0.1 Molecular sets as representations of chemical reactions
A uni-molecular chemical reaction is defined by the natural transformations
specified in the following commutative diagram:
with the states of molecular sets Au = a1,…,an and Bu = b1,…bn being defined as the
endomorphism sets Hom(A,A) and Hom(B,B), respectively. In general, molecular sets MS are
defined as finite sets whose elements are molecules; the 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 molecular class variables is defined as a family of molecular sets [MS]i∈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, 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.
Classification: AMS MSC: 18D35 (category theory; homological algebra :: Categories with
structure :: Structured objects in a category ) 92B05 (Biology and other natural sciences ::
Mathematical biology in general :: General biology and biomathematics) 18E05 (Category theory;
homological algebra :: Abelian categories :: Preadditive, additive categories) 81-00 (quantum
theory :: General reference works )
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.
