The geometry proposition that "four points not in a plane describe one and only one sphere" provides a novel tool for analyzing protein-induced distortions in [4Fe-4S] clusters. A geometrically perfect reference structure comprises interlaced, regular tetrahedra of Fe, S, and S gamma atoms having T(d) symmetry. Three circumspheres are defined by the three sets of four atoms, the circumcenters of which are unique points within the cluster. The structure is thus re-defined by the positions of the circumcenters in xyz space and the r, theta, phi of each atom on its respective sphere. Analysis of 12 high-resolution structures of protein-bound and small molecule [4Fe-4S](SR)(4) clusters revealed: (a) the circumcenters are generally non-coincident by approximately 0.01 to approximately 0.06 A; (b) the Fe radius, r(Fe), is nominally independent of core oxidation state, having values between 1.66 to 1.69 A, whereas r(S) and r(SG), which have ranges of 2.18-2.24 A and 3.87-3.94 A, respectively, both increase by as much as approximately 3% upon reduction from the 3+ to the 1+ core valence; (c) deviation of some atoms from the theta, phi of a perfect tetrahedron can be large, approximately 10 degrees, and sets of atoms can show patterns of motion on their spheres that result from changes in Fe-S bond lengths. Density functional theory calculations suggest that the [4Fe-4S] core itself requires rather little energy to distort (approximately 2 kcal/mol), whereas significantly more energy is required to distort the Sgamma shell (~4 kcal/mol) to that of cluster I in Clostridium acidurici ferredoxin.
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