I realize there are several different mRNA vaccines. I would be happy to know the ballpark figure for any of them.
As a follow-up, is it known about what percentage of injected mRNA strands are successfully translated into proteins?
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Well, we can work this out empirically.
We know that the average molecular mass of a base of a RNA is 339.5 g/mol.
Despite this, to calculate the approximate molecular mass of a single-stranded RNA molecule, you multiply the number of bases by 320.5 (because some bases are more common in sequences than others) and add 159 (same source). This means for a single-stranded molecule of 1000 bases, it will have a molecular mass of 320,659 g/mol. The Pfizer vaccine has a sequence of 4284 bases.
4284 * 320.5 + 159 = 1373181 g/mol.
The Pfizer vaccine has 30 ug/shot (PDF, see page 27 under "Description") or 30x10-6 grams
number of moles = mass/molar mass.
n= 30 x 10-6 grams/1373181 g/mol
n= 2.185 x 10-11
number of molecules = n (above) x Avogadro constant (6.022 x 1023)
Number of molecules = 1.316 x 1013
Now this is a ball-park figure because the RNA isn't actually RNA it has had some special additions to the uridines that make it less susceptible to immune regulation, replacing the uridines with 1-methyl-3'-pseudouridylyl (see Wikipedia link to structure above). However, I believe this alteration to the mass, will not alter the number substantially; it'll still be in the 10 trillion range.