**Random model of protein structure superposition.**
Two four-residue fragments of non-homologous protein chains are shown as spheres,
representing CA atoms, connected by virtual bonds. One fragment, "protein 1",
is colored black, the second, "protein 2", gray.
Gapless equivalences of Cα atoms, depicted by, e.g. the labels b and b' in the figure,
were forced regardless of structural similarity and a minimum RMSD superposition was performed.
Such a forced superposition was termed a "random" superposition. Coordinate difference vector
projections (DVPs) and their standard deviation, σ_{obs}, were computed from the random superposition.
As an example, three projections Δv_{x, y, z} (short thick lines on the x, y, z axes) from one vector Δv
(arrow) are indicated. This procedure was repeated for all pairs of fragments in the dataset (32,004 pairs).
Then two size and shape parameters from each superposition, Rg (gyration radius) and c, were used
in a singular value decomposition (SVD) generalized linear least squares fit to determine coefficients for
a polynomial f. Polynomial f estimated the expected standard deviation of
projections, σ_{exp}, given the size and shape of a superposition, and σ_{exp} was calculated for all
random superpositions in the set. Finally, distributions of σ_{obs} for a narrow range of σ_{exp}
(0.2Å increments) were extracted and approximated with a PDF (Nakagami distribution).
Parameters for this PDF were therefore dependent on size and shape
of the random superpositions contained within the narrow range of σ_{exp}. These parameters
were subsequently fit to a continuous curve with σ_{exp} as the independent variable.
The continuous curve allowed
construction of a random model, and therefore estimation of a p-value, for a superposition
of arbitrary size and shape. As explained in the text but not shown in the figure, PDF
parameters describing a distribution of σ_{obs} simultaneously described the corresponding
distribution of projections Δv_{x, y, z}, using a second PDF (Variance-Gamma distribution).