There was a somewhat equivalent (although as any time when drawing comparisons between modern politics and Rome, one must be very careful -- although things may sound similar, once one starts digging under the surface, the differences can be profound).

The princeps senatus was the (rough translation) "first of the senate". He set various parlimentary rules, spoke first, wrote letters on behalf of the Senate to foreign emissaries, etc.

The princeps senatus was appointed by the Censors and had to be a patrician.

It was a position of immense dignitas but no imperium -- the princeps senatus got no lictors, held no magisterial position in his own right -- and it was one held by some of the most powerful politicians of their day: Scipio Africanus, Marcus Aemilius Lepidus, Marcus Aemilius Scaurus, and others.