category of molecular reactions
A uni-molecular chemical reaction is represented by the natural transformations
, through the following commutative diagram:
![$\displaystyle \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d...
...[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}}$](img2.png) |
(0.1) |
with the states of 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 next. One need to define first the concept
of a molecular class variable. A molecular class variable, or
is defined as a family of molecular sets
with
being an indexing set, or class, defining the molecular range of variation of the
. Most applications in Physics, Chemistry or Biochemistry require that
is a finite set, (that is, without any sub-classes). A homomorphism 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:
![$\displaystyle \xymatrix@M=0.1pc @=4pc{Hom(A,A) \ar[r]^{f} \ar[d]_{e} & Hom(B,B)\ar[d]^{\gamma} \\ {Hom(A,A)} \ar[r]_{\delta} & {R},}$](img21.png) |
(0.2) |
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.
Definition 0.1
The
category of molecular sets is defined as the
category

whose objects are molecular sets

and whose morphisms are molecular transformations

.
Remark 0.1
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.
-
- 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 at cogprints.org with No. 3675.
- 4
-
Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural
and Regulatory Activities in Metabolic and Genetic Networks
FASEB Proceedings 43, 917.
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