category of molecular sets

category of molecular sets

0.1

Molecular sets as representations of chemical reactions

A

uni-molecular chemical reaction

is defined by the

natural transformations

η

:

h

A

⟶

h

B

,

specified in the following

commutative diagram

representing

molecular sets

and their quantum

transformations

:

\xymatrix

⁢

@

⁢

M

=

0.1

⁢

p

⁢

c

⁢

@

=

4

⁢

p

⁢

c

⁢

h

A

⁢

(

A

)

=

H

⁢

o

⁢

m

⁢

(

A

,

A

)

⁢

\ar

⁢

[

r

]

η

A

⁢

\ar

⁢

[

d

]

h

A

⁢

(

t

)

⁢

&

⁢

h

B

⁢

(

A

)

=

H

⁢

o

⁢

m

⁢

(

B

,

A

)

⁢

\ar

⁢

[

d

]

h

B

⁢

(

t

)

⁢

h

A

⁢

(

B

)

=

H

⁢

o

⁢

m

⁢

(

A

,

B

)

⁢

\ar

⁢

[

r

]

η

B

⁢

&

⁢

h

B

⁢

(

B

)

=

H

⁢

o

⁢

m

⁢

(

B

,

B

)

,

(0.1)

with the

states of molecular sets

A

u

=

a

1

,

…

,

a

n

and

B

u

=

b

1

,

…

⁢

b

n

being defined as the

endomorphism

sets

H

⁢

o

⁢

m

⁢

(

A

,

A

)

and

H

⁢

o

⁢

m

⁢

(

B

,

B

)

, respectively. In general,

molecular sets

M

S

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

[

M

S

]

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

,

M

t

:

M

S

→

M

S

of molecular sets, with

t

∈

T

being real time values, is defined as a time-dependent mapping or

function

M

S

⁢

(

t

)

also called a

molecular transformation

,

M

t

.

An

m

.

c

.

v

.

observable

of

B

, characterizing the

products

of chemical type “B” of a chemical reaction is defined as a morphism:

γ

:

H

⁢

o

⁢

m

⁢

(

B

,

B

)

⟶

ℜ

,

where

ℜ

is the set or field of real numbers. This mcv-observable is subject
to the following

commutativity

conditions:

\xymatrix

⁢

@

⁢

M

=

0.1

⁢

p

⁢

c

⁢

@

=

4

⁢

p

⁢

c

⁢

H

⁢

o

⁢

m

⁢

(

A

,

A

)

⁢

\ar

⁢

[

r

]

f

⁢

\ar

⁢

[

d

]

e

⁢

&

⁢

H

⁢

o

⁢

m

⁢

(

B

,

B

)

⁢

\ar

⁢

[

d

]

γ

⁢

H

⁢

o

⁢

m

⁢

(

A

,

A

)

⁢

\ar

⁢

[

r

]

δ

⁢

&

⁢

R

,

(0.2)

with

c

:

A

u

*

⟶

B

u

*

, and

A

u

*

,

B

u

*

being, respectively,
specially prepared

fields of states

of the molecular sets

A

u

, and

B

u

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

B

u

, in the case of a molecular set

A

u

interacting with molecular set

B

u

.

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

C

M

whose objects are molecular sets

M

S

and whose morphisms are molecular transformations

M

t

.

Remark 0.1

.

This is a

mathematical representation

of chemical reaction systems in terms of molecular sets that vary with time
(or

m

⁢

s

⁢

v

’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.

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

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