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A Comparison of the Magnus Expansion and Other Solvers for the Chemical Master Equation with Variable Rates

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Recent Advances in Mathematical and Statistical Methods (AMMCS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 259))

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Abstract

Many traditional approaches for solving the chemical master equation (CME) cannot be used in their basic form when reaction rates change over time, for instance due to cell volume or temperature. One technique is to use the Magnus expansion to represent the solution to the CME as the action of a matrix exponential, for which Krylov-based approximation methods can be applied. In this paper, we compare two variants of the Magnus scheme with some popular ordinary differential equations (ODE) solvers, such as Adams-Bashforth, Runge-Kutta and Backward-differentiation formula (BDF). Our numerical tests show that the Magnus variants are remarkably efficient at computing the transient probability distributions of a transcriptional regulatory system where propensities vary over time due to cell volume increase.

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References

  1. Gillespie, D.: A rigorous derivation of the chemical master equation. Phys. A 188(1–3), 404–425 (1992)

    Article  Google Scholar 

  2. Munsky, B., Khammash, M.: The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys. 124(4), 044104 (2006)

    Article  Google Scholar 

  3. Gillespie, D.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)

    Article  Google Scholar 

  4. Purtan, R., Udrea, A.: A modified stochastic simulation algorithm for time-dependent intensity rates. In: 19th International Conference on Control Systems and Computer Science (2013)

    Google Scholar 

  5. Shampine, L.F., Gordon, M.K.: Computer solution of ordinary differential equations: the initial value problem. W.H. Freeman and Co. (1975)

    Google Scholar 

  6. Brankin, R.W., Gladwell, I., Shampine, L.F.: RKSUITE: a suite of Runge-Kutta codes for the initial value problem for ODEs, Softreport 91–1. Math. Dept., Southern Methodist University, Dallas, TX, USA, Technical report (1991)

    Google Scholar 

  7. Brown, P.N., Byrne, G.D., Hindmarsh, A.C.: VODE: a variable-coefficient ODE solver. SIAM J. Sci. Comput. 10(5), 1038–1051 (1989)

    Article  MathSciNet  Google Scholar 

  8. Magnus, W.: On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7(4), 649–673 (1954)

    Article  MathSciNet  Google Scholar 

  9. Iserles, A., Nørsett, S.P., Rasmussen, A.F.: Time symmetry and high-order Magnus methods. Appl. Numer. Math. 39(3–4), 379–401 (2001)

    Article  MathSciNet  Google Scholar 

  10. MacNamara, S., Burrage, K.: Stochastic modeling of naive T cell homeostasis for competing clonotypes via the master equation. Multiscale Model. Simul. 8(4), 1325–1347 (2010)

    Article  MathSciNet  Google Scholar 

  11. Sidje, R.B.: EXPOKIT: a software package for computing matrix exponentials. ACM Trans. Math. Softw. (TOMS) 24(1), 130–156 (1998)

    Article  Google Scholar 

  12. Sidje, R.B., Stewart, W.J.: A numerical study of large sparse matrix exponentials arising in Markov chains. Comput. Stat. Data Anal. 29(3), 345–368 (1999)

    Article  Google Scholar 

  13. Sidje, R.B., Vo, H.D.: Solving the chemical master equation by a fast adaptive finite state projection based on the stochastic simulation algorithm. Math. Biosci. 269, 10–16 (2015)

    Article  MathSciNet  Google Scholar 

  14. Dinh, K.N., Sidje R.B.: An adaptive Magnus expansion method for solving the Chemical Master Equation with time-dependent propensities. J. Coupled Syst. Multiscale Dyn. https://doi.org/10.1166/jcsmd.2017.1124.

  15. Goutsias, J.: Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122(18), 184102 (2005)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama, Tuscaloosa, USA

    Khanh Dinh & Roger Sidje

Authors
  1. Khanh Dinh
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  2. Roger Sidje
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Corresponding author

Correspondence to Khanh Dinh .

Editor information

Editors and Affiliations

  1. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada

    D. Marc Kilgour

  2. Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada

    Herb Kunze

  3. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada

    Roman Makarov

  4. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada

    Roderick Melnik

  5. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada

    Xu Wang

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Dinh, K., Sidje, R. (2018). A Comparison of the Magnus Expansion and Other Solvers for the Chemical Master Equation with Variable Rates. In: Kilgour, D., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds) Recent Advances in Mathematical and Statistical Methods . AMMCS 2017. Springer Proceedings in Mathematics & Statistics, vol 259. Springer, Cham. https://doi.org/10.1007/978-3-319-99719-3_24

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