Systems Biology (Methods in Molecular Biology)

efficiently established directly at a higher order (i.e., mesoscopic)

level, without the necessity of moving down to the microscopic

scale. Obviously, that strategy does not exclude from considering

the microscopic processes induced on the cells by the presence of

those external stimuli—including genetic expression, physical and

chemical molecular forces—by formulating explicit forms forφ 1 ,

φ 2 , andφ 3 in Eq.4, but this is not mandatory for the success of our

approach.

The minimal requirement for the vectorial model Eq.1 or,

equivalently, for its component-wise version Eq.2 to represent an

acceptable candidate for modeling EMT processes is that they

display three stationary solutions, two stable states, and one unsta-

ble/metastable state (seeFig. 1). This forces the functionΦand its

componentsΦ 1 ,Φ 2 and Φ 3 to satisfy some essential structural

conditions, in order to ensure that

ΦðV;SÞ¼ 0 , V∈fA,B,Mg for anyS ð 5 Þ

or, equivalently,

Φ 1 ðE,F,C;SÞ¼ 0

Φ 2 ðE,F,C;SÞ¼ 0

Φ 3 ðE,F,C;SÞ¼ 0

8

><

>:

,ðE,F,CÞ∈fA,B,Mg for anyS ð 6 Þ

for some (distinct) vectorsA¼(EA,FA,CA),B¼(EB,FB,CB) and

M ¼ (EM,FM,CM) corresponding to biologically relevant

equilibria.

Additional conditions guaranteeing stability forAandB, and

metastability for M, have also to be satisfied (refer to

Subheading3.3).

Since the equilibrium system Eqs.5–6 is multi-dimensional,

the phase-space exhibits a non-trivial geometrical landscape (see

Fig.7), and transitions can occur withsudden changeof values

concerning only some variables (like forcontact discontinuitiesin

continuum physics [50]).

3.3 A Tutorial
Example

We consider a simplified model consisting of a (nonlinear) system

with two coupled first order ODEs for the variablesuandw, that is

τ

du

dt

þu¼w,

dw

dt

þw¼λguðÞ, ð 7 Þ

where the external constraints are given by constant parametersτ,λ

>0 andgis a known structural function whose properties are

detailed later on. In comparison with the general model Eqs.1–2,

order and control parameters correspond toV¼(u,w) andS¼

(τ,λ), respectively, and

112 Chiara Simeoni et al.