  Recent experiments have elucidated the physical properties of kinetoplasts,
which are chain-mail-like structures found in the mitochondria of trypanosome
parasites formed from catenated DNA rings. Inspired by these studies, we use
Monte Carlo simulations to examine the behavior of two-dimensional networks
("membranes") of linked rings. For simplicity, we consider only identical rings
that are circular and rigid and that form networks with a regular linking
structure. We find that the scaling of the eigenvalues of the shape tensor with
membrane size are consistent with the behavior of the flat phase observed in
self-avoiding covalent membranes. Increasing ring thickness tends to swell the
membrane. Remarkably, unlike covalent membranes, the linked-ring membranes tend
to form concave structures with an intrinsic curvature of entropic origin
associated with local excluded-volume interactions. The degree of concavity
increases with increasing ring thickness and is also affected by the type of
linking network. The relevance of the properties of linked-ring model membranes
to those observed in kinetoplasts is discussed.
