synthase system Chock et al., 1980. In recent years,
the mitogen-activated
protein kinase
MAPK signaling pathway has been intensely investigated Hsueh and Law, 1999; Bornfeldt and
Krebs, 1999. The interconnection of several path- ways has also been studied Bray, 1995. Al-
though these structures have a number of similarities with metabolic pathways and net-
works, there is an essential difference in that there is no or only a very small mass flow between the
components of a signaling network.
In living cells, information is frequently pro- cessed in hierarchic systems. In gene expression,
for example, we can discern the levels of DNA, mRNA, and proteins. Between the levels of such
hierarchies, there is flow of information, but again usually no mass flow. In the present paper, we
shall analyze what topological and kinetic proper- ties a system should have so that information can
be transmitted efficiently without or with only a small concomitant mass flow.
The understanding of the regulatory properties of biochemical systems has been greatly improved
by Metabolic Control Analysis MCA. This is a theoretical framework through which the effect of
changes in enzyme activity on the fluxes, concen- trations and other relevant variables characteriz-
ing biochemical systems at steady state can be determined for review see Westerhoff et al., 1995;
Heinrich and Schuster, 1996; Fell, 1997. The largest part of the theory so far developed in
MCA, concerns systems in which the substances are connected by mass flow e.g. Westerhoff and
Van Dam, 1987; Reder, 1988; Brown et al., 1990; Kholodenko et al., 1998; Schuster and Wester-
hoff, 1999. However, also enzyme cascades and other hierarchic systems that are capable of signal
transfer have been analyzed Small and Fell, 1990; Kahn and Westerhoff, 1991; Kholodenko et al.,
1997. In the present paper, we will elaborate on the question as to whether information flow can
be dealt with by MCA. Moreover, we will study the specific topological properties of cellular sig-
naling systems. Special attention will be paid to the fact that knowledge of kinetic parameters of
all the enzymes involved in the network is often incomplete.
2. Metabolic control analysis in the case of imperfect knowledge of kinetic parameters
One of the major achievements of MCA is the derivation of formulas by which the systemic con-
trol properties of a network can be derived from its constituent properties and its structure. In the
absence of conservation relations, these formulae can be written in matrix form Reder, 1988; Hein-
rich and Schuster, 1996:
C
J
= dgJ
− 1
I − v
S N
v S
− 1
N
n
dgJ, 1
C
S
= −
dgS
− 1
v S
N v
S
− 1
NdgJ
2
where C
J
, C
S
, and N denote the matrices of flux
control coefficients and concentration control co- efficients and the stoichiometry matrix. I and
vS stand for the identity matrix and the matrix of non-normalized elasticities, that is, the deriva-
tives of reaction rates with respect to concentra- tions. J denotes the vector of steady state fluxes.
It fulfills the steady-state equation
N J = 0.
3 If the system involves conservation relations for
concentrations, Eqs. 1 and 2 must be slightly modified Reder, 1988; Heinrich and Schuster,
1996. The elasticities contain information about the kinetic properties of the enzymes involved in
the system, while the information about the struc- ture of the biochemical system in terms of how
substances are ‘connected’ to each other by reac- tions, is encompassed in the stoichiometry matrix
N.
Dynamic modeling
and simulation
of metabolism are often hampered by the fact that
the kinetic properties of enzymes are imperfectly known. Furthermore, enzyme activities in vivo are
subject to frequent changes due to inhibition or activation. In contrast, the topology of the system
can be considered constant. Therefore, in many situations, control coefficients cannot be deter-
mined exactly for all enzymes involved. This difficulty has led to the development of a number
of approaches to derive, nevertheless, interesting conclusions about the control and regulation
properties in such situations. In the top-down approach Brown et al., 1990; Brand, 1996 and
modular approach Westerhoff and Van Dam, 1987; Schuster et al., 1993, enzymes are grouped
into blocks so that it is no longer necessary to know all the details within the blocks. The kinetic
properties of the blocks and the control properties of the system are described by the overall elastic-
ities and control coefficients, respectively. More- over, the modular approach is a suitable tool for
describing the control properties of those enzymes that couple exergonic processes to endergonic
processes, such as the various ATPases Schuster and Westerhoff, 1999. Appropriate linear combi-
nation of the overall control coefficients gives the coefficients quantifying the control exerted by the
enzymes as a whole and the control exerted by the slip.
Kinetic parameters are often difficult to mea- sure for very fast reactions. In this situation, it is
often justified to assume these reactions to attain quasi-equilibrium. We were able to show in a
general way that quasi-equilibrium enzymes can be eliminated from the analysis of the control
properties, allowing the calculation of the control coefficients of the slow reactions even if the ki-
netic properties of the fast reactions are unknown Kholodenko et al., 1998.
3. Topological analysis
