Just to summarise: The membrane bound hydrogenase (MBH) of Pyrococcus furiosus pushes a proton outwards through a proton permeable membrane which then returns through a subunit of MBH which looks very much like three quarters of the Mrp antiporter. The Mrp antiporter is derived from a very, very ancient antiporter which appears to have been one of the core systems of the last universal common ancestor (LUCA). Which ion travels in which direction in the modern families of Mrp is currently not particularly clear and may depend on all sorts of factors. OK.
No one has worked out the detailed structure/function of the Mrp antiporter, but there is a very interesting paper from back in 2001 which might give us some insight about function at least.
Mrp‐dependent Na+/H+ antiporters of Bacillus exhibit characteristics that are unanticipated for completely secondary active transporters
The study used modern E coli whose plasma membrane is tight to both protons and Na+. A strain with all of its antiporters deleted was used and then plasmids were engineered to supply a single gene antiporter of the nahA type (also from E coli) or an Mrp from Bacillus pseudofirmus OF4 (yup, that's its name) or one from Bacillus subtilis.
Respectively we have strains ending in -118 endowed with the blank plasmid, -nhaA for the monogenic antiporter, -BSmrp (B subtilis Mrp) or -OFmrp (B pseudofirmus Mrp). They stuck the engineered E coli in to 25mmol of NaCl, fed it and looked at the intracellular Na+ concentration. Very simple.
The first two columns are exactly what you would expect. Having no antiporter gives an intracellular Na+ in equilibrium with culture medium Na+ at around 25mmol, near enough. Any antiporter rescues this, the nhaA monogenic antiporter somewhat better than either of the Mrp antiporters.
Now here comes the clever bit. They added CCCP, a classical protonophore which drops the membrane potential from 150-ish mV to 15-ish mV (obviously the membrane voltage should be zero but the E coli complex I will be working flat out to stop this fatal occurrence plus I suspect the dose of CCCP used was less than supramaximal, though I've not checked this), and then they looked at the intracellular Na+ concentration. So they have suddenly converted a modern E coli to an E coli with a proton permeable, Na+ impermeable cell membrane. Like Pyrococcus but without the boiling water. Or like LUCA. Both organisms to which Mrp antiporting is/was very important.
The simple nhaA is utterly dependent on a proton tight membrane with a transmembrane proton gradient and it fails to antiport anything in the presence of CCCP, as you would expect (line two, both right hand columns). The nhaA is not primordial.
Both of the Mrp type antiporters maintain an intracellular Na+ between 13.4mmol and 16.9mmol, which is "high-but-physiological", using a CCCP proton "leaky" membrane voltage of 15 mV to effectively pump out Na+.
Quite how the Mrp antiporters do this is unclear. Most of the work has been done on the big subunits, A and D. One is probably a proton channel and one a Na+ channel but even this is not completely clear and in Yu et al's Structure of an Ancient Respiratory System they consider both MrpA and D to have proton channels. So it's messy. The combined small subunits, MrpE, MrpF and MrpG, plus the tail end of MrpA appear to be a proton/Na+ antiporter in their own right. I'll refer to this section as a "simple" antiporter.
Clearly, trying to get a Na+ gradient from a proton leaky membrane by making use of a monogenic nhaA type antiporter doesn't work. Using an Mrp antiporter does.
OK, wild speculation time.
I consider the arrangement in Pyrococcus furiosus is necessary because the cell membrane is permeable to protons. Pump a proton outwards and generally it will boomerang back. Pump it directly in to the mouth of an antiporter and it will return while antiporting a Na+ outwards. The Na+ stays outside. It does this using much of the Mrp machinery.
The end game is to drop a precious proton down the throat of the simple antiporter, without losing it through a leaky membrane. I think the Pyrococcus MBH keeps this proton "in-complex" to avoid losing it. Power is probably supplied to the left hand proton channel from the FeNi hydrogenase by the loop cluster and helix HL.
Like this, yellow circles are antiporters:
Looking at Mrp, I think it is the precursor of the MBH and is doing exactly the same thing but geochemically, ie it is an adaptation to a low geochemical proton gradient across a proton leaky membrane. It still takes the proton from a 15mV proton motive force but it initially uses this to antiport another proton outwards, protects this one from loss through the leaky membrane by keeping it "in-complex" and uses this to antiport Na+ outwards, which stays outside:
In both diagrams everything is identical to the right of the FeNi hydrogenase or MrpA N-terminal domain. All that differs is the method for "elevating" the guarded proton to the entrance of the simple antiporter on the right.
TLDR: Mrp is an adaptation to a failing geochemical proton gradient. Membrane bound hydrogenase is the adaptation to a failed geochemical gradient.
It is amazing to me that Mrp keeps its core function today despite the radically different tasks (saline and alkaline tolerance) which it is needed for nowadays. All we need is a proton leaky membrane to return it to do what it did initially. No change.
Okay, it's not an envelope, its the back of a chocolate wrapper. Speculation is such fun, given enough chocolate.