Sensitivity Analysis and Experimental Design - case study of an NF-  B signal pathway Hong Yue Manchester Interdisciplinary Biocentre (MIB) The University of Manchester Fifth International Conference on Sensitivity Analysis of Model Output, June 18-22, 2007, Budapest, Hungary

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

Complexity of NF-  B signal pathway Local and global sensitivity analysis Optimal/robust experimental design Conclusions and future work Outline

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

Complexity of NF-  B 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.

Time-dependent sensitivities (local) Direct difference method (DDM) Sensitivity coefficients Scaled (relative) sensitivity coefficients Sensitivity index

Local sensitivity rankings

Sensitivities with oscillatory output Limit cycle oscillations : Non-convergent sensitivities Damped oscillations: convergent sensitivities

Sensitivities and LS estimation  Assumption on measurement noise: additive, uncorrelated and normally distributed with zero mean and constant variance.  Gradient  Least squares criterion for parameter estimation  Hessian matrix

Sensitivities and LS estimation  Correlation matrix  Fisher information matrix

Understanding correlations from SA cost functions w.r.t. (k 28, k 36 ) and (k 9, k 28 ). Sensitivity coefficients for NF-  B n. Similarity in the shape of sensitivity coefficients: K 28 and k 36 are correlated

Univariate uncertainty range for oscillations Benefit: reduce the searching space for parameter estimation [0.1,12] k36[0.1,1000] k36

Global sensitivity analysis: Morris method  Log-uniformly distributed parameters  Random orientation matrix in Morris Method Max D. Morris, Technometrics, 33, 1991

sensitivity rankingμ-σ plane LSA GSA

Sensitive parameters of NF-  B model k29: I  B  mRNA degradation k36: constituitive I  B  translation k28: I  B  inducible mRNA synthesis k38: I  B  n nuclear import k52: IKKI  B  -NF-  B association k61: IKK signal onset slow adaptation k28, k29, k36, k38 k52, k61 k9, k62 Local sensitive k9: IKKI  B  -NF-  B catalytic k62: IKKI  B  catalyst k19, k42 Global sensitive k19: NF-  B nuclear import k42: constitutive I  B  translation IKK, NF-  B, I  B 

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-  B n in the I  B  -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: x 0 Which states to observe: C Input/excitation signal: u ( k ) Sampling time/rate Fisher Information Matrix Cramer-Rao theory lower bound for the variance of unbiased identifiable parameters

A-optimal D-optimal E-optimal Modified E-optimal design Optimal experimental design Commonly used design principles: 11 22 95% confidence interval The smaller the joint confidence intervals are, the more information is contained in the measurements

Design of IKK activation: intensity 95% confidence intervals when :- IKK=0.01μM (r) modified E-optimal design IKK=0.06μM (b) E-optimal design

Robust experimental design Aim: design the experiment which should valid for a range of parameter values 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 Contribution of measurement states Uncertainty degree

Different insights from local and global SA Importance of SA in systems biology Benefits of optimal/robust experimental design Conclusions Future works SA of limit cycle oscillatory systems Global sensitivity analysis and robust design

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 ”