  The recent discovery of the striking sheet-like multicellular
choanoflagellate species $Choanoeca~flexa$ that dynamically interconverts
between two hemispherical forms of opposite orientation raises fundamental
questions in cell and evolutionary biology, as choanoflagellates are the
closest living relatives of animals. It similarly motivates questions in fluid
and solid mechanics concerning the differential swimming speeds in the two
states and the mechanism of curvature inversion triggered by changes in the
geometry of microvilli emanating from each cell. Here we develop fluid
dynamical and mechanical models to address these observations and show that
they capture the main features of the swimming, feeding, and inversion of
$C.~flexa$ colonies.
