  Using a fast tree-searching algorithm and a Pentium cluster, we enumerated
all the sequences and compact conformations (structures) for a protein folding
model on a cubic lattice of size $4\times3\times3$. We used two types of amino
acids -- hydrophobic (H) and polar (P) -- to make up the sequences, so there
were $2^{36} \approx 6.87 \times 10^{10}$ different sequences. The total number
of distinct structures was 84,731,192. We made use of a simple solvation model
in which the energy of a sequence folded into a structure is minus the number
of hydrophobic amino acids in the ``core'' of the structure. For every
sequence, we found its ground state or ground states, i.e., the structure or
structures for which its energy is lowest. About 0.3% of the sequences have a
unique ground state. The number of structures that are unique ground states of
at least one sequence is 2,662,050, about 3% of the total number of structures.
However, these ``designable'' structures differ drastically in their
designability, defined as the number of sequences whose unique ground state is
that structure. To understand this variation in designability, we studied the
distribution of structures in a high dimensional space in which each structure
is represented by a string of 1's and 0's, denoting core and surface sites,
respectively.
