DEPARTMENT OF COMPUTER SCIENCE DOCTORAL DISSERTATION DEFENSE, 1:30 PM, March 7, 2006

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DEPARTMENT OF COMPUTER SCIENCE DOCTORAL
DISSERTATION DEFENSE, 130 PM, March 7, 2006
Algorithmic Algebraic Model Checking (AAMC)
Hybrid Automata Systems Biology

• Venkatesh Pranesh Mysore
• NYU Bioinformatics Group, Courant Institute Of
  Mathematical Sciences

Advisor
Auditor
Auditor
Reader
Reader
Prof. Jane Hubbard, NYU Biology
Prof. Bud Mishra, NYU CS, Math Cell Biology
Prof. Amir Pnueli, NYU CS
Prof. Thomas Anantharman, NYU
Prof. Vijay Saraswat, IBM
2
Systems Biology Motivation for the Thesis

• Model, Simulate and Analyze Biochemical Systems
  to Test Hypotheses, Validate Predictions and
  Suggest Experiments

3
Numerical Simulation based Analysis is
Insufficient

• New initial condition ? new simulation
• Unknown constants parameter sweep
• Cannot characterize parameter-ranges or initial
  conditions that represent all possible solutions
  to temporal query
• Difficult to handle non-determinism
• Cannot characterize all possible outcomes
• No general proofs

4
Kinetic mass action-based ordinary differential
equations (ODEs) of the biochemical pathway
Hybrid System model with simpler DEs for slow
reactions (using semi-automatic formal methods)
Equilibrium description of fast reactions (using
Gröbner bases)
Hybrid System model with simpler DEs for slow
reactions and new DEs for fast reactions
Algebraic characterization of temporal properties
via Algorithmic Algebraic Model Checking (using
quantifier elimination)
5
Outline of the Defense

• Systems Biology
• Understanding Hybrid Automata
• Semi-Algebraic Hybrid Systems
• Algebraic Analysis of Pseudo-Equilibrium in
  Metabolic Networks
• Tolque A dense-time algebraic model-checker for
  semi-algebraic hybrid automata
• Future Work

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I. Systems Biology

• Meaning, Definition, Scope, Future
• Modeling Biochemical Systems
• Hybrid Systems

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The Meaning of Systems Biology, Kirschner, 2005

• Behavior of complex biological organization and
  processes in terms of the molecular constituents
• Molecular Biology information transfer
• Physiology adaptive states of the cell and
  organism
• Developmental Biology succession of
  physiological states
• Evolutionary Biology, Ecology aspects of the
  organism are products of selection
• Through quantitative measurement, modeling,
  reconstruction and theory

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A Central Problem in Systems BiologyModeling
Analyzing Biochemical Pathways
Answer Assume a model, simulate it and compare
results with the real rat
Question How do you analyze a rat?
?
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Modeling the Biochemical System
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Formalisms Levels of Abstraction

• Logical Modeling Boolean networks, Concurrent
  Transition Systems, Bayesian networks
• Petrinets, Pi-Calculus, Statecharts, Rule-based
  Modeling
• Hybrid Automata (HA) Linear / Affine HA,
  Stochastic HA, Extended S-Systems,
  Semi-Algebraic HA
• Differential Equations (DEs) Qualitative DEs,
  Ordinary DEs, Partial DEs, S-Systems
• Stochastic Master Equation
• Ab Initio Molecular Dynamics

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An Accurate ODE-Model from Chemical Kinetics

• The Law of Kinetic Mass Action (KMA) rate of a
  reaction proportional to product of concentration
  of reactants
• Assumptions high concentration (continuous
  variable), uniform spatial distribution

Cato M Guldberg (1836-1902) Peter Waage
(1833-1900)
-
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Hybrid System Models of Biochemical Networks
Q1 Both OFF
Q2 Delta ON

• Intuitive representation
• Easy approximation
• Hierarchical models
• Model composition
• Well-developed theory

Q3 Notch ON
Q4 Both ON
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Outline of the Defense

• Systems Biology
• Understanding Hybrid Automata
• Semi-Algebraic Hybrid Systems
• Algebraic Analysis of Pseudo-Equilibrium in
  Metabolic Networks
• Tolque A dense-time algebraic model-checker for
  semi-algebraic hybrid automata
• Future Work

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II. Understanding Hybrid Automata

• Numerical Simulation
• Temporal Logic Model-Checking
• Decidability of Reachability
• Hovering by the Decidability Frontier

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1. Simulating a Biochemical Network
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Numerical Integration of ODEs

• Proceed in small time-steps
• Approximate next value using current value of the
  function and its derivatives (Taylor Series)
• Euler forward, Two-way Euler, Runge-Kutta
• f(X,th) f(X,t) h.f(X,t)
• Part of all computer algebra tools like Matlab,
  Mathematica, R

Brook Taylor (1685-1731)
Leonhard Euler (1707-1783)
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2. Ready to Analyze
Temporal Logic
Model Checking
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Temporal Logic

• Linear Temporal Logic (LTL)
• Temporal Operators Next, Eventually, Henceforth,
  Until, and past-counterparts
• (Eventually xgt5)
• Computation Tree Logic (CTL)
• Lack of information ? non-determinism
• Every temporal operator must be prefixed by a
  Path quantifier
• A for all future paths
• E for some future path
• (E Eventually xgt5)

Amir Pnueli
Ed Clarke
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Model Checking

• LTL Model Checking
• Implicit tableau construction
• Büchi Automata
• CTL Model Checking
• Straight-forward Recursive Descent
• Symbolic Model Checking using OBDDs

Zohar Manna
EA Emerson
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3. Decidability Undecidability of Reachability

• The Reachability Problem
• Deciding the reachability problem
• When is a class of automata undecidable?
• How to prove decidability?
• Significance of decidability characterization
• Undecidable Hybrid Automata

Alan Turing (1912-1954)
Marvin Minsky
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Expressivity Computational Power of Hybrid
Systems

• Discrete States have invariants and flows
• Transitions have guards and resets

s1
s2
s3
s4
h1
h2
h3
h4
time
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4. Two-Dimensional Piece-wise Constant Derivative
(PCD) Systems
Oded Maler
Decidable in 2-D, Undecidable in 3-D
Amir Pnueli
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2-dim PCD to 2-dim HPCD

• HPCD
• Flows Constant
• Guards a lt x lt b, ax by c 0
• Resets x x, x ax b
• Invariants Non-overlapping
• Reachability is OPEN no proof yet of
  decidability or undecidability!

X
Eugene Asarin
Gerardo Schneider
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Refining the 2-dim HPCD Class
X

• HPCD
• Flows Constant
• Guards a lt x lt b, ax by c 0
• Resets x x, x ax b
• Invariants Non-overlapping
• Simulation by taking turns ? Comparative guards
  affine resets are unnecessary
• Domain bounded ? Invariants can be made
  non-overlapping

X
X
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(Un)Decidability Frontier
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Summary

• Analysis of hybrid automata
• Numerical simulation of ODEs
• Model-checking temporal logic properties
• Decidability of Reachability
• Characterization of expressivity
• Hybrid Automata easily simulate the Turing or
  Minsky machine
• PCD-like classes are unsuitable for capturing
  biochemical dynamics
• Need more expansive classes and semi-decidable
  approaches

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Outline

• Systems Biology
• Understanding Hybrid Automata
• Semi-Algebraic Hybrid Systems
• Algebraic Analysis of Pseudo-Equilibrium in
  Metabolic Networks
• Tolque A dense-time algebraic model-checker for
  semi-algebraic hybrid automata
• Future Work

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III. Semi-Algebraic Hybrid Systems

• Algorithmic Algebraic Model Checking
• Semi-Algebraic subclass TCTL
• Undecidability in the real Turing Machine
• Approximate Methods Extended Bisimulation
  Partitioning, Polytopes, Grids, Time
  Discretization

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1. Algorithmic Algebraic Model Checking

• Euler forward Numerical integration f(X,th)
  f(X,t) h.f(X,t)
• Expression in X, t and h
• Error integration discretization approximation
• Model Checking iterative process of checking
  what is true now and at next time
• Possible over semi-algebraic sets using
  quantifier elimination

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Quantifier Elimination (QE)

• Semi-Algebraic Set quantifier-free boolean
  formula in polynomial equations and inequalities
• 1940s - Tarski proved QE decidable
• 1973 Collins discovered CAD (cylindrical
  algebraic decomposition)
• Hong implemented Qepcad
• Other Tools Redlog, Maple, Mathematica, AQCS
• Input (Ex) x2 b x c 0
• Output b2 - 4 c gt 0

Alfred Tarski (1902-1983)
Hoon Hong
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2. Semi-Algebraic Hybrid Systems

• The expressions for invariant, initial, guard and
  reset are semi-algebraic
• TCTL semi-decidable
• Undecidability in real Turing Machine
• Efficient approximation algorithms
• Implementation
• Tolque (uses Qepcad)
• Integration with Simpathica in progress
• Extension of Polynomial Hybrid Systems

Martin Fränzle
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Timed Computation Tree Logic (TCTL)

• Dense-Time
• Fix-point characterization of temporal operators
• Continuous-mode Next operator one-step until
  operator
• Model Checking iterative evaluation of
  one-step-until operator

Rajeev Alur
David Dill
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TCTL One-Step Until
State v
State u
ltv,Rgt
ltu,Sgt
Thomas Henzinger
q
p or q
h
q
ltv,Sgt
time
Amenable to quantifier elimination!
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Semi-Decidability Of TCTL

• One-Step Until decidable
• Until fixpoint not guaranteed to converge
• So TCTL is semi-decidable
• Existential segment and negation of Universal
  segment
• Subscripted operators are decidable in non-zeno
  systems

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3. Undecidability in Shub-Blum-Cucker-Smales
Real Turing Machine
Invariant False Flows Null

• Real operations are atomic
• Mandelbrot Set undecidable
• Reachability is undecidable even in real Turing
  Machines

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4. Approximate Methods

• Bisimulation Partitioning
• Polytopes
• Union of Polytopes
• Time Discretization

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a) Extended Bisimulation Partitioning
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b) Polytopes
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c) Union of Polytopes
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Outline

• Systems Biology
• Understanding Hybrid Automata
• Semi-Algebraic Hybrid Systems
• Algebraic Analysis of Pseudo-Equilibrium in
  Metabolic Networks
• Tolque A dense-time algebraic model-checker for
  semi-algebraic hybrid automata
• Future Work

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Multi-Time-Scale Assumption

• Metabolic networks have fast and slow components
• Every chemical reaction reaches equilibrium given
  sufficient time
• So, simulate using a big time-step appropriate
  for slow reactions
• Fast reactions would have recomputed their
  equilibrium point during this time-step

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3 Types of Metabolites
FAST
SLOW
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Pseudo-Equilibrium Simulation

• Fast reactions reach their equilibrium
• Local (involving only this subset of metabolites)
• Momentary (varies with rest of the system)
• Position of the equilibrium is determined given
  the total concentration of the metabolites

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Algebraic Characterization of the Equilibrium

• At equilibrium, all derivatives are zero by
  definition -- a set of simultaneous polynomial
  equations
• Solution of a system of simultaneous multivariate
  polynomial equations with symbolic parameters
  Gröbner Basis algorithm of Buchberger
• Software CoCoA, Macaulay-2,

Bruno Buchberger
Wolfgang Gröbner (1899-1980)
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FAST
SLOW
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Note on Flux Balance Analysis

• Assumes network would have so evolved as to
  optimize physiologically relevant functions
• Relies on numerical optimization to estimate
  equilibrium fluxes
• Algebraic optimization is also decidable -- so
  FBA-based equilibrium characterization can also
  be done symbolically

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Outline

• Systems Biology
• Understanding Hybrid Automata
• Semi-Algebraic Hybrid Systems
• Algebraic Analysis of Pseudo-Equilibrium in
  Metabolic Networks
• Tolque A dense-time algebraic model-checker for
  semi-algebraic hybrid automata
• Future Work

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V. Tolque Proof-of-Concept Tool

• Implemented in C / C
• Accepts the hybrid system specification, with
  flow equation already approximated by user
• Accepts existential until (EU) TCTL query
• Iteratively computes the p one-step until q
  using Qepcad
• Computational complexity of the cylindrical
  algebraic decomposition algorithm
  double-exponential dependence on the number of
  variables
• Qepcad real numbers not supported

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Example Delta-Notch Signaling
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Pattern formation by lateral inhibition with
feedback a mathematical model of Delta-Notch
intercellular signalingJ. R. Collier, N. A. M.
Monk, P. K. Maini and J. H. Lewis (1996)
Rewriting
Where
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One-Cell Delta-Notch Hybrid Automaton
Claire Tomlin Ronojoy Ghosh
Q1 Both OFF
Q2 Delta ON
Q3 Notch ON
Q4 Both ON
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Two-Cell Delta-Notch System
Q1 Neither
Q2 Delta
Q1 Neither
Q2 Delta
X
Q3 Notch
Q4 Both
Q3 Notch
Q4 Both
Cell 1
Cell 2
16 Discrete States
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Linear Approximation
Q2 Delta
Q3 Notch
X
Q2 Delta
Q3 Notch
X
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State Reachability
Q2 Delta
Q3 Notch
X
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State Reachability
Q3 Notch
Q2 Delta
X
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Impossibility Of Reaching Wrong Equilibrium
Q2 Delta
Q3 Notch
X
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Outline

• Systems Biology
• Understanding Hybrid Automata
• Algebraic Analysis of Hybrid Systems
• Algebraic Analysis of Pseudo-Equilibrium in
  Metabolic Networks
• Tolque A dense-time algebraic model-checker for
  semi-algebraic hybrid automata
• Future Work

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Algorithmic Algebraic Model Checking Summary

• Biochemical systems modeled as ODEs, hybrid
  automata or pseudo-equilibrium networks can be
  analyzed algebraically
• Parameters and initial conditions entirely
  symbolic
• Dense time analysis
• Mathematically-grounded, theoretically-sound
  technique
• Can elicit deeper insights about emergent
  universal properties

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Kinetic mass action-based ordinary differential
equations of the biochemical pathway
Hybrid model with simpler DEs for slow reactions
(using semi-automatic formal methods)
Equilibrium description of fast reactions (using
Gröbner bases)
Hybrid model with simpler DEs for slow reactions
and new DEs for fast reactions
Algebraic characterization of temporal properties
via algorithmic algebraic model checking (using
quantifier elimination)
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Future Work

• Integrate symbolic algebra tools
• Optimize, parallelize quantifier elimination
• Determine trade-offs among approximation
  techniques
• Extend temporal operators ( LTL)
• Chaotic dynamical systems, algebraic topology,
  control theory
• Analyze using powerful computers

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Relevant Publications

• C.Piazza, M.Antoniotti, V.Mysore, A.Policriti,
  F.Winkler and B.Mishra, AAMC-I The Case of
  Biochemical Systems and their Reachability
  Analysis, CAV05
• V.Mysore, C.Piazza and B.Mishra, AAMC-II
  Decidability of Semi-Algebraic Model Checking and
  its Applications to Systems Biology, ATVA05
• V.Mysore and B.Mishra, AAMC-III Approximate
  Methods, INFINITY'05
• V.Mysore and A.Pnueli, Refining the
  Undecidability Frontier Of Hybrid Automata,
  FSTTCS05
• A.Casagrande, V.Mysore, C.Piazza and B.Mishra,
  Independent Dynamics Hybrid Automata in Systems
  Biology, AB05

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Relevant Publications in Progress

• V.Mysore and B.Mishra - AAMC-IV Analysis of
  Metabolic Networks
• V.Mysore, J.Hubbard, A.Pnueli and B.Mishra,
  Modeling and Analyzing Biochemical Dynamics A
  Survey
• V.Mysore, M.Antoniotti and B.Mishra, Tolque - A
  Tool for Algorithmic Algebraic Model Checking 
• V.Mysore and A.Pnueli - Chaos and Undecidability
  In Hybrid Automata

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Other Recent Publications

• V.Mysore, G.Narzisi and B.Mishra, Emergency
  Response Planning for a Potential Sarin Gas
  Attack in Manhattan using Agent-based Models,
  ATDM06
• T.Anantharaman, V.Mysore and B.Mishra, Fast and
  Cheap Genome wide Haplotype Construction via
  Optical Mapping, PSB 2005
• V.Mysore, O.Gill, R.S.Daruwala, M.Antoniotti,
  V.Saraswat and B.Mishra, Multi-Agent Modeling and
  Analysis of the Brazilian Food-Poisoning
  Scenario, Agent05
• B. Mishra, R. Daruwala, Y. Zhou, N. Ugel, A.
  Policriti, M. Antoniotti, S. Paxia, M. Rejali, A.
  Rudra, V. Cherepinsky, N. Silver, W. Casey, C.
  Piazza, M. Simeoni, P. Barbano, M. Spivak, J-W.
  Feng, V. Mysore, O. Gill, F. Cheng, B. Sun, I.
  Ioniata, T.S. Anantharaman, E.J.A. Hubbard, A.
  Pnueli, D. Harel, V. Chandru, R. Hariharan, M.
  Wigler, F. Park, S.-C. Lin, Y. Lazebnik, F.
  Winkler, C. Cantor, A. Carbone, and M. Gromov, A
  Sense of Life Computational Experimental
  Investigations with Models of Biochemical
  Evolutionary Processes, OMICS, 2003  

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Acknowledgements
Dr. Jane Hubbard, Professor of Biology, NYU
Dr. Amir Pnueli, Professor of Computer Science,
NYU
Dr. Bud Mishra, Professor of Computer Science,
Mathematics Cell Biology, NYU
Dr. Carla Piazza, Professor of Computer Science,
University of Udine
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Thank You!