Computational Methods in Systems Biology

236 R. Schwieger and H. Siebert

In addition to the (local) interaction graph(s), we attribute tofa second
graph, which describes the dynamics of the components. Here, we use an asyn-
chronous update scheme attributing tofanasynchronous state transitiongraph
Gasync(f)=(Vasync(f),Easync(f)) withVasync(f):={ 0 , 1 }nand

Easync(f)=

{

(s, t)∈Vasync(f)×Vasync(f)|

(

diff(s, t)={i}andfi(s)=ti

)

ors=t=f(s)

}

.

Example 1.Letf(x 1 ,x 2 ,x 3 ,x 4 )=(1−x 4 ,x 1 ,x 2 ·(1−x 4 ), 1 −x 3 ) be a Boolean
function. Its global interaction graph and asynchronous state transition graph is
depicted in Fig. 1. The global interaction graph is given by the adjacency matrix

Σ=

⎛

⎜

⎜

⎝

000 −

+000

0+0−

00 − 0

⎞

⎟

⎟

⎠,

x1

x2

x3

x4

0000

0001

1000

0010

1010

0011

0100

0101

0110

1100

1110

0111

1001

1101

1011

1111

Fig. 1.Left: Global interaction graph of the running example. Right: ASTG of the
running example.

2.2 Monotonic Ensembles and Qualitative Differential Equations

The motivation of our results on sets of Boolean models comes from approaches
to analyze families of ODE models ̇x=f(x) which share some qualitative prop-
erties. Mainly, these are sign constraints on the Jacobi matrix of the right hand
sides of the ODE-System. Instead of the solutionsx(·) of the ODE-systems,
so-called “abstractions” are considered. In the context of this paper, these
abstractions are sequences of sign vectors of the derivatives of the solutions.
A state transition graph on the sign vectors can be constructed based on
the sign matrix, which captures restrictions on the behavior of the solutions.