# Quantum Viruses?

The Twitter conversation that prompted yesterday’s post about composite objects was apparently prompted by a comment somebody made about how a virus left alone would see its quantum wavefunction spread out on a time scale of minutes. This led to wondering about whether a virus could really be considered a particle that would move as a single quantum unit, and then to whether that estimate is reasonable. So, let’s look at that specific question a little more closely. This is going to be one of those swashbuckling physicist estimation deals, in which I’m going to attempt to come up with numbers without even the benefit of an envelope to scribble on.

So, how would we make an estimate of this? Well, the first order of business is to take on the question of whether it’s reasonable to consider a virus a particle. I think the answer is pretty clearly yes, for the reasons described yesterday: as long as the energy scale is such that you don’t have to worry about the composite object falling apart, you can consider it a single particle for estimation purposes. A virus is a fairly hardy thing– if you’re trying to sterilize something, you generally heat it to temperatures of a hundred-odd C and hold it there for some time. And that’s probably not doing real damage to the physical integrity of the virus as a whole, just scrambling some bits of DNA. so I’d say it’s safe to call it a particle.

Given that, how would you estimate the time to spread out from some initial position? Well, you could come up with a detailed model of the initial distribution, and work out what the evolution of the wavefunction should look like, but we’re going for quick and dirty here, which means this is the time to break out one of the two calculations you can actually do using the equation from which this blog derives its name:

$\Delta x \Delta p \geq \frac{\hbar}{2}$

That is, of course, the Heisenberg Uncertainty Principle, relating the uncertainty in position to the uncertainty in momentum. We can safely assume that relativity doesn’t enter into this, meaning that momentum is just mass times velocity, and re-arrange this to find:

$\Delta v \geq \frac{\hbar}{2 m \Delta x}$

This gives you an uncertainty in velocity in terms of the mass of the virus and the initial uncertainty in position. And that’s all we really need to get a sense of the situation we’re interested in.

Note that this isn’t a specific velocity– that would be impossible in quantum physics. Instead it’s a range of velocities. The actual velocity of any one virus would be somewhere within that range, but is impossible to predict. But if you ran the experiment many times– a million, say– you would find that the distribution of measured velocities traced out some overall distribution characterized by a spread given by the formula above: Planck’s constant divided by the mass and the initial position uncertainty.

And over time, that spread of velocities would lead to a spread in position, as the different viruses flew off in all different directions at different speeds. This is described mathematically by a spreading-out of the wavefunction, which is the thing we’re trying to describe. So the spread in velocity really is the thing we’re after.

So, we’ve got an equation, now we need some numbers. So, what’s the scale of a virus? Well, a cell is kind of micron-sized, give or take a bit, and viruses are smaller than that. So let’s say that if we’ve got a single virus, and we’ve pinned down its position reasonably well, we know it to within plus or minus 100nm. That’s probably a reasonable order of magnitude for $\Delta x$.

What’s the mass, then? I could probably pick a type of virus and find this out, provided I was willing to spend a bunch of time on unit conversions, which I’m not. Instead, let’s say that it has around the same density as water, and fits in a cube 100nm on a side– that’s a volume of (10-7m)3 or 10-21m3, and water has a density of 1000 kg per cubic meter, so a mass of an attogram: 10-18 kg.

Then we just plug in to get the velocity, keeping only orders of magnitude. Planck’s constant is 10-34 kg m2/s, so we have:

$\Delta v \geq \frac{10^{-34}kgm^2/s}{(2)(10^{-18}kg)(10^{-7}m)} = \frac{1}{2} 10^{-9}$ m/s.

So the wavefunction would spread out by half a nanometer per second, which means it would take a bit more than three minutes to double the initial position spread of 100nm.

So, yeah, assuming you can call a virus a single quantum particle, it should spread out due to quantum effects on a time scale of minutes.

Now, I’ve made some simplifying assumptions there, and putting more exact numbers in would probably tend to make the velocity spread a bit bigger– the initial position spread and mass are probably both overestimated. But that’s not going to be off by many orders of magnitude, and roughly validates the original comment that kicked all this off. The characteristic time scale for the wavefunction of a single virus to spread out due to quantum-mechanical effects would be tens of seconds.

But is this something you’d actually be able to see? I suppose you might, if you could float a virus in a vacuum and observe its position somehow. A more realistic situation would involve a virus suspended in some sort of fluid– air or water– and in that case, the Brownian motion due to collisions with molecules from the fluid would probably swamp anything you might see from quantum effects. But in the spherical frictionless world of low-energy quantum mechanics, it’s kind of a fun little problem.

(Featured false-color image of an influenza virus from Wikipedia.)

1. #1 CatMat
October 16, 2013

…mass of an attogram: 10-18 kg

Isn’t that a femtogram?

October 16, 2013

Oh, right. It’s an atto-kilo-gram. Stupid SI units.

3. #3 Michael Kelsey
SLAC National Accelerator Laboratory
October 16, 2013

Yup. Historically stupid, probably because it was easier to fabricate a _large_ precision metal ingot than a small one :-/

Hopefully if one of the two replacement methods work (counting silicon atoms or using a watt-balance), SI could move toward specifying the standard unit as “gram”, with “kilo” restored to a prefix.

4. #4 Eric Lund
October 17, 2013

it was easier to fabricate a _large_ precision metal ingot than a small one

It still is. Polishing is just as precise in absolute terms, so the kilogram ingot gives you three extra significant figures for free compared to the gram ingot.

As for relegating the kilo part of kilogram to a suffix: I’m afraid that ship has sailed.Many derived units (newton, joule, watt, pascal, and probably some others I’m overlooking) include the kilogram rather than the gram in their definition. There is a alternative system based on the gram, but the tradeoff is that the length standard is the centimeter rather than the meter, and in that system certain derived units (dyne, erg, etc.) include the centimeter in their definition. The CGS system comes about because many solids and liquids have densities of the order of 1 g/cm^3, and even the densest metals are of the order of 10 g/cm^3, making that system convenient for applications such as chemistry and certain branches of geophysics. The MKS system is often more convenient for experimental physics and engineering (note the names on the derived MKS units I mentioned).

October 17, 2013

The CGS system comes about because many solids and liquids have densities of the order of 1 g/cm^3, and even the densest metals are of the order of 10 g/cm^3, making that system convenient for applications such as chemistry and certain branches of geophysics.

The popular systems of CGS units also tend to redefine charge in a way that simplifies Maxwell’s equations, so they’ve held on there for a long time. Though even Jackson has apparently caved on this.

I recall seeing a talk on gamma-ray bursts back in 1999 that used CGS units throughout, probably because of E&M, though I suspect the chance to bump all their eye-catching exponents up by seven through using ergs rather than joules didn’t hurt.

6. #6 G
Uncertain
October 17, 2013

This is admittedly over my head, but nonetheless: it seems to me that, even with Brownian motion and other factors in any actual biological medium, looking at viruses from the perspective of QM might yield some insights into how they behave. Yes? No? Nuts? Is there any chance that someone out there has relevant interdisciplinary background and could have a go at this?

7. #7 Shawn S
October 17, 2013

I think the idea of changing the SI base unit of mass causes a lot of problems. Unless we have “mystery powers of 10″ floating around (like in chemistry with L=dm3), we have to have new units to replace newtons, joules, watts, etc.

8. #8 RM
October 18, 2013

I know it probably won’t come in handy all that often for you, but I’ll name drop the BioNumbers database (http://bionumbers.hms.harvard.edu/ or just Google “bionumbers”), which is a collection of assorted biological measurments (with references).

For example, searching “diameter of virus” shows that HIV is 126.5 (±16.4) nm in diameter, Cowpea mosaic virus is ~28 nm, and Acanthamoeba polyphaga Mimivirus is 500 nm, and searching for “mass of virus” gives 7.9e-15 to 12.4e-15 g for Vaccinia and 4.08e+7 Da (6.8e-17 g) for Tobacco mosaic virus.

October 18, 2013

I’m not likely to do many calculations with these numbers, but it’s nice to have my order-of-magnitude estimates validated. So, thanks for the link.

10. #10 Jenora Feuer
October 18, 2013

@G:

looking at viruses from the perspective of QM might yield some insights into how they behave. Yes? No? Nuts?

I’d say… maaaaaybe, but not necessarily in the way you’re thinking, and not without a lot of computing requirements behind it.

Some of it is really going to come down to specialized forms of the protein folding problem, which DO have to include a degree of quantum effects. The problem is, to get a full solution, you have to include every fundamental particle, or at least every electron. Heck, I think it wasn’t until the 90s that we got to the point where things as complex as polycyclic aromatic compounds (multiple benzene rings in the same molecule) could be analyzed from a quantum mechanical level enough to get good stabs at volatility from a theoretical standpoint. (I vaguely remember a Scientific American article on this many years ago.) Protein molecules and DNA/RNA can be much more complex.

But figuring out exactly why protein molecules bend and kink the way they do is something that would give useful information on how viruses act, and that does rely on the low-level quantum interactions, albeit in ways that we’re still pinning down. (Hence, Folding@HOME.)

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