The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes stochastic effects into account. In this paper we give a simple argument showing that the solutions of a large class of chemical master equations are bounded in weighted ℓ1-spaces and possess high-order moments. This class includes all equations in which no reactions between two or more already present molecules and further external reactants occur that add mass to the system. As an illustration for the implications of this kind of regularity, we analyze the effect of truncating the state space. This leads to an error analysis for the finite state projections of the chemical master equation, an approximation that forms the basis of many numerical methods.
Mots clés : chemical master equation, existence of solutions and moments, error of finite state projections
@article{M2AN_2014__48_6_1757_0,
author = {Gauckler, Ludwig and Yserentant, Harry},
title = {Regularity and approximability of the solutions to the chemical master equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1757--1775},
publisher = {EDP-Sciences},
volume = {48},
number = {6},
year = {2014},
doi = {10.1051/m2an/2014018},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2014018/}
}
TY - JOUR AU - Gauckler, Ludwig AU - Yserentant, Harry TI - Regularity and approximability of the solutions to the chemical master equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 1757 EP - 1775 VL - 48 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014018/ DO - 10.1051/m2an/2014018 LA - en ID - M2AN_2014__48_6_1757_0 ER -
%0 Journal Article %A Gauckler, Ludwig %A Yserentant, Harry %T Regularity and approximability of the solutions to the chemical master equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1757-1775 %V 48 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014018/ %R 10.1051/m2an/2014018 %G en %F M2AN_2014__48_6_1757_0
Gauckler, Ludwig; Yserentant, Harry. Regularity and approximability of the solutions to the chemical master equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1757-1775. doi : 10.1051/m2an/2014018. http://www.numdam.org/articles/10.1051/m2an/2014018/
[1] , , and , Adaptive discrete Galerkin methods applied to the chemical master equation. SIAM J. Sci. Comput. 30 (2008) 2990-3011. | MR | Zbl
[2] and , Simultaneous state-time approximation of the chemical master equation using tensor product formats. arXiv:1311.3143 (2013).
[3] , Spectral approximation of solutions to the chemical master equation. J. Comput. Appl. Math. 229 (2009) 208-221. | MR | Zbl
[4] , A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22 (1976) 403-434. | MR
[5] , A rigorous derivation of the chemical master equation. Phys. A 188 (1992) 404-425.
[6] , Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem. 58 (2007) 35-55.
[7] , Approximating the solution of the chemical master equation by aggregation. ANZIAM J. 50 (2008) C371-C384. | MR
[8] and , On the numerical solution of the chemical master equation with sums of rank one tensors. ANZIAM J. Electron. Suppl. 52 (2010) C628-C643. | MR
[9] , and , Sparse grids and hybrid methods for the chemical master equation. BIT 48 (2008) 265-283. | MR | Zbl
[10] and , Hybrid method for the chemical master equation. J. Comput. Phys. 227 (2007) 100-122. | Zbl
[11] , Modeling and simulating chemical reactions. SIAM Rev., 50:347-368, 2008. | MR | Zbl
[12] , and , Numerical solution of stochastic models of biochemical kinetics. Can. Appl. Math. Q. 17 (2009) 523-554. | MR | Zbl
[13] , On reduced models for the chemical master equation. Multiscale Model. Simul. 9 (2011) 1646-1676. | MR | Zbl
[14] and , A dynamical low-rank approach to the chemical master equation. Bull. Math. Biol. 70 (2008) 2283-2302. | MR | Zbl
[15] and , Solving chemical master equations by adaptive wavelet compression. J. Comput. Phys. 229 (2010) 5724-5741. | MR | Zbl
[16] , , and , Direct solution of the chemical master equation using quantized tensor trains. PLoS Comput. Biol. 10 (2014) e1003359.
[17] and , Spectral theory for the differential equations of simple birth and death processes. Phil. Trans. R. Soc. A 246 (1954) 321-369. | MR | Zbl
[18] , and , Kolmogorov's differential equations and positive semigroups on first moment sequence spaces. J. Math. Biol. 53 (2006) 642-671. | MR | Zbl
[19] , , and , Hybrid stochastic-deterministic solution of the chemical master equation. Multiscale Model. Simul. 10 (2012) 1232-1262. | MR
[20] and , The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys. 124 (2006) 044104. | Zbl
[21] and , On the differential equations for the transition probabilities of Markov processes with enumerably many states. Proc. Cambridge Philos. Soc. 49 (1953) 247-262. | MR | Zbl
[22] and , An optimal finite state projection method. Procedia Comput. Sci. 1 (2012) 1579-1586.
[23] and , Stochastic semigroups: their construction by perturbation and approximation, in Positivity IV-theory and applications. Tech. Univ. Dresden, Dresden (2006) 135-146. | MR | Zbl
[24] , Numerical methods for the chemical master equation. Doctoral Thesis, Karlsruher Institut für Technologie (2012).
Cité par Sources :
