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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

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

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

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
  • 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

Exploit the network structure to obtain a Bayesian network.

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

Fill up conditional probability tables

S0

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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

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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
  • Sensitivity analysis
  • Parameter estimation
  • Perturbation analysis
  • Parameter-free simulations.

= ??

a case study
A Case Study
  • 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

ODE model

32 species

48 parameters

28 equations

Features:

Large size

Feedback loops

Brown et al. 2004

a case study1
A Case Study
  • Approximate model Construction
    • Settings
      • 5 intervals, 1min time-step, 3 x 106 samples
    • Runtime
      • 4 hours on a cluster of 10 PCs
bn simulation results2
BN-Simulation Results
  • 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
  • Running time
    • ODE based: 22 hours
    • BN based: 0.56 hours
slide26
This is all very well in practice but ...

What about in theory?

Degree of approximations

Sampling technique

Robustness

Approximating the chemical master equation

Probabilistic bounded model checking

slide27
Degree of approximations

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.

slide28
Sample size

What is a good number?

Why is the quality of approximation good?

Dumb luck?

Robustness?

A framework for studying robustness?

slide29
Our Bayesian networks represent finite state Markov chains.

compactly

Formal verification techniques

probabilistic

bounded model checking

Use SAT solvers?

lab members
Lab Members

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

Shazib Pervaiz

Ding Jeak Ling

Hanry Yu

Marie-Veronique Clement

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