Hybrid Systems Modeling and Analysis

of Regulatory Pathways

Rajeev Alur

University of Pennsylvania www.cis.upenn.edu/~alur/

LSB, August 2006

State machines

offon

+ Dynamical systems

dx/dt=kxx<70

dx/dt=-k’xx>60

x>68

x<63

Automotive Robotics AnimationSystemsBiology

CoordinationProtocols

Computer Science Automata/Logic Concurrency Formal verification

+ Control Theory Optimal control Stability analysis Discrete-event

system

Software + Environment

Hybrid Systems

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,] 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

• 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: Architecture and Agents

x

y

j1

j2

L1

(x, y)

v

L2

Walking Model: Behavior and Modes

dx = -vx > stride /2

dy = kv

dy = -kv dx = kvx < stride /2

CHARON ToolkitCHARON 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)

X

Reach(X)

Polyhedral Flow Pipe Approximations

X0

t1

t2

t3

t4

t5t6 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 verificationGiven 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.

t

x

Concrete Space:L x R n

Abstract Space:L x {0,1} k

Overview of the Approach

Safetyproperty

Hybridsystem

Booleanpredicates

Search in abstract space

Analyze counter-example

Propertyholds

No!Counter-example

Realcounter-examplefound

additionalpredicates

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, reuseVisual programming tools

Goal B: Simulation and analysis for better understanding

Classical debugging toolsReachability and stability analysisModel-based experiments to combat the combinatorial explosion due to multiplicity of parameters

What to Model ?

Cellular networks exhibit a complex mix of featuresDiscrete switching as genes are turned on/offHigh degree of concurrencyStochastic behavior (particularly at low concentrations)Chemical reactions

Models possible at different levels of abstractionsDiscrete graph models capturing dependenciesBoolean models capturing qualitative statesPurely continuous modelsHybrid systemsStochastic modelsLocation-aware models

Regulatory Networks

cell-to-cellsignaling

START STOPgene

transcription

translation

regulation

nascentprotein

chemicalreaction

+

-

negative

positive

gene expression

Luminescence / Quorum Sensingin Vibrio Fischeri

Hybrid Modeling

START STOPluxR gene

transcription

translation

regulation

proteinLuxR

chemicalreaction

-+

negative

positive

Ai

Ai

CRP

Traditionally, biological systems are modeled using smooth functions.

Xm

Xm

X

),,X( XmXm

10.5

2swX1

swX

transporttransformdecaysynthesisdt

]x[d

luxRkHluxR

)b)),,AiLuxR(1(

),,CRP([Tdt

)luxR(d

GRNA

AiLuxRAiLuxR

CRPCRPc

LuxRkCorLuxRAir

HLuxR

luxRTdt

)LuxR(d

Gdb

spl

Ai/LuxRAi/LuxR

Hybrid Modeling

At low concentrations, a continuous approximation model might not be appropriate. Instead, a stochastic model should be used.

stochastic model

low conc

continuous model

high conc

In some cases, the biological description of a system is itself hybrid.

Essentiallyhybrid systemDiscrete jump

(mRNA)

Nonlinear dynamics(proteins involved in chemical reactions)

Linear dynamics(proteins not involvedin chemical reactions)

moderegulatoryprotein/complex

Luminescence Regulation

CRP

luxICDABEGluxR

Ai

LuxA

LuxB

luciferase

LuxI

Substrate

LuxR

lux box

CRP binding site

LuxR Ai

OL OR

-

+-

+ cAMP

Reachability

)Co(x8

)Ai(x7

)LuxI(x4

switching surface

lum dynamics

nonlum dynamics

sw88 xx

10bAxx

sw88 xx

lum

00bAxx

sw88 xx

non-lum

sw88 xx

Under what conditionscan the bacterium switch on the light?

sw8x

0ibAxx

8

7

4

x

x

x

x

0

0

)bc(TTH

biclRNA

0i

1c,0c 10

Simulation Results

external Ai

(input)

concentrationsfor various

entities

luminesence(output)

switchhistory

switchhistory

BioSketchPad

Interactive tool for graphical models of biomolecular and cellular networks

Nodes and edges with attributesHierarchical

Intended for use by biologists

Compiler to translate BioSketchPad models to Charon

BioSketchPad Concepts

Species nodesName (e.g. Ca, alcohol dehydrogenase, notch)Type (e.g. gene, protein)Location (e.g. cell membrane, nucleus)N-mer polymerization, electrical chargeInitial concentration

Reaction nodesInput and output connectorsType (e.g. transformation, transcription)Parameters for rate laws

Regulation nodesConnected to species nodes and/or reaction nodes to modulate the rate of reaction by concentration of speciesWeighted 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