Biomedical Engineering Reference
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of 35 elementary modes for a yeast metabolic network fermenting both glucose
and xylose without using experimental data.
4.2 Reduction Based on Thermodynamic Properties
Elementary modes can also be discriminated and reduced on the basis of metabolic
reaction thermodynamics. The main assumption is that metabolic networks have
evolved over time in the sense that cellular regulatory mechanisms were created
that favor efficient pathways with low entropy generation. Wlaschin et al. [ 52 ]
demonstrated with experimentally determined intracellular fluxes that elementary
mode weighting factors are inversely correlated with the entropy generated by the
involved metabolic reactions. Zhao and Kurata [ 53 ] proposed a method for cor-
relating enzyme activity and flux distribution which uses Shannon's maximum-
entropy principle, a measure of system complexity, as an objective function to
estimate the enzyme control flux.
4.3 Reduction Based on Flux Data
Several methods have been proposed to eliminate elementary modes on the basis
of measured flux data. The equation that applies here is Eq. ( 4 ); however, the
number of elementary mode weighting factors is in general much larger than the
number of metabolic fluxes, thus the system is largely undetermined. Palsson and
coworkers [ 54 , 55 ] suggested linear optimization methods to determine how
extreme pathways (the systemically independent subset of elementary modes)
contribute to a given (measured) steady-state flux distribution. There is a range of
possible nonnegative weighting values associated to extreme pathways that pro-
duce a given steady-state flux distribution. This range was calculated by maxi-
mizing and minimizing the extreme pathway weighting factors, resulting in the so-
called a-spectrum. The allowable ranges for the values of k i were computed as
max k i
subject to Eq : ð 4 Þ ;
i ¼ 1 ; ... ; K
0 k i 1
ð 8 Þ
min k i
subject to Eq : ð 4 Þ ;
i ¼ 1 ; ... ; K
0 k i 1
Wang et al. [ 56 ] presented a method to calculate the elementary mode coeffi-
cients for a large set of elementary modes by devising a quadratic program to
explore the possibility and performance of using a subset of the elementary modes
to reconstruct flux distributions. Alternatively, a framework based on elementary
mode analysis and the convex properties of elementary modes was developed to
calculate
flux
regulation
coefficients
(FRC)
corresponding
to
an
appropriate
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