  We study theoretically the replication of Kinetoplast DNA consisting of
several thousands separate mini-circles found in organisms of the class
Kinetoplastida. When the cell is not actively dividing these are topologically
connected in a marginally linked network of rings with only one connected
component. During cell division each mini-circle is removed from the network,
duplicated and then re-attached, along with its progeny. We study this process
under the hypothesis that there is a coupling between the topological state of
the mini-circles and the expression of genetic information encoded on them,
leading to the production of Topoisomerase. This model describes a
self-regulating system capable of full replication that reproduces several
previous experimental findings. We find that the fixed point of the system
depends on a primary free parameter of the model: the ratio between the rate of
removal of mini-circles from the network (R) and their (re)attachment rate (A).
The final topological state is found to be that of a marginally linked network
structure in which the fraction of mini-circles linked to the largest connected
component approaches unity as R/A decreases. Finally we discuss how this may
suggest an evolutionary trade-off between the speed of replication and the
accuracy with which a fully topologically linked state is produced.
