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Parameter sensitivity analysis for biochemical reaction networks

1 School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
2 Mathematics Institute & Zeeman Institute for Systems Biology and Infectious Epidemiology Research, University of Warwick, Coventry CV4 7AL, UK

Special Issues: Mathematical Methods in the Biosciences

Biochemical reaction networks describe the chemical interactions occurring between molecular populations inside the living cell. These networks can be very noisy and complex and they often involve many variables and even more parameters. Parameter sensitivity analysis that studies the effects of parameter changes to the behaviour of biochemical networks can be a powerful tool in unravelling their key parameters and interactions. It can also be very useful in designing experiments that study these networks and in addressing parameter identifiability issues. This article develops a general methodology for analysing the sensitivity of probability distributions of stochastic processes describing the time-evolution of biochemical reaction networks to changes in their parameter values. We derive the coefficients that efficiently summarise the sensitivity of the probability distribution of the network to each parameter and discuss their properties. The methodology is scalable to large and complex stochastic reaction networks involving many parameters and can be applied to oscillatory networks. We use the two-dimensional Brusselator system as an illustrative example and apply our approach to the analysis of the Drosophila circadian clock. We investigate the impact of using stochastic over deterministic models and provide an analysis that can support key decisions for experimental design, such as the choice of variables and time-points to be observed.
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Keywords parameter sensitivity analysis; reaction networks; oscillation; molecular biology

Citation: Giorgos Minas, David A Rand. Parameter sensitivity analysis for biochemical reaction networks. Mathematical Biosciences and Engineering, 2019, 16(5): 3965-3987. doi: 10.3934/mbe.2019196


  • 1. J-T Hwang, E. P. Dougherty, S. Rabitz, et al., The Green's function method of sensitivity analysis in chemical kinetics, J. Chem. Phys., 69 (1978), 5180–5191.
  • 2. J. Stelling, U. Sauer, Z. Szallasi, et al., Robustness of Cellular Functions, Cell, 118 (2004), 675–685.
  • 3. R. N. Gutenkunst, J. J. Waterfall, F. P. Casey, et al., Universally Sloppy Parameter Sensitivities in Systems Biology Models, PLoS Comput. Biol., 3 (2007), e189.
  • 4. M. B. Elowitz, A. J. Levine, E. D. Siggia, et al., Stochastic Gene Expression in a Single Cell,Science, 297 (2002), 1183–1186.
  • 5. J. K. Kim and J. C. Marioni, Inferring the kinetics of stochastic gene expression from single-cell RNA-sequencing data, Genome. Biol., 14 (2013), R7.
  • 6. F. Wimmers, N. Subedi, N. van Buuringen, et al., Single-cell analysis reveals that stochasticity and paracrine signaling control interferon-alpha production by plasmacytoid dendritic cells, Nature Commun., 9 (2018), 3317.
  • 7. S. Plyasunov and A. P. Arkin, Efficient stochastic sensitivity analysis of discrete event systems, J. Comput. Phys., 221 (2007), 724–738.
  • 8. D. F. Anderson, An Efficient Finite Difference Method for Parameter Sensitivities of Continuous Time Markov Chains, SIAM J. Numer. Anal. 50 (2012), 2237–2258.
  • 9. E. S. Wolf and D. F. Anderson, A finite difference method for estimating second order parameter sensitivities of discrete stochastic chemical reaction networks, J. Chem. Phys., 137 (2012), 224112.
  • 10. A. Gupta, M. Rathinam and M. Khammash, Estimation of parameter sensitivities for stochastic reaction networks using tau-leap simulations, SIAM J. Numer. Anal., 56 (2014), 1134–1167.
  • 11. P. Bauer and S. Engblom, Sensitivity estimation and inverse problems in spatial stochastic models of chemical kinetics, Numerical Mathematics and Advanced Applications: ENUMATH 2013, A. AbdulleandS.DeparisandD.KressnerandF.NobileandM.Picasso(eds), 103(2015), 519–527.
  • 12. A. Gupta and M. Khammash, Sensitivity Analysis for Multiscale Stochastic Reaction Networks Using Hybrid Approximations, Bull. Math. Biol., (2018), 1–38.
  • 13. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340–2361.
  • 14. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, Third Edition, Amsterdam: Elsevier. Boston and London: Elsevier; 2007.
  • 15. Y. Cao, D. T. Gillespie and L. R. Petzold, Efficient step size selection for the tau-leaping simulation method, J. Chem. Phys., 124 (2006), 044109.
  • 16. D. T. Gillespie, The chemical Langevin equation, J. Chem. Phys., 113 (2000), 297–306.
  • 17. M. Scott, B. Ingalls and M. Kærn, Estimations of intrinsic and extrinsic noise in models of nonlinear genetic networks, Chaos, 16 (2006), 026107.
  • 18. R. Grima, Linear-noise approximation and the chemical master equation agree up to second-order moments for a class of chemical systems, Phys. Rev. E., 92 (2015), 042124.
  • 19. M. Komorowski, B. Finkenstädt, C. V. Harper, et al., Bayesian inference of biochemical kinetic parameters using the linear noise approximation, BMC Bioinform., 10 (2009), 343.
  • 20. B. Finkenstädt, D. J. Woodcock, M. Komorowski, et al., Quantifying intrinsic and extrinsic noise in gene transcription using the linear noise approximation: An application to single cell data, Ann. Appl. Stat., 7 (2013), 1960–1982.
  • 21. V. Stathopoulos and M. A. Girolami, Markov chain Monte Carlo inference for Markov jump processes via the linear noise approximation, Philos. Transact. A Math. Phys. Eng. Sci., 371 (2013), 20110541.
  • 22. P. Fearnhead, V. Giagos and C. Sherlock, Inference for reaction networks using the linear noise approximation Biometrics, 70 (2014), 457–466.
  • 23. M. Komorowski, M. J. Costa, D. A. Rand, et al., Sensitivity, robustness, and identifiability in stochastic chemical kinetics models P. Natl. Acad. Sci. USA, 108 (2011), 8645–8650.
  • 24. K. Tomita, T. Ohta and H. Tomita, Irreversible Circulation and Orbital Revolution: Hard Mode Instability in Far-from-Equilibrium Situation, Prog. Theor. Phys., 52 (1974), 1744–1765.
  • 25. R. P. Boland, T. Galla and A. J. McKane. How limit cycles and quasi-cycles are related in systems with intrinsic noise, J. Stat. Mech., 09 (2008), P09001.
  • 26. Y. Ito and K. Uchida, Formulas for intrinsic noise evaluation in oscillatory genetic networks, J. Theor. Biol., 267 (2010), 223–234.
  • 27. G. Minas and D. A. Rand, Long-time analytic approximation of large stochastic oscillators: Simulation, analysis and inference, PLoS Comput. Biol., 13 (2017), e1005676.
  • 28. D. A. Rand, Mapping global sensitivity of cellular network dynamics: sensitivity heat maps and a global summation law, J. R. Soc. Interface, 5 (2008), S59–S69.
  • 29. I. Prigogine and R. Lefever, Symmetry Breaking Instabilities in Dissipative Systems. II., J. Chem. Phys., 48 (1968), 1695–1700.
  • 30. D. Gonze, J. Halloy, J. C. Leloup, et al., Stochastic model for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour, CR Biol., 326 (2003), 189–203.
  • 31. T. M. Cover and J. A. Thomas, Elements of information theory (Wiley Series in Telecommunications and Signal Processing), 2nd edition, Wiley-Interscience New York, NY, USA, 2006.
  • 32. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
  • 33. D. T. Gillespie, A rigorous derivation of the chemical master equation, Physica A, 188 (1992), 404–425.
  • 34. D. T. Gillespie, The chemical Langevin equation, J. Chem. Phys. , 113 (2000), 297–306.
  • 35. D. F. Anderson and T. G. Kurtz, Continuous Time Markov Chain Models for Chemical Reaction Networks, in Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology (eds. H. Koeppl, G. Setti, M. di Bernardo, D. Densmore) New York: Springer (2011).
  • 36. T. G. Kurtz, Limit Theorems for Sequences of Jump Markov Processes Approximating Ordinary Differential Processes, J. Appl. Probab., 8 (1971), 344–356.
  • 37. T. G. Kurtz, Approximation of Population Processes, Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics. 36 (1981).
  • 38. A. Gupta and M. Khammash, An efficient and unbiased method for sensitivity analysis of stochastic reaction networks, J. R. Soc. Interface, 11 (2014), 20140979.
  • 39. E. W. J. Wallace, D. T. Gillespie, K. R. Sanft, et al., Linear noise approximation is valid over limited times for any chemical system that is sufficiently large, IET Syst. Biol., 6 (2012), 102–115.
  • 40. T. Philipp, M. Hannes and R. Grima, How reliable is the linear noise approximation of gene regulatory networks? BMC Genomics, 14, S5.


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