A bacterium’s genome, drawn out into a straight thread, is about 1,000 times longer than the cell it came from. If you have one e coli In a gallon-sized jug with some nutrients and after waiting a few hours, the genomes of its descendants were placed, end to end,
reach the moon and back…several times.1
One rarely stops to consider how so much DNA – let alone sugars, proteins, lipids and other molecules – could fit inside such a small vessel. a typical e coli After all, cells measure about a micrometer. Its entire volume is 100 times smaller than 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 vast spaces, where molecules swim in solitary harmony. “But a cell looks like a burrito,” says Caltech biologist Michael Elowitz. All the biochemicals are pushed together, colliding with each other.
It’s a wonder that anything happens at all inside living cells, after all they are loud and crowded places. A painting by David Goodsell, a San Diego biologist who moonlights as a painter, is mesmerizing because it conveys this intensity in visual form.
Such paintings are beautiful, but ultimately simple. They are snapshots of cells at a single moment in time. can paint prompt on complexity, but do not express the profound dynamics of the living cell. All our scientific methods of studying life, similarly, demand that cells be killed or frozen before microscopic images can be taken. Therefore, mathematics and words are our best tools for understanding the active chaos of living cells.
For many years I had a strong dislike for mathematics. Biology was my refuge because it was simple: read the textbook, memorize the facts, and ace the test. (The only reason I majored in biochemistry as a college student was because it didn’t require multivariable calculus.) But then, I got a Ph.D. Started. in bioengineering at Caltech and ended up in the laboratory of Rob Phillips. Rob has a Master of Physical Biology; A man who has spent decades developing 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 statistical mechanics equations on the big whiteboard. I felt like a real biologist for the first time, because suddenly I could think of a question – like “How many ribosomes are there in a typical organism?” e coli
The cell has?” – and figure it out from scratch, using little more than written equations. After all, it was a pleasure to understand the “numbers of biology.”2
Without mathematics, biology is naked; One can understand it at arm’s length. But with numbers, living cells become alive. Mathematics enables one to see a Goodsell painting with new eyes.
Just consider the central dogma. Students learn the basics through words: DNA is transcribed into 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 there in a cell and how? Fast Do they turn and move? Making these calculations reveals both the beauty and strangeness of life on the smallest scales. It provides a deeper appreciation of biology. And to do it all we need is a pencil and paper.
But first, some background. A microbe’s intestines are like a literal Times Square, filled with sugar, protein, and water molecules that move and collide with each other billions of times per second. Space is limited. The inside of a bacterium is 70 percent water by mass, and the other 30 percent consists first of proteins, followed by RNA and lipids. DNA accounts for only one percent of the cell’s mass. and all this Material Fits inside a volume about a quarter of a liter in size. (About 500 billion germs fit inside an aspirin tablet!)
Let us now think about the transcription of DNA into RNA. a typical e coli There are 4,400 genes in total. At any given time, about 25 percent of these genes are being copied into RNA by a large protein called RNA polymerase. Each polymerase protein grabs hold of the DNA and moves at breakneck speed along its length, each second converting about 40 bases of DNA into its corresponding RNA. If an RNA polymerase was scaled up to the size of a human, it would run twice as fast as Usain Bolt’s record-setting speed in the 100-meter race.3
The polymerase makes only one mistake every 100,000 bases.
Less than 30 seconds elapse from the time the polymerase attaches itself to the DNA until the complete RNA is formed. As the RNA is finished, it is released by the protein and spreads out, or floats away. A small army of ribosomes immediately swoops in and captures it. Ribosomes read the letters in the RNA sequence – three at a time – and convert them into amino acids in growing proteins.
Ribosomes also move fast; They make an average-sized protein from RNA in just 24 seconds. A ribosome could translate the first Harry Potter book in two and a half hours, while making only three dozen typos.4
When the ribosome completes this task, its jaws open and the new protein is released. At any given time, there are three or four million proteins floating around in a typical bacterial cell, each protein responsible for breaking down sugar, copying DNA, sending signals to nearby cells, and more. A living cell is an autonomous factory, where machines make machines that build themselves.
On the small scale on which proteins exist, even a subtle difference between two molecules can make a big difference. Diffusion is an example of this. Small molecules, such as water or ions, diffuse rapidly, migrating at about a centimeter per second. (In other words, these molecules move the length of ten thousand bacterial cells in the span of one second.) But larger proteins move more slowly – only a few micrometers (one millionth of a meter) in a single second. The general rule is that it takes four times as long for a molecule to travel twice the distance. Molecules do not move in straight lines, but in three dimensions. Diffusion is described in units of length2/time, meaning that it takes 10 milliseconds for a protein to move across a cell, but 20 days to travel a distance of one centimeter.
Proliferation sets the upper limit of cell size.5
If a cell is too small, not enough “stuff” can fit inside and growth is inhibited. If a cell is too large, nothing gets done because the proteins cannot reach their destination. Life is a search for many small hopes.
As proteins move through a cell, they are also bombarded by water, sugars, and other proteins. Each protein is hit by millions of molecules every second, and its corresponding substrate – the molecule it wants to find – is disappearing. In biology textbooks, one often reads sentences like, “The concentration of the substrate of a protein is 0.5 millimolar.” This means that for every 100,000 water molecules there is one substrate. And yet, even at this rare dilution, the enzyme will find and hit about 500,000 substrates every second!6
Cells are chaotic clumps of energy and random accidents. The central dogma sounds simple in words, but in reality it is a miracle. It’s surprising that cells can do anything.
The first time I did these calculations, I felt a deep appreciation for biology. And now, I want everyone else to feel the same way. We must teach biology students to think like mathematicians: to measure biology carefully, to think in absolute units, and to develop a feeling for the organism.
Throughout this essay, I have depicted cells as dense blobs filled with
Material. This suggests that, if one studied everything in a cell and tallied all its components, we would probably have a complete knowledge of biology. But this is not true.
Some proteins “moonlight” the cell. When their substrate is nearby they perform one function, and when it is not they do something completely different. Many protein signaling pathways also play a specific role in one type of cell, and a slightly different role in another. Biology is extremely strange, and if we ever plan to master it, we will need new scientific methods to measure protein dynamics and the strength of interactions.
When COVID hit in 2020, I left my Ph.D. And went to New York to study journalism. I fell out of touch with Rob, but my appreciation for biological numbers remained. I still enjoy writing accounts in the margins of books. And every day, I feel grateful that I get to learn about biology, a field that is much stranger than anything you might see while scuba diving or traveling to Mars. It is still difficult for me to imagine the microcosm when my mind and experiences are almost entirely confined to the macrocosm. But a pen, paper and imagination are enough.
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