The genesis of teeth in vertebrates

Rücklin and Donoghue 2015 report, 
“Theories on the evolutionary origin of teeth have long been rooted in the condition manifest by chondrichthyans [sharks and ratfish] as the most distant living outgroup to humans and because they exhibit a comparatively simple pattern of tooth replacement. However, their apparent simplicity is secondary given that the extinct placoderms, which constitute the sister lineage(s) to all other jawed vertebrates, exhibit a greater diversity and complexity of dentitions that better inform the nature of an ancestral gnathostome dentition.”

Rücklin and Donoghue 2015 conclude,
“Here, we show that ‘supragnathal’ toothplates from the acanthothoracid placoderm Romundina stellinacomprise (Fig. 1) multi-cuspid teeth, each composed of an enameloid cap and core of dentine.” 

Unfortunately,
Rücklin and Donoghue did not place Romundina in a wide gamut phylogenetic cladogram, like the large reptile tree (LRT, 1519 taxa; subset Fig. 2). In the LRT Romundina has more primitive relatives with similar tooth pads. We’ll meet them now.

The outgroup taxon
for the entire LRT is a small thelodont. Thelodus (Fig. 1) lacks jaws and teeth, but its skin is covered by small tooth-like denticles. Such skin has been compared to shark skin, but overall Thelodus has never been compared specifically to Rhincodon, the giant extant whale shark (Fig. 3), until a few days ago.

Rhincodon is the most primitive taxon in the LRT
(Fig. 2) to have both jaws and teeth, though not the sort of teeth tetrapods have. Tooth carpets in Rhincodon (Fig. 3) resemble shark skin, except they are on the inside of the mouth. Rhincodon tooth carpets also resemble those found in catfish, like Clarias (Fig. 4). Rhincodon tooth carpets contain 300-350 rows of tiny teeth on 20 filter pads, Distinct from higher sharks, like Isurus (Fig. 5) and bony fish, like Stensioella (Fig. 2) and Kenichthys, the jaw margins of Rhincodon and catfish are toothless.

Catfish 
have fewer teeth on smaller tooth pads with fewer rows and columns (Fig. 5). We can expect the catfish-sized Silurian ancestors of giant Rhincodon to be similar.

Other placoderms
like Entelognathus and Romundina (Fig. 1) further reduce tooth plates. Some placoderms lack them entirely. So do other bony fish.

Based on the the tree topology recovered by the LRT,
traditional sharks, like Isurus (Fig. 6), develop marginal teeth by convergence with those found in basal lobefins, like Kenichthys (Fig. 7). Teeth were extremely tiny on this bottom feeder and its kin, like Stensioella. Replacement teeth developed below the marginal teeth. The vomer tooth pads were retained, but vomer teeth were not present. Vomer teeth would return in certain more derived taxa, sometimes as fangs. The basipterygoid had tiny teeth.

In the shark clade
large teeth appear during ontogeny in a different pattern. As everyone knows, each vertically-oriented tooth is replaced by a series of growing teeth ready to rotate into place from the jaw interior.

Other than Polypterus,
(Figs. 2, 8) most lungfish lose their marginal teeth, but develop a vomer/palatine tooth pad for crushing. Polypterus has several sets of teeth arising from various palate bones. Youngolepis develops vomer, palatine and ectopterygoid fangs, larger than their tiny marginal teeth. This pattern is retained by basal tetrapods and lobefins basal to ray-fin fish.

Many teleosts retain the maxilla,
but lose their maxillary teeth. Others lose all their teeth.

Most teeth are used for food capture.
Very few taxa, like elephants and boars, use their teeth for display and as other tools.

References
Rücklin M and Donoghue PCJ 2015. Romundina and the evolutionary origin of teeth. Biology Letters, Royal Society