Princeton University
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
Department of Chemical Engineering and PACM
Princeton University
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
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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
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Fundamentals of Polynomial Chaos
The functional of independent random variables
f (ξ ),
ξ  ( 1( ), 2( ),, n( ))
can be used to represent a random variable, a random field or process.
Spectral expansion (Ghanem and Spanos, 1991)

f (ξ )   ajj (ξ )
j 0
aj’s are PC coefficients, Ψj’s are orthogonal polynomial functions with <Ψi,Ψj>=0
if i≠j. The inner product <·, ·> is defined as  f (ξ ), g (ξ )   f (ξ ) g (ξ )d (ξ ) ,

 (ξ ) is the probability measure of ξ .
Notes
Selection of Ψj is dependent on the probability measure or distribution of ξ ,
e.g., (Xiu and Karniadarkis, 2002)
if  (ξ ) is a Gaussian measure, then Ψj are Hermite polynomials;
if  (ξ ) is a Lesbeque measure, then Ψj are Legendre polynomials.
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Stochastic Galerkin (PC expansion) Method (Ghanem and Spanos, 1991)
Preliminary Formulation
Input: random IC,
BC, parameters
* Model: e.g., ODE
Response: Solution
Model
.
 ( x; K )  x  g ( x; K )  0
* Represent the input in terms of expansion of independent r.v.’s (KL, SVD, POD):
e.g., time-dependent parameter
n
K  K   m m (t )m
m 1
* Represent the response in terms of the truncated PC expansion
P
x(ξ , t )  j (t )j (ξ )
j 0
* The solution process involves solving for the PC coefficients αj(t), j=1,2,…,P
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Stochastic Galerkin Method
Solution technique: Galerkin projection
P
  (j (t )j (ξ ); ξ ), i (ξ )  0
i
j 0
resulting in coupled ODE’s for αj(t),
.
where
A (t )  G ( A (t ))  0
A (t )  ( 0(t ),1(t ),...,P(t ))T
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
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Equation-free Uncertainty Quantification
Coarse time-stepper (Kevrekidis et al., 2003, 2004)
* Lifting (MC, quadrature/cubature):
P
x(ξ i , t 0)  j (t 0)j (ξ i ) , i  1,2,, Ne
j 0
* Microsimulation:
.
* Restriction:
x( ξi , t )  g ( x(ξi , t ); K (ξi ))  0, i  1, 2,
j (t )  x(ξ , t ), j (ξ )  /  j (ξ ), j (ξ )  ,
, Ne
j  0,1,, P
Ne
 x(ξ , t ), j (ξ )   ix(ξ i , t )j (ξ i )
i 1
For Monte Carlo sampling, i  1 / Ne
For quadrature/cubature-points sampling,
with each sampling point.
i
is the weight associated
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Projective Integration (Kevrekidis et al., 2003, 2004)
Fixed-point Computation (Kevrekidis et al., 2003, 2004)

Lifting

Restriction
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Yeast Glycolytic Oscillations (Wolf and Heinrich, Biochem. J. (2000) 345, p321-334)
Notation:
A2 - ADP
A3 - ATP, A2+A3 = A(const)
N1 - NAD+
N2 - NADH, N1+N2 = N(const)
S1 - glucose
S2 - glyceraldehyde-3-P/
dihydroxyacetone-P
S3 - 1,3-bisphospho
-glycerate
S4 - pyruvate/acetaldehyde
S4ex - pyruvate/acetaldehyde ex
J0 - influx of glucose
J - outflux of pyruvate/
acetaldehyde
Reaction scheme for a single cell
glucose
J0
glucose
cytosol
ATP
v1
ADP
glyceraldehyde-3-P/
dihydroxyacetone-P
v2
NADH
NAD+
glycerol
v6
NAD+
NADH
1,3-bisphospho-glycerate
v3
ADP
ATP
v5
NADH
pyruvate/acetaldehyde
J
pyruvate/acetaldehyde ex
NAD+
v4
v7
ethanol
external environment
Reaction rates:
v1 = k1S1A3[1+(A3/KI)q]-1
v2 = k2S2N1
v3 = k3S3A2
v4 = k4S4N2
v5 = k5A3
v6 = k6S2N2
v7 = kS4ex
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Yeast Glycolytic Oscillations
Coupled ODEs for multicellular species concentrations
  A3 , i  q 
dS1, i
 J 0 , i  v1, i  J 0 , i  k 1S 1, iA3 , i 1  
 
dt
  KI  
1
1
  A3 , i  q 
dS 2 , i
 2v1, i  v 2 , i  v6 , i  2k 1S 1, iA3 , i 1  
   k 2 S 2 , i( N  N 2 , i )  k 6 S 2 , iN 2 , i
dt
K
I
 
 
dS 3 , i
 v 2 , i  v 3 , i  k 2 S 2 , i( N  N 2 , i )  k 3 S 3 , i( A  A3 , i )
dt
dS 4 , i
 v 3 , i  v 4 , i  Ji  k 3 S 3 , i( A  A3 , i )  k 4 S 4 , iN 2 , i  Ji
dt
dN 2 , i
 v 2 , i  v 4 , i  v6 , i  k 2 S 2 , i( N  N 2 , i )  k 4 S 4 , iN 2 , i  k 6 S 2 , iN 2 , i
dt
1
  A3 , i  q 
dA3 , i
 2v1, i  2v 3 , i  v 5 , i  2k 1S 1, iA3 , i 1  
   2k 3 S 3 , i( A  A3 , i )  k 5 A3 , i
dt
K
I
 
 
dS 4 , ex  M
 M

 ( S 4 , i  S 4 , ex )  S 4 , ex
 Ji  v7  M 
dt
M i 1
i 1
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Yeast Glycolytic Oscillations
Heterogeneity of the coupled model:
J 0  J 0  J
Polynomial Chaos expansion of the solution:
x(  ,t )  ( S 1(  ,t ),S 2(  ,t ),S 1(  ,t ),S 2(  ,t ),N 2(  ,t ), A3(  ,t ))T
3
x( , t )   α j (t ) j ( )
j 0
Coarse variables: αj and S4ex (25 variables totally)
Lifting:
3
x(i , t )   α j (t ) j (i ),
i  1, 2,..., M
j 0
Fine variables: S1(i), S 2(i), S 3(i), S 4(i), N 2(i), A3(i), S4ex (6M+1 variables)
M – number of cells; M = 1000
Restriction:
3
Minimizing || x ( , t )   αj (t )j ( ) || L2
j 0
to obtain αj
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Yeast Glycolytic Oscillations
Full ensemble simulation
J 0  2.3,
J  0.001
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Yeast Glycolytic Oscillations
Projective integration of PC coefficients
S1
S2
S3
S4
N2
A3
t
A3
t
Time histories of zeroth-order PC coef’s
restricted from the full-ensemble simulation
N2
A phase map of zeroth-order PC coef’s
through projective integration
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Yeast Glycolytic Oscillations
Limit-cycle computation
Poincaré section
A3
fixed point
limit cycle
Poincaré section:
In the space of coarse variables, zeroth-order
PC coef. of N2 is constant
In the space of fine variables, N2 of a single
cell is constant
___ limit cycle in the space of
PC coefficients
xxx restricted PC coefficients of a
limit cycle of the full-ensemble
simulation
N2
Phase maps of zeroth-order PC coef’s
through limit-cycle computation
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Yeast Glycolytic Oscillations
Flow map
T ( x 0)  x (T ; x 0),
T – period of the limit cycle
imaginary
Stability of limit cycles
x - PC coefficients
o - full-ensemble simulation
real
Eigenvalues of Jacobians of the flow maps
in the coarse and fine variable spaces
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Yeast Glycolytic Oscillations
J 0  2.1,
 J  0.08
N2 of the free cell (CPI)
Free oscillator
Zeroth-order PC coefficient of N2 (CPI)
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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
Department of Chemical Engineering and PACM