Many proteins cleave and reseal DNA molecules in precisely orchestrated
ways. Modelling these reactions has often relied on the axis of the DNA
being circular, so these cut-and-seal mechanisms can be
tracked by corresponding changes in the knot type of the DNA axis.
However, when the DNA molecule is linear, or the
protein action does not manifest itself as a change in knot type, or the
knots types are not 4-plats, these knot theoretic models are less germane.
We thus give a taxonomy of local DNA axis configurations. More precisely, we
all rational tangles obtained from a given rational tangle via a rational
replacement (RSR). This builds on work of Berge and Gabai.
We further determine the sites for these RSR of distance greater than 1.
Finally, we classify all knots in lens spaces whose exteriors are
generalised Seifert fibered spaces and their lens space surgeries, extending work of
Biologically then, this classification is endowed with a distance that
determines how many protein reactions
of a particular type (corresponding to steps of a specified size) are
needed to proceed from one local conformation to another.
We conclude by discussing a variety of biological applications of this
Joint work with Ken Baker
- Topology Seminar