Local Parametric Sensitivity Analysis AMATH 882 Lecture 4, Jan 17, 2013

Parametric Sensitivity Analysis Parametric sensitivity analysis investigates the relationship between the variables and parameters in a biochemical network. Variables 1. Concentrations 2. Pathway fluxes 3. Dynamic response 4. Growth rate Parameters 1. Enzyme activity levels 2. Kinetics constants 3. Decay rates 4. Boundary conditions

Parametric Sensitivity Analysis: Example reaction kinetics: steady state:

local sensitivity analysis: effect of perturbation/ intervention: relative sensitivity:

steady state: sensitivity analysis: vector notation implicit differentiation

complete sensitivity analysis:

Sensitivity Analysis: General Computation model: steady state: differentiate: absolute sensitivity:

Application: unregulated chain

sensitivity of flux J to enzyme activities:

Application: product feedback

sensitivity of flux J to enzyme activities: Summation Theorem of Metabolic Control Analysis: conservation law for sensitivities p p=0

Metabolic Control Analysis (MCA) Sensitivity Analysis in the absence of a quantitative model of the network glutamateSuccinate Succinate SemialdehydeGABA Relative response to a change in enzyme activity = Relative response to a direct change in reaction flux (by linearity) ? Control Coefficients: ????????

Utility of MCA 1) If a quantitative (i.e. kinetic) model is available, equates with (local) parametric sensitivity analysis

Utility of MCA 2) In absence of quantitation, allows qualitative analysis of sensitivities, e.g. comparing different topologies The Effect of Feedback Without feedback With feedback S 1 S 2 X 2 X 1 1 EEE 23 ?

Utility of MCA 3) Regardless of quantitation, allows characterization of constraints on sensitivities (sensitivity invariants) The Summation Theorem:  % Relative increase in flux J k glutamateSuccinate Succinate SemialdehydeGABA

The Summation Theorem Similar results for more complex networks:  % General results described in terms of the kernel of the stoichiometry matrix  %

Time-Varying Sensitivities Sensitivities can be addressed over transient or oscillatory behaviour Computation:

Example Perturbation in S 1 (0) Perturbation in k 1

Application to Phototransduction Pathway

Global Sensitivity Analysis Addresses system behaviour over a wide range of parameter values Primarily statistical tools: efficient sampling methods Provides a broader view of behaviour, but… Results often difficult to interpret

Applications of Sensitivity Analysis Trypanosome metabolism. Bakker et al., 1999,J. Biol. Chem Predicting the effect of interventions Drug development

Applications of Sensitivity Analysis Predicting the effect of interventions Drug development Medicine Tumour growth and thiamine, Comin-Anduix et al., 2001, Eur. J. Biochem.

Applications of Sensitivity Analysis Predicting the effect of interventions Drug development Medicine Metabolic engineering Diacetyl production in Lactococcus lactis, Hoefnagel et al. 2002, Microbiology

Applications of Sensitivity Analysis Predicting the effect of interventions Drug development Medicine Metabolic engineering Model construction and analysis Identifying key variables NF-  B pathway. Ihekwaba et al., 2004, IEE Sys. Biol.

Applications of Sensitivity Analysis Predicting the effect of interventions Drug development Medicine Metabolic engineering Model construction and analysis Identifying key variables Model calibration Identifiability. Zak et al. 2003, Genome. Res.