Four-regular graphs with rigid vertices associated to DNA recombination

Abstract

a b s t r a c tGenome rearrangement and homological recombination processes have been modeled by Angeleska et al. [A. Angeleska, N. Jonoska, M. Saito, DNA recombinations through assembly graphs, Discrete Appl. Math. 157 (2009) 3020-3037] as 4-regular spacial graphs with rigid vertices, called assembly graphs. These graphs can also be represented by double occurrence words called assembly words. The rearranged DNA segments are modeled by certain types of paths in the assembly graphs called polygonal paths. The minimum number of polygonal paths visiting all vertices in a graph is called an assembly number for the graph.In this paper, we give formulas for counting certain types of assembly graphs and assembly words. Some of these formulas produce sequences not previously reported at the Online Encyclopedia of Integer Sequences (http://oeis.org). We provide a sharp upper bound for the number of polygonal paths in Hamiltonian sets of polygonal paths, and present a family of graphs that achieves this bound. We investigate changes in the assembly numbers as a result of graph compositions. Finally, we introduce a polynomial invariant for assembly graphs and show properties of this invariant.

Graph Polynomials Motivated by Gene Rearrangements in Ciliates

“…Also, a multimatroid polynomial is proposed as a unified framework to several polynomials for graphs and matroids [5]. The study of gene rearrangements has motivated the introduction of new polynomials, most notably the interlace [2,1] and assembly [7] polynomials, as, e.g., a feature that could measure and compare the complexity of the rearrangement process. Both these polynomials fit in the corpus of existing graph polynomials, and thus techniques and results can be carried over.…”

“…be an arbitrary orientation of C. (i) If, by entering v via e in P, P leaves v via e 1 , then we say that P follows C at v. (ii) If P leaves v via e 3 , then we say that P is orientation-consistent with C at v, and finally (iii) if P leaves v via e 2 , then we say that P is orientationinconsistent with C at v. In [15], the first two are called coherent and the last one is called anticoherent. Moreover, in [7] the last two are called (parallel) p-smoothing and n-smoothing, respectively.…”

“…where the sum is taken over all 2 |V | transition systems s that differ at each vertex from the transition system corresponding to C w (a transition that differs from the transition of C w is called a smoothing in [7]), π(s) equals the number of orientation-consistent transitions of s w.r.t. C w (called p-smoothings in [7]), and c(s) equals the number of circuits in the circuit partition corresponding to s. Example 7.…”

“…Burns et al [7,Section 6] (see also [9]) define the assembly polynomial for a 4-regular graph G w together with an Eulerian circuit C w belonging to a double-occurrence string w over alphabet V . The assembly polynomial of G w w.r.t.…”

“…Possible intermediate results of this recombination can also be read from this 4-regular graph. Based on this observation, Burns et al [7] have proposed the assembly polynomial, a graph polynomial intended to abstract basic features of the assembly process, like the number of molecules excised during this process.…”