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

1. Colloquium on Control in Systems Biology, University of Sheffield, 26th March, 2007 Sensitivity Analysis and Experimental Design- case study of an NF-kB signal pathway Hong Yue Manchester Interdisciplinary Biocentre (MIB) The University of Manchester h.yue@manchester.ac.uk

2. Outline • NF-kB signal pathway • Time-dependant local sensitivity analysis • Global sensitivity analysis • Robust experimental design • Conclusionsand future work

3. NF-kB signal pathway stiff nonlinear ODE model Hoffmann et al., Science, 298, 2002 Nelson et al., Sicence, 306, 2004

4. State-space model of NF-kB • states definition

5. Characteristics of NF-kB signal pathway Important features: • Oscillations ofNF-kBin the nucleus • delayed negative feedback regulation by IkBa • Total NF-kB concentration • Total IKK concentration Control factors: • Initial condition of NF-kB • Initial condition ofIKK

6. About sensitivity analysis • 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 dynamicand static deterministic and stochastic time domain and frequency domain

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

8. Local sensitivity rankings

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

10. Parameter estimation framework based on sensitivities Dynamic sensitivities Correlation analysis Identifiability analysis Robust/fragility analysis Model reduction Parameter estimation Experimental design Yue et al., Molecular BioSystems, 2, 2006

11. 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 • Correlation matrix

12. Understanding correlations K28 and k36 are correlated Sensitivity coefficients for NF-kBn. cost functions w.r.t. (k28, k36) and (k9, k28).

13. 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) large μ: high overall influence (irrelevant input) large σ: input is involved with other inputs or whose effect is nonlinear Max D. Morris, Dept. of Statistics, Iowa State University

14. sensitivity ranking μ-σ plane

15. IKK, NF-kB, IkBa Sensitive parameters of NF-kB model Local sensitive Global sensitive k28, k29, k36, k38 k52, k61 k9, k62 k19, k42 k9: IKKIkBa-NF-kB catalytic k62: IKKIkBa catalyst k19: NF-kB nuclear import k42: constitutive IkBb translation k29: IkBa mRNA degradation k36: constituitiveIkBa translation k28: IkBa inducible mRNA synthesis k38: IkBan nuclear import k52: IKKIkBa-NF-kB association k61: IKK signal onset slow adaptation

16. 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

17. 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

18. q2 q1 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

19. Measurement set selection Forward selection with modified E-optimal design Estimated parameters: x12(IKKIkBb-NF-kB), x21(IkBen-NF-kBn), x13(IKKIkBe) , x19(IkBbn- NF-kBn)

20. Step input amplitude 95% confidence intervals when :- IKK=0.01μM (r) modified E-optimal design IKK=0.06μM (b) E-optimal design

21. Robust experimental design Aim: designthe 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)

22. Robust experimental design Contribution of measurement states Uncertainty degree

23. Conclusions • Importance of sensitivity analysis • Benefits of optimal/robust experimental design Future work • Nonlinear dynamic analysis of limit-cycle oscillation • Sensitivity analysis of oscillatory systems

24. 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 ”