{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T04:49:45Z","timestamp":1791348585499,"version":"4.3.1"},"reference-count":78,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"}],"funder":[{"DOI":"10.13039\/100000057","name":"NIH National Institute of General Medical Sciences","doi-asserted-by":"publisher","award":["R35-GM127084"],"award-info":[{"award-number":["R35-GM127084"]}],"id":[{"id":"10.13039\/100000057","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100017540","name":"National Heart Lung and Blood Institute Division of Intramural Research","doi-asserted-by":"publisher","award":["F30HL176078"],"award-info":[{"award-number":["F30HL176078"]}],"id":[{"id":"10.13039\/100017540","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["cell.com","elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Biophysical Journal"],"published-print":{"date-parts":[[2026,9]]},"DOI":"10.1016\/j.bpj.2026.09.026","type":"journal-article","created":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T15:32:48Z","timestamp":1790177568000},"update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"title":["Truncating the time-evolving discrete chemical master equation for stochastic biological networks"],"prefix":"10.1016","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9729-9656","authenticated-orcid":false,"given":"Ali M.","family":"Farhat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yiyu","family":"Pang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Farid","family":"Manuchehrfar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ying","family":"Tang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jie","family":"Liang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.bpj.2026.09.026_bib1","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1093\/genetics\/149.4.1633","article-title":"Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells","volume":"149","author":"Arkin","year":"1998","journal-title":"Genetics"},{"key":"10.1016\/j.bpj.2026.09.026_bib2","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1126\/science.1070919","article-title":"Stochastic gene expression in a single cell","volume":"297","author":"Elowitz","year":"2002","journal-title":"Science (New York, N.Y.)"},{"key":"10.1016\/j.bpj.2026.09.026_bib3","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.plrev.2005.03.002","article-title":"Laws in Darwinian evolutionary theory","volume":"2","author":"Ao","year":"2005","journal-title":"Phys. Life Rev."},{"key":"10.1016\/j.bpj.2026.09.026_bib4","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1146\/annurev-biophys-050511-102240","article-title":"Cooperativity in Cellular Biochemical Processes: Noise-Enhanced Sensitivity, Fluctuating Enzyme, Bistability with Nonlinear Feedback, and Other Mechanisms for Sigmoidal Responses","volume":"41","author":"Qian","year":"2012","journal-title":"Annu. Rev. Biophys."},{"key":"10.1016\/j.bpj.2026.09.026_bib5","series-title":"Methods in Cell Biology","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/B978-0-12-388403-9.00005-9","article-title":"Chapter 5 - Stochastic Modeling of Cellular Networks","volume":"Volume 110","author":"Stewart-Ornstein","year":"2012"},{"key":"10.1016\/j.bpj.2026.09.026_bib6","doi-asserted-by":"crossref","first-page":"12795","DOI":"10.1073\/pnas.162041399","article-title":"Intrinsic and extrinsic contributions to stochasticity in gene expression","volume":"99","author":"Swain","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"10.1016\/j.bpj.2026.09.026_bib7","doi-asserted-by":"crossref","first-page":"18445","DOI":"10.1073\/pnas.1001455107","article-title":"Probability landscape of heritable and robust epigenetic state of lysogeny in phage lambda","volume":"107","author":"Cao","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"10.1016\/j.bpj.2026.09.026_bib8","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0168-9525(98)01659-X","article-title":"It\u2019s a noisy business! Genetic regulation at the nanomolar scale","volume":"15","author":"McAdams","year":"1999","journal-title":"Trends Genet."},{"key":"10.1016\/j.bpj.2026.09.026_bib9","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1038\/nrg2509","article-title":"Stochastic modelling for quantitative description of heterogeneous biological systems","volume":"10","author":"Wilkinson","year":"2009","journal-title":"Nat. Rev. Genet."},{"key":"10.1016\/j.bpj.2026.09.026_bib10","series-title":"Chemical biophysics: Quantitative analysis of cellular systems. Cambridge Texts in Biomedical Engineering","author":"Beard","year":"2008"},{"key":"10.1016\/j.bpj.2026.09.026_bib11","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1021\/j100540a008","article-title":"Exact stochastic simulation of coupled chemical reactions","volume":"81","author":"Gillespie","year":"1977","journal-title":"J. Phys. Chem."},{"key":"10.1016\/j.bpj.2026.09.026_bib12","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/0378-4371(92)90283-V","article-title":"A rigorous derivation of the chemical master equation","volume":"188","author":"Gillespie","year":"1992","journal-title":"Phys. Stat. Mech. Appl."},{"key":"10.1016\/j.bpj.2026.09.026_bib13","doi-asserted-by":"crossref","DOI":"10.1063\/1.3253798","article-title":"A diffusional bimolecular propensity function","volume":"131","author":"Gillespie","year":"2009","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib14","series-title":"Stochastic Processes in Physics and Chemistry","author":"Kampen","year":"2011"},{"key":"10.1016\/j.bpj.2026.09.026_bib15","doi-asserted-by":"crossref","first-page":"413","DOI":"10.2307\/3212214","article-title":"Stochastic Approach to Chemical Kinetics","volume":"4","author":"McQuarrie","year":"1967","journal-title":"J. Appl. Probab."},{"key":"10.1016\/j.bpj.2026.09.026_bib16","doi-asserted-by":"crossref","first-page":"3315","DOI":"10.1063\/1.1287273","article-title":"An analytical solution of the stochastic master equation for reversible bimolecular reaction kinetics","volume":"113","author":"Laurenzi","year":"2000","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib17","doi-asserted-by":"crossref","first-page":"1727","DOI":"10.1007\/s11538-006-9188-3","article-title":"A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer\u2019s Paradox","volume":"69","author":"Vellela","year":"2007","journal-title":"Bull. Math. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib18","doi-asserted-by":"crossref","first-page":"2145","DOI":"10.1063\/1.1727900","article-title":"Stochastic Models for Second-Order Chemical Reaction Kinetics. The Equilibrium State","volume":"45","author":"Darvey","year":"1966","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib19","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1007\/s12539-010-0072-3","article-title":"Towards predictive stochastic dynamical modeling of cancer genesis and progression","volume":"2","author":"Ao","year":"2010","journal-title":"Interdiscip. Sci."},{"key":"10.1016\/j.bpj.2026.09.026_bib20","series-title":"Energy\/Energetics","author":"Gardiner","year":"1985"},{"key":"10.1016\/j.bpj.2026.09.026_bib21","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1063\/1.481811","article-title":"The chemical Langevin equation","volume":"113","author":"Gillespie","year":"2000","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib22","doi-asserted-by":"crossref","first-page":"5063","DOI":"10.1021\/jp0128832","article-title":"The Chemical Langevin and Fokker-Planck Equations for the Reversible Isomerization Reaction","volume":"106","author":"Gillespie","year":"2002","journal-title":"J. Phys. Chem. A"},{"key":"10.1016\/j.bpj.2026.09.026_bib23","doi-asserted-by":"crossref","first-page":"6959","DOI":"10.1063\/1.1505860","article-title":"Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics","volume":"117","author":"Haseltine","year":"2002","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib24","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/s10955-012-0532-8","article-title":"Relation of a New Interpretation of Stochastic Differential Equations to Ito Process","volume":"148","author":"Shi","year":"2012","journal-title":"J. Stat. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib25","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1139\/p61-056","article-title":"A power series expansion of the master equation","volume":"39","author":"Kampen","year":"1961","journal-title":"Can. J. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib26","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.94.018104","article-title":"Sampling Rare Switching Events in Biochemical Networks","volume":"94","author":"Allen","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"10.1016\/j.bpj.2026.09.026_bib27","doi-asserted-by":"crossref","DOI":"10.1063\/1.4811286","article-title":"Adaptively biased sequential importance sampling for rare events in reaction networks with comparison to exact solutions from finite buffer dCME method","volume":"139","author":"Cao","year":"2013","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib28","doi-asserted-by":"crossref","DOI":"10.1063\/1.3522769","article-title":"Automated estimation of rare event probabilities in biochemical systems","volume":"134","author":"Daigle","year":"2011","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib29","series-title":"Proceedings of the IEEE International Conference on Systems Biology (ISB)","first-page":"27","article-title":"Kinetics of Muller\u2019s ratchet from adaptive landscape viewpoint","author":"Jiao","year":"2011"},{"key":"10.1016\/j.bpj.2026.09.026_bib30","doi-asserted-by":"crossref","DOI":"10.1063\/1.2987701","article-title":"An efficient and exact stochastic simulation method to analyze rare events in biochemical systems","volume":"129","author":"Kuwahara","year":"2008","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib31","doi-asserted-by":"crossref","DOI":"10.1098\/rsfs.2013.0064","article-title":"Quantitative implementation of the endogenous molecular\u2013cellular network hypothesis in hepatocellular carcinoma","volume":"4","author":"Wang","year":"2014","journal-title":"Interface Focus"},{"key":"10.1016\/j.bpj.2026.09.026_bib32","doi-asserted-by":"crossref","DOI":"10.3389\/fgene.2021.645640","article-title":"Exact Probability Landscapes of Stochastic Phenotype Switching in Feed-Forward Loops: Phase Diagrams of Multimodality","volume":"12","author":"Terebus","year":"2021","journal-title":"Front. Genet."},{"key":"10.1016\/j.bpj.2026.09.026_bib33","doi-asserted-by":"crossref","DOI":"10.1063\/1.2145882","article-title":"The finite state projection algorithm for the solution of the chemical master equation","volume":"124","author":"Munsky","year":"2006","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib34","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1016\/j.jcp.2007.05.016","article-title":"A multiple time interval finite state projection algorithm for the solution to the chemical master equation","volume":"226","author":"Munsky","year":"2007","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib35","doi-asserted-by":"crossref","DOI":"10.1088\/1478-3975\/aba1d2","article-title":"An adaptive solution to the chemical master equation using quantized tensor trains with sliding windows","volume":"17","author":"Dinh","year":"2020","journal-title":"Phys. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib36","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1186\/1752-0509-4-42","article-title":"Solving the chemical master equation using sliding windows","volume":"4","author":"Wolf","year":"2010","journal-title":"BMC Syst. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib37","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1038\/s42256-023-00632-6","article-title":"Neural-network solutions to stochastic reaction networks","volume":"5","author":"Tang","year":"2023","journal-title":"Nat. Mach. Intell."},{"key":"10.1016\/j.bpj.2026.09.026_bib38","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1007\/s11538-016-0149-1","article-title":"State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation","volume":"78","author":"Cao","year":"2016","journal-title":"Bull. Math. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib39","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1186\/1752-0509-2-30","article-title":"Optimal enumeration of state space of finitely buffered stochastic molecular networks and exact computation of steady state landscape probability","volume":"2","author":"Cao","year":"2008","journal-title":"BMC Syst. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib40","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1137\/15M1034180","article-title":"Accurate Chemical Master Equation Solution Using Multi-Finite Buffers","volume":"14","author":"Cao","year":"2016","journal-title":"Multiscale Model. Simul."},{"key":"10.1016\/j.bpj.2026.09.026_bib41","series-title":"2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society","first-page":"5228","article-title":"Exact computation of probability landscape of stochastic networks of Single Input and Coupled Toggle Switch Modules","author":"Terebus","year":"2014"},{"key":"10.1016\/j.bpj.2026.09.026_bib42","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1146\/annurev.biophys.37.032807.125928","article-title":"Single-Molecule Approaches to Stochastic Gene Expression","volume":"38","author":"Raj","year":"2009","journal-title":"Annu. Rev. Biophys."},{"key":"10.1016\/j.bpj.2026.09.026_bib43","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1126\/science.1188308","article-title":"Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells","volume":"329","author":"Taniguchi","year":"2010","journal-title":"Science (New York, N.Y.)"},{"key":"10.1016\/j.bpj.2026.09.026_bib44","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1038\/nrn3311","article-title":"title = {Calcium channel auxiliary \u2216(\u2216alpha 2 \u2216gamma\u2216) and \u2216(\u2216beta\u2216) subunits: trafficking and one step beyond}","volume":"13","author":"Dolphin","year":"2012","journal-title":"Nat. Rev. Neurosci."},{"key":"10.1016\/j.bpj.2026.09.026_bib45","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1146\/annurev-nutr-071812-161148","article-title":"AMP-activated protein kinase: maintaining energy homeostasis at the cellular and whole-body levels","volume":"34","author":"Hardie","year":"2014","journal-title":"Annu. Rev. Nutr."},{"key":"10.1016\/j.bpj.2026.09.026_bib46","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1161\/CIRCRESAHA.117.311401","article-title":"Reactive Oxygen Species in Metabolic and Inflammatory Signaling","volume":"122","author":"Forrester","year":"2018","journal-title":"Circ. Res."},{"key":"10.1016\/j.bpj.2026.09.026_bib47","doi-asserted-by":"crossref","first-page":"1564","DOI":"10.1158\/0008-5472.CAN-16-2074","article-title":"Modeling the Genetic Regulation of Cancer Metabolism: Interplay between Glycolysis and Oxidative Phosphorylation","volume":"77","author":"Yu","year":"2017","journal-title":"Cancer Res."},{"key":"10.1016\/j.bpj.2026.09.026_bib48","doi-asserted-by":"crossref","DOI":"10.1002\/advs.202003133","article-title":"A Dimension Reduction Approach for Energy Landscape: Identifying Intermediate States in Metabolism-EMT Network","volume":"8","author":"Kang","year":"2021","journal-title":"Adv. Sci."},{"key":"10.1016\/j.bpj.2026.09.026_bib49","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1038\/387296a0","article-title":"Mdm2 promotes the rapid degradation of p53","volume":"387","author":"Haupt","year":"1997","journal-title":"Nature"},{"key":"10.1016\/j.bpj.2026.09.026_bib50","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1038\/sj.onc.1201432","article-title":"Mdm2: keeping p53 under control","volume":"15","author":"Piette","year":"1997","journal-title":"Oncogene"},{"key":"10.1016\/j.bpj.2026.09.026_bib51","doi-asserted-by":"crossref","first-page":"2216","DOI":"10.3390\/ijms18102216","article-title":"The Role of MDM2 in Promoting Genome Stability versus Instability","volume":"18","author":"Saadatzadeh","year":"2017","journal-title":"Int. J. Mol. Sci."},{"key":"10.1016\/j.bpj.2026.09.026_bib52","first-page":"1001","article-title":"The MDM2-p53 interaction","volume":"1","author":"Moll","year":"2003","journal-title":"Mol. Cancer Res."},{"key":"10.1016\/j.bpj.2026.09.026_bib53","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1016\/j.jad.2019.06.056","article-title":"Significant role of gene-gene interactions of clock genes in mood disorder","volume":"257","author":"Park","year":"2019","journal-title":"J. Affect. Disord."},{"key":"10.1016\/j.bpj.2026.09.026_bib54","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1080\/15622975.2018.1500028","article-title":"Circadian genes in major depressive disorder","volume":"21","author":"Melhuish Beaupre","year":"2020","journal-title":"World J. Biol. Psychiatry"},{"key":"10.1016\/j.bpj.2026.09.026_bib55","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1002\/ajmg.b.30714","article-title":"Clock Genes may Influence Bipolar Disorder Susceptibility and Dysfunctional Circadian Rhythm","volume":"147B","author":"Shi","year":"2008","journal-title":"Am. J. Med. Genet. B Neuropsychiatr. Genet."},{"key":"10.1016\/j.bpj.2026.09.026_bib56","doi-asserted-by":"crossref","first-page":"938","DOI":"10.3390\/ijms18050938","article-title":"Clock Genes and Altered Sleep-Wake Rhythms: Their Role in the Development of Psychiatric Disorders","volume":"18","author":"Charrier","year":"2017","journal-title":"Int. J. Mol. Sci."},{"key":"10.1016\/j.bpj.2026.09.026_bib57","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1038\/35002131","article-title":"Construction of a genetic toggle switch in Escherichia coli","volume":"403","author":"Gardner","year":"2000","journal-title":"Nature"},{"key":"10.1016\/j.bpj.2026.09.026_bib58","doi-asserted-by":"crossref","first-page":"3116","DOI":"10.1016\/S0006-3495(01)75949-8","article-title":"Stochasticity in Transcriptional Regulation: Origins, Consequences, and Mathematical Representations","volume":"81","author":"Kepler","year":"2001","journal-title":"Biophys. J."},{"key":"10.1016\/j.bpj.2026.09.026_bib59","doi-asserted-by":"crossref","first-page":"e60","DOI":"10.1371\/journal.pcbi.0030060","article-title":"Potential Energy Landscape and Robustness of a Gene Regulatory Network: Toggle Switch","volume":"3","author":"Kim","year":"2007","journal-title":"PLoS Comput. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib60","doi-asserted-by":"crossref","DOI":"10.1063\/1.2741544","article-title":"Understanding stochastic simulations of the smallest genetic networks","volume":"126","author":"Schultz","year":"2007","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.bpj.2026.09.026_bib61","first-page":"201","article-title":"The Finite State Projection Approach for the Analysis of Stochastic Noise in Gene Networks","volume":"53","author":"Munsky","year":"2007","journal-title":"IEEE Trans. Circuits Syst. I."},{"key":"10.1016\/j.bpj.2026.09.026_bib62","doi-asserted-by":"crossref","first-page":"2990","DOI":"10.1137\/070689759","article-title":"Adaptive discrete galerkin methods applied to the chemical master equation","volume":"30","author":"Deuflhard","year":"2008","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.bpj.2026.09.026_bib63","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s00791-006-0045-6","article-title":"{Fokker}-{Planck} approximation of the master equation in molecular biology","volume":"12","author":"Sj\u00f6berg","year":"2009","journal-title":"Comput. Vis. Sci."},{"key":"10.1016\/j.bpj.2026.09.026_bib64","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1003359","article-title":"Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains","volume":"10","author":"Kazeev","year":"2014","journal-title":"PLoS Comput. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib65","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1126\/science.1075090","article-title":"Transcriptional regulatory networks in Saccharomyces cerevisiae","volume":"298","author":"Lee","year":"2002","journal-title":"Science"},{"key":"10.1016\/j.bpj.2026.09.026_bib66","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1038\/ng881","article-title":"Network motifs in the transcriptional regulation network of Escherichia coli","volume":"31","author":"Shen-Orr","year":"2002","journal-title":"Nat. Genet."},{"key":"10.1016\/j.bpj.2026.09.026_bib67","doi-asserted-by":"crossref","first-page":"11980","DOI":"10.1073\/pnas.2133841100","article-title":"Structure and function of the feed-forward loop network motif","volume":"100","author":"Mangan","year":"2003","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"10.1016\/j.bpj.2026.09.026_bib68","series-title":"An Introduction to Systems Biology: Design Principles of Biological Circuits","author":"Alon","year":"2019"},{"key":"10.1016\/j.bpj.2026.09.026_bib69","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1016\/j.cell.2005.08.020","article-title":"Core transcriptional regulatory circuitry in human embryonic stem cells","volume":"122","author":"Boyer","year":"2005","journal-title":"Cell"},{"key":"10.1016\/j.bpj.2026.09.026_bib70","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.stemcr.2015.07.004","article-title":"Single-cell analyses of ESCs reveal alternative pluripotent cell states and molecular mechanisms that control self-renewal","volume":"5","author":"Papatsenko","year":"2015","journal-title":"Stem Cell Rep."},{"key":"10.1016\/j.bpj.2026.09.026_bib71","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1016\/j.cell.2015.04.014","article-title":"Making sense of transcription networks","volume":"161","author":"Sorrells","year":"2015","journal-title":"Cell"},{"key":"10.1016\/j.bpj.2026.09.026_bib72","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.molcel.2007.05.018","article-title":"MicroRNA-mediated feedback and feedforward loops are recurrent network motifs in mammals","volume":"26","author":"Tsang","year":"2007","journal-title":"Mol. Cell"},{"key":"10.1016\/j.bpj.2026.09.026_bib73","doi-asserted-by":"crossref","first-page":"854","DOI":"10.1039\/B900177H","article-title":"Genome-wide survey of microRNA\u2013transcription factor feed-forward regulatory circuits in human","volume":"5","author":"Re","year":"2009","journal-title":"Mol. Biosyst."},{"key":"10.1016\/j.bpj.2026.09.026_bib74","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1016\/j.jmb.2005.12.003","article-title":"The incoherent feed-forward loop accelerates the response-time of the gal system of Escherichia coli","volume":"356","author":"Mangan","year":"2006","journal-title":"J. Mol. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib75","doi-asserted-by":"crossref","first-page":"760","DOI":"10.1016\/j.cell.2009.06.013","article-title":"Defining network topologies that can achieve biochemical adaptation","volume":"138","author":"Ma","year":"2009","journal-title":"Cell"},{"key":"10.1016\/j.bpj.2026.09.026_bib76","doi-asserted-by":"crossref","DOI":"10.1088\/1478-3975\/5\/2\/026009","article-title":"A case study of evolutionary computation of biochemical adaptation","volume":"5","author":"Fran\u00e7ois","year":"2008","journal-title":"Phys. Biol."},{"key":"10.1016\/j.bpj.2026.09.026_bib77","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1016\/j.molcel.2009.11.018","article-title":"The incoherent feedforward loop can provide fold-change detection in gene regulation","volume":"36","author":"Goentoro","year":"2009","journal-title":"Mol. Cell"},{"key":"10.1016\/j.bpj.2026.09.026_bib78","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1009623","article-title":"DeepCME: A Deep Learning Framework for Computing Solution Statistics of the Chemical Master Equation","volume":"17","author":"Gupta","year":"2021","journal-title":"PLoS Comput. Biol."}],"container-title":["Biophysical Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0006349526006569?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0006349526006569?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T03:51:50Z","timestamp":1791345110000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0006349526006569"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9]]},"references-count":78,"alternative-id":["S0006349526006569"],"URL":"https:\/\/doi.org\/10.1016\/j.bpj.2026.09.026","relation":{},"ISSN":["0006-3495"],"issn-type":[{"value":"0006-3495","type":"print"}],"subject":[],"published":{"date-parts":[[2026,9]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Truncating the time-evolving discrete chemical master equation for stochastic biological networks","name":"articletitle","label":"Article Title"},{"value":"Biophysical Journal","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.bpj.2026.09.026","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"Published by Elsevier Inc. on behalf of Biophysical Society.","name":"copyright","label":"Copyright"}]}}