symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

symmetric monoidal (bi)categorieswith feedback and biological

networks

E Pareja-Tobes M Manrique R Tobes E Pareja

Era7 bioinformatics

Sysbiol 2008December 1, 2008

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Outline

Introductionwhy categories?

What is category theory?Categories: objects and relationsn-categories: objects, relations, relationsbetween relations, . . .

Symmetric monoidal categories with feedback andbiological networks

Example: Quorum sensing in Vibrio harveyiRelationship with other approaches

Work in progress and future directionsWork in progress

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Outline

Introductionwhy categories?

What is category theory?Categories: objects and relationsn-categories: objects, relations, relationsbetween relations, . . .

Symmetric monoidal categories with feedback andbiological networks

Example: Quorum sensing in Vibrio harveyiRelationship with other approaches

Work in progress and future directionsWork in progress

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Outline

Introductionwhy categories?

What is category theory?Categories: objects and relationsn-categories: objects, relations, relationsbetween relations, . . .

Symmetric monoidal categories with feedback andbiological networks

Example: Quorum sensing in Vibrio harveyiRelationship with other approaches

Work in progress and future directionsWork in progress

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Outline

Introductionwhy categories?

What is category theory?Categories: objects and relationsn-categories: objects, relations, relationsbetween relations, . . .

Symmetric monoidal categories with feedback andbiological networks

Example: Quorum sensing in Vibrio harveyiRelationship with other approaches

Work in progress and future directionsWork in progress

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

why categories?

Systems biology

imposes a

Relational view of biology

emphasis on

processes→ compositionality

mathematical framework?

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

why categories?

Systems biology

imposes a

Relational view of biology

emphasis on

processes→ compositionality

mathematical framework?

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

why categories?

Systems biology

imposes a

Relational view of biology

emphasis on

processes→ compositionality

mathematical framework?

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

why categories?

Systems biology

imposes a

Relational view of biology

emphasis on

processes→ compositionality

mathematical framework?

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

why categories?

Systems biology

imposes a

Relational view of biology

emphasis on

processes→ compositionality

mathematical framework?

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

why categories?

Systems biology

imposes a

Relational view of biology

emphasis on

processes→ compositionality

mathematical framework?

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Categories

I objects

I relations

I composition

I + some axioms

A

B

C

DE

f

g

g  f

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Categories

I objects

I relations

I composition

I + some axioms

A

B

C

DE

f

g

g  f

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Categories

I objects

I relations

I composition

I + some axioms

A

B

C

DE

f

g

g  f

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Categories

I objects

I relations

I composition

I + some axioms

A

B

C

DE

f

g

g  f

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Categories

I objects

I relations

I composition

I + some axioms

A

B

C

DE

f

g

g  f

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Categories

I objects

I relations

I composition

I + some axioms

A

B

C

DE

f

g

g  f

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Categories

I objects

I relations

I composition

I + some axioms

A

B

C

DE

f

g

g  f

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Categories

I objects

I relations

I composition

I + some axioms

A

B

C

DE

f

g

g  f

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)

I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)

I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)

I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)

I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relations

I relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relations

I relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)

I 2 different compositions of 2-cells:I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)

I 2 different compositions of 2-cells:I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequential

I horizontal ≡ parallelI + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequential

I horizontal ≡ parallelI + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Bicategories

I objects (0-cells)I relations (1-cells)I composition of relationsI relations between relations (2-cells)I 2 different compositions of 2-cells:

I vertical ≡ sequentialI horizontal ≡ parallel

I + some (more complex) axioms

A

B

C

DE

f

g

g  f

f

g

α

β

βα

f

α β

f'

gg'

g'   g

f'   f

β*α

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

n-categories

model relations between relations between . . .

I definition: active area of research!

see for example

Higher-Dimensional Categories: an illustrated guide book Cheng, E. Lauda, A.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

n-categories

model relations between relations between . . .

I definition: active area of research!

see for example

Higher-Dimensional Categories: an illustrated guide book Cheng, E. Lauda, A.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

n-categories

model relations between relations between . . .

I definition: active area of research!

see for example

Higher-Dimensional Categories: an illustrated guide book Cheng, E. Lauda, A.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

n-categories

model relations between relations between . . .

I definition: active area of research!

see for example

Higher-Dimensional Categories: an illustrated guide book Cheng, E. Lauda, A.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

symmetric monoidal categories withfeedback

defined by Walters et al as a framework for themodelling of concurrent and distributed processes.

I Bicategories of processes Katis P. Sabadini N. Walters R. 1997

I On the algebra of systems with feedback and boundary Katis P. Sabadini N.

Walters R. 2000

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

symmetric monoidal categories withfeedback

defined by Walters et al as a framework for themodelling of concurrent and distributed processes.

I Bicategories of processes Katis P. Sabadini N. Walters R. 1997

I On the algebra of systems with feedback and boundary Katis P. Sabadini N.

Walters R. 2000

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

symmetric monoidal categories withfeedback

defined by Walters et al as a framework for themodelling of concurrent and distributed processes.

I Bicategories of processes Katis P. Sabadini N. Walters R. 1997

I On the algebra of systems with feedback and boundary Katis P. Sabadini N.

Walters R. 2000

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

symmetric monoidal

There is an operation, ⊗, which acts on

objects:

A, B 7→ A⊗ B

and 1-cells:

(A f−→ B, Cg−→ D) 7→ A⊗C

f⊗g−−→ B ⊗D

≡ parallel composition

+ associativity, unit, and symmetry

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

symmetric monoidal

There is an operation, ⊗, which acts on

objects:

A, B 7→ A⊗ B

and 1-cells:

(A f−→ B, Cg−→ D) 7→ A⊗C

f⊗g−−→ B ⊗D

≡ parallel composition

+ associativity, unit, and symmetry

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

symmetric monoidal

There is an operation, ⊗, which acts on

objects:

A, B 7→ A⊗ B

and 1-cells:

(A f−→ B, Cg−→ D) 7→ A⊗C

f⊗g−−→ B ⊗D

≡ parallel composition

+ associativity, unit, and symmetry

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

interpretation

input source

A1 ⊗ . . .⊗ An

A1

A2A3

An

output target

B1 ⊗ . . .⊗ Bm

B1

B2B3

Bm

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

interpretation

process 1-cell

A1 ⊗ . . .⊗ Anp−→ B1 ⊗ . . .⊗ Bm

A1

A2A3

An

P B1

B2B3

Bm

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

interpretation

sequential composition composition of 1-cells

A1 ⊗ . . .⊗ Anp1 //

p2◦p1 ((QQQQQQQQQQQQ B1 ⊗ . . .⊗ Bm

p2

��C1 ⊗ . . .⊗Ck

A1

A2A3

An

P1 P2B1

B2B3

Bm

C1

C2C3

Ck

parallel composition tensor(A1 ⊗ . . .⊗ An)⊗ (C1 ⊗ . . .⊗Ck)

p1⊗p2

��(B1 ⊗ . . .⊗ Bm)⊗ (D1 ⊗ . . .⊗ Dj)

A1

A2A3

An

P1B1

B2B3

Bm

P2D1

D2D3

Dh

C1

C2C3

Ck

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

interpretation

feedback feedback

A⊗ Up //

��

B ⊗ U

AfbU (f )

// B

A1

A2A3

An

P1B1

B2B3

Bm

UU

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Quorum sensing in Vibrio harveyi

why?

I metabolic, transcriptional and signalingphenomena involved

I data availableI enough complexity as a test for this kind of

approach

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Quorum sensing in Vibrio harveyi

why?

I metabolic, transcriptional and signalingphenomena involved

I data availableI enough complexity as a test for this kind of

approach

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Quorum sensing in Vibrio harveyi

why?

I metabolic, transcriptional and signalingphenomena involved

I data availableI enough complexity as a test for this kind of

approach

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Quorum sensing in Vibrio harveyi

why?

I metabolic, transcriptional and signalingphenomena involved

I data availableI enough complexity as a test for this kind of

approach

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Quorum sensing in Vibrio harveyi

why?

I metabolic, transcriptional and signalingphenomena involved

I data availableI enough complexity as a test for this kind of

approach

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Quorum sensing in Vibrio harveyi

why?

I metabolic, transcriptional and signalingphenomena involved

I data availableI enough complexity as a test for this kind of

approach

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

available models?

At negligible concentrations of AIs, i.e. at low celldensity (LCD), the three sensors act as kinases that trans-fer phosphate through LuxU to LuxO (Freeman andBassler, 1999a,b; Lilley and Bassler, 2000). LuxO~P acti-vates the expression of genes encoding five highly con-served small regulatory RNAs (sRNAs) called Qrr1–5(Tu and Bassler, 2007). The Qrrs pair with the 5′ UTR ofthe luxR mRNA and destabilize it, a process that requiresthe RNA chaperone Hfq (Lenz et al., 2004). LuxR is themaster transcriptional regulator of QS genes in V. harveyi(Showalter et al., 1990; Swartzman et al., 1992). Thus, atLCD, when little LuxR is present, there is no QS andV. harveyi cells act as individuals. At high cell density(HCD), AIs accumulate and bind to their cognate sensors.This event causes the sensors to act as phosphatases,leading to dephosphorylation of LuxO. UnphosphorylatedLuxO is inactive. Transcription of the sRNA-encodinggenes is terminated, causing luxR mRNA to accumulate(Freeman and Bassler, 1999b; Lilley and Bassler, 2000).Newly produced LuxR protein activates and repressesnumerous genes. Most notably, LuxR activates the lux-CDABE operon, encoding luciferase, which is required forbioluminescence (Miyamoto et al., 1994). Thus, at HCD,QS is initiated and V. harveyi cells act as a group.

Most of the regulatory components of the QS circuithave been defined in V. harveyi, allowing us to begin toanalyse the features of the QS-signalling network thatoptimize V. harveyi’s ability to respond to differing com-munity conditions. Here, we report the discovery of anegative feedback loop in the V. harveyi QS regulatory

cascade involving LuxR and the Qrr sRNAs. We show thatLuxR directly binds to and activates transcription of thepromoters preceding qrr2, qrr3 and qrr4, but not qrr1 orqrr5. This leads to increased destabilization of luxRmRNA and downregulation of LuxR production. Mutationof the consensus LuxR-binding sites in the V. harveyi qrr2,qrr3 and qrr4 promoters disrupts the negative feedbackloop and affects the timing of the transition from HCD toLCD mode and vice versa. In the closely related speciesVibrio cholerae, we previously characterized a negativefeedback loop consisting of HapR (the LuxR homologue)and the V. cholerae Qrr sRNAs (Svenningsen et al.,2008). However, in V. cholerae, the mechanism by whichHapR feeds back to activate qrr expression is distinct fromV. harveyi. Together, our studies suggest that LuxR/HapR-sRNA-mediated negative feedback is essential foroptimizing the dynamics of the transitions between indi-vidual and group behaviours in Vibrios.

Results

LuxR binds to the promoters of qrr2, qrr3 and qrr4

HapR, the V. cholerae homologue of V. harveyi LuxR,activates the expression of the V. cholerae qrr genesthrough an indirect mechanism (i.e. HapR does not bindthe qrr promoters directly). The HapR-sRNA-mediatedfeedback loop only operates during the HCD to LCDtransition, when both HapR and LuxO~P are present,and in so doing, it functions to accelerate the transitionof V. cholerae out of social mode into individual cell

Fig. 1. Model of the V. harveyiQuorum-Sensing Circuit. V. harveyi producesand detects three AIs and through modulationof the levels of the master transcriptionalregulator, LuxR, controls downstreamQS-target genes. The three AIs are: CAI-1(circles) which binds to CqsS, HAI-1(pentagons) which binds to LuxN and AI-2(double pentagons) which binds to LuxPQ.At LCD, when LuxO is phosphorylated(LuxO~P), it activates transcription of thegenes encoding the five Qrr sRNAs whichwork in conjuction with Hfq to destabilize themRNA of luxR. At HCD, when LuxO is notphosphorylated, qrr transcription ceases,luxR mRNA is stabilized and LuxR protein isproduced. In a feedback loop, LuxR activatesexpression of qrr2, qrr3 and qrr4, whichaffects the timing of the QS transitions.OM, outer membrane; IM, inner membrane.

Vibrio sRNA-mediated feedback 897

© 2008 The AuthorsJournal compilation © 2008 Blackwell Publishing Ltd, Molecular Microbiology, 70, 896–907

A small-RNA-mediated negative feedback loop controls quorum-sensing dynamics in Vibrio harveyi. Tu KC,

Waters CM, Svenningsen SL, Bassler BL. Mol Microbiol. 2008. 70, 896-907

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

available models?

At negligible concentrations of AIs, i.e. at low celldensity (LCD), the three sensors act as kinases that trans-fer phosphate through LuxU to LuxO (Freeman andBassler, 1999a,b; Lilley and Bassler, 2000). LuxO~P acti-vates the expression of genes encoding five highly con-served small regulatory RNAs (sRNAs) called Qrr1–5(Tu and Bassler, 2007). The Qrrs pair with the 5′ UTR ofthe luxR mRNA and destabilize it, a process that requiresthe RNA chaperone Hfq (Lenz et al., 2004). LuxR is themaster transcriptional regulator of QS genes in V. harveyi(Showalter et al., 1990; Swartzman et al., 1992). Thus, atLCD, when little LuxR is present, there is no QS andV. harveyi cells act as individuals. At high cell density(HCD), AIs accumulate and bind to their cognate sensors.This event causes the sensors to act as phosphatases,leading to dephosphorylation of LuxO. UnphosphorylatedLuxO is inactive. Transcription of the sRNA-encodinggenes is terminated, causing luxR mRNA to accumulate(Freeman and Bassler, 1999b; Lilley and Bassler, 2000).Newly produced LuxR protein activates and repressesnumerous genes. Most notably, LuxR activates the lux-CDABE operon, encoding luciferase, which is required forbioluminescence (Miyamoto et al., 1994). Thus, at HCD,QS is initiated and V. harveyi cells act as a group.

Most of the regulatory components of the QS circuithave been defined in V. harveyi, allowing us to begin toanalyse the features of the QS-signalling network thatoptimize V. harveyi’s ability to respond to differing com-munity conditions. Here, we report the discovery of anegative feedback loop in the V. harveyi QS regulatory

cascade involving LuxR and the Qrr sRNAs. We show thatLuxR directly binds to and activates transcription of thepromoters preceding qrr2, qrr3 and qrr4, but not qrr1 orqrr5. This leads to increased destabilization of luxRmRNA and downregulation of LuxR production. Mutationof the consensus LuxR-binding sites in the V. harveyi qrr2,qrr3 and qrr4 promoters disrupts the negative feedbackloop and affects the timing of the transition from HCD toLCD mode and vice versa. In the closely related speciesVibrio cholerae, we previously characterized a negativefeedback loop consisting of HapR (the LuxR homologue)and the V. cholerae Qrr sRNAs (Svenningsen et al.,2008). However, in V. cholerae, the mechanism by whichHapR feeds back to activate qrr expression is distinct fromV. harveyi. Together, our studies suggest that LuxR/HapR-sRNA-mediated negative feedback is essential foroptimizing the dynamics of the transitions between indi-vidual and group behaviours in Vibrios.

Results

LuxR binds to the promoters of qrr2, qrr3 and qrr4

HapR, the V. cholerae homologue of V. harveyi LuxR,activates the expression of the V. cholerae qrr genesthrough an indirect mechanism (i.e. HapR does not bindthe qrr promoters directly). The HapR-sRNA-mediatedfeedback loop only operates during the HCD to LCDtransition, when both HapR and LuxO~P are present,and in so doing, it functions to accelerate the transitionof V. cholerae out of social mode into individual cell

Fig. 1. Model of the V. harveyiQuorum-Sensing Circuit. V. harveyi producesand detects three AIs and through modulationof the levels of the master transcriptionalregulator, LuxR, controls downstreamQS-target genes. The three AIs are: CAI-1(circles) which binds to CqsS, HAI-1(pentagons) which binds to LuxN and AI-2(double pentagons) which binds to LuxPQ.At LCD, when LuxO is phosphorylated(LuxO~P), it activates transcription of thegenes encoding the five Qrr sRNAs whichwork in conjuction with Hfq to destabilize themRNA of luxR. At HCD, when LuxO is notphosphorylated, qrr transcription ceases,luxR mRNA is stabilized and LuxR protein isproduced. In a feedback loop, LuxR activatesexpression of qrr2, qrr3 and qrr4, whichaffects the timing of the QS transitions.OM, outer membrane; IM, inner membrane.

Vibrio sRNA-mediated feedback 897

© 2008 The AuthorsJournal compilation © 2008 Blackwell Publishing Ltd, Molecular Microbiology, 70, 896–907

A small-RNA-mediated negative feedback loop controls quorum-sensing dynamics in Vibrio harveyi. Tu KC,

Waters CM, Svenningsen SL, Bassler BL. Mol Microbiol. 2008. 70, 896-907

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

available models?

kinase-response regulator two-component system.The kinase and response regulator can exist asseparate proteins or as distinct domains of a hybridprotein. Additionally, the phosphoryl group ispassed from a conserved histidine residue (His1)†on the kinase to the response regulator (Asp1).Following this initial transfer, the response regula-tor then passes the phosphoryl group to a con-served histidine residue (His2) on aphosphotransferase protein. Finally, the phospho-transferase transfers the phosphoryl group toanother response regulator, again into an asparticacid binding pocket (Asp2).13 The added complex-ity of the four-module signaling circuit may allowmore finely tuned responses to stimuli.14

As noted, V. harveyi uses a phosphorelay forquorum sensing signal propagation. In this Gram-negative organism, two parallel systems (twochannels) respond to two different AIs (Scheme 1),AI-1 and AI-2, and converge to regulate theexpression of the luciferase operon luxCDABE.15

AI-1 is 4-hydroxyl butanoyl L-homoserine lactoneproduced by the V. harveyi autoinducer synthaseLuxM16 and is specific to this species. Autoinducer-2 is a furanosyl borate diester 3A-methyl-5,6-dihydro-furo [2,3-D][1,2,3] dioxaborole-2,2,6,6Atetraol17 and is produced by both Gram-negativeand Gram-positive bacteria. AI-1 is recognized bythe membrane-bound sensor protein LuxN, andAI-2 is bound by the periplasmic binding proteinLuxP and the complex interacts with the mem-brane-bound sensor kinase LuxQ. LuxN and LuxQare hybrid proteins containing N-terminal sensorkinase domains as well as response regulatordomains. When AI levels are low, during low celldensity, LuxQ and LuxN act as histidine kinasesand autophosphorylate their response regulatordomains at a conserved aspartate residue (Asp1).

Scheme 1. Representation of the known quorum-sensing pathway in V. harveyi. Autoinducers AI-1 andAI-2 are shown as a hexagon and pentagon, respectively.Phosphorylation sites are identified by H or D represent-ing histidine or aspartic acid residues. As shown, theaspartic acid residues of LuxQ and LuxN are phosphoryl-ated (P). LuxO is shown with the consensus helix-turn-helix (HTH) motif.

Scheme 2. Sequence alignment for LuxU fromV. harveyi (VH), V. cholerae (VC) and two known histidine phosphorelayproteins ArcB-HPt and Ypd1. The degree of homology is indicated by color shading ranging from blue (no homology) todark orange (sequence identity). The conserved active-site histidine residues are enclosed in a box. Sequence alignmentswere performed with the program T-Coffee.68

† The nomenclature used here is to denote which stepeach amino acid plays in the phosphorylation pathway.Thus His1 indicates that a histidine residue participates inthe first phosphorylation reaction. This naming schemedoes not indicate location of the residue in the proteinprimary sequence.

298 NMR Studies of LuxU

Solution structure and dynamics of LuxU from Vibrio harveyi, a phosphotransferase protein involved in bacterial

quorum sensing. Ulrich DL, Kojetin D, Bassler BL, Cavanagh J, Loria JP J Mol Biol. 2005. 347, 297-307.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

available models?

kinase-response regulator two-component system.The kinase and response regulator can exist asseparate proteins or as distinct domains of a hybridprotein. Additionally, the phosphoryl group ispassed from a conserved histidine residue (His1)†on the kinase to the response regulator (Asp1).Following this initial transfer, the response regula-tor then passes the phosphoryl group to a con-served histidine residue (His2) on aphosphotransferase protein. Finally, the phospho-transferase transfers the phosphoryl group toanother response regulator, again into an asparticacid binding pocket (Asp2).13 The added complex-ity of the four-module signaling circuit may allowmore finely tuned responses to stimuli.14

As noted, V. harveyi uses a phosphorelay forquorum sensing signal propagation. In this Gram-negative organism, two parallel systems (twochannels) respond to two different AIs (Scheme 1),AI-1 and AI-2, and converge to regulate theexpression of the luciferase operon luxCDABE.15

AI-1 is 4-hydroxyl butanoyl L-homoserine lactoneproduced by the V. harveyi autoinducer synthaseLuxM16 and is specific to this species. Autoinducer-2 is a furanosyl borate diester 3A-methyl-5,6-dihydro-furo [2,3-D][1,2,3] dioxaborole-2,2,6,6Atetraol17 and is produced by both Gram-negativeand Gram-positive bacteria. AI-1 is recognized bythe membrane-bound sensor protein LuxN, andAI-2 is bound by the periplasmic binding proteinLuxP and the complex interacts with the mem-brane-bound sensor kinase LuxQ. LuxN and LuxQare hybrid proteins containing N-terminal sensorkinase domains as well as response regulatordomains. When AI levels are low, during low celldensity, LuxQ and LuxN act as histidine kinasesand autophosphorylate their response regulatordomains at a conserved aspartate residue (Asp1).

Scheme 1. Representation of the known quorum-sensing pathway in V. harveyi. Autoinducers AI-1 andAI-2 are shown as a hexagon and pentagon, respectively.Phosphorylation sites are identified by H or D represent-ing histidine or aspartic acid residues. As shown, theaspartic acid residues of LuxQ and LuxN are phosphoryl-ated (P). LuxO is shown with the consensus helix-turn-helix (HTH) motif.

Scheme 2. Sequence alignment for LuxU fromV. harveyi (VH), V. cholerae (VC) and two known histidine phosphorelayproteins ArcB-HPt and Ypd1. The degree of homology is indicated by color shading ranging from blue (no homology) todark orange (sequence identity). The conserved active-site histidine residues are enclosed in a box. Sequence alignmentswere performed with the program T-Coffee.68

† The nomenclature used here is to denote which stepeach amino acid plays in the phosphorylation pathway.Thus His1 indicates that a histidine residue participates inthe first phosphorylation reaction. This naming schemedoes not indicate location of the residue in the proteinprimary sequence.

298 NMR Studies of LuxU

Solution structure and dynamics of LuxU from Vibrio harveyi, a phosphotransferase protein involved in bacterial

quorum sensing. Ulrich DL, Kojetin D, Bassler BL, Cavanagh J, Loria JP J Mol Biol. 2005. 347, 297-307.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

available models?

The hypothesis that prevails in literature is thatbacteria use quorum sensing to sense populationdensity (Miller and Bassler, 2001). According to thishypothesis, quorum sensing-regulated genes areexpressed (or repressed) depending on the bacterialcell density. However, this hypothesis has neverbeen proven and is still under debate. Redfield(2002) argued for a more direct function of quorumsensing: the ability to determine whether excretedmolecules rapidly diffuse away from the cell. Thisdiffusion sensing would allow cells to regulateexcretion of degradative enzymes and other geneproducts in such a way as to minimize losses owingto extracellular diffusion and mixing.

Quorum sensing-regulated gene expression ismost often studied in vitro (that is, in bacterialcultures grown in liquid or on solid growthmedium). However, microbiologists are becomingmore and more aware of the fact that this generegulation is linked to and influenced by environ-mental and host-derived signals (Newton and Fray,2004). This mini-review aims at discussing thecurrent knowledge of quorum sensing and quorumquenching in V. harveyi and closely related bacteriain vivo during Vibrio–animal interactions.

The Quorum sensing system of V. harveyi

V. harveyi has been found to use a three-channelquorum sensing system (Figure 1). The first channelof this system is mediated by the harveyi auto-inducer 1 (HAI-1), an acylated homoserine lactone(AHL) (Cao and Meighen, 1989). The second

channel is mediated by the so-called autoinducer 2(AI-2), which is a furanosyl borate diester (Chenet al., 2002). The chemical structure of the thirdautoinducer, called cholerae autoinducer 1 (CAI-1),is still unknown. The autoinducers are detected atthe cell surface by membrane-bound, two-compo-nent receptor proteins that feed a common phos-phorylation/dephosphorylation signal transductioncascade (Taga and Bassler, 2003). Central in thesignal transduction cascade is the LuxO protein.Phosphorylated LuxO indirectly inhibits productionof the transcriptional regulator protein LuxRVh

through the action of five small regulatory RNAs(Tu and Bassler, 2007). LuxRVh directly activates thelux operon (Swartzman et al., 1992), whereas themajority of other quorum sensing-regulated genesappears to be indirectly controlled by LuxRVh

(Waters and Bassler, 2007). Tu and Bassler (2007)recently proposed that the multiple small regulatoryRNAs function to translate increasing autoinducerconcentrations into a precise gradient of LuxRVh,resulting in a gradient of expression of quorumsensing-regulated target genes. In other words, theconcentration of LuxRVh depends on the concentra-tion of the five small regulatory RNAs, which isdetermined by the phosphorylation status of LuxO.The phosphorylation status of LuxO in its turn isdetermined by the net result of the kinase andphosphatase activities of the three receptors andthus dependent on the concentration of the threeautoinducers.

Interestingly, Waters and Bassler (2007) recentlyreported that different quorum sensing-controlled

LuxN LuxSLuxQ

LuxP

CqsS

CqsA

LuxU

LuxO

LuxRVh

sRNAs + Hfq

+ σ54

LuxM

HAI-1

AI-2

CAI-1

P P

P

P

LuxN LuxSLuxQ

LuxP

CqsS

CqsA

LuxU

LuxO

LuxRVh

sRNAs

LuxM

HAI-1

AI-2

CAI-1

P P

P

P

Target genes

ba

Figure 1 Quorum sensing in Vibrio harveyi. The LuxM, LuxS and CqsA enzymes synthesize the autoinducers harveyi autoinducer 1(HAI-1), autoinducer 2 (AI-2) and cholerae autoinducer 1 (CAI-1), respectively. These autoinducers are detected at the cell surface by theLuxN, LuxQ and CqsS two-component receptor proteins, respectively. Detection of AI-2 by LuxQ requires the periplasmic protein LuxP.(a) In the absence of autoinducers, the receptors autophosphorylate and transfer phosphate to LuxO via LuxU. Phosphorylation activatesLuxO, which together with s54 activates the production of five small regulatory RNAs (sRNAs). These sRNAs, together with thechaperone Hfq, destabilize the mRNA encoding the transcriptional regulator LuxRVh. Therefore, in the absence of autoinducers, theLuxRVh protein is not produced. (b) In the presence of high concentrations of the autoinducers, the receptor proteins switch from kinasesto phosphatases, which result in dephosphorylation of LuxO. Dephosphorylated LuxO is inactive and therefore, the sRNAs are notformed and the transcriptional regulator LuxRVh is produced. See text for more details. denotes phosphotransfer.

Quorum sensing in Vibrio harveyi in vivoT Defoirdt et al

20

The ISME Journal

Quorum sensing and quorum quenching in Vibrio harveyi: lessons learned from in vivo work Defoirdt T, Boon N,

Sorgeloos P, Verstraete W, Bossier P ISME J. 2008. 2, 19-26.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

available models?

The hypothesis that prevails in literature is thatbacteria use quorum sensing to sense populationdensity (Miller and Bassler, 2001). According to thishypothesis, quorum sensing-regulated genes areexpressed (or repressed) depending on the bacterialcell density. However, this hypothesis has neverbeen proven and is still under debate. Redfield(2002) argued for a more direct function of quorumsensing: the ability to determine whether excretedmolecules rapidly diffuse away from the cell. Thisdiffusion sensing would allow cells to regulateexcretion of degradative enzymes and other geneproducts in such a way as to minimize losses owingto extracellular diffusion and mixing.

Quorum sensing-regulated gene expression ismost often studied in vitro (that is, in bacterialcultures grown in liquid or on solid growthmedium). However, microbiologists are becomingmore and more aware of the fact that this generegulation is linked to and influenced by environ-mental and host-derived signals (Newton and Fray,2004). This mini-review aims at discussing thecurrent knowledge of quorum sensing and quorumquenching in V. harveyi and closely related bacteriain vivo during Vibrio–animal interactions.

The Quorum sensing system of V. harveyi

V. harveyi has been found to use a three-channelquorum sensing system (Figure 1). The first channelof this system is mediated by the harveyi auto-inducer 1 (HAI-1), an acylated homoserine lactone(AHL) (Cao and Meighen, 1989). The second

channel is mediated by the so-called autoinducer 2(AI-2), which is a furanosyl borate diester (Chenet al., 2002). The chemical structure of the thirdautoinducer, called cholerae autoinducer 1 (CAI-1),is still unknown. The autoinducers are detected atthe cell surface by membrane-bound, two-compo-nent receptor proteins that feed a common phos-phorylation/dephosphorylation signal transductioncascade (Taga and Bassler, 2003). Central in thesignal transduction cascade is the LuxO protein.Phosphorylated LuxO indirectly inhibits productionof the transcriptional regulator protein LuxRVh

through the action of five small regulatory RNAs(Tu and Bassler, 2007). LuxRVh directly activates thelux operon (Swartzman et al., 1992), whereas themajority of other quorum sensing-regulated genesappears to be indirectly controlled by LuxRVh

(Waters and Bassler, 2007). Tu and Bassler (2007)recently proposed that the multiple small regulatoryRNAs function to translate increasing autoinducerconcentrations into a precise gradient of LuxRVh,resulting in a gradient of expression of quorumsensing-regulated target genes. In other words, theconcentration of LuxRVh depends on the concentra-tion of the five small regulatory RNAs, which isdetermined by the phosphorylation status of LuxO.The phosphorylation status of LuxO in its turn isdetermined by the net result of the kinase andphosphatase activities of the three receptors andthus dependent on the concentration of the threeautoinducers.

Interestingly, Waters and Bassler (2007) recentlyreported that different quorum sensing-controlled

LuxN LuxSLuxQ

LuxP

CqsS

CqsA

LuxU

LuxO

LuxRVh

sRNAs + Hfq

+ σ54

LuxM

HAI-1

AI-2

CAI-1

P P

P

P

LuxN LuxSLuxQ

LuxP

CqsS

CqsA

LuxU

LuxO

LuxRVh

sRNAs

LuxM

HAI-1

AI-2

CAI-1

P P

P

P

Target genes

ba

Figure 1 Quorum sensing in Vibrio harveyi. The LuxM, LuxS and CqsA enzymes synthesize the autoinducers harveyi autoinducer 1(HAI-1), autoinducer 2 (AI-2) and cholerae autoinducer 1 (CAI-1), respectively. These autoinducers are detected at the cell surface by theLuxN, LuxQ and CqsS two-component receptor proteins, respectively. Detection of AI-2 by LuxQ requires the periplasmic protein LuxP.(a) In the absence of autoinducers, the receptors autophosphorylate and transfer phosphate to LuxO via LuxU. Phosphorylation activatesLuxO, which together with s54 activates the production of five small regulatory RNAs (sRNAs). These sRNAs, together with thechaperone Hfq, destabilize the mRNA encoding the transcriptional regulator LuxRVh. Therefore, in the absence of autoinducers, theLuxRVh protein is not produced. (b) In the presence of high concentrations of the autoinducers, the receptor proteins switch from kinasesto phosphatases, which result in dephosphorylation of LuxO. Dephosphorylated LuxO is inactive and therefore, the sRNAs are notformed and the transcriptional regulator LuxRVh is produced. See text for more details. denotes phosphotransfer.

Quorum sensing in Vibrio harveyi in vivoT Defoirdt et al

20

The ISME Journal

Quorum sensing and quorum quenching in Vibrio harveyi: lessons learned from in vivo work Defoirdt T, Boon N,

Sorgeloos P, Verstraete W, Bossier P ISME J. 2008. 2, 19-26.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

available models?

In V. harveyi, DPD cyclizes, is hydrated and isconverted into the active AI-2 signal molecule(Chen et al., 2002).

More recent research showed that the active AI-2signal in Salmonella typhimurium has a differentchemical structure when compared to V. harveyiAI-2 (Miller et al., 2004). In S. typhimurium, AI-2induces transcription of the lsrACDBFGE operon.The first four genes in this operon encode an ABCtransporter, through which AI-2 is imported intothe cells (Taga et al., 2003). The other genes ofthe operon, together with the lsrK en lsrR genes,encode proteins that phosphorylate AI-2 and furtherdegrade the signal (Taga and Bassler, 2003).A similar system has been described in Escherichiacoli (Xavier and Bassler, 2005b). It is still unclearwhy these two species produce a signal, which thenactivates its own degradation. The signal might beused as a nutrient, although the bacteria cannotgrow in minimal media containing AI-2 as the solecarbon source (Taga and Bassler, 2003). Another

hypothesis is that by degrading AI-2, these bacteriatrick their competitors into behaving as if there wereno AI-2 (Federle and Bassler, 2003). In an excitingreport, Xavier and Bassler (2005a) studied AI-2 crosstalk between V. harveyi and E. coli and found thatwhen co-cultured, V. harveyi produced only 18% ofthe bioluminescence it produced in pure culture.The effect was shown to be due to internalizationand degradation of AI-2 by E. coli since no reductionoccurred in co-cultures with mutants that aredefective in AI-2 internalization.

The degradation of AI-2 by E. coli is not consti-tutive but is under control of several regulatorymechanisms, such as cAMP-CRP, the repressor LsrRand RpoS (De Keersmaecker et al., 2006), whichmakes it inappropriate for practical applications.Since the different active AI-2 signal molecules areall in equilibrium with each other and with DPD (seeFigure 2), inactivation of one of them will result ina decrease in the concentrations of all the otherforms. However, no other bacteria than E. coli and

methyl acceptor methylated product H2O adenine homocysteine

OH O

OOH

O

HO

OH

O

O

OH

OH

OHOH O OH

OH

OOB

-OH OH

OOH

OHO

OOH

OH

OHOH

DPD

S-DHMF S-THMF S-THMF-borate

R-DHMF R-THMF

a

b

B(OH)4

O OH

OHN

N

N

N

S+

-OOC N+H3

O OH

OHN

N

N

N

H2NH2N

S

-OOC N+H3

O OH

OH

S

-OOC N+H3

OH OH O

OOH

methyltransferasePfs Lux S

SAM SAH

SRH

DPD

Figure 2 Biosynthesis of AI-2. (a) 4,5-dihydroxy-2,3-pentanedione (DPD), the precursor to all AI-2, is synthesized from S-adenosylmethionine (SAM) in three enzymatic steps. SAH, S-adenosylhomocysteine; SRH, S-ribosylhomocysteine. (b) DPD rearrangesand undergoes further reactions (all equilibria) to form distinct biologically active signal molecules that are generically termed AI-2.Vibrio harveyi AI-2 is produced by the upper pathway; Salmonella typhimurium AI-2 by the lower one. S-DHMF: (2S,4S)-dihydroxy-2-methyldihydro-3-furanone; R-DHMF, (2R,4S)-dihydroxy-2methyldihydro-3-furanone; S-THMF, (2S,4S)-2-methyl-2,3,3,4-tetra-hydroxy-tetrahydrofuran; R-THMF, (2R,4S)-2-methyl-2,3,3,4-tetrahydroxytetrahydro-furan (based on Vendeville et al. (2005); De Keersmaeckeret al. (2006)).

Quorum sensing in Vibrio harveyi in vivoT Defoirdt et al

23

The ISME Journal

Quorum sensing and quorum quenching in Vibrio harveyi: lessons learned from in vivo work Defoirdt T, Boon N,

Sorgeloos P, Verstraete W, Bossier P ISME J. 2008. 2, 19-26.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

available models?

In V. harveyi, DPD cyclizes, is hydrated and isconverted into the active AI-2 signal molecule(Chen et al., 2002).

More recent research showed that the active AI-2signal in Salmonella typhimurium has a differentchemical structure when compared to V. harveyiAI-2 (Miller et al., 2004). In S. typhimurium, AI-2induces transcription of the lsrACDBFGE operon.The first four genes in this operon encode an ABCtransporter, through which AI-2 is imported intothe cells (Taga et al., 2003). The other genes ofthe operon, together with the lsrK en lsrR genes,encode proteins that phosphorylate AI-2 and furtherdegrade the signal (Taga and Bassler, 2003).A similar system has been described in Escherichiacoli (Xavier and Bassler, 2005b). It is still unclearwhy these two species produce a signal, which thenactivates its own degradation. The signal might beused as a nutrient, although the bacteria cannotgrow in minimal media containing AI-2 as the solecarbon source (Taga and Bassler, 2003). Another

hypothesis is that by degrading AI-2, these bacteriatrick their competitors into behaving as if there wereno AI-2 (Federle and Bassler, 2003). In an excitingreport, Xavier and Bassler (2005a) studied AI-2 crosstalk between V. harveyi and E. coli and found thatwhen co-cultured, V. harveyi produced only 18% ofthe bioluminescence it produced in pure culture.The effect was shown to be due to internalizationand degradation of AI-2 by E. coli since no reductionoccurred in co-cultures with mutants that aredefective in AI-2 internalization.

The degradation of AI-2 by E. coli is not consti-tutive but is under control of several regulatorymechanisms, such as cAMP-CRP, the repressor LsrRand RpoS (De Keersmaecker et al., 2006), whichmakes it inappropriate for practical applications.Since the different active AI-2 signal molecules areall in equilibrium with each other and with DPD (seeFigure 2), inactivation of one of them will result ina decrease in the concentrations of all the otherforms. However, no other bacteria than E. coli and

methyl acceptor methylated product H2O adenine homocysteine

OH O

OOH

O

HO

OH

O

O

OH

OH

OHOH O OH

OH

OOB

-OH OH

OOH

OHO

OOH

OH

OHOH

DPD

S-DHMF S-THMF S-THMF-borate

R-DHMF R-THMF

a

b

B(OH)4

O OH

OHN

N

N

N

S+

-OOC N+H3

O OH

OHN

N

N

N

H2NH2N

S

-OOC N+H3

O OH

OH

S

-OOC N+H3

OH OH O

OOH

methyltransferasePfs Lux S

SAM SAH

SRH

DPD

Figure 2 Biosynthesis of AI-2. (a) 4,5-dihydroxy-2,3-pentanedione (DPD), the precursor to all AI-2, is synthesized from S-adenosylmethionine (SAM) in three enzymatic steps. SAH, S-adenosylhomocysteine; SRH, S-ribosylhomocysteine. (b) DPD rearrangesand undergoes further reactions (all equilibria) to form distinct biologically active signal molecules that are generically termed AI-2.Vibrio harveyi AI-2 is produced by the upper pathway; Salmonella typhimurium AI-2 by the lower one. S-DHMF: (2S,4S)-dihydroxy-2-methyldihydro-3-furanone; R-DHMF, (2R,4S)-dihydroxy-2methyldihydro-3-furanone; S-THMF, (2S,4S)-2-methyl-2,3,3,4-tetra-hydroxy-tetrahydrofuran; R-THMF, (2R,4S)-2-methyl-2,3,3,4-tetrahydroxytetrahydro-furan (based on Vendeville et al. (2005); De Keersmaeckeret al. (2006)).

Quorum sensing in Vibrio harveyi in vivoT Defoirdt et al

23

The ISME Journal

Quorum sensing and quorum quenching in Vibrio harveyi: lessons learned from in vivo work Defoirdt T, Boon N,

Sorgeloos P, Verstraete W, Bossier P ISME J. 2008. 2, 19-26.

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Example model: quorum sensing in Vibrioharveyi

AI‐2

homocysteine

E LuxS

SAI‐2

AI‐2

phosphorelay

H2O SRH

adenine

E Pfs

h

SAM

Ma SAH

Mam

E Mt

m

LuxPQ

P LuxU

LuxPQp

phosphorylation phosphorelay

LuxPQ

LuxUp

phosphorelay

LuxO LuxOp

LuxU

sigma54

Binding site

LuxO

trans. regulation

small RNAs 

LuxOp

sigma54

Binding site

LuxO

HfQ 

LuxR mRNA destabilization

LuxR mRNA  LuxR mRNA 

destabilized

LuxPQ

LuxPQ.AI‐2

LuxUp

phosphorelay

LuxU

LuxPQp

AI‐2 bindingLuxOp

LuxUp

LuxO

LuxR mRNA 

ribosomeLuxR  

translation

target genes

binding sites  

trans. regulation

small RNAs 

luciferase

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Relationship with other approaches

P/T nets: Petri nets and variants

Representing place/transition nets in Span (Graph) Katis, P. and Sabadini, N. and

Walters, R.F.C. Lecture Notes in Computer Science, 1997

Graph-based frameworks: virtually any frameworkwill fit via free constructions

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Relationship with other approaches

P/T nets: Petri nets and variants

Representing place/transition nets in Span (Graph) Katis, P. and Sabadini, N. and

Walters, R.F.C. Lecture Notes in Computer Science, 1997

Graph-based frameworks: virtually any frameworkwill fit via free constructions

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Relationship with other approaches

P/T nets: Petri nets and variants

Representing place/transition nets in Span (Graph) Katis, P. and Sabadini, N. and

Walters, R.F.C. Lecture Notes in Computer Science, 1997

Graph-based frameworks: virtually any frameworkwill fit via free constructions

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Relationship with other approaches

P/T nets: Petri nets and variants

Representing place/transition nets in Span (Graph) Katis, P. and Sabadini, N. and

Walters, R.F.C. Lecture Notes in Computer Science, 1997

Graph-based frameworks: virtually any frameworkwill fit via free constructions

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Work in progress

General systems theory Mesarovic/Takaharageneral systems theory:

general systems are 1-cells in Rel, compact-closedsymmetric monoidal category

modelling self-organization efficient causation,self-organization as an adjunction:

(minimal realization - behaviour)

An abstract cell model that describes the self-organization of cell function in living

systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007

Quorum sensing in Vibrio harveyi

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Work in progress

General systems theory Mesarovic/Takaharageneral systems theory:

general systems are 1-cells in Rel, compact-closedsymmetric monoidal category

modelling self-organization efficient causation,self-organization as an adjunction:

(minimal realization - behaviour)

An abstract cell model that describes the self-organization of cell function in living

systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007

Quorum sensing in Vibrio harveyi

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Work in progress

General systems theory Mesarovic/Takaharageneral systems theory:

general systems are 1-cells in Rel, compact-closedsymmetric monoidal category

modelling self-organization efficient causation,self-organization as an adjunction:

(minimal realization - behaviour)

An abstract cell model that describes the self-organization of cell function in living

systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007

Quorum sensing in Vibrio harveyi

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Work in progress

General systems theory Mesarovic/Takaharageneral systems theory:

general systems are 1-cells in Rel, compact-closedsymmetric monoidal category

modelling self-organization efficient causation,self-organization as an adjunction:

(minimal realization - behaviour)

An abstract cell model that describes the self-organization of cell function in living

systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007

Quorum sensing in Vibrio harveyi

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Work in progress

General systems theory Mesarovic/Takaharageneral systems theory:

general systems are 1-cells in Rel, compact-closedsymmetric monoidal category

modelling self-organization efficient causation,self-organization as an adjunction:

(minimal realization - behaviour)

An abstract cell model that describes the self-organization of cell function in living

systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007

Quorum sensing in Vibrio harveyi

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Work in progress

General systems theory Mesarovic/Takaharageneral systems theory:

general systems are 1-cells in Rel, compact-closedsymmetric monoidal category

modelling self-organization efficient causation,self-organization as an adjunction:

(minimal realization - behaviour)

An abstract cell model that describes the self-organization of cell function in living

systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007

Quorum sensing in Vibrio harveyi

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Work in progress

General systems theory Mesarovic/Takaharageneral systems theory:

general systems are 1-cells in Rel, compact-closedsymmetric monoidal category

modelling self-organization efficient causation,self-organization as an adjunction:

(minimal realization - behaviour)

An abstract cell model that describes the self-organization of cell function in living

systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007

Quorum sensing in Vibrio harveyi

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Work in progress

General systems theory Mesarovic/Takaharageneral systems theory:

general systems are 1-cells in Rel, compact-closedsymmetric monoidal category

modelling self-organization efficient causation,self-organization as an adjunction:

(minimal realization - behaviour)

An abstract cell model that describes the self-organization of cell function in living

systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007

Quorum sensing in Vibrio harveyi

symmetricmonoidal

(bi)categorieswith feedbackand biological

networks

E Pareja-Tobes, MManrique, R

Tobes, E Pareja

Introductionwhy categories?

Categoriesobjects and relations

objects, relations,relations betweenrelations . . .

symmetricmonoidalcategories withfeedbackexample model: Quorumsensing

Relationship with otherapproaches

the futureWork in progress

Thank you!