# PlanetPhysics/Category of Molecular Sets 4

### Molecular sets as representations of chemical reactionsEdit

A *uni-molecular chemical reaction* is defined by the natural transformations
specified in 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 molecular sets* and
being defined as the endomorphism sets and , respectively. In general, *molecular sets* 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 .

A *molecular class variables* is defined as a family of molecular sets , with being either an indexing set, or a proper class, that defines the variation range of the .
Most physical, chemical or biochemical applications require that is restricted to a finite set, (that is, without any sub-classes). A morphism, or molecular mapping, 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.

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 )

## 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>`

tag; name "ICB2" defined multiple times with different content