Slides Talk 01122008 Sysbiol2008

1. symmetric monoidal (bi)categories with feedback and biological networks symmetric monoidal (bi)categories E Pareja-Tobes, M Manrique, R with feedback and biological Tobes, E Pareja networks Introduction why categories? Categories objects and relations objects, relations, E Pareja-Tobes M Manrique R Tobes E Pareja relations between relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other Era7 bioinformatics approaches the future Work in progress Sysbiol 2008 December 1, 2008

2. symmetric Outline monoidal (bi)categories with feedback and biological networks Introduction E Pareja-Tobes, M why categories? Manrique, R Tobes, E Pareja What is category theory? Introduction why categories? Categories: objects and relations Categories n-categories: objects, relations, relations objects and relations objects, relations, between relations, . . . relations between relations . . . symmetric Symmetric monoidal categories with feedback and monoidal categories with biological networks feedback example model: Quorum Example: Quorum sensing in Vibrio harveyi sensing Relationship with other Relationship with other approaches approaches the future Work in progress Work in progress and future directions Work in progress

6. symmetric why categories? monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Systems biology Manrique, R Tobes, E Pareja imposes a Introduction why categories? Categories Relational view of biology objects and relations objects, relations, relations between relations . . . emphasis on symmetric monoidal categories with processes → compositionality feedback example model: Quorum sensing Relationship with other approaches the future Work in progress mathematical framework?

12. Categories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

13. Categories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? objects Categories objects and relations objects, relations, relations between relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

14. Categories symmetric monoidal (bi)categories with feedback and biological networks A E Pareja-Tobes, M Manrique, R B Tobes, E Pareja C Introduction D why categories? objects E Categories objects and relations objects, relations, relations between relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

15. Categories symmetric monoidal (bi)categories with feedback and biological networks A E Pareja-Tobes, M Manrique, R B Tobes, E Pareja C Introduction D why categories? objects E Categories objects and relations objects, relations, relations between relations relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

16. Categories symmetric monoidal (bi)categories with feedback and biological networks A E Pareja-Tobes, M Manrique, R B Tobes, E Pareja C Introduction D why categories? objects E Categories objects and relations objects, relations, relations between relations relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

17. Categories symmetric monoidal (bi)categories with feedback and biological networks A E Pareja-Tobes, M Manrique, R B Tobes, E Pareja C Introduction D why categories? objects E Categories objects and relations objects, relations, relations between relations relations . . . symmetric monoidal composition categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

18. Categories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? objects Categories objects and relations f objects, relations, relations between relations g relations . . . symmetric monoidal composition categories with feedback example model: Quorum sensing Relationship with other g f approaches the future Work in progress

19. Categories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? objects Categories objects and relations f objects, relations, relations between relations g relations . . . symmetric monoidal composition categories with feedback example model: Quorum sensing Relationship with other g f + some axioms approaches the future Work in progress

20. Bicategories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

21. Bicategories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

22. Bicategories symmetric A monoidal (bi)categories with feedback B and biological networks C E Pareja-Tobes, M D Manrique, R E Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

23. Bicategories symmetric A monoidal (bi)categories with feedback B and biological networks C E Pareja-Tobes, M D Manrique, R E Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

24. Bicategories symmetric A monoidal (bi)categories with feedback B and biological networks C E Pareja-Tobes, M D Manrique, R E Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

25. Bicategories symmetric A monoidal (bi)categories with feedback B and biological networks C E Pareja-Tobes, M D Manrique, R E Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

26. Bicategories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja f Introduction why categories? g Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal g f categories with composition of relations feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

27. Bicategories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum relations between relations (2-cells) sensing Relationship with other approaches the future Work in progress

28. Bicategories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum relations between relations (2-cells) sensing Relationship with other approaches the future Work in progress

29. Bicategories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum relations between relations (2-cells) sensing Relationship with other 2 different compositions of 2-cells: approaches the future Work in progress

30. Bicategories symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum relations between relations (2-cells) sensing Relationship with other 2 different compositions of 2-cells: approaches the future vertical ≡ sequential Work in progress

31. Bicategories symmetric monoidal (bi)categories f with feedback α and biological networks βα β E Pareja-Tobes, M Manrique, R Tobes, E Pareja g Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum relations between relations (2-cells) sensing Relationship with other 2 different compositions of 2-cells: approaches the future vertical ≡ sequential Work in progress

32. Bicategories symmetric monoidal (bi)categories f with feedback α and biological networks βα β E Pareja-Tobes, M Manrique, R Tobes, E Pareja g Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum relations between relations (2-cells) sensing Relationship with other 2 different compositions of 2-cells: approaches the future vertical ≡ sequential Work in progress horizontal ≡ parallel

33. f' f Bicategories symmetric monoidal (bi)categories f f' with feedback and biological β networks α β*α E Pareja-Tobes, M Manrique, R g' Tobes, E Pareja g Introduction why categories? g' g Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum relations between relations (2-cells) sensing Relationship with other 2 different compositions of 2-cells: approaches the future vertical ≡ sequential Work in progress horizontal ≡ parallel

34. f' f Bicategories symmetric monoidal (bi)categories f f' with feedback and biological β networks α β*α E Pareja-Tobes, M Manrique, R g' Tobes, E Pareja g Introduction why categories? g' g Categories objects and relations objects, relations, relations between relations . . . objects (0-cells) symmetric relations (1-cells) monoidal categories with composition of relations feedback example model: Quorum relations between relations (2-cells) sensing Relationship with other 2 different compositions of 2-cells: approaches the future vertical ≡ sequential Work in progress horizontal ≡ parallel + some (more complex) axioms

35. symmetric n-categories monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

36. symmetric n-categories monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja model relations between relations between . . . Introduction why categories? Categories objects and relations objects, relations, relations between relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

37. symmetric n-categories monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja model relations between relations between . . . Introduction why categories? Categories objects and relations deﬁnition: active area of research! objects, relations, relations between relations . . . symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

38. symmetric n-categories monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja model relations between relations between . . . Introduction why categories? Categories objects and relations deﬁnition: active area of research! objects, relations, relations between relations . . . symmetric monoidal categories with see for example feedback example model: Quorum sensing Relationship with other approaches Higher-Dimensional Categories: an illustrated guide book Cheng, E. Lauda, A. the future Work in progress

39. symmetric monoidal categories with symmetric monoidal (bi)categories feedback with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction deﬁned by Walters et al as a framework for the why categories? modelling of concurrent and distributed processes. Categories objects and relations objects, relations, relations between relations . . . symmetric Bicategories of processes Katis P. Sabadini N. Walters R. 1997 monoidal categories with On the algebra of systems with feedback and boundary Katis P. Sabadini N. feedback example model: Quorum Walters R. 2000 sensing Relationship with other approaches the future Work in progress

42. symmetric symmetric monoidal monoidal (bi)categories with feedback and biological There is an operation, ⊗, which acts on networks E Pareja-Tobes, M Manrique, R objects: Tobes, E Pareja Introduction A, B → A ⊗ B why categories? Categories and 1-cells: objects and relations objects, relations, relations between relations . . . g f ⊗g f symmetric (A → B, C − D) → A ⊗ C − → B ⊗ D − → − monoidal categories with feedback example model: Quorum sensing Relationship with other approaches the future ≡ parallel composition Work in progress + associativity, unit, and symmetry

45. interpretation symmetric monoidal (bi)categories with feedback and biological networks input source E Pareja-Tobes, M Manrique, R Tobes, E Pareja A1 Introduction A2 A1 ⊗ . . . ⊗ An A3 why categories? Categories objects and relations An objects, relations, relations between relations . . . symmetric output target monoidal categories with feedback example model: Quorum sensing B1 Relationship with other approaches B1 ⊗ . . . ⊗ Bm B2 the future B3 Work in progress Bm

46. symmetric interpretation monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja process 1-cell Introduction why categories? Categories p objects and relations A1 ⊗ . . . ⊗ An − B1 ⊗ . . . ⊗ Bm → objects, relations, A1 relations between P B1 relations . . . A2 B2 symmetric A3 B3 monoidal categories with feedback An Bm example model: Quorum sensing Relationship with other approaches the future Work in progress

47. symmetric interpretation monoidal (bi)categories with feedback and biological sequential composition composition of 1-cells networks / B1 ⊗ . . . ⊗ Bm p1 E Pareja-Tobes, M A1 ⊗ . . . ⊗ Q n A QQQ Manrique, R A1 B1 C1 QQQ P1 P2 Tobes, E Pareja Q p2 ◦p1 QQQ A2 B2 C2 p2 A3 B3 C3 Q( Introduction An Bm Ck C1 ⊗ . . . ⊗ Ck why categories? Categories objects and relations objects, relations, parallel composition tensor relations between relations . . . (A1 ⊗ . . . ⊗ An ) ⊗ (C1 ⊗ . . . ⊗ Ck ) symmetric A1 B1 monoidal P1 categories with A2 B2 p1 ⊗p2 A3 B3 feedback example model: Quorum An Bm sensing (B1 ⊗ . . . ⊗ Bm ) ⊗ (D1 ⊗ . . . ⊗ Dj ) Relationship with other approaches the future C1 D1 Work in progress P2 C2 D2 C3 D3 Ck Dh

48. interpretation symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja feedback feedback Introduction why categories? / B⊗U p A⊗U A1 B1 Categories P1 objects and relations A2 B2 B3 objects, relations, A3 U U relations between relations . . . An /B Bm symmetric A monoidal fbU (f ) categories with feedback example model: Quorum sensing Relationship with other approaches the future Work in progress

49. symmetric Quorum sensing in Vibrio harveyi monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja why? Introduction why categories? Categories objects and relations metabolic, transcriptional and signaling objects, relations, relations between phenomena involved relations . . . symmetric data available monoidal categories with feedback enough complexity as a test for this kind of example model: Quorum sensing approach Relationship with other approaches the future Work in progress

Slides Talk 01122008 Sysbiol2008

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Slides Talk 01122008 Sysbiol2008