Graph Representations of Monotonic Boolean Model Pools 247Construction of the graphGBooleanQDE (Σ) is easily done based on information
encoded inΣ. At first glance the computational cost for the construction using
Condition ( 4 ) is not cheap. However, in application interaction graphs are often
rather sparse and the condition need only be tested for non-zero entries ofΣ.
Interpretation of the information encoded in the ESTG for a particular func-
tion in the ensemble is hampered by the fact that the nodes represent sets of
preimages of the node state. Since the possibility of these sets being empty is
not excluded, information can not always be transferred in a straight-forward
manner. For this problem, we want to explore two aspects in future research.
First, we would like to clarify whether the existence of an edge in the ESTG
implies the existence of an ensemble functionfwith corresponding edges in its
ASTG. In the QDE setting this statement is true. Second, we want to investigate
which subgraphs ofGBooleanQDE (Σ) arise as quotient graphs of Boolean functions
f∈MB(Σ) and, related, derive constraints from ensemble graph edges for the
models that exhibit them.
In the last section, we shortly touched upon the connection the QDE frame-
work offers between ODE and Boolean models. In the constraint-based view
adopted here, we saw that the ensemble graphsGQDE(Σ)andGBooleanQDE (Σ)are
the same. The interpretation of the nodes as signs of change gives us an alter-
native way of interpreting the Boolean states of the graphGasync(f)
/
fin a
continuous setting, based on differences rather than absolute values, as is often
more natural when processing experimental data. The fact that graphsGQDE(Σ)
andGBooleanQDE (Σ) are identical, but carry different meaning, could in future lead
to new network inference algorithms for predicting edges inΣ, based on network
inference algorithms for Boolean functions. Since QDEs and ODEs are intrin-
sically related, such predictions could be very robust. Beyond this aspect, we
would like to investigate ways to assign a family of ODE systems to a Boolean
functionf∈MB(Σ) such that the subgraph ofGQDE(Σ) induced by this family
via their set of abstractionsS ̃M(Σ)resembles the quotient graphGasync(f)
/
f
more closely. This could provide approaches for validating Boolean models tak-
ing dynamics on a higher resolution level consistent with the qualitative model
properties into account.
Lastly, our results raise the question in how far logic constrains obtained from
different abstractions than Proposition 1 lead to similar results [ 1 ]. For many
biological applications it could make sense to consider a more restricted class of
ODE-systems. For example in [ 5 ] ODE-systems (models of reaction networks)
stemming from different kinetic rate laws are abstracted. It would be interesting
to see in how far such results are comparable to QDEs in general and in how far
they are related to the results in this paper.
References
- Abou-Jaoud ́e, W., Thieffry, D., Feret, J.: Formal derivation of qualitative dynam-
ical models from biochemical networks. Biosystems 149 , 70–112 (2016)