  The encapsulation of active particles, such as bacteria or active colloids,
inside a droplet gives rise to nontrivial shape dynamics and droplet motility.
To understand this behavior, we derive an asymptotic solution for the fluid
flow about a deformable droplet containing an active particle, modeled as a
Stokes-flow singularity, in the case of small shape distortions. Offsetting of
the active particle from the center of the drop breaks symmetry and leads to
excitation of large number of shape modes as well as particle and drop
displacement. Flows due to common singularity representations of active
particles, such as Stokeslets, rotlets, and stresslets, are computed and
compared to results for non-deformable droplets enclosing active particles. The
effect of interfacial properties is also investigated. Surfactants adsorbed at
the droplet interface immobilize the interface and arrest the droplet motion.
Our results highlight strategies to steer the flows of active particles and
create autonomously navigating containers.
