Metabolic Dissipative Structures - Systems Biology of Metabolic and Signaling Networks - page 186

β 0 ¼ f ð βð t λ 1 Þ Þ;

μ ¼ h ð γð t λ 2 Þ Þ:

σ 3 correspond to the maximum activity of the enzymes
E 1 ,E 2 , and E 3 ( V m1 , V m2 , and V m3 ) divided by the Michaelis constants K m1 , K m2 ,
and K m3 , respectively. The z constants are defined as z 1 ¼

The constants σ 1 , σ 2 , and

K m1 / K m2 , z 2 ¼ K m2 / K m3 ,

and z 3 ¼ K m3 / K d3 with K d3 representing the dissociation constant of P 2 by E 3 . L 1

and L 2 are the respective allosteric constant of E 2 and E 3 . The constants d 's are
d 1 ¼

K d3 / K d4 ( K d3 and K d4 are the dissociation
constant of P 2 by E 3 and the dissociation constant of MgATP, respectively). The c
constant is the nonexclusive binding coefficient of the substrate P 1 (De la Fuente
and Cortes 2012 ).
From a dissipative point of view, the essential enzymatic stages are those that
correspond to the biochemical irreversible processes (Ebeling et al. 1986 ). To
simplify the model, we did not consider the intermediate part of glycolysis belong-
ing to the enzymatic reversible stages. In this way, the functions f and h are
supposed to be the identity function:

K m3 / K d2 , d 2 ¼ K m3 / K d3 , and d 3 ¼

f ð βð t λ 1 Þ Þ ¼ βð t λ 1 Þ

h ð γð t λ 2 Þ Þ ¼ γð t λ 2 Þ

The initial functions present a simple harmonic oscillation in the following form:

α 0 ð t Þ¼ A þ B sin ð π t = P Þ

2

β 0 ð t Þ¼ C þ D sin ð π t = P Þ

2

γ 0 ð t Þ¼ E þ F sin ð 2 π t = P Þ

P ¼ 534 s :

The dependent variables α , β , and γ are normalized by dividing them by K m2 ,

K m3 , and K d3 , the parameters λ 2 are time delays which correspond to phase

shifts of the initial functions and P is the period (see for more details De la Fuente
and Cortes 2012 ).
This glycolytic model has been exhaustively analyzed before, revealing a nota-
ble richness of temporal patterns which include the three main routes to chaos
(De la Fuente et al. 1996a , b , 1999b ), as well as a multiple coexisting stable states
(De la Fuente et al. 1998a , 1999b ), and persistent behavior (De la Fuente
et al. 1998b , 1998c ; 1999c ).

λ 1 and