142
Chapter 5
and due to higher costs for maintaining membrane potentials etc. during aerobic growth.
It is interesting to observe that a decreases with
D
in the above table. This may as indicated above be a
result of a changed macromolecular composition. However, this is unlikely as the protein content
normally increases with the specific growth rate and in Section 2.1.4 we noticed that the costs for protein
synthesis accounts for the major ATP costs in synthesizing a cell. The change in
a
with
D
may, however,
also be interpreted differently. If a stays constant at 2.42 moles ATP (C-mole biomass)1
for all 3 dilution
rates P/O would increase from 1.25 to about 1.82 at the highest
D
value. This would imply an increasing
effectiveness of the oxidative phosphorylation with decreasing oxygen uptake. In reality the P/O ratio
may surely vary with the operating conditions, but generally it is believed that the thermodynamic
efficiency decreases with the flux through the respiratory chain.___________________________________
5.3 Simple Metabolic Networks
We will now proceed with the general balance equation (5.1). As discussed earlier there are
J
unknowns in the balance equation, namely the elements of the flux vector v, and there are
K
constraints imposed by the mass balances for the intracellular metabolites. The degrees of
freedom is therefore
F=J-K,
and if exactly
F
rates are measured all the fluxes and the remaining
N+M -F
rates can be calculated. Clearly one can use Gauss elimination to obtain a manual
solution of the set of algebraic equations, but even for relatively simple reaction networks this
becomes cumbersome, and it is much easier to use matrix manipulations. Here we start by
collecting all the measured rates in the vector r,„ and position the remaining non-measured rates
in the vector rc. We now order the equation system such that the equations for the non-measured
rates are given as the upper
N+M -F
equations,
i.e.
(5.24)
Here the matrix T , contains the
N+M -F
rows in TT that correspond to the non-measured rates
and T2 contains the remaining
(N+M+K)-(N+M-F)-K+F=J
rows of TT. Thus, T 2 is a square
matrix, and we can calculate the elements in the flux vector v directly from:
v
(5.25)
Finally the non-measured rates can be calculated from:
r
= T,v
(5.26)
A prerequisite for application of eq. (5.25) is that the matrix T2 is non-singular,
i.e.
that its
determinant is different from zero. There can be three reasons for T 2 being singular:
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