Biochemical fluctuations, optimisation and the linear noise approximation

BMC Systems Biology

Alan J McKane¹  Pedro Mendes²  Joseph D Challenger¹  Jürgen Pahle³ 

[1] Theoretical Physics Division, School of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, UK;Virginia Bioinformatics Institute, Virginia Tech, Washington Street 0477, Blacksburg, VA 24061, USA;School of Computer Science and Manchester Centre for Integrative Systems Biology, The University of Manchester, 131 Princess Street, Manchester M1 7DN, UK

关键词: Systems biology;    Stochastic biochemical models;    Intrinsic noise;    COPASI;    Mitogen-activated kinases signalling;    Optimisation;    Linear noise approximation;   

Others  :  1143849
DOI  :  10.1186/1752-0509-6-86

 received in 2011-08-10, accepted in 2012-03-01,  发布年份 2012

【 摘 要 】

Background

Stochastic fluctuations in molecular numbers have been in many cases shown to be crucial for the understanding of biochemical systems. However, the systematic study of these fluctuations is severely hindered by the high computational demand of stochastic simulation algorithms. This is particularly problematic when, as is often the case, some or many model parameters are not well known. Here, we propose a solution to this problem, namely a combination of the linear noise approximation with optimisation methods. The linear noise approximation is used to efficiently estimate the covariances of particle numbers in the system. Combining it with optimisation methods in a closed-loop to find extrema of covariances within a possibly high-dimensional parameter space allows us to answer various questions. Examples are, what is the lowest amplitude of stochastic fluctuations possible within given parameter ranges? Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species? Unlike stochastic simulation methods, this has no requirement for small numbers of molecules and thus can be applied to cases where stochastic simulation is prohibitive.

Results

We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK. Using our method we were able to quickly find local maxima of covariances between particle numbers in the ERK model depending on the activities of phospho-MKKK and its corresponding phosphatase. With the p38 MAPK model our method was able to efficiently find conditions under which the coefficient of variation of the output of the signalling system, namely the particle number of Hsp27, could be minimised. We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

Conclusions

Our strategy is a practical method for the efficient investigation of fluctuations in biochemical models even when some or many of the model parameters have not yet been fully characterised.

【 授权许可】

   
2012 Pahle et al; licensee BioMed Central Ltd.

【 图 表 】

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

【 参考文献 】

• [1]McAdams HH, Arkin A: It's a noisy business! Genetic regulation at the nanomolar scale. Trends Genet 1999, 15(2):65-69.
• [2]Elowitz MB, Levine AJ, Siggia ED, Swain PS: Stochastic gene expression in a single cell. Science 2002, 297:1183-1186.
• [3]Kummer U, Krajnc B, Pahle J, Green AK, Dixon CJ, Marhl M: Transition from stochastic to deterministic behavior in calcium oscillations. Biophys J 2005, 89(3):1603-1611.
• [4]Rao CV, Wolf DW, Arkin AP: Control, exploitation and tolerance of intracellular noise. Nature 2002, 420:231-237.
• [5]Eldar A, Elowitz MB: Functional roles for noise in genetic circuits. Nature 2010, 467:167-173.
• [6]Munsky B, Khammash M: The finite state projection approach for the analysis of stochastic noise in gene networks. IEEE Trans Automat Contr 2008, 53:201-214.
• [7]Gillespie DT: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comp Phys 1976, 22:403-434.
• [8]Pahle J: Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Brief Bioinform 2009, 10:53-64.
• [9]Elf J, Ehrenberg M: Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res 2003, 13:2475-2484.
• [10]Tomioka R, Kimura H, Kobayashi TJ, Aihara K: Multivariate analysis of noise in genetic regulatory networks. J Theor Biol 2004, 229(4):501-521.
• [11]McKane AJ, Nagy JD, Newman TJ, Stefani MO: Amplified biochemical oscillations in cellular systems. J Stat Phys 2007, 128(1/2):165-191.
• [12]van Kampen N: A power series expansion of the master equation. Can J Phys 1961, 39(4):551-567.
• [13]van Kampen N: The expansion of the master equation. Adv Chem Phys 1976, 34:245-309.
• [14]van Kampen N: Stochastic Processes in Physics and Chemistry. Amsterdam: Elsevier; 1981.
• [15]Hayot F, Jayaprakash C: The linear noise approximation for molecular fluctuations within cells. Phys Biol 2004, 1(4):205-210.
• [16]Orrell D, Ramsey S, de Atauri P, Bolouri H: A method for estimating stochastic noise in large genetic regulatory networks. Bioinformatics 2005, 21(2):208-217.
• [17]Munsky B, Trinh B, Khammash M: Listening to the noise: random fluctuations reveal gene network parameters. Mol Syst Biol 2009, 5:318.
• [18]Mendes P, Kell DB: Non-linear optimization of biochemical pathways: applications to metabolic engineering and parameter estimation. Bioinformatics 1998, 14(10):869-883.
• [19]Moles CG, Mendes P, Banga JR: Parameter estimation in biochemical pathways: a comparison of global optimization methods. Genome Res 2003, 13(11):2467-2474.
• [20]Sahle S, Mendes P, Hoops S, Kummer U: A new strategy for assessing sensitivities in biochemical models. Phil Trans R Soc A 2008, 366(1880):3619-3631.
• [21]Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N, Singhal M, Xu L, Mendes P, Kummer U: COPASI - a COmplex PAthway SImulator. Bioinformatics 2006, 22(24):3067-3074.
• [22]COPASI [http://www.copasi.org] webcite
• [23]Kholodenko BN: Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades. Eur J Biochem 2000, 267:1583-1588.
• [24]Hendriks BS, Hua F, Chabot JR: Analysis of mechanistic pathway models in drug discovery: p38 pathway. Biotechnol Prog 2008, 24:96-109.
• [25]Hucka M, Finney A, Sauro HM, Bolouri H, Doyle JC, Kitano H, the rest of the SBML Forum, Arkin AP, Bornstein BJ, Bray D, Cornish-Bowden A, Cuellar AA, Dronov S, Gilles ED, Ginkel M, Gor V, Goryanin II, Hedley WJ, Hodgman TC, Hofmeyr JH, Hunter PJ, Juty NS, Kasberger JL, Kremling A, Kummer U, Le Novère N, Loew LM, Lucio D, Mendes P, Minch E, Mjolsness ED, Nakayama Y, Nelson MR, Nielsen PF, Sakurada T, Schaff JC, Shapiro BE, Shimizu TS, Spence HD, Stelling J, Takahashi K, Tomita M, Wagner J, Wang J: The systems biology markup language (SBML): a medium for representation and exchange of biochemical network models. Bioinformatics 2003, 19(4):524-531.
• [26]Fano U: Ionization yield of radiations. II. The fluctuations of the number of ions. Phys Rev 1947, 72:26-29.
• [27]Kennedy J, Eberhart R: Particle Swarm Optimization. Proceedings of the Fourth IEEE International Conference on Neural Networks, Perth, Australia 1995, 4:1942-1948.
• [28]Fogel DB, Fogel LJ, Atmar JW: Meta-evolutionary programming. In 25th Asilomar Conference on Signals, Systems and Computers. Volume 1. IEEE Computer Society, Asilomar; 1992::540-545.
• [29]Wolpert DH, Macready WG: No free lunch theorems for optimization. IEEE Trans Evolutionary Comput 1997, 1:67-82.
• [30]Lawrence MC, Jivan A, Shao C, Duan L, Goad D, Zaganjor E, Osborne J, McGlynn K, Stippec S, Earnest S, Chen W, Cobb MH: The roles of MAPKs in disease. Cell Res 2008, 18(4):436-442.
• [31]Kholodenko BN, Birtwistle MR: Four-dimensional dynamics of MAPK information-processing systems. Wiley Interdisciplinary Rev: Syst Biol Med 2009, 1:28-44.
• [32]Chang L, Karin M: Mammalian MAP kinase signalling cascades. Nature 2001, 410:37-40.
• [33]Komorowski M, Costa M, Rand D, Stumpf M: Sensitivity, robustness, and identifiability in stochastic chemical kinetics models. PNAS 2011, 108(21):8645-8650.
• [34]Grima R: An effective rate equation approach to reaction kinetics in small volumes: Theory and application to biochemical reactions in nonequilibrium steady-state conditions. J Chem Phys 2010, 133:035101.
• [35]Bartels RH, Stewart GW: Solution of the matrix equation AX + XB = C [F4]. Comm ACM 1972, 15(9):820-826.
• [36]Reder C: Metabolic control theory: a structural approach. J Theor Biol 1988, 135(2):175-201.
• [37]Funahashi A, Morohashi M, Kitano H, Tanimura N: CellDesigner: a process diagram editor for gene-regulatory and biochemical networks. BIOSILICO 2003, 1(5):159-162.
• [38]Funahashi A, Matsuoka Y, Jouraku A, Morohashi M, Kikuchi N, Kitano H: CellDesigner 3.5: a versatile modeling tool for biochemical networks. Proc of the IEEE 2008, 96(8):1254-1265.