Percolation phenomena in disordered topological networks

Topological polymer networks consist of circular polymers that are topologically linked. Topological networks made of small circular DNA or protein molecules are of great interest in biology and nanotechnology because they are found in living organisms and can be constructed in-vitro. The physical factors that determine the topology of a network as well as the pathways that are followed for its formation remain poorly understood. In our previous work we proposed a novel biophysical/computational approach to model the formation of planar DNA minicircle networks in trypanosomatid parasites. This model suggests that minicircle networks in trypanosomatid parasites emerged from topologically free minicircles upon space confinement through a percolation pathway. Our model however is somewhat idealized because it assumes that the centers of the minicircles in the network are positioned following a regular planar lattice. Here we propose an extension of the model by allowing the centers of the minicircles to be randomly displaced from the vertices of the lattice. We numerically show that networks form following a percolation pathway upon increasing minicircle density. Our model suggests that the critical percolation density increases as D = 0.8357 − 1.4297 exp(0.6439x) with x is the maximum displacement of the centers of the minicircles. Our results therefore show that the plane distribution of minicircles does not dramatically affect the percolation of minicircles and therefore supports they hypothesis that DNA minicircle networks in trypanosomes evolved through a percolation pathway.