Systems Biology (Methods in Molecular Biology)

For the modeling of the chemical network model which

appears in Fig.3, COPASI v. 4.6.32 software was used. On the

other side, numerical integration was performed of the system of

ODEs Eq. (24) through implementation of the Gear algorithm for

hard equations (“stiff”), in Fortran with double precision and

tolerance of 10^8 [57]. It is important to note that most systems

of ordinary differential equations, derived from biological systems,

have rigidity, i.e., stiffness, which is why it is advisable to use

numerical methods such as predictor corrector, for example the

algorithm of Gear [57]. For bifurcation diagram, the package

TISEAN 3.01 [58] was used for Poincare maps, correlation dimen-

sion, and power spectrum.

In Fig.4 dynamical behavior of the proposed chemical network

model for different values of the control parameterIis shown.

When the value of control parameterIis smaller them the EDO

system Eq. (24) exhibit the maximum complexity.

For “high” values of immune surveillance (I¼4) it exists a

population of proliferating tumor cells, wherexcells are the pre-

dominant, which corresponds to the phase state where a growth

avascular stable steady state is reached, the tumor grows to a state

known as dormant state [33, 35].

Table 1
Stability, and complexity for the system of ODEs (Eq.24) for different values of the control parameterI
(N¼5,H¼3). Reprinted from [83]

I

Eigenvalues of the

Jacobian matrix

Lyapunov

exponentsλj LZ complexity DL

4

Sssstable focus

7.2    10 ^2 5.3i

7.2    10 ^2 þ5.3i

7.8

0.0720024

0.0740862

7.82744

–0

3

Limit cycle

(saddle-foci)

þ8.5

3.3e-001 –1.6i

3.3e-001þ1.6i

~0.00

0.227573

5.89284

0.03589 1

2

Saddle-foci

þ6.3

1.3    10 ^1 1.9i

1.3    10 ^1 þ1.9i

~0.00

0.30561

4.05829

0.03888 1

1

Saddle-node

+10

þ3.0

1.0

~0.00

0.779727

1.87158

0.04187 1

0.7

Limit cycle

þ3.5

4.3    10 ^2 1.5i

4.3    10 ^2 þ1.5i

~0.00

~0.00

2.10445

0.06580 2

0.4 Shilnikov’s

chaos

þ2.8

3.3    10 ^2 1.2i

3.3    10 ^2 þ1.2i

þ0.0519588

~0.00

1.71075

0.08972 2.03

Parameters Estimation in Phase-Space Landscape Reconstruction of Cell Fate... 139