# SARS-Cov2 by the numbers

This virus is interesting to think about. Here are some numbers we can immediately derive ‘from first principles’. Please note, as with all my posts, I don’t guarantee accuracy and these ‘guesstimates’ should be considered rough and unpolished and probably accurate to within 1-2 (in some cases 3) orders of magnitude. That’s OK-they should be enough to give us a sense of what it means to be a virus. I encourage everyone to carry out these back of the envelope calculations themselves–it’s an enlightening exercise for me, and the results never cease to amaze me.

## Mass of the virus

Well, 1 H atom weighs ~$10^{-24}$ g. The virus can’t weigh less than 1000 H atoms, and it definitely can’t weigh more than 1ng=$10^{-9}$ g. If we take the geometric mean of these upper and lower bounds, we get a weight of ~$10^{-15}$ g or 1 billion Daltons per virion.

## Total virions in an infected individual:

In an average adult, there are $10^{13}$ cells. A COVID positive individual cannot possibly have less than $10^6$ virions, and it seems unlikely that a virus could make more than 10,000 virions per cell lysed. Therefore, the absolute maximum number of virions a COVID infected individual could produce if all of their cells were infected and lysed is ~$10^{16}$ virions.

Once again, the geometric mean approximation suggests an infected individual may create $10^{11}$ virions in their body.

Is this number reasonable? A previous publication put the total number of viruses in the world at $10^{31}$. If all humans were infected with SARS-Cov2, our upper bound estimate for total virion particles would be $10^{16} * 10^{10} = 10^{26}$. Even if all humans were to infect all cells in all human beings on the planet and lyse them to create 1,000 virion progeny, the resulting virions would barely account for 0.001% of all viruses on Earth.

## Total viral mass if SARS-Cov2 infected all humans on the planet

This number is easy: If the virus weighs $10^{-15}$ g / virion and there are $10^{10}$ human beings on the planet and each human contains $10^{11}$ virions, then there will be $10^{21}$ virion particles once SARS-Cov2 colonizes the planet entirely, and the resulting viral mass will be 1,000kg. In other words, the total viral mass required for a virus of this size to take over the planet is the equivalent of a bear!

## Volume of the virus

1 H atom has a diameter ~ 0.1nm. Since we agreed the virus weighs $10^{-15}$ g this is equivalent to about 100 million H atoms. Therefore, the total volume must be at least 100 million times greater than a single H atom. The volume of one atom is ~ $10^{-3}$ nm $^3$. That means the virus has a volume of at least $10^5$ nm $^3$ = $10^5 * 10^{-24} = 10^{-19}$ L = $10^{-4}$ fL. Of course, this assumes optimal packing, which will definitely not be the case–add an extra order of magnitude for that reason. So, it’s likely the virus measures 0.001fL in volume.

Therefore, it the virus colonizes humanity completely, the total viral volume will be $10^{18}$ fL = 1,000L.

# Conclusion

A single infection creates .1uL of virus – an amount so small it would fit in a (*gasp*) 20uL pipette tip and be barely visible – that weighs 0.1mg. If it infected all of humanity, that virus could be packaged into the approximate volume of a bear. One bear, taking over all of humanity. That is what is happening.

Written on January 4, 2021