生化反应系统的建模与分析
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摘要
很多细胞行为受细胞内基因表达和蛋白质相互作用等生物化学反应的调控.这些细胞内的生化反应表现出明显的随机性,它成为描述细胞行为的可计算建模中不可忽略的因素.该文是关于计算系统生物学中随机模拟的基本理论和新进展的自洽综述.本综述从生化反应系统的基本假设出发,介绍关于生化反应内部噪声、外部随机扰动和包含时间滞后的反应过程的各种数学描述,包括化学主方程、化学福克尔-普朗克方程、化学速率方程、化学郎之万方程等;还介绍了相关的数值模拟方法,包括随机模拟算法和τ跳跃算法.
Many cellular behaviors are regulated by intracellular biochemical reactions such as gene expression and protein-protein interactions. The kinetics of these biochemical reactions are mostly random,and therefore the stochasticity is essential for the modeling of cellular behaviour in the field of computational biology. This paper is a self contained review trying to provide an overview of stochastic modeling and recent advance in computation biology.This paper starts from basic assumptions in biochemical reaction systems,and introduce mathematical formulations for modeling intrinsic noise,external noise,and reactions systems with delay. Multiple dynamical equations are discussed,including chemical master equation,chemical Fokker-Plank equation,chemical reaction rate equation,and chemical Langevin equation,etc. Several numerical methods are introduced,including the stochastic simulation algorithm( SSA) and the Tau-leaping algorithm.
Many cellular behaviors are regulated by intracellular biochemical reactions such as gene expression and protein-protein interactions. The kinetics of these biochemical reactions are mostly random,and therefore the stochasticity is essential for the modeling of cellular behaviour in the field of computational biology. This paper is a self contained review trying to provide an overview of stochastic modeling and recent advance in computation biology.This paper starts from basic assumptions in biochemical reaction systems,and introduce mathematical formulations for modeling intrinsic noise,external noise,and reactions systems with delay. Multiple dynamical equations are discussed,including chemical master equation,chemical Fokker-Plank equation,chemical reaction rate equation,and chemical Langevin equation,etc. Several numerical methods are introduced,including the stochastic simulation algorithm( SSA) and the Tau-leaping algorithm.
引文
[1]Cai Long,Friedman N,Xie X Sunney.Stochastic protein expression in individual cells at the single molecule level[J].Nature Cell Biology,2006,440(7082):358-362.
[2]Choi P J,Cai Long,Frieda K,et al.A stochastic singlemolecule event triggers phenotype switching of a bacterial cell[J].Science,2008,322(5900):442-446.
[3]Elf J,Li Gene Wei,Xie X Sunney.Probing transcription factor dynamics at the single-molecule level in a living cell[J].Science,2007,316(5828):1191-1194.
[4]Xie X Sunney,Choi P J,Li Gene Wei,et al.Single-molecule approach to molecular biology in living bacterial cells[J].Annual Review of Biophysics,2008,37:417-444.
[5]Yu Ji,Xiao Jie,Ren Xiaojia,et al.Probing gene expression in live cells,one protein molecule at a time[J].Science,2006,311(5767):1600-1603.
[6]Taniguchi Y,Choi P J,Li Gene Wei,et al.Quantifying e.coli proteome and transcriptome with single-molecule sensitivity in single cells[J].Science,2010,329(5):533-538.
[7]Bassett A R,Liu Jilong.Crispr/cas9 and genome editing in drosophila[J].Journal of Genetics and Genomics,2014,41(1):7-19.
[8]Mali P,Esvelt K M,Church G M.Cas9 as a versatile tool for engineering biology[J].Nature Methods,2013,10(10):957-963.
[9]Amabile G,Meissner A.Induced pluripotent stem cells:current progress and potential for regenerative medicine[J].Trends in Molecular Medicine,2009,15(2):59-68.
[10]Efe J A,Hilcove S,Kim J,et al.Conversion of mouse fibroblasts into cardiomyocytes using a direct reprogramming strategy[J].Nature Cell Biology,2011,13(3):215-222.
[11]Takahashi K,Yamanaka S.Induction of pluripotent stem cells from mouse embryonic and adult fibroblast cultures by defined factors[J].Cell,2006,126(4):663-676.
[12]Takahashi K,Tanabe K,Ohnuki M,et al.Induction of pluripotent stem cells from adult human fibroblasts by defined factors[J].Cell,2007,131(5):861-872.
[13]Fiering S,Whitelaw E,Martin D.To be or not to be active:the stochastic nature of enhancer action[J].Bioessays,2000,22(4):381-387.
[14]Gillespie D T.Stochastic simulation of chemical kinetics[J].Annu Rev Phys Chem,2007,58:35-55.
[15]Li Hong,Cao Yang,Petzold L R,et al.Algorithms and software for stochastic simulation of biochemical reacting systems[J].Biotech Prog,2008,24(1):56-61.
[16]Mc Adams H H,Arkin A P.Stochastic mechanisms in gene expression[J].Proc Natl Acad Sci,1997,94(3):814-819.
[17]Mc Adams H H,Arkin A P.It’s a noisy business!Genetic regulation at the nanomolar scale[J].Trends Genet,1999,15(2):65-69.
[18]Raj A,van Oudenaarden A.Nature,nurture,or chance:stochastic gene expression and its consequences[J].Cell,2008,135(2):216-226.
[19]Rao C V,Wolf D M,Arkin A P.Control,exploitation and tolerance of intracellular noise[J].Nature,2002,420(6912):231-237.
[20]Samoilov M S,Price G,Arkin A P.From fluctuations to phenotypes:the physiology of noise[J].Sci Stke,2006,2006(366):17.
[21]Shahrezaei V,Swain P S.The stochastic nature of biochemical networks[J].Curr Opin Biotechnol,2008,19(4):369-374.
[22]Pedraza J M,van Oudenaarden A.Noise propagation in gene networks[J].Science,2005,307(5717):1965-1969.
[23]周天寿.生物系统的随机动力学[M].北京:科学出版社,2009.
[24]Kaern M,Elston T C,Blake W J,et al.Stochasticity in gene expression:from theories to phenotypes[J].Nat Rev Genet,2005,6(6):451-464.
[25]Elowitz M B,Levine A J,Siggia E D,et al.Stochastic gene expression in a single cell[J].Science,2002,297(5584):1183-1186.
[26]Paulsson J.Summing up the noise in gene networks[J].Nature,2004,427(6973):415-418.
[27]Golding I,Paulsson J,Zawilski S M,et al.Real-time kinetics of gene activity in individual bacteria[J].Cell,2005,123(6):1025-1036.
[28]Moran U,Phillips R,Milo R.Snapshot:key numbers in biology[J].Cell,2010,141(7):1262.
[29]Lei Jinzhi.Stochasticmodeling in systems biology[J].J Adv Math Appl,2012,1:76-88.
[30]雷锦誌.系统生物学:建模,分析,模拟[M].上海:上海科学技术出版社,2010.
[31]Schlicht R,Winkler G.A delay stochastic process with applications in molecular biology[J].J Math Biol,2008,57(5):613-648.
[32]Oppenheim I,Shuler K E,Weiss G H.Stochastic and deterministic formulation of chemical rate equations[J].J Chem Phys,1969,50(1):460-466.
[33]Gillespie D T.The chemical Langevin equation[J].J Chem Phys,2000,113(1):297-306.
[34]van Kampen N.Stochastic processes in physics and chemistry[M].3rd ed.Amsterdam:North-Holland,2007.
[35]Jahnke T,Huisinga W.Solving the chemical master equation for monomolecular reaction systems analytically[J].J Math Biol,2007,54(1):1-26.
[36]Shahrezaei V,Swain P S.Analytical distributions for stochastic gene expression[J].Proc Natl Acad Sci USA,2008,105(45):17256-17261.
[37]Lipshtat A,Loinger A,Balaban N Q,et al.Genetic toggle switch without cooperative binding[J].Phys Rev Lett,2006,96(18):188101.
[38]Vellela M,Qian Hong.A quasistationary analysis of a stochastic chemical reaction:Keizer’s paradox[J].Bull Math Biol,2007,69(5):1727-1746.
[39]ksendal B.Stochastic differential equations[M].6th ed.Berlin:Springer,2005.
[40]Gillespie D T.Markov processes:an introduction for physical scientists[M].San Diego:Academic Press,1992.
[41]Uhlenbeck G E,Ornstein L S.On the theory of thebrownian motion[J].Physical Review,1930,36:823-841.
[42]van Kampen N.Langevin-like equation with colored noise[J].J Stastis Phys,1989,54(5/6):1289-1308.
[43]Crow E L,Shimizu K.Lognormal distributions:theory and applications[M].New York:Marcel Dekker Inc,1988.
[44]Limpert E,Stahel W A,Abbt M.Log-normal distributions across the sciences:keys and clues[J].Bio Science,2001,51(5):341-352.
[45]Rosenfeld N,Young J W,Alon U,et al.Gene regulation at the single-cell level[J].Science,2005,307(5717):1962-1965.
[46]Sommer S S,Rin N A.The lognormal distribution fits the decay profile[J].Biochem Biophys Res Commu,1979,90(1):135-141.
[47]Tian Tianhai,Burrange K,Burrange P M,et al.Stochastic delay differentiation equations for genetic regulatory networks[J].J Comput Apply Math,2007,205(2):696-707.
[48]Zhang Xuan,Jin Huiqin,Yang Zhuoqin,et al.Effects of elongation delay in transcription dynamics[J].Mathematical Biosciences and Engineering,2014,11(6):1431-1448.
[49]Gillespie D T.A general method for numerically simulating the stochastic time evolution of coupled chemical reactions[J].J Comput Phys,1976,22(4):403-434.
[50]Gillespie D T.Exact stochastic simulation of coupled chemical reactions[J].J Phys Chem,1977,81(25):2340-2361.
[51]Gillespie D T.Approximate accelerated stochastic simulation of chemically reacting systems[J].J Chem Phys,2001,115(4):1716-1733.
[52]Gillespie D T,Petzold L R.Improved leap-size selection for accelerated stochastic simulation[J].J Chem Phys,2003,119(16):8229-8234.
[53]Cao Yang,Gillespie D T,Petzold L R.Efficient step size selection for the tau-leaping simulation method[J].J Chem Phys,2006,124(4):044109.
[54]Chatterjee A,Vlachos D,Katsoulakis M.Binomial distribution basedτ-leap accelerated stochastic simulation[J].J Chem Phys,2005,122(2):024112.
[55]Tian Tianhai,Burrange K.Binomial leap methods for simulating stochastic chemical kinetics[J].J Chem Phys,2004,121(21):10356-10364.
[56]Cao Yang,Gillespie D T,Petzold L R.Avoiding negative populations in explicit poisson tau-leaping[J].J Chem Phys,2005,123(5):054104.
[57]Cao Yang,Gillespie D T,Petzold L R.Accelerated stochastic simulation of the stiff enzyme-substrate reaction[J].J Chem Phys,2005,123(14):144917.
[58]Cao Yang,Gillespie D T,Petzold L R.The slow-scale stochastic simulation algorithm[J].J Chem Phys,2005,122(1):014116.
[59]Goutsias J.Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems[J].J Chem Phys,2005,122(18):184102.
[60]Hu Yucheng,Abdulle A,Li Tiejun.Bossted hybrid method for solving chemical reaction systems with multiple scales in time and population size[J].Commun Comput Phys,2012,12(4):981-1005.
[61]Rao C V,Arkin A P.Stochastic chemical kinetics and the quasi-steady-state assumption:application to the Gillespie algorithm[J].J Chem Phys,2003,118(11):4999-5010.
[62]Samant A,Vlachos D G.Overcoming stiffness in stochastic simulation stemming from partial equilibrium:a multiscale monte carlo algorithm[J].J Chem Phys,2005,123(14):144114.
[63]Alfonsi A,Cancés E,Turinici G,et al.Exact simulation of hybrid stochastic and deterministic models for biochemical systems[R].Paris:Research Report RR-5435 INRIA-00070572,2004.
[64]Erban R,Kevrekidis I G,Adalsteinsson D,et al.Gene regulatory networks:a coarse-grained,equation-free approach to multiscale computation[J].J Chem Phys,2006,124(8):084106.
[65]Haseltine E L,Rawlings J B.Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics[J].J Chem Phys,2002,117(15):6959-6969.
[66]Salis H,Kaznessis Y.Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions[J].J Chem Phys,2005,122(5):054103.
[67]Lv Cheng,Li Xiaoguang,Li Fangting,et al.Constructing the energy landscape for genetic switching system driven by intrinsic noise[J].PLo S One,2014,9(2):e88167.
[68]Deudlhard P,Huisinga W,Jahnke T,et al.Adaptive discrete galerkin methods applied to the chemical master equation[J].SIAM J Sci Comput,2008,30(6):2990-3011.
[69]Engblom S.Galerkin spectral method applied to the chemical master equation[J].Commun Comput Phys,2009,5(5):871-896.
[70]Engblom S.Spectral approximation of solutions to the chemical master equation[J].J Comput Appl Math,2009,229(1):208-221.
[71]Hegland M,Hellander A,Ltstedt P.Sparse grids and hybrid methods for the chemical master equation[J].2008,48(2):265-283.
[72]Jahnke T.An adaptive wavelet method for the chemical master equation[J].SIAM J Sci Comput,2010,31(6):4373-4394.
[73]Jahnke T,Huisinga W.A dynamical low-rank approach to the chemical master equation[J].Bull Math Biol,2008,70(8):2283-2302.
[74]Khoo C F,Hegland M.The total quasi-steady state assumption:its justification by singular perturbation and its application to the chemical master equation[J].ANZIAM J,2008,50:429-443.
[75]Munsky B,Khammash M.A multiple time interval finite state projection algorithm for the solution to the chemical master equation[J].J Comput Phys,2007,226(1):818-835.
[76]Zhou Tianshou,Zhang Jiajun.Analytical results for a multistate gene model[J].SIAM J Appl Math,2012,72(3):789-818.
[77]Kloeden P E,Platen E.Numerical solutions of stochastic differential equation[M].New York:Springer-Verlag,1992.
[78]周天寿.概率主方程的研究综述[J].江西师范大学学报:自然科学版,2015,39(1):1-6.
[79]周天寿.基因表达模型的研究进展:概率分布[J].江西师范大学学报:自然科学版,2012,36(3):221-229.
[2]Choi P J,Cai Long,Frieda K,et al.A stochastic singlemolecule event triggers phenotype switching of a bacterial cell[J].Science,2008,322(5900):442-446.
[3]Elf J,Li Gene Wei,Xie X Sunney.Probing transcription factor dynamics at the single-molecule level in a living cell[J].Science,2007,316(5828):1191-1194.
[4]Xie X Sunney,Choi P J,Li Gene Wei,et al.Single-molecule approach to molecular biology in living bacterial cells[J].Annual Review of Biophysics,2008,37:417-444.
[5]Yu Ji,Xiao Jie,Ren Xiaojia,et al.Probing gene expression in live cells,one protein molecule at a time[J].Science,2006,311(5767):1600-1603.
[6]Taniguchi Y,Choi P J,Li Gene Wei,et al.Quantifying e.coli proteome and transcriptome with single-molecule sensitivity in single cells[J].Science,2010,329(5):533-538.
[7]Bassett A R,Liu Jilong.Crispr/cas9 and genome editing in drosophila[J].Journal of Genetics and Genomics,2014,41(1):7-19.
[8]Mali P,Esvelt K M,Church G M.Cas9 as a versatile tool for engineering biology[J].Nature Methods,2013,10(10):957-963.
[9]Amabile G,Meissner A.Induced pluripotent stem cells:current progress and potential for regenerative medicine[J].Trends in Molecular Medicine,2009,15(2):59-68.
[10]Efe J A,Hilcove S,Kim J,et al.Conversion of mouse fibroblasts into cardiomyocytes using a direct reprogramming strategy[J].Nature Cell Biology,2011,13(3):215-222.
[11]Takahashi K,Yamanaka S.Induction of pluripotent stem cells from mouse embryonic and adult fibroblast cultures by defined factors[J].Cell,2006,126(4):663-676.
[12]Takahashi K,Tanabe K,Ohnuki M,et al.Induction of pluripotent stem cells from adult human fibroblasts by defined factors[J].Cell,2007,131(5):861-872.
[13]Fiering S,Whitelaw E,Martin D.To be or not to be active:the stochastic nature of enhancer action[J].Bioessays,2000,22(4):381-387.
[14]Gillespie D T.Stochastic simulation of chemical kinetics[J].Annu Rev Phys Chem,2007,58:35-55.
[15]Li Hong,Cao Yang,Petzold L R,et al.Algorithms and software for stochastic simulation of biochemical reacting systems[J].Biotech Prog,2008,24(1):56-61.
[16]Mc Adams H H,Arkin A P.Stochastic mechanisms in gene expression[J].Proc Natl Acad Sci,1997,94(3):814-819.
[17]Mc Adams H H,Arkin A P.It’s a noisy business!Genetic regulation at the nanomolar scale[J].Trends Genet,1999,15(2):65-69.
[18]Raj A,van Oudenaarden A.Nature,nurture,or chance:stochastic gene expression and its consequences[J].Cell,2008,135(2):216-226.
[19]Rao C V,Wolf D M,Arkin A P.Control,exploitation and tolerance of intracellular noise[J].Nature,2002,420(6912):231-237.
[20]Samoilov M S,Price G,Arkin A P.From fluctuations to phenotypes:the physiology of noise[J].Sci Stke,2006,2006(366):17.
[21]Shahrezaei V,Swain P S.The stochastic nature of biochemical networks[J].Curr Opin Biotechnol,2008,19(4):369-374.
[22]Pedraza J M,van Oudenaarden A.Noise propagation in gene networks[J].Science,2005,307(5717):1965-1969.
[23]周天寿.生物系统的随机动力学[M].北京:科学出版社,2009.
[24]Kaern M,Elston T C,Blake W J,et al.Stochasticity in gene expression:from theories to phenotypes[J].Nat Rev Genet,2005,6(6):451-464.
[25]Elowitz M B,Levine A J,Siggia E D,et al.Stochastic gene expression in a single cell[J].Science,2002,297(5584):1183-1186.
[26]Paulsson J.Summing up the noise in gene networks[J].Nature,2004,427(6973):415-418.
[27]Golding I,Paulsson J,Zawilski S M,et al.Real-time kinetics of gene activity in individual bacteria[J].Cell,2005,123(6):1025-1036.
[28]Moran U,Phillips R,Milo R.Snapshot:key numbers in biology[J].Cell,2010,141(7):1262.
[29]Lei Jinzhi.Stochasticmodeling in systems biology[J].J Adv Math Appl,2012,1:76-88.
[30]雷锦誌.系统生物学:建模,分析,模拟[M].上海:上海科学技术出版社,2010.
[31]Schlicht R,Winkler G.A delay stochastic process with applications in molecular biology[J].J Math Biol,2008,57(5):613-648.
[32]Oppenheim I,Shuler K E,Weiss G H.Stochastic and deterministic formulation of chemical rate equations[J].J Chem Phys,1969,50(1):460-466.
[33]Gillespie D T.The chemical Langevin equation[J].J Chem Phys,2000,113(1):297-306.
[34]van Kampen N.Stochastic processes in physics and chemistry[M].3rd ed.Amsterdam:North-Holland,2007.
[35]Jahnke T,Huisinga W.Solving the chemical master equation for monomolecular reaction systems analytically[J].J Math Biol,2007,54(1):1-26.
[36]Shahrezaei V,Swain P S.Analytical distributions for stochastic gene expression[J].Proc Natl Acad Sci USA,2008,105(45):17256-17261.
[37]Lipshtat A,Loinger A,Balaban N Q,et al.Genetic toggle switch without cooperative binding[J].Phys Rev Lett,2006,96(18):188101.
[38]Vellela M,Qian Hong.A quasistationary analysis of a stochastic chemical reaction:Keizer’s paradox[J].Bull Math Biol,2007,69(5):1727-1746.
[39]ksendal B.Stochastic differential equations[M].6th ed.Berlin:Springer,2005.
[40]Gillespie D T.Markov processes:an introduction for physical scientists[M].San Diego:Academic Press,1992.
[41]Uhlenbeck G E,Ornstein L S.On the theory of thebrownian motion[J].Physical Review,1930,36:823-841.
[42]van Kampen N.Langevin-like equation with colored noise[J].J Stastis Phys,1989,54(5/6):1289-1308.
[43]Crow E L,Shimizu K.Lognormal distributions:theory and applications[M].New York:Marcel Dekker Inc,1988.
[44]Limpert E,Stahel W A,Abbt M.Log-normal distributions across the sciences:keys and clues[J].Bio Science,2001,51(5):341-352.
[45]Rosenfeld N,Young J W,Alon U,et al.Gene regulation at the single-cell level[J].Science,2005,307(5717):1962-1965.
[46]Sommer S S,Rin N A.The lognormal distribution fits the decay profile[J].Biochem Biophys Res Commu,1979,90(1):135-141.
[47]Tian Tianhai,Burrange K,Burrange P M,et al.Stochastic delay differentiation equations for genetic regulatory networks[J].J Comput Apply Math,2007,205(2):696-707.
[48]Zhang Xuan,Jin Huiqin,Yang Zhuoqin,et al.Effects of elongation delay in transcription dynamics[J].Mathematical Biosciences and Engineering,2014,11(6):1431-1448.
[49]Gillespie D T.A general method for numerically simulating the stochastic time evolution of coupled chemical reactions[J].J Comput Phys,1976,22(4):403-434.
[50]Gillespie D T.Exact stochastic simulation of coupled chemical reactions[J].J Phys Chem,1977,81(25):2340-2361.
[51]Gillespie D T.Approximate accelerated stochastic simulation of chemically reacting systems[J].J Chem Phys,2001,115(4):1716-1733.
[52]Gillespie D T,Petzold L R.Improved leap-size selection for accelerated stochastic simulation[J].J Chem Phys,2003,119(16):8229-8234.
[53]Cao Yang,Gillespie D T,Petzold L R.Efficient step size selection for the tau-leaping simulation method[J].J Chem Phys,2006,124(4):044109.
[54]Chatterjee A,Vlachos D,Katsoulakis M.Binomial distribution basedτ-leap accelerated stochastic simulation[J].J Chem Phys,2005,122(2):024112.
[55]Tian Tianhai,Burrange K.Binomial leap methods for simulating stochastic chemical kinetics[J].J Chem Phys,2004,121(21):10356-10364.
[56]Cao Yang,Gillespie D T,Petzold L R.Avoiding negative populations in explicit poisson tau-leaping[J].J Chem Phys,2005,123(5):054104.
[57]Cao Yang,Gillespie D T,Petzold L R.Accelerated stochastic simulation of the stiff enzyme-substrate reaction[J].J Chem Phys,2005,123(14):144917.
[58]Cao Yang,Gillespie D T,Petzold L R.The slow-scale stochastic simulation algorithm[J].J Chem Phys,2005,122(1):014116.
[59]Goutsias J.Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems[J].J Chem Phys,2005,122(18):184102.
[60]Hu Yucheng,Abdulle A,Li Tiejun.Bossted hybrid method for solving chemical reaction systems with multiple scales in time and population size[J].Commun Comput Phys,2012,12(4):981-1005.
[61]Rao C V,Arkin A P.Stochastic chemical kinetics and the quasi-steady-state assumption:application to the Gillespie algorithm[J].J Chem Phys,2003,118(11):4999-5010.
[62]Samant A,Vlachos D G.Overcoming stiffness in stochastic simulation stemming from partial equilibrium:a multiscale monte carlo algorithm[J].J Chem Phys,2005,123(14):144114.
[63]Alfonsi A,Cancés E,Turinici G,et al.Exact simulation of hybrid stochastic and deterministic models for biochemical systems[R].Paris:Research Report RR-5435 INRIA-00070572,2004.
[64]Erban R,Kevrekidis I G,Adalsteinsson D,et al.Gene regulatory networks:a coarse-grained,equation-free approach to multiscale computation[J].J Chem Phys,2006,124(8):084106.
[65]Haseltine E L,Rawlings J B.Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics[J].J Chem Phys,2002,117(15):6959-6969.
[66]Salis H,Kaznessis Y.Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions[J].J Chem Phys,2005,122(5):054103.
[67]Lv Cheng,Li Xiaoguang,Li Fangting,et al.Constructing the energy landscape for genetic switching system driven by intrinsic noise[J].PLo S One,2014,9(2):e88167.
[68]Deudlhard P,Huisinga W,Jahnke T,et al.Adaptive discrete galerkin methods applied to the chemical master equation[J].SIAM J Sci Comput,2008,30(6):2990-3011.
[69]Engblom S.Galerkin spectral method applied to the chemical master equation[J].Commun Comput Phys,2009,5(5):871-896.
[70]Engblom S.Spectral approximation of solutions to the chemical master equation[J].J Comput Appl Math,2009,229(1):208-221.
[71]Hegland M,Hellander A,Ltstedt P.Sparse grids and hybrid methods for the chemical master equation[J].2008,48(2):265-283.
[72]Jahnke T.An adaptive wavelet method for the chemical master equation[J].SIAM J Sci Comput,2010,31(6):4373-4394.
[73]Jahnke T,Huisinga W.A dynamical low-rank approach to the chemical master equation[J].Bull Math Biol,2008,70(8):2283-2302.
[74]Khoo C F,Hegland M.The total quasi-steady state assumption:its justification by singular perturbation and its application to the chemical master equation[J].ANZIAM J,2008,50:429-443.
[75]Munsky B,Khammash M.A multiple time interval finite state projection algorithm for the solution to the chemical master equation[J].J Comput Phys,2007,226(1):818-835.
[76]Zhou Tianshou,Zhang Jiajun.Analytical results for a multistate gene model[J].SIAM J Appl Math,2012,72(3):789-818.
[77]Kloeden P E,Platen E.Numerical solutions of stochastic differential equation[M].New York:Springer-Verlag,1992.
[78]周天寿.概率主方程的研究综述[J].江西师范大学学报:自然科学版,2015,39(1):1-6.
[79]周天寿.基因表达模型的研究进展:概率分布[J].江西师范大学学报:自然科学版,2012,36(3):221-229.