242
Chapter 7
As shown in Fig. 7.3 the substrates are
Sh
the metabolic products
Pt
and the biomass constituents
Xj.
With these definitions, the stoichiometry for the/th cellular reaction can be specified as
t,o,JISl+ f ifiJIPl+ firJIXl =
0
(
7
.
1
)
(=1
1=1
1=1
In a growth model there will be a stoichiometric equation like (7.1) for each of the
J
cellular
reactions, and it is therefore convenient to write the stoichiometry for all
J
cellular reactions in a
compact form using matrix notation:
AS +
BP
+ TX = 0
(7.2)
where the matrices A,
B
and r are stoichiometric matrices containing stoichiometric coefficients
in the
J
reactions for the substrates, metabolic products and biomass constituents, respectively. In
the matrices A,
B
and r rows represent reactions and columns compounds. Thus, the element in
the/th row and the /th column of A specifies the stoichiometric coefficient for the /th substrate in
the /th reaction. The stoichiometric coefficients may be positive, negative, or zero. Typically they
are negative for substrates (and other compounds consumed in a given reaction) and positive for
metabolic products (and compounds formed in a given reaction). With a stoichiometric
formulation of the general type of (7.2), a large number of the stoichiometric coefficients become
zero, and one may find it cumbersome to specify stoichiometric coefficients for all compounds
and all reactions considered in the model. However, the advantage is that the general matrix
formulation facilitates much of the subsequent analysis because it can be done in the compact
matrix symbolism that at a later stage may be advantageous for computer simulations. When the
stoichiometry has been formulated in matrix notation, it is very easy to spot the participation of a
given compound in the various reactions. One just has to look at the column for this compound
in the appropriate matrix.
7.2.2 Reaction Rates
The stoichiometry specified in Section 7.2.1 defines the relative amounts of the compounds
produced or consumed in each of the
J
intracellular reactions, but does not allow one to calculate
the rates or the relative amounts at which metabolic products are secreted in the medium. This
can be done by introducing the rates of the individual reactions and further coupling them to
determine the overall rates of product secretion. The rate of a given reaction (or process)
considered in the model is now defined by the forward reaction rate (or velocity)
v,
which
specifies that a compound with a stoichiometric coefficient
P
is formed at the rate
Pv
in this
particular reaction. Normally, one of the stoichiometric coefficients in each reaction is arbitrarily
set to be 1, whereby the forward reaction rate becomes equal to the consumption or production
rate of this compound in the particular reaction. For cellular reactions we often use the biomass
as reference to define the so-called
specific rates
, usually with the unit g (g DW h) '. We now
collect the forward reaction rates of the
J
reactions considered in the model in the rate vector v.
Thus, PjVVy specifies the specific rate of formation of the /th metabolic product in the/th reaction.
Because the stoichiometric coefficients for the substrates,
i.e.,
the elements of A, are generally