Filtered finite state projection method for the analysis and estimation of stochastic biochemical reaction networks

Recent advances in fluorescence technologies and microscopy techniques have significantly improved scientists ability to investigate biological processes at the single-cell level. However, fluorescent reporters can only track the temporal dynamics of a limited number of critical components in a cell (e.g., fluorescent proteins), leaving other pivotal dynamic components (such as gene-state) hidden. Moreover, the nature of the interactions among intracellular biomolecular species is inevitably stochastic in the low copy number regime, which adds more difficulties to the investigation of these hidden species dynamics. Therefore developing mathematical and computational tools for inferring the behaviour of stochastic reaction networks from time-course data is urgently needed. Here we develop a finite-dimensional filter for estimating the conditional distribution of the hidden (unobserved) species given continuous-time and noise-free observations of some species (e.g. a fluorescent reporter). It was proposed that in this setting, the conditional distribution evolves in time according to a large or potentially infinite-dimensional system of coupled ordinary differential equations with jumps, known as the filtering equation. We first formally verify the validity of this filtering equation under the non-explosivity condition and then develop a Finite-State Projection method, which provides an approximate solution by truncating the infinite-dimensional system. Additionally, we give computable error bounds for the algorithm. Finally, we present several numerical examples to illustrate our method and compare its performance with an existing particle filtering method for estimating the conditional distribution. ### Competing Interest Statement The authors have declared no competing interest.