Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 57

The stability of the steady states obtains by studying the Jacobian matrix. In this
example, it is clear that the equation of each nullcline, f i ( x 1 ,x 2 )=0, implicitly
defines a function x 2 = f i ( x 1 ). Thus the derivatives of f i and f i are related by the
Implicit Function Theorem:

d f i
dx 1

0= ∂f i
∂x 1

+ ∂f i
∂x 2

dx 2
dx 1

= ∂f i
∂x 1

+ ∂f i
∂x 2

. (2.18)

The Jacobian matrix and its determinant can thus be written:

⎛
⎝ −

⎞

d f 2
dx 1

. (2.19)

d f 1
dx 1

∂f 1
∂x 2

∂f 1
∂x 2

d f 1
dx 1

∂f 1
∂x 2

∂f 2
∂x 2

⎠

J ( x )= and det( J )= −

d f 2
dx 1

∂f 2
∂x 2

∂f 2
∂x 2

−

Therefore, its trace and determinant at a steady state x ∗ are

tr( J ∗ )= − ( γ 1 + γ 2 ) ,

d f 2
dx 1

,

d f 1
dx 1

m 2 θ m 2

( x 2 ) m 2 − 1

det( J ∗ )= γ 2 κ 11

2

( x ∗ ) − ( x ∗ ) (2.20)

( θ m 2
2

+( x 2 ) m 2 ) 2

where d f i /dx 1 ( x ∗ ) denote the slope of the curves f i at x ∗ . It is clear that the trace
is always negative. For the steady states near one of the axis (one of the proteins at
low concentration), it holds that 0 >df 2 /dx 1 ( x ∗ ) >df 1 /dx 1 ( x ∗ ), and therefore
the determinant is positive - these are stable steady states. The middle steady state
is unstable, since the opposite inequality holds and the determinant is negative.
This example is also known as the bistable switch , as only an external stimulus
can force the system to evolve, or switch, from one steady state to the other (see
discussion on Sect. 2.2.4 ).

2.2.3.3 Piecewise Affine Systems for Genetic Network Models

As seen above, the analysis of the dynamics of a dynamical system described by
differential equations can be quite complicated in dimension greater than two. We
are looking for a more algorithmic approach, easily implementable on a computer.
We will consider a qualitative description of the bistable switch, corresponding to
the case m i →∞

where sigmoidal functions h − become step functions. This is an
approximation of the “real” system, done for an easier comprehension.
The formalism is as described in Sect. 2.2.2.4 . The functions f i now represent
the dependence of the rate of synthesis of a protein encoded by gene i on the
concentrations x j of the other proteins in the cell. The term γ i x i represents the
degradation rate of protein x i . The functions f i : R

n

+ → R + can be written as

f i ( x )=
l∈I

κ il b il ( x ) , (2.21)