BMC Systems Biology | |
A multiscale approximation in a heat shock response model of E. coli | |
Hye-Won Kang1  | |
[1] Mathematical Biosciences Institute, Ohio State University, Columbus, OH, USA | |
关键词: Heat shock; Reaction networks; Chemical reaction; Markov chains; Multiscale; | |
Others : 1143458 DOI : 10.1186/1752-0509-6-143 |
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received in 2011-11-17, accepted in 2012-11-07, 发布年份 2012 | |
【 摘 要 】
Background
A heat shock response model of Escherichia coli developed by Srivastava, Peterson, and Bentley (2001) has multiscale nature due to its species numbers and reaction rate constants varying over wide ranges. Applying the method of separation of time-scales and model reduction for stochastic reaction networks extended by Kang and Kurtz (2012), we approximate the chemical network in the heat shock response model.
Results
Scaling the species numbers and the rate constants by powers of the scaling parameter, we embed the model into a one-parameter family of models, each of which is a continuous-time Markov chain. Choosing an appropriate set of scaling exponents for the species numbers and for the rate constants satisfying balance conditions, the behavior of the full network in the time scales of interest is approximated by limiting models in three time scales. Due to the subset of species whose numbers are either approximated as constants or are averaged in terms of other species numbers, the limiting models are located on lower dimensional spaces than the full model and have a simpler structure than the full model does.
Conclusions
The goal of this paper is to illustrate how to apply the multiscale approximation method to the biological model with significant complexity. We applied the method to the heat shock response model involving 9 species and 18 reactions and derived simplified models in three time scales which capture the dynamics of the full model. Convergence of the scaled species numbers to their limit is obtained and errors between the scaled species numbers and their limit are estimated using the central limit theorem.
【 授权许可】
2012 Kang; licensee BioMed Central Ltd.
【 预 览 】
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【 参考文献 】
- [1]Kærn M, Elston T, Blakem W, Collins J: Stochasticity in gene expression: from theories to phenotypes. Nat Rev Genet 2005, 6(6):451-464.
- [2]Gillespie D: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 1976, 22(4):403-434.
- [3]Gillespie D: Exact stochastic simulation of coupled chemical reactions. J Phys Chem 1977, 81(25):2340-2361.
- [4]Rao C, Arkin A: Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J Chem Phys 2003, 118(11):4999-5010.
- [5]Haseltine E, Rawlings J: Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J Chemi Phys 2002, 117(15):6959-6969.
- [6]Cao Y, Gillespie D, Petzold L: Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J Comput Phys 2005, 206(2):395-411.
- [7]Pahle J: Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Briefings Bioinf 2009, 10(1):53-64.
- [8]Ball K, Kurtz T, Popovic L, Rempala G: Asymptotic analysis of multiscale approximations to reaction networks. Ann Appl Probability 2006, 16(4):1925-1961.
- [9]Kang HW, Kurtz T: Separation of time-scales and model reduction for stochastic reaction networks. 2012. arXiv preprint arXiv:1011.1672, to appear in Annals of Applied Probability
- [10]Crudu A, Debussche A, Radulescu O: Hybrid stochastic simplifications for multiscale gene networks. BMC Syst Biol 2009, 3:89. BioMed Central Full Text
- [11]Srivastava R, Peterson M, Bentley W: Stochastic kinetic analysis of the Escherichia coli stress circuit using σ32-targeted antisense. Biotechnol Bioeng 2001, 75(1):120-129.
- [12]Takahashi K, Kaizu K, Hu B, Tomita M: A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics 2004, 20(4):538-546.
- [13]Weinan E, Vanden-Eijnden E: Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J Chem Phys 2005, 123(19):194107.
- [14]Kang HW, Popovic L, Kurtz T: Central limit theorems and diffusion approximations for multiscale Markov chain models. 2012. arXiv preprint arXiv:1208.3783, submitted
- [15]Kurtz T: Averaging for martingale problems and stochastic approximation. Appl Stochastic Anal 1992, 186-209.
- [16]Van Kampen NG: Stochastic processes in physics and chemistry (North-Holland Personal Library). Elsevier; 2007.