Title: DEPARTMENT OF COMPUTER SCIENCE DOCTORAL DISSERTATION DEFENSE, 1:30 PM, March 7, 2006
1DEPARTMENT 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
2Systems Biology Motivation for the Thesis
- Model, Simulate and Analyze Biochemical Systems
to Test Hypotheses, Validate Predictions and
Suggest Experiments
3Numerical 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
4Kinetic 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)
5Outline 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
6I. Systems Biology
- Meaning, Definition, Scope, Future
- Modeling Biochemical Systems
- Hybrid Systems
7The 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
8A 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?
?
9Modeling the Biochemical System
10Formalisms 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
11An 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)
-
12Hybrid 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
13Outline 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
14II. Understanding Hybrid Automata
- Numerical Simulation
- Temporal Logic Model-Checking
- Decidability of Reachability
- Hovering by the Decidability Frontier
151. Simulating a Biochemical Network
16Numerical 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)
172. Ready to Analyze
Temporal Logic
Model Checking
18Temporal 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
19Model 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
203. 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
21Expressivity 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
224. Two-Dimensional Piece-wise Constant Derivative
(PCD) Systems
Oded Maler
Decidable in 2-D, Undecidable in 3-D
Amir Pnueli
232-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
24Refining 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
25(Un)Decidability Frontier
26Summary
- 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
27Outline
- 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
28III. 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
291. 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
30Quantifier 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
312. 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
32Timed 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
33TCTL 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!
34Semi-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
353. 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
364. Approximate Methods
- Bisimulation Partitioning
- Polytopes
- Union of Polytopes
- Time Discretization
37a) Extended Bisimulation Partitioning
38b) Polytopes
39c) Union of Polytopes
40Outline
- 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
41Multi-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
423 Types of Metabolites
FAST
SLOW
43Pseudo-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
44Algebraic 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)
45FAST
SLOW
46Note 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
47Outline
- 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
48V. 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
49Example Delta-Notch Signaling
50Pattern 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
51One-Cell Delta-Notch Hybrid Automaton
Claire Tomlin Ronojoy Ghosh
Q1 Both OFF
Q2 Delta ON
Q3 Notch ON
Q4 Both ON
52Two-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
53Linear Approximation
Q2 Delta
Q3 Notch
X
Q2 Delta
Q3 Notch
X
54State Reachability
Q2 Delta
Q3 Notch
X
55State Reachability
Q3 Notch
Q2 Delta
X
56Impossibility Of Reaching Wrong Equilibrium
Q2 Delta
Q3 Notch
X
57Outline
- 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
58Algorithmic 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
59Kinetic 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)
60Future 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
61Relevant 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
62Relevant 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
63Other 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
64Acknowledgements
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
65Thank You!