Viroids are described as short circular ssRNA with no protein coating.
Are there any analogous infectious particles that contain DNA instead of RNA?
If DNA viroids do not exist, is there an obvious reason for this?
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The definition on the Wikipedia page you recite uses the language 'no protein coating' as opposed to 'protein coating'.
A protein coating would make it a virus.
No protein-coating would make it a Viroid.
As far as we know DNA Viroids do not exist. But this is an advancing field of research and it depends on what structure of genetic material you are referring too.
The general different types of genetic material include - but are not limited to: Single-stranded DNA (ssDNA), Double-stranded DNA (dsDNA), Single-stranded RNA (ssRNA), and double-stranded RNA (dsRNA).
There is more information on Classes I and II DNA viruses here: https://www.ncbi.nlm.nih.gov/books/NBK21523/
If there were no DNA viruses without a protein coating - what is the reason? Viroids only infect plant cells. It may be that Viroids route of entry via microscopic junctions between plant cells (Plasmodesmata) are only large enough to fit single stranded genetic material. Therefore allowing ssDNA and ssRNA, but not dsDNA and dsRNA to act as plant cell infectious agents.