0.1 Molecular Sets and Representations of Chemical Reactions
The uni-molecular chemical reaction is represented by the natural transformations
as specified by the following commutative diagram:
with the states of the molecular sets Au = a1,…,an and Bu = b1,…bn being represented by certain
endomorphisms in Hom(A,A) and Hom(B,B), respectively. In general, molecular sets MS are
defined as finite sets whose elements are ‘molecules’ defined in terms of their molecular observables
that are specified below. molecular class variables, or m.c.v’s are defined as families of molecular
sets [MS]i∈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 Mt : MS → MS of molecular sets, with t ∈ T being real time values, is defined as a
time-dependent mapping or function MS(t) also called a molecular transformation,
Mt.
An m.c.v. observable of B, characterizing the products of chemical type “B” of a chemical reaction
is defined as a morphism:
where ℜ is the set or field of real numbers. This mcv-observable is subject to the following
commutativity conditions:
with c : Au∗→B
u∗, and A
u∗, B
u∗ being, respectively, specially prepared fields of states of the
molecular sets Au, and Bu within a measurement uncertainty range, Δ, 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 Bu, in the case of a molecular set Au interacting with molecular set
Bu.
With these concepts and preliminary data one can now define the category of molecular sets and
their transformations as follows.
0.2 Category of molecular sets and their transformations
Definition 0.1. The category of molecular sets is defined as the category CM whose objects
are molecular sets MS and whose morphisms are molecular transformations Mt.
Remark 0.1. 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.
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 at cogprints.org with No. 3675.
[5] Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural and Regulatory
Activities in Metabolic and Genetic Networks FASEB Proceedings 43, 917.
