Design of Pathway-Level Bioprocess Monitoring and Control Strategies Supported by Metabolic Networks - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 205

fractional operation of this mode within the complete set of elementary modes
[ 57 ].
Schwartz and Kanehisa [ 58 ] showed that a combination of structural and kinetic
modeling in yeast glycolysis significantly constrains the range of possible
behaviors of a metabolic system. All elementary modes are not equal contributors
to physiological cellular states, and this approach may open a direction towards a
broader identification of physiologically relevant elementary modes among the
very large number of stoichiometrically possible modes.
Very recently, Bastin et al. [ 59 ] developed a methodology to compute a
decomposition of admissible flux vectors in a minimal number of elementary
modes without explicitly enumerating all of them. They demonstrated that the
vector of admissible weighting vectors (k) rewritten as
k ¼ X
k

b i 0 X
k

b i h i b i ¼ 1 ð 9 Þ

is necessarily an admissible k satisfying Eq. ( 7 ). In this case, the convex polytope,
H ¼ h 1 h 2 h ½ , contains a number of solutions equal to the number of
measurements p. Each polytope solution represents a minimal flux distribution
given by v i ¼ EM h i and may be viewed as the simplest pathways that satisfy the
pseudo-steady-state assumption and the constraints imposed by the extracellular
measurements defined in Eq. ( 7 ).

4.4 Example: Reduction of the Elementary Modes by Weighting
Factor Minimization

Here we illustrate the method proposed by Schwartz and Kanehisa [ 58 ] for ele-
mentary mode reduction. This method identifies a subset of elementary modes by
minimizing the sum of weighting factors (k):

min X

k

k i

ð 10 Þ

i ¼ 1

subject to Eq : ð 4 Þ

This method was applied to the previously described P. pastoris metabolic
network including 4,119 elementary modes. The results are shown in Table 2 for
three distinct time points. Only 17 elementary modes were obtained with nonzero
weighting factors. Several of these are selected at least twice in the three different
phases of the culture. Note that this method basically selects the elementary modes
which are closest to the actual biological state by minimizing the sum of weighting
factors.