There were several wildcat rounds being used at the time the 444 was introduced by Marlin. A 44 Van Houten was one such round made up using a 30-40 Krag case. But prior to that the bp round 44-77 was a fairly popular round used in Sharps rifles.
In may of 1872 the Ordnance Department started testing a variety of black powder cartridges for possible use as a military round. They compared streight tapered cases as well as bottlenecked case rounds in 40 through 45 caliber using bullets from 290 to 484 grains. This work was summerized in a report fulfilling Special Order Number 107 to determine a proper caliber for small arms. Ultimately the 45-70 was chosen for the army's purpose. In this report there was a round test fired in gun number 15 with a streight tapered 2.3" rimmed case, with a .430 diameter 365 grain bullet that bore a remarkable resemblance to what would later be called the 444 marlin differing only a few thousandths in certain key diamentions. While some have discribed the 444 marlin as a modern 45-70, the historical record shows it to predate the adaptation of the 45-70 as the official military cartridge.
This concept was revisited in 1882 with the Manyard 40-70 cartridge loaded with a 430 grain lead bullet at just over 1300 fps. This was a rimmed, streight .451 diameter case, 2.45" long that shot a .422 diameter bullet. This is slightly longer and narrower than the 444 marlin 2.25" tapered case, but would have been very similar in capacity. Even thought smokeless powder, high pressure loads, muzzle velocities over 2000fps, and modern heat treated steels were well into the future, the precedent for a rimmed, streight walled 21/4" .43 caliber cartridge case was established in the 1870's and 1880's. The facts of the material were gathered from the writings of Glen Fryxell and John Taffin.