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by claude@2026-07, 2026-07-04
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The paper studies stability in Boolean gene regulatory network models by formalizing “attractor coherence,” which measures how often perturbing an attractor state leads to phenotype switching, rather than relying only on robustness measures based on arbitrary transient states. By analyzing 122 expert-curated Boolean network models and performing large-scale simulations of random networks, the authors find a paradoxical “coherence gap”: attractors representing mature cell types are less stable than the trajectories that approach them. They attribute this to canalization, where increased gene-level control stabilizes transient developmental states more than attractors, effectively placing attractors closer to basin boundaries. A key caveat is that the conclusions depend on the structure and assumptions of Boolean network representations and stability is assessed through simulated perturbations. The paper does not explicitly discuss endometriosis or adenomyosis; it was included in the corpus via a keyword match in the upstream search index.
Abstract
Waddington’s epigenetic landscape has served as biology’s 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 gap’s magnitude is almost perfectly predicted by network bias (Spearman’s ρ = -0.997), itself modulated by canalization. These findings revise Waddington’s 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.
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Abstract
Waddington’s epigenetic landscape has served as biology’s 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 gap’s magnitude is almost perfectly predicted by network bias (Spearman’s ρ = -0.997), itself modulated by canalization. These findings revise Waddington’s 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.
Competing Interest Statement
The authors have declared no competing interest.
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