{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T06:20:48Z","timestamp":1768630848457,"version":"3.49.0"},"reference-count":49,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T00:00:00Z","timestamp":1674777600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation","award":["1951583"],"award-info":[{"award-number":["1951583"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Early embryonic development involves forming all specialized cells from a fluid-like mass of identical stem cells. The differentiation process consists of a series of symmetry-breaking events, starting from a high-symmetry state (stem cells) to a low-symmetry state (specialized cells). This scenario closely resembles phase transitions in statistical mechanics. To theoretically study this hypothesis, we model embryonic stem cell (ESC) populations through a coupled Boolean network (BN) model. The interaction is applied using a multilayer Ising model that considers paracrine and autocrine signaling, along with external interventions. It is demonstrated that cell-to-cell variability can be interpreted as a mixture of steady-state probability distributions. Simulations have revealed that such models can undergo a series of first- and second-order phase transitions as a function of the system parameters that describe gene expression noise and interaction strengths. These phase transitions result in spontaneous symmetry-breaking events that generate new types of cells characterized by various steady-state distributions. Coupled BNs have also been shown to self-organize in states that allow spontaneous cell differentiation.<\/jats:p>","DOI":"10.3390\/e25020235","type":"journal-article","created":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T03:56:27Z","timestamp":1675050987000},"page":"235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Emergent Criticality in Coupled Boolean Networks"],"prefix":"10.3390","volume":"25","author":[{"given":"Chris","family":"Kang","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164, USA"}]},{"given":"Madelynn","family":"McElroy","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164, USA"},{"name":"Voiland School of Chemical Engineering and Bioengineering, Washington State University, Pullman, WA 99164, USA"}]},{"given":"Nikolaos K.","family":"Voulgarakis","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164, USA"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1038\/nrg1615","article-title":"Stochasticity in gene expression: From theories to phenotypes","volume":"6","author":"Elston","year":"2005","journal-title":"Nat. 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