Samuel Pastva
A Boolean network is consistent with a signed interaction graph, \(BN \models IG\), when dependencies of \(F_i\) agree with dependencies of \(v_i\).
Many different functions can be consistent with the same influence graph:
...
Not all influences within the IG have to be used by the BN:
Monotonic model pool \(M(IG)\): All Boolean networks \(BN\) s.t. \(BN \models IG\).
Which properties appear due to the influence graph and which are due to the choice of update functions?
Schwieger, Robert, and Heike Siebert. "Structure and behavior in Boolean monotonic model pools." Biosystems 214 (2022): 104610.
2 419 200 networks in the monotonic pool.
No complex attractor. Only fixed-points.
Why?
Paulevé, Loïc, and Adrien Richard. "Static analysis of Boolean networks based on interaction graphs: a survey." Electronic Notes in Theoretical Computer Science 284 (2012): 93-104.
A Boolean network \(\Gamma(IG)\) consisting of functions \(\Gamma_1, \ldots, \Gamma_n\) where:
Schwieger, Robert, and Heike Siebert. "Structure and behavior in Boolean monotonic model pools." Biosystems 214 (2022): 104610.
A transition exists in \(ASTG(\Gamma)\) if \(x_i \nLeftrightarrow x_j\) (for positive dependency) or \(x_i \Leftrightarrow x_j\) (for negative dependency).
Original: The \(ASTG(\Gamma)\) is the union of the quotient graphs of every \(BN \in M(IG)\).
New: The \(ASTG(\Gamma)\) is the union of the \(ASTG(BN)\) for every \(BN \in M_{[!]}(IG)\) (a strict monotonic pool).
Strict pool: If a variable has at least one dependency, it cannot be constant (true/false).
Let BN be an arbitrary network from \(M_{[!]}(IG)\). Corollary:
The approximation is still very broad: For example, every acyclic \(IG\) produces an acyclic \(ASTG\). However, \(ASTG(\Gamma)\) of such \(IG\) often contains cycles.
Skeleton network is a "normal" BN that can be processed using existing tools and algorithms (AEON, GINsim, pystablemotifs, ... ).
In this case, Python interface of AEON is used to run analysis with BDD-based symbolic representation.
The skeleton network has two fixed-point attractors.
These are guaranteed to appear in every network of the (strict) monotonic pool.
For some initial conditions, the network variable never changes its value.
By fixing a (minimal) subset of network variables, all variables become observably stable.
Performing such intervention introduces stability to otherwise unstable variables.
Variable \(v_i\) is observably stable when \(TrapSet(\{ x \in \mathbb{B}^n \mid x_i = true \}) \not= \emptyset\) or \(TrapSet(\{ x \in \mathbb{B}^n \mid x_i = false \}) \not= \emptyset\).
\(TrapSet\): Largest subset that cannot be escaped.
Skeleton network with logical intervention parameters \(p_i\):
Feasible interventions: Projection of \(TrapSet\) results to intervention parameters.
https://github.com/sybila/biodivine-boolean-models
145 real-world Boolean networks from various sources: