a chemical reaction is defined as a morphism:

${\displaystyle \gamma :Hom(B,B)\longrightarrow \Re ,}$ where ${\displaystyle \Re }$ is the set or field of real numbers. This mcv-observable is subject to the following commutativity conditions:

$\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},}$

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with ${\displaystyle c:A_{u}^{*}\longrightarrow B_{u}^{*}}$, and ${\displaystyle A_{u}^{*}}$, ${\displaystyle B_{u}^{*}}$ being, respectively, specially prepared fields of states of the molecular sets ${\displaystyle A_{u}}$, and ${\displaystyle B_{u}}$ within a measurement uncertainty range, ${\displaystyle \Delta }$, which is determined by Heisenberg's uncertainty relation, or the commutator of the observable operators involved, such as ${\displaystyle [A^{*},B^{*}]}$, associated with the observable ${\displaystyle A}$ of molecular set ${\displaystyle A_{u}}$, and respectively, with the obssevable ${\displaystyle B}$ of molecular set ${\displaystyle B_{u}}$, in the case of a molecular set ${\displaystyle A_{u}}$ interacting with molecular set ${\displaystyle B_{u}}$.

With these concepts and preliminary data one can now define the category of molecular sets and their transformations as follows.

Category of molecular sets and their transformations

The category of molecular sets is defined as the category ${\displaystyle C_{M}}$ whose objects are molecular sets ${\displaystyle M_{S}}$ and whose morphisms are molecular transformations ${\displaystyle M_{t}}$.

This is a mathematical representation of chemical reaction systems in terms of molecular sets that vary with time (or ${\displaystyle msv}$'s), and their transformations as a result of diffusion, collisions, and chemical reactions.

Classification: AMS MSC: 18D35 (category theory; homological algebra :: categories with structure :: Structured objects in a category ) 92B05 (Biology and other natural sciences :: Mathematical biology in general :: General biology and biomathematics) 18E05 (Category theory; homological algebra :: abelian categories :: Preadditive, additive categories) 81-00 (quantum theory :: General reference works )

References

1. Bartholomay, A. F.: 1960. Molecular Set Theory. A mathematical representation for chemical reaction mechanisms. Bull. Math. Biophys. , 22 : 285-307.
2. Bartholomay, A. F.: 1965. Molecular Set Theory: II. An aspect of biomathematical theory of sets., Bull. Math. Biophys. 27 : 235-251.
3. Bartholomay, A.: 1971. Molecular Set Theory: III. The Wide-Sense Kinetics of Molecular Sets ., Bulletin of Mathematical Biophysics , 33 : 355-372.
4. Baianu, I. C.: 1983, Natural Transformation Models in Molecular Biology., in Proceedings of the SIAM Natl. Meet ., Denver, CO.; Eprint at cogprints.org with No. 3675. Cite error: Invalid <ref> tag; name "ICB2" defined multiple times with different content