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Comparing Discrete and Piecewise Affine Models of Gene Regulatory Networks Shahrad Jamshidi, Heike Siebert, Alexander Bockmayr Article in Biosystems, 2013 PhD S. Jamshidi, April 2013 Porquerolles, June 2013 Piecewise Affine Differential

  1. Comparing Discrete and Piecewise Affine Models of Gene Regulatory Networks Shahrad Jamshidi, Heike Siebert, Alexander Bockmayr Article in Biosystems, 2013 PhD S. Jamshidi, April 2013 Porquerolles, June 2013

  2. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  3. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . Each variable has thresholds to indicate activation/inhibition Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  4. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . x b Each variable has thresholds to indicate activation/inhibition A B ˙ = λ a ( F a ( x ) − x a ) x a x b ˙ = λ b ( F b ( x ) − x b ) x a � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  5. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . x b Each variable has thresholds to indicate activation/inhibition 2 θ b domain A B 1 θ b ˙ = λ a ( F a ( x ) − x a ) x a x b ˙ = λ b ( F b ( x ) − x b ) x a θ a � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  6. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . Each variable has thresholds to indicate ( F a ( x 0 ) , F b ( x 0 )) activation/inhibition A B x 0 ˙ = λ a ( F a ( x ) − x a ) x a x b ˙ = λ b ( F b ( x ) − x b ) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  7. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . Each variable has thresholds to indicate ( F a ( x 1 ) , F b ( x 1 )) activation/inhibition x 1 A B x 0 ˙ = λ a ( F a ( x ) − x a ) x a x b ˙ = λ b ( F b ( x ) − x b ) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  8. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . focal Each variable has thresholds to indicate ( F a ( x 1 ) , F b ( x 1 )) point activation/inhibition x 1 A B x 0 ˙ = λ a ( F a ( x ) − x a ) x a x b ˙ = λ b ( F b ( x ) − x b ) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  9. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . ( F a ( x 1 ) , F b ( x 1 )) Each variable has thresholds to indicate activation/inhibition x 1 x 2 A B x 0 ˙ = λ a ( F a ( x ) − x a ) x a x b ˙ = λ b ( F b ( x ) − x b ) ( F a ( x 2 ) , F b ( x 2 )) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  10. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . ( F a ( x 1 ) , F b ( x 1 )) Each variable has thresholds to indicate activation/inhibition x 1 x 2 A B x 0 ˙ = λ a ( F a ( x ) − x a ) x a x b ˙ = λ b ( F b ( x ) − x b ) ( F a ( x 2 ) , F b ( x 2 )) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  11. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . ( F a ( x 1 ) , F b ( x 1 )) Each variable has thresholds to indicate activation/inhibition x 1 x 2 A B x 0 ˙ = λ a ( F a ( x ) − x a ) x a x 3 x b ˙ = λ b ( F b ( x ) − x b ) ( F a ( x 3 ) , F b ( x 3 )) ( F a ( x 2 ) , F b ( x 2 )) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  12. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . ( F a ( x 1 ) , F b ( x 1 )) Each variable has thresholds to indicate activation/inhibition x 1 A x 2 B x 0 ˙ = λ a ( F a ( x ) − x a ) x a x 3 x b ˙ = λ b ( F b ( x ) − x b ) ( F a ( x 3 ) , F b ( x 3 )) ( F a ( x 2 ) , F b ( x 2 )) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  13. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . Each variable has thresholds to indicate activation/inhibition A B ˙ = λ a ( F a ( x ) − x a ) x a x b ˙ = λ b ( F b ( x ) − x b ) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  14. Piecewise Affine Differential Equations (PADE) A variable x i represents the protein concentration expressed by gene i . Each variable has thresholds to indicate activation/inhibition A B ˙ = λ a ( F a ( x ) − x a ) x a ?? x b ˙ = λ b ( F b ( x ) − x b ) � piecewise-affine differential equations Glass, L. and Kauffman, S. A.,Co-operative Components, Spatial Localization and Oscillatory Cellular Dynamics (1972)

  15. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  16. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Each variable updates by the function f i ( q ). f a ( q ) f b ( q ) q a q b 00 1 2 01 1 2 A B 02 1 0 10 0 0 11 1 0 12 1 0 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  17. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Each variable updates by the function f i ( q ). 02 12 f a ( q ) f b ( q ) q a q b 00 1 2 01 1 2 02 1 0 01 11 10 0 0 11 1 0 12 1 0 00 10 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  18. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Each variable updates by the function f i ( q ). f cI (00) f cro (00)= 02 12 f a ( q ) f b ( q ) q a q b 00 1 2 01 1 2 02 1 0 01 11 10 0 0 11 1 0 12 1 0 00 10 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  19. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Each variable updates by the function f i ( q ). 02 12 f a ( q ) f b ( q ) q a q b 00 1 2 01 1 2 02 1 0 01 11 10 0 0 11 1 0 12 1 0 00 10 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  20. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Each variable updates by the function f i ( q ). 02 12 f a ( q ) f b ( q ) q a q b 00 1 2 01 1 2 02 1 0 01 11 10 0 0 11 1 0 12 1 0 00 10 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  21. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Each variable updates by the function f i ( q ). 02 12 f a ( q ) f b ( q ) q a q b 00 1 2 01 1 2 02 1 0 01 11 10 0 0 11 1 0 12 1 0 00 10 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  22. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Each variable updates by the function f i ( q ). 02 12 f a ( q ) f b ( q ) q a q b 00 1 2 01 1 2 02 1 0 01 11 10 0 0 11 1 0 12 1 0 00 10 Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  23. Thomas formalism R. Thomas (1973) introduces discrete variables, q i to describe different expression levels of gene i . Each variable updates by the function f i ( q ). 02 12 f a ( q ) f b ( q ) q a q b 00 1 2 01 1 2 02 1 0 01 11 10 0 0 11 1 0 12 1 0 00 10 The state transition graph (STG) describes discrete dynamics. Thomas, R., Boolean Formalization of Genetic Control Circuits (1973)

  24. Discretising a PADE Snoussi (1989) discretises a PADE into a Thomas model. Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

  25. Discretising a PADE Snoussi (1989) discretises a PADE into a Thomas model. 02 12 Assign a discrete state to each domain. A domain and its focal point is like 01 11 a state and its update. 00 10 Snoussi, El H., Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. (1989)

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