Attractors are less stable than their basins: Canalization creates a coherence gap in gene regulatory networks

Abstract

Waddingtons epigenetic landscape has served as biologys central metaphor for cellular differentiation for over half a century, depicting mature cell types as balls resting in stable valley floors. Boolean networks - introduced by Kauffman in 1969 to model gene regulatory dynamics - provide a mathematical formalization of this landscape, where attractors represent phenotypes and basins of attraction correspond to developmental valleys. Traditional stability measures quantify robustness by perturbing arbitrary states, yet biological systems typically reside at attractors rather than in transient states. Here we formalize and systematically analyze attractor coherence - a stability measure Kauffman originally envisioned but never rigorously developed - which quantifies how likely a perturbation of an attractor state causes phenotype switching. Analyzing 122 expertcurated biological Boolean models, we reveal a striking paradox: attractors representing mature cell types are consistently less stable than the developmental trajectories approaching them. Largescale simulations of random networks demonstrate that this coherence gap arises from canalization - a hallmark of biological regulation where individual genes can override others. While canalization increases overall network stability, it disproportionately stabilizes transient states, positioning attractors near basin boundaries. The gaps magnitude is almost perfectly predicted by network bias (Spearmans{rho} = -0.997), itself modulated by canalization. These findings revise Waddingtons landscape: canalization carves deep protective valleys ensuring developmental robustness, yet simultaneously flattens ridges near valley floors, facilitating phenotypic plasticity when multiple fates coexist. This explains how biological systems achieve both reliable development and plasticity, with implications for understanding development, disease-related transitions, and designing robust yet controllable synthetic gene circuits.