Metabolic Dissipative Structures - Systems Biology of Metabolic and Signaling Networks - page 185

Fig. 8.1 Multienzymatic instability-generating system in yeast glycolysis . The irreversible stages
correspond to the enzymes E 1 (hexokinase), E 2 (phosphofructokinase), and E 3 (pyruvate kinase).
Metabolite S (glucose) is transformed by the first enzyme E 1 into the product P 1 (glucose-6-
phosphate). P 0 1 ,P 2 ,P 0 2 , and P 3 denote the concentrations of fructose 6-phospfate, fructose
1,6-bisphospfate, phosphoenolpyruvate, and pyruvate. q 1 is the rate first-order constant for the
removal of P 1 ; q 2 is the rate constant for the sink of the product P 3 . The figure includes the
feedback activation of E 2 and the feedback inhibition of E 3 . The ATP is consumed by E 1 and
recycled by E 3 . Adapted from Fig. 1 in (De la Fuente and Cortes 2012 )

variables. By means of using differential equations with delay it is possible to
consider initial functions (instead of the constant initial values of ODE systems) and
to analyze the consequences of parametric variations (De la Fuente and Cortes
2012 ).
For spatially homogeneous conditions, the time evolution of the glycolytic
system represented in Fig. 8.1 can be described by the following three delay
differential equations:

d d t ¼

z 1 σ 1 Φ 1 ðμÞσ 2 Φ 2 ðα; βÞ q 1 α;

d d t ¼

z 2 σ 2 Φ 2 ðα; βÞσ 3 Φ 3 ðβ; β 0 ; μÞ;

d d t ¼

z 3 σ 3 Φ 3 ðβ; β 0 ; μÞ q 2 γ; (8.4)

where the variables denote the normalized concentrations of glucose-6-

phosphate, fructose 1-6-bisphosphate, and pyruvate, respectively, with the follow-
ing three enzymatic rate functions:

α , β , and γ

Φ 1 ¼ μ SK D 3 = ð K 3 K 2 þ μ K m 1 K D 3 þ SK 2 þ μ SK D 3 Þ ; (8.5)

2

αð 1 þ αÞð 1 þ d 1 βÞ

Φ 2 ¼

2 ;

(8.6)

2 2

L 1 ð 1 þ c αÞ þð 1 þ αÞ ð 1 þ d 1 βÞ

3

d 2 β 0 ð d 2 β 0 Þ

1 þ

Φ 3 ¼ (8.7)

4

4

d 2 β 0 Þ L 2 ð 1 þ d 3 μÞ þð 1 þ

and