  Protein folds are built primarily from the packing together of two types of
structures: alpha-helices and beta-sheets. Neither structure is rigid, and the
flexibility of helices and sheets is often important in determining the final
fold ({\it e.g.}, coiled coils and beta-barrels). Recent work has quantified
the flexibility of alpha-helices using a principal-component analysis (PCA) of
database helical structures (Emberly, 2003). Here, we extend the analysis to
beta-sheet flexibility using PCA on a database of beta-sheet structures. For
sheets of varying dimension and geometry, we find two dominant modes of
flexibility: twist and bend. The distributions of amplitudes for these modes
are found to be Gaussian and independent, suggesting that the PCA twist and
bend modes can be identified as the soft elastic normal modes of sheets. We
consider the scaling of mode eigenvalues with sheet size and find that parallel
beta-sheets are more rigid than anti-parallel sheets over the entire range
studied. Lastly, we discuss the application of our PCA results to modeling and
design of beta-sheet proteins.
