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Hybrid Systems Modeling and Analysis of 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.

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slide1

Hybrid Systems Modeling and Analysisof Regulatory Pathways

Rajeev Alur

University of Pennsylvania

www.cis.upenn.edu/~alur/

LSB, August 2006

slide2

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
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
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
(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
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
slide7

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
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

reachability analysis for dynamical systems

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)
polyhedral flow pipe approximations

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 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)
predicate abstraction

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
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
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
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
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
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
regulatory networks

-

negative

gene

START

STOP

regulation

transcription

positive

+

translation

nascent

protein

cell-to-cell

signaling

chemical

reaction

Regulatory Networks

gene expression

hybrid modeling

+

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

hybrid modeling1

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
luminescence regulation

cAMP

Luminescence Regulation

-

+

CRP

OL

OR

lux box

luxR

luxICDABEG

CRP binding site

+

LuxR

Ai

-

LuxA

LuxI

LuxR

LuxB

Substrate

Ai

luciferase

reachability

non-lum

lum

Reachability

Under what conditions

can the bacterium switch on the light?

lum dynamics

switching surface

nonlum dynamics

simulation results
Simulation Results

switch

history

external Ai

(input)

luminesence

(output)

concentrations

for various

entities

switch

history

biosketchpad
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
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
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
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