Dynamics of attractor transitions in Boolean networks under noise

TL;DR

This paper addresses how persistent stochastic perturbations drive transitions between attractors in Boolean networks. It introduces a Markov-chain-based framework over attractors with a transition matrix $M$ derived from local single-node flips and contrasts this with global randomization, yielding distinct dynamical patterns. The key contributions are (i) that the leading eigenvector $v$ of $M$ predicts stationary attractor occupancy with high accuracy, (ii) a quantitative study of attractor stability via $ ext{Tr}(M)$ and a lower bound on one-step retention, and (iii) entropy-based metrics showing how exploration of attractor space changes with network sensitivity $s$, validated on random regular networks and real-world Cell Collective networks. The findings illuminate how noise sculpt attractor landscapes, offering a practical tool for assessing robustness of biological networks and guiding interpretation of attractor dynamics under stochastic perturbations.

Abstract

Biological systems operate under persistent noise, which can alter system states and induce transitions between attractors. Here, we study the attractor dynamics of Boolean networks focusing on the transitions between attractors induced by noise. By computing transition probabilities between attractors, we present methods at the attractor level to determine dominance, stability, and diversity of attractors, and systematically compare local and global noise. Whereas global noise leads to attractor behavior dictated primarily by basin sizes, local noise produces structured transition patterns characterized by enhanced stability, non-trivial dominance patterns, and broader exploration of the attractor space. Our work offers insight into the dynamics of attractors, showing the importance of transition patterns under noise.

Figures (5)

• Figure 1: (a) Wiring diagram of a Boolean network and its Boolean function are shown. (b) State space of the Boolean network and its attractors are identified. A random flip of a node due to local noise results in the transition of the state $(0,0,0)$, as illustrated in the diagram. (c) The probabilities of remaining within the same attractor or transitioning to a different attractor due to local noise are illustrated.
• Figure 2: Comparison between the empirically measured time fractions spent in attractors, $t_\alpha/t_{\text{tot}}$, and the principal eigenvector components $v_{\alpha}$ of the transition matrix is shown for random regular networks with $N = 20$ and $k = 2,5$. Panels (a,c) show comparisons with eigenvector predictions, while panels (b,d) show comparisons with basin sizes.
• Figure 3: (a) The trace $\mathrm{Tr}(M)$ of transition matrix for random Boolean networks on random regular networks with size $N=20$ and various degrees $k$ with respect to the sensitivity $s$ is shown. (b) The trace normalized by the number of attractors, $\mathrm{Tr}(M)/n_{\mathrm{att}}$ is shown. Inset shows $\mathrm{Tr}(M)/n_{\mathrm{att}}$ as a function of size $N$ for $(k,s)=(2,0.75)$ for squares and $(k,s)=(4,1.5)$ for triangles.
• Figure 4: (a) Entropy $H$ of the principal eigenvector as a function of network sensitivity $s$, (b) the difference $\Delta H$ between entropy from basin sizes and from the eigenvector distribution, and (c) the ratio $R_>/R_<$ comparing transition bias toward larger versus smaller basins. All results are obtained on random regular networks with $N = 20$ and various values of $k$.
• Figure 5: (a) The trace of the transition matrix, $\mathrm{Tr}(M)$, is shown in descending order for real-world Boolean networks. (b) A comparison between the principal eigenvector components $v_{\alpha}$ and basin sizes $b_{\alpha}$ for all attractors across the real-world networks is presented.