  Consider finite sequences $X_{[1,n]}=X_1\dots X_n$ and $Y_{[1,n]}=Y_1\dots
Y_n$ of length $n$, consisting of i.i.d.\ samples of random letters from a
finite alphabet, and let $S$ and $T$ be chosen i.i.d.\ randomly from the unit
ball in the space of symmetric scoring functions over this alphabet augmented
by a gap symbol. We prove a probabilistic upper bound of linear order in
$n^{0.75}$ for the deviation of the score relative to $T$ of optimal alignments
with gaps of $X_{[1,n]}$ and $Y_{[1,n]}$ relative to $S$. It remains an open
problem to prove a lower bound. Our result contributes to the understanding of
the microstructure of optimal alignments relative to one given scoring
function, extending a theory begun by the first two authors.
