molecular set theory

molecular set theory

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Description:

Molecular set theory

(MST)

is a mathematical formulation of the wide-sense chemical kinetics of biomolecular reactions in terms of sets of molecules and their

chemical transformations

represented by set-theoretical

mappings

between

molecular sets

. In a more general sense, MST is the

theory of molecular

categories

defined as categories of molecular sets and their chemical transformations represented as set-theoretical mappings of molecular sets.

Molecular set theory was introduced by Anthony Bartholomay (

[

1

,

2

,

3

]

) and its applications were developed in Mathematical Biology and especially in Mathematical Medicine. The theory has also contributed to biostatistics and the formulation of clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to Physiology, Clinical Biochemistry and Medicine.
A precise mathematical

presentation

of the basic

concepts

in molecular set theory is as follows.

0.1

The Representation of Uni-Molecular Biochemical Reactions as Natural Transformations. Quantum Observables of a Molecular Class Variable

The

uni-molecular chemical reaction

is here represented by the

natural transformations

η

:

h

A

⟶

h

B

, through the following

commutative diagram

:

\xymatrix

⁢

@

⁢

M

=

0.1

⁢

p

⁢

c

⁢

@

=

4

⁢

p

⁢

c

⁢

h

A

⁢

(

A

)

=

H

⁢

(

A

,

A

)

⁢

\ar

⁢

[

r

]

η

A

⁢

\ar

⁢

[

d

]

h

A

⁢

(

t

)

⁢

&

⁢

h

B

⁢

(

A

)

=

H

⁢

(

B

,

A

)

⁢

\ar

⁢

[

d

]

h

B

⁢

(

t

)

⁢

h

A

⁢

(

B

)

=

H

⁢

(

A

,

B

)

⁢

\ar

⁢

[

r

]

η

B

⁢

&

⁢

h

B

⁢

(

B

)

=

H

⁢

(

B

,

B

)

(0.1)

with the states of the molecular sets

A

⁢

u

=

a

1

,

…

,

a

n

and

B

⁢

u

=

b

1

,

…

⁢

b

n

being represented by certain

endomorphisms

in H(A,A) and H(B,B), respectively.

The

observable of an

m

.

c

.

v

,

B

, characterizing the

products

“

B

” of a chemical reaction is defined as a

morphism

:

γ

:

H

⁢

(

B

,

B

)

⟶

R

,

where R is the set of real numbers. This

mcv-observable

is subject
to the following

commutativity

conditions:

\xymatrix

⁢

@

⁢

M

=

0.1

⁢

p

⁢

c

⁢

@

=

4

⁢

p

⁢

c

⁢

H

⁢

(

A

,

A

)

⁢

\ar

⁢

[

r

]

f

⁢

\ar

⁢

[

d

]

e

⁢

&

⁢

H

⁢

(

B

,

B

)

⁢

\ar

⁢

[

d

]

γ

⁢

H

⁢

(

A

,

A

)

⁢

\ar

⁢

[

r

]

δ

⁢

&

⁢

R

,

(0.2)

with

c

:

A

u

*

⟶

B

u

*

, and

A

u

*

,

B

u

*

being
specially prepared

fields of states

, within a measurement uncertainty range,

Δ

.

0.2

An Example of an Emerging Super-Complex Organism as A Quantum–Enzymatic System.

Note that in the case of either uni-molecular or multi-molecular,

reversible

reactions one obtains a

quantum-molecular

groupoid

, QG, defined as above in terms of the mcv-observables. In the case of an enzyme, E, with an activated complex,

(

E

⁢

S

)

*

,
a

quantum biomolecuar groupoid

can be uniquely defined in terms of mcv-observables for the enzyme, its activated complex

(

E

⁢

S

)

*

and the substrate, S. Quantum tunnelling in

(

E

⁢

S

)

*

then leads to the

separation

of the reaction product and the enzyme, E, which enters then a new reaction cycle with another substrate molecule S’,

indistinguishable

–or

equivalent

to–S. By considering a

sequence

of two such reactions coupled together,

Q

⁢

G

1

⇆

Q

⁢

G

2

,

corresponding to an enzyme f, coupled to a ribozyme

ϕ

, one obtains a

quantum-molecular realization of the simplest (

M,R)

-system,

(

f

,

ϕ

)

The non-reductionist caveat here is that the

relational systems

considered above
are open ones, exchanging both energy and mass with the system’s environment in a manner
which is dependent on time, for example in cycles, as the system ‘divides’–reproducing itself; therefore, even though generalized quantum-molecular observables can be defined
as specified above, neither a

stationary

nor a dynamic Schrödinger

equation

holds
for such examples of ‘super-complex’ systems. Furthermore, instead of just energetic constraints–such as the standard quantum

Hamiltonian

–one has the constraints imposed
by the diagram commutativity related to the mcv-observables, canonical

functors

and
natural transformations, as well as to the concentration gradients, diffusion processes,
chemical potentials/activities (molecular Gibbs free energies), enzyme kinetics, and so on. Both the canonical functors and the natural transformations defined above for uni- or
multi- molecular reactions

represent

the relational increase in complexity of the emerging,
super-complex dynamic system, such as, for example, the simplest (

M,R

)-system,

(

f

,

ϕ

)

.

Definition 0.1

.

Multi-Molecular Reactions

are defined by a

canonical functor

:

h

:

M

⟶

[

M

,

𝖲𝖾𝗍

]

which assigns to each molecular set

A

the functor

h

A

, and to each chemical transformation

t

:

A

⟶

B

, the natural transformation

η

A

⁢

B

:

h

A

⟶

h

B

.

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 No. 3675 at

cogprints.org/3675/01

as

Naturaltransfmolbionu6.pdf

.

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

molecular set theory

Canonical name

MolecularSetTheory

Date of creation

2013-03-22 18:11:37

Last modified on

2013-03-22 18:11:37

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

42

Author

bci1 (20947)

Entry type

Topic

Classification

msc 92E99

Classification

msc 92E20

Classification

msc 92B15

Classification

msc 92E10

Synonym

molecular reactions in organisms

Related topic

MolecularSetVariable

Related topic

AbstractRelationalBiology

Related topic

FunctionalBiology

Related topic

SupercategoriesOfComplexSystems

Related topic

GeneticNetsOrNetworks

Related topic

MolecularSetVariable

Related topic

CategoryOfMolecularSets

Related topic

SupercategoryOfMolecularSetVariables

Related topic

SupercategoryOfVariableMolecularSets

Defines

multi-molecular reactions

Defines

molecular class variable

Defines

wide-sense chemical kinetics in living systems and medical applications

Defines

category of molecular sets and their transformations representing chemical reactions

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