category of (M,R)–systems

category of

(

M

,

R

)

–systems

0.1

Metabolic-Replication Systems

http://planetphysics.org/encyclopedia/RobertRosen.html

Robert Rosen

introduced

metabolic–repair models

, or

(

M

,

R

)

-systems in mathematical biology

http://planetphysics.org/encyclopedia/AbstractRelationalBiologyARB.html

(

abstract relational biology

) in 1957 (

[

4

,

5

]

); such systems will be here abbreviated as

M

⁢

R

-systems, (or simply

M

⁢

R

’s). Rosen, then represented the

M

⁢

R

’s in terms of

categories of sets

, deliberately selected without any

structure

other than the

discrete topology of sets

.

Theoreticians of life’s origins

postulate

that Life on Earth has begun with the simplest possible organism, called the

primordial

. Mathematicians interested in biology and this important question of the

minimal

living organism have
attempted to define the functional relations that would have made life possible in a such a minimal system– a grandad
and granma of all living organisms on Earth.

Definition 0.1

.

The simplest

M

⁢

R

-system is a relational model of the primordial organism which is defined by the following

categorical sequence

(or

diagram

) of sets and set-theoretical

mappings

:

f

:

A

→

B

,

ϕ

:

B

→

H

⁢

o

⁢

m

M

⁢

R

⁢

(

A

,

B

)

, where

A

is the set of inputs to the

M

⁢

R

-system,

B

is the set of its outputs, and

ϕ

is the ‘repair map’, or

R

-

component

, of the

M

⁢

R

-system which associates to a certain

product

, or output

b

, the ‘metabolic’ component (such as an enzyme, E, for example)
represented by the set-theoretical mapping

f

. Then,

H

⁢

o

⁢

m

M

⁢

R

⁢

(

A

,

B

)

is defined as the set of all such metabolic (set-theoretical) mappings (occasionally written incorrectly by some authors as

{

f

}

).

Definition 0.2

.

A

general

(

M

,

R

)

-system

was defined by Rosen (1958a,b) as the network or graph of the metabolic and repair components that were specified above in

Definition

0.1

; such components are networked in a complex, abstract ‘organism’ defined by all the abstract

relations

and connecting maps between the sets specifying all the metabolic and repair components of such a general, abstract model of the biological organism. The mappings bettwen

(

M

,

R

)

-systems are defined as the the metabolic and repair set-theoretical mappings, such as

f

and

ϕ

(specified in

Definition 0.1

); moreover, there is also a finite number of sets (just like those that are defined as in

Definition 0.1

):

A

i

,

B

i

, whereas

f

∈

H

⁢

o

⁢

m

M

⁢

R

i

⁢

(

A

i

,

B

i

)

and

ϕ

∈

H

⁢

o

⁢

m

M

⁢

R

i

⁢

[

B

,

H

⁢

o

⁢

m

M

⁢

R

i

⁢

(

A

i

,

B

i

)

]

, with

i

∈

I

, and

I

being a finite

index set

, or

directed set

, with

(

f

,

ϕ

)

being a finite number of distinct metabolic and repair components pairs. Alternatively, one may think of a a general

M

⁢

R

-system as being ‘made of’ a finite number

N

of interconnected

M

⁢

R

i

, metabolic-repair modules with input sets

A

i

and output sets

B

i

. To sum up:
a

general MR-system

can be defined as a

family of interconnected quartets

:

{

(

A

i

,

B

i

,

f

i

,

ϕ

i

)

}

i

∈

I

, where

I

is an index set of integers

i

=

1

,

2

,

…

,

n

.

0.2

Category of (M,R)–systems

Definition 0.3

.

A

category

of

(

M

,

R

)

-system quartet modules

,

{

(

A

i

,

B

i

,

f

i

,

ϕ

i

)

}

i

∈

I

, with I being an index set of integers

i

=

1

,

2

,

…

,

n

, is a

small category

of sets with set-theoretical mappings defined by the MR-morphisms between the quarted modules

{

(

A

i

,

B

i

,

f

i

,

ϕ

i

)

}

i

∈

I

, and also with repair components defined as

ϕ

i

∈

H

⁢

o

⁢

m

M

⁢

R

i

⁢

[

B

,

H

⁢

o

⁢

m

M

⁢

R

i

⁢

(

A

i

,

B

i

)

]

, with the

(

M

,

R

)

-

morphism

composition

defined by the usual composition of functions between sets.

With a few, additional notational changes it can be shown that the category of

(

M

,

R

)

-systems
is a

subcategory

of the

category of automata

(or

sequential machines

),

𝒮

[

M

,

A

]

(

[

7

,

8

]

).

0.2.1

Remarks:

For over two decades, Robert Rosen developed with several coworkers the MR-systems theory and its applications
to life sciences, medicine and

general systems

theory. He also considered biocomplexity to be an ‘emergent’, defining feature of organisms which is

not reducible in terms of the molecular structures

(or molecular components) of the organism and their physicochemical interactions. However, in his last written book in 1997 on

“Essays on Life Itself”, published posthumously in 2000

, Robert Rosen finally accepted the need for representing organisms in terms of

categories with structure

that entail biological functions, both metabolic and repair ones. Note also that, unlike Rashevsky in his theory of

organismic sets

, Rosen did not attempt to extend the

M

⁢

R

s to modeling societies, even though with appropriate modifications of

generalized

(

M

,

R

)

-system categories with structure

(

[

7

,

8

,

13

]

), this is feasible and yields meaningful mathematical and sociological results.
Thus, subsequent publications have generalized MR-system (GMRs) and have studied the fundamental, mathematical

properties

of

algebraic categories

of GMRs that were constructed functorially based on the Yoneda-Grothendieck Lemma
and construction. Then it was shown that such algebraic categories of GMRs are

Cartesian closed

[

7

]

.
Several

molecular biology realizations of GMRs

in terms of DNA, RNAs, enzymes,

R

⁢

N

⁢

A

→

D

⁢

N

⁢

A

-reverse trancriptases, and other biomolecular components were subsequently introduced and discussed in ref.

[

21

,

13

,

14

]

in terms of

non-linear

genetic network

models

in many-valued,

L

⁢

M

n

logic algebras

(or algebraic category

ℒ

⁢

ℳ

of

L

⁢

M

n

logic algebras (

http://planetmath.org/AlgebraicCategoryOfLMnLogicAlgebras

)).

If simple

(

M

,

R

)

-systems are considered as sequential machines or

automata

the category of

(

M

,

R

)

-systems and

(

M

,

R

)

-system

homomorphisms

is a subcategory of the automata category. However, when

(

M

,

R

)

-systems are considered together with their dynamic

representations

the category of dynamic

(

M

,

R

)

-systems is no longer a subcategory of the category of automata.

References

1

Rashevsky, N.: 1965, The Representation of Organisms in Terms of

Predicates

,

Bulletin of

Mathematical Biophysics

27

: 477-491.

2

Rashevsky, N.: 1969, Outline of a Unified Approach to Physics, Biology and Sociology.,

Bulletin of Mathematical Biophysics

31

: 159–198.

3

Rosen, R.: 1985,

Anticipatory Systems

, Pergamon Press: New York.

4

Rosen, R.: 1958a, A Relational Theory of Biological Systems

Bulletin of Mathematical Biophysics

20

: 245-260.

5

Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the
Theory of Categories.,

Bulletin of Mathematical Biophysics

20

: 317-341.

6

Rosen, R.: 1987, On

Complex Systems

,

European Journal of Operational Research

30

:129–134.

7

Baianu, I.C.: 1973, Some Algebraic Properties of

(M,R)

– Systems.

Bulletin of Mathematical Biophysics

35

, 213-217.

8

Baianu, I.C. and M. Marinescu: 1974, On A Functorial Construction of

(M,R)

– Systems.

Revue Roumaine de Mathematiques Pures et Appliquées

19

: 388-391.

9

Baianu, I.C.: 1980,

Natural Transformations of Organismic Structures

.,

Bulletin of Mathematical Biology

,

42

: 431-446.

10

I.C. Baianu: 1977, A Logical Model of Genetic Activities in Łukasiewicz

Algebras

: The Non-linear Theory.

Bulletin of Mathematical Biophysics

,

39

: 249-258.

11

I.C. Baianu: 1983,

Natural Transformation

Models in Molecular Biology., in

Proceedings of the SIAM Natl. Meet

., Denver, CO.;

http://cogprints.org/3675/1/Naturaltransfmolbionu6.pdf

An Eprint is here available .

12

I.C. Baianu: 1984, A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and Genetic Networks.,

FASEB Proceedings

43

, 917.

13

I.C. Baianu: 1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.),

Mathematical Models in Medicine

, vol. 7., Pergamon Press, New York, 1513–1577;

http://documents.cern.ch/cgi-bin/setlink?base=preprint&categ=ext&id=ext-2004-067

CERN Preprint No. EXT-2004-072:.

14

I.C. Baianu: 1987b, Molecular Models of Genetic and Organismic Structures, in

Proceed. Relational Biology Symp.

Argentina;

http://documents.cern.ch/cgi-bin/setlink?base=preprint&categ=ext&id=ext-2004-067

CERN Preprint No.EXT-2004-067:MolecularModelsICB3.doc.

15

I.C. Baianu, Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and
N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks,

(M,R)

–Systems and
Their

Higher Dimensional Algebra

,

http://www.ag.uiuc.edu/fs401/QAuto.pdf

Abstract of Report is here available as a PDF and

http://doc.cern.ch/archive/electronic/other/ext/ext-2004-058/QuantumAutnu3_ICB.pdf

html document

16

R. Brown, J. F. Glazebrook and I. C. Baianu: A

categorical

and higher dimensional algebra framework for complex systems and spacetime structures,

Axiomathes

17

:409–493.
(2007).

17

L. L

o

¨

fgren: 1968. On Axiomatic Explanation of

Complete

Self–Reproduction.

Bull. Math. Biophysics

,

30

: 317–348.

18

Baianu, I.C.: 2004a. Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints–Sussex Univ.

19

Baianu, I.C.: 2004b Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059.

Health Physics and Radiation Effects

(June 29, 2004).

20

Baianu, I. C.: 2006, Robert Rosen’s Work and Complex Systems Biology,

Axiomathes

16

(1–2):25–34.

21

Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos:

Transformations

of Neuronal, Genetic and Neoplastic Networks.,

Axiomathes

,

16

Nos. 1–2: 65–122.

Title

category of

(

M

,

R

)

–systems

Canonical

name

CategoryOfMRsystems

Date of creation

2013-03-22 18:18:21

Last modified on

2013-03-22 18:18:21

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

64

Author

bci1 (20947)

Entry type

Topic

Classification

msc 92B99

Classification

msc 18-00

Classification

msc 93A10

Classification

msc 93A30

Classification

msc 92B20

Classification

msc 92B05

Synonym

MR-systems

Synonym

metabolic-replication systems

Related topic

MathematicalBiology

Related topic

SystemDefinitions

Related topic

ArtificialInteglligence

Related topic

ComplexSystemsBiology

Related topic

IndexOfCategories

Defines

MR-system

Defines

general MR-system

Defines

MR-quartet

Defines

morphism of MR-quartets

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