Dealing with Metabolic Complexity
Chapter 2 Chapter 2 Analysis of Metabolic Reaction Rates
2.2 F'rornaNet MetabolicReactiontoaNetwork of Metabolic Reactions
2.2.2 Aerobic Growth Without Product Formation: A Metabolic Reaction Network
For the values used here, the sensitivity of the yield
Yxls
to RQ becomes infinite as RQ approaches the lower limit. More detailed analysis of the sensitivity issue can be found elsewhere (Grosz et al., 1984).The expressions derived above for the rates of the reactions given by the general stoichiometric equation (2.1) have be used extensively for bioprocess identification and control (Roels, 1983), but they do not provide any information about reaction rates in the intracellular reaction networks.
2.2.2
Aerobic Growth Without Product Formation: A Metabolic
where x denotes the biomass concentration in grams per unit volume biorea.ctor, p is the specific growth rate of the cells, and Yp,,,,,,,, is the biosynthetic requirement of the corresponding precursor (moles of precursor required per gram cells produced).
These precursor yields have been reported for E. coli (Neidhardt, 1987), B. subtilis (Sauer et al., 1996), and many other organisms of industrial importance, and their experimental estimation is a rather standard technique (Sauer et al., 1996).
Mass balances around the various metabolites can easily be constructed. Consider, for example, the mass balance for G6P:
where the subscripts correspond to the reactions as labeled in Figure 1, and the term p[GGP] corresponds to the effects of the dilution due to cell growth. The quasi- steady state assumption is the basic assumption of the metabolic flux balancing. This assumption is based on the fact that metabolic transients are typically rapid compared to cellular growth rates and changes in the environniental conditions. Based on this assumption, equation (2.22) becomes:
The reaction rates Vn and V , can be experimentally determined as follows:
and
where IS] is the concentration of the extracellular carbon source, which in this case is glucose.
If we normalize all of the reaction rates in the network with respect t o the specific
uptake rate (Vn) we can write for the Equation (2.23):
where the lower-case letters denote reaction rates normalized, with respect to V,, rates. In Equation (2.26) the use of the yield coefficient of the biomass on the substrate results from its definition:
This yield can be estimated experimentally. The value of the term
YxIs[G6P]
in the Inass balance equation is the normalized, with respect to V,, dilution term and since it has a mcu smaller value with respect to the main fluxes in the pathway considered here, it will be omitted from the formulation of the mass balances.For the mass balances of the metabolites from G 6 P to T3P we can write:
Here we can identify the first major problem in metabolic flux balancing. These are seven linear relations between the fluxes, but there are eight uknown fluxes. Therefore, the system is underdetermined.
For the rest of the metabolites considered in the network we can formulate the following mass balance equations:
P G A : P E P : 0 =
P y r : 0 = ACoA: 0 =
O G A : 0 = O A A : 0 = s u c : 0 = M a l : 0 =
The above equations for the metabolic fluxes in the central carbon pathways for aerobically growing B. subtilis introduces a problem common in the metabolic flux balances: the unknown reaction rates are more numerous than are the metabolites being balanced, i.e., the number of the equations that can be formulated is less than the number of the unknown fluxes. Most of the metabolic systems are likewise underdetermined (Bonarius et al., 1996; Sauer et al., 1996; Savinell and Palsson, 1992a-c; Varma and Palsson, 1995). In studies that appeared before now in the literature, the investigators introduce a series of assumptions in order to circumvent this problem. Some of these will be examined here. Many research efforts have been devoted to the experimental determination of one or more of the unknown fluxes, so that these assumptions can be avoided or validated. Experimental techniques such as NMR, tracing of radioactive labels, and mass spectroscopy have been successfully used in order to define exact values or strict bounds for certain fluxes such as the flux from G6P to Ru5P (reaction step 1 in Figure 1) and the fluxes in the tricarboxylic acid cycle (TCA) (reaction steps 13 t o 16 in Figure 1) (Mancuso et al., 1994; Reitzer et al., 1980; Walsh and Koshland, 1984).
The general procedure for estimation of the unknown fluxes will be summarized next.
The problem will be formulated as a nonlinear programming problem which, when solved, will provide an estimate for the fluxes that will satisfy the mass balances with minimal error. This objective can be mathematically formulated by defining a new set of variables, r. The number of these variables will be equal to the number of the mass balances. For a metabolic system with n metabolites and m unknown fluxes, the general matrix expression for the mass balances can be written as:
where N is the n x m st~ichiomet~ric matrix, v is the m-dimensional vector of the unknown fluxes, b is the n-dimensional vector of the total sum of the known fluxes for each rnetabolite mass balance, and r is the n-dimensional vector of the residuals frorn the mass balances. The variables of the problem are the fluxes v and the residuals r.
The objective of the problem can be mathematically formulated as follows:
minimize r:
with respect to the fluxes vl, . . . , v,, subject to the following constraints:
I. Mass Balances
The mass balance equations (2.43) will define a set of equality constraints for the reaction rates and the residuals.
11. Bounds on the rates
Many of the reaction rates in any metabolic work are reversible, i.e. they can proceed in both directions. However, there are reactions that are irreversible.
Therefore, if the mass balance network is constructed in such a way that the irreversible reactions will be positive when they proceed only in the allowable di- rection, then the following inequality constraint for the reaction rate is imposed
for every i-th irreversible reaction step.
The above nonlinear optimization problem can have either a unique solution or multiple solutions depending on the constraints. In general, when the number of reactions is larger than the number of mass balance equations, then multiplicities can occur. This is true provided that none of the inequality constraints are active, meaning none of the irreversible reaction rates are zero. In the general case that the rank of the stoichiometric matrix N , is smaller than its smaller dimension, or Rank(N)<m, the system will be called underdetermined. In the case that m