{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T12:07:51Z","timestamp":1791288471019,"version":"4.3.0"},"reference-count":81,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,9,1]],"date-time":"2020-09-01T00:00:00Z","timestamp":1598918400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,9,8]],"date-time":"2020-09-08T00:00:00Z","timestamp":1599523200000},"content-version":"vor","delay-in-days":7,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"DFG","award":["GSC111"],"award-info":[{"award-number":["GSC111"]}]},{"name":"ERS","award":["CLS001"],"award-info":[{"award-number":["CLS001"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Math. Biol."],"published-print":{"date-parts":[[2020,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The central question of systems biology is to understand how individual components of a biological system such as genes or proteins cooperate in emerging phenotypes resulting in the evolution of diseases. As living cells are open systems in quasi-steady state type equilibrium in continuous exchange with their environment, computational techniques that have been successfully applied in statistical thermodynamics to describe phase transitions may provide new insights to the emerging behavior of biological systems. Here we systematically evaluate the translation of computational techniques from solid-state physics to network models that closely resemble biological networks and develop specific translational rules to tackle problems unique to living systems. We focus on logic models exhibiting only two states in each network node. Motivated by the apparent asymmetry between biological states where an entity exhibits boolean states i.e. is active or inactive, we present an adaptation of symmetric Ising model towards an asymmetric one fitting to living systems here referred to as the modified Ising model with gene-type spins. We analyze phase transitions by Monte Carlo simulations and propose a mean-field solution of a modified Ising model of a network type that closely resembles a real-world network, the Barab\u00e1si\u2013Albert model of scale-free networks. We show that asymmetric Ising models show similarities to symmetric Ising models with the external field and undergoes a discontinuous phase transition of the first-order and exhibits hysteresis. The simulation setup presented herein can be directly used for any biological network connectivity dataset and is also applicable for other networks that exhibit similar states of activity. The method proposed here is a general statistical method to deal with non-linear large scale models arising in the context of biological systems and is scalable to any network size.<\/jats:p>","DOI":"10.1007\/s00285-020-01518-6","type":"journal-article","created":{"date-parts":[[2020,9,8]],"date-time":"2020-09-08T04:03:20Z","timestamp":1599537800000},"page":"769-798","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["A modified Ising model of Barab\u00e1si\u2013Albert network with gene-type spins"],"prefix":"10.1007","volume":"81","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3355-0415","authenticated-orcid":false,"given":"Jeyashree","family":"Krishnan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Reza","family":"Torabi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andreas","family":"Schuppert","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Edoardo Di","family":"Napoli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,9,8]]},"reference":[{"issue":"1","key":"1518_CR1","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/S0006-3495(98)77774-4","volume":"74","author":"A Ahsan","year":"1998","unstructured":"Ahsan A, Rudnick J, Bruinsma R (1998) Elasticity theory of the B-DNA to S-DNA transition. Biophys J 74(1):132\u2013137","journal-title":"Biophys J"},{"key":"1518_CR2","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S0076-6879(08)04204-3","volume":"455","author":"T Aksel","year":"2009","unstructured":"Aksel T, Barrick D (2009) Analysis of repeat-protein folding using nearest-neighbor statistical mechanical models. Methods Enzymol 455:95\u2013125","journal-title":"Methods Enzymol"},{"key":"1518_CR3","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","volume":"74","author":"R Albert","year":"2002","unstructured":"Albert R, Barabasi A-L (2002) Statistical mechanics of complex networks Rev. Mod Phys 74:47","journal-title":"Mod Phys"},{"key":"1518_CR4","doi-asserted-by":"crossref","first-page":"066130","DOI":"10.1103\/PhysRevE.70.066130","volume":"70","author":"M Aldana","year":"2004","unstructured":"Aldana M, Larralde H (2004) Phase transitions in scale-free neural networks: departure from the standard mean-field universality class. Phys Rev E 70:066130","journal-title":"Phys Rev E"},{"key":"1518_CR5","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/S0378-4371(02)00740-9","volume":"310","author":"A Aleksiejuk","year":"2002","unstructured":"Aleksiejuk A, Holyst JA, Stauffer D (2002) Ferromagnetic phase transition in Barab\u00e1si-Albert networks. Phys A 310:260\u2013266","journal-title":"Phys A"},{"issue":"2010","key":"1518_CR6","first-page":"6717","volume":"2010","author":"LG Alexppoulos","year":"2010","unstructured":"Alexppoulos LG, Melas IN, Chairakaki AD, Saez-Rodriguez J, Mitsos A (2010) Construction of signaling pathways and identification of drug effects on the liver cancer cell HepG2. Conf Proc IEEE Eng Med Biol Soc 2010(2010):6717\u20136720","journal-title":"Conf Proc IEEE Eng Med Biol Soc"},{"key":"1518_CR7","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.jmb.2006.04.029","volume":"360","author":"S Balaji","year":"2006","unstructured":"Balaji S, Madan Babu M, Iyer L, Luscombe N, Aravind L (2006) Principles of combinatorial regulation in the transcriptional regulatory network of yeast. J Mol Biol 360:213","journal-title":"J Mol Biol"},{"issue":"2","key":"1518_CR8","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.physa.2005.08.034","volume":"362","author":"RH Baran","year":"2006","unstructured":"Baran RH, Ko H (2006) An Ising model of transcription polarity in bacterial chromosomes. Phys A Stat Mecha Appl 362(2):403\u2013422","journal-title":"Phys A Stat Mecha Appl"},{"key":"1518_CR9","first-page":"547","volume":"13","author":"A Barrat","year":"2000","unstructured":"Barrat A, Weigt M (2000) On the properties of small-world network models Eur. Phys J B 13:547","journal-title":"Phys J B"},{"key":"1518_CR10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/PhysRevB.73.224419","volume":"73","author":"M Bartolozzi","year":"2006","unstructured":"Bartolozzi M, Surungan T, Leinweber DB, Williams AG (2006) Spin-glass behavior of the antiferromagnetic Ising model on a scale-free network. Phys Rev B Condens Matter Mater Phys 73:1\u201319","journal-title":"Phys Rev B Condens Matter Mater Phys"},{"key":"1518_CR11","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/S0375-9601(02)01232-X","volume":"303","author":"G Bianconi","year":"2002","unstructured":"Bianconi G (2002) Mean field solution of the Ising model on a Barab\u00e1si-Albert network. Phys Lett A 303:166\u2013168","journal-title":"Phys Lett A"},{"key":"1518_CR12","doi-asserted-by":"crossref","unstructured":"Campbell C, Albert R (2014) Stabilization of perturbed Boolean network attractors through compensatory interactions BMC Systems Biology volume 8. Article number 53","DOI":"10.1186\/1752-0509-8-53"},{"key":"1518_CR13","doi-asserted-by":"crossref","unstructured":"Capitan JA, Cuesta JA (2010) Catastrophic regime shifts in model ecological communities are true phase transitions","DOI":"10.1088\/1742-5468\/2010\/10\/P10003"},{"issue":"2","key":"1518_CR14","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1103\/RevModPhys.81.591","volume":"81","author":"C Castellano","year":"2009","unstructured":"Castellano C, Fortunato S, Loreto V (2009) Statistical physics of social dynamics. Rev Mod Phys 81(2):591\u2013646","journal-title":"Rev Mod Phys"},{"key":"1518_CR15","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.3389\/fphar.2018.01011","volume":"9","author":"A Cesar-Razquin","year":"2018","unstructured":"Cesar-Razquin A, Girardi E, Yang M, Brehme M, Saez-Rodriguez J, Superti-Furga G (2018) In silico prioritization of transporter-drug relationships from drug sensitivity screens. Front Pharmacol 9:1101","journal-title":"Front Pharmacol"},{"key":"1518_CR16","doi-asserted-by":"crossref","unstructured":"Colliva A, Pellegrini R, Testori A, Caselle M (2014) Ising model description of long range correlations in DNA sequences","DOI":"10.1103\/PhysRevE.91.052703"},{"key":"1518_CR17","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s11135-007-9071-9","volume":"41","author":"P Contucci","year":"2007","unstructured":"Contucci P, Ghirlanda S (2007) Modeling society with statistical mechanics: an application to cultural contact and immigration. Qual Quant 41:569\u2013578","journal-title":"Qual Quant"},{"key":"1518_CR18","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1186\/1742-4682-8-30","volume":"8","author":"P Davies","year":"2011","unstructured":"Davies P, Demetrius L, Tuszynski JA (2011) Cancer as a dynamical phase transition. Theor Biol Med Model 8:30","journal-title":"Theor Biol Med Model"},{"key":"1518_CR19","doi-asserted-by":"crossref","first-page":"128301","DOI":"10.1103\/PhysRevLett.122.128301","volume":"122","author":"E DeGiuli","year":"2019","unstructured":"DeGiuli E (2019) Random language model. Phys Rev Lett 122:128301","journal-title":"Phys Rev Lett"},{"key":"1518_CR20","doi-asserted-by":"crossref","first-page":"016104","DOI":"10.1103\/PhysRevE.66.016104","volume":"66","author":"SN Dorogovtsev","year":"2002","unstructured":"Dorogovtsev SN, Godtsev AV, Mendes JFF (2002) Ising model on networks with an arbitrary distribution of connections. Phys Rev E 66:016104","journal-title":"Phys Rev E"},{"key":"1518_CR21","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1103\/RevModPhys.80.1275","volume":"80","author":"SN Dorogovtsev","year":"2008","unstructured":"Dorogovtsev SN, Godtsev AV, Mendes JFF (2008) Critical phenomena in complex networks. Rev Mod Phys 80:1275\u20131335","journal-title":"Rev Mod Phys"},{"key":"1518_CR22","first-page":"3","volume":"60","author":"R Durrett","year":"2013","unstructured":"Durrett R (2013) Cancer modeling: a personal perspective. Not AMS 60:3","journal-title":"Not AMS"},{"key":"1518_CR23","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1073\/pnas.1317876110","volume":"110","author":"MT Facciotti","year":"2013","unstructured":"Facciotti MT (2013) Thermodynamically inspired classifier for molecular phenotypes of health and disease. PNAS 110:48","journal-title":"PNAS"},{"key":"1518_CR24","doi-asserted-by":"crossref","unstructured":"Ferreira AL, Mendes JFF, Ostilli M (2010) First- and second-order phase transitions in Ising models on small world networks, simulations and comparison with an effective field theory","DOI":"10.1103\/PhysRevE.82.011141"},{"key":"1518_CR25","doi-asserted-by":"crossref","unstructured":"Gama-Castro S, Jacinto VJ, Peralta-Gill M, Santos-Zavaleta A, Pealoza-Spindola MI, Contreras-Moreira B, Segura-Salazar J, Muiz-Rascado L, Martinez-Flores I, Salgado H, Bonavides-Martinez C, Abreu-Goodger C, Rodriguez-Penagos C, MirandaRios J, Morett E, Merino E, Huerta AM, Collado-Vides J (2008) Regulondb (version 6.0): gene regulation model of escherichia coli k-12 beyond transcription, active (experimental) annotated promoters and textpresso navigation. Nucleic Acids Research 36, Database issue:D120\u20134","DOI":"10.1093\/nar\/gkm994"},{"issue":"5601","key":"1518_CR26","doi-asserted-by":"crossref","first-page":"2191","DOI":"10.1126\/science.1077809","volume":"298","author":"MM Garcia-Mira","year":"2002","unstructured":"Garcia-Mira MM, Sadqi M, Fischer N, Sanchez-Ruiz JM, Munoz V (2002) Experimental identification of downhill protein folding. Science 298(5601):2191\u20132195","journal-title":"Science"},{"key":"1518_CR27","doi-asserted-by":"crossref","first-page":"8373","DOI":"10.1088\/0305-4470\/33\/47\/304","volume":"33","author":"M Gitterman","year":"2000","unstructured":"Gitterman M (2000) Small-world phenomena in physics: the Ising model. J Phys A 33:8373","journal-title":"J Phys A"},{"key":"1518_CR28","volume-title":"Lectures on phase transition and the renormalization group","author":"N Goldenfeld","year":"1992","unstructured":"Goldenfeld N (1992) Lectures on phase transition and the renormalization group. Perseus Books, Massachusetts"},{"key":"1518_CR29","doi-asserted-by":"crossref","first-page":"066110","DOI":"10.1103\/PhysRevE.65.066110","volume":"65","author":"CP Herrero","year":"2002","unstructured":"Herrero CP (2002) Ising model in small-world networks. Phys Rev E 65:066110","journal-title":"Phys Rev E"},{"key":"1518_CR30","doi-asserted-by":"crossref","first-page":"041102","DOI":"10.1103\/PhysRevE.77.041102","volume":"77","author":"CP Herrero","year":"2008","unstructured":"Herrero CP (2008) Anti-ferromagnetic Ising model in small-world networks. Phys Rev E 77:041102","journal-title":"Phys Rev E"},{"key":"1518_CR31","doi-asserted-by":"crossref","first-page":"032717","DOI":"10.1103\/PhysRevE.87.032717","volume":"87","author":"D Holstein","year":"2013","unstructured":"Holstein D, Goltsv AV, Mendes JFF (2013) Impact of noise and damage on collective dynamics of scale-free neuronal networks. Phys Rev E 87:032717","journal-title":"Phys Rev E"},{"issue":"5","key":"1518_CR32","doi-asserted-by":"crossref","first-page":"2252","DOI":"10.1016\/S0006-3495(00)76472-1","volume":"79","author":"A Irback","year":"2000","unstructured":"Irback A, Sandelin E (2000) On hydrophobicity correlations in protein chains. Biophys J 79(5):2252\u20132258","journal-title":"Biophys J"},{"issue":"18","key":"1518_CR33","doi-asserted-by":"crossref","first-page":"9533","DOI":"10.1073\/pnas.93.18.9533","volume":"93","author":"A Irback","year":"1996","unstructured":"Irback A, Peterson C, Potthast F (1996) Evidence for nonrandom hydrophobicity structures in protein chains. Proc Natl Acad Sci 93(18):9533\u20139538","journal-title":"Proc Natl Acad Sci"},{"key":"1518_CR34","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/BF02980577","volume":"31","author":"E Ising","year":"1925","unstructured":"Ising E (1925) Beitrag zur Theorie des Ferromagnetisms. Z Phys 31:253","journal-title":"Z Phys"},{"key":"1518_CR35","doi-asserted-by":"crossref","first-page":"e0180937","DOI":"10.1371\/journal.pone.0180937","volume":"12","author":"B Jin","year":"2017","unstructured":"Jin B, Liu R, Hao S, Li Z, Zhu C, Zhou X (2017) Defining and characterizing the critical transition state prior to the type 2 diabetes disease. PLoS One 12:e0180937","journal-title":"PLoS One"},{"key":"1518_CR36","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511815881","volume-title":"Statistical physics of fields","author":"M Kardar","year":"2007","unstructured":"Kardar M (2007) Statistical physics of fields. Cambridge University Press, New York"},{"key":"1518_CR37","unstructured":"Krishnan J (2019) Modeling and simulations of complex networks in systems biology Ph.D. thesis, RWTH Aachen"},{"key":"1518_CR38","unstructured":"Krishnan J, Torabi R, Di Napoli E, Schuppert A (2018) Simulations of phase transitions in living systems. Systems Biology of Human Diseases LA, USA"},{"key":"1518_CR39","unstructured":"Krishnan J, Torabi R, Di Napoli E, Schuppert A (2019) A statistical mechanics perspective of phase transitions in living systems. SIAM Conference on Computational Science and Engineering (CSE) , Spokane, Washington, USA"},{"issue":"16","key":"1518_CR40","doi-asserted-by":"crossref","first-page":"6037","DOI":"10.1021\/ja500918k","volume":"136","author":"SG Kubelka","year":"2014","unstructured":"Kubelka SG, Kubelka J (2014) Site-specific thermodynamic stability and unfolding of a de Novo designed protein structural Motif Mapped by 13C isotopically edited IR spectroscopy. J Am Chem Soc 136(16):6037\u20136048","journal-title":"J Am Chem Soc"},{"issue":"48","key":"1518_CR41","doi-asserted-by":"crossref","first-page":"18655","DOI":"10.1073\/pnas.0808600105","volume":"105","author":"J Kubelka","year":"2004","unstructured":"Kubelka J, Henry ER, Cellmer T, Hofrichter J, Eaton WA (2004) Chemical, physical, and theoretical kinetics of an ultrafast folding protein. Proc Natl Acad Sci USA 105(48):18655\u201362","journal-title":"Proc Natl Acad Sci USA"},{"key":"1518_CR42","doi-asserted-by":"crossref","unstructured":"Kumar R, Raghavan P, Rajagopalan D, Sivakumar D, Tomkins A, Upfal E (2000) The Web as a graph.Proceeding of the 9th ACM Symposium on Principles of Database Systems","DOI":"10.1145\/335168.335170"},{"issue":"32","key":"1518_CR43","doi-asserted-by":"crossref","first-page":"9890","DOI":"10.1073\/pnas.1506309112","volume":"112","author":"JK Lai","year":"2015","unstructured":"Lai JK, Kubelka GS, Kubelka J (2015) Sequence, structure, and cooperativity in folding of elementary protein structural motifs. PNAS 112(32):9890\u20139895","journal-title":"PNAS"},{"key":"1518_CR44","volume-title":"Statistical physics","author":"LD Landau","year":"1980","unstructured":"Landau LD, Lifshitz EM (1980) Statistical physics. Pergamon press, Oxford"},{"key":"1518_CR45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1755-8794-6-1","volume":"6","author":"X Liu","year":"2013","unstructured":"Liu X, Liu R, Zhao XM, Chen L (2013) Detecting early-warning signals of type I diabetes and its leading biomolecular networks by dynamical network biomarkers. BMC Med Genomics 6:1\u201310","journal-title":"BMC Med Genomics"},{"issue":"3","key":"1518_CR46","doi-asserted-by":"crossref","first-page":"035004","DOI":"10.1088\/1478-3975\/8\/3\/035004","volume":"8","author":"MY Lobanov","year":"2011","unstructured":"Lobanov MY, Galzitskaya OV (2011) The Ising model for prediction of disordered residues from protein sequence alone. Phys Biol 8(3):035004","journal-title":"Phys Biol"},{"key":"1518_CR47","unstructured":"Lopes JV , Pogorelov YG, dos Santos JMBL (2004) Exact solution of Ising model on a small-world network. cond-mat\/0402138"},{"issue":"4","key":"1518_CR48","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1086\/323419","volume":"69","author":"J Majewski","year":"2001","unstructured":"Majewski J, Hao L, Ott J (2001) The Ising model in physics and statistical genetics. Am J Hum Genet 69(4):853\u2013862","journal-title":"Am J Hum Genet"},{"issue":"4","key":"1518_CR49","doi-asserted-by":"crossref","first-page":"e93616","DOI":"10.1371\/journal.pone.0093616","volume":"9","author":"D Marinazzo","year":"2014","unstructured":"Marinazzo D, Pellicoro M, Wu G, Angelini L, Cortes JM, Stramaglia S (2014) Information transfer and criticality in the Ising model on the human connectome. Plos One 9(4):e93616","journal-title":"Plos One"},{"key":"1518_CR50","doi-asserted-by":"crossref","first-page":"066112","DOI":"10.1103\/PhysRevE.64.066112","volume":"64","author":"RM May","year":"2001","unstructured":"May RM, Lloyd AL (2001) Infection dynamics on scale-free networks. Phys Rev Lett E 64:066112","journal-title":"Phys Rev Lett E"},{"key":"1518_CR51","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","volume":"21","author":"N Metropolis","year":"1953","unstructured":"Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087","journal-title":"J Chem Phys"},{"issue":"9","key":"1518_CR52","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1016\/j.jmb.2016.02.022","volume":"428","author":"C Millership","year":"2016","unstructured":"Millership C, Philips JJ, Main ERG (2016) Ising model reprogramming of a repeat protein\u2019s equilibrium unfolding pathway. J Mol Biol 428(9):1804\u20131817","journal-title":"J Mol Biol"},{"key":"1518_CR53","doi-asserted-by":"crossref","first-page":"e1000591","DOI":"10.1371\/journal.pcbi.1000591","volume":"5","author":"A Mitsos","year":"2009","unstructured":"Mitsos A, Melas IN, Siminelakis P, Chairakaki AD, Saez-Rodriguez J, Alexopoulos LG (2009) Identifying drug effects via pathway alterations using an integer linear programming optimization formulation on phosphoproteomic data. PLOS Comp. Biol 5:e1000591","journal-title":"PLOS Comp. Biol"},{"key":"1518_CR54","doi-asserted-by":"crossref","first-page":"e2000640","DOI":"10.1371\/journal.pbio.2000640","volume":"14","author":"M Mojtahedi","year":"2016","unstructured":"Mojtahedi M, Skupin A, Zhou J, Castano IG, Leong-Quong YR, Chang H, Trachana K, Giuliani A, Huang S (2016) Cell fate decision as high-dimensional critical state transition. PLoS Biol 14:e2000640","journal-title":"PLoS Biol"},{"issue":"20","key":"1518_CR55","doi-asserted-by":"crossref","first-page":"11311","DOI":"10.1073\/pnas.96.20.11311","volume":"96","author":"V Munoz","year":"1999","unstructured":"Munoz V, Eaton WA (1999) A simple model for calculating the kinetics of protein folding from three-dimensional structures. Proc Natl Acad Sci 96(20):11311\u20136","journal-title":"Proc Natl Acad Sci"},{"key":"1518_CR56","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1038\/36626","volume":"390","author":"V Munoz","year":"1997","unstructured":"Munoz V, Thompson PA, Hofrichter J, Eaton WA (1997) Folding dynamics and mechanism of $$\\beta $$-hairpin formation. Nature 390:196\u2013199","journal-title":"Nature"},{"issue":"30","key":"1518_CR57","doi-asserted-by":"crossref","first-page":"8982","DOI":"10.1021\/jp504261g","volume":"118","author":"AN Naganathan","year":"2014","unstructured":"Naganathan AN, Munoz V (2014) Thermodynamics of downhill folding: multi-probe analysis of PDD, a protein that folds over a marginal free energy barrier. J Phys Chem B 118(30):8982\u20138994","journal-title":"J Phys Chem B"},{"issue":"3\u20134","key":"1518_CR58","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1103\/PhysRev.65.117","volume":"65","author":"L Onsager","year":"1944","unstructured":"Onsager L (1944) Crystal statistics. I. a two-dimensional model with an order-disorder transition. Phys Rev 65(3\u20134):117\u2013149","journal-title":"Phys Rev"},{"key":"1518_CR59","doi-asserted-by":"crossref","first-page":"3200","DOI":"10.1103\/PhysRevLett.86.3200","volume":"86","author":"R Pastor-Satorras","year":"2001","unstructured":"Pastor-Satorras R, Vespignani A (2001) Epidemic spreading in scale-free networks. Phys Rev Lett E 86:3200","journal-title":"Phys Rev Lett E"},{"key":"1518_CR60","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1103\/RevModPhys.87.925","volume":"87","author":"R Pastor-Satorras","year":"2015","unstructured":"Pastor-Satorras R, Castellano C, Mieghem PV, Vespignani A (2015) Epidemic processes in complex networks. Rev Mod Phys 87:925\u2013979","journal-title":"Rev Mod Phys"},{"key":"1518_CR61","doi-asserted-by":"crossref","first-page":"057104","DOI":"10.1103\/PhysRevE.64.057104","volume":"64","author":"A Pekalski","year":"2001","unstructured":"Pekalski A (2001) Ising model on a small-world network. Phys Rev E 64:057104","journal-title":"Phys Rev E"},{"key":"1518_CR62","unstructured":"Redner S (2017) Dynamics of voter models on simple and complex networks"},{"issue":"6\u20137","key":"1518_CR63","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1007\/s00285-017-1136-3","volume":"75","author":"UA Rozikov","year":"2017","unstructured":"Rozikov UA (2017) Tree-hierarchy of DNA and distribution of holliday junctions. J Math Biol 75(6\u20137):1715\u20131733","journal-title":"J Math Biol"},{"key":"1518_CR64","first-page":"151","volume":"264","author":"UA Rozikov","year":"2018","unstructured":"Rozikov UA (2018) Holliday junctions for the potts model of DNA algebra, complex analysis and pluripotential theory. Springer Proc Math Stat 264:151\u2013165","journal-title":"Springer Proc Math Stat"},{"issue":"25","key":"1518_CR65","doi-asserted-by":"crossref","first-page":"4271","DOI":"10.1103\/PhysRevLett.71.4271","volume":"71","author":"M Sahimi","year":"1993","unstructured":"Sahimi M, Stauffer D (1993) Ising model above the upper critical dimension: an application to biology. Phys Rev Lett 71(25):4271\u20134273","journal-title":"Phys Rev Lett"},{"key":"1518_CR66","doi-asserted-by":"crossref","unstructured":"Santhanam N, Dingel J, Milenkovic O On modeling gene regulatory networks using Markov random fields. 2009 IEEE Information Theory Workshop on Networking and Information Theory","DOI":"10.1109\/ITWNIT.2009.5158562"},{"key":"1518_CR67","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1038\/35098000","volume":"413","author":"M Scheffer","year":"2001","unstructured":"Scheffer M, Carpenter S, Foley JA, Folke C, Walker B (2001) Catastrophic shifts in ecosystems. Nature 413:591\u2013596","journal-title":"Nature"},{"key":"1518_CR68","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1126\/science.1225244","volume":"338","author":"M Scheffer","year":"2012","unstructured":"Scheffer M, Carpenter S, Timothy L, Bascompte J, Brock W, Dakos V, van de Koppel J, van de Leemput IA, Levin SA, van Nes E, Pascual M, Vandermeer J (2012) Anticipating critical transitions. Science 338:344\u2013348","journal-title":"Science"},{"key":"1518_CR69","first-page":"726","volume":"6","author":"AS Smith","year":"2010","unstructured":"Smith AS (2010) Physics challenged by cells nature. Physics 6:726\u2013729","journal-title":"Physics"},{"issue":"2008","key":"1518_CR70","first-page":"470","volume":"76","author":"D Stauffer","year":"2008","unstructured":"Stauffer D (2008) Social applications of two-dimensional Ising models Am. J Phys 76(2008):470","journal-title":"J Phys"},{"key":"1518_CR71","doi-asserted-by":"crossref","unstructured":"Stauffer D, Hohnisch M, Pittnauer S (2006) The impact of external events on the emergence of social herding of economic sentiment. Phys A 370","DOI":"10.1016\/j.physa.2006.05.033"},{"key":"1518_CR72","doi-asserted-by":"crossref","first-page":"052217","DOI":"10.1103\/PhysRevE.100.052217","volume":"100","author":"R Torabi","year":"2019","unstructured":"Torabi R, Davidsen J (2019) Pattern formation in reaction-diffusion systems in the presence of non-Markovian diffusion. Phys Rev E 100:052217","journal-title":"Phys Rev E"},{"key":"1518_CR73","doi-asserted-by":"crossref","first-page":"052202","DOI":"10.1103\/PhysRevE.94.052202","volume":"94","author":"R Torabi","year":"2016","unstructured":"Torabi R, Rezaei Z (2016) Instability in reaction-superdiffusion systems. Phys Rev E 94:052202","journal-title":"Phys Rev E"},{"key":"1518_CR74","unstructured":"Torquato S (2010) Toward an Ising model of cancer and beyond. arXiv:1010.6284v2 [q-bio.CB]"},{"key":"1518_CR75","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.copbio.2014.11.020","volume":"34","author":"C Trefois","year":"2015","unstructured":"Trefois C, Antony PMA, Goncalves J, Skupin A, Balling R (2015) Critical transitions in chronic disease: transferring concepts from ecology to systems medicine. Curr Opin Biotechnol 34:48\u201355","journal-title":"Curr Opin Biotechnol"},{"key":"1518_CR76","doi-asserted-by":"crossref","unstructured":"Vera J, Urbina F, Palma W (2020) Phase transitions in a decentralized graph-based approach to human language","DOI":"10.1103\/PhysRevE.103.022129"},{"issue":"5","key":"1518_CR77","doi-asserted-by":"crossref","first-page":"055001","DOI":"10.1088\/1478-3975\/9\/5\/055001","volume":"9","author":"RS Wang","year":"2012","unstructured":"Wang RS, Saadatpour A, Albert R (2012) Boolean modeling in systems biology: an overview of methodology and applications. Phys Biol 9(5):055001","journal-title":"Phys Biol"},{"key":"1518_CR78","first-page":"37","volume":"115","author":"YI Wolf","year":"2018","unstructured":"Wolf YI, Katsnelson MI, Koonin EV (2018) Physical foundations of biological complexity. PNAS 115:37","journal-title":"PNAS"},{"issue":"19","key":"1518_CR79","doi-asserted-by":"crossref","first-page":"3845","DOI":"10.1002\/pmic.201100180","volume":"11","author":"L Xumeng","year":"2011","unstructured":"Xumeng L, Feltus FA, Sun X, Wang JZ, Luo F (2011) Identifying differentially expressed genes in cancer patients using a non-parameter Ising model. Proteomics 11(19):3845\u20133852","journal-title":"Proteomics"},{"issue":"42","key":"1518_CR80","doi-asserted-by":"crossref","first-page":"16308","DOI":"10.1073\/pnas.0806447105","volume":"105","author":"R Zhang","year":"2008","unstructured":"Zhang R, Shah MV, Yang J, Nyland SB, Liu X, Yun JK, Albert R, Loughran TP (2008) Network model of survival signaling in large granular lymphocyte leukemia. Proc Natl Acad Sci USA 105(42):16308\u201316313","journal-title":"Proc Natl Acad Sci USA"},{"key":"1518_CR81","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1063\/1.1730390","volume":"31","author":"BH Zimm","year":"1959","unstructured":"Zimm BH, Bragg JK (1959) Theory of the phase transition between Helix and Random Coil in polypeptide chains. J Chem Phys 31:526","journal-title":"J Chem Phys"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-020-01518-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-020-01518-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-020-01518-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,7]],"date-time":"2021-09-07T23:22:47Z","timestamp":1631056967000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-020-01518-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9]]},"references-count":81,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["1518"],"URL":"https:\/\/doi.org\/10.1007\/s00285-020-01518-6","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9]]},"assertion":[{"value":"20 August 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 September 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}