Systems Biology (Methods in Molecular Biology)

can be considered an extension of the “Principle of Maximum

Entropy” [110].

On the other hand, we find that [26] the Prigogine Principle of

“minimal entropy production” [12], understood as an extremal or

variational principle, can be generalized for nonlinear systems far

from thermodynamics equilibrium through the Lyapunov function

[82, 81]. In this sense, the apparent contradiction abovementioned

is explained, as in this context the entropy production rate is used as

extremal principle.

In our case, we use as control parameter,Ω¼f(Pi, Glu), so

S_i¼fðÞPi;Glu. And we can rewrite Eq. (18)as

dS_i

dt

¼

∂S_i

∂Pi

dPi

dt

þ

∂S_i

∂Glu

dGlu

dt

< 0 ð 40 Þ

and considering that in the glycolysis process:

Gluþ2NADþ2ADPþ2Pi¼2Pyrþ2NADHþ2Hþþ2ATP

þ2H 2 O

It can be observed that both Glu and Pi act as reactants∂∂Pit< 0

and∂∂Glut <0, so ∂

S_i

∂Ω 0. Then it fulfills that (seeEq. ( 40)):

dS_i

dt< 0.

Those results allow us to posit that the rate of entropy produc-

tion is a Lyapunov function, i.e., shows its directional character,

behaving as “first-order” phase transition, through a supercritical

bifurcation of Andronov-Hopf. Conclusively, the process of cancer

glycolysis exhibits sustained oscillations leading to self-organization

far from thermodynamics equilibrium.

Therefore, a viable therapeutic approach exploiting abnormal-

ities in cancer glycolytic metabolism would be finding those regions

of control parameters in which self-organization is lost, leading

thus to less complex steady states. That strategy would likely repre-

sent a key to improve and identify new anti-cancer therapies in the

future.

5 Concluding Remarks

The integration of thermodynamics formalism of irreversible pro-

cess, complex systems theory, and systems biology [164] offers a

holistic view of cancer as a self-organized nonlinear dynamical

system far from thermodynamic equilibrium. In this sense, we

show how its evolution of tumor occurs as an emergent phenome-

non, as a phase transition, could be called “biologic phase transi-

tion.” In summary, it was found that:

1. A mechanism for avascular, vascular, and metastasis tumor
   growth based on a chemical network model. Vascular growth
   and metastasis appear as a hard phase transition type, as “first-
   order,” through a supercritical Andronov–Hopf bifurcation,

Parameters Estimation in Phase-Space Landscape Reconstruction of Cell Fate... 163