Systems Biology (Methods in Molecular Biology)

Determinant, i.e., det (J-λI), of Eq.29, and setting it to zero to

solve forλs:

det

k 1 λ 00

k 1 k 2 λ 0

0 k 2 k 3 λ

2

4

3

(^5) ¼ 0
)ðÞk 1 λðÞk 2 λðÞ¼k 3 λ 0
∴λ¼k 1 ,k 2 ,k 3
We note that in biochemical reactions, the rates of reactions
(kvalues) are never negative numbers. Thus,λs for the above
condition are all negative indicating stable nodes. It can be shown
that networks of any complex configurations, connected by first-
order mass-action reactions, are highly stable if k values are
non-negative as is for biological systems.
For systems where nonlinear differential equations are used to
represent the dynamics, such as the Brusselator, the equations are
first linearized using techniques such as Taylor series. Next, stability
is analyzed at specific fixed points around a known equilibrium
point. To illustrate, let us refer back to the generalized kinetic
evolution equation in Sect.2.3. Applying Taylor series to Eq.7 at
X¼a,whereais an equilibrium point:
∂X
∂t
¼
∂FXðÞ
∂X
X¼a
δXþ
∂F^2 ðÞX
∂X^2
X¼a
δX^2 þ
∂F^3 ðÞX
∂X^3
X¼a
δX^3 þ... ð 30 Þ
whereδX¼Xais a small displacement away from the known
equilibrium point at which stability is to be evaluated. Note thatF
(a)¼0, by definition. SinceδXis usually small, higher order terms
δX^2 ,δX^3 , etc., become negligible and often ignored, leaving only
the first-order term.
Considering the species vectorX(¼X 1 ,X 2 ,...,Xn), and rate
of reaction vectorF(¼F 1 ,F 2 ,...,Fn), the Jacobian is
Table 1
Stability criteria
Eigenvalues Evaluated points Type
Real and negative Stable nodes Convergence at all conditions
Real and positive Unstable nodes Divergence at all conditions
Mixed positive and negative real parts Saddle points Mixed or asymptotic conditions
Complex numbers with negative real parts Stable focus Convergence at specific conditions
Complex numbers with positive real parts Unstable focus Divergence at specific conditions
Complex Biological Responses Using Simple Models 191