Fast Adaptive Uniformization for the Chemical Master Equation

Maria-Emanuela-Canini Mateescu, Verena Wolf
Ieee Computer Soc Press, Customer Service Center, Po Box 3014, 10662 Los Vaqueros Circle, Los Alamitos, Ca 90720-1264 Usa, 2009
Article de conférence

Résumé

Within systems biology there is an increasing interest in the stochastic behavior of biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous-time Markov chain (CTMC). Standard Uniformization (SU) is an efficient method for the transient analysis of CTMCs. For systems with very different time scales, such as biochemical reaction networks, SU is computationally expensive. In these cases, a variant of SU, called adaptive uniformization (AU), is known to reduce the large number of iterations needed by SU. The additional difficulty of AU is that it requires the solution of a birth process. In this paper we present an on-the-fly variant of AU, where we improve the original algorithm for AU at the cost of a small approximation error. By means of several examples, we show that our approach is particularly well-suited for biochemical reaction networks.

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Fast adaptive uniformisation of the chemical master equation

Maria-Emanuela-Canini Mateescu, Verena Wolf

Within systems biology there is an increasing interest in the stochastic behaviour of biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous-time Markov chain (CTMC). The uniformisation technique is an efficient method to compute probability distributions of a CTMC if the number of states is manageable. However, the size of a CTMC that represents a biochemical reaction network is usually far beyond what is feasible. In this study, the authors present an on-the-fly variant of uniformisation, where they improve the original algorithm at the cost of a small approximation error. By means of several examples, the authors show that their approach is particularly well-suited for biochemical reaction networks.

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In this thesis we investigate different ways of approximating the solution of the chemical master equation (CME). The CME is a system of differential equations that models the stochastic transient behaviour of biochemical reaction networks. It does so by describing the time evolution of probability distribution over the states of a Markov chain that represents a biological network, and thus its stochasticity is only implicit. The transient solution of a CME is the vector of probabilities over the states of the corresponding Markov chain at a certain time point t, and it has traditionally been obtained by applying methods that are general to continuous-time Markov chains: uniformization, Krylov subspace methods, and general ordinary differential equation (ODE) solvers such as the fourth order Runge-Kutta method. Even though biochemical reaction networks are the main application of our work, some of our results are presented in the more general framework of propagation models (PM), a computational formalism that we introduce in the first part of this thesis. Each propagation model N has two associated propagation processes, one in discrete-time and a second one in continuous-time. These propagation processes propagate a generic mass through a discrete state space. For example, in order to model a CME, N propagates probability mass. In the discrete-time case the propagation is done step-wise, while in the continuous-time case it is done in a continuous flow defined by a differential equation. Again, in the case of the chemical master equation, this differential equation is the equivalent of the chemical master equation itself where probability mass is propagated through a discrete state space. Discrete-time propagation processes can encode methods such as the uniformization method and the fourth order Runge-Kutta integration method that we have mentioned above, and thus by optimizing propagation algorithms we optimize both of these methods simultaneously. In the second part of our thesis, we define stochastic hybrid models that approximate the stochastic behaviour of biochemical reaction networks by treating some variables of the system deterministically. This deterministic approximation is done for species with large populations, for which stochasticity does not play an important role. We propose three such hybrid models, which we introduce from the coarsest to the most refined one: (i) the first one replaces some variables of the system with their overall expectations, (ii) the second one replaces some variables of the system with their expectations conditioned on the values of the stochastic variables, (iii) and finally, the third one, splits each variable into a stochastic part (for low valuations) and a deterministic part (for high valuations), while tracking the conditional expectation of the deterministic part. For each of these algorithms we give the corresponding propagation models that propagate not only probabilities but also the respective continuous approximations for the deterministic variables.

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We present a numerical approximation technique for the analysis of continuous-time Markov chains that describe networks of biochemical reactions and play an important role in the stochastic modeling of biological systems. Our approach is based on the construction of a stochastic hybrid model in which certain discrete random variables of the original Markov chain are approximated by continuous deterministic variables. We compute the solution of the stochastic hybrid model using a numerical algorithm that discretizes time and in each step performs a mutual update of the transient probability distribution of the discrete stochastic variables and the values of the continuous deterministic variables. We implemented the algorithm and we demonstrate its usefulness and efficiency on several case studies from systems biology.

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