Colloquium Presentation MIB

8/13/2019 Colloquium Presentation MIB

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Sensitivity Analysis and Experimental Design

- case study of an NF-kB signal pathway

Hong Yue

Manchester Interdisciplinary Biocentre (MIB)

The University of Manchester

[email protected]

Colloquium on Control in Systems Biology, University of Sheffield, 26thMarch, 2007


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NF-kB signal pathway

Time-dependant local sensitivity analysis

Global sensitivity analysis

Robust experimental design

Conclusionsand future work

Outline


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NF-kB signal pathway

Hoffmann et al., Science, 298, 2002

0 0

1 2

1 2

( , , ), ( )

(state vector)

(parameter vector)

T

n

T

m

X f X t X t X

X x x x

k k k

stiff nonlinear ODE model

0 0.5 1 1.5 2 2.5

x 104

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time/s

NF-kBn

(x15)

Nelson et al., Sicence, 306, 2004


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State-space model of NF-kB

states definition


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Characteristics of NF-kB signal pathway

Important features:

Oscillations ofNF-kBin the nucleus

delayed negative feedback regulation by IkB

Total NF-kB concentration

2 3 5 7 9 12 14 15 17 19 21 0x x x x x x x x x x x

14

61 10

8

i

i

x k x

Total IKK concentration

Control factors:

Initial condition of NF-kB

Initial condition ofIKK


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Determine how sensitive a system is with respect to thechange of parameters

Metabolic control analysis

Identify key parameters that have more impacts on thesystem variables

Applications: parameter estimation, model discrimination &reduction, uncertainty analysis, experimental design

Classification: globaland local

dynamicand static

deterministicand stochastic

time domainand frequency domain

About sensitivity analysis


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0 0

0

0

( , , ), ( )

, ( )j j j j jj j

X f X t X t Xf X f

S J S F S t S X

Time-dependent sensitivities (local)

Direct difference method (DDM)

0

, , 0/ , ( ) ( ) i j i j i j j is x s t x Sensitivity coefficients

Scaled (relative) sensitivity coefficients

, //

ji i ii j

j j j i

x x xsx

Sensitivity index

2

, ,

1

1( )

N

i j i j

k

RS s k

N


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Local sensitivity rankings


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Sensitivities with oscillatory output

Limit cycle oscillations:

Non-convergent sensitivities

Damped oscillations:

convergent sensitivities


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Dynamic sensitivities

Correlationanalysis Identifiabilityanalysis Robust/fragilityanalysis

Parameter estimation framework

based on sensitivities

Yue et al., Molecular BioSystems, 2, 2006

Modelreduction

Parameterestimation

Experimentaldesign


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Sensitivities and LS estimation

Assumption on measurement noise:additive, uncorrelated and normally distributed with zero

mean and constant variance.

Gradient

,

( , )( )) (( )ii i i i

k i k ij

i

j

j

x kJg kr sr k k

Least squares criterion for parameter estimation

2

12( ) ( ) ( , )i i i

k i

J x k x k

2

,

,

,

( )( , ) (( ) ( ) )i i j i l

k ij

i j

i

kl

i

i l

JH j l s k s

srk

kk

Hessian matrix

Correlation matrix ( )cM correlation S


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Understanding correlations

cost functions w.r.t. (k28, k36) and (k9, k28).

Sensitivity coefficients for NF-kBn.

K28and k36are correlated


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Global sensitivity analysis: Morris method

One-factor-at-a-time (OAT)screening method

Global design:covers the entire space over which thefactors may vary

Based on elementary effect (EE). Through a pre-defined

sampling strategy, a number (r) of EEs are gained for eachfactor.

Two sensitivity measures: (mean), (standard deviation)

Max D. Morris, Dept. of Statistics, Iowa State University

large : high overall influence(irrelevant input)

large : input is involved withother inputs or whose effect isnonlinear


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sensitivity ranking -plane


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Sensitive parameters of NF-kB model

k28, k29, k36, k38

k52, k61

k9, k62 k19, k42

Global sensitiveLocal sensitive

k29: IkBamRNA degradation

k36:constituitive

IkBa

translation

k28: IkBa induciblemRNA synthesis

k38: IkBannuclear import

k52: IKKIkBa-NF-kB association

k61: IKK signal onset slow adaptation

k9: IKKIkBa-NF-kB

catalytic

k62: IKKIkBacatalyst

k19: NF-kB nuclear

import

k42: constitutive IkBb

translation

IKK, NF-kB, IkB


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Improved data fitting via estimation

of sensitive parameters

(a) Hoffmann et al., Science (2002) (b) Jin, Yue et al., ACC2007

The fitting result of NF-kBnin the IkBa-NF-kB model


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Optimal experimental design

Basic measure of optimality:

Aim: maximise the identification information while minimizing thenumber of experiments

What to design?

Initial state values: x0

Which states to observe:C

Input/excitation signal: u(k)

Sampling time/rate

Fisher Information Matrix

1T

FIM S Q S

Cramer-Rao theory 2 1

i

FIM

lower bound for the variance of unbiased

identifiable parameters


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A-optimal

D-optimal

E-optimal

Modified E-optimal design

Optimal experimental design

maxdet( )FIM

minmax ( )FIM

1min trace( )FIM

Commonly used design principles:

min cond( )FIM

1

21.96 , 1.96

i ii i

95% confidence interval

The smaller the joint confidence intervals are, the

more information is contained in the measurements


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Measurement set selection

Estimated parameters:

19 29 31 36

38 42 52 61

, , , ,

, , ,

k k k k

k k k k

x12(IKKIkBb-NF-kB),x21(IkBen-NF-kBn),x13(IKKIkBe) ,x19(IkBbn- NF-kBn)

Forward selection with modified E-optimal design


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Step input amplitude

95% confidence intervals when :-IKK=0.01M (r) modified E-optimal

design

IKK=0.06M (b) E-optimal design


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Aim: designthe experiment which should valid for a range ofparameter values

1

11

, ,

, 1, 0,, ,

mm

i i

im

x x

i

01

( , )(nominal)

mT T i

i i i i

i

f xFIM

11

(with uncertainty)

( , , )

mT

i i i ii

m

FIM

blkdiag

This gives a (convex) semi-definite programming problem for

which there are many standard solvers (Flaherty, Jordan,

Arkin, 2006)

Measurement set selection


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Co

ntributionofmeasurementstates

Uncertainty degree

max0 (optimal design) (uniform design)

( ) (robust design)middle


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Importance of sensitivity analysisBenefits of optimal/robust experimental design

Conclusions

Future work

Nonlinear dynamic analysis of limit-cycleoscillation

Sensitivity analysis of oscillatory systems


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Acknowledgement

Dr. Martin Brown, Mr. Fei He, Prof. Hong Wang (Control Systems

Centre)

Dr. Niklas Ludtke, Dr. Joshua Knowles, Dr. Steve Wilkinson, Prof.

Douglas B. Kell (Manchester Interdisciplinary Biocentre, MIB)

Prof. David S. Broomhead, Dr. Yunjiao Wang (School ofMathematics)

Ms. Yisu Jin (Central South University, China)

Mr. Jianfang Jia (Chinese Academy of Sciences)

BBSRC project Constrained optimization ofmetabolic and signalling pathway models:

towards an understanding of the language

of cells

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