  Many imaging techniques for biological systems -- like fixation of cells
coupled with fluorescence microscopy -- provide sharp spatial resolution in
reporting locations of individuals at a single moment in time but also destroy
the dynamics they intend to capture. These snapshot observations contain no
information about individual trajectories, but still encode information about
movement and demographic dynamics, especially when combined with a
well-motivated biophysical model. The relationship between spatially evolving
populations and single-moment representations of their collective locations is
well-established with partial differential equations (PDEs) and their inverse
problems. However, experimental data is commonly a set of locations whose
number is insufficient to approximate a continuous-in-space PDE solution. Here,
motivated by popular subcellular imaging data of gene expression, we embrace
the stochastic nature of the data and investigate the mathematical foundations
of parametrically inferring demographic rates from snapshots of particles
undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward
inference, we rigorously derive a connection between individual particle paths
and their presentation as a Poisson spatial process. Using this framework, we
investigate the properties of the resulting inverse problem and study factors
that affect quality of inference. One pervasive feature of this experimental
regime is the presence of cell-to-cell heterogeneity. Rather than being a
hindrance, we show that cell-to-cell geometric heterogeneity can increase the
quality of inference on dynamics for certain parameter regimes. Altogether, the
results serve as a basis for more detailed investigations of subcellular
spatial patterns of RNA molecules and other stochastically evolving populations
that can only be observed for single instants in their time evolution.
