Simultaneous state-time approximation of the chemical master equation using tensor product formats

Abstract

We study the application of the novel tensor formats (TT, QTT, QTT-Tucker) to the solution of $d$-dimensional chemical master equations, applied mostly to gene regulating networks (signaling cascades, toggle switches, phage-$λ$). For some important cases, e.g. signaling cascade models, we prove good separability properties of the system operator. The Quantized tensor representations (QTT, QTT-Tucker) are employed in both state space and time, and the global state-time $(d+1)$-dimensional system is solved in the structured form by using the ALS-type iteration. This approach leads to the logarithmic dependence of the computational complexity on the system size. When possible, we compare our approach with the direct CME solution and some previously known approximate schemes, and observe a good potential of the newer tensor methods in simulation of relevant biological systems.