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Approximating Bio-Pathways Dynamics. P.S. Thiagarajan School of Computing, National University of Singapore Joint Work with: Liu Bing, David Hsu. Bio-Pathways. Gene regulatory networks Metabolic pathways Signaling pathways. Signaling pathways.

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Approximating bio pathways dynamics

Approximating Bio-Pathways Dynamics

P.S. Thiagarajan

School of Computing, National University of Singapore

Joint Work with: Liu Bing, David Hsu


Bio pathways
Bio-Pathways

Gene regulatory networks

Metabolic pathways

Signaling pathways


Signaling pathways
Signaling pathways

  • To sense external and internal environments of a cell: through a cascade of reactions.

  • A multitude of signaling pathways govern and coordinate the behavior of cells

  • Many disease processes arise from defects in signaling pathways:


The basic model
The Basic Model

  • Signaling pathway

    • A network of bio-chemical reactions

  • Model: A system (network) of ODEs

    • One for each reaction

  • Study the ODE system to understand the dynamics of the signaling pathway

  • Many variations based on this basic model




We want to know
We want to know………..

What is the concentration level of the protein p at time t (steady state)?

Which initial conditions fit the data best?

Sensitivity of reactions/parameters

Effects of perturbations


Many hurdles
Many Hurdles

  • Rate constant values are not known

    • must be estimated

  • Limited noisy data

  • High dimensional system

    • closed form solutions are impossible

    • Must resort to numerical simulations

    • a large number of simulations needed for answering each question


The approximation idea
The Approximation Idea

  • Generate a “sufficiently” large number of “typical” trajectories.

    • View this ensemble as a representation of the dynamics.

    • This leads to a Markov chain model of the ensemble

    • Represent this Markov chain succinctly as a Bayesian network.


The approximation idea1
The Approximation Idea

  • Convert model analysis questions on ODEs to probabilistic inference problems on Bayesian networks.

    • good trade-off between accuracy and efficiency.

  • Pay one-time cost of constructing the Bayesian network.

  • Amortize this cost by performing multiple analysis tasks using the Bayesian network representation.


The technique

Discretize the value and time domains into intervals; A trajectory is a sequence of interval vectors.

The Technique

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Main idea
Main Idea trajectory is a sequence of interval vectors.

The dynamics is the set of all possible trajectories

State transition graph

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Prob(S11→S10)=0.8

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Main idea1
Main Idea trajectory is a sequence of interval vectors.

State transition graph  Markov chain

Pr(S[t+1]|S[t],S[t-1],...,S[1] )= Pr(S[t+1]|S[t])

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Prob(S11→S10)=0.8

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Main idea2
Main Idea trajectory is a sequence of interval vectors.

A trajectory is a sequence of states

The dynamics is the set of all possible trajectories

State transition graph  Markov chain

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Main idea3
Main Idea trajectory is a sequence of interval vectors.

  • A trajectory is a sequence of states

  • The dynamics is the set of all possible trajectories

    • State transition graph  Markov chain

  • But the Markov chain will be huge!

    • 50 binary variables →250 states


Method
Method trajectory is a sequence of interval vectors.

Exploit the network structure to obtain a Bayesian network.

Build BN structure (2 time-slice dynamic BN) directly.

Fill up conditional probability tables

S0

S3

S2

S1

E3

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E0

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ES3

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P1

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... ...

... ...

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P(S1=0|S0=0,E0=0,ES0=0)=0.2

P(S1=0|S0=1,E0=0,ES0=0)=0.4

... ...


Main idea4

time trajectory is a sequence of interval vectors.

X0

X1

X2

X3

X4

y0

y1

y2

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Main Idea

  • Model analysis  Bayesian inference

    • Given initial conditions, what is the probability distribution of Xi at any time T?

    • Use Inference! For instance, the FF algorithm.


Applications
Applications trajectory is a sequence of interval vectors.

  • Sensitivity analysis

  • Parameter estimation

  • Perturbation analysis

  • Parameter-free simulations.

= ??


A case study
A Case Study trajectory is a sequence of interval vectors.

  • The EGF-NGF signaling pathway is important to understand how distinct signals dictate different cellular outcomes by activating the same signaling cascade

Kholodenko 2007


A larger example1
A Larger Example trajectory is a sequence of interval vectors.

ODE model

32 species

48 parameters

28 equations

Features:

Large size

Feedback loops

Brown et al. 2004


A case study1
A Case Study trajectory is a sequence of interval vectors.

  • Approximate model Construction

    • Settings

      • 5 intervals, 1min time-step, 3 x 106 samples

    • Runtime

      • 4 hours on a cluster of 10 PCs


Bn simulation results
BN-Simulation Results trajectory is a sequence of interval vectors.


Bn simulation results1
BN-Simulation Results trajectory is a sequence of interval vectors.


Bn simulation results2
BN-Simulation Results trajectory is a sequence of interval vectors.

  • Running time

    • Generating a stable nominal profile

      • 386.4 seconds

    • A single execution of FF inference

      • 0.29seconds

  • The total computation time will be sharply reduced when many such “queries” need to be answered by model analysis


Global sensitivity analysis
Global Sensitivity Analysis trajectory is a sequence of interval vectors.

  • Running time

    • ODE based: 22 hours

    • BN based: 0.56 hours


This is all very well in practice but ... trajectory is a sequence of interval vectors.

What about in theory?

Degree of approximations

Sampling technique

Robustness

Approximating the chemical master equation

Probabilistic bounded model checking


Degree of approximations trajectory is a sequence of interval vectors.

The flow is continuous and hence measurable

Defines an idealized finite state Markov chain

Infinite time horizon

Discrete probability distributions but real-valued

As number samples increases and the accuracy of the numerical integration improves, the quality of the approximation increases.


Sample size trajectory is a sequence of interval vectors.

What is a good number?

Why is the quality of approximation good?

Dumb luck?

Robustness?

A framework for studying robustness?


Our Bayesian networks represent finite state Markov chains. trajectory is a sequence of interval vectors.

compactly

Formal verification techniques

probabilistic

bounded model checking

Use SAT solvers?


Lab members
Lab Members trajectory is a sequence of interval vectors.

Faculty:

David Hsu

P.S. Thiagarajan

Student s:

Wang Junjie

Geoffrey Koh

Chin Yen Song

Liu Bing

Sucheendra Kumar Palaniappan

Brandon Ooi Nick Sern

Luo Weiwei


Collaborators
Collaborators trajectory is a sequence of interval vectors.

Shazib Pervaiz

Ding Jeak Ling

Hanry Yu

Marie-Veronique Clement


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