A geometric principle may guide self-assembly of fullerene cages from clathrin triskelia and from carbon atoms.
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 <or= 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 <or= 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.
Schein
Stan
S
Department of Psychology, University of California at Los Angeles, Los Angeles, California 90095-1563, USA. schein@ucla.edu
Sands-Kidner
Michelle
M
D003198:Computer Simulation; D003460:Crystallization / Q000379:methods*; D037741:Fullerenes / Q000737:chemistry*; D046911:Macromolecular Substances / Q000737:chemistry; D008956:Models, Chemical; D008958:Models, Molecular; D008968:Molecular Conformation; D049329:Nanostructures / Q000737:chemistry* / Q000648:ultrastructure*
D037741:Fullerenes; D046911:Macromolecular Substances