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A Dynamical Low-Rank Approach to the Chemical Master Equation

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Abstract

Stochastic reaction kinetics have increasingly been used to study cellular systems, with applications ranging from viral replication to gene regulatory networks and to signaling pathways. The underlying evolution equation, known as the chemical master equation (CME), can rarely be solved with traditional methods due to the huge number of degrees of freedom. We present a new approach to directly solve the CME by a dynamical low-rank approximation based on the Dirac–Frenkel–McLachlan variational principle. The new approach has the capability to substantially reduce the number of degrees of freedom, and to turn the CME into a computationally tractable problem. We illustrate the accuracy and efficiency of our methods in application to two examples of biological interest.

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References

  • Alfonsi, A., Cancès, E., Turinici, G., Ventura, B.D., Huisinga, W., 2005. Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems. ESAIM Proc. 14, 1–3.

    MATH  Google Scholar 

  • Arkin, A.P., Ross, J., McAdams, H.H., 1998. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected. Escherichia coli cells. Genetics 149, 1633–648.

    Google Scholar 

  • Beck, M.H., Jäckle, A., Worth, G.A., Meyer, H.-D., 2000. The multiconfiguration time-dependent Hartree method: A highly efficient algorithm for propagating wavepackets. Phys. Rep. 324, 1–05.

    Article  Google Scholar 

  • Burrage, K., Tian, T., 2004. Poisson Runge–Kutta methods for chemical reaction systems. In: Sun, Y.L.W., Tang, T. (Eds.), Advances in Scientific Computing and Applications, pp. 82–6. Science Press, Beijing/New York.

    Google Scholar 

  • Burrage, K., Tian, T., Burrage, P., 2004. A multi-scaled approach for simulating chemical reaction systems. Prog. Biophys. Mol. Biol. 85, 217–34.

    Article  Google Scholar 

  • Burrage, K., Hegland, M., MacNamara, S., Sidje, R.B., 2006. A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems. In: Langville, A.N., Stewart, W.J. (Eds.), Markov Anniversary Meeting: An International Conference to Celebrate the 150th Anniversary of the Birth of A.A. Markov, pp. 21–8. Boson Books.

  • Cao, Y., Gillespie, D., Petzold, L., 2005. The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122(1), 014116.

    Article  Google Scholar 

  • Deuflhard, P., Wulkow, M., 1989. Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable. IMPACT Comput. Sci. Eng. 1(3), 269–01.

    Article  MATH  Google Scholar 

  • Deuflhard, P., Huisinga, W., Jahnke, T., Wulkow, M., 2008. Adaptive discrete Galerkin methods applied to the chemical master equation. SIAM J. Sci. Comput, accepted for publication.

  • Elowitz, M.B., Siggia, E.D., Swain, P.S., Levine, A.J., 2002. Stochastic gene expression in a single cell. Science 297, 1183–186.

    Article  Google Scholar 

  • Engblom, S., 2006. A discrete spectral method for the chemical master equation. Technical Report 2006-036, Uppsala University.

  • Gardiner, C.W., 2004. Handbook of Stochastic Methods, 2rd edn. Springer, Berlin.

    Google Scholar 

  • Gillespie, D.T., 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–34.

    Article  MathSciNet  Google Scholar 

  • Gillespie, D.T., 1977. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–361.

    Article  Google Scholar 

  • Gillespie, D.T., 1992. A rigorous derivation of the chemical master equation. Physica A 188, 404–25.

    Article  Google Scholar 

  • Gillespie, D.T., 2001. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–733.

    Article  Google Scholar 

  • Goutsias, J., 2005. Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122, 184102.

    Article  Google Scholar 

  • Hairer, E., Lubich, C., Wanner, G., 2006. Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin.

    MATH  Google Scholar 

  • Haseltine, E.L., Rawlings, J.B., 2002. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys. 117(15), 6959–969.

    Article  Google Scholar 

  • Hegland, M., Burden, C., Santoso, L., MacNamara, S., Booth, H., 2007. A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math. 205, 708–24.

    Article  MATH  MathSciNet  Google Scholar 

  • Jahnke, T., Huisinga, W., 2007. Solving the chemical master equation for monomolecular reaction systems analytically. J. Math. Biol. 54(1), 1–6.

    Article  MathSciNet  Google Scholar 

  • Koch, O., Lubich, C., 2007. Dynamical low rank approximation. SIAM J. Matrix Anal. Appl. 29, 434–54.

    Article  MathSciNet  Google Scholar 

  • Lathauwer, L.D., Moor, B.D., Vandewalle, J., 2000. A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–278.

    Article  MATH  MathSciNet  Google Scholar 

  • Liu, W.E.D., Vanden-Eijnden, E., 2005. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123, 194107.

    Article  Google Scholar 

  • Lubich, C., 2004. A variational splitting integrator for quantum molecular dynamics. Appl. Numer. Math. 48(3–4), 355–68.

    Article  MATH  MathSciNet  Google Scholar 

  • Lubich, C., 2005. On variational approximations in quantum molecular dynamics. Math. Comput. 74(250), 765–79.

    MATH  MathSciNet  Google Scholar 

  • MacNamara, S., Burrage, K., Sidje, R.B., 2008. Multiscale modeling of chemical kinetics via the master equation. SIAM J. Multiscale Model. Simul. 6(4), 1146–168.

    Article  MATH  MathSciNet  Google Scholar 

  • McAdams, H.H., Arkin, A.P., 1997. Stochastic mechanisms in gene expression. PNAS 94, 814–19.

    Article  Google Scholar 

  • McAdams, H.H., Arkin, A.P., 1999. It’s a noisy business! Genetic regulation at the nanomolar scale. Trends Genet. 15, 65–9.

    Article  Google Scholar 

  • Meyer, H.-D., Manthe, U., Cederbaum, L., 1990. The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165, 73–8.

    Article  Google Scholar 

  • Munsky, B., Khammash, M., 2006. The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys.

  • Nonnenmacher, A., Lubich, C., 2006. Dynamical low-rank approximation: applications and numerical experiments. Technical report, University of Tübingen.

  • Peles, S., Munsky, B., Khammash, M., 2006. Reduction and solution of the chemical master equation using time-scale separation and finite state projection. J. Chem. Phys. 125(20), 204104.

    Article  Google Scholar 

  • Rao, C.V., Arkin, A.P., 2003. Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J. Chem. Phys. 118(11), 4999–010.

    Article  Google Scholar 

  • Raser, J.M., O’Shea, E.K., 2004. Control of stochasticity in eukaryotic gene expression. Science 304, 1811–814.

    Article  Google Scholar 

  • Rathinam, M., Petzold, L., Cao, Y., Gillespie, D., 2003. Stiffness in stochastic chemically reacting systems: the implicit tau-leaping method. J. Chem. Phys. 119, 12784–2794.

    Article  Google Scholar 

  • Roussel, M.R., Zhu, R., 2004. Reducing a chemical master equation by invariant manifold methods. J. Chem. Phys. 121, 8716–730.

    Article  Google Scholar 

  • Salis, H., Kaznessis, Y., 2005. Accurate hybrid simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122.

  • Srivastava, R., You, L., Summers, J., Yin, J., 2002. Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theor. Biol. 218, 309–21.

    Article  MathSciNet  Google Scholar 

  • Steuer, R., 2004. Effects of stochasticity in models of the cell cycle: from quantized cycle times to noise-induced oscillations. J. Theor. Biol. 228, 293–01.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte und Numerische Mathematik, Universität Karlsruhe (TH), Englerstr. 2, 76128, Karlsruhe, Germany

    Tobias Jahnke

  2. Hamilton Institute, NUIM, Maynooth, Co. Kildare, Ireland

    Wilhelm Huisinga

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  1. Tobias Jahnke
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  2. Wilhelm Huisinga
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Correspondence to Tobias Jahnke.

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Jahnke, T., Huisinga, W. A Dynamical Low-Rank Approach to the Chemical Master Equation. Bull. Math. Biol. 70, 2283–2302 (2008). https://doi.org/10.1007/s11538-008-9346-x

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