Linar Mikeev

Thomas A. Henzinger, Maria Mateescu, Linar Mikeev et al.

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.

Aleksandr Andreychenko, Pepijn Crouzen, Linar Mikeev et al.

This paper presents an on-the-fly uniformization technique for the analysis of time-inhomogeneous Markov population models. This technique is applicable to models with infinite state spaces and unbounded rates, which are, for instance, encountered in the realm of biochemical reaction networks. To deal with the infinite state space, we dynamically maintain a finite subset of the states where most of the probability mass is located. This approach yields an underapproximation of the original, infinite system. We present experimental results to show the applicability of our technique.

Alexander Andreychenko, Linar Mikeev, Verena Wolf

The classical problem of moments is addressed by the maximum entropy approach for one-dimensional discrete distributions. The numerical technique of adaptive support approximation is proposed to reconstruct the distributions in the region where the main part of probability mass is located.

Alexander Andreychenko, Linar Mikeev, Verena Wolf

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 is more accurate if conditional distributions are considered.