Hybrid Systems Modeling and Analysis of Regulatory Pathways

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## Hybrid Systems Modeling and Analysis of Regulatory Pathways

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**Hybrid Systems Modeling and Analysisof Regulatory Pathways**Rajeev Alur University of Pennsylvania www.cis.upenn.edu/~alur/ LSB, August 2006**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**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/**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)**(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))**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**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**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**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)**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**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)**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.**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**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**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**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**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**-**negative gene START STOP regulation transcription positive + translation nascent protein cell-to-cell signaling chemical reaction Regulatory Networks gene expression**+**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**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**cAMP**Luminescence Regulation - + CRP OL OR lux box luxR luxICDABEG CRP binding site + LuxR Ai - LuxA LuxI LuxR LuxB Substrate Ai luciferase**non-lum**lum Reachability Under what conditions can the bacterium switch on the light? lum dynamics switching surface nonlum dynamics**Simulation Results**switch history external Ai (input) luminesence (output) concentrations for various entities switch history**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**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**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**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