{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T07:46:10Z","timestamp":1778053570248,"version":"3.51.4"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2014,6,18]],"date-time":"2014-06-18T00:00:00Z","timestamp":1403049600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/2.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["BMC Syst Biol"],"published-print":{"date-parts":[[2014,12]]},"DOI":"10.1186\/1752-0509-8-71","type":"journal-article","created":{"date-parts":[[2014,6,18]],"date-time":"2014-06-18T07:03:12Z","timestamp":1403074992000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method"],"prefix":"10.1186","volume":"8","author":[{"suffix":"Jr.","given":"Tam\u00e1s","family":"Sz\u00e9kely","sequence":"first","affiliation":[]},{"given":"Kevin","family":"Burrage","sequence":"additional","affiliation":[]},{"given":"Konstantinos C","family":"Zygalakis","sequence":"additional","affiliation":[]},{"given":"Manuel","family":"Barrio","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,6,18]]},"reference":[{"key":"1338_CR1","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1093\/genetics\/149.4.1633","volume":"149","author":"A Arkin","year":"1998","unstructured":"Arkin A, Ross J, McAdams HH: Stochastic kinetic analysis of developmental pathway bifurcation in phage \u03bb-infected Escherichia coli cells. Genetics. 1998, 149: 1633-1648.","journal-title":"Genetics"},{"key":"1338_CR2","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1038\/nrg1615","volume":"6","author":"M Kaern","year":"2005","unstructured":"Kaern M, Elston T, Blake W, Collins J: Stochasticity in gene expression: from theories to phenotypes. Nat Rev Genet. 2005, 6: 451-464.","journal-title":"Nat Rev Genet"},{"key":"1338_CR3","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1126\/science.1216379","volume":"336","author":"B Munsky","year":"2012","unstructured":"Munsky B, Neuert G, van Oudenaarden A: Using gene expression noise to understand gene regulation. Science. 2012, 336: 183-","journal-title":"Science"},{"key":"1338_CR4","doi-asserted-by":"publisher","first-page":"1129","DOI":"10.1126\/science.1075988","volume":"297","author":"N Fedoroff","year":"2002","unstructured":"Fedoroff N, Fontana W: Small numbers of big molecules. Science. 2002, 297: 1129-1131.","journal-title":"Science"},{"key":"1338_CR5","doi-asserted-by":"publisher","first-page":"1183","DOI":"10.1126\/science.1070919","volume":"297","author":"M Elowitz","year":"2002","unstructured":"Elowitz M, Levine A, Siggia E, Swain P: Stochastic gene expression in a single cell. Science. 2002, 297: 1183-1186.","journal-title":"Science"},{"key":"1338_CR6","doi-asserted-by":"publisher","first-page":"704","DOI":"10.1063\/1.442111","volume":"75","author":"DT Gillespie","year":"1981","unstructured":"Gillespie DT, Mangel M: Conditioned averages in chemical kinetics. J Chem Phys. 1981, 75: 704-709.","journal-title":"J Chem Phys"},{"key":"1338_CR7","doi-asserted-by":"publisher","first-page":"2350","DOI":"10.1529\/biophysj.106.093781","volume":"92","author":"J Goutsias","year":"2007","unstructured":"Goutsias J: Classical versus stochastic kinetics modeling of biochemical reaction systems. Biophys J. 2007, 92: 2350-2365.","journal-title":"Biophys J"},{"key":"1338_CR8","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1038\/nrg2509","volume":"10","author":"DJ Wilkinson","year":"2009","unstructured":"Wilkinson DJ: Stochastic modelling for quantitative description of heterogeneous biological systems. Nat Rev Genet. 2009, 10: 122-133.","journal-title":"Nat Rev Genet"},{"key":"1338_CR9","doi-asserted-by":"publisher","first-page":"344","DOI":"10.2307\/3211904","volume":"8","author":"TG Kurtz","year":"1971","unstructured":"Kurtz TG: Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Probab. 1971, 8: 344-356.","journal-title":"J Appl Probab"},{"key":"1338_CR10","doi-asserted-by":"publisher","first-page":"2340","DOI":"10.1021\/j100540a008","volume":"81","author":"DT Gillespie","year":"1977","unstructured":"Gillespie DT: Exact stochastic simulation of coupled chemical reactions. J Phys Chem. 1977, 81: 2340-2361.","journal-title":"J Phys Chem"},{"key":"1338_CR11","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1146\/annurev.physchem.58.032806.104637","volume":"58","author":"DT Gillespie","year":"2007","unstructured":"Gillespie DT: Stochastic simulation of chemical kinetics. Annu Rev Phys Chem. 2007, 58: 35-55.","journal-title":"Annu Rev Phys Chem"},{"key":"1338_CR12","doi-asserted-by":"publisher","first-page":"1716","DOI":"10.1063\/1.1378322","volume":"115","author":"DT Gillespie","year":"2001","unstructured":"Gillespie DT: Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys. 2001, 115: 1716-1733.","journal-title":"J Chem Phys"},{"key":"1338_CR13","doi-asserted-by":"publisher","first-page":"8229","DOI":"10.1063\/1.1613254","volume":"119","author":"DT Gillespie","year":"2003","unstructured":"Gillespie DT, Petzold LR: Improved leap-size selection for accelerated stochastic simulation. J Chem Phys. 2003, 119: 8229-8234.","journal-title":"J Chem Phys"},{"key":"1338_CR14","doi-asserted-by":"publisher","first-page":"044109","DOI":"10.1063\/1.2159468","volume":"124","author":"Y Cao","year":"2006","unstructured":"Cao Y, Gillespie DT, Petzold LR: Efficient step size selection for the tau-leaping simulation method. J Chem Phys. 2006, 124: 044109-","journal-title":"J Chem Phys"},{"key":"1338_CR15","doi-asserted-by":"publisher","first-page":"10356","DOI":"10.1063\/1.1810475","volume":"121","author":"TH Tian","year":"2004","unstructured":"Tian TH, Burrage K: Binomial leap methods for simulating stochastic chemical kinetics. J Chem Phys. 2004, 121: 10356-10364.","journal-title":"J Chem Phys"},{"key":"1338_CR16","doi-asserted-by":"publisher","first-page":"024112","DOI":"10.1063\/1.1833357","volume":"122","author":"A Chatterjee","year":"2005","unstructured":"Chatterjee A, Vlachos DG, Katsoulakis MA: Binomial distribution based tau-leap accelerated stochastic simulation. J Chem Phys. 2005, 122: 024112-","journal-title":"J Chem Phys"},{"key":"1338_CR17","doi-asserted-by":"publisher","first-page":"224109","DOI":"10.1063\/1.2741252","volume":"126","author":"X Peng","year":"2007","unstructured":"Peng X, Zhou W, Wang Y: Efficient binomial leap method for simulating chemical kinetics. J Chem Phys. 2007, 126: 224109-","journal-title":"J Chem Phys"},{"key":"1338_CR18","doi-asserted-by":"publisher","first-page":"084101","DOI":"10.1063\/1.2432326","volume":"126","author":"MF Pettigrew","year":"2007","unstructured":"Pettigrew MF, Resat H: Multinomial tau-leaping method for stochastic kinetic simulations. J Chem Phys. 2007, 126: 084101-","journal-title":"J Chem Phys"},{"key":"1338_CR19","doi-asserted-by":"publisher","first-page":"154112","DOI":"10.1063\/1.2894479","volume":"128","author":"Z Xu","year":"2008","unstructured":"Xu Z, Cai X: Unbiased tau-leap methods for stochastic simulation of chemically reacting systems. J Chem Phys. 2008, 128: 154112-","journal-title":"J Chem Phys"},{"key":"1338_CR20","doi-asserted-by":"publisher","first-page":"124109","DOI":"10.1063\/1.3091269","volume":"130","author":"Y Hu","year":"2009","unstructured":"Hu Y, Li T: Highly accurate tau-leaping methods with random corrections. J Chem Phys. 2009, 130: 124109-","journal-title":"J Chem Phys"},{"key":"1338_CR21","doi-asserted-by":"publisher","first-page":"024113","DOI":"10.1063\/1.3609119","volume":"135","author":"Y Hu","year":"2011","unstructured":"Hu Y, Li T, Min B: A weak second order tau-leaping method for chemical kinetic systems. J Chem Phys. 2011, 135: 024113-","journal-title":"J Chem Phys"},{"key":"1338_CR22","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1186\/1752-0509-6-85","volume":"6","author":"T Sz\u00e9kely Jr.","year":"2012","unstructured":"Sz\u00e9kely T, Burrage K, Erban R, Zygalakis KC: A higher-order numerical framework for stochastic simulation of chemical reaction systems. BMC Syst Biol. 2012, 6: 85-","journal-title":"BMC Syst Biol"},{"key":"1338_CR23","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1098\/rsta.1911.0009","volume":"210","author":"LF Richardson","year":"1910","unstructured":"Richardson LF: The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Phil Trans Roy Soc Lond. 1910, 210: 307-357.","journal-title":"Phil Trans Roy Soc Lond"},{"key":"1338_CR24","volume-title":"Solving Ordinary Differential Equations: Nonstiff Problems. 2nd edition","author":"E Hairer","year":"1993","unstructured":"Hairer E, N\u00f8rsett SP, Wanner G: Solving Ordinary Differential Equations: Nonstiff Problems. 2nd edition. 1993, Berlin: Springer-Verlag"},{"key":"1338_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02165234","volume":"8","author":"R Bulirsch","year":"1966","unstructured":"Bulirsch R, Stoer J: Numerical treatment of ordinary differential equations by extrapolation methods. Numerische Mathematik. 1966, 8: 1-13.","journal-title":"Numerische Mathematik"},{"issue":"4","key":"1338_CR26","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1137\/1027140","volume":"27","author":"P Deuflhard","year":"1985","unstructured":"Deuflhard P: Recent progress in extrapolation methods for ordinary differential equations. SIAM Rev. 1985, 27 (4): 505-535.","journal-title":"SIAM Rev"},{"key":"1338_CR27","first-page":"384","volume":"2","author":"WB Gragg","year":"1965","unstructured":"Gragg WB: On extrapolation algorithms for ordinary initial value problems. SIAM J Numer Anal. 1965, 2: 384-403.","journal-title":"SIAM J Numer Anal"},{"key":"1338_CR28","volume-title":"Numerical Recipes in C: The Art of Scientific Computing. 2nd edition","author":"WH Press","year":"1992","unstructured":"Press WH, Teukolsky SA, Vetterling WT, Flannery BP: Numerical Recipes in C: The Art of Scientific Computing. 2nd edition. 1992, Cambridge: Cambridge University Press"},{"key":"1338_CR29","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/0304-4149(78)90020-0","volume":"6","author":"TG Kurtz","year":"1978","unstructured":"Kurtz TG: Strong approximation theorems for density dependent Markov chains. Stochastic Processes Appl. 1978, 6: 223-240.","journal-title":"Stochastic Processes Appl"},{"key":"1338_CR30","volume-title":"Stochastic Processes in Physics and Chemistry. 3rd edition","author":"NG van Kampen","year":"2007","unstructured":"van Kampen NG: Stochastic Processes in Physics and Chemistry. 3rd edition. 2007, Amsterdam: Elsevier"},{"key":"1338_CR31","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.physrep.2013.03.004","volume":"529","author":"J Goutsias","year":"2013","unstructured":"Goutsias J, Jenkinson G: Markovian dynamics on complex reaction networks. Phys Rep. 2013, 529: 199-264.","journal-title":"Phys Rep"},{"key":"1338_CR32","doi-asserted-by":"publisher","first-page":"682","DOI":"10.1214\/aop\/1176994660","volume":"8","author":"TG Kurtz","year":"1980","unstructured":"Kurtz TG: Representations of Markov processes as multiparameter time changes. Ann Probab. 1980, 8: 682-715.","journal-title":"Ann Probab"},{"key":"1338_CR33","doi-asserted-by":"publisher","first-page":"104101","DOI":"10.1063\/1.2771548","volume":"127","author":"T Marquez-Lago","year":"2007","unstructured":"Marquez-Lago T, Burrage K: Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics. J Chem Phys. 2007, 127: 104101-","journal-title":"J Chem Phys"},{"key":"1338_CR34","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1049\/sb:20045021","volume":"1","author":"J Elf","year":"2004","unstructured":"Elf J, Ehrenberg M: Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. Syst Biol. 2004, 1: 230-236.","journal-title":"Syst Biol"},{"issue":"3","key":"1338_CR35","doi-asserted-by":"publisher","first-page":"867","DOI":"10.1137\/040603206","volume":"4","author":"M Rathinam","year":"2005","unstructured":"Rathinam M, Petzold LR, Cao Y, Gillespie DT: Consistency and stability of tau-leaping schemes for chemical reaction systems. Multiscale Model Simul. 2005, 4 (3): 867-895.","journal-title":"Multiscale Model Simul"},{"issue":"2","key":"1338_CR36","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1137\/06066792X","volume":"6","author":"T Li","year":"2007","unstructured":"Li T: Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model Simul. 2007, 6 (2): 417-436.","journal-title":"Multiscale Model Simul"},{"key":"1338_CR37","first-page":"82","volume-title":"Advances in Scientific Computing and Applications","author":"K Burrage","year":"2004","unstructured":"Burrage K, Tian T: Poisson Runge-Kutta methods for chemical reaction systems. Advances in Scientific Computing and Applications. Edited by: Sun YLW, Tang T. 2004, Beijing\/New York: Science Press, 82-96."},{"issue":"6","key":"1338_CR38","doi-asserted-by":"publisher","first-page":"2226","DOI":"10.1214\/10-AAP756","volume":"21","author":"DF Anderson","year":"2011","unstructured":"Anderson DF, Ganguly A, Kurtz TG: Error analysis of tau-leap simulation methods. Ann Appl Probab. 2011, 21 (6): 2226-2262.","journal-title":"Ann Appl Probab"},{"key":"1338_CR39","doi-asserted-by":"publisher","first-page":"965","DOI":"10.4310\/CMS.2011.v9.n4.a2","volume":"9","author":"Y Hu","year":"2011","unstructured":"Hu Y, Li T, Min B: The weak convergence analysis of tau-leaping methods: revisited. Comm Math Sci. 2011, 9: 965-996.","journal-title":"Comm Math Sci"},{"issue":"5","key":"1338_CR40","doi-asserted-by":"publisher","first-page":"1626","DOI":"10.1137\/S0036142999363206","volume":"38","author":"K Burrage","year":"2000","unstructured":"Burrage K, Burrage PM: Order conditions of stochastic Runge\u2013Kutta methods by B-Series. SIAM J Numer Anal. 2000, 38 (5): 1626-1646.","journal-title":"SIAM J Numer Anal"},{"key":"1338_CR41","doi-asserted-by":"publisher","first-page":"2142","DOI":"10.1137\/S0036142996278967","volume":"34","author":"GN Milstein","year":"1997","unstructured":"Milstein GN, Tretyakov MV: Numerical methods in the weak sense for stochastic differential equations with small noise. SIAM J Numer Anal. 1997, 34: 2142-2167.","journal-title":"SIAM J Numer Anal"},{"key":"1338_CR42","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1137\/S1064827594278575","volume":"18","author":"GN Milstein","year":"1997","unstructured":"Milstein GN, Tretyakov MV: Mean-square numerical methods for stochastic differential equations with small noises. SIAM J Sci Comput. 1997, 18: 1067-1087.","journal-title":"SIAM J Sci Comput"},{"key":"1338_CR43","doi-asserted-by":"publisher","first-page":"1789","DOI":"10.1137\/090763275","volume":"32","author":"E Buckwar","year":"2010","unstructured":"Buckwar E, R\u00f6\u00dfler A, Winkler R: Stochastic Runge-Kutta methods for Ito SODEs with small noise. SIAM J Sci Comput. 2010, 32: 1789-1808.","journal-title":"SIAM J Sci Comput"},{"key":"1338_CR44","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1186\/1752-0509-4-110","volume":"4","author":"P Ru\u00e9","year":"2010","unstructured":"Ru\u00e9 P, Villa-Freix\u00e0 J, Burrage K: Simulation methods with extended stability for stiff biochemical kinetics. BMC Syst Biol. 2010, 4: 110-123.","journal-title":"BMC Syst Biol"},{"key":"1338_CR45","doi-asserted-by":"publisher","first-page":"224101","DOI":"10.1063\/1.2745299","volume":"126","author":"Y Cao","year":"2007","unstructured":"Cao Y, Gillespie DT, Petzold LR: The adaptive explicit-implicit tau-leaping method with automatic tau selection. J Chem Phys. 2007, 126: 224101-","journal-title":"J Chem Phys"},{"key":"1338_CR46","doi-asserted-by":"publisher","first-page":"184102","DOI":"10.1063\/1.1889434","volume":"122","author":"J Goutsias","year":"2005","unstructured":"Goutsias J: Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J Chem Phys. 2005, 122: 184102-","journal-title":"J Chem Phys"},{"issue":"4","key":"1338_CR47","doi-asserted-by":"publisher","first-page":"1146","DOI":"10.1137\/060678154","volume":"6","author":"S MacNamara","year":"2008","unstructured":"MacNamara S, Burrage K, Sidje R: Multiscale modeling of chemical kinetics via the master equation. Multiscale Model Simul. 2008, 6 (4): 1146-1168.","journal-title":"Multiscale Model Simul"}],"container-title":["BMC Systems Biology"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1186\/1752-0509-8-71.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1186\/1752-0509-8-71\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1186\/1752-0509-8-71.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,8]],"date-time":"2022-04-08T17:33:29Z","timestamp":1649439209000},"score":1,"resource":{"primary":{"URL":"https:\/\/bmcsystbiol.biomedcentral.com\/articles\/10.1186\/1752-0509-8-71"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,18]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,12]]}},"alternative-id":["1338"],"URL":"https:\/\/doi.org\/10.1186\/1752-0509-8-71","relation":{},"ISSN":["1752-0509"],"issn-type":[{"value":"1752-0509","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,6,18]]},"assertion":[{"value":"22 January 2013","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2014","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2014","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"71"}}