Systems Hybrid 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
x>68
on dx/dt=kx x<70
x<63
off dx/dt=-kx x>60
Computer Science Automata/Logic Concurrency Formal verification + Control Theory Optimal control Stability analysis Discrete-event system Software + Environment Systems Biology
Automotive
Coordination Protocols
Robotics
Animation
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/
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,] s.t. f(0)=x, f()=x, and for all t<=, 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
(x, y)
dy = -kv
dx = kv x < stride /2
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 t2 t1 X0 t4 t5 t6 t7 t8
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 amounts Large 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 regulation positive
+
cell-to-cell signaling
START gene transcription translation nascent protein STOP chemical reaction
Luminescence / Quorum Sensing in Vibrio Fischeri
Hybrid Modeling
d ( LuxR ) Traditionally, biological LuxR modeled using smooth functions. = Tl luxR systems are dt H sp d[ x] = synthesis decay transform transport ( X , , ) dt b d rLuxR / Ai Ai LuxR + rLuxR / Ai Co kG LuxR 1
Xm Xm
CRP
0.5
Xm
negative regulation positive
1 2 X sw Xm X sw
X
+
START luxR gene transcription translation protein LuxR STOP
Ai
chemical reaction Ai
d ( luxR ) = Tc [ ( CRP , CRP , CRP ) dt ( 1 ( LuxR Ai , LuxR Ai , LuxR Ai )) + b ) luxR kG luxR H RNA
Hybrid Modeling
Essentially hybrid system
mode Linear dynamics (proteins not involved in chemical reactions)
Discrete jump (mRNA) regulatory protein/complex Nonlinear dynamics (proteins involved in chemical reactions)
high conc
continuous model
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
+
OL
CRP
cAMP
lux box OR
luxR
CRP binding site
luxICDABEG
LuxR
LuxR
Ai LuxI
+
LuxA
LuxB Ai Substrate
luciferase
Reachability
lum
x = Ax + bi0
x = x7 x8
x8 = x8 sw
non-lum
x = Ax + b00 x8 x8 sw
x8 ( Co )
Un...