  We explore the statistics of assembling soft-matter building blocks to
investigate the uptake and encapsulation of cargo particles by carriers
engulfing their load. While the such carrier-cargo complexes are important for
many applications out of equilibrium, such as drug delivery and synthetic cell
encapsulation, we uncover here the basic statistical physics in minimal
hard-core-like models for particle uptake. Introducing an exactly solvable
equilibrium model in one dimension, we demonstrate that the formation of
carrier-cargo complexes can be largely tuned by both the cargo concentration
and the carriers' interior size. These findings are intuitively explained by
interpreting the internal free space (partition function) of the cargo inside a
carrier as its engulfment strength, which can be mapped to an external control
parameter (chemical potential) of an additional effective particle species.
Assuming a hard carrier membrane, such a mapping can be exactly applied to
account for multiple cargo uptake involving various carrier or cargo species
and even attractive uptake mechanisms, while soft interactions require certain
approximations. We further argue that the Boltzmann occupation law identified
within our approach is broken when particle uptake is governed by
non-equilibrium forces. Speculating on alternative occupation laws using
effective parameters, we put forward a Bose-Einstein-like phase transition
associated with polydisperse carrier properties.
