Computational Methods in Systems Biology

Abduction Based Drug Target Discovery Using Boolean Control Network 61

Aninteraction graph〈X, 〉portrays the causal interactions between vari-
ables of a Boolean network (cf., Fig. 1 ). An interactionxi xjexists if and only
ifxioccurs as literal in a minimaldnfform offj,ie.,

xi xj

def

==xi∈V(dnf(fj)).

2.3 Boolean Control Network

Boolean Control Network (BCN) extends Boolean network by addingcontrol
parameters that are Boolean variables,ui ∈U without equation definition.
Hence, a BCN is defined as a function generating Boolean network parame-
trized by an interpretation of control parametersμ∈SU, called acontrol input:
Fu:SU →(SX→SX). For example, an extension of the Boolean network in
Fig. 1 to a BCN by adding four control parametersu 1 ,u 2 ,u 3 ,u 4 is:

Fu 1 ,u 2 ,u 3 ,u 4 =

⎧

⎨

⎩

x 1 =(x 2 ∧u 1 )∨x 3 ,

x 2 =¬(x 3 ∨¬u 2 ),

x 3 =((¬x 2 ∧x 1 )∨¬u 3 )∧u 4

(2)

The application of a control inputμto a Boolean control networkFμtherefore
reprograms the dynamics. Figure 2 describes the dynamics resulting from the
application^4 of two control inputsμ 1 ={u 1 =0,u 2 =1,u 3 =1,u 4 =1}and
μ 2 ={u 1 =1,u 2 =1,u 3 =1,u 4 =0}.

Fμ 1 =

⎧

⎪⎨

⎪⎩

x 1 =x 3 ,

x 2 =¬x 3 ,

x 3 =¬x 2 ∧x 1

Fμ 2 =

⎧

⎪⎨

⎪⎩

x 1 =x 2 ∨x 3 ,

x 2 =¬x 3 ,

x 3 =1

101 010

000 001

100 011

110 111

x 2

x 3

x^1

x 2

x 3

x^1

x^1

x 3

x 2

x^1

x 2

x 3 110

000 001

100 101 010 011

111

x 2

x 3

x^1

x^1

x 2

x 3

x^1

x^1

x 2

x 3

x 2

x 3

μ 1 ={u 1 =0,u 2 =1,u 3 =1,u 4 =1} μ 2 ={u 1 =1,u 2 =1,u 3 =1,u 4 =0}

Fig. 2.Modification of the dynamics by control inputs for the example of Fig. 1.

(^4) The formulas resulting from the instantiation of the BCN by a control input are
simplified.