Systems Biology (Methods in Molecular Biology)

deterministically and stably. Otherwise, the host will lose its battle

against attacks and their species will not survive for many genera-

tions. In these situations, stable models (like the ones mentioned in

Subheading2.3) can be used to understand the response patterns.

For cell differentiation to occur, on the other hand, a cell needs to

move away from its initial equilibrium state to a new equilibrium

state pertaining to the differentiated cell. If the initial equilibrium

state is “too” stable, it will be difficult to change the cell fate. Thus,

a cell has to move from a stable to an unstable equilibrium state for

the induction of cell differentiation. For example, reprogramming

the key transcription or Yamanaka factors allows a differentiated cell

to dedifferentiate into an inducible pluripotent cell [34].

Similarly, to treat major diseases like cancer or diabetes, the

therapeutic intervention aims to change the equilibrium from an

“unhealthy” to a “healthy” state. However, the new equilibrium

state should be stable; otherwise, the treatment will not be success-

ful and the disease symptoms will persist. Therefore, depending on

the situation, it is necessary to investigate and classify equilibrium

based on stability. Thus, stability analysis can be an important aspect

of biological network modeling.

Linear models, such as those that are made up of first-order

mass-action or M-M kinetics, are always stable as long as their

parameter values are real and positive. For oscillatory or nonlinear

response, as we have seen in the Brusselator and coupled Goodwin

examples, the equilibrium states can vary and can become unstable

depending on the parameter values. Stability analysis, involving

linearization and calculating the eigenvalues of Jacobian matrices,

can be performed to check when a nonlinear model will be stable at

any particular time or for a range of parameter values.

Let us consider again the linear (first-order) mass-action chain

reactions depicted in Eqs.9 to 11. In the Jacobian form, they can be

written as

dX

dt

¼JδX ð 28 Þ

or, in Matrix form

d

dt

X 1

X 2

X 3

2

4

3

(^5) ¼
k 1 00
k 1 k 2 0
0 k 2 k 3
2
4
3
5
X 1
X 2
X 3
2
4
3
(^5) ð 29 Þ
To determine the stability of a system, the eigenvalues (λs) of
the Jacobian matrix are evaluated. If allλs are real and negative, the
reactions will reach a stable node (steady-state level) for each spe-
cies; otherwise, the system can follow a stable focus or become
unstable. Table1 summarizes the conditions for different eigen-
value solutions.
190 Kumar Selvarajoo