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

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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

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

HYBRID SEMANTICS FOR STOCHASTIC π-CALCULUS

Luca Bortolussi1,2 Alberto Policriti3,4

1Dipartimento di Matematica ed Informatica

Università degli studi di Trieste luca@dmi.units.it

2Center for Biomolecular Medicine

Area Science Park, Trieste

3Dipartimento di Matematica ed Informatica

Università degli studi di Udine alberto.policriti@dimi.uniud.it

4Institute for Applied Genomics

Udine Science Park

AB 2008, Castle of Hagenberg, 31th July 2008

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

VIEWS ON SYSTEMS BIOLOGY

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

MODELING IN BIOCHEMISTRY

Stochastic Processes DISCRETE STOCHASTIC Ordinary Differential Equations CONTINUOUS DETERMINISTIC

PROS Physically faithful (for an homogeneous mixture). CONS Analytically unsolvable Simulation is computationally expensive Requires many unknown parameters PROS Well developed theory. Requires usually less parameters. Numerical simulation is faster. CONS Approximates discrete quantities as continuous. Can produce wrong results.

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

MODELING IN BIOCHEMISTRY

Stochastic Processes DISCRETE STOCHASTIC Ordinary Differential Equations CONTINUOUS DETERMINISTIC

PROS Physically faithful (for an homogeneous mixture). CONS Analytically unsolvable Simulation is computationally expensive Requires many unknown parameters PROS Well developed theory. Requires usually less parameters. Numerical simulation is faster. CONS Approximates discrete quantities as continuous. Can produce wrong results.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

In Medias Res Stat Virtus — CICERO

Stochastic processes Ordinary Differential Equations

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

In Medias Res Stat Virtus — CICERO

Stochastic processes Hybrid Automata CONTINUOUS DISCRETE (NON) DETERMINISTIC (or STOCHASTIC) Ordinary Differential Equations

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

OUTLINE

1 INTRODUCTION 2 HYBRID AUTOMATA 3 STOCHASTIC π-CALCULUS 4 EXAMPLES

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

HYBRID AUTOMATA: THE SPIRIT

Many real systems have a double nature. They: evolve in a continuous way, are ruled by a discrete system. MODELING Hybrid Automata have been developed to deal with this hybrid behavior.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

The EXAMPLE ON HYBRID AUTOMATA: THE

THERMOSTAT

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

The EXAMPLE ON HYBRID AUTOMATA: THE

THERMOSTAT

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

The EXAMPLE ON HYBRID AUTOMATA: THE

THERMOSTAT

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

The EXAMPLE ON HYBRID AUTOMATA: THE

THERMOSTAT

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

The EXAMPLE ON HYBRID AUTOMATA: THE

THERMOSTAT

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

STOCHASTIC 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). SYNTAX E ::= 0|X = M, E M ::= 0|π.P ⊕ M P ::= 0|(X|P) π ::= τ r|?xr|!xr CGF ::= (E, P)

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

AN EXAMPLE

SELF-REPRESSING GENE

NETWORK

G + P →kb Gb + P Gb →ku G G →kp G + P P →kd ∅, π-CALCULUS CODE G =?bkb.Gb ⊕ τ kp

p .(G|P)

Gb = τ ku

u .G

P =!bkb.P ⊕ τ kd

d .0

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

AN EXAMPLE

SELF-REPRESSING GENE

NETWORK

G + P →kb Gb + P Gb →ku G G →kp G + P P →kd ∅, π-CALCULUS CODE G =?bkb.Gb ⊕ τ kp

p .(G|P)

Gb = τ ku

u .G

P =!bkb.P ⊕ τ kd

d .0

Stochastic simulation of π-calculus model ODE simulation of π-calculus model

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

CONTROL AUTOMATA

π-CALCULUS SELF-REPRESSING

GENE NETWORK

G =?bkb.Gb ⊕ τ kp

p .(G|P)

Gb = τ ku

u .G

P =!bkb.P ⊕ τ kd

d .0

CONTROL AUTOMATA 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.

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INTRODUCTION HYBRID AUTOMATA

Sπ

EXAMPLES

CONTROL AUTOMATA

π-CALCULUS SELF-REPRESSING

GENE NETWORK

G =?bkb.Gb ⊕ τ kp

p .(G|P)

Gb = τ ku

u .G

P =!bkb.P ⊕ τ kd

d .0

CONTROL AUTOMATA 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.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

FROM Sπ TO HA

PHASE 1 Writing the stochastic π-calculus code. G =?bkb.Gb ⊕ τ kp

p .(G|P)

Gb = τ ku

u .G

P =!bkb.P ⊕ τ kd

d .0

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

FROM Sπ TO HA

PHASE 2 Identify control agents and convert them in an automaton-like form.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

FROM Sπ TO HA

PHASE 3 Generating the HA associated to the single components

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

FROM Sπ TO HA

PHASE 4 Constructing the product automaton.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

ZOOMING ON THE CONSTRUCTION

CONTROL GRAPH It is the graph of the “automaton” associated to π-calculus code.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

ZOOMING ON THE CONSTRUCTION

CONTROL GRAPH It is the graph of the “automaton” associated to π-calculus code.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

ZOOMING ON THE CONSTRUCTION

FLOWS WITHIN MODES They are generated considering only looping edges. Each looping edge is a flux in the ODE.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

ZOOMING ON THE CONSTRUCTION

FLOWS WITHIN MODES They are generated considering only looping edges. Each looping edge is a flux in the ODE.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

ZOOMING ON THE CONSTRUCTION

GUARDS AND RESETS

Guards require that there are enough agents to communicate. Resets describe the effect of an action on the agents.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

ZOOMING ON THE CONSTRUCTION

GUARDS AND RESETS

Guards require that there are enough agents to communicate. Resets describe the effect of an action on the agents.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

ZOOMING ON THE CONSTRUCTION

TIMING CONDITION ON EDGES Clock variables are used to fires discrete transition at their expected time.

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

SELF-REPRESSING GENE NETWORK: SIMULATION OF

THE HYBRID AUTOMATON

SELF-REPRESSING GN G + P →kb Gb + P Gb →ku G G →kp G + P P →kd ∅, π-CALCULUS CODE G =?bkb.Gb ⊕ τ kp

p .(G|P)

Gb = τ ku

u .G

P =!bkb.P ⊕ τ kd

d .0

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

HYBRID REPRESSILATOR

ODE simulation stochastic simulation

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

HYBRID REPRESSILATOR

HA simulation

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INTRODUCTION HYBRID AUTOMATA

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EXAMPLES

THANKS FOR THE ATTENTION Questions?

ADVERTISING: BCI 2008 Fifth International School on Biology, Computation, and Information. Trieste, September 8-12, 2008 http://bci2008.cbm.fvg.it