Gene assembly through cyclic graph decomposition

We present in this paper a graph theoretical model of gene assembly, where (segments of) genes are distributed over a set of circular molecules. This model is motivated by the process of gene assembly in ciliates, but it is more general. In this model a set of circular DNA molecules is represented by a bicoloured and labelled graph consisting of cyclic graphs, and the recombination takes place in two stages: 5rst, by folding ∗ P with respect to a set P of pairs of vertices of the graph (representing pointers in the micronuclear genes of the ciliate), and secondly, by unfolding the so obtained graph to ̃ P with respect to vertices of higher valency. The 5nal graph ̃ P is again a set of bicoloured cyclic graphs, where the genes are present as maximal monochromatic paths. Thus, the process of gene assembly corresponds to the dynamic process of changing cyclic graph decompositions. We show that the operation ̃ is well behaved in many respects, and that there is a sequence of pointer sets P1; : : : ; Pm consisting of one or two pairs such that ̃ P = (· · · (( ̃ P1) ̃ P2) · · · ̃ Pm) and each intermediate step i = (· · · (( ̃P1) ̃P2) · · · ̃Pi) is intracyclic, that is, the segments of a gene that lie in the same connected component of i, will lie in the same connected component of the successor graph i+1. c © 2002 Published by Elsevier Science B.V.