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Hybrid Systems Modeling and Analysis of Regulatory Pathways Rajeev Alur University of Pennsylvania LSB, August 2006

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

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Talk Outline 1. A brief tour of hybrid systems research 2. Application to regulatory pathways Thanks to many colleagues in Penns Bio-Hybrid Group, including Calin Belta (Boston U) Franjo Ivancic (NEC Labs) Vijay Kumar Harvey Rubin Oleg Sokolsky … See

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Hybrid Automata Set L of of locations, and set E of edges Set X of k continuous variables State space: L X R k, Region: subset of R k 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 R k X R k Synchronization labels (communication information)

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(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, ] s.t. f(0)=x, f( )=x, and for all t<=, f(t) satisfies Inv(l) and df(t) satisfies Flow(l)(f(t))

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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

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

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

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CHARON Toolkit

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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) X Reach(X)

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Polyhedral Flow Pipe Approximations X0X0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 t7t7 t8t8 t9t9 divide R [0,T] (X 0 ) into [t k,t k+1 ] segments enclose each segment with a convex polytope R M [0,T] (X 0 ) = union of polytopes

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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)

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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. t x Concrete Space: L x R n Abstract Space: L x {0,1} k

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

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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

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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

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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

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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

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Regulatory Networks cell-to-cell signaling STARTSTOP gene transcription translation regulation nascent protein chemical reaction + - negative positive gene expression

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Luminescence / Quorum Sensing in Vibrio Fischeri

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Hybrid Modeling STARTSTOP luxR gene transcription translation regulation protein LuxR chemical reaction - + negative positive Ai CRP Traditionally, biological systems are modeled using smooth functions

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

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Luminescence Regulation CRP luxICDABEGluxR Ai LuxA LuxB luciferase LuxI Substrate LuxR lux box CRP binding site LuxR Ai OLOL OROR cAMP

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

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Simulation Results external A i (input) concentrations for various entities luminesence (output) switch history switch history

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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

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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

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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

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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

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