Systems Biology (Methods in Molecular Biology)

Φ 1 ðÞ¼u,w;τ,λ

1

τ

ðÞwu, Φ 2 ðÞ¼u,w;τ,λ λguðÞw: ð 8 Þ

We attempt at formulating a hypothetical interpretation of the

dynamical process Eq.7 in terms of biological observations, assum-

ing thaturepresents E-cadherin boundary values andwstands for

the coherency, which is connected with relative E-cadherin density

values along the membrane border with respect to its overall con-

centration. Then, the specific expression forΦ 1 encodes the fact

thatu—describing the E-cadherin boundary distribution of the cell

population—tends to conform to the behavior ofw—accounting

for the system coherency—in a (typically fast) time-scale of orderτ.

Similarly, the expression forΦ 2 entails the convergence ofwtowards

λg(u) in a (slower) time-scale of order 1.

According to the abstract calculations in Eqs.5–6, that now

translate into

Φ 1 ðu,w;τ,λÞ¼ 0

Φ 2 ðu,w;τ,λÞ¼ 0



for the specific functions Eq.8, the stationary solutions to Eq.7 are

given by the points (u,w) which are located at the intersection of

the curves

w¼u and λgðuÞ¼w ð 9 Þ

laying on the phase-plane (i.e., the two-dimensional projection of

the phase-space). As a consequence, the set of equilibria for the

dynamical system Eq.7 is characterized, for any fixedλ>0, as the

zeros of the function (seeFig. 8b)

Fig. 7Hypothetical three-dimensional space-phase diagram depicting the performance of order parameters

Mathematical Modeling of Phase Transitions in Biology 113