%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: category of molecular sets
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%%% Filename: CategoryOfMolecularSets4.tex
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\begin{document}
\subsection{Molecular sets as representations of chemical reactions}
A \emph{\htmladdnormallink{uni-molecular chemical reaction}{http://planetphysics.us/encyclopedia/Molecule.html}} is defined by the \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}
$$\eta: h^A\longrightarrow h^B,$$ specified in the following \htmladdnormallink{commutative diagram}{http://planetphysics.us/encyclopedia/Commutativity.html}:
\begin{equation}
\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)}},
\end{equation}
with the \emph{\htmladdnormallink{states of molecular sets}{http://planetphysics.us/encyclopedia/Molecule.html}} $A_u = a_1, \ldots, a_n$ and
$B_u = b_1, \ldots b_n$ being defined as the endomorphism sets $Hom(A,A)$ and $Hom(B,B)$, respectively. In general, \emph{molecular sets} $M_S$ are defined as finite sets whose elements are \htmladdnormallink{molecules}{http://planetphysics.us/encyclopedia/Molecule.html}; the \emph{molecules} are mathematically defined in terms of their molecular \htmladdnormallink{observables}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} as specified next. In order to define molecular observables one needs to define first the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of a \htmladdnormallink{molecular class variable}{http://planetphysics.us/encyclopedia/Molecule.html} or $m.c.v$.
A \emph{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 \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, 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 \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $M_S (t)$ also called a \emph{\htmladdnormallink{molecular transformation}{http://planetphysics.us/encyclopedia/Molecule.html}}, $M_t$.
An \emph{$m.c.v.$ observable} of $B$, characterizing the products of chemical \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} ``B'' of a chemical reaction is defined as a morphism:
$$\gamma : Hom(B,B) \longrightarrow \Re ,$$
where $\Re$ is the set or \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of real numbers. This mcv-observable is subject
to the following \htmladdnormallink{commutativity}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} conditions:
\begin{equation}
\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},}
\end{equation}~
with $c: A^*_u \longrightarrow B^*_u$, and $A^*_u$, $B^*_u$ being, respectively,
specially prepared \emph{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 \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html}, or the \htmladdnormallink{commutator}{http://planetphysics.us/encyclopedia/Commutator.html} of the observable \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} 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.
\subsection{Category of molecular sets and their transformations}
\begin{definition}
The \emph{category of molecular sets} is defined as the \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $C_M$ whose \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are molecular sets $M_S$ and whose morphisms are molecular transformations $M_t$.
\end{definition}
\begin{remark}
This is a mathematical \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of chemical reaction \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} in terms of molecular sets that vary with time (or $msv$'s), and their transformations as a result of diffusion, \htmladdnormallink{collisions}{http://planetphysics.us/encyclopedia/Collision.html}, and chemical reactions.
\end{remark}
\htmladdnormallink{Classification}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}:
AMS MSC: 18D35 (\htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}; homological algebra :: \htmladdnormallink{categories with structure}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} :: Structured objects in a category )
92B05 (Biology and other natural sciences :: Mathematical biology in general :: General biology and biomathematics)
18E05 (Category theory; homological algebra :: \htmladdnormallink{abelian categories}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} :: Preadditive, \htmladdnormallink{additive categories}{http://planetphysics.us/encyclopedia/DenseSubcategory.html})
81-00 (\htmladdnormallink{quantum theory}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} :: General reference \htmladdnormallink{works}{http://planetphysics.us/encyclopedia/Work.html} )
\begin{thebibliography}{9}
\bibitem{BAF60}
Bartholomay, A. F.: 1960. Molecular Set Theory. A mathematical representation for chemical reaction mechanisms. \emph{Bull. Math. Biophys.}, \textbf{22}: 285-307.
\bibitem{BAF65}
Bartholomay, A. F.: 1965. Molecular Set Theory: II. An aspect of biomathematical theory of sets., \emph{Bull. Math. Biophys.} \textbf{27}: 235-251.
\bibitem{BAF71}
Bartholomay, A.: 1971. Molecular Set Theory: III. The Wide-Sense Kinetics of Molecular Sets ., \emph{Bulletin of Mathematical Biophysics}, \textbf{33}: 355-372.
\bibitem{ICB2}
Baianu, I. C.: 1983, Natural Transformation Models in Molecular
Biology., in \emph{Proceedings of the SIAM Natl. Meet}., Denver,
CO.; Eprint at cogprints.org with No. 3675.
\bibitem{ICB2}
Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural
and Regulatory Activities in Metabolic and Genetic Networks
\emph{FASEB Proceedings} \textbf{43}, 917.
\end{thebibliography}
\end{document}
