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Metabolic flux analysis (MFA) and metabolic control analysis (MCA) are mathematical tools that have become widely applicable in metabolic engineering. Both tools are interrelated and widely used in metabolic engineering research [331, 426, 565]. They are useful in developing models of metabolic activity in a biochemical system. They would be instrumental in developing detailed first principles models of fermentation processes.

The flux is a fundamental determinant of cell physiology and a critical parameter of a metabolic pathway [565]. The pathway is the sequence of feasible and observable biochemical reaction steps linking the input and output metabolites. Consider the linear metabolic pathway in Figure 9.1(a), where A is the input metabolite, B is the output metabolite, Ui denotes the reaction rate of the ith reaction step and E, the corresponding enzyme. The flux J of this linear pathway is equal to the rates of the individual reactions at steady state [565]:

For a branched pathway splitting at intermediate I (Figure 9.1(b)) to produce two output metabolites B and C. two additional fluxes are defined for the branching pathways. The flux of each branch (J2 and J3, in Figure

Figure 9.1. Linear metabolic pathway [565].

Figure 9.1. Linear metabolic pathway [565].

9.1(b)) is equal to individual reaction rates at the corresponding branches. At steady state, J\ — J2 + During a transient, the individual reaction rates are not equal and the pathway flux varies with time. Consequently, MFA can not be used to develop transient first principle models. But any dynamic model proposed to describe the transients in the metabolic pathway has to be consistent with the steady state model based on MFA. This provides a reliable reference for a dynamic model when it is reduced to a steady state description.

MFA is used for studying the properties and capabilities of metabolic networks in microorganisms. It allows stoichiometric studies of biochemical reaction networks and may be used for the determination of stationary metabolic flux distributions, if measurements of uptake and/or excretion rates of a cell culture in steady state are known. The result is a flux map that shows the distribution of anabolic and catabolic fluxes over the metabolic network. Based on such a flux map or a comparison of different flux maps, possible targets for genetic modifications might be identified, the result of an already performed genetic manipulation can be judged or conclusions about the cellular energy metabolism can be drawn. The MFA is also used to optimize the product yield by redirecting fluxes using genetic manipulations [282, 426, 565].

Metabolic control analysis (MCA) applies to steady-state or pseudo-steady-state conditions and relies on the assumption that a stable steady state is uniquely defined by the activities of enzymes catalyzing individual reactions in a metabolic pathway. Enzyme activities are considered to be system parameters along with concentrations of substrate for the first reaction and product of the last reaction in the metabolic pathway, while the flux through the pathway or intermediate metabolite concentrations are considered to be system variables [565]. MCA is a sensitivity analysis framework for the quantitative description of metabolism and physiology that allows the analysis and study of the responses of metabolic systems to changes in their parameters [151, 223, 229, 267, 272]. MCA relies on linear perturbations for the nonlinear problem of enzymatic kinetics of metabolic networks. Hence, MCA predictions are local and any extrapolations should be made with caution. Yet, MCA has been useful in providing measures of metabolic flux control by individual reactions, elucidating the concept of rate-controlling step in enzymatic reaction networks, describing the effects of enzymatic activity on intracellular metabolite concentrations, and coupling local enzymatic kinetics with the metabolic behavior of the system [565].

Consider a two-step pathway where the substrate S is converted to the product P via ail intermediate X and enzymes activities E\ and E->

The flux of conversion of S to P at steady state is denoted by J. The steady-state is uniquely defined by the parameters of the system, the levels of enzyme activities E\ and E2, substrate concentration S and product concentration P [565]. Given the values of these parameters, intermediate metabolite concentration cx and pathway flux J can be determined. If any parameter value is altered, a new steady state is reached and cx and J are changed.

One objective of MCA is to relate the variables of a metabolic system to its parameters and then determine the sensitivity of a system variable to system parameters [565]. These sensitivities summarize the extent of systemic flux control exercised by the activity of an enzyme in the pathway. One can also solve for the concentrations of intracellular metabolites and determine their sensitivities to enzyme activities or other system parameters. The sensitivities are represented by control coefficients that indicate how a parameter affects the behavior of the system at steady state. The flux control coefficients (FCC) are the relative change in steady-state flux resulting from an infinitesimal change in the activity of an enzyme of the pathway divided by the relative change of the enzymatic activity [565]:

Because enzymatic activity is an independent system parameter, its change affects the flux both directly and indirectly through changes caused in other system variables, as indicated by the total derivative symbol in Eq. 9.3. FCCs are dimensionless and for linear pathways they have values from 0 to 1. For branched pathways, FCCs can be generalized to describe the effect of each of the L enzyme activities on each of the L fluxes through various reactions [565]:

where Jk is the steady-state flux through the fcth reaction in the pathway and Ei is the activity of the ¿th enzyme. A similar definition is developed based on the rate of the ¿th reaction (i>l) [565]:

The FCCs for branched pathways may have any positive or negative value. The normalization in the definition of FCCs leads to their sum being equal to unity, the flux-control summation theorem:

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