13th International Conference on Membrane Computing - MTA Sztaki

R. Pagliarini, O. Agrigoroaiei, G. Ciobanu, V. Manca
D F ruc6P = {AT P, AMP }, D Gluc6P = {AT P, AMP }, D F ruc16P 2 = {ADP },
D AT P = ∅, D AMP = ∅, D ADP = ∅.
After that, the set R composed by the rules reported in Table 1 has been
obtained by applying the procedure introduced in Section 4, and the transition
P system Π = (X, R) has been used as starting point to analyze quantitative
causality according with the approach described in Section 2.
r F ruc6P : F ruc6P → F ruc16P 2 + Gluc6P
r F ruc6P :
F ruc16P 2 + Gluc6P → F ruc6P
r Gluc6P : Gluc6P → F ruc6P
r Gluc6P : F ruc6P → Gluc6P
r F ruc16P 2 : F ruc16P 2 → F ruc6P
r F ruc16P 2 : F ruc6P → F ruc16P 2
r AT P = r AMP : AT P → AMP
r AT P = r AMP : AMP → AT P
r F ruc6P,AT P = r F ruc6P,AMP : F ruc6P
→ AT P + AMP
r Gluc6P,AT P = r Gluc6P,AMP : Gluc6P
→ AT P + AMP
r F ruc16P 2,ADP : F ruc16P 2 → ADP
Table 1. The set of rules modelling correlative causality of the yeast glycolytic network.
Let us consider v = AT P + AMP . We look for the possible causes for this
multiset, which corresponds to considering two time-series together. To find its
causes, we start by considering G = 0 R as a potential cause. Then we proceed by
adding rules to 0 R , one by one, until no more are needed to make v appear. More
details regarding this inductive procedure for finding the cause of a multiset can
be found in [2].
For G = 0 R the condition lhs(r) ≤ v\rhs(G) does not take place for r =
r AT P ; from the point of view of correlative causality, this corresponds to saying
that AT P is directly correlated with another time-series therefore it cannot have
an empty cause. We continue by adding to the (now discarded) potential cause
0 R rules r which have in the right hand side rhs(r) at least one common element
with v. The set of these rules is S = {r AT P , r AMP , r F ruc6P,AT P }. For G 1 = G+s,
s ∈ S\{r F ruc6P,AT P } the condition lhs(r) ≤ v\rhs(G 1 ) remains unfulfilled, since
either AT P or AMP will appear in v\rhs(G 1 ) and both of them are left hand
sides of rules. So we choose G 1 = r F ruc6P,AT P and it verifies that it is a cause
for v. To find the other causes, we look at the discarded causes with one element
(i.e., to the rules from S) and add one element from S to each of them, namely
we consider all the multisets with two elements of S. By checking all of them
we find that r AT P + r AMP is a cause for v. Note that r F ruc6P,AT P cannot be a
part of a cause with two elements since that rule alone is sufficient in producing
v. Moreover, there is no cause with more than two elements since having three
or more rules in G would mean that one of them does not contribute to the
appearance of the two elements of v. In the end, we have obtained that the
causes of v are r F ruc6P,AT P and r AT P + r AMP . This corresponds to the time-
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