DNA and Topology

Topology · Axiom Academy

How knot theory and topology save your genetic information Every time a cell divides, it must copy its entire genome—about 3 billion base pairs in humans—and separate two identical DNA molecules into daughter cells. But DNA isn't a simple string: it's a twisted, coiled, knotted molecule that poses profound topological challenges. Enter the mathematics of knot theory and topology, the unsung heroes that prevent your DNA from becoming a catastrophically tangled mess. The iconic double helix structure of DNA—discovered by Watson and Crick—creates an immediate topological problem. The two strands wrap around each other roughly once every 10.5 base pairs. When DNA needs to be replicated, the strands must separate, which means the entire molecule must be unwound. Linking Number: The Topological Invariant The linking number (Lk) is a topological invariant that measures how two closed curves are intertwined. For circular DNA (like bacterial plasmids), the linking number must remain constant unless the DNA is cut. Where Tw is the twist (helical winding of the strands) and Wr is the writhe (supercoiling of the axis). The number of times one strand wraps around the other. For B-form DNA, Tw ≈ N/10.5, where N is the number of base pairs. The supercoiling of the DNA axis in 3D space. Positive writhe creates right-handed supercoils, negative writhe creates left-handed supercoils.

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