Okay you changed the question a few times while I was composing this reply ;o). Have to say i'm still left guessing what the ordering rules are for the angle bracket segments. I picked one ordering that gives the results you're requesting, but you may still have to tweak them a bit. The main idea is to ignore whitespace in the regex using the /x parameter so that you can format the regex for readability:

use strict;
use warnings;

while (<DATA>) {
        print if /
          <MIR-\d+>
          (     <EXP-V-\d+>
                (<ART-\d+>)*
                (<BE-V>)*
             |  <ASSC-PHRASE-\d+>
          )
          <VACCVIRUS-PROP-\d+>/x;
}

__DATA__
<MIR-1><EXP-V-3><VACCVIRUS-PROP-1>
<MIR-1><ASSC-PHRASE-1><VACCVIRUS-PROP-1><PATTERN-1>
<MIR-1><EXP-V-0><ART-0><VACCVIRUS-PROP-1>
<MIR-1><EXP-V-0><ART-0><BE-V><VACCVIRUS-PROP-1>
<MIR-1><EXP-V-0><EXP-N-0><VACCVIRUS-PROP-1>
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