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PlanetPhysics/Category of Molecular Sets 2

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Molecular Sets and Representations of Chemical Reactions

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The uni-molecular chemical reaction is represented by the natural transformations as specified by the following commutative diagram:

Failed to parse (unknown function "\def"): {\displaystyle \def\labelstyle{\textstyle} \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d]_{h^A(t)} & h^B (A) = Hom(B,A)\ar[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}} }

with the states of the 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 below. molecular class variables, or 's are defined as families of molecular sets , with being an indexing set, or class, defining the range of molecular variation of the  ; most applications require that is a proper, finite set, (i.e., without any sub-classes). A morphism 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:

Failed to parse (unknown function "\def"): {\displaystyle \def\labelstyle{\textstyle} \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},} }

~

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.

Category of molecular sets and their transformations

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The category of molecular sets is defined as the category whose objects are molecular sets and whose morphisms are molecular transformations .

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.

All Sources

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[1] [2] [3] [4] [4]

References

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  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. 4.0 4.1 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. Cite error: Invalid <ref> tag; name "ICB2" defined multiple times with different content