my $count = $nucleotideDNA =~ tr/ATCG]//c;  # Remove and count invalids

Sofie:   Note that while this  tr/// (see Quote-Like Operators in perlop) expression counts the number of characters that are not ATCG, it does not remove anything; the string is not changed (update: nor is there any need for change):

c:\@Work\Perl\monks>perl -wMstrict -le
"my $DNA = 'ATATCCCGATCAGG3TT!GCA';
 ;;
 my $nucleotideDNA = $DNA;
 my $count = $nucleotideDNA =~ tr/ATCG//c;
 ;;
 print $DNA;
 print $nucleotideDNA;
 ;;
 print 'sequences are equal' if $DNA eq $nucleotideDNA;
"
ATATCCCGATCAGG3TT!GCA
ATATCCCGATCAGG3TT!GCA
sequences are equal
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However, I agree with the main point that BillKSmith is making: string operations with regexes or with operators like substr and index will tend to be significantly faster (update: and to consume significantly less memory) than equivalent array operations.