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Boolean versus continuous dynamics on small and large model networks vs. 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universitt Darmstadt | 1 Biological background: Gene regulatory networks Replication DNA

  1. Boolean versus continuous dynamics on small and large model networks vs. 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 1

  2. Biological background: Gene regulatory networks Replication DNA Transcription mRNA Translation Protein based on: D. Del Vecchio & E. Sontag Dynamics and Control of Synthetic Bio-molecular Networks Proceedings of Americal Control Conference, 2007 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 2

  3. Biological background: Gene regulatory networks Replication DNA DNA Transcription mRNA mRNA Translation Protein based on: D. Del Vecchio & E. Sontag Dynamics and Control of Synthetic Bio-molecular Networks Proceedings of Americal Control Conference, 2007 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 2

  4. Boolean Networks • Toy-model: on-off states • Parallel update • Deterministic dynamics 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 3

  5. Boolean Networks • Toy-model: on-off states • Parallel update • Deterministic dynamics Dynamics of individual nodes depends on update functions: F C 1 C 2 R In 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 1 0 1 0 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 3

  6. Boolean and continuous dynamics for gene regulatory networks Boolean model σ i = { 0, 1 } σ i ( t + 1) = F i ( σ ( t )) 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 4

  7. Boolean and continuous dynamics for gene regulatory networks Boolean model Continuous model mRNA mRNA P mRNA P P mRNA mRNA mRNA mRNA P P P P mRNA mRNA mRNA mRNA i mRNA i P P P P i σ i = { 0, 1 } ˙ m RNA i = F i ( P ) − α mRNA i σ i ( t + 1) = F i ( σ ( t )) ˙ β mRNA i − δ P i P i = 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 4

  8. Hill function Regulation by single gene 1 n = 1 n = 3 n = 10 mRNA i f + (P) P i 0 0 0.5 1 1.5 P ˙ m RNA i = F i ( P ) − α mRNA i P n F i ( P ) = f + ( P ) = P n + k n Generalization to more inputs: Hill cubes 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 5

  9. Hill cubes Standardized method for converting any Boolean function into a continuous function gene a gene b output F ( P a , P b ) = f + ( P a ) · f − ( P b ) 0 0 0 0 1 0 1 0 1 1 1 0 1 a AND NOT b 1 0 0.5 P b D. Wittmann et al. 0.5 P a Transforming boolean models to continuous models: 1 0 Methodology and application to t-cell receptor signaling. BMC Systems Biology, 3 (1) (2009) 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 6

  10. Comparison: Fixed points and oscillations Continuous Boolean • Fixed points • Fixed points concentration 000... time • Cycles • Oscillations 100... concentration 011... time 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 7

  11. Generalized modelling approach • Steady-state concentrations: mRNA i ∗ , P i ∗ ∗ p i ) f j ( p i ) = F j ( P i mRNA i P i ∗ and functions: ˜ P i • Normalized state variables: r i = mRNA i ∗ , p i = F j ( P ∗ i ) 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 8

  12. Generalized modelling approach • Steady-state concentrations: mRNA i ∗ , P i ∗ ∗ p i ) f j ( p i ) = F j ( P i mRNA i P i ∗ and functions: ˜ P i • Normalized state variables: r i = mRNA i ∗ , p i = F j ( P ∗ i )  � ∂ ˜ ∂ ˜ �  f a f a   � − 1 � α 0 ∂ p a ∂ p b ∂ ˜ − 1 α 0 f b   0   J N =2 =  ∂ p a    β  � � � − 1 �  1 0 0     β − 1 0 1 0 α β ≡ λ : ratio of time scales between mRNA and protein dynamics • 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 8

  13. Generalized modelling approach • Steady-state concentrations: mRNA i ∗ , P i ∗ ∗ p i ) f j ( p i ) = F j ( P i mRNA i P i ∗ and functions: ˜ P i • Normalized state variables: r i = mRNA i ∗ , p i = F j ( P ∗ i )  � ∂ ˜ ∂ ˜ �  f a f a   � − 1 � α 0 ∂ p a ∂ p b ∂ ˜ − 1 α 0 f b   0   J N =2 =  ∂ p a    β  � � � − 1 �  1 0 0     β − 1 0 1 0 α β ≡ λ : ratio of time scales between mRNA and protein dynamics • � ∈ [ 0, n ] if protein i is an activator ∂ ˜ f j ∂ p i ≡ ˜ • f j p i = ∈ [ − n , 0] if protein i is an inhibitor T. Gross, U. Feudel Generalized models as a universal approach to the analysis of nonlinear dynamical systems Physical Review E 73 (1) (2006) 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 8

  14. Simple Loops • Even loops : even number of inhibitors • Odd loops : odd number of inhibitors 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 9

  15. Simple Loops Example: Three gene network Even loop: f 3 p 2 ~ Odd loop: ~ f 2 p 1 ~ f 1 p 3 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 10

  16. Simple Loops Example: Three gene network Even loop: even even odd f 3 p 2 ~ Odd loop: odd odd ~ f 2 p 1 ~ f 1 p 3 even 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 10

  17. Simple Loops Example: Three gene network Even loop: even even odd f 3 p 2 ~ Odd loop: odd odd ~ f 2 p 1 ~ f 1 p 3 even 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 10

  18. Loops with an additional input x = 1 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 11

  19. Loops with an additional input x = 3 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 11

  20. Loops with an additional input Example: Two gene network 4 gene a 2 gene b f � a p b 0 � 2 � 4 � 2 � 4 0 f � a p a 4 E. Gehrmann, B. Drossel 2 2 Boolean versus continuous dynamics 0 on simple two-gene modules � 2 Physical Review E 82 (4) (2010) 4 f � b p a � 4 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 12

  21. Loops with an additional input Example: Two gene network 4 gene a 2 gene b f � a p b 0 � 2 � 4 � 2 � 4 0 f � a p a 4 E. Gehrmann, B. Drossel 2 2 Boolean versus continuous dynamics 0 on simple two-gene modules � 2 Physical Review E 82 (4) (2010) 4 f � b p a � 4 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 12

  22. Loops with an additional input Example: Two-gene network � ∈ [ 0, n ] if protein i is an activator ˜ f j p i = ∈ [ − n , 0] if protein i is an inhibitor + gene a Oscillations gene b f a p b ~ E. Gehrmann, B. Drossel Boolean versus continuous dynamics on simple two-gene modules Physical Review E 82 (4) (2010) ~ f a p a 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 13

  23. Loops with an additional input Example: Two-gene network � ∈ [ 0, n ] if protein i is an activator ˜ f j p i = ∈ [ − n , 0] if protein i is an inhibitor + gene a Oscillations Boolean cycle gene b f a p b ~ b R O N E. Gehrmann, B. Drossel a Boolean versus continuous dynamics on simple two-gene modules Physical Review E 82 (4) (2010) ~ f a p a 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 13

  24. Loops with an additional input Example: Two-gene network � ∈ [ 0, n ] if protein i is an activator ˜ f j p i = ∈ [ − n , 0] if protein i is an inhibitor + gene a Oscillations Boolean cycle gene b b D N A a f a p b ~ b R O N E. Gehrmann, B. Drossel a Boolean versus continuous dynamics on simple two-gene modules Physical Review E 82 (4) (2010) ~ f a p a 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 13

  25. Loops with an additional input Example: Two-gene network � ∈ [ 0, n ] if protein i is an activator ˜ f j p i = ∈ [ − n , 0] if protein i is an inhibitor + gene a Oscillations b Boolean cycle D gene b b N D A N ) a A T a O N ( f a p b ~ b b T R O O N N E. Gehrmann, B. Drossel D N a Boolean versus continuous dynamics A on simple two-gene modules a Physical Review E 82 (4) (2010) ~ f a p a 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 13

  26. N-gene networks with self-input or crosslink � ∈ [ 0, n ] x = 1 if protein i is an activator f i = ∈ [ − n , 0] if protein i is an inhibitor HB SNB f Neff HB E. Ackermann, E.M. Weiel, T. Pfaff, B. Drossel Boolean versus continuous dynamics im modules with two feedback loops In preparation f xeff 14.05.2012 | Eva Christina Ackermann and Barbara Drossel | Technische Universität Darmstadt | 14

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