Identification and Estimation of the Influential Parameters in Bioreaction Systems

Mordechai Shacham, Chem. Eng. Dept., Ben-Gurion University, Beer-Sheva, Israel
Neima Brauner, School of Engineering, Tel-Aviv University, Tel-Aviv, Israel
In recent years there has been a rising interest in mathematical modeling in systems biology in particular in dynamic modeling of bioreaction systems. The goal of the mathematical modeling is to obtain expressions that describe the dynamic behavior of the systems under consideration. The resultant models can be applied for analysis, design optimization and control of the biological systems. The mathematical models are often described by systems of ordinary differential equations (ODE's). The models usually contain unknown parameters that need to be determined using a set of measurements (experimental data). Typically, the estimation of the parameter values is performed using a maximum likelihood approach, where the objective is to minimize the weighted squared error between the set of the measured data and the corresponding model predictions. Sun et al. (2012) provide a recent review of the methods used for parameter estimation. They classify the optimization algorithms used as deterministic global, metaheuristic global (such as evolutionary algorithms, EA) and deterministic local (gradient-based) algorithms. They point out that deterministic global algorithms are very time consuming and usually cannot obtain satisfactory solutions in a reasonable time frame when applied to practical problems of realistic size. Consequently, the use of metaheuristics for identifying near optimal parameter values is recommend, followed by gradient based methods in order to refine these values.
Kravaris et al. (2013) point out that parameter estimation for practical, complex systems is an ill-conditioned problem. The main reasons mentioned for the ill-conditioning are excessive number of unknown parameters in the fundamental models, uncertainty regarding the suitability of the mathematical model to represent the data and insufficient amount and/or low precision, noisy experimental data. In order to overcome these limitations Kravaris et al. (2013) recommend that the parameter estimation problem of complex systems should not deal solely with the determination of the optimal solution, but concentrate on computing a solution which is robust to variations in the experimental data.
We have developed a new algorithm whose objective is the identification of the maximal number of influential parameters and their optimal values so that the resultant dynamic model is robust to the variations of the experimental data. The new algorithm involves the use of a stepwise regression procedure for identifying the influential parameters. The procedure starts with a minimal set of parameters that can yield a feasible solution (thus satisfying the state variable and their derivative values at time = 0) and continues with adding more parameters until the increased number of parameters results in an unstable set of parameters. The optimal parameter values obtained in the earlier step are used as initial estimates for next step, to reduce the parameter search space and speed-up the computation. The stability of the parameters is determined using two sets of data, one is the "true" set of the experimental data and the other one is obtained by smoothing the experimental data using cubic smoothing splines. The (relative) difference between corresponding parameter values obtained with the two sets of data is used as a measure for the reliability of the parameter set and the model stability. The stability is further verified by the confidence interval- -to- parameter value ratios. The optimization and determination of approximate parameter values is carried out first using the fast collocation method suggested by Liu and and Wang (2009). These approximate values
are used as first estimates for rigorous computations, using the more time consuming, variable stepsize Runge-Kutta or Gear's methods of integration.
The proposed method is demonstrated by identifying the maximal set of influential parameters and determination of their optimal values for the model of the "ethanol fermentation" process presented by Liu and Wang (2009) using the data provided by Gennemark and Wedelin (2009). It is shown that the proposed algorithm converges rapidly and the list of the influential parameters (and their optimal values) matches the one found by Liu and Wang (2009) who used ad-hoc considerations to discard parameters from the model.

References

1. Gennemark, P. and Wedelin, D., Benchmarks for identification of ordinary differential equations from time series data, Bioinformatics, 25, 780-786, 2009.

2. Kravaris, C., Hahn, J. and Chu, Y., 2013, Advances and selected recent developments in state and parameter estimation, Computers and Chemical Engineering, 51, 111-123, 2013.

3. Liu, P.-K. and Wang, F.-S., Hybrid Differential Evolution with Geometric Mean Mutation in Parameter Estimation of Bioreaction Systems with Large Parameter Search Space, Computers and Chemical Engineering, 33, 1851-1860, 2009.

4. Sun, J., Jonathan M. Garibaldi, J. M.and Hodgman, C., Parameter Estimation Using Metaheuristics in

Systems Biology: A Comprehensive Review, IEEE/ACM Transactions on Computational Biology and

Bioinformatics, 9, 185-202, 2012.