  The intricate shapes of biological membranes such as tubules and membrane
stacks are induced by proteins. In this article, we systematically investigate
the membrane shapes induced by arc-shaped scaffolds such as proteins and
protein complexes with coarse-grained modeling and simulations. We find that
arc-shaped scaffolds induce membrane tubules at membrane coverages larger than
a threshold of about 40%, irrespective of their arc angle. The membrane
morphologies at intermediate coverages below this tubulation threshold, in
contrast, strongly depend on the arc angle. Scaffolds with arc angles of about
60 degree akin to N-BAR domains do not change the membrane shape at coverages
below the tubulation threshold, while scaffolds with arc angles larger than
about 120 degree induce double-membrane stacks at intermediate coverages. The
scaffolds stabilize the curved membrane edges that connect the membrane stacks,
as suggested for complexes of reticulon proteins. Our results provide general
insights on the determinants of membrane shaping by arc-shaped scaffolds.
