Definition
A genetic alphabet \(\langle \Sigma,\sim \rangle\) is an algebraic structure on an alphabet \(\Sigma\) with a unary operation \(\sim\) verifying the following properties:
Note
To stay consistent with the biology lexicon, we will be referring to a word over a genetic alphabet as a sequence, only explicitly naming a mathematical sequence when needed to.
Examples
Definition
A circular word over an alphabet \(\Sigma\) is a finite word with no end. It can be noted \(w^{(c)}\), where \(w\) is a finite word of \(\Sigma^\star\).
Definition: Cardinality
Given a circular sequence \(s^{(c)}\), the cardinal of \(s^{(c)}\), noted \(\lvert s^{(c)} \rvert\), is defined as:
Definition: Equality
Given two sequences \(a^{(c)}\) and \(b^{(c)}\) with
let the \(=\) relation be defined as:
where \(\sigma\) is the circular shift defined as:
Property
\(=\) is a relation of equivalence over \(\Sigma^{(c)}\)
Demonstration
Given the set of circular sequences \(\Sigma^{(c)}\) using an alphabet \(\Sigma\):
Reflexivity:
Symetry: \(\forall s_1^{(c)}, s_2^{(c)} \in \Sigma^{(c)} \times \Sigma^{(c)}\):
Transitivity: \(\forall s_1, s_2, s_3 \in \Sigma^{(c)} \times \Sigma^{(c)} \times \Sigma^{(c)}\)
Definition: Automaton acception
Given a finite automaton \(A\) over an alphabet \(\Sigma\), and \(u^{(c)}\) a sequence of \(\Sigma^{(c)}\), \(A\) accepts \(u^{(c)}\) iff there exist a sequence \(v\) of \(\Sigma^\star\) such that:
Definition
Given a genetic alphabet \(\langle \Sigma, \sim \rangle\), a restriction enzyme \(e\) can be defined as a tuple \((S, n, k)\) where:
Note
This definition only covers single-cut restriction enzymes found in vivo, but we don’t need to cover the case of double-cut restriction enzymes since they are not used in modular cloning.
Definition: Enzyme types
A restriction enzyme \((S, n, k)\) is:
Definition
An assembly is a function of \(\mathcal{P}(\Sigma^\star \cup \Sigma^{(c)}) \times \mathcal{P}(E)\) to \(\mathcal{P}(\Sigma^\star \cup \Sigma^{(c)})\), which to a set of distinct sequences \(\{d_1, \dots, d_m\}\) and a set of restriction enzymes \(\{e_1, \dots, e_n\}\) associates the set of digested/ligated sequences \(A = \{a_1, \dots a_k\}\).
The notation for an assembly is: