identification and estimation of the influential parameters in bioreaction systems mordechai shacham...

Identification and Estimation of the Influential Parameters in Bioreaction Systems

Mordechai ShachamBen Gurion University of the Negev

Beer-Sheva, Israel

Neima BraunerTel-Aviv University

Tel Aviv, Israel

Identifiability of Kinetic Model Parameters

*Kadam et al., Biotechnol. Prog. 20, 698-705 (2004)

Closed-system enzymatic hydrolysis of lignocellulosic biomass*

32

312

21

053.1111.1

056.1

rrdt

dG

rrdt

dG

rrdt

dS

233

3

2233

2222

2

2122

1121

2

111

1

1

1

GK

X

K

GK

GEkr

K

X

K

G

K

GSREEk

r

K

X

K

G

K

GSREk

r

IXIG

M

Fr

IXIGIG

SBBr

IXIGIG

SBr

12 Model parameters to be determined by regression of experimental data

S, G2, G –substrate, cellobiose and glucose concentrations

A Motivating Example


*Sin et al., Computers and Chemical Engineering, 34, 1385 – 1392 (2010)

Kadam et. al and Sin et al.* used the same experimental data set in order to identify the 12 parameters of the model.

Observe large differences in parameter values Confidence intervals greater (in absolute value) than the parameters

Kadam et al.

No. Symbol Definition Value Value Conf. Int.

1 k 1r Reaction rate constant 22.3 19.16 ±830.39

2 K 1IG 2 Inhibition constant for cellobiose 0.015 0.07 ±3.3

3 K 1IG Inhibition constant for glucose 0.1 0.03 ±1.214 K 1IX Inhibition constant for xylose 0.1 0.16 ±6.84


6 K 2IG 2 Inhibition constant for cellobiose 132 91.14 ±11265.3

7 K 2IG Inhibition constant for glucose 0.04 0.07 ±1.63

8 K 2IX Inhibition constant for xylose 0.2 0.93 ±23.11


10 K 3M Cellobiose saturation constant 24.3 104.86 ±2335.93

11 K 3IG Inhibition constant for glucose 3.9 0.77 ±3.64

12 K 3IX Inhibition constant for xylose 201 326.62 ±14306

Sin et al.*


Confidence interval (CIi) greater (in absolute value) than the parameter (PVi) suggests that PVi=0 is statistically justified.

CIi ≥ PVi the corresponding parameters are unidentifiable.

For this problem Sin et al (2010) found (based on CIi / PVi ratios) that there are 45 Identifiable Parameter Subsets (IPS) containing only 6 (out of 12) parameters.

Relying solely on statistical considerations for determining the optimal IPS for nonlinear regression problems is insufficient and may result in an unacceptable physical model.

Demonstration of Some Model Structure Constrains in IPS selection

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053.1111.1

056.1

rrdt

dG

rrdt

dG

rrdt

dS

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2122

1121

2

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1

GK

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GK

GEkr

K

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G

K

GSREEk

r

K

X

K

G

K

GSREk

r

IXIG

M

Fr

IXIGIG

SBBr

IXIGIG

SBr

H(ierarchy) = 1 params

H = 2 parameters

H =3 parameters

Parameters that cannot be set at 0

SREEkSREkdtdy SBBrSBr 212111 /

2 1 1/ 1.056 r B SdG dt k E R S

2 1 2/ 1.111 r B B SdG dt k E E R S

An initial feasible model with 2 parameters

Stepwise Regression-in Dynamic Parameter Estimation Problems

Using stepwise regression, a complex model is built sequentially by adding one parameter (with the associated terms of the explanatory variables) to the model at every step, until a stopping criterion is satisfied.

Initially, parameters constrained to nonzero values and parameters (and the associated terms) required to represent the available data at t = 0 are included in the model.

Consequently the parameter whose addition causes the greatest reduction of the objective function value is selected for inclusion in the model at every step.

To assure convergence to a feasible physical model, the search should be constrained to a physically valid parameter values space

Stopping criterion- When for all remaining non-basic CIi ≥ PVi

Initial Model Analysis and Modification

for Nonlinear, Constrained Parameter Estimation Problems

Definition of new parameters to enable addition of one parameter at a time to the model and setting the rest of the parameters at zero

Classification of the parameters’ hierarchy within the model’s various expressions

Keeping parameters constrained to nonzero values in explicit forms

Selection of initial feasible set of parameter values from amongst the 1st hierarchy and nonzero constrained parameters, which is sufficient for representing the initial trend of the state variables and ensures that none of the state variables remains constant during the integration.

Description of the example problem: Kinetic Model of ethanol fermentation1

x - concentration of cell mass

s - concentration of glucose

p1 - concentration of ethanol

p2 - concentration of glycerol

0.1 ≤ Yp1/s ≤ 0.51 - ethanol yield factor

0.1 ≤ Yp2/s ≤ 0.2 - glycerol yield factor.

1 Liu and Wang, Comput. Chem. Eng. 33, 1851-1860, 2009.

xqdt

dp

xqdt

dp

Y

q

Y

q

dt

ds

xdt

dx

p

p

sp

p

sp

p

22

11

/2

2

/1

1

1 2

2 2 21 1 1 1 2 2 2 2

'1 1

1 ' 2 ' ' 2 '1 1 1 1

'2 2

2 '' 2 '' ' 2 '2 1 1 2

/ / /

/ /

/ /

p pm

s SI p p I p p I

p pp

s SI p p I

p pp

s SI p p I

K Ks

K s s K K p p K K p p K

s Kq

K s s K K p p K

s Kq

K s s K K p p K

19 parameters

Part of the Data Used for Parameter Identification1,2

1Wang, et al., I&EC Research, 40, 2876-2885, 2001. 2 Gennemark and Wedelin, Bioinformatics, 25, 780-786, 2009.

Point No.: i t (h) xi(g/L) si(g/L) p1i(g/L) p2i(g/L)

1 0 0.25794 197.938 0 0

2 1 0.35163 196.3455 0 0

3 2 0.50495 195.443 0 0

4 3 0.91191 192.0625 3.0597 0.2169

5 4 1.512 187.3675 4.8155 0.83731

6 5 2.8031 178.5835 7.5538 1.1603

7 6 4.4029 166.403 13.0554 1.6974

8 7 6.2635 144.1565 19.5166 2.4533

9 8 8.6109 124.5265 25.3135 3.0111

10 9 9.6147 102.8158 35.0076 3.8146

11 10 11.2518 85.5552 43.1744 4.5128

12 11 12.1692 68.7319 50.5293 5.1625

13 12 13.1241 51.4266 59.1247 5.8394

14 13 14.2433 36.2531 66.1442 6.2358

15 14 15.0539 24.549 71.8272 6.574

16 15 16.1507 8.4547 78.4983 6.8424

17 16 16.5711 2.1814 81.2294 6.9718

18 17 16.9961 0.52 85.8299 7.142

19 18 17.2255 0 85.9216 7.2436

x - cell mass concentration

s - glucose concentration

p1 - ethanol concentration

p2 - glycerol concentration

Example Problem - Initial Model Analysis and Modification

xqdt

dp

xqdt

dp

xqxqdt

ds

xdt

dx

p

p

pp

322

211

8271

2218217

2166

2

2119115

2145

1

2213212

2111110

294

1

1

1

1

1

1

1

1

1

1

1

1

ppss

sq

ppss

sq

ppppss

s

p

p

Definition of new parameters to enable addition of one parameter at a time to the model and setting the rest of the parameters at zero.

The 19 original parameters were replaced by new parameters π1, π2,… π19.

H(ierarchy) = 1 params

0.1 ≤ Yp1/s ≤ 0.51

0.1 ≤ Yp2/s ≤ 0.2π7 = π2/Yp1/s

π8 = π3/Yp2/s

The initial minimal parameter set includes Yp1/s, Yp2/s, π1 , π2 , π3.

Search for Optimal Parameter Values for a Particular Parameter Set

Subject to:

2

, ,1 1

*minn m

ci i i

i

x x W

maxmin

0)0(

,,

xx

xFx

tdt

d

1,{ }maxi iW x

0.06

0.07

0.08

0.09

0.1

0.11

0 5 10 15 20

Parameter No.

Ob

ject

ive

Fu

nct

ion

Initial Basic Parameter Set and Search for an Additional Parameter to Enter the Set

π12

Initial basis

Selected optimal parm.

No. Parameter

Estimate Optimal PV CI

1 Y p 1/s 0.51 0.50785 0.51 -

2 Y p 2/s 0.2 0.19606 0.2 -

3 π 1 0.002 0.0022795 2.580E-03 1.176E-04

4 π 2 0.01 0.010761 9.036E-03 6.049E-04

5 π 3 0.0008 0.0009852 8.386E-04 5.140E-05

6 π 12 1.125E-01 4.811E-02

0.72787 0.098133 0.067776 -

Initial basis

Objective function

1st stage

All PVi > CIi

Parameters Identifiable with Statistical JustificationNo. Parameter

PV CI PV CI

1 Y p 1/s 0.51 - 0.51 -

2 Y p 2/s 0.2 - 0.2 -

3 π 1 2.447E-02 1.257E-02 2.302E-02 1.214E-02

4 π 2 1.699E-02 3.047E-03 1.514E-01 1.742E-01

5 π 3 7.771E-04 3.346E-05 7.810E-04 3.114E-05

6 π 12 6.907E-01 1.452E-01 6.691E-01 1.381E-01

7 π 5 9.123E-03 3.452E-03 1.139E-01 1.322E-01

8 π 4 3.405E-02 1.842E-02 3.206E-02 1.808E-02

9 π 19 - - 3.107E-04 1.366E-04

0.029413 - 0.02465 -

3rd stage 4th stage

Objective function

xqdt

dp

xqdt

dp

xqxqdt

ds

xdt

dx

p

p

pp

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211

8271

CIi > PVi

sq

s

sq

ps

s

p

p

2

51

2124

1

1

1

1

1

π7 = π2/Yp1/s

π8 = π3/Yp2/s

Plot of the Experimental Data and Calculated Values Using the Optimal 8 Parameters Model

Very good agreement between the calculated and experimental values

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Time (min)

Co

nc

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g/L

)

s

p1

sc

p1c



Plot of the Experimental Data and Calculated Values Using the Optimal 8 Parameters Model

Very good agreement between the calculated and experimental values



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Time (min)

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ConclusionsIn many parameter estimation problems only a subset of the parameters can be identified (i.e., assigned significant value) using the available experimental data.

Considerations related to the model’s structure, parameter hierarchy, constrains on parameter values, initial trend of the data and reduction of the objective function value need to be employed in order to estimate the optimal values of the Identifiable Parameter Subset (IPS).

A new stepwise regression procedure for nonlinear models, based on these principles has been developed for detection of the IPS and estimating the optimal parameter values.

The proposed procedure has proven to be successful in identification of a kinetic model of ethanol fermentation

Plot of the Experimental Data and Calculated Values Using the Initial 5 Parameters Model

The initial model already follows well the trend of the data



0

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200

250

0 5 10 15 20

Time (min)

Co

nc

en

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g/L

)

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p1

sc

p1c

Plot of the Experimental Data and Calculated Values Using the Initial 5 Parameters Model



0

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Time (min)

Co

nc

en

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tio

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p2

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p2c

The initial model already follows well the trend of the data

identification and estimation of the influential parameters in bioreaction systems mordechai shacham...

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