Systems Biology (Methods in Molecular Biology)

The appearance of new structures in nature far from thermody-

namic equilibrium appears similar to a “phase transition,” generi-

cally called bifurcation [17]. Due to the nonlinear nature of the

dynamical system and the feedback processes, the fluctuations grow

and amplify at the macroscopic level which leads to the appearance

of the system’s self-organization and consequently the complexity

that it exhibits [18].

The seminal work of Landau [19] established the theoretical

foundation of the phase transition. According to Landau formal-

ism, the potentialΦor “Landau potential” is defined in terms of the

variables that describe the systemγ, and the order parameterμ[20].

The order parameter is defined empirically. As shown in a

previous work, in the case of cancer evolution [21], the order

parameterμis taken as the difference between the fractal dimension

of normal dfN and tumor cells dfC, thus μ¼dfNdfC ,where

μ¼0 in the symmetric phase (normal cells) andμ 6 ¼0 for tumor

cells. The order parameterμis called the “morphological degree of

complexity” [22].

The Landau potentialΦcan be written as

ΦðÞ¼γ;μ Φ 0 ðÞþμ αγðÞμ^2 þβγðÞμ^4 þð 14 Þ

In the neighborhood of the transition pointμ¼0:

∂ΦðÞγ;μ

∂μ



γ

¼ 4 Cμ^3 þ 2 Aμ¼ 0 ð 15 Þ

the stability condition is fulfilled as

∂^2 ΦðÞγ;μ

∂μ^2



γ

¼ 2 Aþ 12 Cμ^2 > 0 ð 16 Þ

Considering Eqs. (15) and (16), there are three possibilities:

1.γ>γC)A>0,

2.γ<γC)A<0,

3. By continuity we have:γ¼γC)A¼0.

Although a satisfactory description is achieved through the

dissipation functionΨof tumor growth [22], Landau formalism

is limited since it is based on a mean field theory [23]; in other

words, it does not consider correlations or fluctuations.

For this reason, it is convenient to use instead, in order to

describe the process of tumor growth the so-called Lyapunov

function [24]. At the end of the nineteenth century, Lyapunov

developed a method for studying the stability of equilibrium posi-

tions that bears his name. This method allows knowing the global

stability of the dynamics of a system [25]. Letpbe a fixed point,

steady state, of the flowdxdtx_¼fxðÞ. A functionV(x) is called a

132 Sheyla Montero et al.