How calculus helps to unveil the secrets of DNA

Calculation has traditionally been applied to "hard" sciences such as physics, astronomy and chemistry. But in recent decades, he has taken a step forward in the fields of biology and medicine, in areas such as epidemiology, population biology, neuroscience, and medical imaging . We have seen examples of mathematical biology throughout our history, ranging from the use of calculus to predict the results of facial surgery to modeling HIV to fighting the immune system. But all these examples concerned one aspect of the mystery of change, the most modern obsession with calculation. In contrast, the following example is drawn from the ancient mystery of the curves, which was relaunched by a puzzle on the three-dimensional path of DNA.

The puzzle was how DNA, an extremely long molecule that contains all the genetic information needed to make a person, is packaged in cells. Each of your ten billion cells contains about two meters of DNA. If laid end to end, this DNA would reach the sun and the back dozens of times. Nevertheless, a skeptic might say that this comparison is not as impressive as it looks; it simply reflects the number of cells that each of us has. A more informative comparison is with the size of the cell's nucleus, the container that contains the DNA. The diameter of a typical nucleus is about five millionths of a meter, and so it is four hundred thousand times smaller than the DNA that must be found there. This compression factor is equivalent to stuffing twenty kilometers of string into a tennis ball.

In addition to this, the DNA can not be inserted into the nucleus at random. Do not get confused. The packaging should be done in an orderly manner so that the DNA can be read by enzymes and translated into proteins necessary for maintaining the cell. Orderly packaging is also important so that the DNA can be copied cleanly when the cell is about to divide.

Evolution has solved the problem of packaging with reels, the same solution as when we need to store a long thread. The DNA in the cells is wrapped around molecular reels composed of specialized proteins called histones. For further compaction, the coils are tied end to end, like beads on a necklace, and then the necklace is wrapped in string-like fibers, themselves wound into chromosomes. These reel bobbins compact the DNA enough to integrate it into the cramped quarters of the nucleus.

But the reels were not the original solution of nature to the problem of packaging. The first creatures on Earth were unicellular organisms lacking nuclei and chromosomes. They did not have reels, just like the bacteria and viruses of today. In such cases, the genetic material is compacted by a mechanism based on geometry and elasticity. Imagine pulling a rubber band and twisting it on one side and holding it between your fingers. At first, each successive turn of the elastic introduces a twist. The twists accumulate and the elastic stays upright until the accumulated torsion crosses a threshold. Then, the elastic is suddenly deformed in the third dimension. He begins to wrap on himself, as he writhes in pain. These contortions cause the elastic to bend and compact. The DNA does the same thing.

This phenomenon is known as super winding. It is spread in circular loops of DNA. Although we tend to think of DNA as a straight-edged helix with free ends, it often closes on itself to form a circle. When this happens, it is as if you were removing your belt, put some twists in it, and close it again. After that, the number of twists in the belt can not change. It's blocked. If you try to twist the belt somewhere along its length without removing it, counter-twists will form elsewhere to compensate. There is a conservation law at work here. The same thing happens when you store a garden hose by piling it on the floor with several coils stacked on top of each other. When you try to pull the pipe straight, it twists between your hands. The coils are transformed into twists. The conversion can also go in the opposite direction, from twists to reels, such as when an elastic twists when it is crooked. The DNA of primitive organisms uses this thrill. Some enzymes can cut DNA, twist it and close it again. When the DNA relaxes its forces to reduce its energy, the law of conservation requires it to become more clogged and therefore more compact. The resulting path of the DNA molecule is no longer in a plane. It twists in three dimensions.

In the early 1970s, an American mathematician, Brock Fuller, gave the first mathematical description of this three-dimensional contortion of DNA. He invented a quantity that he nicknamed the twisted number of DNA. He derived formulas using integrals and derivatives and proved some theorems about the deforming number that formalized the law of conservation of twists and coils. The study of the geometry and topology of DNA has since flourished. Mathematicians have used the theory of knots and entanglement calculus to elucidate the mechanisms of certain enzymes that can twist DNA, cut it or introduce nodes and links to it. These enzymes modify the topology of the DNA and are therefore called topoisomerases. They can break DNA strands and reseal them, and they are essential for cells to divide and grow. They have proven to be effective targets for anticancer drugs for chemotherapy. The mechanism of action is not entirely clear, but it is thought that by blocking the action of topoisomerases, drugs (called topoisomerase inhibitors) can selectively damage the DNA of cancer cells, which makes them commit suicide. Good news for the patient, bad news for the tumor.

In the application of computation to super-coiled DNA, the double helix is ​​modeled as a continuous curve. As usual, the calculus likes to work with continuous objects. In reality, DNA is a discrete collection of atoms. There is nothing really continuous about it. But basically, it can be treated as if it's a continuous curve, like an ideal rubber band. The advantage of doing this is that the elasticity theory and differential geometry apparatus, two derivatives of the computation, can then be applied to calculate how the DNA is deformed under the effect of the forces of the proteins. environment and interactions with itself.

The most important point is that calculus takes its usual creative license, dealing with discrete objects as they were continuous to clear up their behavior. The modeling is approximate but useful. In any case, it's the only game in town. Without assumption of continuity, the principle of infinity can not be deployed. And without the principle of the infinite, we have no computation, no differential geometry, no theory of elasticity.

I expect in the future to see even more examples of calculations and continuous mathematics applied to intrinsically distinct biology actors: genes, cells, proteins and other actors in the field. biological drama. It is just too difficult to take advantage of the continuum approximation to not use it. Until we develop a new form of computation that works for discrete systems as well as traditional computation for continuum systems, the infinity principle will continue to guide us in the mathematical modeling of human beings. living.

Excerpt from INFINITE POWERS by Steven Strogatz. Copyright © 2019 by Steven Strogatz. Reprinted with permission of Houghton Mifflin Harcourt Publishing Company. All rights reserved.