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Approximating Bio-Pathways Dynamics

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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

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

- 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………..

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

- 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

- 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 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.

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

The Technique3

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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 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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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 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 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

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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

... ...

time trajectory is a sequence of interval vectors.

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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 trajectory is a sequence of interval vectors.

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

= ??

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 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 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

- Settings

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

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

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

- Generating a stable nominal profile
- The total computation time will be sharply reduced when many such “queries” need to be answered by model 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 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 trajectory is a sequence of interval vectors.

Shazib Pervaiz

Ding Jeak Ling

Hanry Yu

Marie-Veronique Clement

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