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.

We propose a numerical technique for parameter inference in Markov models of biological processes. Based on time-series data of a process we estimate the kinetic rate constants by maximizing the likelihood of the data. The computation of the likelihood relies on a dynamic abstraction of the discrete state space of the Markov model which successfully mitigates the problem of state space largeness. We compare two variants of our method to state-of-the-art, recently published methods and demonstrate their usefulness and efficiency on several case studies from systems biology.

Based on the theory of stochastic chemical kinetics, the inherent randomness of biochemical reaction networks can be described by discrete-state continuous-time Markov chains. However, the analysis of such processes is computationally expensive and sophisticated numerical methods are required. Here, we propose an analysis framework in which we integrate a number of moments of the process instead of the state probabilities. This results in a very efficient simulation of the time evolution of the process. To regain the state probabilities from the moment representation, we combine the fast moment-based simulation with a maximum entropy approach for the reconstruction of the underlying probability distribution. We investigate the usefulness of this combined approach in the setting of stochastic chemical kinetics and present numerical results for three reaction networks showing its efficiency and accuracy. Besides a simple dimerization system, we study a bistable switch system and a multiattractor network with complex dynamics. ACM Reference Format:Alexander Andreychenko, Linar Mikeev, and Verena Wolf. 2015. Model reconstruction for moment-based stochastic chemical kinetics. ACM Trans. Model.

Recent experimental imaging techniques are able to tag and count molecular populations in a living cell. From these data mathematical models are inferred and calibrated. If small populations are present, discrete-state stochastic models are widely-used to describe the discreteness and randomness of molecular interactions. Based on time-series data of the molecular populations, the corresponding stochastic reaction rate constants can be estimated. This procedure is computationally very challenging, since the underlying stochastic process has to be solved for different parameters in order to obtain optimal estimates. Here, we focus on the maximum likelihood method and estimate rate constants, initial populations and parameters representing measurement errors.

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The stochastic dynamics of biochemical reaction networks can be accurately described by discrete-state Markov processes where each chemical reaction corresponds to a state transition of the process. Due to the largeness problem of the state space, analysis techniques based on an exploration of the state space are often not feasible and the integration of the moments of the underlying probability distribution has become a very popular alternative. In this paper the focus is on a comparison of reconstructed distributions from their moments obtained by two different moment-based analysis methods, the method of moments (MM) and the method of conditional moments (MCM). We use the maximum entropy principle to derive a distribution that fits best to a given sequence of (conditional) moments. For the two gene regulatory networks that we consider we find that the MCM approach is more suitable to describe multimodal distributions and that the reconstruction of marginal distributions is more accurate if conditional distributions are considered.

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