colloquium presentation mib
TRANSCRIPT
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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
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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Robust experimental design
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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Robust experimental design
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