#!/usr/bin/perl -w
use strict;
use Data::Dumper;

...

while ( my $line = <STDIN> ) { # No need to declare $line until here.
    my @AA = $line;            # No need to declare @AA until here. 
                               # Might need to split the line, though ...

    print Dumper \@AA; # See what's really in that array. Note the backslash.

    ...

##</code><code>##

foreach $AA(@AA) 
{
#I want now the sum of protein's molecular weight.
$sum += $MWs{ "$AA(@AA)" };
             # ^^
             # incorrect syntax
}

##</code><code>##

# Sum the proteins' molecular weights.
my $sum;  # no need to declare $sum until here.
          # Also no need to initialize to zero; Perl will do that when you add to it.
foreach my $AA ( @AA ) {
    print "AA: $AA Weight: $MWs{ $AA }\n";
    $sum += $MWs{ $AA };
}