  At the heart of the structured architecture and complex dynamics of
biological systems are specific and timely interactions operated by
biomolecules. In many instances, biomolecular agents are spatially confined to
flexible lipid membranes where, among other functions, they control cell
adhesion, motility and tissue formation. Besides being central to several
biological processes, \emph{multivalent interactions} mediated by reactive
linkers confined to deformable substrates underpin the design of
synthetic-biological platforms and advanced biomimetic materials. Here we
review recent advances on the experimental study and theoretical modelling of a
heterogeneous class of biomimetic systems in which synthetic linkers mediate
multivalent interactions between fluid and deformable colloidal units,
including lipid vesicles and emulsion droplets. Linkers are often prepared from
synthetic DNA nanostructures, enabling full programmability of the
thermodynamic and kinetic properties of their mutual interactions. The coupling
of the statistical effects of multivalent interactions with substrate fluidity
and deformability gives rise to a rich emerging phenomenology that, in the
context of self-assembled soft materials, has been shown to produce exotic
phase behaviour, stimuli-responsiveness, and kinetic programmability of the
self-assembly process. Applications to (synthetic) biology will also be
reviewed.
