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Slides of a 15 min talk at Sysbiol 2008 Valencia: Symmetric monoidal (bi)categories with feedback and biological networks

Slides of a 15 min talk at Sysbiol 2008 Valencia: Symmetric monoidal (bi)categories with feedback and biological networks

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- 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
- 3. 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
- 4. 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
- 5. 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?
- 7. 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?
- 8. 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?
- 9. 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?
- 10. 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?
- 11. 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
- 40. 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
- 41. 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
- 43. 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
- 44. 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
- 50. 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
- 51. 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
- 52. 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
- 53. 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
- 54. 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
- 55. symmetric available models? monoidal (bi)categories with feedback and biological Vibrio sRNA-mediated feedback 897 networks Fig. 1. Model of the V. harveyi Quorum-Sensing Circuit. V. harveyi produces E Pareja-Tobes, M and detects three AIs and through modulation of the levels of the master transcriptional Manrique, R regulator, LuxR, controls downstream Tobes, E Pareja QS-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. Introduction At LCD, when LuxO is phosphorylated (LuxO~P), it activates transcription of the why categories? genes encoding the ﬁve Qrr sRNAs which work in conjuction with Hfq to destabilize the mRNA of luxR. At HCD, when LuxO is not Categories phosphorylated, qrr transcription ceases, luxR mRNA is stabilized and LuxR protein is objects and relations produced. In a feedback loop, LuxR activates objects, relations, expression of qrr2, qrr3 and qrr4, which relations between affects the timing of the QS transitions. relations . . . OM, outer membrane; IM, inner membrane. symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches At negligible concentrations of AIs, i.e. at low cell cascade involving LuxR and the Qrr sRNAs. We show that density (LCD), the three sensors act as kinases that trans- LuxR directly binds to and activates transcription of the the future fer phosphate through LuxU to LuxO (Freeman and promoters preceding qrr2, qrr3 and qrr4, but not qrr1 or Work in progress Bassler, 1999a,b; Lilley and Bassler, 2000). LuxO~P acti- qrr5. This leads to increased destabilization of luxR vates the expression of genes encoding ﬁve highly con- mRNA and downregulation of LuxR production. Mutation of the consensus LuxR-binding in Vibrio harveyi. Tu KC, A small-RNA-mediated negative feedback loop controls quorum-sensing dynamics sites in the V. harveyi qrr2, served small regulatory RNAs (sRNAs) called Qrr1–5 (Tu and Bassler, 2007). The Qrrs pair with the 5′ UTR of qrr3 and qrr4 promoters disrupts the negative feedback the luxR mRNA and destabilize it, a process that requires BL. Mol Microbiol. 2008. 70, transition from HCD to Waters CM, Svenningsen SL, Bassler loop and affects the timing of the 896-907 the RNA chaperone Hfq (Lenz et al., 2004). LuxR is the LCD mode and vice versa. In the closely related species master transcriptional regulator of QS genes in V. harveyi Vibrio cholerae, we previously characterized a negative (Showalter et al., 1990; Swartzman et al., 1992). Thus, at feedback loop consisting of HapR (the LuxR homologue) LCD, when little LuxR is present, there is no QS and and the V. cholerae Qrr sRNAs (Svenningsen et al., V. harveyi cells act as individuals. At high cell density 2008). However, in V. cholerae, the mechanism by which
- 56. symmetric available models? monoidal (bi)categories with feedback and biological Vibrio sRNA-mediated feedback 897 networks Fig. 1. Model of the V. harveyi Quorum-Sensing Circuit. V. harveyi produces E Pareja-Tobes, M and detects three AIs and through modulation of the levels of the master transcriptional Manrique, R regulator, LuxR, controls downstream Tobes, E Pareja QS-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. Introduction At LCD, when LuxO is phosphorylated (LuxO~P), it activates transcription of the why categories? genes encoding the ﬁve Qrr sRNAs which work in conjuction with Hfq to destabilize the mRNA of luxR. At HCD, when LuxO is not Categories phosphorylated, qrr transcription ceases, luxR mRNA is stabilized and LuxR protein is objects and relations produced. In a feedback loop, LuxR activates objects, relations, expression of qrr2, qrr3 and qrr4, which relations between affects the timing of the QS transitions. relations . . . OM, outer membrane; IM, inner membrane. symmetric monoidal categories with feedback example model: Quorum sensing Relationship with other approaches At negligible concentrations of AIs, i.e. at low cell cascade involving LuxR and the Qrr sRNAs. We show that density (LCD), the three sensors act as kinases that trans- LuxR directly binds to and activates transcription of the the future fer phosphate through LuxU to LuxO (Freeman and promoters preceding qrr2, qrr3 and qrr4, but not qrr1 or Work in progress Bassler, 1999a,b; Lilley and Bassler, 2000). LuxO~P acti- qrr5. This leads to increased destabilization of luxR vates the expression of genes encoding ﬁve highly con- mRNA and downregulation of LuxR production. Mutation of the consensus LuxR-binding in Vibrio harveyi. Tu KC, A small-RNA-mediated negative feedback loop controls quorum-sensing dynamics sites in the V. harveyi qrr2, served small regulatory RNAs (sRNAs) called Qrr1–5 (Tu and Bassler, 2007). The Qrrs pair with the 5′ UTR of qrr3 and qrr4 promoters disrupts the negative feedback the luxR mRNA and destabilize it, a process that requires BL. Mol Microbiol. 2008. 70, transition from HCD to Waters CM, Svenningsen SL, Bassler loop and affects the timing of the 896-907 the RNA chaperone Hfq (Lenz et al., 2004). LuxR is the LCD mode and vice versa. In the closely related species master transcriptional regulator of QS genes in V. harveyi Vibrio cholerae, we previously characterized a negative (Showalter et al., 1990; Swartzman et al., 1992). Thus, at feedback loop consisting of HapR (the LuxR homologue) LCD, when little LuxR is present, there is no QS and and the V. cholerae Qrr sRNAs (Svenningsen et al., V. harveyi cells act as individuals. At high cell density 2008). However, in V. cholerae, the mechanism by which
- 57. symmetric available models? monoidal (bi)categories 298 NMR Studies of LuxU with feedback and biological kinase-response regulator two-component system. networks The kinase and response regulator can exist as separate proteins or as distinct domains of a hybrid protein. Additionally, the phosphoryl group E Pareja-Tobes, M is passed from a conserved histidine residue (His1)† Manrique, R on the kinase to the response regulator (Asp1). Following this initial transfer, the response regula-Tobes, E Pareja tor then passes the phosphoryl group to a con- served histidine residue (His2) on a phosphotransferase protein. Finally, the phospho- transferase transfers the phosphoryl group Introduction to another response regulator, again into an aspartic categories? why acid binding pocket (Asp2).13 The added complex- ity of the four-module signaling circuit may allow Categories more ﬁnely tuned responses to stimuli.14 As noted, V. harveyi uses a phosphorelay for objects and relations quorum sensing signal propagation. In this Gram- objects, relations, negative organism, two parallel systems (two relations between channels) respond to two different AIs (Scheme 1), relations . . . AI-1 and AI-2, and converge to regulate the expression of the luciferase operon luxCDABE.15 symmetric AI-1 is 4-hydroxyl butanoyl L-homoserine lactone monoidal produced by the V. harveyi autoinducer synthase LuxM16 and is speciﬁc to this species. Autoinducer- categories with 2 is a furanosyl borate diester 3A-methyl-5,6- feedback dihydro-furo [2,3-D][1,2,3] dioxaborole-2,2,6,6A tetraol17 and is produced by both Gram-negative example model: Quorum and Gram-positive bacteria. AI-1 is recognized by sensing the membrane-bound sensor protein LuxN, and Scheme 1. Representation of the known quorum- Relationship with other AI-2 is bound by the periplasmic binding protein sensing pathway in V. harveyi. Autoinducers AI-1 and approaches LuxP and the complex interacts with the mem- AI-2 are shown as a hexagon and pentagon, respectively. brane-bound sensor kinase LuxQ. LuxN and LuxQ Phosphorylation sites are identiﬁed by H or D represent- are hybrid proteins containing N-terminal sensor future the ing histidine or aspartic acid residues. As shown, the aspartic acid residues of LuxQ and LuxN are phosphoryl- kinase domains as well as response regulator Work in progress ated (P). LuxO is shown with the consensus helix-turn- domains. When AI levels are low, during low cell helix (HTH) motif. density, LuxQ and LuxN act as histidine kinases and autophosphorylate their response regulator domains at a conserved aspartate residue (Asp1). 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.
- 58. symmetric available models? monoidal (bi)categories 298 NMR Studies of LuxU with feedback and biological kinase-response regulator two-component system. networks The kinase and response regulator can exist as separate proteins or as distinct domains of a hybrid protein. Additionally, the phosphoryl group E Pareja-Tobes, M is passed from a conserved histidine residue (His1)† Manrique, R on the kinase to the response regulator (Asp1). Following this initial transfer, the response regula-Tobes, E Pareja tor then passes the phosphoryl group to a con- served histidine residue (His2) on a phosphotransferase protein. Finally, the phospho- transferase transfers the phosphoryl group Introduction to another response regulator, again into an aspartic categories? why acid binding pocket (Asp2).13 The added complex- ity of the four-module signaling circuit may allow Categories more ﬁnely tuned responses to stimuli.14 As noted, V. harveyi uses a phosphorelay for objects and relations quorum sensing signal propagation. In this Gram- objects, relations, negative organism, two parallel systems (two relations between channels) respond to two different AIs (Scheme 1), relations . . . AI-1 and AI-2, and converge to regulate the expression of the luciferase operon luxCDABE.15 symmetric AI-1 is 4-hydroxyl butanoyl L-homoserine lactone monoidal produced by the V. harveyi autoinducer synthase LuxM16 and is speciﬁc to this species. Autoinducer- categories with 2 is a furanosyl borate diester 3A-methyl-5,6- feedback dihydro-furo [2,3-D][1,2,3] dioxaborole-2,2,6,6A tetraol17 and is produced by both Gram-negative example model: Quorum and Gram-positive bacteria. AI-1 is recognized by sensing the membrane-bound sensor protein LuxN, and Scheme 1. Representation of the known quorum- Relationship with other AI-2 is bound by the periplasmic binding protein sensing pathway in V. harveyi. Autoinducers AI-1 and approaches LuxP and the complex interacts with the mem- AI-2 are shown as a hexagon and pentagon, respectively. brane-bound sensor kinase LuxQ. LuxN and LuxQ Phosphorylation sites are identiﬁed by H or D represent- are hybrid proteins containing N-terminal sensor future the ing histidine or aspartic acid residues. As shown, the aspartic acid residues of LuxQ and LuxN are phosphoryl- kinase domains as well as response regulator Work in progress ated (P). LuxO is shown with the consensus helix-turn- domains. When AI levels are low, during low cell helix (HTH) motif. density, LuxQ and LuxN act as histidine kinases and autophosphorylate their response regulator domains at a conserved aspartate residue (Asp1). 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.
- 59. The phosphorylation status of LuxO in its turn is The Quorum sensing system of V. harveyi determined by the net result of the kinase and phosphatase activities of the three receptors and V. harveyi has been found to use a three-channel symmetric available models? thus dependent on the concentration of the three quorum sensing system (Figure 1). The first channel monoidal autoinducers. of this system is mediated by the harveyi auto- (bi)categories Interestingly, Waters and Bassler (2007) recently inducer 1 (HAI-1), an acylated homoserine lactone reported that different quorum sensing-controlled (AHL) (Cao and Meighen, 1989). The second with feedback and biological networks a b E Pareja-Tobes, M AI-2 AI-2 Manrique, R HAI-1 HAI-1 CAI-1 CAI-1 Tobes, E Pareja LuxP LuxP LuxQ LuxQ LuxN LuxN LuxS LuxS CqsS CqsS Introduction P P P P P P why categories? LuxU LuxU LuxM LuxM P P CqsA CqsA Categories 54 +σ LuxO LuxO objects and relations sRNAs + Hfq sRNAs objects, relations, Target genes LuxRVh LuxRVh relations between relations . . . symmetric Figure 1 Quorum sensing in Vibrio harveyi. The LuxM, LuxS and CqsA enzymes synthesize the autoinducers harveyi autoinducer 1 monoidal (HAI-1), autoinducer 2 (AI-2) and cholerae autoinducer 1 (CAI-1), respectively. These autoinducers are detected at the cell surface by the LuxN, LuxQ and CqsS two-component receptor proteins, respectively. Detection of AI-2 by LuxQ requires the periplasmic protein LuxP. categories with (a) In the absence of autoinducers, the receptors autophosphorylate and transfer phosphate to LuxO via LuxU. Phosphorylation activates feedback LuxO, which together with s54 activates the production of five small regulatory RNAs (sRNAs). These sRNAs, together with the chaperone Hfq, destabilize the mRNA encoding the transcriptional regulator LuxRVh. Therefore, in the absence of autoinducers, the example model: Quorum LuxRVh protein is not produced. (b) In the presence of high concentrations of the autoinducers, the receptor proteins switch from kinases sensing to phosphatases, which result in dephosphorylation of LuxO. Dephosphorylated LuxO is inactive and therefore, the sRNAs are not formed and the transcriptional regulator LuxRVh is produced. See text for more details. denotes phosphotransfer. Relationship with other approaches The ISME Journal the future Work in progress 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.
- 60. The phosphorylation status of LuxO in its turn is The Quorum sensing system of V. harveyi determined by the net result of the kinase and phosphatase activities of the three receptors and V. harveyi has been found to use a three-channel symmetric available models? thus dependent on the concentration of the three quorum sensing system (Figure 1). The first channel monoidal autoinducers. of this system is mediated by the harveyi auto- (bi)categories Interestingly, Waters and Bassler (2007) recently inducer 1 (HAI-1), an acylated homoserine lactone reported that different quorum sensing-controlled (AHL) (Cao and Meighen, 1989). The second with feedback and biological networks a b E Pareja-Tobes, M AI-2 AI-2 Manrique, R HAI-1 HAI-1 CAI-1 CAI-1 Tobes, E Pareja LuxP LuxP LuxQ LuxQ LuxN LuxN LuxS LuxS CqsS CqsS Introduction P P P P P P why categories? LuxU LuxU LuxM LuxM P P CqsA CqsA Categories 54 +σ LuxO LuxO objects and relations sRNAs + Hfq sRNAs objects, relations, Target genes LuxRVh LuxRVh relations between relations . . . symmetric Figure 1 Quorum sensing in Vibrio harveyi. The LuxM, LuxS and CqsA enzymes synthesize the autoinducers harveyi autoinducer 1 monoidal (HAI-1), autoinducer 2 (AI-2) and cholerae autoinducer 1 (CAI-1), respectively. These autoinducers are detected at the cell surface by the LuxN, LuxQ and CqsS two-component receptor proteins, respectively. Detection of AI-2 by LuxQ requires the periplasmic protein LuxP. categories with (a) In the absence of autoinducers, the receptors autophosphorylate and transfer phosphate to LuxO via LuxU. Phosphorylation activates feedback LuxO, which together with s54 activates the production of five small regulatory RNAs (sRNAs). These sRNAs, together with the chaperone Hfq, destabilize the mRNA encoding the transcriptional regulator LuxRVh. Therefore, in the absence of autoinducers, the example model: Quorum LuxRVh protein is not produced. (b) In the presence of high concentrations of the autoinducers, the receptor proteins switch from kinases sensing to phosphatases, which result in dephosphorylation of LuxO. Dephosphorylated LuxO is inactive and therefore, the sRNAs are not formed and the transcriptional regulator LuxRVh is produced. See text for more details. denotes phosphotransfer. Relationship with other approaches The ISME Journal the future Work in progress 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.
- 61. symmetric available models? monoidal (bi)categories with feedback Quorum sensing in Vibrio harveyi in vivo and biological T Defoirdt et al networks 23 E Pareja-Tobes, M a SAM SAH Manrique, R N+H3 N+H3 -OOC -OOC Tobes, E Pareja SRH N+H3 -OOC S+ S Introduction why categories? OH OH O O DPD S Categories OH N N OH OH O OH O N N objects and relations N N objects, relations, OH HO relations between N N H2N OH O H2N methyltransferase relations . . . Pfs Lux S symmetric monoidal methyl acceptor methylated product H2O adenine homocysteine categories with HO O H b feedback - B O O example model: Quorum OH OH B(OH)4 O O sensing OH Relationship with other HO HO O OH approaches OH O HO the future S-DHMF S-THMF S-THMF-borate OH O Work in progress OH O Quorum sensing and quorum quenching in Vibrio harveyi: lessons learned from in vivo work Defoirdt T, Boon N, OH O HO OH Sorgeloos P, Verstraete W, Bossier P ISME J. 2008.OH 19-26. 2, DPD O HO O HO R-DHMF R-THMF Figure 2 Biosynthesis of AI-2. (a) 4,5-dihydroxy-2,3-pentanedione (DPD), the precursor to all AI-2, is synthesized from S-
- 62. symmetric available models? monoidal (bi)categories with feedback Quorum sensing in Vibrio harveyi in vivo and biological T Defoirdt et al networks 23 E Pareja-Tobes, M a SAM SAH Manrique, R N+H3 N+H3 -OOC -OOC Tobes, E Pareja SRH N+H3 -OOC S+ S Introduction why categories? OH OH O O DPD S Categories OH N N OH OH O OH O N N objects and relations N N objects, relations, OH HO relations between N N H2N OH O H2N methyltransferase relations . . . Pfs Lux S symmetric monoidal methyl acceptor methylated product H2O adenine homocysteine categories with HO O H b feedback - B O O example model: Quorum OH OH B(OH)4 O O sensing OH Relationship with other HO HO O OH approaches OH O HO the future S-DHMF S-THMF S-THMF-borate OH O Work in progress OH O Quorum sensing and quorum quenching in Vibrio harveyi: lessons learned from in vivo work Defoirdt T, Boon N, OH O HO OH Sorgeloos P, Verstraete W, Bossier P ISME J. 2008.OH 19-26. 2, DPD O HO O HO R-DHMF R-THMF Figure 2 Biosynthesis of AI-2. (a) 4,5-dihydroxy-2,3-pentanedione (DPD), the precursor to all AI-2, is synthesized from S-
- 63. Example model: quorum sensing in Vibrio symmetric monoidal (bi)categories harveyi with feedback and biological networks E Pareja-Tobes, M Manrique, R phosphorelay Tobes, E Pareja LuxOp LuxO m phosphorelay AI‐2 binding Mam SAM LuxUp LuxU LuxUp LuxPQ E Mt h LuxPQp LuxPQ.AI‐2 Ma SAH adenine Introduction SAI‐2 E Pfs AI‐2 AI‐2 why categories? SRH H2O E LuxS homocysteine Categories objects and relations objects, relations, phosphorylation phosphorelay relations between LuxPQp LuxPQ LuxPQ relations . . . phosphorelay P LuxU LuxUp LuxU symmetric trans. regulation monoidal LuxO LuxOp LuxOp categories with sigma54 sigma54 feedback Binding site Binding site LuxR mRNA destabilization LuxO LuxO example model: Quorum small RNAs sensing Relationship with other HfQ approaches destabilized translation trans. regulation LuxR mRNA LuxR mRNA the future ribosome small RNAs LuxR Work in progress LuxR mRNA luciferase target genes binding sites
- 64. symmetric Relationship with other approaches monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja P/T nets: Petri nets and variants Introduction why categories? Categories Representing place/transition nets in Span (Graph) Katis, P. and Sabadini, N. and objects and relations objects, relations, Walters, R.F.C. Lecture Notes in Computer Science, 1997 relations between relations . . . symmetric monoidal categories with feedback Graph-based frameworks: virtually any framework example model: Quorum sensing will ﬁt via free constructions Relationship with other approaches the future Work in progress
- 65. symmetric Relationship with other approaches monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja P/T nets: Petri nets and variants Introduction why categories? Categories Representing place/transition nets in Span (Graph) Katis, P. and Sabadini, N. and objects and relations objects, relations, Walters, R.F.C. Lecture Notes in Computer Science, 1997 relations between relations . . . symmetric monoidal categories with feedback Graph-based frameworks: virtually any framework example model: Quorum sensing will ﬁt via free constructions Relationship with other approaches the future Work in progress
- 66. symmetric Relationship with other approaches monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja P/T nets: Petri nets and variants Introduction why categories? Categories Representing place/transition nets in Span (Graph) Katis, P. and Sabadini, N. and objects and relations objects, relations, Walters, R.F.C. Lecture Notes in Computer Science, 1997 relations between relations . . . symmetric monoidal categories with feedback Graph-based frameworks: virtually any framework example model: Quorum sensing will ﬁt via free constructions Relationship with other approaches the future Work in progress
- 67. symmetric Relationship with other approaches monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja P/T nets: Petri nets and variants Introduction why categories? Categories Representing place/transition nets in Span (Graph) Katis, P. and Sabadini, N. and objects and relations objects, relations, Walters, R.F.C. Lecture Notes in Computer Science, 1997 relations between relations . . . symmetric monoidal categories with feedback Graph-based frameworks: virtually any framework example model: Quorum sensing will ﬁt via free constructions Relationship with other approaches the future Work in progress
- 68. symmetric Work in progress monoidal (bi)categories with feedback and biological networks ˇ General systems theory Mesarovic /Takahara E Pareja-Tobes, M Manrique, R general systems theory: Tobes, E Pareja general systems are 1-cells in Rel, compact-closed Introduction why categories? symmetric monoidal category Categories objects and relations objects, relations, modelling self-organization efﬁcient causation, relations between relations . . . self-organization as an adjunction: symmetric monoidal categories with (minimal realization - behaviour) feedback example model: Quorum sensing An abstract cell model that describes the self-organization of cell function in living Relationship with other approaches systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007 the future Work in progress Quorum sensing in Vibrio harveyi
- 69. symmetric Work in progress monoidal (bi)categories with feedback and biological networks ˇ General systems theory Mesarovic /Takahara E Pareja-Tobes, M Manrique, R general systems theory: Tobes, E Pareja general systems are 1-cells in Rel, compact-closed Introduction why categories? symmetric monoidal category Categories objects and relations objects, relations, modelling self-organization efﬁcient causation, relations between relations . . . self-organization as an adjunction: symmetric monoidal categories with (minimal realization - behaviour) feedback example model: Quorum sensing An abstract cell model that describes the self-organization of cell function in living Relationship with other approaches systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007 the future Work in progress Quorum sensing in Vibrio harveyi
- 70. symmetric Work in progress monoidal (bi)categories with feedback and biological networks ˇ General systems theory Mesarovic /Takahara E Pareja-Tobes, M Manrique, R general systems theory: Tobes, E Pareja general systems are 1-cells in Rel, compact-closed Introduction why categories? symmetric monoidal category Categories objects and relations objects, relations, modelling self-organization efﬁcient causation, relations between relations . . . self-organization as an adjunction: symmetric monoidal categories with (minimal realization - behaviour) feedback example model: Quorum sensing An abstract cell model that describes the self-organization of cell function in living Relationship with other approaches systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007 the future Work in progress Quorum sensing in Vibrio harveyi
- 71. symmetric Work in progress monoidal (bi)categories with feedback and biological networks ˇ General systems theory Mesarovic /Takahara E Pareja-Tobes, M Manrique, R general systems theory: Tobes, E Pareja general systems are 1-cells in Rel, compact-closed Introduction why categories? symmetric monoidal category Categories objects and relations objects, relations, modelling self-organization efﬁcient causation, relations between relations . . . self-organization as an adjunction: symmetric monoidal categories with (minimal realization - behaviour) feedback example model: Quorum sensing An abstract cell model that describes the self-organization of cell function in living Relationship with other approaches systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007 the future Work in progress Quorum sensing in Vibrio harveyi
- 72. symmetric Work in progress monoidal (bi)categories with feedback and biological networks ˇ General systems theory Mesarovic /Takahara E Pareja-Tobes, M Manrique, R general systems theory: Tobes, E Pareja general systems are 1-cells in Rel, compact-closed Introduction why categories? symmetric monoidal category Categories objects and relations objects, relations, modelling self-organization efﬁcient causation, relations between relations . . . self-organization as an adjunction: symmetric monoidal categories with (minimal realization - behaviour) feedback example model: Quorum sensing An abstract cell model that describes the self-organization of cell function in living Relationship with other approaches systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007 the future Work in progress Quorum sensing in Vibrio harveyi
- 73. symmetric Work in progress monoidal (bi)categories with feedback and biological networks ˇ General systems theory Mesarovic /Takahara E Pareja-Tobes, M Manrique, R general systems theory: Tobes, E Pareja general systems are 1-cells in Rel, compact-closed Introduction why categories? symmetric monoidal category Categories objects and relations objects, relations, modelling self-organization efﬁcient causation, relations between relations . . . self-organization as an adjunction: symmetric monoidal categories with (minimal realization - behaviour) feedback example model: Quorum sensing An abstract cell model that describes the self-organization of cell function in living Relationship with other approaches systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007 the future Work in progress Quorum sensing in Vibrio harveyi
- 74. symmetric Work in progress monoidal (bi)categories with feedback and biological networks ˇ General systems theory Mesarovic /Takahara E Pareja-Tobes, M Manrique, R general systems theory: Tobes, E Pareja general systems are 1-cells in Rel, compact-closed Introduction why categories? symmetric monoidal category Categories objects and relations objects, relations, modelling self-organization efﬁcient causation, relations between relations . . . self-organization as an adjunction: symmetric monoidal categories with (minimal realization - behaviour) feedback example model: Quorum sensing An abstract cell model that describes the self-organization of cell function in living Relationship with other approaches systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007 the future Work in progress Quorum sensing in Vibrio harveyi
- 75. symmetric Work in progress monoidal (bi)categories with feedback and biological networks ˇ General systems theory Mesarovic /Takahara E Pareja-Tobes, M Manrique, R general systems theory: Tobes, E Pareja general systems are 1-cells in Rel, compact-closed Introduction why categories? symmetric monoidal category Categories objects and relations objects, relations, modelling self-organization efﬁcient causation, relations between relations . . . self-organization as an adjunction: symmetric monoidal categories with (minimal realization - behaviour) feedback example model: Quorum sensing An abstract cell model that describes the self-organization of cell function in living Relationship with other approaches systems Wolkenhauer, O. & Hofmeyr, J Journal of Theoretical Biology, 2007 the future Work in progress Quorum sensing in Vibrio harveyi
- 76. symmetric monoidal (bi)categories with feedback and biological networks E Pareja-Tobes, M Manrique, R Tobes, E Pareja Introduction why categories? Thank you! 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

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