On the number of asynchronous attractors in AND-NOT Boolean networks
Van-Giang Trinh, Samuel Pastva, Jordan Rozum, Kyu Hyong Park, Réka Albert
TL;DR
This paper investigates the number of asynchronous attractors in AND-NOT Boolean networks under asynchronous update, establishing two new upper bounds based on the structure of the global influence graph. The first bound requires a set $U$ that intersects every strong even cycle of $G(f)$, yielding at most $2^{|U|}$ attractors; the second bound uses a dominating set of $G(f)$ to obtain the same type of exponential bound. The proofs leverage a percolation-based reduction and the concept of delocalizing triples to show that breaking all strong even cycles collapses multi-stability, with a key intermediate result showing that absence of strong even cycles implies a unique attractor. Examples from the ERBB receptor network demonstrate the tightened bounds compared to prior results, and the framework connects to logic programs and abstract argumentation through encoded Boolean networks. Overall, the results advance attractor counting for AND-NOT BNs and have potential implications for network design and control in biology and symbolic AI.
Abstract
Boolean Networks (BNs) describe the time evolution of binary states using logic functions on the nodes of a network. They are fundamental models for complex discrete dynamical systems, with applications in various areas of science and engineering, and especially in systems biology. A key aspect of the dynamical behavior of BNs is the number of attractors, which determines the diversity of long-term system trajectories. Due to the noisy nature and incomplete characterization of biological systems, a stochastic asynchronous update scheme is often more appropriate than the deterministic synchronous one. AND-NOT BNs, whose logic functions are the conjunction of literals, are an important subclass of BNs because of their structural simplicity and their usefulness in analyzing biological systems for which the only information available is a collection of interactions among components. In this paper, we establish new theoretical results regarding asynchronous attractors in AND-NOT BNs. We derive two new upper bounds for the number of asynchronous attractors in an AND-NOT BN based on structural properties (strong even cycles and dominating sets, respectively) of the AND-NOT BN. These findings contribute to a more comprehensive understanding of asynchronous dynamics in AND-NOT BNs, with implications for attractor enumeration and counting, as well as for network design and control.