Important examples of matrices K with a block-
diagonal structure are provided by the ubiquitous hierarchic systems in biology. Fig. 1A depicts
such a system, for which the null-space matrix can be chosen so as to involve two diagonal blocks
K = Á
à Ã
à Ä
1 1
1 1
 Ã
à Ã
Å .
7
In addition to intermediary metabolism and protein metabolism, more levels such as those of
DNA and mRNA can be included. If reactions 1 enzyme synthesis and 2 enzyme degradation
are both activated in such a way that the enzyme level remains constant, the flux through the upper
level increases without any change in the flux through the lower level. Conversely, by changing
the corresponding parameters in reactions 3 or 4 or both, the flux through the lower level can be
changed without any change in the upper level.
Interestingly, the property of matrix K to be
block-diagonal is retained even if the system is depicted on a more detailed level, as in Fig. 1B. In
this representation, the stoichiometry matrix can no longer be block-diagonalized. The substances
in Fig. 1B cannot be grouped in such a way that the concentration changes in each group are due
to distinct chemical reactions, whereas the steady state fluxes can be arranged in two groups of
distinct fluxes. This example shows that the null- space matrix K rather than the stoichiometry ma-
trix N is the more accurate mathematical concept to analyze the subsystem structure.
4. Zero flux control
Another approach that is helpful in the situa- tion where kinetic parameters are incompletely
known concerns the decision which enzymes can, or cannot at all, control which fluxes irrespective
of the values of kinetic parameters. The situation where some flux cannot be controlled by some
reaction may arise if
1. some reactions are irreversible, 2. some
enzymes are
saturated with
their substrates,
3. some enzymes are very fast quasi-equilibrium enzymes,
4. the topology of the network is such that some subsystems are independent of each other with
respect to fluxes. In the case of incomplete knowledge of kinetic
parameters, often
qualitative knowledge
of whether or not a reaction rate depends on some
concentration is available. This knowledge which is, for example, given in most cases on the
Boehringer chart, and nowadays in the several metabolic databases in the WWW can be com-
piled in a qualitative elasticity matrix,
o
ji qual
= 0,
if 6
j
S
i
= 0 for any
admissible concentration vector x,
otherwise. 8
From the knowledge of this matrix and the stoichiometry matrix, one can deduce which fluxes
are unsusceptible to control by which enzymes Schuster and Schuster, 1992; Heinrich and Schus-
ter, 1996. In the above-mentioned case d, the block-diagonal structure of the null-space matrix
plays a central role. For example, in the system shown in Fig. 1A, reactions 3 and 4 do not exert
any control on the flux through reactions 1 and 2. Such a situation has been called ‘dictatorial con-
trol’ Westerhoff et al., 1990. It can also occur in enzyme cascades Fig. 2 if there is no feedback
from a lower level to a higher level in the cascade Kahn and Westerhoff, 1991. However, the con-
trol need not be dictatorial in these situations: If Í
Á Ä
Fig. 1. Simple hierarchic system in simplified A and more detailed B representations. Abbreviations: E, free enzyme;
ES, enzyme – substrate complex; S, substrate; P, product.
Fig. 2. Multi-level enzyme cascade. Abbreviations: S, signal; E
i
and E
i
, active and inactive forms, respectively, of the enzyme at level i; T and T, active and inactive forms, respectively, of
the target.
This matrix cannot be block-diagonalized, while the corresponding null-space matrix can. The latter
has formally the same form as given in Eq. 7. By the formalism developed in Schuster and
Schuster 1992, it can be shown that reactions 3 and 4 would not exert any control on the flux
through reactions 1 and 2 which is, in fact, the glycolytic
flux if
fructose-2,6-bisphosphate F2,6P
2
were no effector of enzymes 1 and 2. However, F2,6P
2
is known to be a potent activator of 6-phosphofructokinase in many cell types and to
play an important role in the regulation of glycol- ysis Yuan et al., 1990; Lefebvre et al., 1996. This
will be analyzed in more detail at the end of the following section.
In a mutant where activation by F2,6P
2
is absent, reactions 1 and 2 do exert a dictatorial control on
the F2,6P
2
cycle irrespective of the values of kinetic parameters. Note that the feature of flux control
insusceptibility is asymmetric; it may happen that a subsystem 1 is able to control a subsystem 2 while
the latter cannot control the former.
If not only information is available about which elasticities are zero and which are not, but also the
signs of the elasticities are known, qualitative conclusions about the sign pattern of control coeffi-
cients can be drawn Lundy and Sen, 1995. Gen- erally, however, only the signs of a subset of the
control coefficients can be determined in this situ- ation.
5. Control of signal transduction
