  In systems biology, Boolean networks (BNs) aim at modeling the qualitative
dynamics of quantitative biological systems. Contrary to their (a)synchronous
interpretations, the Most Permissive (MP) interpretation guarantees capturing
all the trajectories of any quantitative system compatible with the BN, without
additional parameters. Notably, the MP mode has the ability to capture
transitions related to the heterogeneity of time scales and concentration
scales in the abstracted quantitative system and which are not captured by
asynchronous modes. So far, the analysis of MPBNs has focused on Boolean
dynamical properties, such as the existence of particular trajectories or
attractors. This paper addresses the sampling of trajectories from MPBNs in
order to quantify the propensities of attractors reachable from a given initial
BN configuration. The computation of MP transitions from a configuration is
performed by iteratively discovering possible state changes. The number of
iterations is referred to as the permissive depth, where the first depth
corresponds to the asynchronous transitions. This permissive depth reflects the
potential concentration and time scales heterogeneity along the abstracted
quantitative process. The simulation of MPBNs is illustrated on several models
from the literature, on which the depth parametrization can help to assess the
robustness of predictions on attractor propensities changes triggered by model
perturbations.
