Computational Methods in Systems Biology

Temporal Reprogramming of Boolean Networks 187

c 1

tc1

c 2

p 0 p 1

t 1 , 1

(^11)
(^10)
t 2 , 0
(^2120)
t 3 , 0
(^31)
(^30)
(^31)
lock3 0 lock3 1
t 3 , 1 ′
p 0 p 1
t 3 , 1
t′¬ 3 ,{¬x 1 }
(^1030)
Fig. 4.(top) Excerpt of the encoding of temporary perturbations. (bottom) Excerpt
of the encoding of permanent perturbations.
Definition 6.Given a Boolean network f of dimension n, the Petri net
(P, T, A, M 0 )modelling itskpermanent perturbations is given by

• P=Pf∪{p 0 ,p 1 ,c 1 ,...,ck}∪

⋃

i∈[n]{locki^0 ,locki^1 }

• TandAare the smallest sets which satisfy
  BN transitions∀tl,c∈Tf,withl=iorl=¬i,i∈[n],
  t′l,c∈Twith•t′l,c=•tl,c∪{locki 0 }andt′l,c•=tl,c•∪{locki 0 }
  Perturbation transitionsfori∈[n],
  ti, 0 ∈Twith•ti, 0 ={i 1 ,p 0 ,locki 0 }andti, 0 • ={i 0 ,p 1 ,locki 1 }
  ti, 0 ′∈Twith•ti, 0 ′={i 0 ,p 0 ,locki 0 }andti, 0 ′•={i 0 ,p 1 ,locki 1 }
  ti, 1 ∈Twith•ti, 1 ={i 0 ,p 0 ,locki 0 }andti, 1 • ={i 1 ,p 1 ,locki 1 }
  ti, 1 ′∈Twith•ti, 1 ′={i 1 ,p 0 ,locki 0 }andti, 1 ′•={i 1 ,p 1 ,locki 1 }
  Perturbation enablingforj∈[k−1],
  tcj∈Twith•tcj={p 1 ,cj}andtcj•={p 0 ,cj+1}