13th International Conference on Membrane Computing - MTA Sztaki

An analysis of correlative and quantitative causality in P systems
2. { r i : α i → x i
}
, for each xi ∈ X such that C xi ≠ ∅, where α i is equal to the
one introduced in the previous point;
3. {r ij : x i → β j }, for each x i ∈ X and each x j ∈ D xi such that D xi ≠ ∅, where
β j is the multiset corresponding to the set {x j } ∪ C xj , with multiplicity one
for each element.
Note that there can be rules which have different labels but are identical, for
example if C x = {y} then r x = r y and r x = r y .
By applying the above procedure to preliminary case studies, we inferred the
network topology of the reactions and the regulative mechanisms from time-series
of reaction models modelling synthetic metabolic phenomena. In particular, since
experiments conducted under identical conditions do not necessarily lead to identical
results, we also focused on different factors causing this variability, such as,
enzymatic variability, intrinsic variability, and environmental variability 2 . This
is due to the fact that the rules that constitute the set R reflect the meanings of
the statistical indexes that we introduced in Section 3. The rules introduced at
the first two points represent the fact that correlation and cross-correlation do
not give information about the direction of cause-effect interactions, and then we
have to consider both the verses of the possible relationships. From a biological
point of view, these types of cause-effects interactions can be the result of regulative
mechanisms governing the behaviours of the system under investigation.
Differently, the rules introduced at the third point model the causality relationships
due both to the biological network topology and regulative mechanisms.
This combination induces dynamic changes and variability in species concentrations
which have inherent delays, giving us knowledge about the existence of
directed relations of cause-effect.
In a second phase, the sets X and R are used for building up a model useful to
associate correlative causality and quantitative causality by means of membrane
systems, namely a transition P system Π = (X, R, u 0 ) with one membrane.
Using it we can analyze the situations which lead to at least one x i to appear
and compare them to their correlative counterpart.
Proposition 1. Consider x i ∈ X. Then any possible cause for x i is either 0 R
(the empty multiset of rules) or it is a multiset r with just one element.
Proof. We show that any cause G for the multiset x i in Π has at most one
element. Suppose that G has at least two elements. From Definition 1 we have
that rhs(G) ∩ x i > rhs(G − r) ∩ x i for any r ∈ G. By the definition of ∩,
rhs(G) ∩ x i is either x i or 0. From the previous inequality, rhs(G) cannot be 0;
thus rhs(G) = x i . Hence x i is an element of the right hand side of some rule
2 To mimic enzymatic variability a random variation of approximately ±10% has been
introduced by multiplying each metabolic flux values with a random number from a
normal distribution with unit mean and 0.05 standard deviation. To induce intrinsic
variability we add a stochastic term to each substance of the system. This term is a
random number from unit Normal distribution. In order to generate data subject to
environmental variability, we add a stochastic term only to the flux associated with
the reactions which introduce matter in the system.
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