# T:A:L:K:S

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 title: On hybrid models for stochastic reaction kinetics name: Jahnke first name: Tobias location/conference: SPP-JT09 WWW-link: http://www.mathematik.uni-karlsruhe.de/ianm3/~jahnke/ PREPRINT-link: http://www.mathematik.uni-karlsruhe.de/ianm3/~jahnke/ PRESENTATION-link: http://dfg-spp1324.de/download/jt09/talks/jahnke.pdf abstract: Many processes in systems biology can be modelled as reaction systems in which $d\in\mathbb{N}$ different species interact via $r\in\mathbb{N}$ reaction channels. In most applications the time evolution of such a system can be accurately described in terms of classical \emph{deterministic reaction kinetics}: The reaction system is translated into a system of $d$ ordinary differential equations (the reaction-rate equations), and the solution $y(t)\in\mathbb{R}^d$ indicates how the concentration or amount of each of the $d$ species changes in time. The traditional model is simple and computationally cheap, but fails in situations where the influence of stochastic noise cannot be ignored, and where the amount of certain species is so small that it must 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. \emph{Stochastic reaction kinetics} provides a more accurate description because it respects the discreteness and randomness of the system. The time evolution is modeled by a random variable $X(t)\in\mathbb{N}_0^d$ which evolves according to a Markov jump process. If $X(t)=z$ for some state $z=(z_1, \ldots, z_d)\in\mathbb{N}_0^d$ then exactly $z_i$ particles of the $i$-th species exist at time $t$. The object of interest is the probability $p(t,z) = \mathbb{P}(X(t)=z)$ that at time $t$ the system is in state $z\in\mathbb{N}^d$. The corresponding distribution $p$ is the solution of the chemical master equation, but solving this equation is a highly nontrivial problem because the solution has to be computed in each state of a high-dimensional state space. The idea of \emph{hybrid deterministic-stochastic models} is to interpolate between the accurate but computationally costly stochastic reaction kinetics and the simple but rather coarse deterministic description. In hybrid models, some part of the system (e.g. some of the species) is treated stochastically while the other part is represented in the deterministic setting. A particularly appealing hybrid model has recently been proposed by Andreas Hellander and Per L\"otstedt. In this talk, we discuss the properties of the Hellander-L\"otstedt model and sketch an extension which allows to overcome certain limitations.