Metabolic pathways modeling

Another two key concepts in metabolic engineering are metabolic pathway analysis and metabolic pathway modeling. The former is used for assessing inherent network properties in the complete biochemical reaction networks. It involves identification of the metabolic network structure (or pathway topology), quantification of the fluxes through the branches of the metabolic network, and identification of the control structures within the metabolic network. 

Metabolic pathway modeling Metabolic flux analysis... 

R. C. Jackson, Toxicology, 102, 197 (1995). Toxicity Prediction from Metabolic Pathway Modelling. 

This is not the place to expose in detail the problems and the solutions already obtained in studying biochemical reaction networks. However, because of the importance of this problem and the great recent interest in understanding metabolic networks, we hope to throw a little light on this area. Figure 10.3-23 shows a model for the metabolic pathways involved in the central carbon metabolism of Escherichia coli through glycolysis and the pentose phosphate pathway [22]. 

A model can be defined as a set of relationships between the variables of interest in the system being investigated. A set of relationships may be in the form of equations the variables depend on the use to which the model is applied. Therefore, mathematical equations based on mass and energy balances, transport phenomena, essential metabolic pathway, and physiology of the culture are employed to describe the reaction processes taking place in a bioreactor. These equations form a model that enables reactor outputs to be related to geometrical aspects and operating conditions of the system. 

Here the effects of any one fractionating step can be expressed in a change in isotopic composition in a wider range of body tissue components, including the product as well as the precursor of a (reversible) reaction. The details depend on the explicit model, for example how rates depend on metabolite concentrations. Therefore, where a metabolic pathway is, or becomes, reversible, the effect on isofractionation on measured body components can be more widespread. 

PBPK models have also been used to explain the rate of excretion of inhaled trichloroethylene and its major metabolites (Bogen 1988 Fisher et al. 1989, 1990, 1991 Ikeda et al. 1972 Ramsey and Anderson 1984 Sato et al. 1977). One model was based on the results of trichloroethylene inhalation studies using volunteers who inhaled 100 ppm trichloroethylene for 4 horns (Sato et al. 1977). The model used first-order kinetics to describe the major metabolic pathways for trichloroethylene in vessel-rich tissues (brain, liver, kidney), low perfused muscle tissue, and poorly perfused fat tissue and assumed that the compartments were at equilibrium. A value of 104 L/hour for whole-body metabolic clearance of trichloroethylene was predicted. Another PBPK model was developed to fit human metabolism data to urinary metabolites measured in chronically exposed workers (Bogen 1988). This model assumed that pulmonary uptake is continuous, so that the alveolar concentration is in equilibrium with that in the blood and all tissue compartments, and was an expansion of a model developed to predict the behavior of styrene (another volatile organic compound) in four tissue groups (Ramsey and Andersen 1984). 

Sato et al. (1991) expanded their earlier PBPK model to account for differences in body weight, body fat content, and sex and applied it to predicting the effect of these factors on trichloroethylene metabolism and excretion. Their model consisted of seven compartments (lung, vessel rich tissue, vessel poor tissue, muscle, fat tissue, gastrointestinal system, and hepatic system) and made various assumptions about the metabolic pathways considered. First-order Michaelis-Menten kinetics were assumed for simplicity, and the first metabolic product was assumed to be chloral hydrate, which was then converted to TCA and trichloroethanol. Further assumptions were that metabolism was limited to the hepatic compartment and that tissue and organ volumes were related to body weight. The metabolic parameters, (the scaling constant for the maximum rate of metabolism) and (the Michaelis constant), were those determined for trichloroethylene in a study by Koizumi (1989) and are presented in Table 2-3. 

Morgan, J.A. and Rhodes, D., Mathematical modeling of plant metabolic pathways, Metabol. Eng. 4, 80, 2002. 

Certainly, the identification of the degradation products responsible for the cytotoxic effects and their metabolic pathways require a thorough elucidation, and a cultured RPE offers a good model for these investigations. 

The Caco-2 cell line was isolated from a human colon carcinoma, and has been characterized as one of the best in vitro models of intestinal epithelium. Indeed, in contrast to other intestinal cell lines, Caco-2 cells are able to constitute a homogenous monolayer and to spontaneously differentiate into polarized cells, highly similar to human mature enterocytes, after approximately 2 weeks of culture. Furthermore, the Caco-2 cells present microvillosities at the apical side and have a high transmembrane resistivity, which confirms the fact that the cells are confluent and link to one another via gap junctions. Finally, they can absorb different compounds, express many enzymes involved in intestinal metabolic pathways (Pinto et al. 1983, Musto et al. 1995, Salvini et al. 2002), and give reproducible in vitro results consistent with results obtained in in vivo studies (Artursson and Karlsson 1991). 

It is in principle possible for a free enzyme to promote reaction in a geochemical system, but enzyme kinetics are invoked in geochemical modeling most commonly to describe the effect of microbial metabolism. Microbes are sometimes described from a geochemical perspective as self-replicating enzymes. This is of course a considerable simplification of reality, as we will discuss in the following chapter (Chapter 18), since even the simplest metabolic pathway involves a series of enzymes. 

Recent work in our laboratories has confirmed the existence of a similar pathway in the oxidation of vindoline in mammals (777). The availability of compounds such as 59 as analytical standards, along with published mass spectral and NMR spectral properties of this compound, served to facilitate identification of metabolites formed in mammalian liver microsome incubations. Two compounds are produced during incubations with mouse liver microsome preparations 17-deacetylvindoline, and the dihydrovindoline ether dimer 59. Both compounds were isolated and completely characterized by spectral comparison to authentic standards. This work emphasizes the prospective value of microbial and enzymatic transformation studies in predicting pathways of metabolism in mammalian systems. This work would also suggest the involvement of cytochrome P-450 enzyme system(s) in the oxidation process. Whether the first steps involve direct introduction of molecular oxygen at position 3 of vindoline or an initial abstraction of electrons, as in Scheme 15, remains unknown. The establishment of a metabolic pathway in mammals, identical to those found in Strep-tomycetes, with copper oxidases and peroxidases again confirms the prospective value of the microbial models of mammalian metabolism concept. 

In the above-mentioned examples, the prediction of CYP-mediated compound interactions is a starting point in any metabolic pathway prediction or enzyme inactivation. This chapter presents an evolution of a standard method [1], widely used in pharmaceutical research in the early-ADMET (absorption, distribution, metabolism, excretion and toxicity) field, which provides information on the biotransformations produced by CYP-mediated substrate interactions. The methodology can be applied automatically to all the cytochromes whose 3 D structure can be modeled or is known, including plants as well as phase II enzymes. It can be used by chemists to detect molecular positions that should be protected to avoid metabolic degradation, or to check the suitability of a new scaffold or prodrug. The fully automated procedure is also a valuable new tool in early-ADMET where metabolite- or mechanism based inhibition (MBI) must be evaluated as early as possible. 

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. 

Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructo-kinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization v3 (ATP) is modeled by a saturable Michaelis Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm ), the Hill coefficient n, and the total concentration, 4 / = [ATP] + [ADP]. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways. 

In this section, we describe a recently proposed approach that aims overcome some of the difficulties [23, 84, 296, 325] Structural Kinetic Modeling (SKM) seeks to provide a bridge between stoichiometric analysis and explicit kinetic models of metabolism and represents an intermediate step on the way from topological analysis to detailed kinetic models of metabolic pathways. Different from approximative kinetics described above, SKM is based on those properties that are a priori independent of the functional form of the rate equation. 

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

H. G. Holzhtitter, G. Jacobasch, and A. Bisdorff, Mathematical modeling of metabolic pathways affected by an enzyme deficiency. A mathematical model of glycolysis in normal and pyruvate kinase deficient red blood cells. Eur. J. Biochem. 149(1), 101 111 (1985). 

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