Sensitivity Analysis and Experimental Design - case study of an NF-kB signal pathway

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Sensitivity Analysis and Experimental Design-
case study of an NF-kB signal pathway
Fifth International Conference on Sensitivity
Analysis of Model Output, June 18-22, 2007,
Budapest, Hungary

• Hong Yue
• Manchester Interdisciplinary Biocentre (MIB)
• The University of Manchester
• h.yue_at_manchester.ac.uk

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Motivation
Sensitivity analysis
Correlation analysis
Identifiability analysis
Robust/uncertainty analysis
Model reduction
Parameter estimation
Experimental design
Yue et al., Molecular BioSystems, 2, 2006
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Outline

• Complexity of NF-kB signal pathway
• Local and global sensitivity analysis
• Optimal/robust experimental design
• Conclusions and future work

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NF-kB signal pathway
stiff nonlinear ODE model
Hoffmann et al., Science, 298, 2002
Nelson et al., Sicence, 306, 2004
Sen and Baltimore, Cell, 46, 1986
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Complexity of NF-kB signal pathway

• Nonlinearity linear, bilinear, constant terms
  

• Large number of parameters and variables, stiff
  ODEs
• Different oscillation patterns
• stamped and limit-cycle oscillations
• Stochastic issues, cross-talks, etc.

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Time-dependent sensitivities (local)

• Sensitivity coefficients

• Direct difference method (DDM)

• Scaled (relative) sensitivity coefficients

• Sensitivity index

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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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Sensitivities and LS estimation

• Assumption on measurement noise

additive, uncorrelated and normally distributed
with zero mean and constant variance.

• Least squares criterion for parameter estimation

• Gradient

• Hessian matrix

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Sensitivities and LS estimation

• Correlation matrix

• Fisher information matrix

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Understanding correlations from SA
Similarity in the shape of sensitivity
coefficients K28 and k36 are correlated
Sensitivity coefficients for NF-kBn.
cost functions w.r.t. (k28, k36) and (k9, k28).
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Univariate uncertainty range for oscillations
0.1,12 k36
0.1,1000 k36
Benefit reduce the searching space for
parameter estimation
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Global sensitivity analysis Morris method

• Log-uniformly distributed parameters
• Random orientation matrix in Morris Method

Max D. Morris, Technometrics, 33, 1991
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sensitivity ranking
µ-s plane
GSA
LSA
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Sensitive parameters of NF-kB model
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Improved data fitting via estimation of sensitive
parameters
(b) Jin, Yue et al., ACC2007
(a) Hoffmann et al., Science (2002)
The fitting result of NF-kBn in the IkBa-NF-kB
model
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Optimal experimental design
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

Basic measure of optimality
Fisher Information Matrix
Cramer-Rao theory
lower bound for the variance of unbiased
identifiable parameters
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Optimal experimental design
Commonly used design principles

• A-optimal
• D-optimal
• E-optimal
• Modified E-optimal design

95 confidence interval
The smaller the joint confidence intervals are,
the more information is contained in the
measurements
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Design of IKK activation intensity
95 confidence intervals when - IKK0.01µM (r)
modified E-optimal design IKK0.06µM (b)
E-optimal design
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Robust experimental design
Aim
design the experiment which should valid for a
range of parameter values
Measurement set selection
This gives a (convex) semi-definite programming
problem for which there are many standard solvers
(Flaherty, Jordan, Arkin, 2006)
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Robust experimental design
Contribution of measurement states
Uncertainty degree
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Conclusions

• Different insights from local and global SA
• Importance of SA in systems biology
• Benefits of optimal/robust experimental design

Future works

• SA of limit cycle oscillatory systems
• Global sensitivity analysis and robust design

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Acknowledgement
Prof. Douglas B. Kell principal investigator
(Manchester Interdisciplinary Biocentre, MIB)
Dr. Martin Brown, Mr. Fei He, Prof. Hong Wang
(Control Systems Centre) Dr. Niklas Ludtke
(MIB) Prof. David S. Broomhead (School of
Mathematics) Ms. Yisu Jin (Central South
University, China)
BBSRC project Constrained optimization of
metabolic and signalling pathway models towards
an understanding of the language of cells