  Is protein secondary structure primarily determined by local interactions
between residues closely spaced along the amino acid backbone, or by non-local
tertiary interactions? To answer this question we have measured the entropy
densities of primary structure and secondary structure sequences, and the local
inter-sequence mutual information density. We find that the important
inter-sequence interactions are short ranged, that correlations between
neighboring amino acids are essentially uninformative, and that only 1/4 of the
total information needed to determine the secondary structure is available from
local inter-sequence correlations. Since the remaining information must come
from non-local interactions, this observation supports the view that the
majority of most proteins fold via a cooperative process where secondary and
tertiary structure form concurrently. To provide a more direct comparison to
existing secondary structure prediction methods, we construct a simple hidden
Markov model (HMM) of the sequences. This HMM achieves a prediction accuracy
comparable to other single sequence secondary structure prediction algorithms,
and can extract almost all of the inter-sequence mutual information. This
suggests that these algorithms are almost optimal, and that we should not
expect a dramatic improvement in prediction accuracy. However, local
correlations between secondary and primary structure are probably of
under-appreciated importance in many tertiary structure prediction methods,
such as threading.
