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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
Hybrid Systems
State machines + Dynamical systems
on
dx/dt=kx
x<70
Automotive
x>68
x<63
Coordination
Protocols
off
dx/dt=-k’x
x>60
Robotics
Computer Science
 Automata/Logic
 Concurrency
 Formal verification
+ Control Theory
 Optimal control
 Stability analysis
 Discrete-event system
Software + Environment
Animation
Systems
Biology
Talk Outline
1.
A brief tour of hybrid systems research
2.
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 X Rk
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
L2
j1
j2
y
dy = -kv
dx = kv
x < stride /2
(x, y)
CHARON Toolkit
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)
Reach(X)
X
Polyhedral Flow Pipe Approximations
t3
t4
t5
t6
t7
t8
t2
t1
X0
t9
• divide R[0,T](X0) into [tk,tk+1] segments
• enclose each segment with a convex polytope
• 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)
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.
x
t
Concrete Space:
LxRn
Abstract Space:
L x {0,1} k
Overview of the Approach
Hybrid
system
Safety
property
Boolean
predicates
additional
predicates
Search in abstract space
No!
Counter-example
Real
counterexample
found
Property
holds
Analyze counter-example
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
Regulatory Networks
gene expression
-
negative
gene
START
regulation
positive
+
transcription
translation
nascent
protein
cell-to-cell
signaling
chemical
reaction
STOP
Luminescence / Quorum Sensing
in Vibrio Fischeri
Hybrid Modeling
d ( LuxR ) biological systems
LuxR are modeled using smooth functions.
Traditionally,
 Tl luxR 

dt
H sp
d[ x]
 synthesis  decay  transform
 transport
 ( X , , )
dt
b
d
 rLuxR
Ai
LuxR

r
Co  kG LuxR
1

/ Ai
LuxR / Ai
Xm
CRP
+
negative
0.5
Xm
Xm
1  Xm
2
X sw
X sw
START
luxR gene
X
STOP
regulation
positive
transcription
translation
-
protein
LuxR
Ai
chemical
reaction
d ( luxR )
 Tc [  ( CRP , CRP , CRP )
dt
( 1   ( LuxR  Ai , LuxR Ai , LuxR Ai ))  b )

Ai
luxR
 kG luxR
H RNA
Hybrid Modeling
Essentially
hybrid
system
Discrete jump
(mRNA)
regulatory
protein/complex
mode
high conc
continuous model
Nonlinear dynamics
(proteins involved in
chemical reactions)
Linear dynamics
(proteins not involved
in chemical reactions)
At low
concentrations,
a continuous
approximation
model might
not be
appropriate.
Instead, a
stochastic
model should
be used.
low conc
stochastic model
In some cases,
the biological
description of
a system is
itself hybrid.
Luminescence Regulation
+
CRP
cAMP
OL
OR
lux box
luxR
luxICDABEG
CRP binding site
LuxR
-
+
Ai
LuxI
LuxA
LuxR
LuxB
Ai
Substrate
luciferase
Reachability
lum
x  Ax  b10
x8  x8 sw
non-lum
x8  x8 sw
x  Ax  b00
x8  x8 sw
x8  x8 sw
x  Ax  bi0
c0  0 , c1  1
Under what conditions
x4 bacteriumswitch
H RNATlon
Tc (the
ci light?
b )
can the
x   x7 
 
 x8 
bi0  


0
0



x8 ( Co )
lum dynamics
x8 sw
switching surface
nonlum dynamics
x4 ( LuxI )
x7 ( Ai )
Simulation Results
external Ai
(input)
luminesence
(output)
switch
history
switch
history
concentrations
for various
entities
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