  $\alpha$-helices stand out as common and relatively invariant secondary
structural elements of proteins. However, $\alpha$-helices are not rigid bodies
and their deformations can be significant in protein function ({\it e.g.}
coiled coils). To quantify the flexibility of $\alpha$-helices we have
performed a structural principal-component analysis of helices of different
lengths from a representative set of protein folds in the Protein Data Bank. We
find three dominant modes of flexibility: two degenerate bend modes and one
twist mode. The data are consistent with independent Gaussian distributions for
each mode. The mode eigenvalues, which measure flexibility, follow simple
scaling forms as a function of helix length. The dominant bend and twist modes
and their harmonics are reproduced by a simple spring model, which incorporates
hydrogen-bonding and excluded volume. As an application, we examine the amount
of bend and twist in helices making up several coiled-coil proteins.
Incorporation of $\alpha$-helix flexibility into structure refinement and
design is discussed.
