150
Chapter 5
< v
r—
0.5
0
0
0
0
0 '
-]
M
- 2
r s
V 2
0
1
0
0
0
0
rl
rl
V 3
0
0
0
1
0
0
rf
~ 2rg - n - rf
V 4
1
-1
-1
-1
0
0
0
rf
V 5
0
0
1
1
-1
-1
0
0.5rf
u ,
v
1
-1
1
0
0
- 2 j
, 0 .
V")
©
1
C
1
0 0
<N
1
(3)
It is observed that the formation of 2 mol of formate is accompanied by the formation of 1 mol of acetate
(according to Eq. (2) the rate of acetate formation is equal to v5). This fixed ratio between the rates of
formation of two extracellular metabolites is explained by the NADH balance: When acetyl-CoA is
formed together with formate, it is necessary to regenerate exactly 1 mol NAD+
per mole of acetyl-CoA
formed (namely, the NAD+
used in the EMP pathway), and the flux from acetyl-CoA is therefore split
equally between formation of ethanol (where two NAD' molecules are regenerated) and acetate.
Similarly, if no formate is produced, the conversion of pyruvate to acetyl-CoA occurs solely via the
pyruvate dehydrogenase route, which leads to additional NADH production. In this case, all the acetyl-
CoA must be channeled toward ethanol, where the required NAD+
is regenerated.
With measurements of glucose, lactate, and formate, there are no difficulties in determining all the fluxes
and hereby also the non-measured rates. However, if the measurements of glucose, formate and acetate
are selected instead T; becomes:
- 0 . 5
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
- 1
- 1
- 1
0
0
0
0
1
1
- 1
- 1
1
- 1
1
0
0
- 2
The matrix is singular and it cannot be inverted. This can be seen from the fact that subtraction of the
third column from the second column gives a column that is identical with the last column. Thus, the
columns 2, 3 and 6 are linearly dependent, and this results in a singularity in the matrix. In other words
measurements of glucose, formate and acetate cannot be used to calculate uniquely the remaining three
unkown pathway fluxes to lactate, CG;, and ethanol. The reason, of course, is that with formate and
acetate as the only measurements of metabolic products there is no information about the rate of NADH
consumption (one of the rates
rlt rc
or
re
has to be measured) and the NADH balance is of little use. Since
ra = 0.5
r,
then any set of 3 measurements that includes the pair (ra
rf)
will lead to a non-observable
system. In the general case one will of course search for sets of measurements that leads to observability.
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