Sensitivity Analysis and Experimental
Design
- case study of an NF-B signal pathway
Hong YueManchester Interdisciplinary Biocentre (MIB)
The University of [email protected]
Colloquium on Control in Systems Biology, University of Sheffield, 26th March, 2007
NF-B signal pathway
Time-dependant local sensitivity analysis
Global sensitivity analysis
Robust experimental design
Conclusions and future work
Outline
NF-B 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
-B
n (
x15
)
Nelson et al., Sicence, 306, 2004
State-space model of NF-kB
states definition
Characteristics of NF-B signal pathway
Important features:
Oscillations of NF-B in the nucleus
delayed negative feedback regulation by IB Total NF-B concentration
2 3 5 7 9 12 14 15 17 19 21 0x x x x x x x x x x x
14
61 108
ii
x k x
Total IKK concentration
Control factors: Initial condition of NF-B
Initial condition of IKK
Determine how sensitive a system is with respect to the change of parameters
Metabolic control analysis
Identify key parameters that have more impacts on the system variables
Applications: parameter estimation, model discrimination & reduction, uncertainty analysis, experimental design
Classification: global and local
dynamic and static
deterministic and stochastic
time domain and frequency domain
About sensitivity analysis
0 0
00
( , , ), ( )
, ( )j j j j jj j
X f X t X t X
f X fS 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 i
i jj j j i
x x xs
x
Sensitivity index
2
, ,1
1( )
N
i j i jk
RS s kN
Local sensitivity rankings
Sensitivities with oscillatory output
Limit cycle oscillations:
Non-convergent sensitivities
Damped oscillations:
convergent sensitivities
Dynamic sensitivities
Correlation analysis
Identifiability analysis
Robust/fragility analysis
Parameter estimation framework based on sensitivities
Yue et al., Molecular BioSystems, 2, 2006
Model reduction
Parameter estimation
Experimental design
Sensitivities and LS estimation
Assumption on measurement noise:
additive, uncorrelated and normally distributed with zero mean and constant variance.
Gradient
,
( , )( )) (( )i
i i i ik i k ij
ij
j
x kJg kr sr k k
Least squares criterion for parameter estimation
212( ) ( ) ( , )i i i
k i
J x k x k
2
,,
,
( )( , ) (( ) ( ) )i i j i l
k ij
i ji
kli
i l
JH j l s k s
srk
kk
Hessian matrix
Correlation matrix ( )cM correlation S
Understanding correlations
cost functions w.r.t. (k28, k36) and (k9, k28).
Sensitivity coefficients for NF-Bn.
K28 and k36 are correlated
Global sensitivity analysis: Morris method
One-factor-at-a-time (OAT) screening method
Global design: covers the entire space over which the factors may vary
Based on elementary effect (EE). Through a pre-defined sampling strategy, a number (r) of EEs are gained for each factor.
Two sensitivity measures: μ (mean), σ (standard deviation)
Max D. Morris, Dept. of Statistics, Iowa State University
large μ: high overall influence (irrelevant input)
large σ: input is involved with other inputs or whose effect is nonlinear
sensitivity ranking μ-σ plane
Sensitive parameters of NF-B model
k28, k29, k36, k38
k52, k61k9, k62 k19, k42
Global sensitive
Local sensitive
k29: IB mRNA degradation
k36: constituitive IB translation
k28: IBinducible mRNA synthesis
k38: IBn nuclear import
k52: IKKIB-NF-B association
k61: IKK signal onset slow adaptation
k9: IKKIB-NF-B catalytic
k62: IKKIB catalyst
k19: NF-B nuclear import
k42: constitutive IB translation
IKK, NF-B, IB
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-Bn in the IB-NF-B model
Optimal experimental design
Basic measure of optimality:
Aim: maximise the identification information while minimizing the number 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 1TFIM S Q S
Cramer-Rao theory 2 1
iFIM
lower bound for the variance of unbiased identifiable parameters
A-optimal
D-optimal
E-optimal
Modified E-optimal design
Optimal experimental design
max det( )FIM
minmax ( )FIM
1min trace( )FIM
Commonly used design principles:
min cond( )FIM
1
2
1.96 , 1.96i i
i i
95% confidence interval
The smaller the joint confidence intervals are, the more information is contained in the measurements
Measurement set selection
Estimated parameters:
19 29 31 36
38 42 52 61
, , , ,
, , ,
k k k k
k k k k
x12(IKKIB-NF-B), x21(IBn-NF-Bn), x13(IKKIB) , x19(IBn- NF-Bn)
Forward selection with modified E-optimal design
Step input amplitude
95% confidence intervals when :-
IKK=0.01μM (r) modified E-optimal designIKK=0.06μM (b) E-optimal design
Robust experimental design
Aim: design the experiment which should valid for a range of parameter values
1
11
, ,, 1, 0,
, ,
mm
i iim
x xi
01
( , )(nominal)
mT T i
i i i ii
f xFIM
1
1
(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
Robust experimental design
Con
trib
utio
n of
mea
sure
men
t st
ates
Uncertainty degree
max0 (optimal design) (uniform design)
( ) (robust design)middle
Importance of sensitivity analysis
Benefits of optimal/robust experimental design
Conclusions
Future work
Nonlinear dynamic analysis of limit-cycle oscillation
Sensitivity analysis of oscillatory systems
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 of Mathematics)
Ms. Yisu Jin (Central South University, China)
Mr. Jianfang Jia (Chinese Academy of Sciences)
BBSRC project “Constrained optimization of metabolic and signalling pathway models: towards an understanding of the language of cells ”
