Topological Process of Splitting DNA-Links

A DNA replicon is modeled by a special type of 2-component link, called a DNA-link, in which two circles form a double helix around a trivial center core curve. The DNA replication process is semi-conservative, which is interpreted as a splitting process of the DNA-link. To split this non-trivial link, the linking number must become zero, and thus an unknotting operation is necessary. Some families of enzymes act as the unknotting operation. The present paper considers two topological problems; one is to know how the linking number is reduced and the other, how the enzymes are allocated at appropriate places. For the first problem, we suggest a reduction system of the linking number of a DNA-link. From this system, the number of repetitions of the procedure is obtained and this could be reduced when the DNA is previously relaxed by type I topoisomerases. For the second problem, we propose a possible conformation of the DNA-link in which the unknotting operation does not change the knot type of the core curve but decreases the writhe. This conformation could allocate type II topoisomerases to appropriate places. These models suggest that the combination of type I and type II topoisomerases efficiently reduces the linking number and it is possible to allocate enzymes by the conformation of DNA strands.