# PlanetPhysics/Category of Molecular Sets 2

### Molecular Sets and Representations of Chemical ReactionsEdit

The *uni-molecular chemical reaction* is represented by the natural transformations
as specified by the following commutative diagram:

Failed to parse (unknown function "\def"): {\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 the *molecular sets* and
being represented by certain endomorphisms
in and , respectively. In general, *molecular sets* are defined as
finite sets whose elements are `molecules' defined in terms of their molecular observables that are specified below.
*molecular class variables*, or 's are defined as *families of molecular sets* ,
with being an indexing set, or class, defining the *range of molecular variation of the * ;
most applications require that is a proper, finite set, (i.e., without any sub-classes). A morphism of molecular sets, with being real time values, is defined as a time-dependent mapping or function also called a *molecular transformation*, .

An * observable* of , characterizing the products of chemical type "B" of a chemical reaction is defined as a morphism:

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

Failed to parse (unknown function "\def"): {\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},} }

~

with , and , being, respectively,
specially prepared *fields of states* of the molecular sets , and within a measurement uncertainty range, , which is determined by Heisenberg's uncertainty relation, or the commutator of the observable operators involved, such as , associated with the observable of molecular set , and respectively, with the obssevable of molecular set , in the case of a molecular set interacting with molecular set .

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 transformationsEdit

The *category of molecular sets* is defined as the category whose objects are molecular sets and whose morphisms are molecular transformations .

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

## All SourcesEdit

^{[1]}^{[2]}^{[3]}^{[4]}^{[4]}

## ReferencesEdit

- ↑
Bartholomay, A. F.: 1960. Molecular Set Theory. A mathematical representation for chemical reaction mechanisms.
*Bull. Math. Biophys.*,**22**: 285-307. - ↑
Bartholomay, A. F.: 1965. Molecular Set Theory: II. An aspect of biomathematical theory of sets.,
*Bull. Math. Biophys.***27**: 235-251. - ↑
Bartholomay, A.: 1971. Molecular Set Theory: III. The Wide-Sense Kinetics of Molecular Sets .,
*Bulletin of Mathematical Biophysics*,**33**: 355-372. - ↑
^{4.0}^{4.1}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>`

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