How to Weigh a Cell

Microbes are small. Tens of thousands of them fit in the period at the end of this sentence. And yet, despite their tinyness, it is possible to weigh individual microbes with remarkable precision.

A single yeast cell weighs about 100 picograms. An E. coli bacterium weighs 0.55 picograms, or 100 million times less than a grain of sand.¹ With masses so small, weighing a single cell seems an impossible task. After all, a normal kitchen scale resolves down to 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 a typical pantry can provide.

Over the last few decades, scientists have created wondrous devices to weigh individual cells with femtogram precision.² Before those devices existed, though, scientists instead made do with whatever was 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 Nencki. While working in Germany, Nencki grew cells to saturation in large flasks (using liquid media made from rotting meat), and let the flask sit for a few hours until cells settled to the bottom.³ Nencki then poured this broth through a filter to catch the cells, washed them with water to remove sugars and salts, and weighed whatever was left. He called this the "wet weight."

Next, Nencki put those wet cells onto an elevated shelf inside of a sealed glass jar. He filled the bottom of this jar with concentrated sulfuric acid, which attracts and absorbs water vapor swirling inside. After several days, the sulfuric acid reacts with all the water in the jar, thus turning the cells into a bone-dry pellet. Nencki weighed these pellets and subtracted the dry weight from the wet weight. From this experiment, he estimated that bacteria are 82.42 percent water by mass.⁴

Nencki's attempts to weigh cells stopped there. He never took the extra step of calculating how many cells were in the dried pellet and, therefore, what the mass of each cell might be. That step was later taken by Carl von Nägeli, a German naturalist who is mostly forgotten today, but was considered an intellectual giant in his day.⁵ Nägeli was likely the first person to publish an estimate for the mass of a single bacterium.⁶

Sometime around 1877, Nägeli peered at living cells beneath his microscope and studied their dimensions. Yeast cells, he noted, measured about ten micrometers across (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 estimates of their volumes. If a yeast measures ten micrometers across and is roughly spherical, for example, then its radius is five micrometers and its volume is given by:

$$ V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (5\ \mu m)^3 \approx 524\ \mu m^3 $$

In other words, Nägeli estimated that a yeast cell has a volume of about 500 cubic micrometers.⁸ And with volumes in-hand, Nägeli could then estimate cell mass.

He knew from Nencki's work that cells are about 80 percent water, and he also knew that the density of water is one gram per cubic centimeter. Therefore, a cell's wet mass is roughly its volume times the density of water, and its dry mass is the 20 percent that remains after the water is removed. Nägeli crunched these numbers for his small bacteria, estimating their weights at 0.1 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, built on scattered observations and assumptions, rather than a direct measurement of a single cell. That changed in 1953, when two biologists at Southern Illinois University (partly funded by the Anheuser-Busch brewery)¹⁰ invented one of the first methods to weigh a single yeast cell. And they did it, incredibly, with only a microscope, some sugar water, and a camera.

To understand how these biologists did it, though, we first need to take a short detour through Stokes' law.

When a small sphere falls through a fluid, the fluid pushes back with a drag force that grows with speed. The drag eventually matches gravity, and the sphere falls at a constant, terminal velocity. In 1845, George Stokes (an Irish mathematician who held the Lucasian chair at Cambridge University for longer than even Isaac Newton) showed that this drag force depends on the sphere's radius r, the fluid's viscosity η, and its velocity v:

$$ F_{drag} = 6\pi \eta r v $$

This is the equation for drag force, or the amount of resistance that the fluid exerts on the sphere. But the sphere is also acted on by gravity, which pulls it down, and by buoyancy, which pushes it up. The downward pull is given by the sphere's weight minus the buoyant force, where d₁ is the sphere's density, d₀ is the fluid's density, and g is gravitational acceleration:

$$ F_{net} = \frac{4}{3}\pi r^3 (d_1 - d_0) g $$

At terminal velocity, the drag force balances the downward force. Setting the two equations equal to each other, then, yields an expression with a single unknown parameter (for velocity):

$$ \frac{4}{3}\pi r^3 (d_1 - d_0) g = 6\pi \eta r v $$

This equation can be used to calculate the mass of cells. Simply drop the cells (or "spheres") into a fluid and watch them sink. Record how long it takes each one to fall a known distance, and calculate the velocity, v, from those images. Every other term in the equation is already known. The sphere's radius can be easily measured; the fluid's density and viscosity are known properties; and gravitational acceleration is a constant. So all a biologist needs to do is plug these values in and solve for the one thing that cannot be measured directly, which is the sphere's mass. Getting there just requires a small rearrangement of terms, since mass (volume times density) is hiding inside the left-hand side of the above equation. After rearranging terms, we get:

$$ m = \frac{4}{3}\pi r^3 \left( d_0 + \frac{9 \eta v}{2 r^2 g} \right) $$

For the 1953 experiment, the two biologists suspended yeast cells in water with one percent sugar, dropped the mixture onto a microscope slide, and then propped the slide up vertically. Then, they used their microscope and camera to film time-lapse videos of the yeast cells falling, frame by frame. They did this for 67 cells at 400× magnification.

Using the photos, they measured the distance each cell traveled over a fixed time and determined the radius of each cell using a scale bar. And finally, they used Stokes' law to calculate the weight of individual cells: 79 picograms on average; a value remarkably close to estimates made as recently as 2022.

The "falling yeast" experiment worked because the researchers assumed, much like Nägeli, that each yeast cell is spherical. This is an important assumption, because Stokes' law only works for spheres moving through a fluid; not for cylinders or other shapes.

But, of course, not all cells are spheres! E. coli cells look more like cylinders. If you were to repeat the 1953 experiment on such bacteria, then, the cells would create turbulence, throwing off Stokes' equations. Weighing a single bacterium required a completely different strategy.

In 1998, biologists figured out how to weigh single bacteria using a transmission electron microscope. When a beam of electrons passes through a sample, some will make it through and strike a detector on the other side. Many other electrons, though, will be absorbed or scattered away. How many electrons get lost depends on how much material they must pass through. A heavier cell deflects more electrons than a smaller one, which shows up as a weaker signal in the detector. By measuring how many electrons hit a detector after passing through a cell, then, researchers can work backward to calculate the cell's mass.

To do this, the researchers first calibrated an electron microscope by firing beams at tiny latex beads with known dimensions and densities, and therefore known mass. In each case, they measured how many electrons struck the detector and plotted these numbers as a function of each bead's mass to create a "calibration curve."

Next, they repeated the procedure on E. coli. Electron microscopes only work under vacuum, so the cells were first dried to remove moisture. The team shot electron beams at 678 different E. coli cells and plotted each measurement against the calibration curve. They found that the cells' dry weights ranged from 83 to 1,172 femtograms, depending on their growth phase. The median weight was 222 femtograms, which is about a trillion-times smaller than a grain of rice.¹¹

The electron microscope method was highly accurate but also inherently limited, since each cell could only be weighed once (and only while dead, under vacuum). To weigh a single cell as it grows — arguably the holy grail for these kinds of measurements — biologists needed to create an instrument that could weigh the same cell again and again, while it was still living, growing, and dividing.

This problem was solved in 2010, when MIT scientists unveiled a suspended microchannel resonator that could measure the mass of any bacterial cell in liquid, while living and dividing. The device is a "hollow microcantilever beam containing an embedded fluidic microchannel" or, said another way, a little beam that has a U-shaped tunnel running through its interior and vibrates very quickly, similar to a guitar string.¹²

As a single cell floats through the channel in this beam, the beam's vibration changes ever so slightly. Heavier cells shift the beam's vibrations more than lighter ones.¹³ Those frequency shifts are what allow researchers to figure out the cell's "buoyant mass," or how much heavier the cell is compared to water.¹⁴ Such measurements have femtogram precision, meaning they can detect changes between two masses 1,000 times smaller than a bacterium's weight.

The device can measure the same cell again and again. After a cell passes through the vibrating beam and its mass is recorded, researchers reverse the flow of liquid and send it back through the channel in the opposite direction. In this way, one can weigh a cell at every phase of its lifecycle.

When the authors measured 48 individual cells and averaged the results, they found that a typical E. coli weighs about 0.55 picograms. But this number is constantly in flux because the growth rate increases in proportion to cell size. At 37°C, the smallest cells added about 0.06 picograms of mass per hour, while the largest gained closer to 0.14 picograms per hour.

Efforts to weigh cells have spanned more than a century. The tools have changed enormously during that time, but every scientist in this story shared a common belief that nothing is impossible to measure.

Lord Rayleigh had this same instinct in 1890, when he set out to calculate the length of a single oil molecule. Rayleigh dropped a small amount of olive oil into a dish of water in his home laboratory, and then watched the oil spread into a thin, circular film. He knew the volume of the oil applied, and then measured the area it covered upon the water, to calculate the film's thickness: 1.6 × 10⁻⁷ centimeters, the length of a single molecule. Amazingly, Rayleigh was off by just 18.5 percent compared to modern X-ray measurements, despite using only that dish of water, a dropper, and a ruler.

Nencki, Nägeli, and the scientists who followed them achieved similarly accurate estimates with equally simple equipment. They all understood that a cell is not some irreducibly mysterious entity, but rather a physical object; a collection of molecules with a real mass, volume, and density. Once they accepted that, the question of how to weigh them became a physics problem, solvable with mathematics and ingenuity.