lular reaction systems, they involve, e.g., (1) maximization of steady-state fluxes,
(2) minimization of the concentrations of metabolic intermediates, (3) minimization
of transition times, (4) maximization of sensitivity to an external signal, and (5) opti-
mization of thermodynamic efficiencies (Heinrich and Schuster 1998).
Evolution and optimization of cellular systems are subject to physical constraints.
Such constraints include, for example, differences in free energy for participants of a
reaction expressed by the equilibrium constant, diffusion limitations in the move-
ment of compounds through the cell, structural requirements in the composition of
macromolecules, or the stoichiometry of metabolic systems.
In order to take into account biological constraints, the concept of a cost function
has been introduced (Reich 1983). The following have been suggested as cost func-
tions: the total amount of enzyme in a cell or the pathway under consideration
(Reich 1983), the total energy utilization (Stucki 1980), or the evolutionary effort
(Heinrich and Holzhutter 1985) counting the number of mutations or events neces-
sary to attain a certain state.
In the following three sections we will study how metabolic networks should be
designed if they were designed according to optimality principles. We investigate the
consequences of the demand for rapid conversion of substrate into product on the
catalytic properties of single enzymes and on the appropriate amount of enzymes in
a metabolic pathway. In the first two sections, we determine conditions that yield
maximal steady-state fluxes. In the third section, an example for temporal regulation
of pathway properties is studied.
10.3.1
Optimization of Catalytic Properties of Single Enzymes
An important function of enzymes is to increase the rate of a reaction. Therefore,
evolutionary pressure should lead towards a maximization of the reaction rate
v ? max (Pettersson 1989, 1992; Heinrich and Hoffmann 1991; Wilhelm et al.
1994). High reaction rates may be achieved only if the kinetic properties of the en-
zymes are suitably adapted. We identify the optimal kinetic parameters that maxi-
mize rates for the reversible conversion of substrate S into product P (Klipp and
Heinrich 1994).
There are two constraints to be considered. First, the action of an enzyme cannot
alter the thermodynamic equilibrium constant for the conversion of S to P (Eq. (5-13)).
Changes of kinetic properties must obey the thermodynamic constraint. Second, the
values of the kinetic parameters are limited by physical constraints even for the best
enzymes. Their maximal possible values are denoted by k
max
, and we consider all
rate constants to be normalized by their respective k
max
, such that the maximal
values of the normalized kinetic constants are one.
For a reaction that can be described with linear kinetics
(10-31)
with the thermodynamic equilibrium constant
356
10 Evolution and Self-organization