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category of molecular sets
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The uni-molecular chemical reaction is represented by the natural transformations
 , through the following commutative diagram:
![$\displaystyle \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d... ...[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}}$](http://images.planetmath.org:8080/cache/objects/10868/l2h/img2.png "$\displaystyle \xymatrix@M=0.1pc @=4pc{h^A(A) = Hom(A,A) \ar[r]^{\eta_{A}} \ar[d... ...[d]^{h^B (t)} \\ {h^A (B) = Hom(A,B)} \ar[r]_{\eta_{B}} & {h^B (B) = Hom(B,B)}}$") |
(0.1) |
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
![$ [M_S]_{i \in I}$](http://images.planetmath.org:8080/cache/objects/10868/l2h/img9.png "$ [M_S]_{i \in I}$") , with  being an indexing set, or class, defining the range of molecular variation of the  ; most applications require that  is a proper, finite set, (i.e., without any sub-classes). A morphism
 of molecular sets, with  being real time values, is defined as a time-dependent mapping or function  also called a  molecular transformation.
An  observable of  , characterizing the products of chemical type “B” of a chemical reaction is defined as a morphism:
where  is the set or field of real numbers. This mcv-observable is subject to the following commutativity conditions:
![$\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},}$](http://images.planetmath.org:8080/cache/objects/10868/l2h/img21.png "$\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},}$") |
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with
 , and  ,  being, respectively, specially prepared fields of states of the molecular sets  , and  within a measurement uncertainty range,  , which is determined by Heisenberg's uncertainty relation,
or the commutator of the observable operators involved, such as
![$ [A^*, B^*]$](http://images.planetmath.org:8080/cache/objects/10868/l2h/img28.png "$ [A^*, B^*]$") , 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: 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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"category of molecular sets" is owned by bci1.
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See Also: molecular set and molecular class variables, molecular set theory, categories and supercategories in relational biology, complex systems biology, abstract relational biology, supercategory of variable molecular sets
| Other names: |
class of molecular set variables and their transformations |
| Also defines: |
category of molecular sets, molecular set, chemical transformation, molecular set variable, molecular transformation |
| Keywords: |
molecular set theory, molecular set variables and their transformations |
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Cross-references: mathematical representation, objects, category, transformations, operators, commutator, relation, commutativity, field, type, products, function, mapping, real, morphism, applications, variation, range, class, indexing set, molecular class variables, terms, finite sets, endomorphisms, commutative diagram, natural transformations
There are 8 references to this entry.
This is version 23 of category of molecular sets, born on 2008-07-25, modified 2008-10-11.
Object id is 10868, canonical name is CategoryOfMolecularSets.
Accessed 746 times total.
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 ) |
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