How to mathematically model a biological enzyme:
Problem: DNA gets knotted during replication. Knots prevent gene expression and cell division. Topoisomerase II unknots DNA by cutting both strands, passing another strand through, and resealing.
Mathematical model: "Tangle surgery"
• Represent DNA segment as rational tangle (horizontal + vertical twists)
• Enzyme action = remove one tangle, insert another
• Question: what sequence of surgeries unknots the DNA?
Verification: Use knot invariants (Jones polynomial, Alexander polynomial) to check if you've reached the unknot.
This isn't just theory - it's used for drug design. Many chemotherapy drugs target topoisomerases to stop cancer cells from dividing. Understanding the math of unknotting → better enzyme inhibitors → more effective cancer treatment.
Pure topology becomes practical medicine.
Problem: DNA gets knotted during replication. Knots prevent gene expression and cell division. Topoisomerase II unknots DNA by cutting both strands, passing another strand through, and resealing.
Mathematical model: "Tangle surgery"
• Represent DNA segment as rational tangle (horizontal + vertical twists)
• Enzyme action = remove one tangle, insert another
• Question: what sequence of surgeries unknots the DNA?
Verification: Use knot invariants (Jones polynomial, Alexander polynomial) to check if you've reached the unknot.
This isn't just theory - it's used for drug design. Many chemotherapy drugs target topoisomerases to stop cancer cells from dividing. Understanding the math of unknotting → better enzyme inhibitors → more effective cancer treatment.
Pure topology becomes practical medicine.