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Homemolecular set and molecular class variables

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# molecular set and molecular class variables

###### Definition 0.1.

*Molecular sets* $M_{S}$ are defined as finite sets of molecules that are being represented as elements of such sets.

###### Definition 0.2.

*molecular set variables* (or *variable molecular sets*), $S_{{mv}}$’s, are *mathematical representations of chemical reaction systems* in terms of *an indexed family $([M]_{t})_{{t\in T}}$, or class of molecular sets* that vary with time, $t$, as a result of diffusion, collisions, and chemical reactions.

###### Definition 0.3.

*Molecular class variables*, or $m.c.v$’s are defined as *families of molecular sets* $[M_{S}]_{{i\in I}}$,
with $I$ being an indexing set, or class, defining the *range of molecular variation of the $m.c.v$*;
most applications require that $I$ is a proper, finite set, (i.e., without any sub-classes).
A morphism $M_{t}:M_{S}\to M_{S}$ 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 $M_{t}$ *molecular transformation*.

An alternative definition is available in terms of natural transformations of organismic structures or quantum *functorial morphisms*, as further specified next.

###### Definition 0.4.

An *$mcv$ 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:

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

with $c:A^{*}_{u}\longrightarrow B^{*}_{u}$, and $A^{*}_{u}$, $B^{*}_{u}$ being
specially prepared *fields of states*, within a measurement uncertainty range, $\Delta$, specified by the
observable *operator commutation relation* as generally defined by the *Heisenberg Uncertainty Principle* in Quantum Mechanics.

Remark:
The family $([M]_{t})_{{t\in T}}$ and the associated class of its molecular transformations
can be thus employed to define a *category of molecular sets*, with composition defined by the
concatenation of sequential molecular transformations.

# 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 No. 3675 at cogprints.org/3675/01 as “Naturaltransfmolbionu6.pdf”. - 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

18E05*no label found*18-00

*no label found*

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