Incorporating spatial diffusion into models of bursty stochastic transcription

The dynamics of gene expression are both stochastic and spatial at the molecular scale. Mechanistic models of mRNA count distributions have revealed countless insights but largely neglect the frontier of subcellular spatial resolution. The spatial distribution of mRNA encodes their dynamics, including inherently spatial processes like transport to the nuclear boundary for export. Due to the technical challenges of spatial stochastic processes, tools for studying these subcellular spatial patterns are still limited. Here, we introduce a spatial stochastic model of nuclear mRNA with telegraph transcriptional dynamics. Observations of the model can be concisely described as following a spatial Cox process driven by a stochastically switching partial differential equation (PDE). We derive analytical solutions for spatial and demographic moments and validate them with simulations. We show that the distribution of mRNA counts can be accurately approximated by a Poisson-Beta distribution with tractable parameters, even with complex spatial dynamics. This observation allows for efficient parameter inference demonstrated on synthetic data. Altogether, our work adds progress toward a new frontier of subcellular spatial resolution in inferring the dynamics of gene expression from static snapshot data.