P1252428682NfwgS

1
Hybrid Systems Modeling and Analysisof
Regulatory Pathways
Rajeev Alur University of Pennsylvania
www.cis.upenn.edu/alur/
LSB, August 2006
2
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 Penns 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 X Rk
• 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-gt (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-gt (l,x) where f is a
  continuous function from 0,d s.t. f(0)x,
  f(d)x, and for all tltd, 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 gt stride /2
dy kv
L1
j1
j2
L2
(x, y)
y
dx kv x lt stride /2
dy -kv
9
CHARON Toolkit
10
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
Polyhedral Flow Pipe Approximations
X0

• RM0,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
Predicate Abstraction

• Input is a hybrid automaton and a set of k
  boolean predicates, e.g. xy gt 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
Regulatory Networks
gene expression
20
Luminescence / Quorum Sensingin Vibrio Fischeri
21
Hybrid Modeling

• Traditionally, biological systems are modeled
  using smooth functions.

CRP
22
Hybrid Modeling
23
Luminescence Regulation
-

CRP
OL
OR
lux box
luxICDABEG
luxR
CRP binding site

LuxR
Ai
-
LuxA
LuxI
LuxR
LuxB
Substrate
Ai
luciferase
24
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