Metabolic Dissipative Structures - Systems Biology of Metabolic and Signaling Networks - page 187

8.2.3 Numerical Integration of the Glycolytic Model Under
Sinusoidal Substrate Input Flux Values

In cell-free extracts, a sinusoidal input of glucose can produce periodic, quasiperi-
odic, or chaotic behavior (Markus et al. 1985a , b ). Monitoring NADH fluorescence
from glycolyzing baker's yeast cell-free extracts subjected to sinusoidal glucose
input flux has shown that quasiperiodic dynamics is common at low input
amplitudes whereas chaotic behavior happens at high amplitudes (Markus
et al. 1985a , b ). We adopted a similar approach, by analyzing the dynamic behavior
of the glycolytic system considered under periodic input flux with a sinusoidal
source of glucose S ¼ S 0 þ A sin ðω t Þ . Assuming the experimental value of

S 0 ¼

h (Markus et al. 1984 ), after dividing by K m2 , the Michaelis constant of
phosphofructokinase, a normalized input flux S 0 ¼

=

6mM

:

033 Hz was obtained (De la
Fuente and Cortes 2012 ). Under these conditions, different types of dynamic
patterns can be observed as a function of the amplitude A of the sinusoidal glucose
input flux, as bifurcation parameter (De la Fuente et al. 1996b , 1999b ).
Figure 8.2 shows different time series of the fructose 1,6-bisphospfate concen-
tration after numerical integration of the glycolytic model ( 8.4 ) at different glucose
input flux. A quasiperiodic route to chaos is observed (cf. left panel in Fig. 8.2 ). For
A

0

0.001 the biochemical oscillator exhibits a periodic pattern (Fig. 8.2a ). After
increasing the amplitude to A

¼

0.005 another Hopf bifurcation appears along with
quasiperiodic behavior (Fig. 8.2b ). Above A

¼

0.021, complex quasiperiodic
oscillations take place (Fig. 8.2c ), and after a new Hopf bifurcation the resulting
dynamic behavior evolves into deterministic chaos ( A

¼

0.023, Fig. 8.2d ), as
predicted (Ruelle and Takens 1971 ). This behavior corresponds to a typical quasi-
periodic route to chaos, in agreement with experimental data (Markus et al. 1985b ).

¼

8.2.4 Measure of the Effective Functional Structure of
Glycolysis

The time series of enzymatic activity obtained were analyzed using the nonlinear
technique of transfer entropy in order to quantify the effective connectivity between
glycolytic irreversible enzymes. We reasoned that the oscillatory patterns exhibited
by metabolic intermediates might have information which can be captured by the
TE measure. TE measures the influences between pairs of time series of catalytic
activity, thus resulting in an asymmetric quantity that defines directionality in time
and cause to effect, allowing to quantify the flow of functional information between
processes behaving nonlinearly (Schreiber 2000 ).
For the calculation of TE, the different states of each dynamic variable were
obtained simply by rounding the value of the variable at time t to the nearest integer.
Thus, the state probabilities were computed by counting the number of times that