  $\beta$-barrel membrane proteins are found in the outer membrane of
gram-negative bacteria, mitochondria, and chloroplasts. We have developed
probabilistic models to quantify propensities of residues for different spatial
locations and for interstrand pairwise contact interactions involving strong
H-bonds, side-chain interactions, and weak H-bonds. The propensity values and
p-values measuring statistical significance are calculated exactly by
analytical formulae we have developed. Contrary to the ``positive-inside'' rule
for helical membrane proteins, $\beta$-barrel membrane proteins follow a
significant albeit weaker ``positive-outside'' rule, in that the basic residues
Arg and Lys are disproportionately favored in the extracellular cap region and
disfavored in the periplasmic cap region. Different residue pairs prefer strong
backbone H-bonded interstrand pairings (e.g. Gly-Aromatic) or non-H-bonded
pairings (e.g. Aromatic-Aromatic). In addition, Tyr and Phe participate in
aromatic rescue by shielding Gly from polar environments. These propensities
can be used to predict the registration of strand pairs, an important task for
the structure prediction of $\beta$-barrel membrane proteins. Our accuracy of
44% is considerably better than random (7%) and other studies. Our results
imply several experiments that can help to elucidate the mechanisms of in vitro
and in vivo folding of $\beta$-barrel membrane proteins. See supplementary
material after the bibliography for detailed techniques.
