A bacterium's genome, pulled into a straight thread, is nearly 1,000 times longer than the cell from which it came. If you placed one E. coli into a gallon-sized jug with some nutrients and waited a few hours, the genomes of its descendants, placed end-to-end, would reach to the moon and back...several times.¹

One rarely pauses to ponder how so much DNA — let alone sugars, proteins, lipids, and other molecules — can fit inside such a small vessel. A typical E. coli cell, after all, measures about one micrometer across. Its entire volume is 100 times smaller than that of a red blood cell, and about 100 million times smaller than a grain of sand.

The truth is that biochemistry textbooks often depict cells as spacious places, where molecules float in secluded harmony. "But a cell looks more like a burrito," says Michael Elowitz, a biologist at Caltech. All the biochemicals are pushed together, bumping into each other.

It's a wonder that anything gets done inside of living cells at all, for they are fast and crowded places. A painting by David Goodsell, a biologist in San Diego who moonlights as an illustrator, enamors because it conveys this denseness in visual form.

Such paintings are beautiful, but ultimately simplistic. They are snapshots of cells at a single moment in time. Paintings can hint at complexity, but don't convey the intense dynamics of a living cell. All of our scientific methods to study life, similarly, demand that cells be killed or frozen before a microscopic image is taken. Therefore, mathematics and words are our best tools to understand the active chaos of living cells.

For many years, I had an intense aversion to mathematics. Biology was my refuge because it was simple: Read the textbook, memorize the facts, and ace the exam. (The only reason I majored in biochemistry as a college student was because it didn't have a multivariable calculus requirement.) But then, I started a Ph.D. in Bioengineering at Caltech and landed in Rob Phillips' lab. Rob is a master of physical biology; a person who has spent decades building a numerical intuition for biology.

And suddenly, I was thrown into the deep end of biophysics. I took courses like Physical Biology of the Cell and wrote out statistical mechanics equations on big whiteboards. I felt like a real biologist for the first time, because suddenly I could think of a question — like "How many ribosomes does a typical E. coli cell have?" — and figure it out from scratch, using little more than scribbled equations. It was a joy, at last, to finally grasp the "numbers of biology."²

Without mathematics, biology is naked; one can only comprehend it at arm's length. But with numbers, living cells come alive. Mathematics enables one to see a Goodsell painting with fresh eyes.

Just consider the Central Dogma. Students learn the basics through words: DNA is transcribed to RNA, which is translated into proteins. But what does this really mean? How fast does DNA become RNA, or RNA a protein? How many proteins are in a cell, and how fast do they fold and move? Doing these calculations reveals both the beauty and weirdness of life at the smallest scales. It lends a deeper appreciation to biology. And all we need to do it is a pencil and paper.

But first, some background. A microbe's guts are a veritable Times Square, crowded with sugars, proteins, and water molecules that ricochet and smash into each other billions of times each second. Space is limited. A bacterium's insides are 70 percent water by mass, and the other 30 percent is dominated by proteins first, followed by RNA and lipids. DNA accounts for just one percent of a cell's mass. And all of this stuff fits inside a volume that is one-quadrillionth the size of a liter. (About 500 billion microbes fit inside of an aspirin tablet!)

Now let's think about the transcription of DNA into RNA. A typical E. coli has 4,400 genes in total. At any given moment, about 25 percent of these genes are being copied into RNA by a large protein called RNA polymerase. Each polymerase protein grabs onto the DNA and zips along its length at breakneck speeds, converting about 40 bases of DNA into its corresponding RNA each second. If an RNA polymerase were scaled up to the size of a human, it would move twice as fast as Usain Bolt's record-setting pace in a 100-meter dash.³ The polymerase only makes about one mistake every 100,000 bases.

Fewer than 30 seconds pass from the time polymerase binds to the DNA to the time it makes a full RNA. As soon as the RNA is finished, it is released by the protein and diffuses, or floats, away. A small army of ribosomes quickly swoops in and grabs it. The ribosomes read the letters in the RNA sequence — three at a time — and convert them into amino acids in a growing protein.

Ribosomes also move quickly; they make an average-sized protein from RNA in just 24 seconds. A single ribosome could translate the first Harry Potter book in two-and-a-half hours while making just three dozen typos along the way.⁴

When a ribosome finishes this task, its jaws unclench and the new protein is released. At any given time, a typical bacterial cell has three or four million proteins floating around, each responsible for breaking down sugar, copying DNA, sending signals to nearby cells, and much more. A living cell is an autonomous factory, where machines build machines that build themselves.

At the small scales in which proteins exist, even a subtle difference between two molecules can make a big difference. Diffusion is one example of this. Small molecules, such as water or ions, diffuse quickly, migrating about one centimeter per second. (In other words, these molecules travel the length of ten thousand bacterial cells in the span of one second.) But large proteins move more slowly — only a few micrometers (one-millionth of a meter) in the same second. A rule of thumb is that, for a molecule to move twice as far, it takes four times as long. Molecules don't move in straight lines, but jostle about in three dimensions. Diffusion is described in units of length²/time, meaning it takes a protein 10 milliseconds to move across a cell, but 20 days to travel one centimeter.

Diffusion sets an upper limit on a cell's size.⁵ If a cell is too small, not enough "stuff" fits inside and evolution is constrained. If a cell is too large, nothing ever gets done because proteins cannot reach their destinations. Life is a search for many little optimas.

As proteins move through a cell, they are also bombarded by water, sugars, and other proteins. Every protein collides with millions of molecules every second, and its corresponding substrate — the molecule it means to find — is vanishingly rare. In biology textbooks, one often reads sentences like, "A protein's substrate has a concentration of 0.5 millimolar." All this means is that there's one substrate for every 100,000 water molecules. And yet, even at this sparse dilution, the enzyme will find, and collide with, about 500,000 substrates every second!⁶

Cells are chaotic swarms of energy and fortuitous accidents. The Central Dogma sounds simple in words, but is a miracle in reality. It's a wonder that cells get anything done at all.

The first time I did these calculations, I felt an intense appreciation for biology. And now, I want everyone else to feel the same. We ought to teach students of biology to think as mathematicians: to carefully quantify biology, to think in absolute units, and to develop a feeling for the organism.

Throughout this essay, I've depicted cells as dense blobs filled with lots of stuff. This insinuates that, if one studied everything in a cell and tallied all of its components, then perhaps we'd have a complete knowledge of biology. But this isn't true.

Some proteins "moonlight" in the cell. They carry out one function when their substrate is around, and do something entirely different when it isn't. Many protein signaling pathways also play a specific role in one type of cell, and something different in another. Biology is infinitely weird, and if we ever plan to master it, we will need new scientific methods to measure protein dynamics and interaction strengths.

When COVID came in 2020, I left my Ph.D. and moved to New York to study journalism. I fell out of contact with Rob, but my appreciation for biological numbers remained. I still enjoy jotting down calculations in the margins of books. And every day, I feel grateful that I get to learn about biology, a field that is far stranger than anything one could see while scuba diving or traveling to Mars. It is still difficult for me to imagine the microscopic world when my mind and experiences are almost wholly confined to the macroscopic world. But a pen, paper, and imagination seem to suffice.