Sensitivity Analysis and Experimental Design - case study of an NF-  B signal pathway Hong Yue Manchester Interdisciplinary Biocentre (MIB) The University of Manchester Colloquium on Control in Systems Biology, University of Sheffield, 26 th 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 stiff nonlinear ODE model 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 I  B  Total NF -  B concentration 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

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

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  Least squares criterion for parameter estimation  Hessian matrix  Correlation matrix

Understanding correlations cost functions w.r.t. (k 28, k 36 ) and (k 9, k 28 ). Sensitivity coefficients for NF-  B n. K 28 and k 36 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 rankingμ-σ plane

Sensitive parameters of NF-  B model k28, k29, k36, k38 k52, k61 k9, k62k19, k42 Global sensitiveLocal sensitive 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 k9: IKKI  B  -NF-  B catalytic k62: IKKI  B  catalyst 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

Measurement set selection Estimated parameters: x 12 (IKKI  B  -NF-  B), x 21 (I  B  n -NF-  B n ), x 13 (IKKI  B  ), x 19 (I  B  n - NF-  B n ) Forward selection with modified E-optimal design

Step input amplitude 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

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 ”