Abstract. We present an on-the-fly abstraction technique for infinite-state continuous-time Markov chains. We consider Markov chains that are specified by a finite set of transition classes. Such models naturally represent biochemical reactions and therefore play an important role in the stochastic modeling of biological systems. We approximate the transient probability distributions at various time instances by solving a sequence of dynamically constructed abstract models, each depending on the previous one. Each abstract model is a finite Markov chain that represents the behavior of the original, infinite chain during a specific time interval. Our approach provides complete information about probability distributions, not just about individual parameters like the mean. The error of each abstraction can be computed, and the precision of the abstraction refined when desired. We implemented the algorithm and demonstrate its usefulness and efficiency on several case studies from systems biology. 1

Abstract. Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discrete-state continuous-time stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive. We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required. 1

Abstract. We compare several languages for specifying Markovian population models such as queuing networks and chemical reaction networks. These languages —matrix descriptions, stochastic Petri nets, stochastic process algebras, stoichiometric equations, and guarded command models — all describe continuous-time Markov chains, but they differ according to important properties, such as compositionality, expressiveness and succinctness, executability, ease of use, and the support they provide for checking the well-formedness of a model and for analyzing a model. 1

neuronal fluctuations in a recurrent

Cataloguing-in-Publication entry

Extensive numerical simulation of a reaction--diffusion--system reveal an unusual system size dependence of the fluctuation magnitude. If \Omega denotes the system size parameter, e. g. particle number, fluctuations are usually predicted to be of order\Omega 0:5 (stable case) or\Omega 1 (diffusion--type case). In contrast, a scaling like\Omega 0:84 is observed in a combined birth--death and random--walk process, which is described by a multivariate chemical master equation and corresponds to the Fisher equation in the macroscopic limit. PACS: 05.40.+j; 87.10.+e; 82.20.-w The system. In this work we consider the following simple nonlinear reaction-- diffusion--system: Particles are distributed along one spatial coordinate and move by way of diffusion. They react according to the scheme A * ) 2A. Thus, the presence of A-particles at some location leads to further production, and at the same time reactions of two A-particles will destroy one of them. Macroscopic Description. Mea...

A measurement on a quantum system is said to cause the “collapse” of the quantum state vector or density matrix. An analogous collapse occurs with measurements on a classical stochastic process. This paper addresses the question of describing the response of a classical stochastic process when there is feedback from the output of a measurement to the input, and is intended to give a simplified model for quantum-mechanical processes that occur along a space-like reaction coordinate. The classical system can be thought of in physical terms as two counterflowing probability streams, which stochastically exchange probability currents in a way that the net probability current, and hence the overall probability, suitably interpreted, is conserved. The proposed formalism extends the mathematics of those stochastic processes describable with linear, singlestep, unidirectional transition probabilities, known as Markov chains and stochastic matrices. It is shown that a certain rearrangement and combination of the input and output of two stochastic matrices of the same order yields another matrix of the same type. Each measurement causes the partial collapse of the probability current distribution in the midst of such a process, giving rise to calculable, but non-Markov, values for the ensuing modification of the system’s output probability distribution. The paper concludes with an analysis of a simple classical probabilistic version of a so-called grandfather paradox. 1

the mathematical structure and algorithmic

Stochastic approaches for modelling