princeton university department of chemical engineering and pacm equation-free uncertainty...

Princeton University

Department of Chemical Engineering and PACMDepartment of Chemical Engineering and PACM

Equation-Free Uncertainty Quantification:An Application to Yeast Glycolytic Oscillations

Katherine A. Bold, Yu Zou, Ioannis G. KevrekidisDepartment of Chemical Engineering and PACM


Michael A. HensonDepartment of Chemical EngineeringUniversity of Massachusetts, Amherst

WCCM VII, LAJuly 16-22, 2006



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



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



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) aj’s are PC coefficients, Ψj’s are orthogonal polynomial functions with <Ψi,Ψj>=0 if i≠j. The inner product <·, ·> is defined as , is the probability measure of .

Notes Selection of Ψj is dependent on the probability measure or distribution of , e.g., (Xiu and Karniadarkis, 2002) if is a Gaussian measure, then Ψj are Hermite polynomials; if is a Lesbeque measure, then Ψj are Legendre polynomials.

0

)()(j

jja ξξf

)()()()(),( ξξξξξ dgfgf

))(,),(),((),( 21 nξξf

)(ξ ξ

ξ

)(ξ

Fundamentals of Polynomial Chaos

)(ξ



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 αj(t), j=1,2,…,P

1

( )n

m m mm

K K t

ModelInput: random IC, BC, parameters

Response: Solution

P

j

jj ttx0

)()(),( ξξ

0);();( KxxKx g.

Stochastic Galerkin (PC expansion) Method (Ghanem and Spanos, 1991)



Solution technique: Galerkin projection

resulting in coupled ODE’s for αj(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

it i

P

j

jj

0)(),;)()((0

ξξξ

0))(()(.

tt AGA

TP tttt ))(),...,(),(()( 10 A

Stochastic Galerkin Method



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.

e

P

jijji Nittx ,,2,1,)()(),(

0

00

ξξ

( , ) ( ( , ); ( )) 0, 1,2, ,ξ ξ ξ ei i ix t x t K i N .

g

Pjtxt jjjj ,,1,0,)(),(/)(),,()( ξξξξ

)(),()(),,(1

ij

N

iiij

e

txtx ξξξξ

ei N/1i

Equation-free Uncertainty Quantification



Projective Integration (Kevrekidis et al., 2003, 2004)

Lifting Restriction

Fixed-point Computation (Kevrekidis et al., 2003, 2004)



Yeast Glycolytic Oscillations (Wolf and Heinrich, Biochem. J. (2000) 345, p321-334)

glucose

J0

glucose

glyceraldehyde-3-P/dihydroxyacetone-P

NADH NAD+

glycerol

v1

v2

NAD+

NADH

1,3-bisphospho-glycerate

v3

ATP

ADP

ADP

ATP

v5

pyruvate/acetaldehyde

pyruvate/acetaldehyde ex

J

NADH NAD+

v6

v4ethanol

external environmentv7

cytosol

Notation:A2 - ADPA3 - ATP, A2+A3 = A(const)N1 - NAD+

N2 - NADH, N1+N2 = N(const)S1 - glucoseS2 - glyceraldehyde-3-P/ dihydroxyacetone-PS3 - 1,3-bisphospho -glycerateS4 - pyruvate/acetaldehydeS4

ex - pyruvate/acetaldehyde ex

J0 - influx of glucoseJ - outflux of pyruvate/ acetaldehyde

Reaction rates:v1 = k1S1A3[1+(A3/KI)q]-1

v2 = k2S2N1

v3 = k3S3A2

v4 = k4S4N2

v5 = k5A3

v6 = k6S2N2

v7 = kS4ex

Reaction scheme for a single cell



ex,4

M

1i

ex,4i,47

M

1i

iex,4

i,35i,3i,33

1q

I

i,3i,3i,11i,5i,3i,1

i,3

i,2i,26i,2i,44i,2i,22i,6i,4i,2i,2

ii,2i,44i,3i,33ii,4i,3i,4

i,3i,33i,2i,22i,3i,2i,3

i,2i,26i,2i,22

1q

I

i,3i,3i,11i,6i,2i,1

i,2

1q

I

i,3i,3i,11i,0i,1i,0

i,1

S)SS(M

vJMdt

dS

Ak)AA(Sk2K

A1ASk2vv2v2

dt

dA

NSkNSk)NN(Skvvvdt

dN

JNSk)AA(SkJvvdt

dS

)AA(Sk)NN(Skvvdt

dS

NSk)NN(SkK

A1ASk2vvv2

dt

dS

K

A1ASkJvJ

dt

dS

Coupled ODEs for multicellular species concentrations

Yeast Glycolytic Oscillations



Heterogeneity of the coupled model:


J00 JJ

Polynomial Chaos expansion of the solution:

T322121 ))t,(A),t,(N),t,(S),t,(S),t,(S),t,(S()t,( x

3

0

( , ) ( ) ( )x α j jj

t t

Lifting:

3

0

( , ) ( ) ( ), 1, 2,...,x αi j j ij

t t i M

Fine variables: M – number of cells; M = 1000

Coarse variables: αj and S4ex (25 variables totally)

exiiiiii SANSSSS 4324321 ),(),(),(),(),(),( (6M+1 variables)

Restriction:

2||)()(),(3

0L

j

jj tt

αx||Minimizing to obtain αj




Full ensemble simulation

001.0,3.2J J0




Projective integration of PC coefficients

S1 S2

S3 S4

N2A3

t t N2

A3

A phase map of zeroth-order PC coef’s through projective integration

Time histories of zeroth-order PC coef’srestricted from the full-ensemble simulation




Limit-cycle computation

Poincaré section

fixed point

limit cycle

___ limit cycle in the space of PC coefficientsxxx restricted PC coefficients of a limit cycle of the full-ensemble simulation

A3

N2

Phase maps of zeroth-order PC coef’s through limit-cycle computation

Poincaré section:In the space of coarse variables, zeroth-orderPC coef. of N2 is constantIn the space of fine variables, N2 of a single cell is constant




Stability of limit cycles

Eigenvalues of Jacobians of the flow mapsin the coarse and fine variable spaces

Flow map ),;()( 00 xxx TT T – period of the limit

cycle

real

imag

inar

y

x - PC coefficientso - full-ensemble simulation




Free oscillator 0 2.1, 0.08JJ

Zeroth-order PC coefficient of N2 (CPI)

N2 o

f th

e fr

ee c

ell

(CP

I)



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

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