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# category of molecular sets

# 0.1 Molecular sets as representations of chemical reactions

A *uni-molecular chemical reaction* is defined by the natural transformations

$\eta:h^{A}\longrightarrow h^{B},$ |

specified in the following commutative diagram representing molecular sets and their quantum transformations:

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

with the *states of molecular sets* $A_{u}=a_{1},\ldots,a_{n}$ and
$B_{u}=b_{1},\ldots b_{n}$ being defined as the endomorphism sets $Hom(A,A)$ and $Hom(B,B)$, respectively. In general, *molecular sets* $M_{S}$ 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 $m.c.v$.

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

An *$m.c.v.$ observable* of $B$, characterizing the products of chemical type “B” of a chemical reaction is defined as a morphism:

$\gamma:Hom(B,B)\longrightarrow\Re,$ |

where $\Re$ is the set or field of real numbers. This mcv-observable is subject to the following commutativity conditions:

$\xymatrix@M=0.1pc@=4pc{Hom(A,A)\ar[r]^{{f}}\ar[d]_{{e}}&Hom(B,B)\ar[d]^{{% \gamma}}\\ {Hom(A,A)}\ar[r]_{{\delta}}&{R},}$ | (0.2) |

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

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

# 0.2 Category of molecular sets and their transformations

###### Definition 0.1.

###### Remark 0.1.

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

# 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. - 5
Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural
and Regulatory Activities in Metabolic and Genetic Networks
*FASEB Proceedings*43, 917.

## Mathematics Subject Classification

81-00*no label found*18E05

*no label found*92B05

*no label found*18D35

*no label found*

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