  Compartmentalization of biochemical processes underlies all biological
systems, from the organelle to the tissue scale. Theoretical models to study
the interplay between noisy reaction dynamics and compartmentalization are
sparse, and typically very challenging to analyze computationally. Recent
studies have made progress towards addressing this problem in the context of
concrete biological systems but general approaches remain lacking. In this work
we propose a mathematical framework based on counting processes that allows us
to study compartment populations with arbitrary interactions and internal
biochemistry. We provide an efficient description of the population dynamics in
terms of differential equations which capture moments of the population and
their variability. We demonstrate the relevance of our approach using several
case studies inspired by biological systems at different scales.
