Apparatus is provided for use with a native valve of a heart of a subject. The apparatus includes: (1) an annular upstream support portion, comprising an expandable first frame, the upstream support portion configured to be placed against an upstream surface of the native valve; (2) a flexible polyester connector; and (3) an anchoring element, flexibly coupled to the upstream support portion by the connector, and configured to anchor the upstream support portion to the native valve by engaging tissue of the native valve.