1. Princeton University Department of Chemical Engineering and PACM Equation-Free Uncertainty Quantification: An Application to Yeast Glycolytic Oscillations Katherine A. Bold, Yu Zou, Ioannis G. Kevrekidis Department of Chemical Engineering and PACM Princeton University Michael A. Henson Department of Chemical Engineering University of Massachusetts, Amherst WCCM VII, LA July 16-22, 2006

2. Princeton University Department of Chemical Engineering and PACM Outline 1.Background for Uncertainty Quantification 2.Fundamentals of Polynomial Chaos 3.Stochastic Galerkin Method 4.Equation-Free Uncertainty Quantification 5.Application to Yeast Glycolytic Oscillations 6.Remarks

3. Princeton University Department of Chemical Engineering and PACM Background for Uncertainty Quantification Uncertain Phenomena in science and engineering * Inherent Uncertainty: Uncertainty Principle of quantum mechanics, Kinetic theory of gas, … * Uncertainty due to lack of knowledge: randomness of BC, IC and parameters in a mathematical model, measurement errors associated with an inaccurate instrument, … Scopes of application * Estimate and predict propagation of probabilities for model variables: chemical reactants, biological oscillators, stock and bond values, structural random vibration,… * Design and decision making in risk management: optimal selection of parameters in a manufacturing process, assessment of an investment to achieve maximum profit,... * Evaluate and update model predictions via experimental data: validate accuracy of a stochastic model based on experiment, data assimilation, … Modeling Techniques * Sampling methods (non-intrusive): Monte Carlo sampling, Quasi Monte Carlo, Latin Hypercube Sampling, Quadrature/Cubature rules * Non-sampling methods (intrusive) : perturbation methods, higher-order moment analysis, stochastic Galerkin method

4. Princeton University Department of Chemical Engineering and PACM The functional of independent random variables can be used to represent a random variable, a random field or process. Spectral expansion (Ghanem and Spanos, 1991) a j ’s are PC coefficients, Ψ j ’s are orthogonal polynomial functions with =0 if i≠j. The inner product is defined as, is the probability measure of. Notes Selection of Ψ j is dependent on the probability measure or distribution of, e.g., (Xiu and Karniadarkis, 2002) if is a Gaussian measure, then Ψ j are Hermite polynomials; if is a Lesbeque measure, then Ψ j are Legendre polynomials. Fundamentals of Polynomial Chaos

5. Princeton University Department of Chemical Engineering and PACM Preliminary Formulation * Model: e.g., ODE * Represent the input in terms of expansion of independent r.v.’s (KL, SVD, POD): e.g., time-dependent parameter * Represent the response in terms of the truncated PC expansion * The solution process involves solving for the PC coefficients α j (t), j=1,2,…,P Model Input: random IC, BC, parameters Response: Solution Stochastic Galerkin (PC expansion) Method (Ghanem and Spanos, 1991)

6. Princeton University Department of Chemical Engineering and PACM Solution technique: Galerkin projection resulting in coupled ODE’s for α j (t), where Advantages and weakness * PC expansion has exponential convergence rate * Model reduction * Free of moment closure problems ? The coupled ODE’s of PC coefficients may not be obtained explicitly Stochastic Galerkin Method

7. Princeton University Department of Chemical Engineering and PACM Coarse time-stepper (Kevrekidis et al., 2003, 2004) * Lifting (MC, quadrature/cubature): * Microsimulation: * Restriction: For Monte Carlo sampling, For quadrature/cubature-points sampling, is the weight associated with each sampling point. Equation-free Uncertainty Quantification

8. Princeton University Department of Chemical Engineering and PACM Projective Integration (Kevrekidis et al., 2003, 2004) LiftingRestriction Fixed-point Computation (Kevrekidis et al., 2003, 2004)

9. Princeton University Department of Chemical Engineering and PACM Yeast Glycolytic Oscillations (Wolf and Heinrich, Biochem. J. (2000) 345, p321-334) glucose J0J0 glyceraldehyde-3-P/ dihydroxyacetone-P NADHNAD + glycerol v1v1 v2v2 NAD + NADH 1,3-bisphospho-glycerate v3v3 ATP ADP ATP v5v5 pyruvate/acetaldehyde pyruvate/acetaldehyde ex J NADHNAD + v6v6 v4v4 ethanol external environment v7v7 cytosol Notation: A 2 - ADP A 3 - ATP, A 2 +A 3 = A(const) N 1 - NAD + N 2 - NADH, N 1 +N 2 = N(const) S 1 - glucose S 2 - glyceraldehyde-3-P/ dihydroxyacetone-P S 3 - 1,3-bisphospho -glycerate S 4 - pyruvate/acetaldehyde S 4 ex - pyruvate/acetaldehyde ex J 0 - influx of glucose J - outflux of pyruvate/ acetaldehyde Reaction rates: v 1 = k 1 S 1 A 3 [1+(A 3 /K I ) q ] -1 v 2 = k 2 S 2 N 1 v 3 = k 3 S 3 A 2 v 4 = k 4 S 4 N 2 v 5 = k 5 A 3 v 6 = k 6 S 2 N 2 v 7 = kS 4 ex Reaction scheme for a single cell

10. Princeton University Department of Chemical Engineering and PACM Coupled ODEs for multicellular species concentrations Yeast Glycolytic Oscillations

11. Princeton University Department of Chemical Engineering and PACM Heterogeneity of the coupled model: Yeast Glycolytic Oscillations Polynomial Chaos expansion of the solution: Lifting: Fine variables: M – number of cells; M = 1000 Coarse variables: α j and S 4 ex (25 variables totally) (6M+1 variables) Restriction: Minimizingto obtain α j

12. Princeton University Department of Chemical Engineering and PACM Yeast Glycolytic Oscillations Full ensemble simulation

13. Princeton University Department of Chemical Engineering and PACM Yeast Glycolytic Oscillations Projective integration of PC coefficients S1S1 S2S2 S3S3 S4S4 N2N2 A3A3 tt N2N2 A3A3 A phase map of zeroth-order PC coef’s through projective integration Time histories of zeroth-order PC coef’s restricted from the full-ensemble simulation

14. Princeton University Department of Chemical Engineering and PACM Yeast Glycolytic Oscillations Limit-cycle computation Poincaré section fixed point limit cycle ___ limit cycle in the space of PC coefficients xxx restricted PC coefficients of a limit cycle of the full-ensemble simulation A3A3 N2N2 Phase maps of zeroth-order PC coef’s through limit-cycle computation Poincaré section: In the space of coarse variables, zeroth-order PC coef. of N 2 is constant In the space of fine variables, N 2 of a single cell is constant

15. Princeton University Department of Chemical Engineering and PACM Yeast Glycolytic Oscillations Stability of limit cycles Eigenvalues of Jacobians of the flow maps in the coarse and fine variable spaces Flow map T – period of the limit cycle real imaginary x - PC coefficients o - full-ensemble simulation

16. Princeton University Department of Chemical Engineering and PACM Yeast Glycolytic Oscillations Free oscillator Zeroth-order PC coefficient of N 2 (CPI) N 2 of the free cell (CPI)

17. Princeton University Department of Chemical Engineering and PACM Remarks EF UQ is applied to the biological oscillations. The case of only one random parameter is studied. The work can be possibly extended to situations with multiple random parameters or random processes. More advantageous sampling techniques, such as cubature rules and Quasi Monte Carlo, may be used. Reference Bold, K.A., Zou, Y., Kevrekidis, I.G., and Henson, M.A., Efficient simulation of coupled biological oscillators through Equation-Free Uncertainty Quantification, in preparation, available at http://arnold.princeton.edu/~yzou