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A statistical Bayesian framework for the identification of biological networks from perturbation experiments

ECCB 2010, Ghent, Belgium Nicole Radde

Institute for Systems Theory and Automatic Control University of Stuttgart

September 26, 2010

Parameter estimation as inverse problem

Optimization problem

Given: model m characterized by a parameter vector θ dataset y Wanted: Find ˆ θ that optimizes an objective function F(θ, y) ˆ θ = arg min

θ F(θ, y)

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Content

1

Statistical approaches for parameter estimation

2

Bayesian regularization

3

Application results

4

Conclusions

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Statistical approaches

Statistical approaches

y : Random variables, sampling distribution p(y|θ) Standard objective function: log-likelihood − log Ly(θ) = − log p(y|θ) can directly include noise can handle latent variables → marginalization: p(y|θ) =

• X

p(y, x|θ) with y :

• bservables

x : latent variables

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Identifiability

Practical non-identifiability: Sparse data (low time resolution, hidden states) Flat likelihood: Fisher information I(θML) has small eigenvalues → Normal approximation N(θML, I −1(θML)) has large covariance becomes better with increasing dataset size Structural non-identifiability: Independent of dataset size Correlation between parameters

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Structural non-identifiability

Example: A

k−1

− − ⇀ ↽ − −

k1

B Parameters θ = (k1, k−1) ˙ [A] = −k1[A] + k−1[B] ˙ [B] = −˙ [A] [A] + [B] = N Measure steady state: y = [¯ A] [¯ B] = k−1 k1 Likelihood function: p(k1, k−1) ∝ exp

• −1

2

• k−1

k1 − 1

2

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Regularization

Idea

Problem: Data does not contain enough information to identify parameter values Large variance of ML/MSE estimates across different experiments Regularization: Add additional data-independent regularization term in

• bjective function

e.g. Tikhonov-regularization: ˆ θTR = arg min

θ

• i

yi − xi(θ) 2

• data-term

+ α θ 2

• regularization term

(1)

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Bayesian regularization

θ, y are both random variables joint distribution: p(y, θ) = p(y|θ)p(θ) = p(θ|y)p(y) Objective function is the posterior distribution: p(θ|y) = p(y|θ)p(θ) p(y) with p(y|θ) Likelihood function p(θ) Prior distribution p(y) Evidence

• 8
• 6
• 4
• 2

2 4 6 8

• 8
• 6
• 4
• 2

2 4 6 8 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 prior posterior θ1 θ2 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

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Bayesian regularization

− log p(θ|y) = − log p(y|θ)

• data-term

− log p(θ)

• regularization term

+ logp(y) independent of θ The posterior distribution does not only provide point estimates, but also contains information about confidence intervals and identifiability Information-theoretic concepts can be used as measures for the information content of the posterior distribution

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Stochastic embedding of ODEs

Measurement noise models

System: ˙ x = f (x, θ1) → x(t, x0, θ1) deterministic Observations: yt

i = xi(t, x0, θ2) + ǫ(θ2), ǫ: noise

stochastic Independence graph: Sampling distribution: p(y|x, θ) =

n

• i=1

T

• t=1

p(yt

i |xi(t, x0, θ1), θ2)

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Sampling schemes: Rejection sampling

Rejection sampling

• 1. Sample θt from prior p(θ)
• 2. Accept θt with p = p(θ|y)

M·p(θ)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 10
• 5

5 10 θ M Acceptance rate: p(x)=posterior(x)/M*prior(x) posterior prior envelope

Uncorrelated samples, but low (1/M) acceptance rate!

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MCMC sampling

Markov Chain Monte Carlo sampling

Can be used if acceptance rate of rejection or importance sampling is low Produces correlated samples, but has higher acceptance rate Computationally expensive if mixing is slow (long time of MC to converge to equilibrium distribution) Sampling scheme:

1 Sample θt+1 from a Markov chain p(θt+1|θt) 2 Accept θt+1 with

p = min

• 1, p(θt+1|y)p(θt|θt+1)

p(θt|y)p(θt+1|θt)

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Hamiltonian Monte Carlo

• 1. Write target density p(θ) ∼ exp (−V (θ))
• 2. Extend sampling space θ by auxiliary momentum vector η:

p(θ, η) ∼ exp

• −1

2ηTη − V (θ)

• = exp (−H(θ, η))
• 3. Start with random momentum drawn from a Gaussian

distribution

• 4. Create trajectory in the space θ according to

˙ θ = η ˙ η = −∇V (θ)

• 5. Accept new θ with PA = min(1, exp (−∆H(θ, η)))

Faster mixing time by producing less correlated samples (larger steps), but harder to tune and implement.

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Posterior summaries

Posterior samples can be used for: Posterior density estimation: Estimation of posterior summaries: Entropy Posterior information content about θ KLD(prior posterior) Information content of data y about θ Mode Maximum a-posteriori point estimator Mean Point estimator Experimental design: Choose experiment that maximizes the expected information content of the posterior distribution

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Secretory pathway control

Regulation of secretion at the TGN via the protein kinase D and the ceramide transfer protein CERT Cooperation with the Institute of Immunology and Cell Biology: Angelika Hausser Monilola Olayioye

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Modeling framework

Model for secretory pathway control

1 2 3 4 5 6 ⋆ ⋆ ˙ x1 = θ1x6 − x1 PKD ˙ x2 = θ2x1 − x2 PI(4)KIIIβ ˙ x3 = θ3x2 − x3 PI(4)P ˙ x4 = θ4x3 − x4 − θ8 x1x4

1+x4

CERT ˙ x5 = θ5x4 − θ6x5 − θ9

x5 1+x5

ceramide ˙ x6 = −θ7x6 + θ9

x5 1+x5

DAG In the following: Estimation of θ = (θ2, θ8)

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Perturbation experiments

1 2 3 4 5 6 ⋆ ⋆ 1 2 3 4 5 6 ⋆ ⋆

Measurements: Relative steady states of two components under different network perturbations: ¯ yP

i

= ¯ xP ¯ xU + ǫ, ǫ ∼ N(0, σ2) Prior: Gamma distributions

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Posterior

0.5 1 1.5 2 2.5 3 3.5 4 4.5 θ2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 θ8 0.5 1 1.5 2 2.5 3 3.5 4 4.5

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Simple ODE Example

1 2 3 θ1 θ2 System: ˙ x1 = a1 − uθ1x1 − θ2x2 − b1x1 ˙ x2 = a2 + uθ1x1 − h(c1, x2)c1 − b2x2 ˙ x3 = a3 − θ2x1 + h(c1, x2) − b3x3 h(c, x) = c x2 1 + x2 Measurements: yi = ¯ xi(θ, u) ¯ xi(θ, ˆ u) + ǫ , ǫ ∼ N(0, (0.01yi)2) i ∈ I = {1, 3} Parameters: a = 1 2(2, 3, 1)T, b = 1 2(1, 4, 2)T, c = 0.7, α = 0.1, θ⋆ = (1, 0.1)T Posterior: p(θ|y) ∝ exp

• −1

2 y⋆−y(θ)2 σ2

− αθ0.5

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Sampling tests: MCMC vs. HMC

prior: MCMC posterior: HMC posterior: Parameters are identifiable HMC performs better in this example with the same efficient sampling size, but is also computationally more expensive We expect that HMC outperforms MCMC in higher dimensions

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Sampling-Summary

Correlation time τint,A: average amount of Markov Chain steps after which we have a new independent point. It varies depending

• n the observable A. It reduces the effective sample size:

NA

eff . =

N 2τint,A . Example: N = 1000 t in s τint,θ1 τint,θ2 efficiency Hybrid Monte Carlo 1479.8 23 10 0.020 Metropolis Monte Carlo 315.83 103 84 0.017

the efficiency is the number of independent points per second. t is the computation time (duration of the whole sampling procedure).

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Conclusions

Summary Statistical Bayesian approaches for parameter estimation of ODE models for biological networks Identifiability problems Investigation of posterior distributions via sampling Future work Parameter estimation for more realistic model of secretory pathway control and biological data Needs further improvement of sampling schemes Efficient estimation of posterior summaries Design of experiments for optimal parameter identification

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Acknowledgements

Andrei Kramer Patrick Weber Thomas Hamm References:

• 1. Kramer A, Hasenauer J, Allg¨
• wer F, Radde N. (2010). Computation
• f the posterior enrtopy in a Bayesian framework for parameter

estimation in biological networks. IEEE Multi-Conference on Systems and Control (MCS 2010), September 8-10, Yokohama, Japan.

• 2. Kramer A, Radde N. (2010). Towards experimental design using a

Bayesian framework for parameter identification in dynamic intracellular network models. Procedia Comput Sci 1(1), 1639–1647.

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Thanks for your interest. Any questions?

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