1. Introduction

   Mathematical modeling is a powerful approach for understanding the complexity of biological systems. I’ve been constructing the simulation model of higher plants on the E-Cell system (Takahashi et al., 2004), which is the software for precise whole cell simulation. Firstly, I constructed a simple model of cystein biosynthesis pathway and examined the state of nutrient deficiencies. More precise investigation was restricted within the limited information of kinetic parameters of enzymes and levels of metabolites.
   Comprehensive data analyzed with recent "omics" technologies (genomics, transcriptomics, proteomics and metabolomics) makes it possible to compensate the lack of information by means of parameter estimation technique. I constructed an artificial biochemical network and examined whether the parameters of enzyme kinetics can be estimated by real-coded genetic algorithm from given metabolic time-courses. For construction of large-scaled model, power-law representation such as S-system and generalized mass action system (GMA) (Voit, 2000; Torres and Voit, 2002) becomes a powerful means. Here I developed the modeling tool that can automatically generate a model description for the E-Cell system.

2. Simple Model of Cystein Biosynthesis Pathway

   A simple mathematical model of the cystein synthesis pathway in higher plants was constructed on the E-Cell system. The prototype model consisted of 18 metabolites and 6 enzymatic reactions based on known enzyme kinetics (Fig. 1). Computer simulation of the model gave time-courses of 7 metabolite levels in the cystein synthesis pathway. The response to nutrient deficiencies was simulated, and the metabolite levels were compared among various extents of sulfate starvation in quasi steady state. The results suggested that cystein synthesis was mainly determined by the change of two metabolite levels, hydrogen sulfide (H₂S) and o-acetylserine (OAS) (Fig. 2).

Figure 1 Cystein biosynthesis pathway.

Figure 2 Relative concentrations of metabolic intermediates in response to sulfate deficiency (50% and 20%). The level of cysteine synthesis was stabilized by the change of two metabolite levels, H₂S and OAS.

3. Parameter Estimation of Enzyme Kinetics Model by Real-Coded Genetic Algorithm

   The realcoded-genetic algorithm, which were improved DIDC algorithm (Kimura et al., 2003), was examined to estimate unknown enzyme kinetics for construction of more detailed model. An artificial biochemical network consisted of 10 metabolites and 4 enzymatic reactions was constructed, reffuring the reaction schemes in the cystein biosyntehesis pathway (Fig. 3). Serine o-acetyl transferase was competitively and non-competitively inhibited by cysteine, cysteine synthase included Hill coefficients in the rate equation, and acetyl-CoA synthase catalyzed three-substrate reaction. The parameter estimation was performed using metabolic time-courses generated in the simulation with given kinetics parameters. Extremely good results were obtained when the GA was applied to estimate the parameters of enzyme kinetics. The average values of relative error between given parameters and estimated parameters were under 2.74% (Table 1).

Figure 3 Artificial biochemical pathway.

Table 1 Estimated parameters of enzyme kinetics in the artificial pathway.

4. Power-Law Model Generator Based on Time-Courses for E-Cell System

   The model construction based on the power-law representation with time-courses is more advantageous than the construction method of traditional Michaelis-Menten type rate equations involving complex schemes of individual enzymatic reaction, because all parameters in the power-law rate equations as object functions can be estimated by informatics optimization technique with the information of network structure of systems from which the data was obtained. Here I developed the modeling tool that can automatically generate a model description of power-law model for the E-Cell system.

4.1. Structure of Model Generator

   The tool is composed of four parts, 1) pre-processing of time-courses obtained by biochemical experiments to eliminate inherent measurement errors and biological noises, 2) automatic generation of objective functions of power-law rate equations from information of network structure, 3) optimization of the parameters of rate equations by real-coded genetic algorithm, 4) conversion to the E-Cell model files using estimated parameters (Fig. 4).

Figure 4 Structure of the model generator.

4.2. Numerical Experiment

   Consider the branched pathway with feedback inhibition in Fig. 5. The model pathway includes four dependent variables, and one of the end products, C, inhibits the production of A. When parameters are given as follows, the system behaves steady oscillation (Voit, 2000).

dA/dt = 5C^-16 - 5A^0.5,       A₀=1.1,
dB/dt = 5A^0.5 - 10B^0.5,      B₀=0.5,
dC/dt = 2B^0.5 - 1.25C^0.5,    C₀=0.9,
dD/dt = 8B^0.5 - 5D^0.5,       D₀=0.75.

   The model was constructed and simulated with E-Cell system and the time-courses of four dependent variables were obtained. As the result of application the time-courses to the Model Generator, the parametes were estimated and following differential equations were obtained. The relative error of each parameter was under 2%.

dA/dt = 5.034C^-16.06 - 5.050A^0.5,
dB/dt = 4.886A^0.510 - 9.910B^0.5,
dC/dt = 1.971B^0.516 - 1.207C^0.5,
dD/dt = 7.870B^0.519 - 4.852D^0.5.

   The simulation results between these two models were compared in Fig. 6. Slight difference was appeared but almost same fluctuations were observed.

Figure 5 Branched pathway with feed back inhibition (Voit, 2000).

Figure 6 Comparison of simulation results between two models constructing defined parameters and estimated parameters.

Acknowledgements

I thank Dr. Yasuhiro Naito, Dr. Masaaki Noji, Prof. Kazuki Saito and Prof. Masaru Tomita for the supports of this research.

References

Kimura, S. and Konagaya, A. (2003) Trans Jpn Soc. Artif. Intell. 18: 193-202.

Takahashi, K., Kaizu, K., Hu, B. and Tomita, M. (2004) Bioinformatics 20: 538-546.

Torres, N. V. and Voit, E. O. (2002) Pathway analysis and optimization in metabolic engineering, Cambridge University Press, Cambridge, UK.

Voit, E. O. (2000) Computational analysis of biochemical systems, Cambridge University Press, Cambridge, UK.

Poster Presentations in Academic Conferences (Apr. 2005 〜 Mar. 2006)

Sato, S., Naito, Y., Noji, M., Saito, K. and Tomita, M. (2005) A mathematical model of sulfur metabolism in higher plants., Metabolomics 2005, The First Annual Meeting of the Metabolomics Society, Poster presentation, Institute for Advanced Bioscienses, Keio university, Tsuruoka, Japan.

Sato, S., Naito, Y., Noji, M., Saito, K. and Tomita, M. (2005) A mathematical model of sulfur metabolism in higher plants using E-Cell system., 6th International Workshop on Plant Sulfur Metabolism, Poster presentation, Kazusa Akademia Center, Kisarazu, Japan.

Sato, S., Naito, Y and Tomita, M. (2005) A modeling tool for E-Cell system; Power-law model generator based on time-course data., Genome Informatics 2005, Poster presentation, Pacifico Yokohama, Yokohama, Japan.