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F ROM S TO HA P HASE 1 Writing the stochastic -calculus code. G =? - PowerPoint PPT Presentation

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I NTRODUCTION H YBRID A UTOMATA S E XAMPLES H YBRID S EMANTICS FOR S TOCHASTIC - CALCULUS Luca Bortolussi 1 , 2 Alberto Policriti 3 , 4 1 Dipartimento di Matematica ed Informatica Universit degli studi di Trieste luca@dmi.units.it 2 Center

  1. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES H YBRID S EMANTICS FOR S TOCHASTIC π - CALCULUS Luca Bortolussi 1 , 2 Alberto Policriti 3 , 4 1 Dipartimento di Matematica ed Informatica Università degli studi di Trieste luca@dmi.units.it 2 Center for Biomolecular Medicine Area Science Park, Trieste 3 Dipartimento di Matematica ed Informatica Università degli studi di Udine alberto.policriti@dimi.uniud.it 4 Institute for Applied Genomics Udine Science Park AB 2008, Castle of Hagenberg, 31 th July 2008

  2. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES V IEWS ON S YSTEMS B IOLOGY

  3. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES M ODELING IN B IOCHEMISTRY Stochastic Processes Ordinary Differential Equations DISCRETE CONTINUOUS STOCHASTIC DETERMINISTIC P ROS P ROS Well developed theory. Physically faithful (for an homogeneous mixture). Requires usually less parameters. C ONS Numerical simulation is faster. Analytically unsolvable C ONS Simulation is computationally expensive Approximates discrete quantities as continuous. Requires many unknown parameters Can produce wrong results.

  4. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES M ODELING IN B IOCHEMISTRY Stochastic Processes Ordinary Differential Equations DISCRETE CONTINUOUS STOCHASTIC DETERMINISTIC P ROS P ROS Well developed theory. Physically faithful (for an homogeneous mixture). Requires usually less parameters. C ONS Numerical simulation is faster. Analytically unsolvable C ONS Simulation is computationally expensive Approximates discrete quantities as continuous. Requires many unknown parameters Can produce wrong results.

  5. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES In Medias Res Stat Virtus — C ICERO Ordinary Stochastic Differential processes Equations

  6. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES In Medias Res Stat Virtus — C ICERO Hybrid Automata CONTINUOUS Ordinary Stochastic DISCRETE Differential processes Equations (NON) DETERMINISTIC (or STOCHASTIC)

  7. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES O UTLINE 1 I NTRODUCTION 2 H YBRID A UTOMATA 3 S TOCHASTIC π - CALCULUS 4 E XAMPLES

  8. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES H YBRID A UTOMATA : THE S PIRIT Many real systems have a double nature. They: evolve in a continuous way, are ruled by a discrete system. M ODELING Hybrid Automata have been developed to deal with this hybrid behavior.

  9. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES The EXAMPLE ON H YBRID A UTOMATA : THE THERMOSTAT

  10. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES The EXAMPLE ON H YBRID A UTOMATA : THE THERMOSTAT

  11. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES The EXAMPLE ON H YBRID A UTOMATA : THE THERMOSTAT

  12. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES The EXAMPLE ON H YBRID A UTOMATA : THE THERMOSTAT

  13. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES The EXAMPLE ON H YBRID A UTOMATA : THE THERMOSTAT

  14. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES S TOCHASTIC π - CALCULUS STOCHASTIC π - CALCULUS There are two basic entities: agents and channel names Agents interact by exchanging channel names through channels (message-based computation). Computation resides in the evolution of the status of the agents (behavioral computation). Communications take an exponentially distributed time to be completed. (stochastic evolution). S YNTAX E ::= 0 | X = M , E M ::= 0 | π. P ⊕ M P ::= 0 | ( X | P ) π ::= τ r | ? x r | ! x r CGF ::= ( E , P )

  15. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES A N EXAMPLE S ELF -R EPRESSING GENE π - CALCULUS CODE NETWORK G + P → k b G b + P G =? b k b . G b ⊕ τ k p p . ( G | P ) G b → k u G G b = τ k u u . G G → k p G + P P =! b k b . P ⊕ τ k d d . 0 P → k d ∅ ,

  16. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES A N EXAMPLE S ELF -R EPRESSING GENE π - CALCULUS CODE NETWORK G + P → k b G b + P G =? b k b . G b ⊕ τ k p p . ( G | P ) G b → k u G G b = τ k u u . G G → k p G + P P =! b k b . P ⊕ τ k d d . 0 P → k d ∅ , Stochastic simulation of π -calculus ODE simulation of π -calculus model model

  17. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES C ONTROL AUTOMATA π - CALCULUS SELF - REPRESSING GENE NETWORK G =? b k b . G b ⊕ τ k p p . ( G | P ) G b = τ k u u . G P =! b k b . P ⊕ τ k d d . 0 C ONTROL A UTOMATA The identification of multi-state components is connected with conservation laws of the system, and it can be performed by solving a zero-one integer programming problem.

  18. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES C ONTROL AUTOMATA π - CALCULUS SELF - REPRESSING GENE NETWORK G =? b k b . G b ⊕ τ k p p . ( G | P ) G b = τ k u u . G P =! b k b . P ⊕ τ k d d . 0 C ONTROL A UTOMATA The identification of multi-state components is connected with conservation laws of the system, and it can be performed by solving a zero-one integer programming problem.

  19. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES F ROM S π TO HA P HASE 1 Writing the stochastic π -calculus code. G =? b k b . G b ⊕ τ k p p . ( G | P ) G b = τ k u u . G P =! b k b . P ⊕ τ k d d . 0

  20. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES F ROM S π TO HA P HASE 2 Identify control agents and convert them in an automaton-like form.

  21. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES F ROM S π TO HA P HASE 3 Generating the HA associated to the single components

  22. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES F ROM S π TO HA P HASE 4 Constructing the product automaton.

  23. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES Z OOMING ON THE CONSTRUCTION C ONTROL G RAPH It is the graph of the “automaton” associated to π -calculus code.

  24. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES Z OOMING ON THE CONSTRUCTION C ONTROL G RAPH It is the graph of the “automaton” associated to π -calculus code.

  25. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES Z OOMING ON THE CONSTRUCTION F LOWS WITHIN MODES They are generated considering only looping edges. Each looping edge is a flux in the ODE.

  26. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES Z OOMING ON THE CONSTRUCTION F LOWS WITHIN MODES They are generated considering only looping edges. Each looping edge is a flux in the ODE.

  27. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES Z OOMING ON THE CONSTRUCTION G UARDS AND R ESETS Guards require that there are enough agents to communicate. Resets describe the effect of an action on the agents.

  28. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES Z OOMING ON THE CONSTRUCTION G UARDS AND R ESETS Guards require that there are enough agents to communicate. Resets describe the effect of an action on the agents.

  29. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES Z OOMING ON THE CONSTRUCTION T IMING CONDITION ON EDGES Clock variables are used to fires discrete transition at their expected time.

  30. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES S ELF -R EPRESSING GENE NETWORK : SIMULATION OF THE HYBRID AUTOMATON π - CALCULUS CODE S ELF -R EPRESSING GN G + P → k b G b + P G =? b k b . G b ⊕ τ k p p . ( G | P ) G b → k u G G b = τ k u G → k p G + P u . G P =! b k b . P ⊕ τ k d P → k d ∅ , d . 0

  31. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES H YBRID R EPRESSILATOR ODE simulation stochastic simulation

  32. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES H YBRID R EPRESSILATOR HA simulation

  33. I NTRODUCTION H YBRID A UTOMATA S π E XAMPLES T HANKS FOR THE ATTENTION Questions? A DVERTISING : BCI 2008 Fifth International School on Biology, Computation, and Information. Trieste, September 8-12, 2008 http://bci2008.cbm.fvg.it

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