Differential Evolutionary Algorithms for In Vivo Dynamic Analysis of Glycolysis and Pentose Phosphate Pathway in Escherichia coli - Parallel Computing for Bioinformatics and Computational Biology - page 67

Steady-state concentrations for all intermediate metabolites were needed for V m
calculation. Some of these had to be estimated because no experimental results were
available. Owing to our measured values of fdp and gap, it was impossible to esti-
mate the steady-state concentration of dhap from a near-equilibrium assumption for
triosephosphate isomerase (TIS). Likewise a near-equilibriumassumption for aldolase
and TIS is impossible either. Therefore, the same mass action ratio as experimentally
measured by Schaefer et al. [18] was employed ( 3mM / 2.3mM ) .

Thus a concentration of 0.167mM was calculated for dhap based on a measured
value of 0.218mM for gap. For the estimation of pgp, 3pg, and 2pg concentrations, a
near-equilibrium assumption for the enzymes PGK, PGluMu, and ENO was reason-
able. The concentrations of the three metabolites were calculated using the measured
pep concentration and the Keq values of the different enzymes as taken from Ni and
Savageau [17] for PGK, and from Bakker et al. [13] for PGluMu and ENO, respec-
tively. The resulting ratio of the intracellular concentration of pep and 3pg was found
to be close to the values measured by Schaefer et al. [18] and Bhattacharya et al. [38]
for E. coli . A similar ratio was also found by Bakker et al. [13] in the case of try-
panosome and by Mulquinay [14] for human erythrocyte. Thus, there is substantial
experimental evidence that the assumption of near-equilibrium conditions is valid.
Concordant with Vaseghi et al. [35], near-equilibrium conditions were assumed for
the five reactions of the non-oxidative part of the ppp (Ru5P, R5PI, TK1, TK2, and
TAL). The metabolite concentrations of ribu5p, rib5p, xyl5p, sed7p, and e4p were
estimated from measured steady-state concentrations of f6p and gap.
Table 3.1 summarizes all the estimated and measured steady-state concentrations.

[ gap ] / [ dhap ]=

3.3 ESTIMATION OF THE PARAMETERS OF THE MODEL BY
DIFFERENTIAL EVOLUTIONARY ALGORITHM

The kinetic parameters of these equations were fit to the measurements by minimizing
the sum of relative squared errors using differential evolutionary algorithm. Differ-
ential evolution (DE) is one of evolutionary algorithms (EAs), which are a class of
stochastic search and optimization methods including genetic algorithms (GAs), evo-
lutionary programming, evolution strategies, genetic programming, and all methods
based on genetic and evolution. The DE algorithm was introduced by Rainer Storn
and Kenneth Price. DE is a higher implementation of GAs. Owing to its fast conver-
gence and robustness properties, it seems to be a promising method for optimizing
real-valued multi-modal objective functions. Compared with traditional search and
optimization methods, the EAs are more robust and straightforward to use in complex
problems: they are capable of workingwithminimumassumptions about the objective
functions. Only the value is required to guide the research process of parameters.
The generation of the vectors containing the parameters of the model is attained
by an independent procedure:

X i , G = X 1, i ··· X D , i (3.23)