# 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