This Article Full Text Full Text (PDF) All Versions of this Article: biophysj.107.110817v1 94/3/958    most recent Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Schein, S. Articles by Sands-Kidner, M. Search for Related Content PubMed PubMed Citation Articles by Schein, S. Articles by Sands-Kidner, M.

A Geometric Principle May Guide Self-Assembly of Fullerene Cages from Clathrin Triskelia and from Carbon Atoms

Stan Schein ^*   and Michelle Sands-Kidner ^*

^* Department of Psychology, Brain Research Institute, and California NanoSystems Institute, University of California at Los Angeles, Los Angeles, California

Correspondence: Address reprint requests to Stan Schein, University of California at Los Angeles, Mailcode 951563, Los Angeles, CA 90095-1563. Tel.: 310-825-0505; Fax: 310-206-5895; E-mail: schein{at}ucla.edu.

Clathrin triskelia and carbon atoms alike self-assemble into a limited selection of fullerene cages (with n three connected vertices, 3n/2 edges, 12 pentagonal faces, and (n–20)/2 hexagonal faces). We show that a geometric constraint—exclusion of head-to-tail dihedral angle discrepancies (DADs)—explains this limited selection as well as successful assembly into such closed cages in the first place. An edge running from a pentagon to a hexagon has a DAD, since the dihedral angles about the edge broaden from its pentagon (tail) end to its hexagon (head) end. Of the 21 configurations of a central face and surrounding faces, six have such DAD vectors arranged head-to-tail. Of the 5770 mathematically possible fullerene cages for n 60, excluding those with any of the six configurations leaves just 15 cages plus buckminsterfullerene (n = 60), among them the known clathrin cages. Of the 216,739 mathematically possible cages for 60 < n 84, just the 50 that obey the isolated-pentagon rule, among them known carbon cages, pass. The absence of likely fullerenes for some n (30,34,46,48,52–58,62–68) explains the abundance of certain cages, including buckminsterfullerene. These principles also suggest a "probable roads" path to self-assembly in place of pentagon-road and fullerene-road hypotheses.