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

We apply the novel tensor product formats (tensor train, quantized TT [QTT], and QTT-Tucker) to the solution of d-dimensional chemical master equations for gene regulating networks (signaling cascades, toggle switches, and phage- λ). For some important cases, for example, signaling cascade models, we prove analytical tensor product representations 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 tensor product form by the alternating minimal energy iteration, the ALS-type algorithm. This approach leads to the logarithmic dependence of the computational complexity on the volume of the state space. We investigate the proposed technique numerically and compare it with the direct chemical master equation solution and some previously known approximate schemes, where possible. We observe that the newer tensor methods demonstrate a good potential in simulation of relevant biological systems.