Biochemical Reaction Networks
153
formation o f biom ass directly from p recursor m etabolites.
In the second approach the synthesis o f m ost build in g blocks is considered in the m odel, and the biom ass
equation is described as a reaction w here build in g blocks are converted into biom ass. T he stoichiom etric
coefficients for the b uilding blocks are identified from know ledge o f the am ount o f different building
blocks that is needed for biom ass form ation. T his approach typically results in a substantial increase in
the m odel com plexity, since a large num ber o f reactions leads to the m any different building blocks.
Perhaps lum ping o f reactions that lead to m any o f the b uilding blocks can be done but still the num ber o f
reactions considered in the m odel is large. T h e advantage is, how ever, that it is relatively easy to identify
the different elem ents o f the biom ass equation.
In the last approach reactions for synthesis o f the different m acrom olecules are included in the m odel,
e.g. reactions for synthesis o f proteins, lipids, D N A , R N A and carbohydrates. T he biom ass equation is
described as a reaction w here the m acrom olecules are co nverted into biom ass, and the stoichiom etric
coefficients for the m acrom olecules are given by the m acrom olecular com position o f the biom ass. W ith
this approach it is relatively easy to study the influence on the calculated fluxes o f the m acrom olecular
composition w hich directly appears in the biom ass equation.
Which ever approach is used requires a substantial inform ation about how biom ass is synthesized and on
the m etabolic costs o f the d ifferent precu rso r m etabo lites/b u ild in g blocks/m acrom olecules. In addition
the costs o f A T P , N A D P H and N A D H for biom ass form ation m ust also be available, and this requires
information about the biom ass com position. In recent years this type o f inform ation has becom e
available for m any m icroorganism s as part o f flux analysis studies. If no inform ation is available for the
investigated system one m ay use data from related organism s. It is already recom m endable to calculate
the sensitivities o f the calculated fluxes to variations in the estim ated (or experim entally determ ined)
biomass equation.__________________________________________________________________________________________
5.4.1 Use of Measurable Rates
When
F
or more rates are measured all the fluxes can be estimated using matrix inversion as
discussed in Section 5.3. Eq. (5.25) directly gives the solution for the flux vector v when exactly
F
rates are measured. This is often referred to as a
determined system.
If more than
F
rates are
measured the system is
over-determined.
Here the matrix T 2 is not quadratic and it cannot be
inverted directly. To circumvent this problem there are two possibilities:
(i)
One may use a sub-set of the measured rates and calculate the fluxes using eq. (5.25)
and the other rates using eq. (5.26). Through comparison of the calculated rates and
the measured rates that are not used to calculate the elements of the flux vector one
may check the consistency of the model (and model predicted values may be found
for the measured rates if the model is believed to have the correct structure).
(ii)
One may use a statistical procedure on the whole set of data to obtain good estimates
for the elements in the flux vector v and obtain new (and better) estimates for the
measured rates.
In the first case the solution is found as for a determined system. In the other case a statistical
procedure similar to that described in Section 3.6 has to be applied, but there may be different
approaches (see Stephanopoulos
et al.
(1998) for details). Often one will, however, simply use
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