category of molecular sets

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 diagramMathworldPlanetmath representing molecular sets and their quantum transformationsPlanetmathPlanetmath:

\xymatrix@M=0.1pc@=4pchA(A)=Hom(A,A)\ar[r]ηA\ar[d]hA(t)&hB(A)=Hom(B,A)\ar[d]hB(t)hA(B)=Hom(A,B)\ar[r]ηB&hB(B)=Hom(B,B), (0.1)

with the states of molecular sets Au=a1,,an and Bu=b1,bn being defined as the endomorphismPlanetmathPlanetmathPlanetmath sets Hom(A,A) and Hom(B,B), respectively. In general, molecular sets MS are defined as finite setsMathworldPlanetmath 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]iI, with I being either an indexing set, or a proper classMathworldPlanetmath, 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 morphismMathworldPlanetmath, or molecular mapping, Mt:MSMS of molecular sets, with tT 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 productsPlanetmathPlanetmathPlanetmath 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:

\xymatrix@M=0.1pc@=4pcHom(A,A)\ar[r]f\ar[d]e&Hom(B,B)\ar[d]γHom(A,A)\ar[r]δ&R, (0.2)

with c:Au*Bu*, and Au*, Bu* 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 relationMathworldPlanetmathPlanetmathPlanetmath, or the commutatorPlanetmathPlanetmath of the observable operators involved, such as [A*,B*], associated with the observable A of molecular set Au, 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.


  • 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 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.
Title category of molecular sets
Canonical name CategoryOfMolecularSets
Date of creation 2013-03-22 18:16:02
Last modified on 2013-03-22 18:16:02
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 38
Author bci1 (20947)
Entry type Topic
Classification msc 81-00
Classification msc 18E05
Classification msc 92B05
Classification msc 18D35
Synonym class of molecular set variables and their transformations
Synonym mcv-observable
Related topic MolecularSetVariable
Related topic MolecularSetTheory
Related topic SupercategoriesOfComplexSystems
Related topic ComplexSystemsBiology
Related topic SupercategoryOfVariableMolecularSets
Related topic IndexOfCategories
Related topic CategoryOfQuantumAutomata
Defines category of molecular sets
Defines molecular set
Defines molecular set mapping
Defines states of molecular sets
Defines uni-molecular chemical reaction
Defines chemical transformation
Defines molecular set variable
Defines molecular transformation
Defines m.c.v. observable
Defines mcv-observable
Defines states of molecular se