You have to make the connect string more elaborate, the simple "just the database name" won't do any more. The normal format is a set of name/value pairs coupled with an =" sign, and the pairs separated by a semicolon. Their order of appearance is irrelevant. So, try something like

my $dsn = 'dbi:mysql:database=genedb;host=rat';
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(as per the example that can be found here.)

p.s. You were right, it is in the docs.