81-00-CategoryOfMolecularSets.tex

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\pmsynonym{class of molecular set variables and their transformations}{CategoryOfMolecularSets}
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%\pmkeywords{molecular set theory}
%\pmkeywords{molecular set variables and their transformations}
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\pmrelated{SupercategoriesOfComplexSystems}
\pmrelated{ComplexSystemsBiology}
\pmrelated{SupercategoryOfVariableMolecularSets}
\pmrelated{IndexOfCategories}
\pmrelated{CategoryOfQuantumAutomata}
\pmdefines{category of molecular sets}
\pmdefines{molecular set}
\pmdefines{molecular set mapping}
\pmdefines{states of molecular sets}
\pmdefines{uni-molecular chemical reaction}
\pmdefines{chemical transformation}
\pmdefines{molecular set variable}
\pmdefines{molecular transformation}
\pmdefines{$m.c.v.$ observable}
\pmdefines{mcv-observable}
\pmdefines{states of molecular se}

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\begin{document}
\subsection{Molecular sets as representations of chemical reactions}

A \emph{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:
\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{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, \emph{molecular sets} $M_S$ are defined as finite sets whose elements are molecules; the \emph{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 \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 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  \emph{molecular transformation}, $M_t$.

An \emph{$m.c.v.$ 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:
\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 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.

\subsection{Category of molecular sets and their transformations}
\begin{definition}
 The \emph{category of molecular sets} is defined as the category $C_M$ whose objects are molecular sets $M_S$ and whose morphisms are molecular transformations $M_t$. 
\end{definition}

\begin{remark}
 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. 
\end{remark}

\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}

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