  This paper is concerned with minimization of a fourth-order linearized
Canham-Helfrich energy subject to Dirichlet boundary conditions on curves
inside the domain. Such problems arise in the modeling of the mechanical
interaction of biomembranes with embedded particles. There, the curve
conditions result from the imposed particle--membrane coupling. We prove
almost-$H^{\frac{5}{2}}$ regularity of the solution and then consider two
possible penalty formulations. For the combination of these penalty
formulations with a Bogner-Fox-Schmit finite element discretization we prove
discretization error estimates which are optimal in view of the solution's
reduced regularity. The error estimates are based on a general estimate for
linear penalty problems in Hilbert spaces. Finally, we illustrate the
theoretical results by numerical computations. An important feature of the
presented discretization is that it does not require to resolve the particle
boundary. This is crucial in order to avoid re-meshing if the presented problem
arises as subproblem in a model where particles are allowed to move or rotate.
