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.