Incorporating spatial diffusion into models of bursty stochastic transcription.

The dynamics of gene expression are stochastic and spatial at the molecular scale, with messenger RNA (mRNA) transcribed at specific nuclear locations and then transported to the nuclear boundary for export. Consequently, the spatial distributions of these molecules encode their underlying dynamics. While mechanistic models for molecular counts have revealed numerous insights into gene expression, they have largely neglected now-available subcellular spatial resolution down to individual molecules. Owing to the technical challenges inherent in spatial stochastic processes, tools for studying these subcellular spatial patterns are still limited. Here, we introduce a spatial stochastic model of nuclear mRNA with two-state (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. 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 towards a new frontier of subcellular spatial resolution in inferring the dynamics of gene expression from static snapshot data.