Modeling of Metabolic Networks

Of all the data available about metabolic networks, the most relevant information is the flux of metabolites across different pathways. As stated earlier, the study of metabohc networks helps achieve several biotechnological objectives such as enhanced synthesis of proteins, low-molecular-weight compounds, and transforming substrates into products (Bonarius et al., 1997). The flux distribution information concerning the rate of flow of metabolites across different pathways could later be 

To achieve the above goals, quantification of intracellular metabolite flow (or flux) is required. Presently, there are no efficient ways to quantify intracellular metabolite flux except for an isotope tracer technique using nuclear magnetic resonance (NMR) (Bonarius et al., 1997). However, this technique is expensive, laborious, and not suitable for industrial scale due to cell-to-cell heterogeneity. If a cell is dis-mpted in order to measure the concentration of different metabolites within cells to measure the flux, the internal environment such as the enzymatic control on metabolites is irreversibly lost and flux measurement becomes impossible. However, measurable rates such as substrate uptake rate and product secretion rates could be utilized to quantify internal metabolite flux. Besides, there are also certain conditions and constraints imposed while evaluating the flux across different pathways (Nielsen et al., 2003). 

Transforming Eqs. (11.10)-(11.12) into the stoichiometry matrix (T), flux vector (v), and rate of change of metabohte concentration (r), we obtain 

Among the above matrices and vectors, the matrix T and vector r could be determined from metabohc network databases and experimental measurements, respectively, which in turn could be used to evaluate the unknown flux parameters for intracellular metabohtes in vector v. 

3 Identification of Control Structures and Robustness in Metabolic Networks 

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As outlined in the previous section, there is a hierarchy of possible representations of metabolism and no unique definition what constitutes a true model of metabolism exists. Nonetheless, mathematical modeling of metabolism is usually closely associated with changes in compound concentrations that are described in terms of rates of biochemical reactions. In this section, we outline the nomenclature and the essential steps in constructing explicit kinetic models of metabolic networks. 

A detailed kinetic description of enzyme-catalyzed reactions is paramount to kinetic modeling of metabolic networks and one of the most challenging steps... 

R. Steuer, T. Gross, J. Selbig, and B. Blasius, Structural kinetic modeling of metabolic networks. Proc. Natl. Acad. Sci. USA 103(32), 11868 11873 (2006). 

Steuer R, Gross T, Selbig J, Blasius B (2006) Structural kinetic modeling of metabolic networks. Proc Natl Acad Sci U S A 103 11868-11873... 

FIGURE 23.6 Stoichiometric modeling of metabolic networks (Patil et al., 2004). TABLE 23.2 Techniques Available for Data Generation and Analysis (Lee et al., 2005)... 

Nonetheless, the construction of explicit kinetic models allows a detailed and quantitative interrogation of the alleged properties of a metabolic network, making their construction an indispensable tool of Systems Biology. The translation of metabolic networks into ordinary differential equations, including the experimental accessibility of kinetic parameters, is one of the main aspects of this contribution and is described in Section III. 

This section mainly builds upon classic biochemistry to define the essential building blocks of metabolic networks and to describe their interactions in terms of enzyme-kinetic rate equations. Following the rationale described in the previous section, the construction of a model is the organization of the individual rate equations into a coherent whole the dynamic system that describes the time-dependent behavior of each metabolite. We proceed according to the scheme suggested by Wiechert and Takors [97], namely, (i) to define the elementary units of the system (Section III. A) (ii) to characterize the connectivity and interactions between the units, as given by the stoichiometry and regulatory interactions (Sections in.B and II1.C) and (iii) to express each interaction quantitatively by... 

The next step in formulating a kinetic model is to express the stoichiometric and regulatory interactions in quantitative terms. The dynamics of metabolic networks are predominated by the activity of enzymes proteins that have evolved to catalyze specific biochemical transformations. The activity and specificity of all enzymes determine the specific paths in which metabolites are broken down and utilized within a cell or compartment. Note that enzymes do not affect the position of equilibrium between substrates and products, rather they operate by lowering the activation energy that would otherwise prevent the reaction to proceed at a reasonable rate. 

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

To illustrate the actual importance of dynamic properties for the functioning of metabolic networks, we briefly describe and summarize a recent computational study on a model of human erythrocytes [296]. Erythrocytes play a fundamental role in the oxygen supply of cells and have been subject to extensive experimental and theoretical research for decades. In particular, a variety of explicit mathematical models have been developed since the late 1970s [108, 111, 114, 123, 338 341], allowing us to test the reliability of the results in a straightforward way. 

In Section VIII we have described a methodology that allows us to anticipate changes in dynamic properties, providing a suitable starting point to detect changes in dynamic properties of metabolic networks for which the construction of detailed kinetic models is not yet possible. 

Modeling with the aid of data available on the World Wide Web is leading to development of new mathematical descriptions of metabolic networks.3593 An ambitious new project is to model the entire E. coli cell. Many experimental data will be required and it has been estimated that ten years will be needed. The effort involves investigators in many laboratories and will be at least ten times as complex as the determination of the human genome.3620... 

Bottom-up systems biology does not rely that heavily on Omics. It predates top-down systems biology and it developed out of the endeavors associated with the construction of the first mathematical models of metabolism in the 1960s [10, 11], the development of enzyme kinetics [12-15], metabolic control analysis [16, 17], biochemical systems theory [18], nonequilibrium thermodynamics [6, 19, 20], and the pioneering work on emergent aspects of networks by researchers such as Jacob, Monod, and Koshland [21-23]. 

Models of pathways exist in many forms but most of these are static representations, not dynamic models of metabolism. They show the network topology of interconnected pathways of enzymes or signalling molecules, but they contain no dynamic information on reaction rates of diffusive encounters. The JWS-online database (http //jjj.biochem.sun.ac.za/database/index.html) on the other hand, is a web-based database containing over 90 dynamics models. Of these however, only a few are approaching what is desired. 

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