On the number of asynchronous attractors in AND-NOT Boolean networks

2025-03-24

AI Summary

This study addresses the problem of establishing tight upper bounds on the number of attractors in asynchronous AND-NOT Boolean networks, aiming to characterize the interplay between long-term dynamical diversity and structural constraints. Methodologically, it introduces two novel graph-theoretic features—strong even cycles and dominating sets—and integrates them with strong connectivity analysis, even-cycle theory, and Boolean dynamical systems techniques. The result is two computationally tractable, tight upper bounds on the number of asynchronous attractors. These bounds break from prior conservative paradigms reliant solely on network size or topological diameter, achieving significantly improved precision and interpretability. The contributions provide new theoretical tools and computational foundations for attractor enumeration, controllability analysis of Boolean networks, and designability modeling in synthetic biology.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Study asynchronous attractors in AND-NOT Boolean networks
• Derive upper bounds for attractor count using structural properties
• Enhance understanding of dynamics for network design and control

Innovation

• Derives upper bounds for asynchronous attractors
• Analyzes AND-NOT Boolean networks structurally
• Uses strong even cycles and dominating sets