How to Weigh a Cell
Microbes are small. Tens of thousands of them fit in the period at the end of this<br>sentence. And yet, despite their tinyness, it is possible to weigh individual microbes with<br>remarkable precision.
A single yeast cell weighs about 100 picograms. An E. coli bacterium weighs 0.55<br>picograms, or 100 million times less than a grain of sand.1 With masses so small, weighing a single cell seems an impossible task.<br>After all, a normal kitchen scale resolves down to<br>0.1 grams, whereas an E. coli weighs 100 billion times less than that. Weighing a cell, then, demands eleven orders of magnitude more precision than the scale in<br>a typical pantry can provide.
Over the last few decades, scientists have created wondrous devices to weigh individual<br>cells with femtogram<br>precision.2<br>Before those devices existed, though, scientists instead made do with whatever was<br>lying around the lab; often just microscopes, centrifuges, and scraps of paper.
Attempts to weigh cells probably began in the 1800s with a Polish-Russian chemist, Marceli<br>Nencki. While working in Germany, Nencki grew cells to saturation in large flasks (using liquid<br>media made from rotting meat), and let the flask sit for a few hours until cells settled to<br>the bottom.3 Nencki then poured this broth through a filter to catch the cells, washed them with<br>water to remove sugars and salts, and weighed whatever was left. He called this the<br>"wet weight."
Next, Nencki put those wet cells onto an elevated shelf inside of a sealed glass jar. He filled the<br>bottom of this jar with concentrated sulfuric acid, which attracts and absorbs water vapor swirling inside.<br>After several days, the sulfuric acid reacts with all the water in the jar, thus<br>turning the cells into a bone-dry pellet. Nencki weighed these pellets and subtracted<br>the dry weight from the wet weight. From this experiment, he estimated that bacteria are<br>82.42 percent water by<br>mass.4
Nencki's attempts to weigh cells stopped there. He never took the extra step of<br>calculating how many cells were in the dried pellet and, therefore, what the mass of each<br>cell might be. That step was later taken by<br>Carl von<br>Nägeli, a German naturalist who is mostly forgotten today, but was considered an<br>intellectual giant in his<br>day.5<br>Nägeli was likely the first person to publish an estimate for the mass of a single<br>bacterium.6
Sometime around 1877, Nägeli peered at living cells beneath his microscope and<br>studied their dimensions. Yeast cells, he noted, measured about ten micrometers across<br>(or one-one hundredth of a millimeter). Large bacteria measured about two micrometers.7
And then, Nägeli took a leap of genius.
By approximating the cells as spheres, Nägeli began calculating rough<br>estimates of their volumes. If a yeast measures ten micrometers across and is roughly<br>spherical, for example, then its radius is five micrometers and its volume is given by:
$$<br>V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (5\ \mu m)^3 \approx 524\ \mu m^3<br>$$
In other words, Nägeli estimated that a yeast cell has a volume of about 500 cubic micrometers.8<br>And with volumes in-hand, Nägeli could then estimate cell mass.
He knew from Nencki's<br>work that cells are about 80 percent water, and he also knew that the<br>density of water is one gram per cubic centimeter. Therefore, a cell's<br>wet mass is roughly its volume times the density of water, and<br>its dry mass is the 20 percent that remains after the water is<br>removed. Nägeli crunched these numbers for his small bacteria, estimating their weights at 0.1<br>picograms wet and 0.033 picograms dry. (This is, strikingly, within an order-of-magnitude of modern measurements.)9
Nägeli's estimate went unchallenged for decades. But it was a calculation,<br>built on scattered observations and assumptions, rather than a direct measurement of a<br>single cell. That changed in 1953, when two biologists at Southern Illinois University<br>(partly funded by the Anheuser-Busch<br>brewery)10<br>invented one of the first methods to<br>weigh<br>a single yeast cell. And they did it, incredibly, with only a microscope,<br>some sugar water, and a camera.
To understand how these biologists did it, though, we first need to take a short detour through<br>Stokes' law.
When a small sphere falls through a fluid, the fluid pushes back with a drag<br>force that grows with speed. The drag eventually matches gravity, and the sphere falls<br>at a constant, terminal velocity. In 1845, George Stokes (an Irish mathematician who held<br>the Lucasian chair at Cambridge University for longer than even Isaac Newton) showed that<br>this drag force depends on the sphere's radius r, the fluid's viscosity<br>η, and its velocity v:
$$<br>F_{drag} = 6\pi \eta r v<br>$$
This is the equation for drag force, or the amount of resistance that the<br>fluid exerts on the sphere. But the sphere is also acted on by gravity, which<br>pulls it down, and by buoyancy, which pushes it up. The downward pull is given by the<br>sphere's weight minus the buoyant force, where d1 is the sphere's density, d0 is...