LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

204 Tero Harju, Ion Petre, and Grzegorz Rozenberg
Let Σ and Γ be two alphabets. A function ϕ: Σ ∗ → Γ ∗ is a morphism, if
ϕ(uv) =ϕ(u)ϕ(v) for all strings u, v ∈ Σ ∗ . A morphism ϕ: Σ � → Γ � between
signed strings is required to satisfy ϕ(u) =ϕ(u) for all u, v ∈ Σ � .Notethat,
for a morphism ϕ, the images ϕ(a) of the letters a ∈ Σ determine ϕ, i.e., if
the images of the letters a ∈ Σ are given, then the image of a string over Σ is
determined.
Example 2. Let Σ = {a, b} and Γ = {0, 1, 2}, andletϕ: Σ � → Γ � be the
morphism defined by ϕ(a) = 0 and ϕ(b) = 2. Then for u = abb, wehave
ϕ(u) =022 and ϕ(u) =ϕ(bba) =220 =ϕ(u). ✷
Signed strings u ∈ Σ � and v ∈ Γ � are isomorphic, if there exists an injective
morphism ϕ: Σ � → Γ � such that ϕ(Σ) ⊆ Γ and ϕ(u) =v. Hence two strings
are isomorphic if each of the strings can be obtained from the other by renaming
letters.
A signed string v = a1a2 ...an is a signed permutation of another signed
string u = b1b2 ...bn, if there exists a permutation i1,i2,...,in of [1,n]such
that aj ∈{bij , bij } for each j ∈ [1,n].
Example 3. Let u = a a bcb ∈{a, b, c} � .Thenu is a signed permutation of
acabb,aswellasofa babc (and many other signed strings). Also, u is isomorphic
to the signed string v = a a cbc. The isomorphism ϕ in question is defined
by: ϕ(a) =a, ϕ(c) =b, andϕ(b) =c. Hence also ϕ(a) =a, ϕ(c) =b, and
ϕ(b) =c. ✷
3 MDS Arrangements and Legal Strings
The structural information about the micronuclear or an intermediate precursor
of a macronuclear gene can be given by the sequence of MDSs forming the
gene. The whole assembly process can be then thought of as a process of assembling
MDSs through splicing, to obtain the MDSs in the orthodox order
M1,M2,...,Mκ. Thus one can represent a macronuclear gene, as well as its
micronuclear or an intermediate precursor, by its sequence of MDSs only.
Example 4. The actin I gene of Sterkiella nova has the following MDS/IES micronuclear
structure:
M3I1M4I2M6I3M5I4M7I5M9I6M 2I7M1I8M8
(see Fig. 1(a)). The ‘micronuclear pattern’ is obtained by removing the IESs Ii.
For our purposes, the so obtained string
α = M3M4M6M5M7M9M 2M1M8
has the same information as the structure given in the representation (1).
The macronuclear version of this gene is given in Fig. 1(b). There the IESs
have been excised and the MDSs have been spliced (by overlapping of their ends)
in the orthodox order M1,M2,...,M9. ✷
(1)