Identification and Estimation of the Influential Parameters in Bioreaction Systems

1. Identification and Estimation of the Influential Parameters in Bioreaction Systems Mordechai Shacham Ben Gurion University of the Negev Beer-Sheva, Israel Neima Brauner Tel-Aviv University Tel Aviv, Israel

2. A Motivating Example Identifiability of Kinetic Model Parameters Closed-system enzymatic hydrolysis of lignocellulosic biomass* 12 Model parameters to be determined by regression of experimental data S, G2, G –substrate, cellobiose and glucose concentrations *Kadam et al., Biotechnol. Prog. 20, 698-705 (2004)

3. Identifiability of Kinetic Model Parameters 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 *Sin et al., Computers and Chemical Engineering, 34, 1385 – 1392 (2010)

4. Identifiability of Kinetic Model Parameters 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 / PViratios) that there are 45 Identifiable Parameter Subsets (IPS) containingonly 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.

5. Demonstration of Some Model Structure Constrains in IPS selection H(ierarchy) = 1 params H = 2 parameters H =3 parameters Parameters that cannot be set at 0 An initial feasible model with 2 parameters

6. 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. Toassure 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

7. 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.

8. 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. 19 parameters 1 Liu and Wang, Comput. Chem. Eng. 33, 1851-1860, 2009.

9. Part of the Data Used for Parameter Identification1,2 x - cell mass concentration s - glucose concentration p1 - ethanol concentration p2 - glycerol concentration 1Wang, et al., I&EC Research, 40, 2876-2885, 2001. 2 Gennemark and Wedelin, Bioinformatics, 25, 780-786, 2009.

10. Example Problem - Initial Model Analysis and Modification 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≤ 0.2 π7 = π2/Yp1/s π8 = π3/Yp2/s The initial minimal parameter set includes Yp1/s, Yp2/s, π1 , π2 , π3.

11. Search for Optimal Parameter Values for a Particular Parameter Set Subject to:

12. Initial Basic Parameter Set and Search for an Additional Parameter to Enter the Set All PVi > CIi Initial basis Selected optimal parm. π12

13. Parameters Identifiable with Statistical Justification CIi > PVi π7 = π2/Yp1/s π8 = π3/Yp2/s

14. Plot of the Experimental Data and Calculated Values Using the Optimal 8 Parameters Model s - concentration of glucose p1 - concentration of ethanol Very good agreement between the calculated and experimental values

15. Plot of the Experimental Data and Calculated Values Using the Optimal 8 Parameters Model x - concentration of cell mass p2 - concentration of glycerol Very good agreement between the calculated and experimental values

16. Conclusions • In 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

17. Plot of the Experimental Data and Calculated Values Using the Initial 5 Parameters Model s - concentration of glucose p1 - concentration of ethanol The initial model already follows well the trend of the data

18. Plot of the Experimental Data and Calculated Values Using the Initial 5 Parameters Model x - concentration of cell mass p2 - concentration of glycerol The initial model already follows well the trend of the data