Biochemical Reactions - A First Look
73
3.5 Systematic Analysis of Black Box Stoichiometries
In all the examples of Section 3.4 there was a single carbon and energy source, a single nitrogen
source such as NH3
(or NH4N 03) that was recovered in the biomass and usually a single
metabolic product apart from C02. The yield coefficients could be calculated manually using a
carbon - and a degree of reduction balance. In the general case with
N
substrates and
M
metabolic products, some of which might contain nitrogen, a more systematic procedure to
calculate the stoichiometry is, however, needed.
Let the general stoichiometry be
- C ff„ O* AT, - X
YVJ SJ - rv 0 2
+
Yv x
+ £
Y„p, Pj
+
Y „ C 0 2 + Y ^ H f t
=
0
(3.30)
7=2
j=\
For this stoichiometry the balance has to close for each of the four elements as illustrated in
several examples in Section 3.4 where the carbon balance was set up using the yield coefficients.
The elemental balances can, however, also be set up in terms of the volumetric production rates
q, and generally this can be specified as:
N
M
+y*<
i* + l l y p i<
ipi
= 0
(3-31)
7=1
H
where the coefficient
y
specifies the content of the element in the given compound, e.g. for
glucose
y
is 1
for carbon, 2 for hydrogen and 1 for oxygen. For biomass given by eq. (3.20)
y
is
1
for carbon, 1.8 for hydrogen, 0.5 for oxygen and 0.2 for nitrogen. If an element does not appear
in the compound
y
is zero, e.g. in watery is zero for carbon. Equation (3.31) is one relation
between the
M
+
N+
1 reaction rates, just as equation (3.30) is a chemical equation showing how
chemically specified substrates are converted to different specified products. To generalize the
four elemental balances we define a matrix E with 4 rows and N + M + 1
columns. In each
column the elemental composition of one of the reaction species is written, i.e. the y’s for the
different reaction species are given. Now (3.31) can be written in compact notation
E q = 0
(3.32)
where q is the
N + M + l
column vector of volumetric reaction rates. Equation (3.32) provides
four constraints between the
A/+1 rates in q, and 4 of these can consequently be calculated
from the remaining
N+ M-3
rates. If other elements - such as S - appear in some of the reactant
compositions (cystein would be a case in point) the extension of (3.32) to contain more rows is
obvious, but then there are typically an additional substrate included and the degrees of freedom
is not affected.
Let the
N + M -3
measured rates be placed in the first
N + M
-3 positions qm
of q and the
remaining 4 elements in qc. Now equation (3.32) can be rewritten