  Polymer composites are ubiquitous in biology and industry alike, owing to
their emergent desirable mechanical properties not attainable in single-species
systems. At the same time, polymer topology has been shown to play a key role
in tuning the rheology of polymeric fluids. However, how topology impacts the
rheology of composites remains poorly understood. Here, we create composites of
rigid rods (microtubules) polymerized within entangled solutions of flexible
linear and ring polymers (DNA). We couple linear and nonlinear optical tweezers
microrheology with confocal microscopy and scaled particle theory to show that
composites of linear DNA and microtubules exhibit a strongly non-monotonic
dependence of elasticity and stiffness on microtubule concentration due to
depletion-driven polymerization and flocculation of microtubules. In contrast,
composites of ring DNA and microtubules show a much more modest monotonic
increase in elastic strength with microtubule concentration, which we
demonstrate arises from the increased ability of rings to mix with
microtubules.
