LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

(a)
(b)
Formal Properties of Gene Assembly 205
M3 M4 M6 M5 M7 M9 M2 M1 M8
I1 I2 I3 I4 I5 I6 I7 I8
M1 M2 M3 M4 M5 M6 M7 M9
Fig. 1. (a) The micronuclear version of the actin I gene in Sterkiella nova. (b)
The macronuclear version of the actin I gene of Sterkiella nova. Thevertical
lines describe the positions where the MDSs have been spliced together (by
overlapping of their ends)
For each κ ≥ 1, let
Θκ = { Mi | 1 ≤ i ≤ κ}
be an alphabet representing elementary MDSs, i.e., each Mi denotes an MDSs
are MDS arrange-
that is present in the micronucleus. The signed strings in Θ� κ
ments (of size κ). An MDS arrangement α ∈ Θ∗ κ is orthodox, ifitisoftheform
M1M2 ...Mκ. Note that an orthodox MDS arrangement does not contain any
inversions of MDSs, and the MDSs are in their orthodox order. A signed permutation
of an orthodox MDS arrangement α is a realistic MDS arrangement.
A nonempty signed string v ∈ Σ� is a legal string over ∆ if every letter
a ∈ dom(v) occurs exactly twice in v. (Recall that an occurrence of a letter can
be signed.)
A letter a is positive in a legal string v, ifbothaand a are substrings of v,
otherwise, a is negative in v.
Let ∆ = {2, 3,...} be a designated alphabet of pointers.
Example 5. Let v = 2432 5345 be a legal string over ∆. Pointers 2 and 5 are
positive in u, while 3 and 4 are negative in v. On the other hand, the string
w =2432 5 3 5 is not legal, since 4 has only one occurrence in w. ✷
We shall now represent the MDS arrangements by legal strings over the
pointer alphabet ∆. In this way legal strings become a formalism for describing
the sequences of pointers present in the micronuclear and the intermediate
molecules.
Let ϱκ : Θ � κ → ∆� be the morphism defined by:
ϱκ(M1) =2, ϱκ(Mκ) =κ, ϱκ(Mi) =ii+1 for2