Solving the chemical master equation for monomolecular reaction systems analytically the date of receipt and acceptance should be inserted later Abstract The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce any property of the solution. The model class includes many interesting examples and may also be used as the starting point for the design of new numerical methods for the CME of more complex reaction systems.

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Stochastic reaction kinetics has increasingly been used to study cellular systems, with applications ranging from viral replication to gene regulatory networks and to signalling 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.

Abstract. The chemical master equation plays a fundamental role for the understanding of gene regulatory networks and other discrete stochastic reaction systems. Solving this equation numerically, however, is usually extremely expensive or even impossible due to the huge size of the state space. Thus, the chemical master equation must often be replaced by a reduced model which operates with a considerably smaller number of degrees of freedom but hopefully still provides the essential information about the dynamics of the full system. We prove error bounds for two reduced models which have previously been proposed in the literature. Based on the error analysis, an alternative model reduction approach for the chemical master equation is introduced and discussed, and its advantage is illustrated by numerical examples.Key words. Chemical master equation, model reduction, hybrid models, error bounds AMS subject classifications. 34K60, 65C20, 60J27, 92D251. Introduction. Many processes in nature can be considered as reaction systems in which d ∈ N different species interact via r ∈ N reaction channels. The time evolution of such a system is usually modeled by the traditional reaction-rate equations, a set of d coupled ordinary differential equations which indicate how the concentrations of the d species change in time. This approach is simple and computationally cheap, but fails in situations where the influence of inherent stochastic noise cannot be ignored, and where certain species have to be described in terms of integer particle numbers instead of real-valued, continuous concentrations. This is the case in gene regulatory networks, viral kinetics with few infectious individuals, and many other biological systems.The chemical master equation (CME) respects the discreteness and randomness of the problem and thus provides a more accurate model. The system is considered as a random variable Z(t) which evolves according to a Markov jump process on N

The convergence of an alternating direction implicit method for Maxwell's equations on product domains is investigated. Unlike the classical Yee scheme and most other integrators proposed in the literature, this method is both unconditionally stable and computationally cheap. We prove second-order convergence of the time-discretization in the framework of operator semigroup theory. In contrast to formal considerations based on Taylor expansions, our convergence analysis respects the unboundedness of the involved differential operators. The proofs are based on results concerning the regularity of the Cauchy problems, which then allow to apply an abstract convergence proof by Hansen and Ostermann [13].