

The first cyclization step is a Claisen rearrangement - three bonds all move in a circle, which causes the allyl group to migrate to the aromatic ring. The resulting ketone tautomerizes to the more stable phenol structure.
After an olefin cross-metathesis to introduce two methyl groups, the next step was in effect a double cyclization. First a stereoselective Shi epoxidation. Then the epoxide is opened during the second cyclization to form columbianetin.
This last cyclization is an an example of a 5-exo-tet cyclization according to Baldwin's Rules for ring formation. The resulting ring is a 5-membered ring. The bond broken to make the new ring is not part of the new ring (exo). And when the new ring is formed, the nucleophile is forming a bond to a tetrahedral atom (tet).

To get around this, they first converted it to a malonate ester under mild conditions. This was reacted with p-hydroxybenzaldehyde and catalytic piperidine to generate the final product by way of a Doebner-Knoevenagel condensation. In effect building the hydroxycinnamate group instead of adding it as a single piece.

Magolan, J., & Coster, M. (2009). Total Synthesis of (+)-Angelmarin The Journal of Organic Chemistry, 74 (14), 5083-5086 DOI: 10.1021/jo900613u
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