Biological metrics on DNA knots and catenanes

Molecular biologists use circular DNA to probe the action of enzymes which are important in cellular metabolism. Ordinary cellular processes such as replication and recombination can cause this circular DNA to become knotted and linked. Although human DNA is linear, it is extremely long and kept well organized by being attached at many places to a protein scaffold. Thus, human DNA is also subject to topological entanglement problems caused by cellular processes, and enzymes exist in the cell to solve these entanglement problems.; Topoisomerases are enzymes that can change crossings of a knot or link by passing one strand of DNA through another (figure 1a). They are responsible for keeping the DNA unknotted and unlinked and are the targets of many anti-cancer and anti-bacterial drugs. This enzyme action can be studied using a generalization of the unknotting number: a strand passage metric on knots (links) (Mk1) in which one computes the minimum number of strand passages necessary to inter-convert a pair of knots (links). There is unfortunately no algorithm for computing this metric. There are, however, several methods for computing lower bounds involving signature, linking number, and Dehn surgery on the double branched cover of the knot (link). Use of the cyclic surgery theorem allows the complete classification of distance one 4-plats (DS), (To). 4-plats are knots and links of the form:*; In order to obtain chiral strand passage information, one can define the positive distance between two knots {dollar}Ksb1{dollar} and {dollar}Ksb2{dollar} as the minimum number of positive crossing changes (changing a positive crossing to a negative crossing) needed to change {dollar}Ksb1{dollar} into {dollar}Ksb2{dollar}, where negative crossing changes are not allowed. The distance is defined to be infinity if one cannot change {dollar}Ksb1{dollar} into {dollar}Ksb2{dollar} using only positive crossing changes. Recombination distances can be used to study recombinases which are enzymes which break two segments of DNA and interchange the ends (figure 2b), which has the biological result of an exchange of genetic information. Tables of many of these distances are generated by computer calculation for 4-plats and composites of 4-plats (which includes all knots of seven or fewer crossings).* ftn*Please refer to the dissertation for diagrams.