Is God a Mathematician_ - Mario Livio [100]
Deoxyribonucleic acid, or DNA, is the genetic material of all cells. It consists of two very long strands that are intertwined and twisted around each other millions of times to form a double helix. Along the two backbones, which can be thought of as the sides of a ladder, sugar and phosphate molecules alternate. The “rungs” of the ladder consist of pairs of bases connected by hydrogen bonds in a prescribed fashion (adenine bonds only with thymine, and cytosine only with guanine; figure 58). When a cell divides, the first step is replication of DNA, so that daughter cells can receive copies. Similarly, in the process of transcription (in which genetic information from DNA is copied to RNA), a section of the DNA double helix is uncoiled and only one DNA strand serves as a template. After the synthesis of RNA is complete, the DNA recoils into its helix. Neither the replication nor the transcription process is easy, however, because DNA is so tightly knotted and coiled (in order to compact the information storage) that unless some unpacking takes place, these vital life processes could not proceed smoothly. In addition, for the replication process to reach completion, offspring DNA molecules must be unknotted, and the parent DNA must eventually be restored to its original configuration.
Figure 58
The agents that take care of the unknotting and disentanglement are enzymes. Enzymes can pass one DNA strand through another by creating temporary breaks and reconnecting the ends differently. Does this process sound familiar? These are precisely the surgical operations introduced by Conway for the unraveling of mathematical knots (represented in figure 56). In other words, from a topological standpoint, DNA is a complex knot that has to be unknotted by enzymes to allow for replication or transcription to occur. By using knot theory to calculate how difficult it is to unknot the DNA, researchers can study the properties of the enzymes that do the unknotting. Better yet, using experimental visualization techniques such as electron microscopy and gel electrophoresis, scientists can actually observe and quantify the changes in the knotting and linking of DNA caused by an enzyme (figure 59 shows an electron micrograph of a DNA knot). The challenge to mathematicians is then to deduce the mechanisms by which the enzymes operate from the observed changes in the topology of the DNA. As a byproduct, the changes in the number of crossings in the DNA knot give biologists a measure of the reaction rates of the enzymes—how many crossings per minute can an enzyme of a given concentration affect.
But molecular biology is not the only arena in which knot theory found unforeseen applications. String theory—the current attempt to formulate a unified theory that explains all the forces in nature—is also concerned with knots.
Figure 59
The Universe on a String?
Gravity is the force that operates on the largest scales. It holds the stars in the galaxies together, and it influences the expansion of the universe. Einstein’s general relativity is a remarkable theory of gravity. Deep within the atomic nucleus, other forces and a different theory reign supreme. The strong nuclear force holds particles called quarks together to form the familiar protons and neutrons, the basic constituents of matter. The behavior of the particles and the forces in the subatomic world is governed by the laws of quantum mechanics. Do quarks and galaxies play by the same rules? Physicists believe they should, even though they don’t yet quite know why. For decades, physicists have been searching for a “theory of everything”—a comprehensive description of the laws of nature. In particular, they want to bridge the gap between the large and the small with a quantum theory of gravity—a reconciliation of general relativity with quantum mechanics. String theory appears to be the current best bet for such a theory of everything. Originally developed and discarded as a theory for the nuclear force itself, string theory was revived from obscurity in 1974 by physicists