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Hybrid Systems Modeling and Analysis of Regulatory Pathways

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  1. Hybrid Systems Modeling and Analysisof Regulatory Pathways Rajeev Alur University of Pennsylvania www.cis.upenn.edu/~alur/ LSB, August 2006

  2. State machines + Dynamical systems on off x>68 dx/dt=-k’x x>60 dx/dt=kx x<70 x<63 Coordination Protocols Systems Biology Automotive Robotics Animation Hybrid Systems Computer Science • Automata/Logic • Concurrency • Formal verification + Control Theory • Optimal control • Stability analysis • Discrete-event system Software+ Environment

  3. Talk Outline • A brief tour of hybrid systems research • Application to regulatory pathways Thanks to many colleagues in Penn’s Bio-Hybrid Group, including Calin Belta (Boston U) Franjo Ivancic (NEC Labs) Vijay Kumar Harvey Rubin Oleg Sokolsky … See http://www.cis.upenn.edu/biocomp/

  4. Hybrid Automata • Set L of of locations, and set E of edges • Set X of k continuous variables • State space: L X Rk, Region: subset of Rk • For each location l, • Initial states: region Init(l) • Invariant: region Inv(l) • Continuous dynamics: dX in Flow(l)(X) • For each edge e from location l to location l’ • Guard: region Guard(e) • Update relation over Rk XRk • Synchronization labels (communication information)

  5. (Finite) Executions of Hybrid Automata • State: (l, x) such that x satisfies Inv(l) • Initialization: (l,x) s.t. x satisfies Init(l) • Two types of state updates • Discrete switches: (l,x) –a-> (l’,x’) if there is an a-labeled edge e from l to l’ s.t. x satisfies Guard(e) and (x,x’) satisfies update relation Jump(e) • Continuous flows: (l,x) –f-> (l,x’) where f is a continuous function from [0,d] s.t. f(0)=x, f(d)=x’, and for all t<=d, f(t) satisfies Inv(l) and df(t) satisfies Flow(l)(f(t))

  6. CHARON Language Features • Individual components described as agents • Composition, instantiation, and hiding • Individual behaviors described as modes • Encapsulation, instantiation, and Scoping • Support for concurrency • Shared variables as well as message passing • Support for discrete and continuous behavior • Differential as well as algebraic constraints • Discrete transitions can call Java routines

  7. Walking Model: Architecture and Agents • Input • touch sensors • Output • desired angles of each joint • Components • Brain: control four legs • Four legs: control servo motors • Instantiated from the same pattern

  8. Walking Model: Behavior and Modes v x dx = -v x > stride /2 dy = kv L1 j1 L2 j2 (x, y) y dx = kv x < stride /2 dy = -kv

  9. CHARON Toolkit

  10. Reach(X) X Reachability Analysis for Dynamical Systems • Goal: Given an initial region, compute whether a bad state can be reached • Key step: compute Reach(X) for a given set X under dx/dt = f(x)

  11. t6 t5 t7 t4 t3 t8 t2 t9 t1 • divide R[0,T](X0) into [tk,tk+1] segments • enclose each segment with a convex polytope Polyhedral Flow Pipe Approximations X0 • RM[0,T](X0) = union of polytopes

  12. Abstraction and Refinement • Abstraction-based verification • Given a model M, build an abstraction A • Check A for violation of properties • Either A is safe, or is adequate to indicate a bug in M, or gives false negatives (in that case, refine the abstraction and repeat) • Many projects exploring abstraction-based verification for hybrid systems • Predicate abstraction (Charon at Penn) • Counter-example guided abstraction refinement (CEGAR at CMU) • Qualitative abstraction using symbolic derivatives (SAL at SRI)

  13. x t Concrete Space: L x Rn Abstract Space: L x {0,1} k Predicate Abstraction • Input is a hybrid automaton and a set of k boolean predicates, e.g. x+y > 5-z. • The partitioning of the concrete state space is specified by the user-defined k predicates.

  14. Overview of the Approach Hybrid system Boolean predicates additional predicates Search in abstract space Safety property No! Counter-example Property holds Analyze counter-example Real counter- example found

  15. Hybrid Systems Wrap-up • Efficient simulation • Accurate event detection • Symbolic simulation • Computing reachable state-space • Many new techniques emerging: level sets, Zenotopes, dimensionality reduction.. • Scalability still remains a challenge

  16. Cellular Networks • Networks of interacting biomolecules carry out many essential functions in living cells (gene regulation, protein production) • Both positive and negative feedback loops • Design principles poorly understood • Large amounts of data is becoming available • Beyond Human Genome: Behavioral models of cellular networks • Modeling becoming increasingly relevant as an aid to narrow the space of experiments

  17. Model-based Systems Biology • Goal A: Provide notations for describing complex systems in a modular, structured manner • Principles of concurrency theory (e.g. compositionality) • Hierarchy, encapsulation, reuse • Visual programming tools • Goal B: Simulation and analysis for better understanding • Classical debugging tools • Reachability and stability analysis • Model-based experiments to combat the combinatorial explosion due to multiplicity of parameters

  18. What to Model ? • Cellular networks exhibit a complex mix of features • Discrete switching as genes are turned on/off • High degree of concurrency • Stochastic behavior (particularly at low concentrations) • Chemical reactions • Models possible at different levels of abstractions • Discrete graph models capturing dependencies • Boolean models capturing qualitative states • Purely continuous models • Hybrid systems • Stochastic models • Location-aware models

  19. - negative gene START STOP regulation transcription positive + translation nascent protein cell-to-cell signaling chemical reaction Regulatory Networks gene expression

  20. Luminescence / Quorum Sensingin Vibrio Fischeri

  21. + negative luxR gene START STOP regulation transcription positive - translation protein LuxR chemical reaction 1 0.5 Ai Ai Hybrid Modeling Traditionally, biological systems are modeled using smooth functions. CRP

  22. At low concentrations, a continuous approximation model might not be appropriate. Instead, a stochastic model should be used. Essentially hybrid system Discrete jump (mRNA) regulatory protein/complex mode Linear dynamics (proteins not involved in chemical reactions) In some cases, the biological description of a system is itself hybrid. high conc low conc continuous model stochastic model Nonlinear dynamics (proteins involved in chemical reactions) Hybrid Modeling

  23. cAMP Luminescence Regulation - + CRP OL OR lux box luxR luxICDABEG CRP binding site + LuxR Ai - LuxA LuxI LuxR LuxB Substrate Ai luciferase

  24. non-lum lum Reachability Under what conditions can the bacterium switch on the light? lum dynamics switching surface nonlum dynamics

  25. Simulation Results switch history external Ai (input) luminesence (output) concentrations for various entities switch history

  26. BioSketchPad • Interactive tool for graphical models of biomolecular and cellular networks • Nodes and edges with attributes • Hierarchical • Intended for use by biologists • Compiler to translate BioSketchPad models to Charon

  27. BioSketchPad Concepts • Species nodes • Name (e.g. Ca, alcohol dehydrogenase, notch) • Type (e.g. gene, protein) • Location (e.g. cell membrane, nucleus) • N-mer polymerization, electrical charge • Initial concentration • Reaction nodes • Input and output connectors • Type (e.g. transformation, transcription) • Parameters for rate laws • Regulation nodes • Connected to species nodes and/or reaction nodes to modulate the rate of reaction by concentration of species • Weighted sum, tabular, product forms

  28. Summary • Hybrid systems are useful to model some biological regulatory networks. • The simulation/reachability results of the luminescence control in Vibrio fischeri are in accordance with phenomena observed in experiments. • Modeling concepts such as hierarchy, concurrency, reuse, are relevant for modular specifications • BioSketchPad integrates many of these ideas

  29. Challenges • Finding all the information needed to build a model is difficult • Finding people who can build models is even more difficult • Finding a common format for exchanging models among tools can make more models available • Scalability of analysis