Computational Methods in Systems Biology

Graph Representations of Monotonic Boolean Model Pools 239

points in the sequence the sign of the solution derivative is then constant. We
are now interested in the sequence of those signs. Formally, we get the following
definition.

Definition 4 ([ 4 , p. 23]).Forasolutionx∈E, consider the ordered sequence
(tj)in[0,T]consisting of 0 and all boundary points of the closure of all sets
{t∈[0,T]

∣

∣[ ̇x(t)] =v}withv∈{− 1 , 1 }n. A sequence(τj)s.t.τj∈(tj,tj+1)

gives rise to the sequencex ̃=( ̃xj):=([ ̇x(τj)])which is calledabstractionof
x(·). We denote theset of abstractionsof the solutions of a monotonic ensemble
M(Σ)by

S ̃M(Σ):={x ̃| ̃xis the abstraction ofx(·)for somex∈SM(Σ)}.

To illustrate the notion, we extract the abstraction for a solution of the running
example.

Example 3.Consider the solution of the ODE depicted in Fig. 2. Its abstraction
is given by ((− 1 ,− 1 ,− 1 ,−1),(1,− 1 ,− 1 ,−1),(1, 1 ,− 1 ,−1)), since on the begin-
ning of the trajectory all components are decreasing. Then the first component
starts increasing followed by the second one.

Based on the abstractions a state transition graph can now be constructed. The
states correspond to the signs of ̇x(·) and edges indicate subsequent sign vectors
in some abstraction.

Definition 5 ([ 4 , p. 23]).Thedirected state transition graphGQDE(Σ)ofthe
monotonic ensembleis defined by the vertex set

VQDE(Σ):={− 1 , 1 }n,

calledqualitative states, and the edge set

EQDE(Σ):={(v, w)|∃ ̃x∈S ̃M(Σ),j∈N: ̃xj=vand ̃xj+1=w},

calledqualitative transitions.

In the following, we indicate the edge relation with→.

Example 4.The state transition graph of our running example is depicted
in Fig. 3. We see that we can find the trajectory (− 1 ,− 1 ,− 1 ,−1) →
(1,− 1 ,− 1 ,−1)→(1, 1 ,− 1 ,−1) of Example 3 in this graph.

Naturally, this graph would not be very helpful in application if we would
need to solve all ODEs in the ensemble to construct it. The following proposition
constitutes a different approach. It basically says that a change in sign of a
componentimust be caused by a consistent dependency on a componentjin
the right hand side functionf, as captured in thei, j-th entryσijof the sign
matrixΣ.