A Boolean encoding of the Most Permissive semantics for Boolean networks

Abstract

Boolean networks are widely used to model biological regulatory networks and study their dynamics. Classical semantics, such as the asynchronous semantics, do not always accurately capture transient or asymptotic behaviors observed in quantitative models. To address this limitation, the Most Permissive semantics was introduced by Paulevé et al., extending Boolean dynamics with intermediate activity levels that allow components to transiently activate or inhibit their targets during transitions. In this work, we provide a Boolean encoding of the Most Permissive semantics: each component of the original network is represented by a triplet of Boolean variables, and we derive the extended logical function governing the resulting network. We prove that the asynchronous dynamics of the encoded network exactly reproduces the attainability properties of the original network under Most Permissive semantics. This encoding is implemented as a modifier within the bioLQM framework, making it directly compatible with existing tools such as GINsim. To address scalability limitations, we further extend the tool to support partial unfolding, restricted to a user-defined subset of components.

Figures (9)

• Figure 1: Example A. Left: logical rules. Right: associated regulatory graph.
• Figure 2: State transition graphs from configuration $111$ for Example A. Left: synchronous STG. Middle: asynchronous STG. Right: Generalized asynchronous STG.
• Figure 3: Left: Schema of transitions between levels for a Boolean variable. Right: Schema of transitions between levels for a variable in Most Permissive semantics. Black arrows indicate a guaranteed transition, green arrows indicate a transition under positive control, and red arrows indicate a transition under negative control.
• Figure 4: STG from initial configuration $111$ for Example A in Most Permissive semantics. Only Boolean configurations are shown. Configurations in red are those reachable with Most Permissive semantics, but not with classical semantics. Solid arrows represent transitions that are possible with classical semantics, while dotted arrows are those that pass through the increasing or decreasing level of Most Permissive.
• Figure 5: Encoding with three Boolean variables of the Most Permissive semantics. Green and red edges represent controlled transitions, black edges are guaranteed transitions.