Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-08T00:16:26.477Z Has data issue: false hasContentIssue false

Regularity and approximability of the solutions to the chemicalmaster equation

Published online by Cambridge University Press:  03 October 2014

Ludwig Gauckler
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany. . gauckler@math.tu-berlin.de; yserentant@math.tu-berlin.de
Harry Yserentant
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany. . gauckler@math.tu-berlin.de; yserentant@math.tu-berlin.de
Get access

Abstract

The chemical master equation is a fundamental equation in chemical kinetics. It underliesthe classical reaction-rate equations and takes stochastic effects into account. In thispaper we give a simple argument showing that the solutions of a large class of chemicalmaster equations are bounded in weighted 1-spaces and possess high-ordermoments. This class includes all equations in which no reactions between two or morealready present molecules and further external reactants occur that add mass to thesystem. As an illustration for the implications of this kind of regularity, we analyze theeffect of truncating the state space. This leads to an error analysis for the finite stateprojections of the chemical master equation, an approximation that forms the basis of manynumerical methods.

Type
Research Article
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Deuflhard, P., Huisinga, W., Jahnke, T. and Wulkow, M., Adaptive discrete Galerkin methods applied to the chemical master equation. SIAM J. Sci. Comput. 30 (2008) 29903011. Google Scholar
S.V. Dolgov and B.N. Khoromskij, Simultaneous state-time approximation of the chemical master equation using tensor product formats. arXiv:1311.3143 (2013).
Engblom, S., Spectral approximation of solutions to the chemical master equation. J. Comput. Appl. Math. 229 (2009) 208221. Google Scholar
Gillespie, D.T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22 (1976) 403434. Google Scholar
Gillespie, D.T., A rigorous derivation of the chemical master equation. Phys. A 188 (1992) 404425. Google Scholar
Gillespie, D.T., Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem. 58 (2007) 3555. Google Scholar
Hegland, M., Approximating the solution of the chemical master equation by aggregation. ANZIAM J. 50 (2008) C371C384. Google Scholar
Hegland, M. and Garcke, J., On the numerical solution of the chemical master equation with sums of rank one tensors. ANZIAM J. Electron. Suppl. 52 (2010) C628C643. Google Scholar
Hegland, M., Hellander, A. and Lötstedt, P., Sparse grids and hybrid methods for the chemical master equation. BIT 48 (2008) 265283. Google Scholar
Hellander, A. and Lötstedt, P., Hybrid method for the chemical master equation. J. Comput. Phys. 227 (2007) 100122. Google Scholar
D.J. Higham, Modeling and simulating chemical reactions. SIAM Rev., 50:347–368, 2008.
Ilie, S., Enright, W.H. and Jackson, K.R., Numerical solution of stochastic models of biochemical kinetics. Can. Appl. Math. Q. 17 (2009) 523554. Google Scholar
Jahnke, T., On reduced models for the chemical master equation. Multiscale Model. Simul. 9 (2011) 16461676. Google Scholar
Jahnke, T. and Huisinga, W., A dynamical low-rank approach to the chemical master equation. Bull. Math. Biol. 70 (2008) 22832302. Google ScholarPubMed
Jahnke, T. and Udrescu, T., Solving chemical master equations by adaptive wavelet compression. J. Comput. Phys. 229 (2010) 57245741. Google Scholar
Kazeev, V., Khammash, M., Nip, M. and Schwab, Ch., Direct solution of the chemical master equation using quantized tensor trains. PLoS Comput. Biol. 10 (2014) e1003359. Google ScholarPubMed
Ledermann, W. and Reuter, G.E.H., Spectral theory for the differential equations of simple birth and death processes. Phil. Trans. R. Soc. A 246 (1954) 321369. Google Scholar
Martcheva, M., Thieme, H.R. and Dhirasakdanon, T., Kolmogorov’s differential equations and positive semigroups on first moment sequence spaces. J. Math. Biol. 53 (2006) 642671. Google ScholarPubMed
Menz, S., Latorre, J.C., Schütte, C. and Huisinga, W., Hybrid stochastic-deterministic solution of the chemical master equation. Multiscale Model. Simul. 10 (2012) 12321262. Google Scholar
Munsky, B. and Khammash, M., The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys. 124 (2006) 044104. Google Scholar
Reuter, G.E.H. and Ledermann, W., On the differential equations for the transition probabilities of Markov processes with enumerably many states. Proc. Cambridge Philos. Soc. 49 (1953) 247262. Google Scholar
Sunkara, V. and Hegland, M., An optimal finite state projection method. Procedia Comput. Sci. 1 (2012) 15791586. Google Scholar
H.R. Thieme and J. Voigt, Stochastic semigroups: their construction by perturbation and approximation, in Positivity IV-theory and applications. Tech. Univ. Dresden, Dresden (2006) 135–146.
T. Udrescu, Numerical methods for the chemical master equation. Doctoral Thesis, Karlsruher Institut für Technologie (2012).