Computational Methods in Systems Biology

Graph Representations of Monotonic Boolean Model Pools 235

2.1 Boolean Networks

We consider an arbitrary Boolean functionf:{ 0 , 1 }n→{ 0 , 1 }n,n∈Ncap-
turing the dynamics ofninteracting components represented by 0/1 variables.

Definition 1.Thediscrete derivative of the functionf:{ 0 , 1 }n→{ 0 , 1 }nis
defined by

(∂jfi)(x):=

fi(x⊕ej)−fi(x)

(xj⊕1)−xj

∈{− 1 , 0 , 1 },

where( ⊕is the addition modulo 2. Furthermore, we denote with∇fithe vector
∂ 1 fi,..., ∂nfi

)t

.

As is standard, we then derive an interaction graph from the derivatives that
captures dependencies between the components, either locally, i.e., in a given
state, or globally summarizing all possible interactions between components.

Definition 2.Thelocal interaction graph IGf(x):=(V, E),x∈{ 0 , 1 }nof
a Boolean functionf:{ 0 , 1 }n→{ 0 , 1 }nconsists ofnverticesV:={ 1 ,...,n}
andasignededge-setE

(

IGf(x)

)

, which is defined as

(i, j, )∈E

(

IGf(x)

)

⇔(∂ifj)(x)=

with∈{− 1 , 1 }. We denote withIGglobal(f) the global interaction graph
defined as the union of all local interaction graphs, i.e.,

IGglobal(f)=

⋃

x∈{ 0 , 1 }n

IGf(x)

For convenience, we often identify an interaction graph with its signed adjacency
matrix.
In general, the global interaction graph can contain two edges with oppo-
site signs between two components. However, here we consider only func-
tionsf, which lead to interaction graphs with maximally one edge between
two components. That is, in the following we only consider Boolean functions
f:{ 0 , 1 }n→{ 0 , 1 }n, where

∀x, y∈{ 0 , 1 }n∀∈{ 1 ,− 1 }:(s, t, )∈E

(

IGf(x)

)

⇒(s, t,−)∈E

(

IGf(y)

)

holds. We call such functionsmonotonic. This should not be confused with
the notion of monotone functions as defined for example in [ 8 ], which is more
restrictive. The assumption poses no severe restriction for application since most
models of bioregulatory systems lack parallel edges of different signs (cf. model
repositories as e.g. for PyBoolNet [ 7 ]^1 ).

(^1) https://github.com/hklarner/PyBoolNet/tree/master/PyBoolNet/Repository.